Issues of Identity and Individuality in Quantum Mechanics Adam Caulton Darwin College, University of Cambridge 26th September 2011 This dissertation is submitted for the degree of Doctor of Philosophy. Abstract This dissertation is ordered into three Parts. Part I is an investigation into identity, indiscernibility and individuality in logic and metaphysics. In Chapter 2, I investigate identity and discernibility in classical first-order logic. My aim will will be to define four different ways in which objects can be discerned from one another, and to relate these definitions: (i) to the idea of symmetry; and (ii) to the idea of individuality. In Chapter 3, the four kinds of discernibility are put to use in defining four rival metaphysical theses about indiscernibility and individuality. Part II sets up a philosophical framework for the work of Part III. In Chapter 4, I give an account of the rational reconstruction of concepts, in- spired chiefly by Carnap and Haslanger. I also offer an account of the interpretation of physical theories. In Chapter 5, I turn to the specific problem of finding candidate concepts of particle. I present five desiderata that any putative explication ought to satisfy, in order that the proposed concept is a concept of particle at all. Part III surveys three rival proposals for the concept of particle in quantum mechanics. In Chapter 6, I define factorism and distinguish it from haecceitism. I then propose an amendment to recent work by Saunders, Muller and Seevinck, which seeks to show that factorist particles are all at least weakly discernible. I then present reasons for rejecting factorism. In Chapter 7, I investigate and build on recent heterodox proposals by Ghirardi, Marinatto and Weber about the most natural concept of entanglement, and by Zanardi about the idea of a natural decomposition of an assembly. In Chapter 8, I appraise the first of my two heterodox proposals for the con- cept of particle, varietism. I define varietism, and then compare its performance against the desiderata laid out in Chapter 5. I argue that, despite its many merits, varietism suffers a fatal ambiguity problem. In Chapter 9, I present the second heterodox proposal: emergentism. I argue that emergentism provides the best concept of particle, but that it is does so im- perfectly; so there may be no concept of particle to be had in quantum mechanics. If emergentism is true, then particles are (higher-order) properties of the assembly, itself treated as the basic bearer of properties. Preface I am grateful to many people for discussions, criticisms and suggestions regarding the claims and arguments of this dissertation. I cannot possibly name all the people who have taught and inspired me, but any attempt at such a list must include Nazim Bouatta, Tim Button, Eric Curiel, Newton da Costa, Foad Dizadji- Bahmani, John Earman, Steven French, Katherine Hawley, Jules Holroyd, Leon Horsten, Nick Huggett, Nick Jones, Jeff Ketland, Brian King, Eleanor Knox, James Ladyman, Øystein Linnebo, Fraser MacBride, Kerry McKenzie, Fred Muller, Sam Nicholson, Tom Pashby, Oliver Pooley, Miklo´s Re´dei, Simon Saunders, Michael Seevinck, Rob Spekkens, Nic Teh, Lee Walters, and Nathan Wildman. I would also like to thank Margrit Edwards, Charlie Evans, Heather Sanderson and Lesley Lancaster for all their help over the last four years. Finally, I wish to thank Jeremy Butterfield for his unwavering encouragement, infinite patience, and for being the most supportive supervisor a graduate student could hope for. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration, except some aspects of Chapters 3 and 4, which have appeared as an article co-authored with Jeremy Butterfield. This dissertation does not exceed the 80,000 word limit set by the Philosophy Degree Committee. 1 Contents 1 Introduction 7 1.1 Prospectus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 I Identity in Logic and Metaphysics 17 2 Identity and indiscernibility in logic 18 2.1 Stipulations about jargon . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.1 Haecceitism . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 A logical perspective on identity . . . . . . . . . . . . . . . . . . . . 26 2.2.1 The Hilbert-Bernays account . . . . . . . . . . . . . . . . . . 26 2.2.2 Permutations on domains . . . . . . . . . . . . . . . . . . . 32 2.3 Four kinds of discernment . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Three preliminary comments . . . . . . . . . . . . . . . . . . 37 2.3.2 The four kinds defined . . . . . . . . . . . . . . . . . . . . . 38 2.4 Absolute indiscernibility: some results . . . . . . . . . . . . . . . . 42 2.4.1 Invariance of absolute indiscernibility classes . . . . . . . . . 44 2.4.2 Illustrations and a counterexample . . . . . . . . . . . . . . 48 2.4.3 Finite domains: absolute indiscernibility and the existence of symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2 3 Four metaphysical theses 57 3.1 The four theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 The theses’ verdicts about what is possible . . . . . . . . . . . . . . 64 3.2.1 Haecceitism . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.2 QII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.3 WPII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.4 SPII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.5 A glance at the classification for structures with three objects 73 3.3 Structuralism and intrinsicalism . . . . . . . . . . . . . . . . . . . . 74 3.3.1 Relation to the four metaphysical theses . . . . . . . . . . . 75 3.3.2 The semantics of the structuralist . . . . . . . . . . . . . . . 76 II Representing particles 80 4 Concepts and representation 81 4.1 Concepts: analysis, explication and reform . . . . . . . . . . . . . . 81 4.1.1 The Moorean and Carnapian views . . . . . . . . . . . . . . 82 4.1.2 Haslanger’s scheme . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Physics, mathematics and language . . . . . . . . . . . . . . . . . . 91 4.2.1 Two realms: the physical and the mathematical . . . . . . . 91 4.2.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2.3 Representation . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.4 A note about physical properties . . . . . . . . . . . . . . . 100 5 Particles: what’s in a name? 102 5.1 Five desiderata for the concept of particle . . . . . . . . . . . . . . 103 5.1.1 Being physical . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3 5.1.2 Being located . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1.3 Persisting over time . . . . . . . . . . . . . . . . . . . . . . . 109 5.1.4 Composing assemblies . . . . . . . . . . . . . . . . . . . . . 110 5.1.5 Being applicable across different theories . . . . . . . . . . . 115 5.2 The desiderata for a single quantum system . . . . . . . . . . . . . 117 5.2.1 Satisfying the desiderata for “single-system” Hilbert spaces . 117 5.2.2 Natural decompositions and dressing . . . . . . . . . . . . . 120 III What is a quantum particle? 123 6 Against factorism 124 6.1 Factorism defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2 Factorism and haecceitism . . . . . . . . . . . . . . . . . . . . . . . 126 6.2.1 Haecceitism defined . . . . . . . . . . . . . . . . . . . . . . . 127 6.2.2 Factorism and haecceitism distinguished . . . . . . . . . . . 130 6.3 Factorism and discernibility . . . . . . . . . . . . . . . . . . . . . . 132 6.3.1 The old orthodoxy . . . . . . . . . . . . . . . . . . . . . . . 132 6.3.2 The new orthodoxy? . . . . . . . . . . . . . . . . . . . . . . 133 6.3.3 Muller and Saunders on discernment . . . . . . . . . . . . . 134 6.3.4 Muller and Seevinck on discernment . . . . . . . . . . . . . 139 6.3.5 A better way to discern factorism’s particles . . . . . . . . . 143 6.4 The defects of factorism . . . . . . . . . . . . . . . . . . . . . . . . 157 6.4.1 Losing individuality . . . . . . . . . . . . . . . . . . . . . . . 158 6.4.2 Losing limits . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.4.3 Analogy with the average taxpayer . . . . . . . . . . . . . . 160 6.4.4 A haecceitistic response . . . . . . . . . . . . . . . . . . . . 162 4 7 Entanglement and individuation for anti-factorists 164 7.1 Subtleties of entanglement . . . . . . . . . . . . . . . . . . . . . . . 165 7.1.1 Entanglement for two distinguishable systems . . . . . . . . 166 7.1.2 GM-entanglement for two indistinguishable systems . . . . . 169 7.2 Qualitatively individuating quantum systems . . . . . . . . . . . . . 182 7.2.1 Qualitative individuation and natural decompositions . . . . 183 7.2.2 Russellian vs. Strawsonian approaches to individuation . . . 192 7.2.3 Qualitatively individuated systems on their own . . . . . . . 193 7.3 Qualitative individuation over time . . . . . . . . . . . . . . . . . . 199 8 Against varietism 204 8.1 Varietism defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.1.1 What varietism says about non-GM-entangled fermions . . . 206 8.1.2 What varietism says about non-GM-entangled bosons . . . . 210 8.1.3 What varietism says about GM-entangled states . . . . . . . 211 8.2 The merits of varietism . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.2.1 Varietist particles are physical . . . . . . . . . . . . . . . . . 224 8.2.2 Varietist particles are locational . . . . . . . . . . . . . . . . 226 8.2.3 Varietist particles do not (always) persist over time . . . . . 227 8.2.4 Varietist particles compose assemblies . . . . . . . . . . . . . 228 8.2.5 Varietist particles have inter-theoretic applicability . . . . . 229 8.2.6 Are varietist particles discernible? . . . . . . . . . . . . . . . 237 8.3 A preferred basis problem for varietism . . . . . . . . . . . . . . . . 240 8.3.1 The problem of basis arbitrariness . . . . . . . . . . . . . . . 241 8.3.2 The ‘One size fits all’ response . . . . . . . . . . . . . . . . . 243 8.3.3 The ‘Complicate’ response . . . . . . . . . . . . . . . . . . . 245 5 8.3.4 The ‘Coalesce’ response . . . . . . . . . . . . . . . . . . . . 248 8.3.5 The ‘Multiply’ response . . . . . . . . . . . . . . . . . . . . 250 8.3.6 The ‘Overlap’ response . . . . . . . . . . . . . . . . . . . . . 252 9 Emergentism: winner in a poor field? 255 9.1 Emergentism defined . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.1.1 Mode realism . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.1.2 The assembly is the object . . . . . . . . . . . . . . . . . . . 261 9.1.3 Regaining particles under assembly realism . . . . . . . . . . 263 9.2 Conclusion: Is emergentism good enough? . . . . . . . . . . . . . . 265 6 Chapter 1 Introduction What is a quantum particle? What properties and relations do they possess? It is the aim of this dissertation to make some progress in finding out. There has been discussion over many decades about the treatment of indis- tinguishable (also known as ‘identical’) particles.1 This discussion has been in- vigorated in the last few years, principally by Saunders’ (2003b) revival of the Hilbert-Bernays account of identity (also briefly discussed by Quine) and his, and later Muller’s and Seevinck’s (Muller and Saunders (2008), Muller and Seevinck (2009)), application of it to quantum theory. In short, Saunders saw that there is an error in the consensus of the previous philosophical literature. That literature had shown that for any assembly of indis- tinguishable quantum particles (fermions or bosons), and any state of the assembly (appropriately (anti-) symmetrized), and any two particles in the assembly: the re- duced density matrices (reduced states) of the particles (and so all probabilities for single-particle measurements) were equal; and so were appropriate corresponding two-particle conditional probabilities. This result strongly suggests that quantum theory endemically violates the principle of the identity of indiscernibles: for any two particles in the assembly are surely indiscernible.2 1Castellani (1998) is a valuable collection of classic and contemporary articles. French and Krause (2006) is a thorough recent monograph. 2This consensus seems to have been first stated by Margenau (1944, pp. 201-3) and it is endorsed and elaborated by e.g. French and Redhead (1988, p. 241), Butterfield (1993, p. 464), 7 Saunders’ basic insight (2006, p. 7) is that the Hilbert-Bernays account provides ways that two objects can be distinct, which are not captured by these orthodox quantum probabilities: and yet which are instantiated by quantum theory. Thus on the Hilbert-Bernays account, two objects can be distinct, even while sharing all their monadic properties and their relations to all other objects. For they can be distinguished by either: (i) a relation R between them holding one way but not the other (which is called ‘being relatively discernible’); (ii) a relation R between them holding in both directions but neither object having R to itself (‘being weakly discernible’). Saunders, and later Muller and Saunders (2008), argue that the second case (ii) is instantiated by fermions: the prototype example being the relation R = ‘. . . has opposite value for spin (in any spatial direction) to . . . ’ for two spin-1 2 fermions in the singlet state. More recently, Muller and Seevinck (2009) have argued that (ii) is instantiated by all particles. (I will propose a refinement of these results in Section 6.3.) Saunders’ proposals have led to several developments, including exploring the parallel between quantum particles and spacetime points (e.g. Pooley (2006), Caulton and Butterfield (2011), Muller (forthcoming)). But this is work for an- other day. Here there will be enough to do focussing solely on quantum theory. The Hilbert-Bernays account is a reductive account, reducing identity to a con- junction of statements of indiscernibility. So this account is controversial: many philosophers hold that identity is irreducible to any sort of qualitative facts, but nevertheless wholly unproblematic because completely understood.3 This latter view is certainly defensible, perhaps orthodox, once one sets aside issues about and French (2006, §4). Massimi (2001, pp. 326-7) questions these authors’ emphasis on monadic properties, but agrees that quantum mechanics violates the identity of indiscernibles when the quantum state is taken to codify purely relational properties. 3For example, Lewis (1986, pp. 192-193). More generally: in the philosophy of logic, the Fregean tradition that identity is indefinable, but understood, remains strong—including as a response to the Hilbert-Bernays account: cf. e.g. Ketland (2006, p. 305 and Sections 5, 7). 8 diachronic identity—as I will. But this is not a problem for me: for I am not com- mitted to the Hilbert-Bernays account (and nor is Saunders’ basic insight about weak discernibility in quantum theory)—in fact, I will support an interpretation of quantum mechanics in which particles may be indiscernible. Besides, these con- troversies do not undercut the rationale for my investigation (in Chapter 2) into the kinds of discernibility; essentially because anyone, whatever their philosophy of identity or their attitude to the identity of indiscernibles, will accept that dis- cernibility is a sufficient condition for diversity (the ‘non-identity of discernibles’). It cannot be denied that, with the resources of the Hilbert-Bernays account, two objects previously thought indiscernible may yet be discernible (albeit merely ‘relatively’ or ‘weakly’). Nevertheless, the quantum particles discussed in very nearly all of the philosophical literature (including the recent work by Saunders, Muller and Seevinck) possess a feature which is intuitively like indiscernibility. The consensus (shared by Saunders, Muller and Seevinck) is that quantum particles of the same species—electrons, say—possess exactly the same monadic properties: they are, as I shall say (in Section 2.3) ‘absolutely indiscernible’. The unfortu- nate upshot of this is that no single one of them can be ‘picked out’ in language or in thought, without appeal to some sort of ‘underlying’ or ‘non-qualitative’ property—namely, an haecceity. But this result is intolerable. In the face of such a result, the correct response can only be that something, somewhere has gone wrong. Thus I claim that a fundamental mis-interpretation of quantum theory pervades the consensus. That is, I claim that the consensus has mis-identified the representational relationships between the quantum formalism and the physical world. The interpretative proposal that leads to this intolerable result I call factorism. Despite its unfamilar name, factorism is a familiar proposal. It says that particles are the physical correlates of the labels of factor Hilbert spaces. I will distinguish this from another doctrine, familiar in philosophy: haecceitism (Section 6.2). I will agree that factorism is right for distinguishable systems—i.e. systems for which the Indistinguishability Postulate is not imposed.4 Since such systems differ in, say, 4For a thorough discussion of the Indistinguishability Postulate, see French and Krause (2006, pp. 131-149). 9 intrinsic charge or mass, the labels can be taken as short for such distinguishing properties. But this construal is not available for indistinguishable systems. Here, the correlates of labels make very poor particles, for the reasons just given; and thus I reject factorism. Indeed, as I will argue (Section 6.4), the factorist’s particles are like that familiar abstraction of high-school statistics, ‘the average taxpayer’, and factorism makes a reification error (what Whitehead (1925) called ‘the fallacy of misplaced concreteness’) analogous to those who take the average taxpayer as a real person. I will therefore seek a successor to factorism. That is: I will seek a new account of what a quantum particle is, and of how particles are represented in the formalism of quantum theory. But how does this project proceed? How can one look for a new concept without already knowing what one is looking for? The answer is that we do, at least in part, know what we are looking for. (Oth- erwise we could not even have suspected that the consensus was in such error.) We have an intuitive notion of what a particle is, and there are other theories—namely, classical mechanics and quantum field theory—whose interpretations provide, in each of their own domains, a more precise and complete account. With an eye on our cloudy, pre-theoretic notion, and on “neighbouring” theories’ accounts, we may search the quantum formalism for objects more worthy of the name ‘particle’. It is the work of Chapters 4 and 5 to set up the philosophical framework that will make these rough ideas precise. I will discuss two heterodox interpretative proposals for what these natural candidates are. I call them varietism and emergentism. They both propose that the real particles are the things of which the factorist’s “particles” are the average, just as the average taxpayer encodes statistical information about a population of real people. Furthermore, both proposals make a break from the current debate in phi- losophy of physics about whether quantum particles are discernible. We will see that particles in the senses of varietism and emergentism will often be absolutely discernible, contra the received consensus. But, regrettably, the weak discernibil- ity results that in all states any two particles are weakly discernible (mentioned above) will not carry over either to varietism or emergentism. Therefore, for both 10 varietism and emergentism, bosons and paraparticles may fail to be discernible at all. An important difference between varietism and emergentism is whether parti- cles exist in all states of the assembly. According to varietism, there are particles always, i.e. in all states of the assembly. But according to emergentism, there are particles only sometimes, i.e. only in some states of the assembly. This prompts a revision in our understanding of the relationship between an assembly and its particles. I investigate these matters in Chapter 9. To explain and assess my two anti-factorist proposals, I need first to present (Chapter 7) some little-noted subtleties of entanglement for indistinguishable sys- tems. Here I follow an analysis by Ghirardi, Marinatto and Weber (2002). The leading idea is that the definition of entanglement must not appeal to factor- space labels, since it must respect the requirement that all physical quantities be symmetric (i.e. permutation invariant). This can be done, while meshing with the familiar definition for distinguishable systems (namely, that a non-entangled state is a separable, or product state). One defines a state of the assembly to be non-entangled iff the symmetrization of some 1-dimensional projector on a factor space has probability 1. Then the main result (Section 7.1) is, roughly speak- ing, that a state is non-entangled iff it is the appropriate symmetry projection of a product state (taken as including, for bosons, the case of a product of identi- cal factors). (Here, ‘appropriate symmetry projection’ means, for fermions and bosons, the familiar (anti)-symmetrization; and for paraparticles, projection onto their subspace.) This implies, in particular, that fermionic states do not always count as entangled; and bosonic states need not be product states to count as non-entangled. Later, in Section 7.2, I extend Ghirardi and Marinatto’s work by developing a means of individuating—i.e., picking out from other constituents of the assembly— a system, or collection of systems, qualitatively—i.e., in way that appeals not to factor-space labels, but to single-system states. This development dovetails nicely with recent work by Zanardi (2001). I then provide (Section 7.2.3) a recipe for calculating reduced density operators for qualitatively individuated systems. With Chapter 7’s results in hand, I can state and assess my two anti-factorist 11 proposals: starting (in Chapter 8) with varietism. Roughly speaking, varietism proposes that particles are those objects whose statistics are given by pure states whenever (and only whenever) the assembly’s state is non-entangled in Ghirardi and Marinatto’s sense. The proposal has many merits (reviewed in Section 8.2) in terms of satisfying Chapter 5’s desiderata for the concept of particle. However, this proposal founders on an ineliminable ambiguity (Section 8.3), for fermionic or paraparticle states, in the specification of what these objects could be. This ambiguity is troubling in three ways. 1. It is not a matter of philosophical argument or controvertible interpretation, but follows from the results reviewed in Chapter 7. 2. It is an ambiguity, not between some handful of alternatives, but between continuum-many: for a pair of fermions it is parameterized by points on the Riemann sphere, i.e. by an extended complex number z ∈ C ∪ {∞}. 3. The ambiguity cannot be assimilated to the philosophically familiar cases, such as reference to macroscopic bodies with vague spatial boundaries (Geach’s (1980) and Unger’s (1980) ‘problem of the many’). Such cases are less trou- bling because the rival disambiguations almost coincide, and we know the parts whose shared containment underpins the near-total overlap of the rival disambiguations. Neither of these features carry over for varietism’s disam- biguated rival particles. This leaves my third proposal, emergentism (Chapter 9). This has two leading ideas, one philosophical and one physical. I believe the physical idea reflects the practice of physics; the philosophical idea is more idiosyncratic. The physical idea arises from my admission that factorism is correct for dis- tinguishable systems. This prompts one to require that indistinguishable particles should also be ‘label-able’. That is: which properties of the assembly count as par- ticles is governed by which properties can act as labels. In general, these properties must be qualitative, i.e. they correspond to states belonging to some basis in the single-particle Hilbert space. Specifically which single-particle states they are will, in general, depend on the state of the assembly; but they must be states in which 12 no two separate degrees of freedom are entangled. In short: there is a preferred basis problem which emergentism seeks to solve in the same way that nowadays proponents of the Everett interpretation of quantum mechanics seek to solve the traditional preferred basis problem: by appeal to the particular dynamical features of the situation at hand (cf. e.g. Wallace (2003, 2010)). The philosophical idea is that the basic objects of predication are not particles, but something else: perhaps spatial regions or the entire assembly itself. Emer- gentism then saves particles from being an additional ontological extravagance by identifying them with higher-order properties of the basic object or objects, whether spatial regions or the assembly. This proposal: (i) accommodates the ambiguity about fermions and paraparticles which besets varietism; but also (ii) violates the desideratum that particles be the building blocks of assemblies, in the sense (cf. Chapter 5) that particles’ properties and relations determine all properties of the assembly: for the assembly may be in a state in which there are no particles. I conclude that emergentism is the winner in a poor field. There are two main regrets. First, there seems to be no way of preserving the recent results about weak discernibility (by Saunders, Muller, Seevinck, and here in Section 6.3) which recently revived the debate about the discernibility of quantum particles. And second, there seems to be no way to think of assemblies as being “built out of” the emergentist’s particles. In this way, the emergentist sees the assembly as many see the field in quantum field theory. Indeed: under emergentism, the assembly of elementary quantum mechanics appears as nothing but the quantum field in a particular limit of (what the textbooks call) conserved total particle number. This leads to two surprises. First, particles under emergentism are also closer to classical particles than the consensus has it. And second, if the arguments in the following Chapters are right, then, in non-relativistic elementary quantum mechanics, we already have reason enough to adopt an basic ontology of fields, rather than particles. 13 I emphasize at the outset that my focus is orthodox quantum mechanics. More specifically, I set aside: (i) heterodox cousins such as pilot-wave theory, with its distinctive (and at- tractively straightforward!) meaning of ‘particle’, and dynamical reduction models, as developed by Ghirardi and others (reviewed by Bassi and Ghirardi (2003)); (ii) some programmatic responses in defence of varietism that would involve changing the formalism (details in Section 8.3); and (iii) thorough investigations of quantum field theory (QFT), although I will briefly mention QFT in several places, particularly in Chapters 8 and 9. I also give notice that I will not shed light on the measurement problem (as the exclusion (i) above hints). What I will do is combine considerations about logic, metaphysics, philosophical methodology and philosophy of language with subtleties about entanglement and individuation—odd bedfellows, you may say!— to state and assess three proposals of what is a quantum particle. 1.1 Prospectus Here I give a short summary of the content of each Chapter. This thesis is ordered into three Parts. Part I is a rather self-contained inves- tigation into identity and indiscernibility in logic and metaphysics. In Chapter 2, I investigate identity and discernibility in classical first-order logic. My aim will will be to define four different ways in which objects can be discerned from one another, and to relate these definitions: (i) to the idea of symmetry; and (ii) to the idea of individuality. In Chapter 3, the kinds of discernibility defined in the previous Chapter are put to use in defining four rival metaphysical theses about identity and individuality. These theses are linked to various positions in the extant literature on identity and indiscernibility, and are compared by their commitments as to what, for each them, 14 is possible. The Chapter concludes with a discussion of a heterodox semantics that is more congenial to three of the four rival metaphysical theses. Part II sets up a philosophical framework for the work of Part III. It gives an account of what I am doing by asking the question: What is the best concept of particle for quantum mechanics? In Chapter 4 I give an account of the rational reconstruction of concepts, in- spired chiefly by Carnap (1950) and Haslanger (2006). I also propose a way of understanding the interpretation of physical theories. The idea of a representation relation between mathematical and physical realms explains how a theory’s math- ematical formalism is afforded physical content. This unifies the two projects of theory interpretation and rational reconstruction. In Chapter 5, I turn to the specific problem of finding candidate concepts of particle. I present five desiderata that any putative concept ought to satisfy, in order that the concept is a concept of particle at all. The role of these desiderata is then demonstrated for single-system Hilbert spaces. Part III surveys the three rival proposals for the concept of particle in quantum mechanics. In Chapter 6, I define factorism and distinguish it from haecceitism. I then propose an amendment to recent work by Saunders, Muller and Seevinck, which seeks to show that factorist particles are all at least weakly discernible. I then present reasons for rejecting factorism. In Chapter 7, I turn to more formal matters, and investigate recent heterodox proposals by Ghirardi, Marinatto and Weber about the most natural concept of entanglement. I link this work to Zanardi’s proposed conditions for natural de- compositions of an assembly’s Hilbert space. I build on this work to develop a means of ‘qualitatively individuating’ a system, and propose a recipe for calculat- ing expectation values and reduced density operators for such systems. In Chapter 8, I appraise the first of my two heterodox proposals for the con- cept of particle, varietism. I define varietism, and then compare its performance against the desiderata laid out in Chapter 5. I argue that, despite its many merits, varietism suffers a fatal flaw. 15 In Chapter 9, I present the second of the two heterodox proposals for the concept of particle, emergentism. I argue that emergentist provides the best con- cept of particle, but that it is does so imperfectly; so there may be no concept of particle to be had in quantum mechanics. If emergentism is true, then particles are (higher-order) properties of the assembly, itself treated as the basic bearer of properties. 16 Part I Identity in Logic and Metaphysics 17 Chapter 2 Identity and indiscernibility in logic The main aim of this Chapter is to define four kinds of discernibility, inspired by Quine (1976a) and Saunders (2003b), and investigate their formal interrelations. These kinds of discernibility, and the formal results regarding them, are important in their own right, and they will be useful in later Chapters. In Section 2.1, I begin by laying down some stipulations about the philosophical terms I will use in this Chapter and thereafter. Then in Section 2.2, I briefly discuss identity in classical first-order predicate logic. In Section 2.3, I define the four kinds of dis- cernibility, and investigate further what I call individuality. Finally (Section 2.4), I present some interesting formal results which link certain kinds of discernibility to permutations on the domain of quantification. 2.1 Stipulations about jargon Here I make some stipulations about philosophical terms. I think that all of them are natural and innocuous, though the last one, about ‘individual’, is a bit idiosyncratic. ‘Object’, ‘identity’, ‘discernibility’ — I will use ‘object’ for the broad idea, in the tradition of Frege and Quine, of a potential referent of a singular term, or 18 value of a variable. The negation of the identity relation on objects I will call (indifferently): ‘non-identity’, ‘distinctness’, ‘diversity’; (and hence use cognate words like ‘distinct’, ‘diverse’). When I have in mind that a formula applies to one of two objects but not the other, I will say that they are ‘discerned’, or that the formula ‘discerns’ them. I will also use ‘discernment’ and ‘discernibility’: these are synonyms; (though the former usefully avoids connotations of modality, the latter often sounds better). Their negation I will call ‘indiscernibility’. ‘Individual’ — Following a recent tradition started by Muller & Saunders (2008), I will also use ‘individual’ for a narrower notion than ‘object’, viz. an object that is discerned from others by a strong, and traditional, form of difference—which I will call ‘absolute discernibility’. Anticipating the following section, this usage may be illustrated by the fact that a haecceitist (in my terms) would demand that all objects be individuals, in virtue of each possessing its own unique property. It is worth distinguishing cases according to whether the discernment is by an arbitrary language or an ‘ideal’ one. That is: since the Hilbert-Bernays account will be cast in a formal first-order language, and such languages can differ as to their non-logical vocabulary (primitive predicates), our discussion will sometimes be relative to a choice of such vocabulary. So, for example, an object that fails to be an individual by the lights of an impoverished language may yet be an individual in an ideal language adequate for expressing all facts (especially all facts about identity and diversity). Therefore, one might use the term ‘L-individual’ for any object which is absolutely discerned from all others using the linguistic resources of the language L. The un-prefixed term ‘individual’ may then be reserved for the case of ideal language. However, I will stick to the simple term ‘individual’, since it will always be obvious which language is under consideration. In Chapter 3, I will use ‘individual’ in the strictly correct sense just proposed, since there I envisage a language which is taken to be adequate for expressing all facts. ‘Haecceitism’ — Though the details will not be needed until Chapter 3, I should say what we will mean by ‘haecceitism’ (a venerable doctrine going back to Kaplan (1975) if not Duns Scotus!). The core meaning is advocacy of haecceities, i.e. non-qualitative thisness properties: almost always associated with the claim that every object has its own haecceity. But this core meaning is itself ambiguous, 19 and authors differ about the implications and connotations of ‘haecceity’—about how ‘thick’ a notion is advocated. Some discussion is therefore in order. 2.1.1 Haecceitism At first sight, there are (at least) three salient ways to construe haecceitism.1 For each, I give a description and one or more proponents. All three will refer to possible worlds, and will use the language of Lewis’s (1986) metaphysics (though it will not require a commitment to his form of modal realism, or to any form of modal realism for that matter). The first is the weakest and the second and third are equivalent; I favour the second and third (and prefer the formulation in the third). 1. Haecceitistic differences. Following Lewis (1986, p. 221), we may take haec- ceitism to be about the way possible worlds represent the modal properties of objects. It is the denial of the following supervenience thesis: a world’s repre- sentation of de re possibilities (that is, possibilities pertaining to particular objects) supervenes on the qualitative mosaic, i.e. the pattern of instanti- ation of qualitative properties and relations, within that world.2 Thus a haecceitist allows that two worlds, exactly alike in their qualitative features, may still disagree as to which object partakes in which property or relation.3 This version of haecceitism makes no further claims as to what exactly is left out of the purely qualitative representations, and so it is the weakest of the three haecceitisms. But since two qualitatively identical worlds may disagree about what they represent de re, they must differ in some non-qualitative 1I owe much of the discussion here to my conversations with Jeremy Butterfield and Fraser MacBride. 2This prompts the question, ‘What is a qualitative property or relation?’ That is not a question I will here attempt to answer. 3Note that this is a stronger position than one that just allows duplicate worlds, that is, several worlds exactly alike in their qualitative features. The existence of duplicate worlds does not entail haecceitism in Lewis’s sense, since, for each individual, every world in a class of mutual duplicates may represent the same dossier of de re possibilities. Lewis (1986, p. 224) himself refrains from committing either to duplicate worlds or a “principle of identity of indiscernibles” applied to worlds; though the question seems moot for anyone who does not take possible worlds to be concretely existing entities. 20 ways: ways which somehow represent (actual) objects in a way that does not rely on how things are qualitatively (whether accidentally or essentially) with those objects. There are two natural candidates for these representatives: the objects themselves (or some abstract surrogate for them), divorced from their qualitative clothing; or some non-qualitative properties which suitably track the objects across worlds. These two candidates prompt the second and third kinds of haecceitism (which, we argue below, are in fact equiva- lent). I see no sensible alternative to these two candidates for the missing representatives—though perhaps the difference could be taken as a primitive relation between worlds. But haecceitism in my first sense does not entail either of the haecceitisms below, though they each entail haecceitism in my first sense of the acceptance of haecceitistic differences. 2. Combinatorial independence. Lewis’s definition was inspired by a definition by Kaplan (1975, pp. 722-3); but Kaplan’s is stronger. It is phrased explic- itly in terms of trans-world identification.4 But I believe there is a version of haecceitism, clearer than Kaplan’s, defined in terms of combinatorial possi- bility; a version which is still stronger than Lewis’s, but does not commit one to claims about non-spatiotemporal overlap between worlds or trans-world mereological sums. According to this version of haecceitism, objects partake independently— that is, independently of each other and of the qualitative properties and relations—in the exhaustive recombinations which generate the full space of possible worlds. For example: with a domain of N objects there are: 2N - many possible property extensions (each of them distinct); 2N 2 -many distinct binary relation extensions (each of them distinct); so the number of distinct worlds5 containing N -objects, k monadic properties and l binary relations (and no other relations), is 2kN+lN 2 . 4Kaplan: haecceitism is ‘the doctrine that holds that it does make sense to ask—without reference to common attributes and behavior—whether this is the same individual in another possible world, that individuals can be extended in logical space (i.e., through possible worlds) in much the way we commonly regard them as being extended in physical space and time, and that a common “thisness” may underlie extreme dissimilarity or distinct thisnesses may underlie great resemblance.’ 5Or distinct equivalence classes of world-duplicates! 21 Combinatorial independence would appear to favour the doctrine that ob- jects “endure” identically through possible worlds, since it seems sensible to identify property extensions with sets of objects, and the same objects are added to or subtracted from the extensions in the generation of new worlds. But trans-world “perdurance”—the doctrine that trans-world “identity” (in fact a misnomer, according to the doctrine) is a relation holding between parts of the same trans-world continuant—can be accommodated without difficulty. (It is the commitment to there being a unique, objective trans- world continuation relation, and therefore something for a rigid designator to get a grip on, which distinguishes the perdurantist from those, like Lewis, who favour modal talk of world-bound objects: cf. Lewis (1986, pp. 218-20).) 3. Haecceitistic properties. Perhaps the most obvious form of haecceitism—and perhaps prima facie the least attractive—is the acceptance of haecceitis- tic properties. According to this view, for every object there is a property uniquely associated with it, which that object and no other necessarily pos- sesses, and which is not necessarily co-extensive with any (perhaps complex) qualitative property. An imprecise (and even quasi-religious) reading of these properties is as “inner essences” or “souls” (hence the view’s unpopularity). Perhaps a more precise, and less controversial, reading is that each thing pos- sesses some property which “makes” it that thing and no other: a property “in virtue of which” it is that thing (cf. Adams (1981, p. 13)). Whether or not that is in fact more precise, this reading implies a notion of ontological primacy—that the property comes in some sense “before” the thing—about which I remain silent. But besides, I disavow this implication. I intend our third version of haecceitism to be no more controversial than combinatorial independence; in fact I take this version of haecceitism to be equivalent to combinatorial independence. When I say that, according to this view, there is a haecceitistic property corresponding to each object, by ‘object’ is not meant ‘world-bound ob- ject’, which would entail a profligate multiplication of properties. Rather, these haecceitistic properties are envisaged as an alternative means to secur- ing trans-world continuation (understood according to either endurantism or 22 perdurantism). The motivation is as follows. Throughout this thesis, I will be a quidditist, meaning that I take for granted the trans-world identity of properties and relations. (I remain agnostic as to whether this trans-world identity is genuine identity, i.e. endurance of qual- ities; or whether there is instead a similarity relation applying to qualities between worlds (either second-order, applying directly to the qualities them- selves; or else first-order, applying to the qualities indirectly, via the objects that instantiate them), i.e. perdurance.) So for the sake of mere uniformity, haecceitism may be accommodated into my quidditistic framework by let- ting a trans-world monadic property do duty for each trans-world object. Haecceitism and anti-haecceitism alike can then be discussed neutrally in terms of world-bound objects and trans-world properties and relations, the difference between them reconstrued in terms of whether or not there are primitive monadic properties which allow one to simulate rigid designation.6 This position may be characterized syntactically as one that demands that, in a language adequate for expressing all facts, the primitive vocabulary contains a 1-place predicate ‘Nax’ for each envisaged trans-world object a.7 That a monadic property can do duty for a trans-world object—or, to rephrase in terms of the object-language, that a monadic predicate can do duty for a rigid designator—without any loss (or gain!) in expressive adequacy, is well known (cf. Quine (1960, §38). Given a haecceitistic pred- icate ‘Nax’, one can introduce the corresponding rigid designator by defi- nition: a := ιx.Nax; and conversely, given a rigid designator ‘a’ and the 6Anti-quidditists (like Black 2000) can still use our framework, by populating the domain of objects with properties and relations, now treated as the value of first-order variables, and using the new predicate or predicates ‘has’, so that ‘Fa’ becomes ‘a has F ’ (cf. Lewis 1970b). Anti-quidditism may then amount to what I later (Chapter 3) call ‘anti-haecceitism’, but applied to these hypostatized qualities. However, this purported anti-quidditism must be “quidditist” about the ‘has’ relation(s), a situation that clearly cannot be remedied by hypostatizing again, on pain of initiating a Bradley-like regress. There is no space to pursue the issue here. But we endorse the view of Lewis (2002) that ‘has’ is a ‘non-relational tie’, and speculate that, if it is treated like a relation in the logic, then it is better off treated as one whose identity across worlds is not in question—that is: treated “quidditistically”. 7To simplify, I assume that a haecceitist is a haecceitist about every object. This leaves some (uninteresting) logical space between haecceitism and what we later characterise as anti- haecceitism (see the end of Section 3.1). 23 identity predicate ‘=’, one can likewise introduce the corresponding haec- ceitistic predicate: ∀x(Nax ≡ x = a). We therefore urge the view that the difference between the two stronger versions of haecceitism is merely no- tational, meaning that the addition of the haecceitistic predicate ‘Nax’ to one’s primitive vocabulary ontologically commits one to no more, and no less, than the addition of the name ‘a’, together with all the machinery of rigid designation.8 From now on, I will take haecceitism in a sense stronger than the first, Lewisian version. The second and third, stronger versions are equivalent, but I favour the notational trappings of the third version, i.e. the acceptance of haecceitistic properties. Thus when I later ban names from the object-language (in Section 2.2.1), this ought to be seen not as a substantial restriction against the haecceitist, but merely a convenient narrowing of notational options for the sake of a more unified presentation. We simply require the haecceitist to express her position though the adoption of haecceitistic predicates, though all are at liberty to— indeed, all should—read each instance of the haecceitist’s ‘Nax’ as ‘x = a’. A concern may remain: how can ontological commitment to a property be equivalent to ontological commitment to a (trans-world) object? There is no mys- tery, once we lay down some principles for what ontological commitment to prop- erties involves. I take it that ontological commitment to a collection of properties, relations and objects entails a commitment to all the logical constructions thereof, because instantiation of the logical constructions can be defined away without residue in terms of instantiation and the existence of the original collection. (For example, for any object, the complex predicate ‘Fx∧Gx’ is satisfied by that object 8By saying that I intend the acceptance of haecceitistic properties to be equivalent to com- binatorial independence, I do not intend to rule out as counting as a (strong) version of the acceptance of haecceitistic properties the view described above, viz. that the haecceities are in some sense ontologically “prior” to the objects, and serve to “ground their identity” (whatever that may mean). That view would entail combinatorial independence between objects and other objects, and between objects and the qualitative properties and relations (though not, of course, between objects and their haecceities), and would equally well be served by rigid designators as by the addition of haecceitistic predicates to the primitive vocabulary. The point is that the second and third versions of haecceitism match completely in logical strength, so they are equally consistent with more metaphysically ambitious views which seek to “ground” trans-world identity in non-qualitative properties. 24 just in case both ‘Fx’ and ‘Gx’ are satisfied by it.) So ontological commitment to logical constructions is no further commitment at all. Now, ontological com- mitment to certain objects is revealed clearly enough: one need only peer into the domain of quantification to see if they are there. Ontological commitment to complex properties and relations is equally straightforward, being a matter of commitment to their components. But what about ontological commitment to the properties and relations taken as primitive, i.e. not as logical constructions—what does that involve? Well, since ontological commitment to objects is clear enough, let us use that: let us say that ontological commitment to the primitive proper- ties and relations is a commitment to their being instantiated by some object or objects. With these principles laid down, we can now prove that commitment to the trans-world continuant a and commitment to the haecceitistic property, being a, entail each other. Left to right: We can take two routes. First route: Commitment to any object at all (i.e. a non-empty domain) entails commitment to the identity relation, which is instantiated by everything. So commitment to the trans-world continuant a entails commitment to the identity relation, since a = a. But the property being a is just a logical construction out of the identity relation and the trans-world continuant a, so commitment to the trans-world continuant a entails commitment to the property being a. Second route: Commitment to the trans- world continuant a entails commitment to the property being a being instantiated. But that is just to say that commitment to the trans-world continuant a entails commitment to the property being a. Right to left: Commitment to the property being a entails either: (i) a commitment to its being instantiated (if taken as primitive); or (ii) a commitment to the entities of which it is a logical construction (if taken as a logical construction). If (i), then we are committed to something’s being a, that is, the existence of a. If (ii), and ‘being a’ is understood properly as containing a rigid designator, not as an abbreviated definite description a` la Russell, then we are committed to its logical components, i.e. the identity relation, and a itself. QED. To sum up: the acceptance of haecceitistic differences (the first version of haec- ceitism, above) need not commit one to either combinatorially independent trans- 25 world objects (our second version), or to non-qualitative properties that could do duty for them (our third version), though the two latter doctrines are perhaps the most natural way of securing haecceitistic differences, and may themselves be considered as notational variants of each other. I stipulate that we mean by ‘haec- ceitism’ one of the stronger versions, and for reasons purely to do with uniformity of presentation, I stipulate that the advocacy of this stronger version of haecceitism be expressed through the acceptance of haecceitistic properties. This version of haecceitism will be further developed, along with three other metaphysical theses, in Chapter 3. 2.2 A logical perspective on identity So to sum up, our aim is to use the Hilbert-Bernays account as a spring-board so as to give a precise “logical geography” of discernibility. This logical geography will be in terms of the syntax of a formal first-order language. But we will also relate our definitions to the idea of permutations on the domain of quantification, and to the idea of these permutations being symmetries. These relations seem not to have been much studied in the recent philosophical literature about the Hilbert-Bernays account; and we will see that they turn out to be subtle—some natural conjectures are false.9 2.2.1 The Hilbert-Bernays account What I will call the Hilbert-Bernays account of the identity of objects, treated as (the values of variables) in a first-order language, goes as follows; (cf. Hilbert and Bernays (1934, §5) : who in fact do not endorse it!), Quine (1970, pp. 61-64) and Saunders (2003a, p. 5)). The idea is that there being only finitely many primitive predicates enables us to capture the idea of identity in a single axiom. In fact, the axiom is a biconditional in which identity is equivalent to a long conjunction of statements that predicates are co-instantiated. The conjunction exhausts, in 9We thank N. da Costa and J. Ketland for alerting us to their results in this area, which we (culpably!) had missed. Further references below. 26 a natural sense, the predicates constructible in the language; and it caters for quantification in predicates’ argument-places other than the two occupied by x and y. In detail, we proceed as follows. Suppose that F 1i is the ith 1-place predicate, G2j is the jth 2-place predicate, and H 3 k is the kth 3-place predicate; (we will not need to specify the ranges of i, j, k). Suppose that the language has no names, or function symbols, so that predicates are the only kind of non-logical vocabulary. Then the biconditional will take the following form: ∀x∀y { x = y ≡ [ . . . ∧ (F 1i x ≡ F 1i y) ∧ . . . . . . ∧ ∀z ((G2jxz ≡ G2jyz) ∧ (G2jzx ≡ G2jzy)) ∧ . . . . . . ∧ ∀z∀w ( (H3kxzw ≡ H3kyzw) ∧ (H3kzxw ≡ H3kzyw) ∧ (H3kzwx ≡ H3kzwy) ) ∧ . . . ]} (HB) (For primitive predicates, I will usually omit the brackets and commas often used to indicate argument-places.) Note that for each two-place predicate, there are two biconditionals to include on the right-hand side; and similarly for a three-place predicate. The general rule is: n biconditionals for an n-place predicate.10 This definition of the Hilbert-Bernays account prompts three comments. 1. Envisaging a rich enough language:— The main comment is the obvious one: since the right hand side (in square brackets) of (HB) defines an equivalence relation—which from now on I will call ‘indiscernibility’ (or for emphasis: ‘indiscernibility by the primitive vocabulary’)—discussion is bound to turn on the issue whether this relation is truly identity of the objects in the do- main. Someone who advocates (HB) is envisaging a vocabulary rich enough 10We do not need to include explicitly on the right-hand side clauses with repeated instances of x or y, such as (G2jxx ≡ G2jyy), since these clauses are implied by the conjunction of other relevant biconditionals. For example, from ∀z(G2jxz ≡ G2jyz) we have, in particular, that (G2jxx ≡ G2jyx). And from ∀z(G2jzx ≡ G2jzy) we have, in particular, that (G2jyx ≡ G2jyy). It follows that (G2jxx ≡ G2jyy). A similar chain of arguments applies for an arbitrary n-place predicate. 27 (or: a domain of objects that is varied enough, by the lights of the vocabulary chosen) to discern any two distinct objects, and thereby force the equivalence relation given by the right hand side of (HB) to be identity. This comment can be developed in six ways. (i) Indiscernibility has the formal properties of identity expressible in first- order logic, i.e. being an equivalence relation and substitutivity. (Cf. Equa- tion (2.1) in (i) of comment 3 below; and for details, Ketland (2009, Lemma 12)). (ii) Let us call an interpretation of the language, comprising a domain D and various subsets of D,D2 etc. a ‘structure’; (since in the literature on identity, ‘interpretation’ also often means ‘philosophical interpreta- tion’). Then: if in a given structure, the identity relation is first-order definable, then it is defined by indiscernibility, i.e. the right hand side of (HB) (Ketland (2009, Theorem 16)). (iii) On the other hand: assuming ‘=’ is to be interpreted as the identity relation (cf. (i) in 3 below), there will in general be structures in which the leftward implication of (HB) fails; i.e. structures with at least two objects indiscernible from each other. We say ‘in general’ since from the view-point of pure logic, ‘=’ might itself be one of the 2-place predicates G2j : in which case, the leftward implication trivially holds. (iv) The last sentence leads to the wider question whether the interpreta- tion given to some of the predicates F,G,H etc. somehow presupposes identity, so that the Hilbert-Bernays account’s reduction of identity is, philosophically speaking, a charade. This question will sometimes crop up below (e.g. for the first time, in footnote 12); but I will not need to address it systematically. Here I just note as an example of the question, the theory of pure sets. It has only one primitive predicate, ‘∈’, and the axiom (HB) for it logically entails the axiom of extensionality. But one might well say that this only gives a genuine reduction of identity if one can understand the intended interpretation of ‘∈’ without a prior understanding of ‘=’. 28 (v) There is the obvious wider question, why I restrict my discussion to first-order languages. Here my main reply is twofold: (a) first-order lan- guages are favoured by the incompleteness of higher-order logics, and I am anyway sympathetic to the view that first-order logic suffices for the formalisation of physical theories (cf. Boolos & Jeffrey (1974, p. 197); Lewis (1970b, p. 429)); and (b) the Hilbert-Bernays account is thus restricted—and though, as I emphasized, I do not endorse it, it forms a good spring-board for discussing kinds of indiscernibility. There are also some basic points about identity in second-order logic, which I should register at the outset. Many discussions (especially textbooks: e.g. van Dalen (1994, p. 151); Boolos & Jeffrey (1974, p. 280) take the principle of the identity of indiscernibles to be expressed by the second-order for- mula (∀P (Px ≡ Py) ⊃ x = y): which is a theorem of (any deductive system for) second-order logic with a sufficiently liberal comprehension scheme. But even if one is content with second-order logic, this result does not diminish the interest of the Hilbert-Bernays account (or of classifications of kinds of discernibility based on it). For the second- order formula is a theorem simply because the values of the predicate variable include singleton sets of elements of the domain (cf. Ketland (2006, p. 313)). And allowing such singleton sets as properties of course leads back both to haecceitism, discussed in Section 2.1, and to (iv)’s question of whether understanding the primitive predicates requires a prior understanding of identity. In any case, I will discuss the princi- ple of the identity of indiscernibles from a philosophical perspective in Chapter 3. (vi) Finally, there is the question what the proponent of the Hilbert-Bernays account is to do when faced with a language with infinitely many primi- tive predicates. The right hand side of (HB) is constrained to be finitely long, so in the case of infinitely many primitives it is prevented from cap- turing all the ways two objects can differ. Here I see two roads open to the Hilbert-Bernays advocate: (a) through parametrization, infinitely many n-adic primitives, say R1xy,R2xy, . . . may be subsumed under a single, new (n+ 1)-adic primitive, say Rxyz, where the extra argument 29 place is intended to vary over an index set for the previous primitives; and (b) one may resort to infinitary logic, which allows for infinitely long formulas—suggesting, as regards (HB), in philosophical terms, a supervenience of identity rather than a reduction of it. (I will consider infinitary logic just once below, towards the end of Section 2.4.3.) To sum up these six remarks: I see the Hilbert-Bernays account as intending a reduction of identity facts to qualitative facts—as proposing that there are no indiscernible pairs of objects. This theme will recur in what follows. Indeed, the next two comments relate to the choice of language. 2. Banning names:— From now on, it will be clearest to require the language to have no individual constants, nor function symbols, so that the non-logical vocabulary contains only predicates. But this will not affect my arguments: they would carry over intact if constants and function symbols were allowed. As I see it, only the haecceitist is likely to object to this apparent limita- tion in expressive power. But here, Section 2.1’s discussion of haecceitism comes into its own. For even with no names, the haecceitist has to hand her thisness predicates Nax,Nbx etc., with which to refer to objects by defi- nite descriptions. Thus I propose, following Quine (1960, §§37-39), that we, and in particular the haecceitist, replace proper names by 1-place predicates; (each with an accompanying uniqueness axiom; and with the predicates then shoe-horned into the syntactic form of singular terms, by invoking Russell’s theory of descriptions).11 And as emphasised in Section 2.1, Nax etc. are to be thinly construed: the predicate Nax commits one to nothing beyond what the predicate a = x commits one to. Thus the presence of these predicates in the non-logical vocabulary means that the haecceitist should have no qualms about endorsing the Hilbert-Bernays account—albeit in letter, rather than in spirit.12 11Then any sentence S =: Φ(a) containing the name a, where the 1-place formula Φ(x) contains no occurrence of a, may be replaced by the materially equivalent sentence ∀x (Nax ⊃ Φ(x)), which contains no occurrence of the name a. Note also that Saunders (2006, p. 53) limits his inquiry to languages without names; but no Quinean trick is invoked. Robinson (2000, p. 163) calls such languages ‘suitable’. 12For the spirit of the Hilbert-Bernays account is a reduction of identity facts to putatively 30 A point of terminology: Though I ban constants from the formal language, I will still use a and b as names in the meta-language (i.e. the language in which I write!) for the one (or two!) objects, with whose identity or diversity we are concerned. I will also always use ‘=’ in the meta-language to mean identity! 3. ‘=’ as a logical constant?:— It is common in the philosophy of logic to dis- tinguish two approaches by which formal languages and logics treat identity: (i) ‘=’ is a logical constant in the sense that it is required, by the definition of the semantics, to be interpreted, in any domain of quantification, as the identity relation. Then for any formula (open sentence) Φ(x) with one free variable x, the formula ∀x∀y (x = y ⊃ (Φ(x) ≡ Φ(y))) (2.1) is a logical truth (i.e. is true in every structure). Thus on this approach, the rightward implication in (HB) is a logical truth. But the leftward implication in (HB) is not. For as discussed in com- ment 1, the language may not be discerning enough. More precisely: there are structures in which (no matter how rich the language!) the leftward implication fails.13 (ii) ‘=’ is treated like any other 2-place predicate, so that its properties flow entirely from the theory with which we are concerned, in particular its axioms if it is an axiomatized theory. Of course, we expect our theory to impose on ‘=’ such properties as being an equivalence relation qualitative facts, about the co-instantiation of properties. Indeed: according to the approach in which ‘=’ is not a logical constant (cf. comment 3(ii)), acceptance of (HB) entails that the language with the equality symbol ‘=’ is a definitional extension of the language without it. However, representing each haecceity by a one-place primitive predicate, accompanied by a uniqueness axiom, assumes the concept of identity through the use of ‘=’ in the axiom: making this reduction of identity, philosophically speaking, a charade. By contrast, if only genuinely qualitative properties are expressed by the non-logical vocabulary, the philosophical reduction of identity facts to qualitative facts can succeed—provided, of course, that the language is rich enough or the domain varied enough. 13Such is Wiggins’ (2001, pp. 184-185) criticism of the Hilbert-Bernays account as formulated by Quine (1960, (1970). 31 (cf. Ketland (2009, Lemmas 8-10)). We can of course go further towards capturing the intuitive idea of identity by imposing every instance of the schema, eq. 2.1. So on this approach, the Hilbert-Bernays account can be regarded as a proposed finitary alternative to imposing eq. 2.1, a proposal whose plausibility depends on the language being rich enough. Of course, independently of the language being rich: imposing (HB) in a language with finitely many primitives implies as a theorem the truth of eq. 2.1.14 To sum up this Section: the conjunction on the right hand side of (HB) makes vivid how, on the Hilbert-Bernays account, objects can be distinct for different reasons, according to which conjunct fails to hold. In Section 2.3, I will introduce a taxonomy of these kinds. In fact, this taxonomy will distinguish different ways in which a single conjunct can fail to hold. The tenor of that discussion will be mostly syntactic. So I will complement it by first discussing identity, and the Hilbert-Bernays account, in terms of permutations on the domain of quantification. 2.2.2 Permutations on domains I will now discuss how permutations on a domain of objects can be used to express qualitative similarities and differences between the objects. More precisely: I will define what it is for a permutation to be a symmetry of an interpretation of the language, and relate this notion to the Hilbert-Bernays account. Definition of a symmetry Let D be a domain of quantification, in which the predicates F 1i , G 2 j , H 3 k etc. get interpreted. So writing ‘ext’ for ‘extension’, ext(F 1i ) is, for each i, a subset of D; and ext(G2j) is, for each j, a subset of D ×D = D2; and ext(H3k) is, for each k, a subset of D3 etc. For the resulting interpretation of the language, i.e. D together 14Hilbert and Bernays (1934, p. 186) show that ‘=’ as defined by (HB) is (up to co- extension!) the only (non-logical) two-place predicate to imply reflexivity and every instance of the schema, eq. 2.1. The argument is reproduced in Quine (1970, pp. 62-63). 32 with these assigned extensions, we write I. I shall also call such an interpretation, a ‘structure’. Now let pi be a permutation of D.15 I now define a symmetry as a permutation that “preserves all properties and relations”. So I say that pi is a symmetry (aka automorphism) of I iff all the extensions of all the predicates are invariant under pi. That is, using o1, o2, ... as meta-linguistic variables: pi is a symmetry iff: ∀o1, o2, o3 ∈ D, ∀i, j, k : o1 ∈ ext(F 1i ) iff pi(o1) ∈ ext(F 1i ); and 〈o1, o2〉 ∈ ext(G2j) iff 〈pi(o1), pi(o2)〉 ∈ ext(G2j); and 〈o1, o2, o3〉 ∈ ext(H3k) iff 〈pi(o1), pi(o2), pi(o3)〉 ∈ ext(H3k); (2.2) and similarly for predicates with four or more argument-places.16 Relation to the Hilbert-Bernays account Let me now compare this definition with the Hilbert-Bernays account. For the moment, I make just two comments, (1) and (2) below. They dispose of natural conjectures, about symmetries leaving invariant the indiscernibility equivalence classes. That is, the conjectures are false: the first conjecture fails because of the somewhat subtle notions of weak and relative discernibility, which will be central later; and the second is technically a special case of the first.17 In Section 2.4, I will discuss how the conjectures can be mended: roughly speaking, we need to replace indiscernibility by a weaker and “less subtle” notion, called ‘absolute indiscernibility’. 15I note en passant that since a permutation is a bijection, the definition of permutation involves the use of the ‘=’ symbol; so that which functions are considered to be permutations is subject to one’s treatment of identity. But no worries: as I noted in comment 2 of Section 2.2.1, this definition is cast in the meta-language! 16So a haecceitist (cf. comment 2 in Section 2.2.1) will take only the identity map as a symmetry—unless they stipulate that haecceitistic properties are exempt from the definition of symmetry. 17My two comments agree with Ketland’s results and examples (2006, Theorem (iii) and example in footnote 17) or (2009, Theorem 35(a) and example). But I will not spell out the differences in jargon or examples, except to report that Ketland calls a structure ‘Quinian’ iff the leftward implication of (HB) holds in it, i.e. if the identity relation is first-order definable, and so (cf. comment 1(ii) of Section 2.2.1) defined by indiscernibility, i.e. by the right hand side of (HB). 33 (1): All equivalence classes invariant?:— Suppose that we adopt the relativiza- tion to an arbitrary language: so we do not require the language to be rich enough to force indiscernibility to be identity. Then one might conjecture that, for each choice of language, a permutation is a symmetry iff it leaves invariant (also known as: fixes) each indiscernibility equivalence class. That is: pi is a symmetry of the structure I iff each member of each equivalence class ⊂ dom(I) is sent by pi to a member of that same equivalence class. In fact, this conjecture is false. The condition, leaving invariant the indiscerni- bility equivalence classes, is stronger than being a symmetry. I first prove the true implication, and then give a counterexample to the converse. So suppose pi leaves invariant each indiscernibility class; and let 〈o1, o2, ..., on〉 be in the extension ext(Jn) of some n-place predicate Jn. Since pi leaves invariant the indiscernibility class of o1, it follows that 〈pi(o1), o2, ..., on〉 is also in ext(Jn). (For if not, (HB)’s corresponding conjunct, i.e. the conjunct for the first argument-place of the predicate Jn, would discern o1 and pi(o1)). From this, it follows similarly that since pi fixes the indiscernibility class of o2, 〈pi(o1), pi(o2), ..., on〉 is also in ext(Jn). And so on: after n steps, we conclude that 〈pi(o1), pi(o2), ..., pi(on)〉 is in ext(Jn). Therefore pi is a symmetry. Philosophical remark: one way of thinking of the Hilbert-Bernays account triv- ializes this theorem. That is: according to comment 1 of Section 2.2.1, the propo- nent of this account envisages that the indiscernibility classes are singletons. So only the identity map leaves them all invariant, and trivially, it is a symmetry. (Compare the discussion of haecceitism in footnotes 12 and 16.) Counterexample to the converse: Consider the following structure, whose do- main comprises four objects, which we label a to d. The primitive non-logical vocabulary consists of just the 2-place relation symbol R. R is interpreted as having the extension ext(R) = {〈a, b〉, 〈a, c〉, 〈a, d〉, 〈b, a〉, 〈b, c〉, 〈b, d〉} . Now let ‘=’ be defined by the Hilbert-Bernays axiom (HB). From this the reader 34 can check that ‘=’ has the extension ext(=) = {〈a, a〉, 〈b, b〉, 〈c, c〉, 〈d, d〉, 〈d, c〉, 〈c, d〉} ; i.e. c and d are indiscernible. So ‘=’ is not interpreted as identity (cf. the relativiza- tion of ‘=’ to the language, discussed in 3(ii) of §2.2.1). The relation ‘=’ carves the domain into three equivalence classes: [a] = {a}, [b] = {b} and [c] = [d] = {c, d}. Now consider the permutation pi, whose only effect is to interchange the objects a and b; i.e. pi := ( abcd bacd ) ≡ (ab). This permutation is a symmetry, since it preserves the extension of R according to the requirement (2.2); yet it does not leave invariant the equivalence classes [a] and [b] (Fig. 2.1).18 Figure 2.1: A counterexample to the claim that symmetries leave invariant the indiscernibility classes. In this structure, drawn on the left, the permutation pi which swaps a and b is a symmetry; but a and b form their own separate equivalence classes under ‘=’, as shown on the right. (2): Only the trivial symmetry?:— Suppose now that in I, the predicate ‘=’ is interpreted as identity; and that (HB) holds in I. So we are supposing that the objects are various enough, the language rich enough, that indiscernibility in I is identity. Or in other words: the indiscernibility classes are singletons. On these suppositions, one might conjecture that that the only symmetry is the trivial one, i.e. the identity map id : D → D. (Such structures are often called ‘rigid’ (Hodges 18Two remarks. (1): Agreed, this counterexample could be simplified. A structure with just a and b, with ext(R) = {〈a, b〉, 〈b, a〉} has a, b discernible, but the swap a 7→ b, b 7→ a is a symmetry. But this example will also be used later. (2) Accordingly, in the counterexample, (ab)(cd) would work equally well: i.e. it also is a symmetry that does not leave invariant [a] and [b]. Looking ahead: Theorem 1 in Section 2.4.1 will imply that {a, b} and {c, d} are each subsets of absolute indiscernibility classes; in fact, each is an absolute indiscernibility class. 35 (1997, p. 94)).) In fact, this is false. Comment (1) has just shown that there are symmetries that do not leave invariant the indiscernibility classes. Our present suppositions have now collapsed these indiscernibility classes into singletons. So there will be symmetries which do not leave invariant the singletons, i.e. are not the identity map on D. To be explicit, consider the structure, and its indiscernibility classes, drawn in Fig. 2.2. Figure 2.2: A counterexample to the claim that if the indiscernibility classes are singletons, the only symmetry is the identity map. In this structure, the indis- cernibles of Figure 2.1 have been identified. a and b are still distinct (they are discernible) but are swapped by the symmetry pi. 2.3 Four kinds of discernment I turn to defining the different ways in which two distinct objects can be discerned in a structure. These kinds of discernment will be developed (indirectly) in terms of which conjuncts on the RHS of the Hilbert-Bernays axiom (HB) are false in the structure.19 19Other authors, notably Muller and Saunders 2008), consider the discernment of two objects by a theory (say T ), so that the RHS of (HB), applied to the two objects in question, is a theorem of T ; whereas our concern is the discernment of two objects in a structure. My focus on structures is necessary: given our ban on names, the sentence expressing the satisfaction of the RHS of (HB) by the two objects cannot even be written! 36 2.3.1 Three preliminary comments (1) Precursors: Broadly speaking, my four kinds of discernment follow the dis- cussions by Quine (1960, 1970, 1976a) and Saunders (2003a, 2003b, 2006). Quine (1960, p. 230) endorses the Hilbert-Bernays account of identity and then distin- guishes what he calls absolute and relative discernibility. His absolute discernibility will correspond to (the disjunction of) my first two kinds—and I will follow him by calling this disjunction ‘absolute’. Besides, his relative discernibility will corre- spond to (the disjunction of) my third and fourth kinds. But I will follow Saunders ((2003a, p. 5); (2003b, pp. 19-20); (2006, p. 5)) by reserving ‘relative’ for the third kind, and using ‘weak’ for the fourth. (So for me ‘non-absolute’ will mean ‘relative or weak’.)20 (2) Suggestive labels: I will label these kinds with words like ‘intrinsic’ which are vivid, but also connote metaphysical doctrines and controversies (e.g. Lewis 1986, pp. 59-63). I disavow the connotations: the official meaning is as defined, and so is relative to the interpretation of the non-logical vocabulary. (3): Two pairs yield four kinds: The four kinds of discernment arise from two pairs. We begin by distinguishing between a formula with one free variable (labelled 1) and a formula with two free variables (labelled 2).21 Each of these cases is then broken down into two subcases (labelled a and b) yielding four cases in all: labelled 1(a) to 2(b). We will also give the four cases mnemonic labels: e.g. 1(a) will also be called (Int) for ‘intrinsic’. The intuitive idea that distinguishes sub-cases will be different for 1 and 2. For 1, the idea is to distinguish whether discernibility depends on a relation to another object; while for 2, the idea is to distinguish whether discernibility depends on an asymmetric relation. Both these ideas are semantic, and even a bit vague. But the 20Quine (1976a, p. 113) defines what he calls grades of discriminability, which is a spectrum of strength. Saunders (2006, pp. 19-20) agrees that there is such a spectrum of strength, although in his (2003a, p. 5) he makes the three categories ‘absolute’, ‘relative’ and ‘weak’ mutually exclusive. I say ‘kinds’ not ‘categories’ or ‘grades’ to avoid the connotation of mutual exclusion or a spectrum of strength. 21We thank Leon Horsten for the observation that the notion of discernibility may be pa- rameterized to other objects (so that, e.g., we might say that a is discernible from b relative to c, d, . . .), which would involve formulas with more than two variables. The idea seems to us workable, but we will not pursue it here. 37 definitions of the sub-cases will be syntactic, and precise—and we will therefore remark that they do not completely match the intuitive idea. As announced in comment 2 of Section 2.2.1, a and b will be names in the meta-language (in which I am writing) for the one or two objects with which we are concerned. 2.3.2 The four kinds defined 1(a) (Int) 1-place formulas with no bound variables, which apply to only one of the two objects a and b. This of course covers the case of primitive 1-place predicates, e.g. Fx: so that for example, a ∈ ext(F ) but b /∈ ext(F ), or vice versa. But we also intend this case to cover 1-place formulas arising by slotting into a polyadic formula more than one occurrence of a single free variable, while the polyadic formula nevertheless does not contain any bound variables. The intent here is to exclude formulas which quantify over objects other than the two we are concerned with. So this case will include formulas such as: Rxx and Fx ∧ Hxxx; (R,H primitive 2-place and 3-place predicates, respectively; not abbreviations of more complex open sentences). But it will exclude formulas such as ∀z(Fz ⊃ Rzx), which contain bound variables.22 I will say that two objects that do not share some monadic formula in this sense are discerned intrinsically, since their distinctness does not rely on any relation either object holds to any other. An everyday example, taking ‘is spherical’ as a primitive 1-place predicate, is given by a ball and a die. Another example, with Rxy the primitive 2-place predicate ‘loves’ (so that Rxx is the 1-place predicate ‘loves his- or herself’) is Narcissus ∈ ext(R ∗ ∗), 22Recall my ban on individual constants ((2) in Section 2.2.1). If we had instead allowed them, this sub-case 1(a) would be defined so as to also exclude all formulas containing any constant, including a and b. The exclusion of formulas such as Rcx, which refer to a third object, is obviously desirable, given the intuitive idea of discernment by intrinsic properties. However, the exclusion of formulas involving only the constants a and-or b may be more puzzling. My rationale is that, for any formula of the type Rax, Rbx, etc. which is responsible for discerning two objects, there will be an alternative formula (either Rxx or Rxy) which we would instead credit for the discernment, and which falls under one of the three other kinds. 38 but (alas) Echo /∈ ext(R ∗ ∗). I shall say that any pair of objects discerned other than by 1(a)—i.e. dis- cerned by 1(b), or by 2(a), or by 2(b) below—are extrinsically discerned. 1(b) (Ext) 1-place formulas with bound variables, which apply to only one of the two objects a and b. That is: this kind contains polyadic formulas that do contain bound variables. So it contains formulas such as ∀z(Fz ⊃ Rzx). And a and b are discerned by such formulas if: for example, a ∈ ext(∀z(Fz ⊃ Rz∗)) but b /∈ ext(∀z(Fz ⊃ Rz∗)). I will say that they have been discerned externally. An example, taking F = ‘is a man’, R = ‘admires’ is: Cleopatra ∈ ext(∀z(Fz ⊃ Rz∗)) but Caesar /∈ ext(∀z(Fz ⊃ Rz∗)). (Recall that I reserve the term extrinsic to cover all three kinds 1(b), 2(a), 2(b): so external discernment is more specific than extrinsic.) The intent is that in this kind of discernment, diversity follows from the relations the two objects a and b have to other objects. However, as I said in (3) of Section 2.3.1, the precise syntactical definition cannot be expected to match exactly the intuitive idea. And indeed, there are examples of external discernment where the relevant value of the bound variable in the discerning formula is in fact a or b, even though this is invisible from the syntactic perspective. (In the example just given, the universal quantifier in ∀z(Fz ⊃ Rzx) quantifies over a domain that includes Caesar himself.)23 I will say that two objects discerned by a formula either of kind (Int) or of kind (Ext) are absolutely discerned. Note that a pair of objects could be both intrinsically and externally discerned. But since (Ext) is intuitively a “weaker” form of discernment, I shall sometimes say that a pair of objects that are externally, but not intrinsically, discerned, are merely externally discerned. Interlude: Individuality and absolute discernment. I will also say that an object that is absolutely discerned from all other objects is an individual or has individu- 23As to my ban on individual constants: if we had instead allowed them, this sub-case 1(b) would be defined so as to also exclude formulas containing a and b, but to allow other constants c, d etc., so as to capture the idea of discernment by relations to other objects. But as in the case of bound variables, it could turn out that the “third” object picked out is in fact a or b. Cf. footnote 22. 39 ality. Note that if an object is an individual, some or all of the other objects might themselves fail to be individuals (cf. figure 2.3). Figure 2.3: An object’s being an individual requires its being absolutely discerned from all others, but not their being absolutely discerned from anything else. Here, c is absolutely discerned from both a and b, e.g. by the formula ∀z(Rzx ⊃ Rxz), while a and b are themselves non-individuals. Being an individual is tantamount to being the bearer of a uniquely instantiated definite description: where ‘tantamount to’ indicates a qualification. The idea is: given an individual, we take seriatim the formulas that absolutely discern it from the other objects in the structure, and conjoin them and so construct a definite description that is instantiated only by the given individual. The qualification is that in an infinite domain, there could be infinitely many ways that a given individual was absolutely discerned from all the various others: think of how a finite vocabulary supports arbitrarily long formulas, and so denumerably many of them. Thus in an infinite domain the above “seriatim” procedure might yield an infinite conjunction—preventing a finitely long uniquely instantiated definite description.24 I will examine absolute discernibility in Section 2.4. For the moment, I return to our four kinds of discernibility: i.e. to presenting the last two kinds. End of Interlude. 2(a) (Rel) Formulas with two free variables, which are satisfied by the two objects a and b in one order, but not the other. For example, for the formulas Rxy and ∃zHxzy, we have: 〈a, b〉 ∈ ext(R), but 〈b, a〉 /∈ ext(R); and 〈a, b〉 ∈ 24A terminological note: Saunders (2003b, p. 10) says that an object that is the bearer of a uniquely instantiated definite description is ‘referentially determinate’, and Quine (1976a, p. 113) calls such an object ‘specifiable’. So, modulo my qualification about infinite domains, these terms correspond to my (and Muller & Saunders’ (2008)) use of ‘individual’. Cf. also comment 1(vi) in Section 2.2.1. 40 ext(∃zH ∗ z•), but 〈b, a〉 /∈ ext(∃zH ∗ z•). Here the diversity of a and b is an extrinsic matter (both intuitively, and according to my definition of ‘extrinsic’, which is discernment by any means other than (Int)), since it follows from their relation to each other. But it is not a matter of a relation to any third object. Following Quine (1960, p. 230), I will say that objects so discerned are relatively discerned. And as above, I will say that objects that are relatively discerned but neither intrinsically nor externally discerned, are merely relatively discerned. Merely relatively discerned objects are never individuals in my sense (viz. absolutely discerned from all other objects).25 2(b) (Weak) Formulas with two free variables, which are satisfied by the two objects a and b taken in either order, but not by either object taken twice. For example, for the formulas Rxy and ∃zHxzy, we have: 〈a, a〉, 〈b, b〉 /∈ ext(R), but 〈a, b〉, 〈b, a〉 ∈ ext(R); and 〈a, a〉, 〈b, b〉 /∈ ext(∃zH ∗ z•), but 〈a, b〉, 〈b, a〉 ∈ ext(∃zH ∗ z•). (We say ‘but not by either object taken twice’ to prevent a and b being intrinsically discerned.) Again, the diversity of a and b is extrinsic, but does not depend on a third object; rather diversity follows from their pattern of instantiation of the relation R. I call objects so discerned weakly discerned. And I will say that objects that are weakly discerned but neither intrinsically nor externally nor relatively discerned (i.e. fall outside (Int), (Ext), (Rel) above), are merely weakly discerned. Objects which are discerned merely weakly are not individuals, in my sense (since they are not absolutely discerned). Max Black’s famous example of two spheres a mile apart (1952, p. 156) is an example of two such objects. For the two spheres bear the relation ‘is a mile away from’, one to another; but not each to itself. The irony is that Black, apparently unaware of weak discernibility, proposes his duplicate spheres as a putative example of two ob- 25Note that this kind of discernment does not require the discerning formula, for example Gxy, to be asymmetric for all its instances; i.e., we do not require ∀x∀y(Gxy ⊃ ¬Gyx). In this I agree with Saunders (2003a, p. 5) and Quine (1960, p. 230). My rationale is that intuitively, this kind of discernment does not require anything about the global pattern of instantiation of the relation concerned. The same remark applies to (Weak) below, where now I differ from Saunders (2003a, p. 5), who demands that the discerning relation be irreflexive. Nevertheless, I adopt Saunders’/Quine’s word ‘weak’. 41 jects that are qualitatively indiscernible (and therefore as a counterexample to the principle of the identity of indiscernibles).26 I will also say that two objects that are not discerned by any of our four kinds (i.e. by no 1-place or 2-place formula whatsoever) are indiscernible. For emphasis, I will sometimes call such a pair utterly indiscernible. In particular, I will say ‘utter indiscernibility’ when contrasting this case with the failure of only one (or two or three) of our four kinds of discernibility. Of course, utter indiscernibles are only accepted by someone who denies the Hilbert-Bernays account. 2.4 Absolute indiscernibility: some results In Section 2.2.2, we saw that a permutation leaving invariant the indiscernibility classes must be a symmetry; then we gave a counterexample to the converse state- ment, and to a related conjecture that if indiscernibility is identity, there is only the trivial symmetry. But now that I have defined absolute discernibility (viz. as the disjunction, (Int) or (Ext)), we can ask about the corresponding claims that use instead the absolute concept. That is the task of this Section. (But its results are hardly needed for the discussions and results in later Sections.) I will prove that with the absolute concept, Section 2.2.2’s converse statement is “resurrected”, i.e. a symmetry leaves invariant the absolute indiscernibility classes (Section 2.4.1). Then I will give some illustrations, including a counterexample to the converse of this statement (Section 2.4.2). Then I will show that for a finite domain of quantification, absolute indiscernibility of two objects is equivalent to the existence of a symmetry mapping one object to the other (Section 2.4.3).27 26That is, assuming that space is not closed. In a closed universe, an object may be a non-zero distance from itself, so the relation ‘is one mile away from’ is not irreflexive, and cannot be used to discern. French’s (2006, §4) and Hawley’s (2009, p. 109) charge of circularity against Saunders (2003a) enters here: it seems that the irreflexivity of ‘is one mile away from’ relies on the prior guarantee that the two spheres are indeed distinct; but their distinctness is supposed, in turn, to be grounded by that very relation being irreflexive. The openness or closedness of space would decide the matter, of course, but that too seems to stand or fall with the irreflexivity or otherwise of distance relations—between spatial or spacetime points, if not material objects. Cf. also comment 1(iv) in Section 2.2.1. 27Absolute discernibility and individuality are closely related to definability in a formal lan- 42 But first, beware of an ambiguity of English. For relations of indiscernibility, we have a choice of two usages. Should we use ‘absolute indiscernibility’ for just ‘not absolutely discernible’ (which will therefore include pairs of objects that are discernible, albeit by other means than absolutely)? Or should we use ‘absolute indiscernibility’ for some kind (species) of indiscernibility—as, indeed, the English adjective ‘absolute’ connotes? (And if so, which kind should we mean?)28 I stipulate that I mean the former. Then: since absolute discernibility is a kind of (implies) discernibility, we have, by contraposition: indiscernibility implies absolute indiscernibility (in my usage). Since both indiscernibility and absolute indiscernibility are equivalence relations, this implies that the absolute indiscerni- bility classes are unions of the indiscernibility classes; cf. Figure 2.4. With this definition, Section 2.4.1’s theorem will be: a symmetry leaves invariant the abso- lute indiscernibility classes. Figure 2.4: Preview to Section 2.4.1’s theorem. A generic symmetry pi acting on a domain must preserve the absolute indiscernibility classes (thick broken lines), but may break the indiscernibility classes (thin broken lines). Note also that the object at centre-bottom, alone in its absolute indiscernibility class and therefore an individual, must be sent to itself under pi. Besides, for later use, I make the corresponding stipulation about the phrases ‘intrinsic indiscernibility’, ‘external indiscernibility’ etc. That is: by ‘intrinsic guage; (for example, an object that is definable is an individual in the sense of Section 2.3.2’s Interlude). The interplay between definability (and related notions) and invariance under sym- metries is given a sophisticated treatment by da Costa & Rodrigues (2007), who consider higher- than-first-order structures. Some of their results have close affinities with our two theorems; in particular their theorems 7.3-7.7. As in footnote 9, I thank N. da Costa. 28This sort of ambiguity is of course not specific to discernment: it is common enough: should we read ‘recalcitrant immobility’ as ‘not-(recalcitrant mobility)’ or as ‘recalcitrant not-mobility’? 43 indiscernibility’ and ‘intrinsically indiscernible’, I will mean ‘not-(intrinsic dis- cernibility)’ and ‘not-(intrinsically discernible)’, respectively; and so on for other phrases. 2.4.1 Invariance of absolute indiscernibility classes Recall that in Section 2.2.2, we saw that leaving invariant (fixing) the indiscerni- bility classes was sufficient, but not necessary, for being a symmetry. That is: Section 2.2.2’s counterexample showed that being a symmetry is not sufficient for leaving invariant the indiscernibility classes. This situation prompts the question, what being a symmetry is sufficient for. More precisely: is there a natural way to weaken Section 2.2.2’s sufficient condition for being a symmetry—viz. indis- cernibility invariance—into being instead a necessary condition? In other words: one might conjecture that leaving invariant some supersets of the indiscernibility classes yields a necessary condition of being a symmetry. In fact, my concept of absolute indiscernibility is the natural weakening. (N.B. The ban on names is essential to its being a weakening: allowing names in a discerning formula makes (the natural redefinition of) absolute discernment equiv- alent to weak discernment.) That is: being a symmetry implies leaving invariant the absolute indiscernibility classes. Cf. Figure 2.4. I will first prove this, and then give a counterexample to the converse statement: it will be similar to the counterexample used in Section 2.2.2 against that Section’s converse statement. Theorem 1: For any structure (i.e. interpretation of a first-order language): if a permutation is a symmetry, then it leaves invariant the absolute indiscernibility classes. Proof: I will prove the contrapositive: I assume that there is some element a of the domain which is absolutely discernible from its image b := pi(a) under the permutation pi, and I prove that pi is not a symmetry. (Remember that ‘a’ and ‘b’ are names in the metalanguage only; we stick to our ascetic object-language demands set down in comment 2 of Section 2.2.1.29) So our assumption is that 29I am extremely grateful to Leon Horsten for making me aware of the problems with a previous version of this proof, in which names were reintroduced into the object language. 44 for some object a in the domain, with b := pi(a), there is some formula Φ(x) with one free variable for which a ∈ ext(Φ) while b /∈ ext(Φ), or vice versa (a /∈ ext(Φ) while b ∈ ext(Φ)): The proof proceeds by induction on the logical complexity of the absolutely discerning formula Φ. From the assumption that a ∈ ext(Φ) iff b /∈ ext(Φ), I will show that, whatever the main connective or quantifier used in the last stage of the stage-by-stage construction of Φ, there is some logically simpler open formula, perhaps with more than one (n, say) free variable, Ψ(x1, . . . xn), and some ob- jects o1, . . . on ∈ D (not necessarily including a, and not necessarily n in number, since maybe oi = oj for some i 6= j) such that we have: 〈o1, . . . on〉 ∈ ext(Ψ) iff 〈pi(o1), . . . pi(on)〉 /∈ ext(Ψ), where pi(oi) is the image of oi under the permutation pi. That is, we continue to break Ψ down to logically simpler formulas until we obtain some atomic formula whose differential satisfaction by some sequence of objects and the sequence of their images under the permutation pi directly contradicts pi’s being a symmetry. The proof begins by setting Ψ := Φ(x) (so to start with, the adicity n of our formula equals 1 and our objects oi comprise only a). We then reiterate the procedure until we reach an atomic formula. Thus:— Step one. We have that 〈o1, . . . on〉 ∈ ext(Ψ) iff 〈pi(o1), . . . pi(on)〉 /∈ ext(Ψ). (Remember that to start with, we set n = 1 and o1 = a. And the ‘iff’ means only material equivalence.) Step two. Proceed by cases: • If Ψ(x1, . . . xn) = ¬ξ(x1, . . . xn), then we have 〈o1, . . . on〉 ∈ ext(¬ξ) iff 〈pi(o1), . . . pi(on)〉 /∈ ext(¬ξ); that is, 〈o1, . . . on〉 ∈ ext(ξ) iff 〈pi(o1), . . . pi(on)〉 /∈ ext(ξ); so ξ is our new, simpler Ψ. • If Ψ is a conjunction, then we can write Ψ(x1, . . . xn) = ( ξ(xi(1), . . . xi(l)) ∧ η(xj(1), . . . xj(m) ) , where l,m 6 n and l + m > n, and i : {1, 2, . . . l} → {1, 2, . . . n} and j : {1, 2, . . .m} → {1, 2, . . . n} are injective maps. First of all, we recognise that 45 〈o1, . . . on〉 ∈ ext(Ψ) iff 〈oi(1), . . . oi(l)〉 ∈ ext(ξ) and 〈oj(1), . . . oj(m)〉 ∈ ext(η). Then, given step one, namely 〈o1, . . . on〉 ∈ ext(Ψ) iff 〈pi(o1), . . . pi(on)〉 /∈ ext(Ψ), this is equivalent to 〈oi(1), . . . oi(l)〉 ∈ ext(ξ) and 〈oj(1), . . . oj(m)〉 ∈ ext(η) iff 〈pi (oi(1)) , . . . pi (oi(l))〉 /∈ ext(ξ) or 〈pi (oj(1)) , . . . pi (oj(m))〉 /∈ ext(η). That is: 〈oi(1), . . . oi(l)〉 ∈ ext(ξ) iff 〈pi ( oi(1) ) , . . . pi ( oi(l) )〉 /∈ ext(ξ), or 〈oj(1), . . . oj(m)〉 ∈ ext(η) iff 〈pi ( oj(1) ) , . . . pi ( oj(m) )〉 /∈ ext(η). So either the formula ξ(x1, . . . xl) or η(x1, . . . xm) is our new formula Ψ; with adicity l, respectively m, replacing the adicity n; and the objects oi(1), . . . oi(l), respectively oj(1), . . . oj(m) replacing the objects o1, . . . on. (This is an inclu- sive ‘or’: if either formula suffices, imagine that only one is chosen to continue the inductive procedure. Heuristic remarks: (i) It is only in this clause that the process can reduce the number of variables occurring in Ψ, and hence the number n of objects under consideration. (ii) The next three cases can be dropped in the usual way, if we suppose the language to use just ¬,∧ as primitive connectives.) • If Ψ = (ξ ∨ η), then continue with Ψ = ¬(¬ξ ∧ ¬η). • If Ψ = (ξ ⊃ η), then continue with Ψ = ¬(ξ ∧ ¬η). • If Ψ = (ξ ≡ η), then continue with Ψ = (¬(ξ ∧ ¬η) ∧ ¬(η ∧ ¬ξ)). • If Ψ(x1, . . . xn) = ∃zξ(z, x1, . . . xn), then we have, using ‘∗’ to mark the n- component argument-place, 〈o1, . . . on〉 ∈ ext(∃zξ(z, ∗)) iff 〈pi(o1), . . . pi(on)〉 /∈ ext(∃zξ(z, ∗)). So we have 〈o1, . . . on〉 ∈ ext(∃zξ(z, ∗)) and 〈pi(o1), . . . pi(on)〉 ∈ ext(¬∃zξ(z, ∗)), or 〈o1, . . . on〉 ∈ ext(¬∃zξ(z, ∗)) and 〈pi(o1), . . . pi(on)〉 ∈ ext(∃zξ(z, ∗)). That is: 〈o1, . . . on〉 ∈ ext(∃zξ(z, ∗)) and 〈pi(o1), . . . pi(on)〉 ∈ ext(∀z¬ξ(z, ∗)), or 〈o1, . . . on〉 ∈ ext(∀z¬ξ(z, ∗)) and 〈pi(o1), . . . pi(on)〉 ∈ ext(∃zξ(z, ∗)). 46 – The first disjunct entails that there is some object in D—call it c—for which 〈c, o1, . . . on〉 ∈ ext(ξ) and 〈pi(c), pi(o1), . . . pi(on)〉 ∈ ext(¬ξ), i.e. 〈c, o1, . . . on〉 ∈ ext(ξ) and 〈pi(c), pi(o1), . . . pi(on)〉 /∈ ext(ξ). (The second conjunct holds for pi(c), since it holds for all objects in D.) – The second disjunct entails that there is some object in D—call it d— for which 〈pi−1(d), o1, . . . on〉 ∈ ext(¬ξ) and 〈d, pi(o1), . . . pi(on)〉 ∈ ext(ξ), i.e. 〈pi−1(d), o1, . . . on〉 /∈ ext(ξ) and 〈d, pi(o1), . . . pi(on)〉 ∈ ext(ξ). (The first conjunct holds for pi−1(d), since it holds for all objects in D.) But we can give pi−1(d) the name c; so that we can recombine the dis- juncts and conclude that, for some object c in D, 〈c, o1, . . . on〉 ∈ ext(ξ) iff 〈pi(c), pi(o1), . . . pi(on)〉 /∈ ext(ξ). So the formula ξ(x1, . . . xn+1) is our new Ψ, n + 1 is our new adicity, and c, o1, . . . on are our new objects. (Heuristic remark: It is only in this clause that the process can increase, by one, the adicity of Ψ, and hence the number of objects under consideration.) • If Ψ(x1, . . . xn) = ∀zξ(z, x1, . . . xn), then continue with Ψ(x1, . . . xn) = ¬∃z¬ξ(z, x1, . . . xn). (Heuristic remark: This case can be dropped in the usual way, if we suppose the language to use just ∃ as the primitive quantifier.) • If Ψ is an atomic formula, then: – either Ψ = F 1i for some primitive 1-place predicate F 1 i , in which case: o1 ext(F 1 i ) iff pi(o1) /∈ ext(F 1i ); – or Ψ = G2j for some primitive 2-place predicate G 2 j , in which case: 〈o1, o2〉 ∈ ext(G2j) iff 〈pi(o1), pi(o2)〉 /∈ ext(G2j) (I emphasize that this case includes the 2-place predicate G2j being ‘=’, i.e. equality); – and so on for any 3- or higher-place predicates. 47 Each case directly contradicts the original assumption that pi is a symmetry (cf. Equation (2.2)). End of proof Corollary 1: An individual is sent to itself by any symmetry. Proof: Section 2.3.2 defined an individual as an object that is absolutely discerned from every other object. So its absolute indiscernibility class is its singleton set. QED. This implies, as a special case, the “resurrection” of Section 2.2.2’s second conjecture. That is, we have Corollary 2: If all objects are individuals, the only symmetry is the identity map. 2.4.2 Illustrations and a counterexample I will illustrate Theorem 1 and Corollary 2, with examples based on those in Section 2.2.2. Roughly speaking, these examples will show how Section 2.2.2’s counterexamples to its two conjectures are “defeated” once we consider absolute indiscernibility instead of utter indiscernibility. Then I will give a counterexample to the converse of Theorem 1. Theorem 1 illustrated:— In Section 2.2.2’s counterexample (1), a and b are absolutely indiscernible. Thus Figure 2.5 illustrates the theorem. Figure 2.5: Illustration of Theorem 1, viz. that a symmetry leaves invariant the equivalence classes for the relation ‘is absolutely indiscernible from’. Here, the symmetry pi from Fig. 2.1 leaves invariant the absolute indiscernibility classes, shown on the right. 48 Corollary 2 illustrated:— I similarly illustrate Corollary 2 by modifying Section 2.2.2’s second counterexample, i.e. counterexample (2) (Figure 2.2) to Section 2.2.2’s second conjecture. The rough idea is to identify absolute indiscernibles; rather than just utter indiscernibles (as is required by (HB)). But beware: iden- tifying absolute indiscernibles that are not utter indiscernibles will lead to a con- tradiction. Figure 2.2 (and also Figure 2.5) is a case in point: it makes true Rab and ¬Raa, so that if one identifies a and b, one is committed to the contradiction between Raa and ¬Raa. But by increasing slightly the extension of R, turn- ing absolute indiscernibles into utter indiscernibles, we can give an illustration of Corollary 2, based on Figure 2.2, which avoids contradiction. Namely, we require that Raa and Rbb; this makes a and b utterly indiscernible, not merely absolutely indiscernible. Then we identify a and b, yielding Figure 2.6. Figure 2.6: Illustration of Corollary 2. When ‘is not absolutely discernible from’ is taken as identity, the only symmetry for each structure is the identity map. Against the Theorem’s converse:— I turn to showing that Theorem 1’s converse does not hold: there are structures for which there are permutations which pre- serve the absolute indiscernibility classes, yet which are not symmetries. Consider the structure in Figure 2.7.30 In this structure the relation R has the extension ext(R) = {〈a, b〉, 〈b, a〉, 〈a, c〉, 〈b, d〉}. So as in Fig 2.1 (i.e. the counterexample in (1) of Section 2.2.2), a, b are weakly discernible, and c, d are indiscernible. But a and b are absolutely indiscernible. (Proof using Theorem 1: the permutation (ab)(cd) is a symmetry, so {a, b} and {c, d} must each be (subsets of) absolute indiscernibility classes.) Then the familiar permutation pi, which just swaps a and b, clearly preserves the absolute indiscernibility classes. Yet pi is not a symmetry, since e.g. 〈a, c〉 ∈ ext(R), but 〈pi(a), pi(c)〉 = 〈b, c〉 /∈ ext(R). Figure 2.7 illustrates the general reason why the class of symmetries is a sub- set of the class of permutations that leave invariant the absolute indiscernibility classes. Namely: for a permutation to be a symmetry, it is not enough that it map 30I thank Tim Button for convincing me of this, and for giving this counterexample. 49 Figure 2.7: A counterexample to the claim that if the absolute indiscernibility classes are left invariant by the permutation pi, then pi is a symmetry. each object to one absolutely indiscernible from it; it must, so to speak, drag all the related objects along with it. For example, in Figure 2.7, it is not enough to swap a and b; the objects “connected” to them, namely c and d respectively, must be swapped too. (In more complex structures, we would then have to investigate the objects “connected” to these secondary objects, and so on). To sum up: this counterexample, together with Theorem 1 and the results of Section 2.2.2, place symmetries on a spectrum of logical strength, between two varieties of permutations defined using our notions of utter indiscernibility and absolute indiscernibility. That is: for a given structure, we have: pi leaves invariant the indiscernibility classes (cf. §2.2.2) ======⇒ pi is a symmetry (cf. Th. 1)======⇒ pi leaves invariant the absolute indiscernibility classes 2.4.3 Finite domains: absolute indiscernibility and the ex- istence of symmetries For structures with a finite domain of objects, there is a partial converse to Section 2.4.1’s Theorem 1: viz. that if a and b are absolutely indiscernible, then there is a symmetry that sends a to b. To prove this, we will temporarily expand the language to contain a name for each object. I will also use the Carnapian idea of a state-description of a structure (Carnap (1950, p. 71)). This is the conjunction of all the true atomic sentences, together with the negations of all the false ones. But for our purposes, the state-description should also include the conjunction of 50 all the true statements of non-identity between the objects in the domain. This will ensure that a map that I will need to define in terms of the state-description is a bijection (and thereby a symmetry). I see no philosophical or dialectical weakness in the proof’s adverting to these non-identity statements. But I agree that the reason it is legitimate to include them is different, according to whether you adopt the Hilbert-Bernays account of identity or not. Thus the opponent to the Hilbert-Bernays account will include in the state-description all the non-identity sentences holding between any two ob- jects in the domain; for the state-description is to be a complete description of the structure, so these non-identity facts should be included. For the proponent of the Hilbert-Bernays account, on the other hand, facts about identity and non-identity are entailed by the qualitative facts, in accordance with (HB). So a description of a structure (in particular, a Carnapian state-description) can be complete, i.e. ex- press all the facts, without explicitly including all the true non-identity sentences. But it is also harmless to include them as conjuncts in the state-description.31 I will state the Theorem as a logical equivalence, although one implication (the leftward one) is just a restatement of Section 2.4.1’s Theorem 1, and so does not need the assumption of a finite domain. Theorem 2: In any finite structure, for any two objects x and y: x and y are absolutely indiscernible ⇐⇒ there is some symmetry pi such that pi(x) = y. Proof: Leftward : This direction is an instance of Section 2.4.1’s Theorem 1, that symmetries leave invariant the absolute indiscernibility classes. Rightward: Consider an arbitrary finite structure with n distinct objects, o1, o2, . . . oi, . . . on in its domain D, and any two absolutely indiscernible objects in that domain, o1, o2 (so we set x = o1, y = o2). We temporarily expand the 31Harmless, that is, provided the HB-advocate is clear-headed. Recall from comment 1 of Section 2.2.1, that the proponent of the Hilbert-Bernays account assumes that the language is rich enough, or that the domain is varied enough, for each object to be discerned in some way (maybe: relatively or weakly) from every other. Thus one could also argue that this assumption involves no loss of generality: for if it does not hold for a structure, then the indiscernible objects are to be identified; or else—on pain of contradiction for the HB-advocate—the primitive vocabulary is to be expanded so as to discern them, and the proof is then run again with a structure of discerned objects. 51 language to include a name oˆi for each object oi. 32 Then: 1. Construct the state-description S of the structure. Thus for example, if the language has just one primitive 1-place predicate F and one primitive 2-place predicate R, define: S := ∧ i,j Sij ∧ ∧ i Si ∧ ∧ i 2, for every state |Ψ〉 ∈ A(H⊗H), ∑ i,j P (1) ij P (2) ij |Ψ〉 = ∑ i,j P (2) ij P (1) ij |Ψ〉 = −2|Ψ〉 (6.2) and ∑ i,j ( P (1) ij )2 |Ψ〉 = ∑ i,j ( P (2) ij )2 |Ψ〉 = 2(d− 1)|Ψ〉 , (6.3) where d = dim(H). Thus every state of the assembly is an eigenstate of the operators used in the definition of Rt; and so we do not need to assume the Born rule. Rt therefore promises to provide categorical discernment. To see that Rt discerns the particles weakly for some t, note that Rt(1, 1) and Rt(2, 2) iff t = 2(d − 1), whereas Rt(1, 2) and Rt(2, 1) only if t = −2.5 So the relations R2(d−1) and R−2 both serve to weakly discern particles 1 and 2. Finally, it remains to be shown that Rt is a physical relation. I turn to Muller and Saunder’s criteria (2008, pp. 527-8): (Req1) Physical meaning. All properties and relations should be trans- parently defined in terms of physical states and operators that corre- spond to physical magnitudes, as in [the weak projection postulate],6 in order for the properties and relations to be physically meaningful. (Req2) Permutation invariance. Any property of one particle is a prop- erty of any other; relations should be permutation-invariant, so binary relations are symmetric and either reflexive or irreflexive. 5Remember that ‘1’ and ‘2’ serve as particle labels in the expressions ‘Rt(1, 2)’, etc. 6The weak projection postulate is effectively Einstein, Podolsky and Rosen’s (1935) reality condition that the assembly’s being in an eigenstate of any self-adjoint operator Q with eigen- value q is a sufficient condition for the assembly’s possessing the property corresponding to the quantity’s Q having value q. This is an interpretative principle, which, like Muller and Saunders (2008) and Muller and Seevinck (2009), I take for granted. 136 (Req2) is clearly true of Rt. (Req1) is also true of Rt, provided that: (i) the projec- tors Ei are physically meaningful; and (ii) the physical meaningfulness of operators is preserved under mathematical operations; for our purposes these must include: arithmetical operations, i.e. addition and multiplication; and tensor multiplication with the identity. (Note: Muller and Saunders take (i) (along with (Req2)) to be sufficient to establish that Rt is a physical relation (2008, pp. 534-5). However, it is clear that (ii) is also required.)  Commentary. I take no issue with Muller and Saunders’ claim that their rela- tions Rt provide categorical and weak discernment. However, I question whether the relations Rt may properly be considered physical. I take no issue with the idea that projectors per se be physically meaningful (like Muller and Saunders, I agree that these can be considered to represent specific experimental questions with a yes/no answer); but Rt is defined in terms of non-symmetric projectors Ei ⊗ 1, etc. Yet, being anti-haecceitists, I take it as compulsory—that is, as a necessary condition for representing a physical quantity—that the quantities obey the Indistinguishability Postulate. This brings us to my criticism of (Req2). My criticism has two components. First: it misapplies the correct idea that physical quantities must be symmetric. By requiring only that the relations defined from the quantum mechanical quantities be symmetric, (Req2) fails to rule out quantum mechanical quantities which may themselves be non-symmetric. To take a simple illustration of this point: ‘x is particle 1 and y is particle 2’ clearly fails to be a physical relation, both in the proper sense, and in terms of (Req2). But the relation ‘x is particle 1 and y is particle 2, or x is particle 2 and y is particle 1’ is equally unphysical, yet it satisfies (Req2). It may be replied: this is where (Req1) comes in. But this brings us to the second component of my criticism of (Req2): it is redundant. For it is anyway nec- essary for a quantity to be symmetric to satisfy (Req1), since any non-symmetric quantity contravenes IP, and therefore cannot represent a ‘physical magnitude’. Indeed: since (Req1) already demands that the quantities be physical, why do we need another requirement at all? 137 It may be objected on behalf of Muller and Saunders that, while the quantities P (1) ij and P (2) ij indeed fail to be symmetric, the quantities defined in terms of them— namely, the ∑ i,j P (x) ij P (y) ij —are symmetric. This is indeed true: ∑ i,j ( P (1) ij )2 =∑ i,j ( P (2) ij )2 = 2(d− 1)1⊗ 1 and ∑i,j P (1)ij P (2)ij = ∑i,j P (2)ij P (1)ij = 2(∑iEi ⊗Ei − 1⊗1), where 1 is the identity on H. (Note that the restrictions of both quantities to the anti-symmetric sector, A(H ⊗ H), are multiples of the identity on that sector.) But I see no force in the objection: the physical significance of these quantities was supposed to rest on their being constructions out of quantities like Ei ⊗ 1; yet it is precisely these quantities which run afoul of IP. Without any convincing account of the physical significance of the building blocks of the ∑ i,j P (x) ij P (y) ij , these quantities must be assessed for their physical significance on their own terms. But, since they are all multiples of the identity on the assembly’s state space, this significance is trivial: they all correspond to experimental questions which yield the same answer on every physical state. This triviality is a problem for Muller and Saunders, since it blocks the Rt from being physical relations. If we now attempt to redefine the Rt in a way that avoids misleading reference to the chimerically physical quantities P (x) ij we obtain: Rt(x, y) iff (x = y and 2(d− 1)ρ = tρ) or (x 6= y and (−2)ρ = tρ) (6.4) This is equivalent to: Rt(x, y) iff (x = y and t = 2(d− 1)) or (x 6= y and t = −2). (6.5) So long as we have a definition of the Rt in terms of quantities that seems (i.e. from the point of view of the syntax) to treat the x = y and x 6= y cases equally, the fact that a different quantity (i.e. a different multiple of the identity) underlies each of these two cases is tolerable. (In just the same way, Rxy and Rxx are strictly speaking different predicates—since one refers to a relation while the other refers to a monadic property—yet it is normal to treat any instance of Rxx as a (special) instance of Rxy. Indeed: weak discernment relies on this being legitimate.) But since the quantities ∑ i,j P (x) ij P (y) ij must be taken at face value—that is, as nothing 138 but multiples of the identity—we must adopt definition (6.5) over definition (6.1), and definition (6.5) is hopelessly gerrymandered and unphysical. Thus Muller and Saunders’ proof that any two fermions are physically discernible does not go through. In Section 6.3.5, I propose an alternative relation which will discern fermions physically and weakly, though not categorically. But first let me address the main results in Muller and Seevinck (2009). 6.3.4 Muller and Seevinck on discernment Muller and Seevinck use a similar framework to Muller and Saunders (2008): specifically, they carry over the three distinctions between kinds of discernment presented above, and the two requirements for physical significance, (Req1) and (Req2).7 There are two main results to discuss: the first concerns spinless parti- cles with infinite-dimensional Hilbert spaces; the second concerns spinning systems with finite-dimensional Hilbert spaces. I begin with their Theorem 1. (Note that I rephrase their Theorems; cf. Muller and Seevinck (2009, pp. 189).) (SMS2) In an assembly with Hilbert space ⊗N L2(R3) and the associated algebra of quantities B(⊗N L2(R3)), any two particles are categor- ically, weakly, physically discernible. Reconstruction of proof (cf. Muller and Seevinck (2009, p. 189)): Again, for simplicity’s sake, I restrict attention to the case of two particles (N = 2). Let Q be the position operator for a single particle in some dimension (say x), and let be P be the momentum operator in that same dimension. (So Q and P are (partially) defined on L2(R3); and I shall not go into detail about the partialness of the domains of definition, which are adequately discussed by Muller and Seevinck.) 7Muller and Seevinck (2009, pp. 185-6) entertain adding a third requirement, to the effect that discernment by a relation is ‘authentic’ only if it is irreducible to monadic properties, and discernment by a monadic property is ‘authentic’ only if it is irreducible to relations. They reject this extra requirement, as do I; but my reasons are different. For me, physical meaning (embodied in (Req1)) is all one could, and should, reasonably ask for—so long as that is taken to entail the requirement that IP is satisfied; cf. my commentary of Muller and Saunders’ proof in Section 6.3.3. 139 Now define Q(1) := Q⊗1 and Q(2) := 1⊗Q, and P (1) := P ⊗1 and P (2) := 1⊗P , where 1 is the identity on L2(R3). We may now define a relation C as follows: C(x, y) iff [P (x), Q(y)]ρ = cρ, for some c 6= 0 , (6.6) where ρ is the density operator representing the state of the assembly. Now for every state we have C(1, 1) and C(2, 2), since [P (1), Q(1)] = [P (2), Q(2)] = −i~1⊗1. And we also have ¬C(1, 2) and ¬C(2, 1), since [P (1), Q(2)] = [P (2), Q(1)] = 0. Thus C weakly discerns particles 1 and 2. This discernment is categorical, since C holds or not categorically, i.e. without probabilistic assumptions. And the discernment is physical, since C meets (Req1) and (Req2).  Commentary. First of all I note that the condition in (SMS2), that each par- ticles’ state space be L2(R3), I grant, of course, since I take it to be a compulsory requirement that particles be located (cf. Sections 5.1.2 and 5.2.1). Second: since the discernment is categorical, it is no restriction that the full (i.e. un-symmetrized) Hilbert space is used in the proof: the proof carries over for all restrictions to sym- metry sectors. As in Section 6.3.3, again I take no issue with the claim that the discernment is weak or that it is categorical, but I deny that it is physical. The reason is the same as for Muller and Saunders (2008): namely, the proof uses unphysical quantities. (Thus I deny that (Req1) is satisfied.) Again we demand not just that the discerning relation be symmetric, but also that it be defined using only physical—a fortiori, only symmetric—quantities. And Q(x) and P (x), despite their tantalising intuitive interpretation, do not count as physical quantities. I now turn to Muller and Seevinck’s second main Theorem: (SMS3) In an assembly with a finite-dimensional Hilbert space ⊗N C2s+1, where s ∈ {1 2 , 1, 3 2 , . . .} and the associated algebra of quantities B(⊗N C2s+1), any two particles are categorically, weakly, physically discernible using only their spin degrees of freedom. Reconstruction of proof (cf. Muller and Seevinck (2009, p. 193-7)): Again I restrict attention to the case of two particles (N = 2). Let S = σxi + σyj + σkk 140 be the quantity representing a single particle’s spin (so S acts on C2s+1). Then we define S1 := S⊗ 1 and S2 := 1⊗ S, and the relation T as follows: T (x, y) iff for all ρ ∈ D(C2s+1 ⊗ C2s+1), |(Sx + Sy)|2ρ = 4s(s+ 1)~2ρ. (6.7) I note that |S|2 = s(s+1)~21; this entails that |2S1|2 = |2S2|2 = 4s(s+1)~21⊗1; so T (1, 1) and T (2, 2) both hold. Meanwhile, |(S1+S2)|2 = |S|2⊗1+1⊗|S|2+2S⊗S = 2s(s + 1)~21 ⊗ 1 + 2S ⊗ S. But the eigenvalues of |(S1 + S2)|2 never exceed (2s)(2s+1)~2 < 4s(s+1)~2, so ¬T (1, 2) and ¬T (2, 1) both hold. This discernment is clearly weak. It is categorical, since it relies on no probabilistic assumptions, and it is physical, since T satisfies (Req1) and (Req2).  Commentary. I note that, in order to put the physical significance of T on firmer ground, Muller and Seevinck extend the EPR reality condition (cf. footnote 6) to a necessary and sufficient condition, which they call the ‘strong property postulate’. According to this postulate, the assembly possesses the property cor- responding to the quantity’s Q having value q if and only if the assembly’s state is an eigenstate of the self-adjoint operator Q, with eigenvalue q. This strengthening is required to establish that the assembly does not possess combined total spin√ 4s(s+ 1)~ when it is not in an eigenstate of the total spin operator. Freedom from this stronger reality condition can be bought at the price of a concession to settle for probabilistic rather than categorical discernment. For we may define the new relation T ′: T ′(x, y) iff Tr ( ρ|(Sx + Sy)|2 ) = 4s(s+ 1)~2. (6.8) It is clear that T ′ discerns iff the “de-modalized” version of T discerns. But the definition of T ′ involves a commitment to the Born rule, so T ′’s discernment is probabilistic. This trade-off between the strong reality condition and the Born rule will also be a feature of my proposed discerning relations in the following Section. My previous objection, which I levelled against (SMS1) and (SMS2), appears to be valid here too. For, even though |(S1 +S2)|2 and |2S1|2 = |2S2|2 are symmetric, once again their building blocks (S1 and S2) are not, and (it may be argued) it is 141 only when defined in terms of these components that T is not a gerrymandered relation. However, our usual objection does not hold in this case. On the contrary, it seems reasonable to take T (x, y) as a natural physical relation, even though its explicit mathematical form depends on whether x = y or x 6= y. To see this, it should be enough that T can be parsed in English as the relation: ‘the combined total spin of x and y has the magnitude √ 4s(s+ 1)~ in all states’. Combined total spin is a symmetric quantity, and it has obvious physical significance. Therefore I do not take issue with the discerning relation being physical. I have two further objections in this case: one mild, the other more serious. The mild problem is that the relation T is different in a significant way from the previous relations Rt and C. While Rt and C both applied to a given state of the assembly, the definition of T involves quantification over all states of the assembly. It is therefore a modal relation. But appeal to modal relations in this context is problematic, since it threatens to trivialise the search for a discerning relation for every state. It would turn out that PII is necessarily true if it is possibly true: a result that is at best controversial. (Note, incidentally, that we cannot quite criticise the use of modal relations on the grounds that it assumes haecceitism. The natural thing to do for a factorist is to use the Hilbert space labels to cross-identify, and this seems to have a whiff of haecceitism about it. However, the factorist strategy need not commit one to haecceitism, since the quantification over states may be restricted to the (anti-) symmetric sectors, in which all states are permutation-invariant.) This mild problem is easily addressed. We simply drop the quantification over states in the definition of T . If we do this, then the (unquantified) right-hand side of the definition (6.7) is still satisfied iff x = y, for all states ρ. We thereby drop the modal involvement. Thus we define a new relation, to be parsed as ‘the combined total spin of x and y has the magnitude √ 4s(s+ 1)~’. The discernment remains categorical, since no probabilistic assumptions have been made. The serious problem is that (SMS3) is clearly only applicable to assemblies whose constituent particles have non-zero spin. This might seem to be only a mild 142 omission, since as a matter of fact no zero-spin particles actually exist (except those that are composed of particles with non-zero spin, and might therefore be discerned by their internal structure). However, it would be nice to establish the discernibility of quantum particles for all values of spin, whether or not they happen to be realised. To sum up: the same problem beleaguers the first two results (SMS1) and (SMS2), which aim to demonstrate the discernibility of (respectively) fermions and any particles with spatial degrees of freedom. The problem is that they both appeal to quantities which, in virtue of contravening IP, are non-physical. The third result, (SMS3), avoids this problem (modulo dropping some unnecessary modal involvement). However, it does not apply to particles with zero-spin. I now turn to my proposal for discerning any species of particle, for any value of spin. 6.3.5 A better way to discern factorism’s particles Muller and Saunders Theorem 3 (pp. 539-40) contains the germ of a better way to secure discernment; i.e. a way free of the criticisms discussed in Sections 6.3.3 and 6.3.4. This Section develops the germ. I proceed in stages. First I outline the basic idea, and propose a relation which weakly and physically discerns two particles in any two-particle assembly, using statistical variance. Then I investigate discern- ment for heterodox state spaces, in which particles may have definite position, and give a relation that will weakly and physically discern there too. Finally, I propose a relation that weakly and physically discerns any two particles in an assembly of any number of particles. Stage A: The basic idea. My basic idea is that particles may be discerned by taking advantage of anti- correlations between single-particle states. In the case of fermions, this is ‘easy’ because of Pauli exclusion: in any basis the occupation number for any single- particle state never exceeds one. In the case of the other particles, it is more tricky, due to the fact that states for non-fermionic particles may have as terms product states with equal factors. In these states, two or more particles are fully correlated, so there does not seem to be any quantum property or relation which 143 would discern them. The solution is to change the basis to one in which anti- correlations appear with non-zero amplitude; the quantity associated with this new basis can then form the basis of a discerning relation. Thus my strategy is discernment through anti-correlations, and the finding of anti-correlations through dispersion. For any state in which two particles are fully correlated, there will be dispersion in some other basis; in particular, the dispersion will involve branches with non-zero-amplitudes in which the particles are anti-correlated. Stage B: The variance operator. For simplicity, I focus exclusively on the two particle case. We may take the assembly Hilbert space to be L2(R3) ⊗ L2(R3), but my results still carry over if we restrict to a symmetry sector, or add additional (e.g. spin) degrees of freedom. Anti-correlations between single-particle states in an eigenbasis for some single- particle quantity Amay be indicated by means of the following ‘standard deviation’ operator: ∆A := 1 2 (A⊗ 1− 1⊗ A) . (6.9) Actually, I will use its square ∆2A, the ‘variance’ operator, which, like ∆A, is self- adjoint (since A is). Unlike ∆A, ∆ 2 A is a symmetric operator, so it is in line with the Indistinguishability Postulate (IP), and is therefore eligible to represent a physical quantity. I also introduce the symmetrized quantity A, which may be viewed as a mean of A taken over the two particles: A := 1 2 (A⊗ 1 + 1⊗ A) . (6.10) Note that the over-line does not indicate an expectation value: A is an operator. By similarly defining A2 = 1 2 (A2 ⊗ 1 + 1⊗ A2) we can express the variance 144 operator more suggestively: ∆2A = 1 4 (A⊗ 1− 1⊗ A)2 = 1 4 ( A2 ⊗ 1 + 1⊗ A2 − 2A⊗ A) = 1 2 ( A2 − A⊗ A ) (6.11) and ∆2A = 1 4 ( A2 ⊗ 1 + 1⊗ A2 − 2A⊗ A) = 1 2 ( A2 ⊗ 1 + 1⊗ A2)− 1 4 ( A2 ⊗ 1 + 2A⊗ A+ 1⊗ A2) = A2 − A2 . (6.12) It is the latter equation (6.12) which justifies the term ‘variance’ for ∆2A. But note again that it is not the (c-numbered) statistical variance of A over a given wavefunction; it is the variance of the operator A over the two particles: ∆2A is itself still an operator. The former equation (6.11) makes it most clear that ∆2A measures the anti-correlation between each of the two particles’ A-eigenstates. In particular, for any state all of whose terms are product states with equal factors in the A-basis: |Ψ〉 = ∑ k ck|φk〉 ⊗ |φk〉 , (6.13) where A|φk〉 = ak|φk〉 , (6.14) the variance has eigenvalue zero: ∆2A|Ψ〉 = 1 4 ∑ k ck ( A2 ⊗ 1 + 1⊗ A2 − 2A⊗ A) |φk〉 ⊗ |φk〉 = 1 4 ∑ k ck ( a2k + a 2 k − 2a2k ) |φk〉 ⊗ |φk〉 = 0 . (6.15) 145 In general, however, a state with anti-correlations will not be an eigenstate of ∆2A. For a generic state-vector |Φ〉 = ∑ ij cij|φi〉 ⊗ |φj〉 (6.16) we have ∆2A|Φ〉 = 1 4 ∑ ij cij ( A2 ⊗ 1 + 1⊗ A2 − 2A⊗ A) |φi〉 ⊗ |φj〉 = 1 4 ∑ ij cij ( a2i + a 2 j − 2aiaj ) |φi〉 ⊗ |φj〉 = 1 4 ∑ ij cij (ai − aj)2 |φi〉 ⊗ |φj〉 (6.17) so that 〈 ∆2A 〉 := 〈Φ|∆2A|Φ〉 = 1 4 ∑ ijkl c∗klcij (ai − aj)2 〈φk|φi〉〈φl|φj〉 = 1 4 ∑ ij |cij|2 (ai − aj)2 . (6.18) If we assume that A is non-degenerate (ai = aj implies i = j), then it is clear from (6.18) that there is a positive contribution to the value of 〈 ∆2A 〉 from every anti-correlation that has a non-zero amplitude. Stage C: Variance provides a discerning relation. If a two-particle state has anti-correlations in a single-particle quantity A, we can build a symmetric, irreflexive relation which discerns them. The main idea is: if the expectation of the variance operator is non-zero, then this can be expressed as a relation between the two particles which neither particle bears to itself. Following Muller & Saunders (2008) and Muller & Seevinck (2009), we define the operators A(1) := A⊗ 1 ; A(2) := 1⊗ A . (6.19) 146 These quantities, being non-symmetric, are unphysical, but they can be used to define physical quantities: note, for example, that ∆A ≡ 12 ( A(1) − A(2)) and A ≡ 1 2 ( A(1) + A(2) ) . We then define the relation R as follows: R(A, x, y) iff 1 4 ( A(x) − A(y))2 ρ 6= 0 . (6.20) In English: R(A, x, y) holds for the state ρ if and only if ρ is not an eigenstate of the absolute difference between x’s and y’s operator A, with eigenvalue zero. Here the variable A ranges over single-particle quantities, x and y range over the particles 1 and 2, and t ranges over the reals. This definition implies that R(A, 1, 2) iff R(A, 2, 1), iff ∆2Aρ 6= 0. And ¬R(A, 1, 1) and ¬R(A, 2, 2). So R(A, x, y) is symmetric and irreflexive for each A. If ∆2A does not annihilate ρ, then we have R(A, 1, 2) and R(A, 2, 1); so in this case R(A, x, y) weakly discerns particles 1 and 2. Moreover, the discernment is categorical. The question remains whether this discernment is physical. I claim that it is, on the assumption of factorism, since the quantity 1 4 ( A(x) − A(y))2, which is symmetric, can be understood as a measure of anti-correlations between x and y for the single-particle quantity A—i.e., a measure of difference between x’s and y’s values for A. Thus it is no surprise that 1 4 ( A(x) − A(y))2 = 0 for x = y; for no object can take a value for any quantity that is different from itself. So long as the single-particle operator A has physical significance, so does 1 4 ( A(x) − A(y))2 = 0. I emphasize that the physical meaning of 1 4 ( A(x) − A(y))2 = 0 should not be thought of as depending on A(x)’s having physical meaning. There is an important analogy here with relative distance. The relative distance between particle x and particle y need not be thought of as deriving its meaning from the absolute positions of x and y, even though the mathematical formalism of our theory may indeed allow us to define the relative distance in terms of these absolute positions. We need not take these mathematical definitions as representative of any physical fact, since we are not forced to admit that an element of the theory’s formalism which has a physical correlate also has physical correlates for all of its mathematical building blocks. This is because these mathematical building blocks may contain redundant structure which is not transmitted to all 147 of their by-products. Such is the case of relative distance. And in fact, relative distance is more than an analogy: for (squared) relative distance is an instance of ∆2A, if we set A = Q, the single-particle position operator. Note that an additional assumption is required to transmit physical significance from 1 4 ( A(x) − A(y))2 = 0 to R(A, x, y): we need to assume Muller and Seevinck’s ‘strong property postulate’. Recall that this states that any physical quantity of the assembly takes a certain value if and only if the assembly is in the appropriate eigenstate for that physical quantity’s corresponding operator. What is important here is the ‘only if’ component of the biconditional: this enables us to say that the difference in x’s and y’s values for A is non-zero just in case the assembly is not in the eigenstate with eigenvalue zero—including when the assembly is not in an eigenstate at all. I summarise the foregoing discussion in the following Lemmas: Lemma 1 For all two-particle assemblies, and all single-particle quantities A, the relation R(A, x, y) has physical significance if A does, on the assumption of the strong property postulate. Lemma 2 For each state ρ of an assembly of two particles, and each single-particle quantity A, the relation R(A, x, y) discerns particles 1 and 2 weakly, cate- gorically and physically if and only if ∆2Aρ 6= 0, on the assumption of the strong property postulate. Proofs: See above.  As with (SMS3), in the previous Section, we can instead forego the strong property postulate and instead take advantage of the Born rule, thereby settling for probabilistic discernment. We then define the relation R′ as follows: R′(A, x, y) iff 1 4 Tr [ ρ ( A(x) − A(y))2] 6= 0 . (6.21) Similar considerations to those above entail thatR′(A, 1, 2) iffR′(A, 2, 1), iff 〈∆2A〉 6= 0. And ¬R(A, 1, 1) and ¬R(A, 2, 2). So R(A, x, y) weakly discerns particles 1 and 2 just in case 〈∆2A〉 6= 0. Thus: 148 Lemma 3 For all single-particle quantities A, the relation R′(A, x, y) has physical significance if A does, on the assumption of the Born rule. Lemma 4 For each state ρ of the assembly, and each single-particle quantity A, the relation R′(A, x, y) discerns particles 1 and 2 weakly, probabilistically and physically, if and only if 〈∆2A〉 6= 0 for that state. Proofs: See above.  Stage D: Discernment for all two-particle states. So far we have seen that two particles in a state with non-zero variance in some single-particle quantity A—i.e. two particles which are anti-correlated in A—may be discerned. To guarantee discernment in all two-particle states it remains to be shown that, for any such state, there will be some single-particle quantity whose eigenbasis has anti-correlations. In fact I prove a stronger result: namely that there is some single-particle quantity which discerns the two particles in all states of the assembly. Moreover, this quantity is familiar: it is position; and since I require all particles to have a location (cf. Section 5.1.2), it will be a quantity that will always be available to discern. Theorem 1 For each state ρ of an assembly of two particles, the relationR(Q, x, y) discerns particles 1 and 2 weakly, categorically and physically; where Q is the single-particle position operator; on the assumption of the strong property postulate. Proof: We assume the strong property postulate. From Lemma 2, we know that R(Q, x, y) discerns particles 1 and 2 weakly, categorically and physically, in the state ρ if and only if ∆2Qρ 6= 0. Let us first consider only pure states, and later generalise to all states. Pure states. Since we are working in the position representation, we use wave- functions rather than state-vectors or density operators. The most general form for a wavefunction for the assembly is Ψ(x,y) = ∑ ij cijφi(x)φj(y) , (6.22) 149 where the φi are an orthonormal basis for L 2(R3). (We assume zero spin, but the proof is trivially extended for any non-zero value for spin.) Now (∆2QΨ)(x,y) = ∑ ij cij ( x2φi(x)φj(y) + φi(x)y 2φj(y)− 2xφi(x).yφj(y) ) = (∑ ij cijφi(x)φj(y) ) (x− y)2 = Ψ(x,y)(x− y)2 (6.23) (cf. Equation (6.17)). This is the zero function only if Ψ(x,y) = 0 whenever x 6= y. But then it cannot be represented in L2(R3) ⊗ L2(R3), since it is not a function. (We essentially appeal to the fact that no wavefunction is infinitely peaked at the diagonal points of the configuration space. The necessary Ψ can be written as a measure: Ψ(x,y) = f(x)δ(3)(x− y), for some function f ∈ L2(R3). I return to this point in Theorem 3, below.) Therefore we conclude that (∆2QΨ)(x,y) 6= 0. It follows that ∆2Q|Ψ〉〈Ψ| 6= 0. Mixed states. We extend to density operators by taking convex combinations of (not necessarily othogonal) projectors. We have that ∆2Q (∑ i pi|Ψi〉〈Ψi| ) = ∑ i pi∆ 2 Q|Ψi〉〈Ψi| 6= 0 (6.24) since both the pi and the spectrum of ∆ 2 Q are positive. From Lemma 2, we conclude that R(Q, x, y) discerns particles 1 and 2 weakly. The discernment is categorical since we made no probabilistic assumptions. Fi- nally, the discernment is physical, as follows from Lemma 1, the strong property postulate, and the fact that Q is physical.  We can now also prove Theorem 2 For each state ρ of an assembly of two particles, the relationR′(Q, x, y) discerns particles 1 and 2 weakly, probabilistically and physically; where Q is the single-particle position operator; on the assumption of the Born rule. Proof. We assume the Born rule. Then for any state ρ we have (cf. Equations 150 6.23, 6.24): 〈 ∆2Q 〉 = Tr ( ρ∆2Q ) = ∑ i pi〈Ψi|∆2Q|Ψi〉 = ∑ i pi ∫ d3x ∫ d3y |Ψi(x,y)|2(x− y)2 , (6.25) which is positive, i.e. non-zero (cf. Equation (6.18)). From Lemma 4, R′(Q, x, y) therefore discerns weakly. The discernment is probabilistic, since we assumed the Born rule. Finally, the discernment is physical, as follows from Lemma 3, the Born rule, and the fact that Q is physical.  It may be objected against the proofs of our two Theorems that we rely too heavily on a technical feature of the assembly’s Hilbert space, namely that it contains no states which exhibit no spread in (x− y)2. Effectively, unfavourable cases for discernment have been ruled out of the assembly’s Hilbert space a priori. But this objection is easily dealt with. Theorem 3 If we permit two-particle states to be represented by measures as well as by functions, then for all such states, either R(Q, x, y) or R(P, x, y) discerns particles 1 and 2 weakly, categorically and physically; where Q is the single-particle position operator, P is the single-particle momentum operator; on the assumption of the strong projection postulate. Proof. The guiding idea is that any state will exhibit spread in either relative position or relative momentum, so no state is annihilated by both ∆2Q and ∆ 2 P. We now allow measures, as well as functions, to represent states of the assembly. Recall from the proof of Theorem 1 that (∆2QΨ)(x,y) = 0 only if Ψ(x,y) = 0 whenever x 6= y. In this case Ψ(x,y) = f(x)δ(3)(x− y), for some measure f(x). Note at this point that the two particles cannot be fermions, since Ψ(x,y) − Ψ(y,x) = 0. We now move to the momentum basis by performing a Fourier 151 transform on Ψ: Ψ(k, l) = ∫ d3x ∫ d3y Ψ(x,y)e−ik.xe−il.y = ∫ d3x ∫ d3y f(x)δ(3)(x− y)e−i(k.x+l.y) = ∫ d3x f(x)e−i(k+l).x = f(k + l). (6.26) This yields (∆2PΨ)(k, l) = (k− l)2f(k + l) , (6.27) which is the zero function only if f(k + l) = 0 whenever k 6= l. But we can only satisfy this requirement if f is the zero function. But in that case Ψ(x,y) is the zero function, and so does not represent a state. So if (∆2QΨ)(x,y) is the zero function, then (∆2PΨ)(k, l) can’t be. This result is easily extended to mixed states. With this result and Lemma 2 we conclude that either R(Q, x, y) or R(P, x, y) (or both) discerns particles 1 and 2 weakly. The discernment is categorical, since we made no probabilistic assumptions. Finally, the discernment is physical, given Lemma 1, the strong property postulate, and the fact that both Q and P are physical.  It only remains to state our Theorem 4 If we permit two-particle states to be represented by measures as well as by functions, then for all such states, either R(Q, x, y) or R(P, x, y) discerns particles 1 and 2 weakly, probabilistically and physically; where Q is the single-particle position operator, P is the single-particle momentum operator; on the assumption of the Born rule. Proof: Left to the reader.  So we have established the weak discernibility of indistinguishable particles in any two-particle assembly. But my results are restricted to the two particle case. Therefore, I now turn to the many-particle case, and present Theorems for assemblies of any number of particles. 152 Stage E: Discernment for all many-particle states. I begin by defining a generalized N -particle variance operator for each single- particle quantity A, for any N > 2. For any single-particle quantity A we define( ∆ (N) A )2 := A2 − A2 = 1 N N∑ i 1⊗ · · ·A2i ⊗ · · ·1− ( 1 N N∑ i 1⊗ · · ·Ai ⊗ · · ·1 )2 = 1 N N∑ i 1⊗ · · ·A2i ⊗ · · ·1 − 1 N2 ( N∑ i 1⊗ · · ·A2i ⊗ · · ·1 + 2 N∑ i 2, and also to paraparticles, will in most cases be obvious; I will highlight any subtleties as they arise. 7.1.1 Entanglement for two distinguishable systems In the case of two distinguishable systems (S1 and S2, say), the assembly’s Hilbert space is simply H1⊗H2, where H1 is the Hilbert space for S1, etc. (Thus factorism rules for distinguishable systems.) IP is not imposed, so we do not concentrate on the symmetric or anti-symmetric subspaces. Entanglement for distinguishable systems is then defined in terms of non- separability. However, to show more clearly the naturalness of Ghirardi, Marinatto and Weber’s extension of the concept of entanglement to the case of indistinguish- able systems, and to follow their presentation more closely, we define: The system S1, constituent of the assembly S = S1 + S2, described by the pure density operator ρ, is non-entangled with subsystem S2 iff there exists a projection operator P onto a one-dimensional subspace of H1 such that: Tr[ρ P ⊗ 1] = 1. (7.1) where 1 is the identity on H2. This is equivalent to each of the following familiar conditions (assuming pure ρ): 1. The reduced density operator ρ1 = Tr (2)(ρ) of subsystem S1 is a projection operator onto a one-dimensional subspace of H1 (i.e. S1’s state is pure); 2. Writing ρ = |ψ〉〈ψ|: the state vector |ψ〉 is factorizable (i.e. separable); i.e., there exist a state |φ〉 ∈ H1 and a state |χ〉 ∈ H2 such that |ψ〉 = |φ〉 ⊗ |χ〉. 166 The support of a density operator ρ is defined as the smallest (“logically strongest”) projector P that ρ makes certain: Tr(ρP ) = 1 and there is no Q < P with Tr(ρQ) = 1. The support of ρ is the projector onto the range ran(ρ). So: for a system in the state ρ, a quantity A whose spectral decomposition contains the projector onto ran(ρ) has (with certainty) the corresponding value. But no refinement of A in the corresponding part of its spectrum does so. I again adopt the usual eigenvalue-eigenstate link (cf. Section 4.2.3). Then only such a quantity A, or a function of it, has a definite value. (We may also say that a quantity B for which the projector onto ran(ρ) is a sum of spectral projectors has an ‘unsharp’ value in the corresponding range.) So: S1 is entangled iff the reduced density operator ρ1 = Tr (2) (ρ) of subsystem S1 has a multi-dimensional support, PM1 say, projecting onto a multi-dimensional subspace M1 ⊂ H1. With the eigenvalue-eigenstate link: this is so iff no one- dimensional projector has a value. The extreme case (‘total’ or ‘maximal’ entanglement) is where M1 = H1, i.e. PM1 = 1. Then there is no self-adjoint operator on H1 for which one can claim with certainty that the outcome of its measurement will belong to any proper subset of its spectrum. A phrase which is vivid, and is adopted by Ghirardi and Marinatto (2004, p. 2), is ‘complete set of properties’: a system with density operator ρ has a complete set of properties iff ρ is a one-dimensional projector, i.e. ran(ρ) is one-dimensional; and similarly, they say a system ‘has the complete set of properties identified by ρ’ etc. To avoid connotations of completions of quantum mechanics (hidden variable theories etc.), we will prefer to say: the system (or its state) is maximally specific; and similarly, I will say the system is maximally specific a` la ρ. And similarly, if ρ = |ψ〉〈ψ|, then I will say that the system is maximally specific a` la |ψ〉. Note that for two distinguishable systems, one system is maximally specific iff the other is. We will see in Section 7.1.2 that this is not true for indistinguishable bosons! Example 1 (Ghirardi and Marinatto (2004, p. 3)): Suppose that an e−e+ system is described by the state vector (with obvious notation): 167 |ψ〉 = 1√ 2 (|↑〉e− ⊗ |↓〉e+ − |↓〉e− ⊗ |↑〉e+)⊗ |R〉e− ⊗ |L〉e+ , (7.2) where |R〉 and |L〉 are two orthonormal states, whose coordinate representations have compact disjoint supports at the spatial regions Right and Left, respectively. (Note the use of the tensor product to bind both states of different systems and states of the same system associated with different degrees of freedom.) The reduced density operator describing the electron e− acts on He− = C2 ⊗ L2(R3) and has the following form: ρe− = 1 2 (|↑〉〈↑ |+ |↓〉〈↓ |)⊗ |R〉〈R| = 1 2 1⊗ |R〉〈R|. (7.3) Although there is no value of the spin along any direction, the electron is, with certainty, inside the right region R; (and an analogous statement can be made concerning the positron, i.e. that it is inside L). In short: some ‘definite properties’ are possessed because the range of the density operator (7.3) is a proper subspace of He− , i.e. the two-dimensional subspace spanned by |↑〉 ⊗ |R〉 and |↓〉 ⊗ |R〉. Example 2: We now consider instead, for the e−e+ system: |ψ〉 = 1√ 2 (|↑〉e− ⊗ |↓〉e+ − |↓〉e− ⊗ |↑〉e+)⊗ (∑ i ci|φi〉e− ⊗ |χi〉e+ ) (7.4) where ∀i, ci 6= 0, and {|φi〉} and {|χi〉} are two complete orthonormal sets of the Hilbert spaces L2(R3) associated to the spatial degrees of freedom of the electron and positron, respectively. The reduced density operator for the electron is: ρe− = Tr (2) (|ψ〉〈ψ|) = 1 2 1⊗ ∑ i |ci|2|φi〉〈φi|. (7.5) In Equation (7.5), 1 is the identity operator on the spin space of the electron. Since the range of ρe− is now the whole Hilbert space of the first particle, all we can say with certainty about the measurement of any self-adjoint operator is that the outcome will be in its spectrum. 168 Finally, we recall the familiar fact that non-entanglement corresponds to fac- torization of all probabilities for all joint measurements: An assembly’s pure state |ψ〉 is non-entangled (i.e. its constituent systems are not entangled with one an- other) iff the following equation holds for any pair of quantities A of H1 and B of H2 such that |ψ〉 belongs to the domains of A⊗ 1 and 1⊗B: 〈ψ|A⊗B|ψ〉 = 〈ψ|A⊗ 1|ψ〉〈ψ|1⊗B|ψ〉. (7.6) Since |ψ〉 is non-entangled iff |ψ〉 = |ς〉 ⊗ |χ〉 for some |ς〉 ∈ H1, |χ〉 ∈ H2, the above expectation value may be expressed as 〈ψ|A⊗B|ψ〉 = 〈ς|A|ς〉〈χ|B|χ〉. (7.7) We will return to this last feature of non-entangled states for distinguishable sys- tems much later (in Stage E of Section 7.1.2 and Section 7.2). Before that, we turn to the definition of GM-entanglement. 7.1.2 GM-entanglement for two indistinguishable systems I now explore the consequences of Ghirardi and Marinatto’s adaptation of the previous definition (Equation (7.1) above) of non-entanglement. The main results will be: (i) Since IP must be obeyed for indistinguishable systems, the basic idea of non-entanglement—viz. some 1-dimensional single-system projector being certain—needs to be made precise using symmetric quantities. This revi- sion of the definition will involve a projector representing, for a factorist, the idea that at least one of the systems is in the state associated with that projector. (ii) GM-entanglement for indistinguishable systems shares many of the attractive features of entanglement for distinguishable systems. In particular: (a) there is a sense in which an assembly’s state is non-GM-entangled iff it supervenes on the states of its constituents; and (b) analogues of Bell’s Theorem (1964) 169 and Gisin’s Theorem (1991) hold for the GM-entanglement of two-system assemblies. (This latter result will be proved much later, once we have introduced the idea of qualitative individuation, in Section 7.2.) (iii) The relation between maximal specificity and non-entanglement is weaker for bosons and paraparticles than for fermions or for distinguishable systems. To be more precise: the state of two distinguishable systems is non-entangled iff either system is maximally specific, iff both systems are maximally spe- cific. This is because for distinguishable systems and fermions we have an equivalence: one system is maximally specific iff the other is. We will see that this is not true for bosons or paraparticles.1 Since we will want non- GM-entanglement to be a symmetric relation between the systems—amongst other things, this will allow us to treat non-entanglement as a property of states of the assembly—non-GM-entanglement is defined in terms of both systems of the assembly being maximally specific. (iv) Non-GM-entangled states of fermions and paraparticles possess a feature which is not shared by either non-entangled distinguishable systems, nor non-GM-entangled bosons. This feature is that, for any non-GM-entangled state of the assembly, decomposition into maximally specific systems is not unique. This result will be especially important for Chapter 8, below. This Section is divided into five Stages. In Stage A, I define in one go, both for bosons and fermions: first non-entanglement; second, at least one system being maximally specific. In Stage B, I show that at least one system being maximally specific is equivalent to the composite’s state-vector being obtained by symmetriz- ing or anti-symmetrizing a factorized state. In Stage C, I specialize this theorem to fermions. In Stage D, I specialize this theorem to bosons and paraparticles. In Stage E, I turn to entanglement and correlations—connecting with the examples at the end of Section 7.1.1. I emphasise that these results are not novel: they are all borrowed from Ghirardi and Marinatto (2003). Stage A: Non-GM-entanglement, and being maximally specific 1The exception arises for multiply occupied single-system states, so it does not apply to fermions. 170 To represent an assembly of two indistinguishable systems, we usually begin with the Hilbert space H ⊗ H, where H is the Hilbert space for one such system. As is familiar (cf. e.g. French and Krause (2006, Ch. 4)), the Indistinguishability Postulate induces a decomposition of the assembly’s Hilbert space into symmetry sectors—in this case the symmetric subspace S(H ⊗ H) (bosons) or the anti- symmetric subspace A(H⊗H) (fermions). Inspired by Ghirardi, Marinatto and Weber (2002), I define: The indistinguishable constituents of a two-system assembly are non- GM-entangled iff both systems are maximally specific. (The assembly is then defined as GM-entangled iff it is not non-GM-entangled.) And we define: Given an assembly of two indistinguishable systems, one of the systems is maximally specific iff there exists a one-dimensional projector P , defined on H, such that: Tr(ρE) = 1 (7.8) where E := P ⊗ 1 + 1⊗ P − P ⊗ P. (7.9) Indeed, extending my terminology from Section 7.1.1, we may say that one of the systems is maximally specific a` la P . E is invariant under action by the symmetric group S2, and it is a projection operator: E2 = E. Furthermore Tr(ρE) is the probability of finding at least one of the two systems in the state onto which the one-dimensional operator P projects. For E can also be written as E = (1− P )⊗ 1 + 1⊗ (1− P ) + P ⊗ P ; (7.10) so this definition of ‘one of the systems is maximally specific’ is really a definition of ‘at least one system is maximally specific’. Stage B: ‘At least one being maximally specific’: a general theorem 171 The definition just given is equivalent to the assembly’s state taking a certain form: Theorem (cf. Ghirardi and Marinatto (2003, Theorems 4.2 & 4.3)): At least one of the systems in a two-system assembly is maximally specific iff the assembly’s state is obtained by symmetrizing or anti-symmetrizing a separable (i.e. product) state. Sketch of Proof: Right to left (easy half): If |ψ〉 is obtained by symmetrizing or anti-symmetrizing a factorized state of two indistinguishable constituents, then: |ψ〉 = N (|φ〉 ⊗ |χ〉 ± |χ〉 ⊗ |φ〉) . (7.11) By expressing the state |χ〉 as |χ〉 = α|φ〉+ β|φ⊥〉, 〈φ|φ⊥〉 = 0 (7.12) and choosing P = |φ〉〈φ|, one gets immediately Tr(ρE) ≡ 〈ψ|E|ψ〉 = 2(1± |α| 2) 2(1± |α|2) = 1. (7.13) Left to Right (hard half): If one chooses a complete orthonormal set of single- system states whose first element |φ0〉 := |φ〉 spans the range of P , writing |ψ〉 = ∑ ij cij|φi〉 ⊗ |φj〉, ∑ ij |cij|2 = 1, (7.14) and, using the explicit expression for E in terms of P , one obtains: E|ψ〉 = |φ0〉 ⊗ (∑ j 6=0 c0j|φj〉 ) + (∑ j 6=0 cj0|φj〉 ) ⊗ |φ0〉+ c00|φ0〉 ⊗ |φ0〉. (7.15) Imposing condition (7.8) implies that E|ψ〉 = |ψ〉 (since E is a projector), which implies that cij = 0 for i, j 6= 0. Taking into account that for indistinguishable 172 systems c0j = ±cj0, the normalization condition of the state |ψ〉 becomes |c00|2 + 2 ∑ j 6=0 |c0j|2 = 1. (7.16) We have then shown that: |ψ〉 = |φ0〉 ⊗ (∑ j 6=0 c0j|φj〉 ) + (∑ j 6=0 cj0|φj〉 ) ⊗ |φ0〉+ c00|φ0〉 ⊗ |φ0〉. (7.17) In the case of fermions c00 = 0. Then, introducing a normalized vector |ξ〉 :=√ 2 ∑ j 6=0 c0j|φj〉 we obtain |ψ〉 = 1√ 2 ( |φ0〉 ⊗ |ξ〉 − |ξ〉 ⊗ |φ0〉) , (7.18) where 〈φ0|ξ〉 = 0. For bosons, defining the following normalized vector |θ〉 = √ 4 2− |c00|2 (∑ j 6=0 c0j|φj〉+ c00 2 |φ0〉 ) , (7.19) the two-particle state vector (7.17) becomes |ψ〉 = √ 2− |c00|2 4 (|φ0〉 ⊗ |θ〉+ |θ〉 ⊗ |φ0〉) . (7.20) Note that in this case the states |φ0〉 and |θ〉 are orthogonal iff c00 = 0, in which case |θ〉 = |ξ〉.  I now deal separately with the case of fermions and bosons. I will denote the appropriate restrictions of the operator E of equation (7.9) as Ef and Eb in the two cases, respectively. Stage C: ‘At least one being maximally specific’: the theorem for fermions Since P ⊗ P = 0 on the space of anti-symmetric states A(H ⊗H), one can drop such a term in all previous formulae. Accordingly, Ef = P ⊗ 1 + 1⊗ P . 173 In accordance with the definition of maximal specificity in Stage A: due to the orthogonality of |φ0〉 and |ξ〉, for the state in equation (7.18), we conclude not only that there is one fermion that is maximally specific a` la the state |φ0〉 (in Ghirardi and Marinatto’s jargon: one fermion possessing the complete set of properties identified by the state |φ0〉), but also that the other fermion is maximally specific a` la |ξ〉. So according to Stage A’s definition of non-entanglement, we have proved: The fermions of a two-system assembly, whose (pure) state is given by |ψ〉, are non-GM-entangled iff |ψ〉 is obtained by anti-symmetrizing a separable state. We may now extend the definition of GM-entanglement to apply to the states of an assembly: a two-fermion assembly’s state may then be described as GM-entangled iff its constituent fermions are GM-entangled. We can say more. If in expression (7.18), we write P = |φ0〉〈φ0| and Q = |ξ〉〈ξ| and we define the operators Ef := P ⊗ 1 + 1 ⊗ P and Ff := Q ⊗ 1 + 1 ⊗ Q, we see that Tr (Ef |ψ〉〈ψ|) = 1, Tr (Ff |ψ〉〈ψ|) = 1,  (7.21) Moreover, EfFf = FfEf = P ⊗ Q + Q ⊗ P (since P ⊥ Q) is a one-dimensional projector on the Hilbert space of the assembly, and it satisfies: 〈ψ|EfFf |ψ〉 ≡ 〈ψ| (P ⊗Q+Q⊗ P ) |ψ〉 = 1. (7.22) It is this identity that allows us to say, not only that one fermion is maximally specific a` la P and one fermion is maximally specific a` la Q, but that one fermion is maximally specific a` la P and the other fermion is maximally specific a` la Q. This crucial difference will be important in the next stage, where we consider bosons and paraparticles, since for those symmetry types the two formulations come apart. 174 I note here an important feature of fermionic states, which will be crucial later, in Chapter 8. There is an arbitrariness, up to two dimensions, about the states a` la which the two fermions are maximally specific. For suppose the state |ψ〉 is (7.18), and consider the two-dimensional subspace of H spanned by the single-system states |φ0〉 and |ξ〉: clearly, if one chooses any two other orthogonal single-constituent states |κ〉 and |λ〉 spanning the same subspace, then |ψ〉 can also be written (up to an overall phase factor) as: |ψ〉 = 1√ 2 (|κ〉 ⊗ |λ〉 − |λ〉 ⊗ |κ〉) . (7.23) Thus the two fermions are also maximally specific a` la |κ〉 and |λ〉. Since this arbitrariness involves fixing an orthonormal basis in C2, it may be parameterized by CP1 ∼= C ∪ {∞}, the Riemann sphere. In fact, each orthonormal basis is represented by a pair of antipodal points {z, 1 z } on the Riemann sphere, since we can permute the two basis vectors without change. A vivid visual metaphor of this basis freedom is provided by the fact that a pair of antipodal points on the Riemann sphere specifies a line through its centre, and a plane orthogonal to that line that divides the sphere into two hemispheres. Thus the arbitrariness in the single-system states a` la which two non-GM-entangled fermions are maximally specific corresponds to the continuum-many ways one may divide a sphere into two hemispheres; cf. Figure 7.1. In the case of three or more fermions, the above results all apply. Specifically: as is true for distinguishable systems, one fermion in the assembly is maximally specific iff they all are, iff the assembly’s state is non-GM-entangled, iff the state is obtained by anti-symmetrizing a separable state. Furthermore, the “preferred basis problem” applying to two-fermion assemblies is only exacerbated by the presence of more fermions. To see this, it is enough to notice that total anti-symmetrization is constituted by all possible pair-wise anti-symmetrizations. For example, in the non-GM-entangled N = 3 state |Ψ〉, 175 Figure 7.1: Two ways to halve the Riemann sphere, and two pairs of orthogonal states, a` la which two non-entangled fermions may be maximally specific. where we define s(i) := (i+ 1) mod 3 (so s3(i) ≡ i), |Ψ〉 = 1√ 6 ∣∣∣∣∣∣∣∣∣∣∣∣ |α〉1 |β〉1 |γ〉1 |α〉2 |β〉2 |γ〉2 |α〉3 |β〉3 |γ〉3 ∣∣∣∣∣∣∣∣∣∣∣∣ ≡ 1√ 6 3∑ i=1 |α〉i ⊗ ( |β〉s(i) ⊗ |γ〉s2(i) − |γ〉s(i) ⊗ |β〉s2(i) ) ≡ 1√ 6 3∑ i=1 |β〉i ⊗ ( |γ〉s(i) ⊗ |α〉s2(i) − |α〉s(i) ⊗ |γ〉s2(i) ) ≡ 1√ 6 3∑ i=1 |γ〉i ⊗ ( |α〉s(i) ⊗ |β〉s2(i) − |β〉s(i) ⊗ |α〉s2(i) ) (7.24) (where we have added labels simply to make the ordering in the tensor prod- uct clear), all of the bracketed two-fermion states are subject to the usual basis arbitrariness (each in a different two-dimensional subspace of H). 176 Generally, any non-GM-entangled state of N fermions: 1√ N ! ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ |φ1〉1 |φ2〉1 · · · |φN〉1 |φ1〉2 |φ2〉2 · · · |φN〉2 ... ... . . . ... |φ1〉N |φ2〉N · · · |φN〉N ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (7.25) (where {|φ1〉, |φ2〉, . . . |φN〉} is an orthonormal basis of the single-system Hilbert space) is the unique totally anti-symmetric state in the Hilbert space ⊗N h, where h := span({|φ1〉, |φ2〉, . . . |φN〉}) ∼= CN . Therefore the same state (7.25) is picked out when the single-system Hilbert space h is decomposed into any other orthonor- mal basis. Therefore the basis arbitrariness for any non-GM-entangled state of a N -fermion assembly corresponds to the arbitrariness in selecting a basis in CN . Each basis—and therefore each set of N single-particle states a` la which N systems may be said to be maximally specific—corresponds to a point in the manifold ( CPN−1 × CPN−2 × · · · × CP1) /SN (7.26) (where we quotient by the natural group action of SN , since a permutation of basis vectors does not change the basis). This manifold has (N − 1)!2N−1 real dimensions! Finally, since in the state of an assembly of any type of paraparticles, the states of at least two systems are pair-wise anti-symmetrized (Tung (1985, Ch. 5)), this basis arbitrariness applies as much to paraparticles as to fermions, though parameterizing this arbitrariness is a more subtle matter. Stage D: ‘At least one being maximally specific’: the theorem for bosons The broad similarity to fermions is clear, especially from Equations (7.11) and (7.20). As for fermions, the requirement that one of the two bosons is maximally specific implies that the state is obtained by symmetrizing a factorized state. How- ever, there are some remarkable differences from the fermion case. For bosons, 177 three cases are possible, according to the single-system states that are the factors of the separable state: 1. |θ〉 ∝ |φ0〉, i.e. |c00| = 1. Then the state is |ψ〉 = |φ0〉 ⊗ |φ0〉 and one can infer that there are two bosons each maximally specific a` la (each with the complete set of properties associated to) P = |φ0〉〈φ0|. It may checked that for this state 〈ψ|P ⊗ P |ψ〉 = 1. 2. 〈θ|φ0〉 = 0, i.e. c00 = 0. One can then consider the operators Eb and Fb, defined as I did for fermions in Stage C, and their product EbFb = FbEb = P ⊗ Q + Q ⊗ P . Then exactly the same argument as for fermions implies that one of the two bosons is maximally specific a` la P and the other of the two bosons is maximally specific a` la Q. That is, 〈ψ|EbFb|ψ〉 = 〈ψ|(P ⊗Q+ Q⊗ P )|ψ〉 = 1. 3. Finally, it can happen that 0 < |〈θ|φ0〉| < 1. In this case, it is true that there is a boson maximally specific a` la P—i.e., 〈ψ|Eb|ψ〉 = 1—and it is true that there is a boson maximally specific a` la Q—i.e., 〈ψ|Fb|ψ〉 = 1. But it is not true that one is maximally specific a` la P and the other is maximally specific a` la Q. (Note that in this case EbFb 6= FbEb, and neither are equal to P ⊗Q+Q⊗ P .) For there is a non vanishing probability of finding both particles in the same state, since 〈ψ|P ⊗ P |ψ〉 = 〈ψ|Q⊗Q|ψ〉 = |c00|2 > 0.2 According to our our definition of non-GM-entanglement, both bosons are non- GM-entangled for the first two cases. But in the last case we cannot say that both bosons are non-GM-entangled, even though we may say that one system is maximally specific a` la one projector, and one system is maximally specific a` la another, distinct (but not orthogonal) projector. The worry, of course, is that we are counting contributions from the same system each time, so we must resist the plural article ‘both’. So we count any state of this third type as GM-entangled, being a superposition of states of the first and second types. To sum up: 2A plausible candidate projector to associate with the proposition, ‘One system is maximally specific a` la P and the other is maximally specific a` la Q’ is G := P ⊗Q+Q⊗P − (P ⊗P +Q⊗ Q−PQ⊗PQ), where we must take advantage of the fact that 〈ψ|PQ⊗PQ|ψ〉 = 〈ψ|QP⊗QP |ψ〉 for |ψ〉 in (7.20). Then 〈ψ|G|ψ〉 = 2−4|c00|2+3|c00|42−|c00|2 , which is less than 1 iff 0 < |〈θ|φ0〉| < 1. 178 The bosons of a two-system assembly, whose (pure) state is given by |ψ〉, are non-GM-entangled iff either: (i) |ψ〉 is obtained by symmetriz- ing a factorized product of two orthogonal states; or (ii) |ψ〉 is a product state of identical factors. We may then again extend the definition to apply to states of the assembly in the obvious way: the assembly’s state is non-GM-entangled iff both systems are. Note that in the boson case, for any non-GM-entangled state, the two states a` la which the two systems are maximally specific are uniquely determined— contrary to what happens for fermions (or paraparticles). That is: there are no other orthogonal states |κ〉 and |λ〉, differing from |φ0〉 and |θ〉, such that one can write |ψ〉 in the form |ψ〉 = 1√ 2 (|κ〉 ⊗ |λ〉+ |λ〉 ⊗ |κ〉) . (7.27) Finally, the foregoing results carry over to the case of three or more bosons. The results and definitions also apply to paraparticles, since for any assembly of paraparticles of any single type, at least two constituent systems may occupy the same state (Tung (1985, Ch. 5)). However, we must, of course, replace the phrase ‘symmetrized state’ in the above with the phrase ‘state with symmetry type µ’, where µ is the appropriate paraparticle type. To sum up the previous two stages, for non-GM-entanglement, in our sense that all constituents are maximally specific, it must be the case that: (i) the state for the assembly is obtained, by the appropriate symmetry projec- tion, from a separable state in ⊗N H; and (ii) the factors of the separable state in question must be orthogonal in the fermion case, and they can be either orthogonal or equal in the boson or paraparticle case. Stage E: Entanglement and correlations I turn at last to entanglement! GM-entangled states can very well occur; and here we connect with the examples at the end of Section 7.1.1. Thus consider the 179 following state of two spin-1/2 constituents: |ψ〉 = 1√ 2 (|↑〉 ⊗ |↓〉 − |↓〉 ⊗ |↑〉)⊗ |ω12〉 (7.28) with |ω12〉 a symmetric state of L2(R3)⊗L2(R3). State (7.28) cannot be written as a symmetrized product of two orthogonal states, and, consequently no constituent is maximally specific (in Ghirardi-argot: possesses any complete set of (internal and spatial) properties). This sort of example also returns us to the topic of non-entanglement cor- responding to factorization of probabilities for joint measurements. To connect with the Bell’s theorem literature, let us consider case in which the two con- stituents are in different spatial regions. Let us then consider two indistinguish- able constituents with space and internal degrees of freedom and let us denote as Hsp and Hint the corresponding single-system Hilbert spaces. The Hilbert space for the whole system is the appropriate symmetric or antisymmetric subspace, S(H(1)int ⊗H(1)sp ⊗H(2)int ⊗H(2)sp ) or A(H(1)int ⊗H(1)sp ⊗H(2)int ⊗H(2)sp ), respectively. Let us also assume that the pure state associated to the composite system is obtained by (anti-)symmetrizing a factorized state of the two particles corresponding to their having different spatial locations. To be explicit, we start from a state: |ψfact〉 = |ς, R〉 ⊗ |χ, L〉, (7.29) where e.g. |ς, R〉 is an abbreviation for the single-system state |ς〉 ⊗ |R〉, and |ς〉 and |χ〉 are two arbitrary states of the internal space of a single systems and |R〉 and |L〉 are two orthogonal states whose spatial supports are compact, disjoint and far away from each other. This situation is the one of interest for all experiments about the non-local features of quantum states. From the state (7.29) we pass now to the properly (anti-)symmetrized state: |ψ〉 = 1√ 2 (|ς, R〉 ⊗ |χ, L〉 ± |χ, L〉 ⊗ |ς, R〉) . (7.30) 180 We already know that for the operators E = P ⊗ 1 + 1⊗ P − P ⊗ P, P = |ς, R〉〈ς, R| F = Q⊗ 1 + 1⊗Q−Q⊗Q, Q = |χ, L〉〈χ, L| (7.31) the following equations hold: Tr (E|ψ〉〈ψ|) = 1, Tr (F |ψ〉〈ψ|) = 1, (7.32) which imply that the constituents are maximally specific a` la projectors P and Q. However, here we are interested in the correlations between the outcomes of measurement processes on the constituents—this will offer a taster for the results in Section 7.2. For any two operators A,B ∈ B(Hint), we construct the operators AR := (A⊗ |R〉〈R|)1 ⊗ 12 + 11 ⊗ (A⊗ |R〉〈R|)2 BL := (B ⊗ |L〉〈L|)1 ⊗ 12 + 11 ⊗ (B ⊗ |L〉〈L|)2 } (7.33) (where I use labels for clarity, and 1 is the identity on Hint ⊗ Hsp). Now it may be checked that AR and BL are symmetric—that is, they obey IP—for any A,B ∈ B(Hint). Therefore, any Hermitian A,B satisfy the necessary condition for representing a physical quantity under IP. In fact, I wish to interpret AR as an operation on any system whose spatial wavefunction’s support overlaps R (and similarly for BL and the region L). It may then be checked that 〈ψ|ARBL|ψ〉 = 〈ς|A|ς〉〈χ|B|χ〉 . (7.34) Equation (7.34) shows that the probabilities referring to the internal degrees of freedom factorize, just as in the case of two distinguishable constituents; cf. Equa- tions (7.6) and (7.7). The same conclusion does not hold when the state is a genuinely entangled state, such as: |ψ′〉 = 1 2 (|ς〉1 ⊗ |χ〉2 − |χ〉1 ⊗ |ς〉2)⊗ (|R〉1 ⊗ |L〉2 ± |L〉1 ⊗ |R〉2) , (7.35) 181 for which 〈ψ′|ARBL|ψ′〉 = 1 2 〈ς|A|ς〉〈χ|B|χ〉+ 1 2 〈χ|A|χ〉〈ς|B|ς〉 − Re [〈ς|A|χ〉〈χ|B|ς〉] , (7.36) in which quantum interference is clearly manifested. (Compare the expectation value 〈ψ′′|A ⊗ B|ψ′′〉 for the entangled state of two distinguishable systems, with only internal degrees of freedom: |ψ′′〉 = 1√ 2 (|ς〉1 ⊗ |χ〉2 − |χ〉1 ⊗ |ς〉2).) The upshot should be obvious: from the point of view of the correlations, and consequently of the implications concerning nonlocality, the non-GM-entangled states of two indistinguishable systems have some of the (rather nice) features as the non-entangled states of two distinguishable systems. This prompts the suggestion (made in (ii)(a) at this beginning of this Section) that non-GM-entangled states supervene on the states of constituent, maximally specific systems—just as non-entangled states supervene on the states of con- stituent, distinguishable systems. Indeed, we see that this true, so long as the symmetry type of the assembly is determined by the states of the constitutent systems.3 For, given a symmetry type, any collection of N single-system states determines at most one non-GM-entangled state, of that symmetry type, for the N -system assembly for which there are N systems maximally specific a` la one of the N single-system states. I return to this in Section 8.2.4. This concludes my reconstruction of the Ghirardi, Marinatto and Weber re- sults. I now turn to a novel idea, which is inspired by these results: qualitative individuation of quantum systems. 7.2 Qualitatively individuating quantum systems A central idea that may be taken from Ghirardi et al ’s idea of a maximally spe- cific system is picking out a system according to the state it occupies; for any maximally specific system is maximally specific a` la some state. Let me use the term individuation for this act of picking out a object, or collection of objects, 3The spin-statistics theorem would provide the necessary connection; but I emphasise that, strictly speaking, this theorem lies outside the realm of elementary quantum mechanics. 182 according to some property that it may have; and let me call the property in question the individuation criterion. Thus an object need not be an individual (cf. Section 2.3’s Interlude) in order to be individuated in the present sense. But it will be uniquely picked out only if it is an individual. In general, an object is individuated by some individuation criterion iff every other object in its absolute indisernibility equivalence class is likewise individuated (cf. Section 2.4). That is: any individuation criterion succeeds in uniquely picking out, not single objects, but absolute indiscernibility classes of objects. Let us further say that an object, or class of objects, is qualitatively individ- uated iff its individuation criterion is a qualitative property. In quantum me- chanics, I submit, qualitative properties are represented by projectors. (Factorist individuation—i.e. individuation according to factor Hilbert space labels—may be considered non-qualitative individuation.) Thus maximally specific systems, since they are individuated using (one-dimensional) projectors, are qualitatively indi- viduated systems. And since the individuating projectors are one-dimensional, the corresponding qualitative property is logically strong, and maximally rich. But qualitatively individuated systems need not be maximally specific. I will propose individuating systems using multi -dimensional projectors. This is the guiding idea of the remaining Sections of this Chapter. In this Section, I will first introduce the main ideas behind qualitative individuation, and link it to natural decompositions of the assembly’s Hilbert space (Section 7.2.1). Then (Section 7.2.2) I will propose two reasonable ways to deal with failure of individuation. Finally (Section 7.2.3), I will give a general prescription for calculating the reduced density operator for a qualitatively individuated system. 7.2.1 Qualitative individuation and natural decompositions One may cash out the idea of a constituent of an assembly in terms of natu- ral decompositions of the assembly’s Hilbert space. In the case of distinguish- able systems—for which I endorse factorism—the natural decomposition is given ab initio: it is the decomposition into the factor Hilbert spaces corresponding to each distinguishable system. Even in this case, the systems are qualitatively 183 individuated—the difference is that the individuation criteria are state-invariant (‘intrinsic’) properties, so they are not represented in the formalism by projectors. In the case of indistinguishable systems, since I deny factorism, we must work in the opposite direction: i.e., we are given the assembly’s Hilbert space and we must search for its natural decompositions. I will argue here that subspaces of the assembly’s Hilbert space may be naturally decomposed into spaces which represent the possibilities for qualitatively individuated systems. What counts as a “natural decomposition”? The answer, suggested by Zanardi (2001, p. 1), is provided by the algebra of quantities defined for the assembly. To be more specific (Zanardi (2001, p. 3)): When is it legitimate to consider a pair of observable algebras as de- scribing a bipartite quantum system? Suppose that A1 and A2 are two commuting ∗-subalgebras of A := End(H) such that the subalge- bra A1 ∨ A2 they generate, i.e., the minimal ∗-subalgebra containing both A1 and A2, amounts to the whole A, and moreover one has the (noncanonical) algebra isomorphism, A1 ∨ A2 ∼= A1 ⊗A2 (7.37) The standard, genuinely bipartite, situation is of course H = H1 ⊗ H2,A1 = End(H1)⊗1, and A2 = 1⊗End(H2). If A′i := {X | [X,A1] = 0} denotes the commutant of A1, in this case one has A′i = A2. Thus Zanardi’s proposal is to work by analogy with the distinguishable case: we look for commuting subalgebras whose tensor product is isomorphic to the entire algebra for the assembly’s Hilbert space. The requirements of commutativity and isomorphism with the entire algebra are tantamount to two of Zanardi et al ’s (2004, p. 1) three necessary and jointly sufficient conditions for a natural decomposition: what he calls subsystem independence and completeness, respectively. The remain- ing requirement, local accessibility, is that the subalgebras be “controllable”. For us, this is tantamount to satsifying IP. Since we will only consider subalgebras of the algebra of symmetric quantities, we can take this requirement to be fulfilled 184 by default. However, unlike for distinguishable systems, in the case of indistinguishable systems, prospects seem dim for finding natural decompositions of the assembly’s entire Hilbert space. For the Hilbert space can even have a prime number of dimensions. (E.g., the Hilbert space for a pair of two-level bosons has three di- mensions.) My basic idea in this Section is that we may instead look for natural decompo- sitions of subspaces of the assembly’s Hilbert space. The collection of constituents corresponding to these decompositions must then be interpreted as co-existing only in those states belonging to the given subspace. But this is not objectionable per se. Agreed: in the case of distinguishable systems—and even for haecceitis- tic factorists—there are means of individuating systems which will suffice for all states. But if one is not a haecceitist, why should one demand or expect this across the board? Now that we have limited our search for natural decompositions to subspaces of the assembly’s Hilbert space, I will show that qualitatively individuated systems provide the natural decompositions being sought. So let us consider what algebra of operators we may associate with a quali- tatively individuated system. For simplicity I will concentrate on the two-system case. Recall that qualitative individuation is individuation by projectors. So sup- pose that our two individuation criteria (one for each of the two systems) are rep- resented by the projectors Eα, Eβ, each of which acts on the single-system Hilbert space. I require that Eα ⊥ Eβ, i.e. EαEβ = EβEα = 0, so that none of the two systems is individuated by the other’s criterion. (The importance of this condition will soon become clear.) Call the system individuated by Eα the α-system, and the system individuated by Eβ, the β-system. Now consider the subspace of the assembly’s Hilbert space Mλ(α, β) := ran(Eα ⊗ Eβ + Eβ ⊗ Eα) , (7.38) where λ ∈ {s, a} indicates whether the subspace lies in the symmetric or anti- symmetric sector; i.e., whether the assembly consists of bosons or fermions. (Again, 185 this is an assembly of two systems, so there are no paraparticle states.) In every state in this subspace, and in only these states, every term of the state has a single-system state in the range of Eα and a single-system state in the range of Eβ. In the case that Eα and Eβ are both one-dimensional, this subspace is one-dimensional, and is spanned by the unique non-GM-entangled state for the symmetry type λ in which one system is maximally specific a` la Eα and the other is maximally specific a` la Eβ. In general I will say that, for all and only states whose support lies solely in Mλ(α, β), the individuation is successful, or that it succeeds. The condition of success is equivalent to Tr [ρ(Eα ⊗ Eβ + Eβ ⊗ Eα)] = 1 . (7.39) I now claim that, for both bosonic and fermionic two-system assemblies, the subspaceMλ(α, β) may be naturally decomposed into two spaces: one correspond- ing to the α-system and one corresponding to the β-system. To prove this, we need to fulfil Zanardi’s requirements of completeness and subsystem independence. That is, we need to find two commuting subalgebras Aα and Aβ (one for the α-system and one for the β-system) whose tensor product is isomorphic to the algebra of symmetric operators onMλ(α, β). SinceMλ(α, β) is either symmetric or anti-symmetric, the latter is the full set of bounded operators on Mλ(α, β), i.e. B(Mλ(α, β)). First of all, we can limit our search forAα andAβ to subalgebras of B(H), where H is the single-system Hilbert space. This is because our individuation criteria Eα, Eβ are single-system projectors, so we expect the Hilbert spaces associated with the α-system and β-system to be no larger than H. Second, we ought to demand that every operator A ∈ Aα commute with the individuation criterion Eα: i.e., ∀A ∈ Aα, [A,Eα] = 0. (And similarly for the β-system.) The reason is so we do not lose track of the α-system by operating on it with operators from its own algebra. As we shall see, this condition is crucial in order to secure the required algebraic structure for Aα (and Aβ). We may now use Schur’s Lemma to establish that, when Eα is not trivial 186 (i.e. not the identity), the representation, on the single-system Hilbert space, of the algebra of operators fulfilling both of these conditions, i.e. A := {A ∈ B(H) | [A,Eα] = 0} must be reducible. I.e., A = Aα ⊕ A′, where H =M⊕M⊥, and M = ran(Eα) supports a representation of Aα and M⊥ = ran(1 − Eα) sup- ports a representation of A′. Therefore, EαAEα = Aα ⊕ 0. Furthermore, since Eα is the identity on M, Aα’s representation on M is irreducible; consequently Aα = B(M), the algebra of all bounded linear operators on M. Similar results hold for the β-system. Let us make the identifications Aα = Aα ≡ B(ran(Eα)) and Aβ = B(ran(Eβ)). Then Aα and Aβ commute, since Eα ⊥ Eβ, and so the representations of Aα and Aβ on H are disjoint. This satisfies Zanardi’s first condition for the decomposition being natural. It remains to be shown that Aα ⊗ Aβ and B(Mλ(α, β)) are isomorphic. For this I define a linear map piλ. It acts on the assembly Hilbert space, its dual space, the algebra of operators on the assembly Hilbert space, and (consequently) matrix elements of such operators. It is defined as follows. For all |φ〉 ∈ ran(Eα), |χ〉 ∈ ran(Eβ); and all |Ψ〉, |Φ〉 ∈ ran(Eα)⊗ ran(Eβ); and all A ∈ Aα, B ∈ Aβ; and all P,Q ∈ Aα ⊗Aβ; and all a, b ∈ C: |φ〉 ⊗ |χ〉 piλ7−→ 1√ 2 (|φ〉 ⊗ |χ〉 ± |χ〉 ⊗ |φ〉) (7.40) piλ (a|Ψ〉+ b|Φ〉) := apiλ(|Ψ〉) + bpiλ(|Φ〉) (7.41) (and similarly for the dual space, and where the ‘±’ in (7.40) corresponds to whether λ is symmetric or anti-symmetric); and A⊗B piλ7−→ A⊗B +B ⊗ A (7.42) piλ(aP + bQ) := apiλ(P ) + bpiλ(Q) (7.43) piλ(〈Ψ|Q|Φ〉) := piλ(〈Ψ|)piλ(Q)piλ(|Φ〉) . (7.44) 187 Then we have, for example, for any A,C ∈ Aα and B,D ∈ Aβ, piλ(A⊗B)piλ(C ⊗D) := (A⊗B +B ⊗ A) (C ⊗D +D ⊗ C) = AC ⊗BD +BD ⊗ AC + AD ⊗BC +BC ⊗ AD = AC ⊗BD +BD ⊗ AC = piλ(AC ⊗BD), (7.45) where we use the fact that AD = BC = 0, since the pairs A,D and B,C have dis- joint representations. Note also that the ranges of C of D are the domains of A and B, respectively. This all relies on our original stipulation that a qualitatively indi- viduated system’s algebra commute with its individuation criterion. It follows from all this that piλ(ran(Eα)⊗ran(Eβ)) =Mλ(α, β) and piλ(Aα⊗Aβ) = B(Mλ(α, β)).4 To see that piλ is an isomorphism, note that it is one-to-one, and that it pre- serves the matrix elements of all operators in Aα ⊗ Aβ. Since for all |φi〉, |φk〉 ∈ ran(Eα); and all |χj〉, |χl〉 ∈ ran(Eβ); and all A ∈ Aα and all B ∈ Aβ: piλ (〈φi|A|φk〉〈χj|B|χl〉) ≡ piλ (〈φi| ⊗ 〈χj|A⊗B|φk〉 ⊗ |χl〉) = piλ(〈φi| ⊗ 〈χj|)piλ(A⊗B)piλ(|φk〉 ⊗ |χl〉) = 1 2 〈φi|A|φk〉〈χj|B|χl〉+ 1 2 〈φi|B|φk〉〈χj|A|χl〉 + 1 2 〈χj|A|φk〉〈φi|B|χl〉+ 1 2 〈χj|B|φk〉〈φi|A|χl〉 + 1 2 〈φi|A|χl〉〈χj|B|φk〉+ 1 2 〈φi|B|χl〉〈χj|A|φk〉 + 1 2 〈χj|A|χl〉〈φi|B|φk〉+ 1 2 〈χj|B|χl〉〈φi|A|φk〉 = 〈φi|A|φk〉〈χj|B|χl〉 (7.46) (since A|χj〉 = A|χl〉 = B|φi〉 = B|φk〉 = 0). The linearity of piλ covers all linear combinations of the above, and so all states in ran(Eα)⊗ran(Eβ) and all operators in Aα ⊗Aβ. Thus the algebra B(Mλ(α, β)) has a natural decomposition into commuting 4For greater clarity, one can imagine each A ∈ Aα flanked on both sides by the projector Eα, and each B ∈ Aβ flanked on both sides by Eβ . This is harmless, since A ≡ EαAEα and B ≡ EβBEβ . The above results then manifestly follow from Eα ⊥ Eβ and [A,Eα] = [B,Eβ ] = 0. 188 Figure 7.2: (a) The (anti-) symmetric projection of the tensor product of two Hilbert spaces may be decomposed into spaces which exhibit a tensor product structure. (Light grey squares indicate condensed states, which remain under symmetrization but not anti-symmetrization.) (b) If the two Hilbert spaces are decomposed into eigensubspaces of only one degree of freedom, then the “off- diagonal” elements of the decomposition serve as irreps for the full algebra of operators for the other degrees of freedom. single-system algebras Aα and Aβ, corresponding to the systems qualitatively in- dividuated by Eα and Eβ, respectively, where Eα ⊥ Eβ. The result is easily generalised to assemblies of more than two systems (and therefore also to para- particles). The above results apply to any subspace B(Mλ(α, β)) of the assembly Hilbert space, as defined in (7.38), so long as Eα ⊥ Eβ. I call such subspaces off-diagonal, since they contain no even partially condensed states—i.e., states in which the same single-system state is multiply occupied. We may now decompose the entire assembly Hilbert space into diagonal and off-diagonal subspaces, and give natural decompositions for each of the off-diagonal subspaces. Each off-diagonal subspace is associated with its own pair of qualitatively individuated systems, and thus behaves, in its own right, like a Hilbert space for an assembly of distinguishable systems; cf. Figure 7.2(a). In more detail: We may decompose the single-system Hilbert space H using a 189 complete family of projectors {Ei}, ∑ iEi = 1: H = (∑ i Ei ) H = ⊕ i Ei(H) =: ⊕ i hi (7.47) Then, with Sλ the appropriate symmetry projector (boson, fermion, etc.), the assembly Hilbert space is Sλ(H⊗H) = Sλ [(⊕ i hi ) ⊗ (⊕ i hi )] (7.48) = Sλ [⊕ i (hi ⊗ hi)⊕ ⊕ i 2. Note also that there is no analogue of Gisin’s Theorem for paraparticles, since paraparticle states only arise for three or more systems, and Gisin’s Theorem cannot be extended beyond the two-system case. (See Z˙ukowski et al (2002) for more details). 190 where eα and eβ act on H1 and 1 is the identity on H2. From the results above, it follows that Aα = B(ran(Eα)) = B(ran(eα)) ⊗ B(H2) and Aβ = B(ran(Eβ)) = B(ran(eβ))⊗B(H2). Thus the full algebra B(H2) is available to both qualitatively individuated systems.6 (Cf. Figure 7.2(b).) To conclude this Subsection, I will say something briefly about Huggett and Imbo’s (2009, pp. 313) recent claim that it is not necessary to impose the In- distinguishability Postulate (IP) on systems with identical intrinsic (i.e. state- independent) properties. This is because, they claim, systems may be distinguished according to their ‘trajectories’ (i.e. single-system states). If they are correct, this would entail that factorism is, after all, a viable interpretative position for such systems—so long as we understand factor Hilbert space labels as representing these trajectories (just as, in the case of distinguishable systems, we use factor Hilbert space labels to represent distinct state-independent properties of the systems). The results of this Subsection show that Huggett and Imbo are partly correct. In my jargon: they are right that an un-symmetrised Hilbert space is an equally adequate (since isomorphic) means to represent an assembly of qualitatively in- dividuated systems—for those states in which the individuation criteria succeed ; and that therefore there is no practical need to impose IP, or, therefore, to repudi- ate factorism when representing those states and those states alone. But they are wrong to claim that it is not necessary to impose IP to represent all of the available states for systems with identical intrinsic properties. For the isomorphism result above, on which Huggett and Imbo’s claim depends, holds only for the appropriate off-diagonal subspace. As soon as the assembly’s state has components that lie outside of this subspace, the isomorphism breaks down. I must emphasise too that the breakdown of isomorphism outside of the rel- evant subspace does not just mean that, for states outside this subspace, the two formalisms yield conflicting empirical claims; empirical claims that confirm IP. Rather, the quasi-factorist formalism ceases to make physical sense for states outside of the relevant subspace. For, outside of this subspace, the systems no 6This case corresponds to Huggett and Imbo’s (2009, pp. 315-6) ‘approximately distinguish- able’ systems. It also corresponds the the example I presented at the end of Stage E of Section 7.1.2, for which eα = |R〉〈R| and eβ = |L〉〈L|. 191 longer occupy the states upon which their individuation—and therefore the entire quasi-factorist formalism—was based. Huggett and Imbo mistakenly suppose that imposing IP prevents one from qualitatively individuating systems. (As Huggett and Imbo (2009, p. 315) put it: ‘IP ⇒ trajectory indistinguishability’.) But that assumes what I deny: namely, factorism. Without factorism, we can agree with Huggett and Imbo that systems may be qualitatively individuated, without contravening IP. Moreover: without factorism but with IP, we may represent all of the states available to the assembly, without fear that our representational apparatus will break down. 7.2.2 Russellian vs. Strawsonian approaches to individua- tion All of the results of the previous Subsection apply only to ‘the relevant’ subspace of the assembly Hilbert space. This is the subspace for which the individuation criteria for the systems succeeds; i.e. for which the projector E(α, β) := Eα ⊗ Eβ + Eβ ⊗ Eα (7.52) has expectation value 1. What about states for which individuation does not succeed? The question is important, since we want a procedure for calculating ex- pectation values of quantities which belong to the joint algebra of the qualitatively individuated quantities; and we want that procedure to be as general as possible. The way one proceeds depends on one’s stance toward reference failure for individuation criteria. I see two equally acceptable routes, which may be associ- ated (perhaps tenuously) with the classic debate over reference failure for definite descriptions. With a little poetic licence, I call the two routes Russellian and Strawsonian. The Russellian route (cf. Russell 1905) takes the claim of success of the indi- viduation criteria Eα and Eβ to be an implicit tag-along claim in addition to any explicit claim which implements those criteria. Thus the expectation value for any A ∈ B(Mλ(α, β)) is given by Tr(E(α, β)ρE(α, β)A), which uses the usual quantum 192 mechanical specifications for a conjunction. But A commutes with E(α, β), since it is a sum of products of single-system quantities, each of which commutes with Eα and Eβ. So we may simplify to Tr(ρE(α, β)A). The Strawsonian route (cf. Strawson 1950) instead takes the joint success of the individuation criteria Eα and Eβ as a presupposition of any claim which uses that strategy. Therefore any expectation values calculated under the presupposition of the success of E(α, β) must be renormalized by conditionalizing on that success. This is done using the usual Lu¨der rule ρ 7→ ραβ := E(α, β)ρE(α, β) Tr(ρE(α, β)) . (7.53) The expectation value of any quantity A ∈ B(Mλ(α, β)) is then given simply by Tr(ραβA). Note that conditionalization requires that Tr(ρE(α, β)) > 0, which means that the state must have some terms for which individuation succeeds. The fact that Tr(ραβA) is undefined when Tr(ρE(α, β)) = 0 meshes rather nicely with Strawson’s famous claim that statements containing failed definite descriptions do not possess a truth value. It will have been noted that the difference between the Russellian and Strawso- nian routes for expectation values lies only in the multiplicative factor 1Tr(ρE(α,β)) . A point in favour of the Strawsonian approach is that the identity operator in- dexed to the two qualitatively individuated systems, piλ(1 ⊗ 1), has expectation value 1 for all normalizable states, while under the Russellian route the identity’s expectation value is equal to E(α, β)’s expectation value. A point in favour of the Russellian approach is that expectation values may be defined for all states. On this approach, if the assembly has no terms for which the individuation succeeds, then expectation value for every A ∈ B(Mλ(α, β)) is zero. 7.2.3 Qualitatively individuated systems on their own In this Subsection, I turn away from the problem of completely decomposing an assembly into natural constituent systems, and turn instead to the problem of 193 picking out a single constituent system from the assembly. I seek a means to calculate expectation values for quantities associated with a single qualitatively individuated system, whose individuation criterion we may choose. The way I will proceed is inspired in part by the Strawsonian approach to individuation in Section 7.2.2. The main idea, there and here, is to conditionalize upon the success of the individuation. As usual, I work, for the sake of simplicity, in the N = 2 case (unless otherwise stated); the generalization to N > 2 will be obvious. We begin with a chosen individuation criterion, a projector Eα, which acts on the single-system Hilbert space. Then it may be checked that the operator nα := Eα ⊗ 1 + 1⊗ Eα (7.54) is a number operator for the two-system assembly’s Hilbert space. That is, it “counts” the number of systems which are picked out by Eα. I now define a linear map piα from the single-system algebra B(H) into a par- ticular subalgebra of the assembly’s algebra. This subalgebra will be the operators which are associated with the α-system. I define piα(A) := EαAEα ⊗ 1 + 1⊗ EαAEα . (7.55) (Note that, if A = EαAEα, then piα is just the symmetrizer for A.) I now claim that the expectation value for any single-system quantity A, asso- ciated with the α-system is 〈A〉α := 〈piα(A)〉〈nα〉 . (7.56) I will establish this claim by considering a few examples. 1. The state of the assembly |ψ〉 = 1√ 2 (|α〉 ⊗ |β〉 ± |β〉 ⊗ |α〉), where Eα|α〉 = |α〉 and Eα|β〉 = 0, and Q|α〉 = q|α〉. Then 〈nα〉 = 1 and 〈piα(Q)〉 = q; so 〈Q〉α = q. That is, the system individuated by Eα takes as its expectation for Q the value q, associated with the state |α〉, for which individuation succeeds 194 (i.e., the state that it is in the range of Eα). (Indeed, the α-system is in an eigenstate for Q, since 〈Q2〉α = q2.) 2. |ψ〉 = c1 1√2 (|α1〉 ⊗ |β1〉 ± |β1〉 ⊗ |α1〉)+c2 1√2 (|α2〉 ⊗ |β2〉 ± |β2〉 ⊗ |α2〉), where |c1|2 + |c2|2 = 1; and for all i = 1, 2: Eα|αi〉 = |αi〉 and Eα|βi〉 = 0, and Q|αi〉 = qi|αi〉. Then 〈nα〉 = 1 and 〈piα(Q)〉 = |c1|2q1 + |c2|2q2; so 〈Q〉α = |c1|2q1 + |c2|2q2. That is, the system individuated by Eα takes as its expectation for Q the average for all single-system states |αi〉, for which the individuation succeeds. The weights for this average are given by the relative amplitudes of the non-GM-entangled terms. 3. |ψ〉 = |α〉 ⊗ |α〉, for |α〉 as above. Then 〈nα〉 = 2 and 〈piα(Q)〉 = 2q; so 〈Q〉α = q. In this case, Eα individuates two systems, and the expectation (indeed, eigenvalue) for Q for both of them is q. 4. |ψ〉 = 1√ 2 (|α1〉 ⊗ |α2〉 ± |α2〉 ⊗ |α1〉), for |α1〉, |α2〉 as above. Then 〈nα〉 = 2 and 〈piα(Q)〉 = q1 + q2; so 〈Q〉α = 12(q1 + q2). In this case, Eα again individuates two systems, one whose expectation value for Q is q1, and one whose value is q2; thus we take the average. However, the weights for this average are not given, as above, by relative amplitudes for non-GM-entangled terms; (the entire state is non-GM-entangled). Rather, they are given by the relative frequency, in a single non-GM-entangled state, of each single-system state for which individuation succeeds. 5. |ψ〉 = c1Sλ (|α1〉 ⊗ |α2〉 ⊗ |β1〉) + c2Sλ (|α3〉 ⊗ |β1〉 ⊗ |β2〉), where Sλ is the (anti-) symmetrizer on the assembly Hilbert space, and the single-system states are defined as before. (So N = 3; and for simplicity we set aside paraparticles.) Then 〈nα〉 = 2|c1|2+|c2|2 and 〈piα(Q)〉 = |c1|2(q1+q2)+|c2|2q3; so 〈Q〉α = |c1|2(q1+q2)+|c2|2q32|c1|2+|c2|2 . In this case, the weights for the average are determined jointly by relative amplitudes and relative frequencies. If |c1| = |c2|, then 〈Q〉α = 13(q1 + q2 + q3); thus each of the three states in the range of Eα are afforded equal weight, whether or not they belong to the same non-GM-entangled term. Thus my claim—that the expectation value of any single-system quantity A, for 195 the system qualitatively individuated by Eα, is given by (7.56)—yields the right results, at least for cases 1 to 4. Case 5 seems to me less clear cut, since one might favour a different way to calculate statistical weights from the relative amplitudes and relative frequencies. However, I submit, there are no clear intuitions to rely on in this case, and I can see no objection to the way given by (7.56). It may have been noticed that piα is not an isomorphism between the single- system algebra B(H)—or indeed any subalgebra thereof—and the range of piα. It is not even a homomorphism. For it may be checked that piα(AB) 6= piα(A)piα(B) does not hold, even for all those A,B ∈ B(H) that commute with Eα. This is not an objection to (7.56), and should come as no surprise. For there are states of the assembly in which Eα fails to individuate a unique system (cf. exam- ples 3-5, above). In these states, we should not expect that piα(AB) = piα(A)piα(B). To perform the operation B, followed by A, on a given α-system (corresponding to piα(AB)) relies on a re-identification of that system (and that system alone) after we have operated with B. But the individuation criterion Eα cannot be guaranteed to pick out that very same system, if more than one system is picked out by Eα. 7 On the other hand, it may be checked that, for any two states |ψ〉 of the assembly that are eigenstates of nα with eigenvalue 1—i.e., for all states in which exactly one system is individuated by Eα—we have 〈ψ|piα(AB)|ψ〉 = 〈ψ|piα(A)piα(B)|ψ〉, as expected. Thus we have a recipe for calculating the expectation value of any single-system quantity for a qualitatively individuated system or systems. It remains for me to give a general prescription for calculating the reduced density operator for such a system. We require that the reduced density operator ρα satisfy the condition that, for all A ∈ B(H): Tr(ραA) = 〈A〉α, as given in (7.56). We know from Gleason’s Theorem that a unique such operator exists. As usual, I work by analogy with the case of “distinguishable” systems. The usual prescription for the reduced density operator of a constituent system, say 7It is worth emphasising that the possibility of multiple α-systems does not arise only for bosons, since Eα need not be a one-dimensional projector. 196 the kth, of the assembly is (with ρ the state of the assembly) ρk := Trk (ρ) , (7.57) where Trk denotes a partial trace over all but the kth factor Hilbert space. Now this prescription is obviously no use to anti-factorists; but an equivalent formulation to (7.57) exists that will be of far more use. First we choose a complete orthobasis {|φi〉} for the single-system Hilbert space H. Then ρk := ∑ i,j Tr (ρ|φj〉〈φi|k) |φi〉〈φj| (7.58) where |φj〉〈φi|k := k−1⊗ 1⊗ |φj〉〈φi| ⊗ N−k⊗ 1 (7.59) and we now perform a full trace on the assembly Hilbert space. We may adapt (7.58) for anti-factorist, qualitatively individuated systems in the following way. First, we replace each operator |φj〉〈φi|k, which is indexed to a factor Hilbert space, with piα(|φj〉〈φi|), as given in (7.55). And second, we “conditionalize” by dividing by 〈nα〉 = Tr(ρnα). Thus ρα = 1 〈nα〉 ∑ i,j Tr [ρ piα(|φj〉〈φi|)] |φi〉〈φj| (7.60) Written out in full, and for any N , we have ρα = ∑ i,j |φi〉〈φj| Tr [ ρ ( N∑ k=1 k−1⊗ 1⊗ Eα|φj〉〈φi|Eα ⊗ N−k⊗ 1 )] Tr [ ρ ( N∑ k=1 k−1⊗ 1⊗ Eα ⊗ N−k⊗ 1 )] (7.61) It may then be shown (as required) that for any A ∈ B(H), Tr(ραA) = 〈A〉α. 197 For this, let {|ξi〉} be a complete eigenbasis for A, where A|ξi〉 = ai|ξi〉. Then Tr(ραA) = 1 〈nα〉 ∑ i,j,k Tr [ρ piα(|ξj〉〈ξi|)] 〈ξk|ξi〉〈ξj|A|ξk〉 (7.62) = 1 〈nα〉 ∑ i,j,k akTr [ρ piα(|ξj〉〈ξi|)] δkiδjk (7.63) = 1 〈nα〉 ∑ k Tr [ρ ak piα(|ξk〉〈ξk|)] (7.64) = 1 〈nα〉Tr [ ρ piα (∑ k ak |ξk〉〈ξk| )] (7.65) = 1 〈nα〉Tr (ρ piα (A)) (7.66) =: 〈A〉α . (7.67) Remember that ρα as given in (7.60) and (7.61) is the average state of any system individuated by Eα. So long as the state ρ is an eigenstate of nα with eigenvalue 1—or even a superposition of nα = 0 and nα = 1 eigenstates—then ρα yields the state of the α-system. However, if ρ contains eigenstates with nα > 1, then the interpretation of ρα as the state of the α-system can no longer be sustained, since in those terms we effectively average all systems picked out by Eα. For this reason, I note as an aside that it seems appropriate to favour something like a Lewisian counterpart theory for claims involving qualitatively individuated systems, over any doctrine of “trans-state” identity. In Lewis’s theory, it is per- fectly in order for an object to have multiple counterparts in some possible worlds, just as there may be multiple α-systems in some states. Additionally, the obvious freedom in the choice of the individuating criterion Eα meshes well with other flexibilities in Lewis’s counterpart relation (cf. Lewis 1968, pp. 115-6). This sug- gestion warrants a much more in-depth treatment, but there is no space to do that here. To conclude this Section, I note two important limiting cases of Equation (7.60). The first is when we are maximally discriminating in our individuation; i.e., where 198 Eα is a one-dimensional projector. Let |α〉 be the state for which Eα|α〉 = |α〉. Then, so long as 〈nα〉 > 0, ρα = |α〉〈α|, which is to be expected. The second limiting case lies at the other extreme, in which we individuate with maximum indiscriminateness, i.e. with Eα = 1. In this case 〈nα〉 = N and piα(A) = ∑N k=1 ⊗k−1 1⊗ A⊗⊗N−k 1; so ρα = 1 N ∑ i,j |φi〉〈φj| Tr [ ρ ( N∑ k=1 k−1⊗ 1⊗ |φj〉〈φi| ⊗ N−k⊗ 1 )] (7.68) = 1 N N∑ k=1 ∑ i,j |φi〉〈φj| Tr [ ρ ( k−1⊗ 1⊗ |φj〉〈φi| ⊗ N−k⊗ 1 )] (7.69) = 1 N N∑ k=1 ∑ i,j |φi〉〈φj| Tr [ρ (|φj〉〈φi|k)] (from (7.59)) (7.70) = 1 N N∑ k=1 ρk (from (7.58)). (7.71) So with maximum indiscriminateness ρα is the “average” of the standard reduced density operators obtained by partial tracing. In the case of indistinguishable systems (i.e. when IP is imposed), we of course have ρ1 = ρ2 = . . . = ρk =: ρ, in which case ρα = ρ. This vindicates my claim in Section 6.4.3 that, in the context of indistinguishable systems, standard reduced density operators obtained by partial tracing codify only the state of the average system, and not the state of any particular system. 7.3 Qualitative individuation over time In this Section I take a brief look at individuation criteria which evolve over time. The investigation here will be all too brief, but will hopefully give a flavour of the direction of future investigation. Let E := Eα⊗Eβ+Eβ⊗Eα represent our individuation criteria for two particles at time t = 0. The obvious way to turn this into evolving individuation criteria is to use the usual Heisenberg prescription for time-dependent quantities. Thus, 199 if U(t) = e−it H ~ is the evolution operator for the assembly, with Hamiltonian H, then we may define time-dependent operator E(t) = U(t)EU †(t) (7.72) The expectation value of E(t) is a constant of the motion: 〈E(t)〉t = Tr(U(t)ρU †(t)U(t)EU †(t)) = Tr(ρE) = 〈E〉0, (7.73) so if the individuation criteria succeed at t = 0, then the dynamics preserves this successfulness over time. This proposal may be seen as analogous to individuation procedures in the clas- sical mechanics of point particles, where we quotient by the symmetric group (Belot 2001). Working in the reduced phase space, by evolving any equivalence class of system points along the Hamiltonian flow we achieve natural trans-temporal iden- tifications for the point particles by demanding continuous trajectories for each particle. However, returning to the quantum case, we cannot guarantee, unlike in the classical case, that the time-evolute of E has the right features to count as a pair of individuation criteria for the two particles. For this it is necessary and sufficient that U(t) (Eα ⊗ Eβ + Eβ ⊗ Eα)U †(t) = Eα(t)⊗ Eβ(t) + Eβ(t)⊗ Eα(t), (7.74) where, at any time t, Eα(t) ⊥ Eβ(t). That is, it is necessary and sufficient that E ’s evolution may be expressed in terms of a piecemeal evolution of the single-system projectors Eα and Eβ. Too see that condition (7.74) does not hold generally, one need only consider an evolution that takes a heterogeneous state of two bosons at t = Ti to a product state at t = Tf . In this case, at time Ti, Eα(Ti) ⊥ Eβ(Ti); but by t = Tf , we have Eα(Tf ) = Eβ(Tf ), which contradicts the requirement of Section 7.2.1 that individuation criteria for distinct systems be orthogonal. I know of no general result, which gives conditions on either the evolution U(t) or the assembly’s state, 200 for the satisfaction of (7.74). However, it is easily seen that condition (7.74) is satisfied if the dynamical evolution is factorizable; i.e. if U(t) = W (t) ⊗W (t), where W (t) is a continuous one-parameter family of unitaries on the single-system Hilbert space H. Under this evolution, the purity of single-system states is preserved over time. But factorizable evolutions present a problem. For there may be no unique pair of time-dependent individuation criteria Eα(t), Eβ(t) for which condition (7.74) is satisfied. This result is as bad as there being no such pair, if our goal is to find uniquely natural trans-temporal identity conditions. (I return to this point in Section 8.2.3.) The fact that uniqueness is not guaranteed for factorizable evolutions is illus- trated by the following simple example. Consider the singlet state for two spin-1 2 fermions: |ψ〉 = 1√ 2 (|↑〉 ⊗ |↓〉 − |↓〉 ⊗ |↑〉) , (7.75) and the trivial evolution U(t) = eiγ(t), where γ : R→ R is any real-valued function of the time for which γ(0) = 0, so that |ψ(t)〉 = eiγ(t)|ψ〉. And suppose that at time t = 0 we individuate two systems using the projectors | ↑〉〈↑ | and | ↓〉〈↓ |. (Remember from Stage C of Section 7.1.2 that non-GM-entangled fermionic states suffer a basis arbitrariness, so that any pair of orthogonal projectors associated with the same two-dimensional subspace spanned by |↑〉 and |↓〉 would have been equally good individuators.) We now seek time-dependent individuation criteria for these two systems. Condition (7.74) suggests the time-dependent projectors W (t)| ↑〉〈↑ |W †(t) and W (t)| ↓〉〈↓ |W †(t), where U(t) = W (t) ⊗ W (t) for some unitary W (t) on H. But W (t) is under-determined by this requirement. We make use of the group isomorphism U(2)⊗ U(2) ∼= U(3)⊕ U(1). Let w(t) be any continuous one-parameter family of unitary 2× 2 matrices w(t) = ( α(t)eiφ(t) β(t)eiφ(t) −β∗(t) α∗(t) ) (7.76) 201 where |α(t)|2 + |β(t)|2. Then w(t)⊗w(t) =  α2(t)e2iφ(t) α(t)β(t)e2iφ(t) α(t)β(t)e2iφ(t) β2(t)e2iφ(t) −α(t)β∗(t)eiφ(t) |α(t)|2eiφ(t) −|β(t)|2eiφ(t) α∗(t)β(t)eiφ(t) −α(t)β∗(t)eiφ(t) −|β(t)|2eiφ(t) |α(t)|2eiφ(t) α∗(t)β(t)eiφ(t) [β∗(t)]2 −α∗(t)β∗(t) −α∗(t)β∗(t) [α∗(t)]2  (7.77) With a suitable change of basis this becomes w(t)⊗w(t) =  α2(t)e2iφ(t) α(t)β(t)e2iφ(t) β2(t)e2iφ(t) 0 −α(t)β∗(t)eiφ(t) (|α(t)|2 − |β(t)|2)eiφ(t) α∗(t)β(t)eiφ(t) 0 [β∗(t)]2 −α∗(t)β∗(t) [α∗(t)]2 0 0 0 0 eiφ(t)  (7.78) where the change of basis is such that w(t)⊗ w(t) is decomposed into symmetric components (the 3 × 3 matrix at top-left) and anti-symmetric components (the c-number at bottom-right). Therefore, the restriction of w(t)⊗w(t) to the anti-symmetric sectorA(C2⊗C2), spanned by |ψ〉, is eiφ(t). So if we make the identification W (t) = w(t), we have U(t) = eiφ(t) on A(C2 ⊗ C2). Thus W (t) may be any unitary 2× 2 matrix of the form given in Equation (7.76), subject only to the requirement that φ(t) = γ(t). The upshot is that the trans-temporal identity conditions for constituent sys- tems in the trajectory |ψ(t)〉 are subject to somewhat the same problems as face the parts of a rotating sphere of uniform, continuous matter (cf. e.g. Zimmerman (1998), Butterfield (2006b)). In both cases no uniquely natural trans-temporal identity conditions (which, in the case of the rotating sphere, would determine its angular velocity) appears to be available. To sum up: Factorizable evolutions give favourable conditions under which trans-temporal individuation criteria for constituent systems may be defined; but the conditions are too favourable: the criteria may not be unique. At the other end of the spectrum, non-factorizable evolutions, which do not preserve non-GM- entanglement, may not even allow time-dependent individuation criteria to be defined. For there is no guarantee that the condition (7.74) can be satisfied. 202 This concludes our tour of the technicalia associated with a general anti- factorist approach to permutation-invariant quantum mechanics. The foregoing results prompt a new understanding of particles in quantum mechanics, but we are far from a complete picture. We now turn our attention to the attempt to put some metaphysical meat on these mathematical bones. 203 Chapter 8 Against varietism My claim that factorism makes an interpretative error similar to the reification of the average person prompts the question: If the factorists’ ‘particles’ are statistical constructs, of what are they statistical constructs? With the results of Section 7 in hand, we are now in a position to make positive attempts at an answer. In the first Section of this Chapter, I will outline the first such attempt, a doctrine I call varietism, because it attempts to “transfer” the qualitative variety in an assembly’s state to the monadic properties of its constituent particles. Then (Section 8.2), I will assess the degree to which varietism satisfies the desiderata for the concept of particle outlined in Section 5.1, and argue that it fares well. However, varietism suffers from a problem that may be fatal: I outline this problem in Section 8.3. 8.1 Varietism defined Recall that I endorse factorism in the case of distinguishable systems. So, as usual, I proceed by analogy with that case. (Of course, factorism’s associating particles with factor Hilbert spaces was a strategy that proceeded by analogy with the case of distinguishable systems! But extending a different feature to the case of indistinguishable particles will give us a different result.) For an assembly of distinguishable systems, a state is non-entangled iff it is separable. And, by definition, a separable state is one in which the constituent 204 systems are in pure states; in Section 7.1.1, we called such systems maximally specific. This general rule, which applies to any distinguishable quantum systems what- ever, can be adapted to the specific case of an assembly of distinguishable particles by ensuring that the assembly’s Hilbert space, and associated algebra of operators, is of the right kind. A particle is then just a consituent system of an assembly of “the right kind”. “The right kind” is determined by the desiderata of Section 5.1: the Hilbert space must be decomposable into single-systems Hilbert spaces, each of which supports a representation of the spacetime symmetry group (usually the Galilei group). Given the discussion in Section 5.2.1, an assembly of N dis- tinguishable particles will have a Hilbert space H = ⊗N (L2(R3)⊗Hint), where Hint allows for more (internal) degrees of freedom (we may set Hint = C). Returning to non-entanglement, we may say that, in the case of distinguish- able particles, the assembly’s state is non-entangled iff the constituent particles are maximally specific. We now have a template functional definition of ‘particle’ that we can apply to the case of indistinguishable particles, bearing in mind Ghi- rardi and Marinatto’s heterodox definition of entanglement for indistinguishable systems. So the varietist says that particles are those systems that are maximally specific just in case the state of an assembly of the right kind is non-GM-entangled. In this case, an assembly is “of the right kind” iff its Hilbert space is a sym- metry sector of the corresponding Hilbert space for an assembly of distinguishable particles. That is, iff H = Sµ [⊗N (L2(R3)⊗Hint)], where Sµ is a projector onto the symmetry sector associated with the irreducible representation µ of SN . Let us investigate this functional definition a bit further: we say that ‘particles are those systems that . . . ’, but what exactly is a ‘system’ for varietists? Recall that being maximally specific is that same as being successfully qualitatively indi- viduated by a one-dimensional projector. (The fact that we appeal, even in the dis- tinguishable case, to qualitative individuation is precisely what allows us naturally to extend the functional definition to the indistinguishable case.) But factorist particles may also be individuated non-qualitatively, by the appropriate factor Hilbert space label. This permits us to latch on to (i.e. individuate) a particle in a non-GM-entangled state of the assembly by appealing to whichever single-particle 205 state a´ la which it is maximally specific (i.e. whichever single-particle state makes its qualitative individuation successful), and then go on to re-identify that particle in other states of the assembly: other states in which it may occupy a different single-particle state altogether—pure or mixed. For the factorist, non-qualitative individuation is king: we might say that, for a factorist, being associated with a particular Hilbert space label is an essential property of a distinguishable particle; whichever single-particle state it may occupy is merely accidental. In the indistinguishable case, the varietist rejects factorism, so she has no non-qualitative means of individuating systems. Therefore she can only appeal to single-particle states to cross-identify systems between different states of the assembly. In metaphysicians’ jargon, the varietist must adopt a qualitative essen- tialism. Here a multitude of possible routes present themselves for the varietist. I will now investigate these routes. I begin by restricting ourselves to non-GM-entangled states (Sections 8.1.1 and 8.1.2); I then turn to GM-entangled states in Section 8.1.3. I will conclude with a more specific commitment to what, for a varietist, particles are. 8.1.1 What varietism says about non-GM-entangled fermions Let us take the simplest example, i.e. a non-GM-entangled state of an assembly of two fermions, belonging to the Hilbert space A(H⊗H): |ψ〉 = 1√ 2 (|φi〉 ⊗ |φj〉 − |φj〉 ⊗ |φi〉) . (8.1) where 〈φi|φj〉 = 0. (It will be obvious how to generalize these considerations for assemblies of three or more particles.) |ψ〉 is non-GM-entangled, since we have Tr(Ei|ψ〉〈ψ|) = 1, where Ei := |φi〉〈φi| ⊗ 1 + 1 ⊗ |φi〉〈φi|, and Tr(Ej|ψ〉〈ψ|) = 1 for Ej defined similarly; thus we have two maximally specific systems. But which systems are they? Here the varietist stays silent. Since the varietist can only individuate qualitatively, and since she has already individuated using the most specific criteria possible (i.e. one-dimensional projectors), there is nothing 206 more for her to say. In words: ‘|ψ〉 is a non-entangled state of two fermions, one of which is in the pure state |φi〉, and the other of which is in the pure state |φj〉.’ This is what we would say, in the distinguishable case (where we are all factorists), about the product state |φi〉 ⊗ |φj〉 (except we would not describe these particles as fermions), with the notable difference that we can also add to this description which particle occupies which state.1 The varietist can make no such addition: the permutation-invariant information is all the information. Note that |ψ〉 can be obtained from the product state by anti-symmetrization: recall (cf. Section 7.1.2) that any non-GM-entangled state can be obtained from a product state by the appropriate symmetry projection. We may therefore begin our characterization of varietism by laying down as a general rule: (V0) The varietist description of a non-GM-entangled fermionic state |Ψ〉 reads like the factorist description of the permutation-invariant information of the corresponding product state from which |Ψ〉 is obtained, upon anti- symmetrization. An immediate problem arises for fermions. As we saw in Stage C of Section 7.1.2, any non-GM-entangled fermionic state is obtainable by anti-symmetrization from a variety of different product states. There are as many pairs of varietist particles for the state |ψ〉 as there are ways to halve a sphere: the particles in |φi〉 and |φj〉 are only one such pair. This problem is significant, and I will argue later that it may be fatal for varietism. But for now we bracket the problem, and continue our exposition. There are two reasons for this: (i) despite this problem, there is something intuitively appealing about varietism, so it is worth developing a full account of it for its own sake; and (ii) the problem may yet be overcome, in which case a full account of varietism is all the more valuable. I return to this “preferred basis problem” for fermions in Section 8.3; in the meantime we return to the exposition. 1Remember that this does not make our factorist a haecceitist, for the particle labels may represent distinct intrinsic qualitative properties of the particles, such as mass or charge. 207 What would the factorist say about |ψ〉 in (8.1), above? The answer depends on the subspecies of factorist. A haecceitistic factorist describes |ψ〉 in the following way: ‘Set aside Ghirardi’s understanding of “entanglement”! |ψ〉 is an entan- gled state, with two non-entangled terms. In one of the terms, particle 1 is in the pure state |φi〉 and particle 2 is in the pure state |φj〉; these states are transposed in the other term. The two terms have a relative amplitude of −1; this means that the two particles are fermions.’ There is something here for the varietist to agree with: namely that |ψ〉 is an eigenstate of having one fermion in |φi〉 and one in |φj〉. But the varietist takes this as an exhaustive characterisation of |ψ〉, while for the haecceitistic factorist there is (in principle) more to ask: in particular, which particle is in which state.2 This makes the varietist sound like an anti-haecceitist: indeed, varietism is anti-haecceitistic in the sense of Section 6.2.1, i.e. in taking the group action of SN on the assembly’s Hilbert space to represent no physical change, i.e. in taking the permutation-invariant information to be exhaustive. But that is anti-haecceitism about the factorist’s particles: it is not the sort of “anti-haecceitism” that matters most to a varietist. Recall, from Section 6.2.2, that the question of haecceitism properly so-called comes after the question, ‘What are the objects?’ The varietist’s particles do not correspond to factor Hilbert space labels, so she would not con- sider permutations of factor Hilbert space labels (i.e. the representations of SN) to correspond to a genuine swapping of particles among the single-particle states. Hence it is a separate question whether possibilities for the varietist’s particles supervene on the qualitative character of physical states. But the formalism as it stands compels the answer Yes, i.e. anti-haecceitism for the varietist’s particles too. A specification of single-particle pure states, together with a specification of symmetry type, suffices to determine a unique (non-GM-entangled) state for 2The haecceitistic factorist would have to admit that, in practice, no measurement could be performed which would answer this question; otherwise the projection postulate would come into conflict with the Indistinguishability Postulate. The haecceitist, of course, takes the Indistin- guishability Postulate to be a contingent fact. 208 the assembly (cf. Stage E of Section 7.1.2);3 therefore there are never two distinct non-GM-entangled states which do not differ with regard to qualitative character. (I will return to this point below.) So, barring any claim that the formalism is incomplete (which I will consider in Section 8.2.6), anti-haecceitism in the sense appropriate to varietism is forced upon the varietist. As we have said, varietists are also anti-haecceitistic about the factorist’s parti- cles, but they are of course not anti-haecceitistic factorists. Admittedly, varietists and anti-haecceitistic factorists alike will be uncomfortable about talking, as the haecceitistic factorist does, of the two separable terms of |ψ〉 as if each were a pos- sible pure state of the assembly. (For an anti-haecceitist factorist, heterogeneous separable states represent mixed states of the assembly.) Thus here is what the anti-haecceitist factorist says about the state |ψ〉: ‘Set aside Ghirardi’s understanding of “entanglement”! |ψ〉 is an entan- gled state, since its particles are not in pure states. Both particles are in the same mixed state 1 2 |φi〉〈φi|+ 12 |φj〉〈φj|; but don’t take those “12”s as probabilities for the assembly occupying the corresponding product states, like the haecceitist does. These states are not available to the assembly—in fact their mathematical representations don’t really make sense to me. The minus sign between the two terms in |ψ〉 tells us that the particles in question are fermions; but again, don’t think of |ψ〉 as a superposition of physically possible product states.’ (Note how the anti-haecceitistic factorist has managed to describe the state |ψ〉 in a way that relies on associating particles with factor Hilbert spaces, yet without mentioning Hilbert space labels. This trick relies on the fact that both particles are in the same mixed state.) The contrast with what the varietist would say is stark. For the varietist, the particles are in pure states; therefore |ψ〉 is not GM-entangled. 3Strictly speaking, that is only true for fermionic and bosonic states. For paraparticle states one would also have to specify certain relations between the particles; but these are qualitative re- lations (i.e., they are permutation-invariant), so the spirit of my claim here holds for paraparticles too. 209 8.1.2 What varietism says about non-GM-entangled bosons Section 8.1.1’s account carries over mutatis mutandis, i.e. with judicious substitu- tions of ‘boson’ for ‘fermion’, etc., for heterogeneous bosonic states, i.e. states such as |ψ′〉 = 1√ 2 (|φi〉 ⊗ |φj〉+ |φj〉 ⊗ |φi〉) (8.2) where 〈φi|φj〉 = 0. But bosons may also exist in homogeneous product states, such as |ψ′′〉 = |φi〉 ⊗ |φi〉. (8.3) What might the varietist say about these? The case is interesting, since, despite there being two maximally specific systems, the systems in question are maximally specific a` la the same one-dimensional projector. Therefore, it cannot be said that the systems have been successfully qualitatively individuated. Nevertheless, we may say that both systems are maximally specific without actually having to individuate them. It is enough that (|φi〉〈φi| ⊗ |φi〉〈φi|) |ψ′′〉 = 2|ψ′′〉, i.e. that |ψ′′〉 is an eigenstate of there being exactly 2 particles in state |φi〉. That is, it is enough so long as we do not impose any form of the identity of indiscernibles (cf. Chapter 3) on the particles (we will come back to this point in Section 8.2). Homogeneous product states offer a rare opportunity for consensus between factorists, of both haecceitistic and anti-haecceitistic persuasion, and varietists. All three would describe |ψ′′〉 as a pure state of two particles in which both particles were in the pure state |φi〉. The consensus is no surprise: in these states (and only these states) factor Hilbert space labels—which the factorist uses to individuate particles—align perfectly with single-particle states—which the varietist uses to individuate particles. The only remaining case to consider for bosons are states such as |ψ′′′〉 = 1√ 2(1 + |〈φi|θj〉|2) (|φi〉 ⊗ |θj〉+ |θj〉 ⊗ |φi〉) (8.4) where 0 < |〈φi|θj〉| < 1. But recall from Stage D of Section 7.1.2 that this state is 210 in fact GM-entangled, being (perhaps) more perspicuously written as |ψ′′′〉 = 2〈φi|θj〉|φi〉 ⊗ |φi〉+ ∑ k 6=i〈φk|θj〉 (|φi〉 ⊗ |φk〉+ |φk〉 ⊗ |φi〉)√ 2(1 + |〈φi|θj〉|2) . (8.5) We are therefore led to consider varietism’s account for entangled states. But first I sum up Sections 8.1.1 and 8.1.2 by laying down an extension of (V0) to apply to all non-GM-entangled states: (V1) The varietist description of a non-GM-entangled state |Ψ〉 reads like some factorist description of the permutation-invariant information of the corre- sponding product state from which |Ψ〉 is obtained, by the appropriate sym- metry projection. The modifications have been italicized: I say ‘some factorist description’, to bracket the basis arbitrariness problem for fermions and paraparticles; and we now incor- porate all symmetry types (including paraparticles) by referring to the ‘appropriate symmetry projection’. This concludes the preliminary exposition of varietism for non-GM-entangled states. 8.1.3 What varietism says about GM-entangled states The functional definition of ‘particle’ given at the beginning of Section 8 applied only to non-entangled states. For a factorist, this is easily extended to entangled states by appealing to non-qualitative individuation with factor Hilbert space la- bels. The anti-factorists—of which varietists are a subspecies—can make no such appeal in the case of GM-entangled states. The available routes ahead for the varietist may be placed into two broad options. (One of these options further bifurcates, as we shall see.) The first option is to be cagey, so I call it ‘cagey varietism’. Cagey varietists avoid the problem of GM-entangled states by claiming that particles exist only in non-GM-entangled states of the assembly. That would be to admit that the concept—the intension—of particle failed to pick out an extension in some (indeed most) quantum mechanical states. The second option is to be heroic. Heroic varietists maintain that particles 211 exist in every state of the assembly, including the GM-entangled ones. I consider the two options in turn.4 Cagey varietism. The problem with denying the existence of particles for GM- entangled states is that the cagey varietist is then faced with the question of what does exists when the assembly’s state is GM-entangled. Presumably the assembly continues to exist, yet without its supposedly constituent particles. So a cagey varietist believes an assembly can exist without its particles. She must therefore give up on the idea that particles always compose the assembly. This contradicts the strong version of our compositionality desideratum (cf. Section 5.1.4), since according to that desideratum an assembly must be the mereological sum of its particles. It also contradicts the weak version of our compositionality desideratum, since (trivially) any two distinct GM-entangled states differ without there being any corresponding difference in the states of the particles (of which there are none). (However, supervenience still holds between the non-GM-entangled states of the assembly and the states of the particles.) But matters are worse for the cagey varietist. If the assembly always exists, and may exist even though its particles don’t, why do we need particles at all? The assembly’s state alone is enough to make any statement about it true or false—nevermind the particles! The cagey varietist has a response. There is a natural way, she argues, to ad- mit that the assembly’s state suffices to make any statement about it true or false without threatening the particles with redundancy. For the particles could them- selves be features—i.e., properties—of the assembly’s non-GM-entangled states. They are not extra idle objects, but rather ontological free-riders. That way, talk of particles just is convenient talk about the assembly when it is in a non-GM- entangled state. The cagey varietist need not give up on particles altogether, but she must give up on them as objects. The view that particles are not objects but properties is part of the view I call emergentism. I discuss this view below in Chapter 9, so we will say no more about it here. It is enough to note that cagey varietism collapses into a particular version 4Of course, these options are not exhaustive: one may maintain that particles exist in some but not all GM-entangled states. This option is unmotivated and arbitrary, so I exclude it. 212 of it. Henceforth, ‘varietism’ will always mean the heroic kind. Heroic varietism. The second option is to claim that particles exist for all states of the assembly. We might therefore hope to retain the familiar principle that the assembly is composed of particles, at least in the weak sense of Section 5.1.4. But once again we find that our way ahead is not determined. Prima facie, there are (at least) two natural ways to proceed. 1. Weaken. Inspired by the technical results in Sections 7.2, the varietist may consider weakening the individuation criteria in order to successfully indi- viduate across several non-GM-entangled terms. In this case a particle need no longer be a maximally specific system, but a system which is specific a` la some (possibly multi-dimensional) single-system projector P , i.e. Tr(ρE) = 1, (8.6) where ρ is the state of the assembly and E := P ⊗ 1 + 1⊗ P − P ⊗ P . 2. Relativize. An alternative suggestion is to retain maximum specificity, but take advantage of the fact that any GM-entangled state is a superposition of non-GM-entangled states. Thus the varietist may consider treating any two non-GM-entangled terms as representing distinct collections of particles, related by being superposed. Under this proposal, particles are “branch- bound” entities, where by “branch” we mean a non-GM-entangled state. So under this suggestion, a particle is a system that is maximally specific on at least one of the branches of the assembly’s state. This is equivalent to Tr(F |Ξ〉〈Ξ|) = 1, Tr(ρ|Ξ〉〈Ξ|) > 0. (8.7) for some non-GM-entangled state |Ξ〉, where ρ is the (pure) state of the assembly and F := Q ⊗ 1 + 1 ⊗ Q − Q ⊗ Q, for some one-dimensional projector Q. An example may help to illustrate these two suggestions. Consider the following 213 state for a two-fermion assembly: |ψ〉 = α 1√ 2 (|φ1〉 ⊗ |φ2〉 − |φ2〉 ⊗ |φ1〉) + β 1√ 2 (|φ3〉 ⊗ |φ4〉 − |φ4〉 ⊗ |φ3〉) (8.8) where |α|2 + |β|2 = 1. I assume that the states |φi〉 belong to a single-system Hilbert space that is appropriate for a particle interpretation, in accordance with Section 5.2.1. What, according to our two new species of (heroic) varietist, are the constituent particles in this state? Individuating particles under Weaken The proponent of Weaken needs to find single-system projectors which satisfy Equation (8.6). I set aside for now the basis arbitrariness problem (cf. Stage C in Section 7.1.2) for fermion states by considering only projectors in the {|φi〉} basis. Still there are many options. They are: P13 = |φ1〉〈φ1|+ |φ3〉〈φ3| P14 = |φ1〉〈φ1|+ |φ4〉〈φ4| P23 = |φ2〉〈φ2|+ |φ3〉〈φ3| P24 = |φ2〉〈φ2|+ |φ4〉〈φ4| P123 = |φ1〉〈φ1|+ |φ2〉〈φ2|+ |φ3〉〈φ3| P124 = |φ1〉〈φ1|+ |φ2〉〈φ2|+ |φ4〉〈φ4| P134 = |φ1〉〈φ1|+ |φ3〉〈φ3|+ |φ4〉〈φ4| P234 = |φ2〉〈φ2|+ |φ3〉〈φ3|+ |φ4〉〈φ4| P1234 = |φ1〉〈φ1|+ |φ2〉〈φ2|+ |φ3〉〈φ3|+ |φ4〉〈φ4|  (8.9) and any other projector Q such that Q > Pλ for any Pλ listed above. (It may be checked that Equation (8.6) holds for all such projectors.) Must the advocate of Weaken therefore say that the state |ψ〉 in Equation (8.8) above contains at least 9 (and potentially infinitely many—depending on the dimension of the single-particle Hilbert space) particles? She must, but be careful not to misunderstand her. For it is normally an unspoken rule (except perhaps for analytic metaphysicians!) that when counting a collection of objects, the objects in question are taken to be wholly distinct. If this unspoken rule is explictly relaxed, and the objects in question may overlap, the 214 question, ‘How many are there?’ can have a surprisingly large answer—consider the ‘How many triangles are there?’ puzzles in old IQ tests.5 So the proponent of Weaken sanguinely admits that, yes, at least 9 particles are described by |ψ〉, but many of them overlap many of the others. Which overlap which? This may be answered by calculating probabilities for being in the states corresponding to the projectors listed in (8.9), for each par- ticle qualitatively individuated6 using those same projectors. We use the results of Section 7.2.3 (in particular Equation (7.56)). It follows that for the particle individuated by Pλ and the single-particle state Pµ we have p(µ|λ) := p(Pµ|Pλ) = 〈Pµ〉λ = 〈PλPµPλ ⊗ 1 + 1⊗ PλPµPλ〉〈Pλ ⊗ 1 + 1⊗ Pλ〉 (8.10) The resulting probabilities for the state |ψ〉 in (8.8) are shown in Table 8.1. These probabilities may be interpreted as a measure of degree of overlap.7 For example, p(13|24) = p(24|13) = 0, so the particle that is specific a` la P13 is wholly distinct from the particle that is specific a` la P24; and similarly for P14 and P23. (We may use the definite article in all these cases, since the denominator in Equation (8.10) is equal to 1 for all these projectors.) Meanwhile, p(123|13) = 1 but p(13|123) = 1 1+|α|2 < 1, so the particle that is specific a` la P13 is a proper part of the sum of particles (note the plural!—the denominator in Equation (8.10) is more than 1 for Pλ = P123) that are specific a` la P123. And every particle or sum of particles that is/are specific a` la any one of the projectors in (8.9) is a part of the sum of the particles that are specific a` la P1234, since p(µ|1234) = 1 for all Pµ in (8.9). There are exactly two particles that are specific a` la P1234 (the denominator of Equation (8.10) is equal to 2 for Pλ = P1234); we may identify their sum with the assembly itself. 5A more concrete example: How many pairs of socks are there for 4 similar socks? The answer is 42C = 6, not 4 2 = 2. 6Recall from Section 7.2.3 that more than one system may satisfy the same individuation criterion. In this case, as usual, we average over all such systems. 7This measure satisfies the necessary conditions for giving the fraction of overlap, so that p(µ|λ) gives the fraction of the sum of particles specific a` la Pλ that are also specific a` la Pµ. It may checked that e.g. p(23|1234) = p(23|123)p(123|1234) + p(23|234)p(234|1234). So, for example, p(13|1234) = 12 may then be interpreted as the particle specific a` la P13 overlapping exactly half of the two particles specific a` la P1234. 215 Single-particle state P13 P14 P23 P24 P123 P124 P134 P234 P1234 In di vi du at io n cr it er io n P13 1 |α|2 |β|2 0 1 |α|2 1 |β|2 1 P14 |α|2 1 0 |β|2 |α|2 1 1 |β|2 1 P23 |β|2 0 1 |α|2 1 |α|2 |β|2 1 1 P24 0 |β|2 |α|2 1 |α|2 1 |β|2 1 1 P123 1 1+|α|2 |α|2 1+|α|2 1 1+|α|2 |α|2 1+|α|2 1 2|α|2 1+|α|2 1 1+|α|2 1 1+|α|2 1 P124 |α|2 1+|α|2 1 1+|α|2 |α|2 1+|α|2 1 1+|α|2 2|α|2 1+|α|2 1 1 1+|α|2 1 1+|α|2 1 P134 1 1+|β|2 1 1+|β|2 |β|2 1+|β|2 |β|2 1+|β|2 1 1+|β|2 1 1+|β|2 1 2|β|2 1+|β|2 1 P234 |β|2 1+|β|2 |β|2 1+|β|2 1 1+|β|2 1 1+|β|2 1 1+|β|2 1 1+|β|2 2|β|2 1+|β|2 1 1 P1234 1 2 1 2 1 2 1 2 1+|α|2 2 1+|α|2 2 1+|β|2 2 1+|β|2 2 1 Table 8.1: Single-particle probabilities for the various single-particle states for the various qualitatively “individuated” particles in state |ψ〉 in (8.8), associated with the projectors in (8.9). N.B. |α|2 + |β|2 = 1. If we count only strictly non-overlapping particles, then we recover the reas- suring result that state |ψ〉 in (8.8) contains two particles. The particles must not overlap, and their sum must be identical to the assembly. So we require ei- ther two projectors Pκ and Pλ from (8.9) such that p(κ|λ) = p(λ|κ) = 0 and 〈Pκ ⊗ Pλ + Pλ ⊗ Pκ〉 = 1; or else just one projector Pκ from (8.9) such that 〈Pκ ⊗ Pκ〉 = 1. There are three ways to satisfy this requirement: we may use the pair P13 and P24 (there is one particle specific a` la each of this pair), or the pair P14 and P23 (similarly), or the single projector P1234 (of which there are two corresponding particles). Doesn’t this mean now that there are six wholly distinct 216 particles and not two? No: since it may be checked that the sum of any one of the three pairs is identical to the sum of any other.8 Thus the advocate of Weaken recovers the results we expect by appealing to mereology. The weakening from ‘maximally specific’ to merely ‘specific’ has allowed constituent particles to overlap, and be parts of, one another. What does the advocate of Relativize have to say about state |ψ〉 in (8.8)? What, for her, are the constituent particles? Individuating particles under Relativize The proponent of Relativize must express |ψ〉 as a superposition of non-GM- entangled “branches”, all of which satisfy Equation (8.7). The constituent particles are then the maximally specific systems on each branch. But how are the branches determined? It turns out that, while it is a basis-independent matter whether or not a state is GM-entangled, it is not determined which non-GM-entangled states superpose to yield a given GM-entanged state. This is a kind of basis arbitrariness that affects systems of all symmetry types, not just fermions and paraparticles. I address this problem below. For now, we will work in the {|φi〉} product basis to give a flavour of the account given by the proponent of Relativize. The state |ψ〉 in (8.8) is expressed as the superposition of two non-GM-entangled states in the {|φi〉} product basis: 1√ 2 (|φ1〉 ⊗ |φ2〉 − |φ2〉 ⊗ |φ1〉) and 1√ 2 (|φ3〉 ⊗ |φ4〉 − |φ4〉 ⊗ |φ3〉) . (8.11) Each non-GM-entangled state in (8.11)—each a branch of |ψ〉 in (8.8)—is identified with a separate collection of particles, each of which is maximally specific a` la some single-particle state, just as in Section 5.2.1 each branch was associated with a different particle. In my example we have two branches. One consists of the pair of particles maximally specific a` la P1 = |φ1〉〈φ1| and P2 = |φ2〉〈φ2|; the other consists of the pair of particles maximally specific a` la P3 = |φ3〉〈φ3| and P4 = |φ4〉〈φ4|. (It may be checked that Equation (8.7) is satisfied for these branches and these single-particle projectors.) 8I.e., p([P13⊗P24+P24⊗P13]|[P14⊗P23+P23⊗P14]) = p([P14⊗P23+P23⊗P14]|P1234⊗P1234) = p(P1234 ⊗ P1234|[P13 ⊗ P24 + P24 ⊗ P13]) = 1. 217 The superposition of these two branches in (8.8) is then understood as co- existing (in a world, at a time) pairs of particles. The relative amplitude β α between the two branches may be interpreted as a relation—itself a determinate of a single dyadic determinable—holding between the two pairs of branch-bound particles. The assembly itself may be identified with the sum of these two superposed pairs, so related. As peculiar as it sounds, this suggestion is akin to the by-now-familiar account of macroscopic objects suggested by many proponents of the Everettian response to the measurement problem (Wallace (2003), Butterfield (2001)). Macroscopic objects, according to this account, are high-level patterns described by the univer- sal wavefunction. However, these patterns are instantiated only in some branches of the universal quantum state and not others; one should therefore not expect to be able to find the same macroscopic objects in each branch. We may even say that macroscopic objects exist only in some branches and not others, so long as that is not taken to imply any sort of semantic indeterminacy of the existence claim. On the contrary: for the Everettian, existence in a branch entails existence simpliciter (just as, say, existence in Leicester entails existence simpliciter). And so as for macroscopic objects under the Everettian’s suggestion, so too for the varietist’s particles, under my suggestion. (Note, however, that the varietist need not be an Everettian: so far I have only considered states of microscopic assemblies, and have said nothing about the microscopic/macroscopic boundary or the measurement process. Nevertheless, it cannot be denied that the varietist’s account holds promise for a simple Everettian story for how particles compose macroscopic objects. Perhaps mereology—which is composition in the strong sense in Section 5.1.4—will do after all.) It is also worth emphasising at this point that the ontological picture recom- mended by the proponent of Relativize is easily extended to accommodate states which are superpositions of different numbers of particles. Such states do not arise in elementary quantum mechanics, but are typical (indeed, characteristic!) of the theory of quantum fields, and are taken by some philosophers to preclude the possibility of a particle interpretation.9 9Of course, in QFT unsharp particle number is not the only problem thought to face particle 218 However, according to the proponent of Relativize, particles in different branches are strictly distinct, so there is no need for each branch to contain the same number of particles. Any (Fock space) state of the quantum field could still be understood in terms of (branch-bound) particles possessing certain properties and relations, so long as we include also relations that encode relative amplitudes between branches. Therefore, unsharp particle number is no special problem: indeed the effective re- striction, in elementary quantum mechanics, to a particular summand of the Fock space, now has no special ontological significance. (Of course, the restriction is perfectly natural, practically speaking, if the interactions are such as to constrain the assembly’s state to a particular Fock space summand, as indeed they do for low energies.) Let us now pursue the problem raised at the outset for the proponent of Rel- ativize, namely: How do we determine, for a given GM-entangled states, which non-GM-entangled states are to be its branches? To see the problem, let us con- sider a slightly different state: |ψ′±〉 = α 1√ 2 (|φ1〉 ⊗ |φ2〉 ± |φ2〉 ⊗ |φ1〉) + β 1√ 2 (|φ2〉 ⊗ |φ3〉 ± |φ3〉 ⊗ |φ2〉) + γ 1√ 2 (|φ3〉 ⊗ |φ4〉 ± |φ4〉 ⊗ |φ3〉) . (8.12) (We consider both the bosonic and fermionic version, to show that the problem is not peculiar to any single symmetry type.) Now we define two new single-particle states |χ±1 〉 := 1√|α|2+|β|2 (α|φ1〉 ± β|φ3〉) ; |χ±3 〉 := 1√|α|2+|β|2 (β ∗|φ1〉 ∓ α∗|φ3〉)  (8.13) Note that 〈χ±1 |φ2〉 = 〈χ±3 |φ2〉 = 〈χ±1 |φ4〉 = 〈χ±3 |φ4〉 = 0. The state |ψ′±〉 may now interpretations. Other problems facing any proponent of particles, include the Unruh effect and difficulties of localization. For a survey of these problems, see Baker (2009). 219 be re-expressed as |ψ′±〉 = α′± 1√ 2 (|φ2〉 ⊗ |χ±1 〉 ± |χ±1 〉 ⊗ |φ2〉)+ β′ 1√ 2 (|χ±1 〉 ⊗ |φ4〉 ± |φ4〉 ⊗ |χ±1 〉) + γ′ 1√ 2 (|φ4〉 ⊗ |χ±3 〉 ± |χ±3 〉 ⊗ |φ4〉) (8.14) where α′± := ± √|α|2 + |β|2 β′ := αγ√|α|2+|β|2 γ′ := β ∗γ√ |α|2+|β|2  (8.15) It can be seen from (8.12) and (8.14) that the state |ψ′±〉 is of the same form in the product basis induced by the single-particle basis {|φ1〉, |φ2〉, |φ3〉, |φ4〉} as in the product basis induced by {|φ2〉, |χ±1 〉, |φ4〉, |χ±2 〉}. Therefore it is hard to see what consideration could favour the former basis without equally favouring the latter, and vice versa. But although we may not be able to solve the problem, we can convert into another that I have already acknowledged. For the arbitrariness in basis for the fermionic GM-entangled state |ψ′−〉 would be overcome if the arbitrariness in basis for the constituent non-GM-entangled states, first discussed in Stage C of Section 7.1.2 were overcome. A breaking of the under-determination in the latter case induces a breaking of the under-determination of the former. The proponent of Relativize may therefore delegate the solving of her basis arbitrariness problem, for fermionic GM-entangled states, to any varietist, who must solve the basis arbitrariness problem for fermionic non-GM-entangled states. Besides, maybe the proponent of Relativize will shirk their responsibility even for bosonic GM-entangled states. Recall that bosonic non-GM-entangled states do not suffer a basis arbitrariness problem. But if they did, it seems plausible that any solution for fermionic states would be exportable to bosonic states: after all, in the former problem we seek a privileged basis for the single-particle Hilbert space. This privileged single-particle basis, if such there be, would induce a privileged product basis for states of any symmetry type. Thus I allow the proponent of Relativize to see her basis arbitrariness problem for any GM-entangled state as 220 not especially fatal to her project. The real problem is the basis arbitrariness for non-GM-entangled states, and that faces every varietist. Weaken or Relativize? We have seen what each of the two kinds of heroic varietism has to say about GM- entangled states. So which particular species of heroic varietism should we prefer: one that has particles as possibly less than maximally specific, but as constituents always of an entire state; or one that has particles as branch-bound systems, but as always maximally specific? The satisfying answer is that we need not choose. For there is no good reason to take Weaken and Relativize as mutually exclusive options. On the contrary: the mereological machinery that both endorse can be used to establish their con- cordance. The proposal for unification is simple: The (maximally specific, branch- bound) particles according to Relativize are parts of the (typically not maximally specific, trans-branch) particles according to Weaken. For an illustration of this proposal, consider again the state |ψ〉 in (8.8). The constituent particles of this state under Relativize are four, and each is individuated by the projector Pi := |φi〉〈φi|, where i = {1, 2, 3, 4}. The constituent particles under Weaken are individuated by the projectors shown in (8.9). But just as, under Weaken, we were encouraged to think of the particle specific a` la P13 (say) as a proper part of the particle specific a` la P123, why not think of the particle maximally specific—in the relevant branch—a` la P1 as a proper part of the particle specific a` la P13? This identification has advantages beyond reconciling Weaken and Relativize: it serves to explain the probabilities in Table 8.1. For example, in that Table p(14|13) = |α|2: this can be understood as the particle specific a` la P14 overlapping the particle specific a` la P13, where the overlap is identical to the particle maximally specific a` la P1, which exists on a branch with amplitude α. Conversely, p(24|13) = 0, so the particle specific a` la P24 has no common part with the particle specific a` la P13: i.e. there is no particle maximally specific on some branch that is a part of both. Overlap is part-identity, and part-identity is identity of parts. The particles according to Relativize provide the parts whose identity grounds the overlap of the 221 particles according to Weaken. Thus I return to an observation I made in Section 7.2.3, that we are led to something like a quantum counterpart theory (Lewis (1968; 1986, Ch. 4)) in which the basic entities are not world-bound, but rather branch-bound particles. These particles are maximally specific on their branch; that is, each may be qualitatively “individuated” on that branch with a one-dimensional projector that acts on the single-particle Hilbert space. With single-particle projectors—which may be multi- dimensional—we define counterpart relations, which select an (integer) number of particles (possibly zero) in each non-GM-entangled branch.10 As with familiar counterpart relations, the projector may fail to individuate a unique particle in each branch. In the example above, this occurs for projectors P123, P124, P134, P234 and P1234 (and more besides). In these cases the counterpart relation will not do as a surrogate for a “trans-branch” identity relation. How- ever, by arbitrarily selecting one of the selected branch-bound particles from each branch, we define a “trans-branch individual”, consisting of at most one particle from each branch. These trans-branch individuals typically overlap one another, and a single counterpart relation (a single projector) may define several such. If a trans-branch individual has a branch-bound particle in every branch, we may call it ubiquitous. Ubiquitous trans-branch individuals are precisely what were called ‘particles’ by the original proponent of Weaken (cf. Figure 8.1). Branch-bound particles must, of course, occupy pure states—the states a` la which they are max- imally specific. Trans-branch individuals, on the other hand, may occupy mixed states, so long as they are individuated by a multi-dimensional projector. But which of the two better deserve the term ‘particle’: the branch-bound kind or the trans-branch kind? There is no sensible answer to this question and none is needed. Better to let context decide. If we are talking about constitution, we are likely to want to talk about the non-overlapping, maximally specific, branch-bound objects which are the parts of all the others. If we are talking about modality, or if we are qualitatively individuating (perhaps for the purposes of calculating an 10Note that, while maximally specific particles are branch-bound, even a one-dimensional projector may succeed in selecting some particle in several branches. A single branch-bound particle is uniquely selected only by giving maximally specific individuation criteria for every particle on its branch, using the individuation methods in Section 7.2.1. 222 &% '$ P1 &% '$ P2 &% '$ P3 &% '$ P4 P13 P24 P14 J J J J JJ J J JJ P23 ' & P134 α β Figure 8.1: Branch-bound particles and some counterpart relations for the state |ψ〉 in (8.8). The counterpart relation induced by P134 defines two overlapping trans-branch individuals, which are the trans-branch individuals uniquely defined by P13 and P14. All of these trans-branch individuals are ubiquitous. expectation value), we are likely to want to talk about the (possibly overlapping) objects that have parts in more than one branch, and more than one state. This concludes my outline of varietism. I sum up with a final explicit statement of what, for any state |Ψ〉 of the assembly, the varietist’s particles are: (V2) Any state |Ψ〉 is a sum of terms, each representing a non-GM-entangled “branch”. (Which sum of terms is determined by whatever solution we find for the basis arbitrariness problem.) Each branch is composed exhaustively of (i.e. is the mereological sum of) wholly distinct maximally specific systems (which therefore must be in pure states); these are the most basic particles. Most generally, a particle is any mereological sum of basic particles, where at most one constituent basic particle belongs to each branch (a “trans-branch” individual). Any two branches are related by a relative amplitude. The full state |Ψ〉 is the mereological sum of these branches, so related. I now turn to the desiderata for the concept of particle laid out in Section 5.1, and argue that the varietist’s particles satisfy these desiderata to an acceptable de- gree. In the following Section (Section 8.3) I finally address the basis arbitrariness 223 problem which threatens varietism’s viability. 8.2 The merits of varietism In this Section, I will address to what degree varietism satisfies Section 5.1’s desiderata for the concept of particle. I will argue that varietism provides a toler- able target concept for particle—so long as we may assume the basis arbitrariness problem solved—a challenge I postpone until Section 8.3. In a final Section (8.2.6), we address the question whether varietist particles are discernible. But first I ad- dress, in order, the various desiderata for the concept of particle. 8.2.1 Varietist particles are physical Factorism erroneously affords physical existence to a statistical construct, and va- rietism is the most natural proposal for what the factorist’s particles are statistical constructs of. We saw in Section 7.2.3 that we recover the factorist’s prescription for calculating the reduced density operator of a constituent particle by setting the qualitative individuation criterion as broad as possible (cf. Equation (7.71)). Under a varietist interpretation, this is tantamount to laying down a counterpart relation which will select every branch-bound particle in every branch; the result- ing density operator is therefore a statistical average over the states of all of these branch-bound particles. That covers changeable properties. The unchangeable properties—what are often called ‘intrinsic’ or ‘kinematical’ properties—such as mass, spin and charge, are the same for any one of a collection of varietist particles of the same species as for the factorist particles which are statistical constructs of them.11 It might be objected that the varietist attributes more unchangeable properties to her particles than the factorist does to his, since it is with single-particle projec- tors that the varietist individuates her particles in the first place. Although these 11There is of course one notable exception; namely the unchangeable property of existing in the same mixed state as all other particles of the same species. This is, strictly speaking, a state-invariant property for an anti-haecceitistic factorist. 224 projectors represent quantities that are changeable for a factorist particle—insofar as their expectation values vary from state to state—they are unchangeable for a varietist particle—insofar as they constitute essential properties for that particle. This claim has a more than passing resemblance to an erroneous potential criticism of counterpart theory (cf. Lewis (1986, pp. 9-13)). This potential criticism is that, since under counterpart theory every object exists only on one world, all objects have all of their properties essentially. The correct response is to point out that the modal properties of a world-bound object—say real-world Humphrey—are represented by the occurrent properties of certain objects in other worlds—other- world Humphreys, or Humphrey counterparts—which need not share the same properties as our original, real-world Humphrey. Thus world-bound objects have modal properties ‘vicariously’ (Lewis (1986, p. 10)). This response need only be slightly modified to suit the varietist. The slight modification registers the fact that particles are not only world-bound; they are branch-bound too. So a varietist particle may not only have modal properties vicariously, but may have occurrent (but other-branchly) properties vicariously too. They are represented by branch- bound particles that exist in other branches of the same state. The claim that the varietist’s particles are physical must be judged on whether they behave in a way befitting of physical entities. This judgement must be in- formed, in part, by their satisfaction of certain of our other strands of meaning for the concept of particle: namely, compositionality and inter-theoretic applicability. For, if the varietist’s particles are good candidates for the constituents (in the broad sense; cf. Section 5.1.4) of familiar, macroscopic, physical objects, then they are good candidates for being physical entities. And if the varietist’s particles look like classical particles in the classical limit, then we have a defeasible reason to extend our belief in the physicality of classical particles to the varietist’s particles, even outside the classical limit. (Remember that it was here that the factorist proposal failed.) We must therefore look to these other desiderata, as we shall do below, in Sections 8.2.4 and 8.2.5. Identity over time does not speak for or against the varietist’s particles being physical, since plenty of physical things (like table-stages or events) do not exist over time. Locationality does not help either: if you don’t believe that locationality 225 entails physicality, then it is clearly no help; and if you believe that locationality does entail physicality, then presumably there would be no way to convince you that something was locational before you were already convinced that it was phys- ical. All agree that there are plenty of unphysical things (e.g. centres of mass) which at least seem to be locational, or are treated as such. Therefore, let me say no more about physicality here, and turn to the other strands of meaning. 8.2.2 Varietist particles are locational That the varietist’s particles are locational is clear (given that they are physical). For either one of the following two claims will always hold: 1. Location is used to qualitatively individuate the particles. This need not mean individuating by a precise location, as in e.g. ‘The particle occupying the lo- cation with co-ordinates (x, y, z)’. (Just as well, given that Hilbert spaces do not contain eigenstates of position!) Rather, it means that particles are in- dividuated by spatial wavefunctions, upon which there is no such restriction. (Since spatial wavefunctions determine momentum-space wavefunctions and vice versa, we count it as an instance of locational individuation even when the wavefunctions used yield sharply peaked expectation values for momen- tum and not position.) However, we demanded in Section 5.1.2 that locationality be cashed out in the following way: the state space for any particle must support a represen- tation of the spacetime symmetry group. But particles that are individuated by projectors whose support is less than all of co-ordinate space simply can- not be found outside of that region of support. So the state space for such a particle do not support a representation of the spacetime symmetry group because it is not closed under action by that group. Despite this, it remains true that individuating criteria may appeal to any state in a state-space that does support such a representation. And because of this, varietist particles that are individuated by spatial wavefunctions may 226 be said to fulfil the spirit of the locationality desideratum. That is to say, they satisfy the desideratum well enough. 2. Location is not used to individuate the particles. From Section 7.2.1 we know that this case requires the single-particle Hilbert space to contain at least one internal degree of freedom. So suppose we individuate with the projector Eσ := eσ ⊗ 1sp, where eσ acts on the factor single-particle Hilbert space corresponding to the internal degree of freedom, and 1sp is the identity on the spatial degree of freedom. By applying Equation (7.55), we find that the operator representing the position of the particle specific a` la eσ is piσ(Q) = N∑ k=1 [ k−1⊗ 1⊗ (eσ ⊗Q)⊗ N−k⊗ 1 ] (8.16) where Q is the single-particle operator representing position, and 1 is the identity on the full Hilbert space. In this case the particle individuated by eσ does have a state space which supports a representation of the full spacetime symmetry group. The crucial fact is that, since eσ only acts on the internal degree of freedom, the individ- uation criterion Eσ commutes with every generator of the group action on the single-particle Hilbert space. 8.2.3 Varietist particles do not (always) persist over time The varietist countenances trans-branch individuals—objects that have as parts at most one basic particle per branch. And if one is not averse to countenancing objects at other times than the present, or objects in other worlds than the actual, then there is no reason to restrict these to actual, present branches. Thus, depend- ing on one’s ontological commitment to other times and other possible worlds, a trans-branch individual may also be a trans-temporal and trans-world individual. According to varietism, an actual, present trans-branch individual deserves the name ‘particle’ as much as the branch-bound particles that are its parts. Therefore the varietist may countenance particles that exist over time. 227 However, as pointed out in Section 5.1.3, the commitment to trans-temporal individuals comes to more than a commitment to arbitrary mereological sums of objects existing at different times. The commitment is not to the existence of such entities—which is all too easy, given that one is happy to countenance non- present objects at all—but to the naturalness of such entities (cf. also Lewis (1986, p. 213)). The naturalness of trans-temporal particles is threatened by the preliminary considerations in Section 7.3. There we found that for many possible quantum histories, either no uniquely natural trans-temporal individuation strategies exist, or else no trans-temporal individuation strategy exists at all. In fact matters are worse for the varietist, since even a single individuation strategy may (under-) determine several trans-branch individuals (cf. Section 8.1.3). If we rule that any trans-temporal individual must be defined by a natural trans-temporal individuation strategy to earn the name ‘particle’, then the quan- tum world admits too many pathologies for it to be true that whenever there are branch-bound particles, then there are also particles of the trans-temporal kind. This means that varietism fails to satisfy the trans-temporal desideratum. But trans-temporal persistence was a non-compulsory constraint: trans-temporal indi- viduation problems are not enough to prevent varietism’s proposed target concept of particle being viable; they simply suggest that, if varietism is right, we are often better off to talk about particles of the branch-bound, therefore time-bound, kind. 8.2.4 Varietist particles compose assemblies According to the recommended ontological picture in Section 8.1.3, the state of the assembly always supervenes on the properties and relations of the constituent par- ticles, so long as we also include the relations which encode relative amplitudes be- tween non-GM-entangled branches—these must be interpreted as multi-grade rela- tions between the particles themselves. Therefore weak compositionality—i.e., su- pervenience of the assembly’s properties on the particles’ properties and relations— is satisfied. In fact, strong compositionality—i.e., mereological compositionality— is satisfied, since in any state of the assembly, the assembly is a mereological sum 228 of suitably related branch-bound particles. As we saw in Stage E of Section 7.1.2, there is an interesting supervenience result for varietism that is analogous to a more familiar supervenience result for factorism. The result for factorism is that an assembly’s properties supervene on its constituent particles’ properties alone iff the assembly is non-entangled. The corresponding result for varietism is that an assembly’s properties supervene on its constituent particles’ properties and symmetry type iff the assembly is non-GM- entangled.12 There can be no doubt that varietist particles satisfy the compositionality desideratum. Whether varietist particles might even compose macroscopic objects mereologically is an interesting question, but one that may be decoupled from our interests here. For the answer depends not on anything specific to quantum mechanics, but rather on whether macroscopic objects may be said to have precise characteristics. 8.2.5 Varietist particles have inter-theoretic applicability Recall from Section 5.1.5 that the inter-theoretic applicability of a QM-local par- ticle concept is a matter of ontological continuity in the limits of successful partial reduction of the other theories in question. ‘Ontological continuity’ is used here rather elastically: it is enough for me that the relevant objects in each theory behave alike in the appropriate limit. Of course, to make sense of ‘behaving alike’ I need a language to describe this behaviour that stands astride the different the- ories. And I need ‘behaving alike’ to come to more than just ‘satisfies the other desiderata for any target concept of particle’, since that much is already guaran- teed. What I want to know is whether, in some reasonable sense, the varietist’s particles become classical particles in the classical limit, and whether they become 12The supervenience result does not hold for paraparticles, since for any specific paraparticle type, distinct assembly states (that is, distinct generalized rays) exist which yield identical oc- cupation numbers for single-particle states. We may regain supervenience by allowing relations between constituent particles into the picture which encode, for any two particles, whether their states are symmetrized or anti-symmetrized in the assembly’s state. (This information deter- mines a unique standard Young tableau, and the standard Young tableaux are in one-to-one correspondence with states of all assemblies of all symmetry types; cf. Tung (1985, Ch. 5).). 229 Fock space quanta in a limit of a QFT, of conserved total particle number. Making ‘behaving alike’ precise in a way that is sufficiently general is—thankfully!— not a task I need undertake here.13 I have a specific cases to consider, so let me appeal to the details of those cases. I consider classical mechanics and QFT in turn. Varietist particles in the classical limit Here I will only consider the ~ → 0 limit, and my discussion will be somewhat elementary.14 This limit is typically modelled by a sequence of coherent states that are ever-narrowing Gaussians on the system’s configuration and momentum spaces (Landsman (2007, §5)). These Gaussians, parameterized by values for ~, are used to calculate expectation values for the various quantities. In the ~ → 0 limit, we effectively obtain a Dirac delta function centred at a point in the system’s phase space, which acts as a surrogate for the classical state associated with that point. That is: lim~→0 (〈Q2〉~ − 〈Q〉2~) = 0 and lim~→0 (〈P2〉~ − 〈P〉2~) = 0. The case we are concerned with is more complicated, for two reasons. The first complication is that we are dealing with systems that are themselves assemblies of constituent systems. For distinguishable systems, this is easily handled by defining the appropriate coherent states as products of single-constituent coherent states. Thus in the classical limit we take for granted that the assembly’s state is not entangled. However, the second complication is that we are dealing with indistinguishable systems, and the resulting superselection rule means that we are not at liberty to consider the behaviour of arbitrary products of coherent states. This second complication is easily overcome by simply acting on any given prod- uct state with a projector of the appropriate symmetry type. Thus the assembly’s wavefunction will be a superposition of multivariate Gaussians (i.e. products of single-constituent Gaussians), each centred at an image, under action by SN , of some point in the configuration space. (The wavefunction has the same charac- ter in the momentum representation too.) No information is lost—so long as we 13Optimism that the right sense of ‘behaving alike’ can be made sufficiently precise for general purposes surely underpins Ladyman’s (1998, 2002) ontic structural realism. 14An indispensable and thorough introduction to the classical limit in quantum mechanics is given by Landsman (2007). 230 also know the symmetry type of the state—if we instead represent the state as a Gaussian centred at a single point in the assembly’s reduced configuration space, formed by quotienting by SN . Given the results in Stages C and D of Section 7.1.2, this means that in the classical limit we take for granted that the assembly’s state is non-GM-entangled. That quantum states in the classical limit are taken to be non-GM-entangled is the key to the varietist’s success in establishing ontological continuity between her particles and classical particles. For, as we have seen in Sections 8.1.1 and 8.1.2, in any non-GM-entangled state of the assembly, the constituent varietist particles are all in pure states. In the approach to the classical limit, these pure states are ever-narrowing Gaussians in both co-ordinate and momentum representations, so that in the classical limit itself, the varietist particles may be attributed a definite location and momentum. In more detail: Recall from Sections 7.2.1 that qualitative individuation of quantum systems will be successful so long as the quantum state has its support restricted to some off-diagonal block of the reduced configuration space (RCS) of the assembly. For example, consider two particles on the real line. Then each off-diagonal block of the RCS is defined by a pair of intervals 〈∆1,∆2〉, such that ∆1 and ∆2 are connected open regions of R, and for every x1 ∈ ∆1 and every x2 ∈ ∆2, x1 < x2. Now for any state Ψ(x, y), if∫ ∆1 dx ∫ ∆2 dy |Ψ(x, y)|2 + ∫ ∆2 dx ∫ ∆1 dy |Ψ(x, y)|2 = 1 (8.17) then 〈E∆1 ⊗ E∆2 + E∆2 ⊗ E∆1〉 = 1 (8.18) where (E∆1f)(x) := { f(x), for x ∈ ∆1 0, for x /∈ ∆1 ; (E∆2f)(x) := { f(x), for x ∈ ∆2 0, for x /∈ ∆2 (8.19) Therefore, if condition (8.17) holds, we may say that there is one system specific a` la E∆1 and one specific a` la E∆2 . And since E∆1 ⊥ E∆2 , we may further say that 231 these systems are distinct. If, further, the state Ψ(x, y) is non-GM-entangled—as it will be when considering the classical limit—then each of these individuation criteria succeeds in picking out a unique branch-bound particle. One will occupy a state whose support lies in ∆1; the other’s state will have its support in ∆2. Moreover, these two particles exhaustively constitute the assembly itself. When considering the classical limit, we restrict attention to states Ψ (~) (q,p)(x, y), where Ψ (~) (q,p)(x, y) is a Gaussian centred at q = (q1, q2) (where q1 6 q2) in the RCS, and its Fourier transform is a Gaussian centred at p = (p1, p2) (where p1 6 p2) in the reduced momentum space. Now let us set ∆ () 1 := (q1 − , q1 + ),∆()2 ≡ (q2 − , q2 + ) for some  > 0. Then for any  > 0, Equation (8.17) holds for Ψ(x, y) ≡ Ψ(~)(q,p)(x, y) for some value of ~ > 0, and all smaller values. Thus, given the comments in the previous paragraph, for any  > 0, at some point along the way to the classical (~→ 0) limit, if q2−q1 > 2, then the projectors defined from ∆ () 1 and ∆ () 2 as in Equation (8.19) will each succeed in individuating a branch- bound particle, whose spatial wavefunctions are centred at q1 and q2 respectively, and whose momentum wavefunctions are centred at p1 and p2 respectively, and which together exhaustively compose the assembly. Therefore, by selecting the correct  we can, for any state Ψ (~) (q,p)(x, y), such that q2 > q1, individuate branch-bound particles for some value ~ > 0 and thereafter, along the way toward the classical limit. These branch-bound particles will have increasingly definite locations and momenta as ~→ 0. Thus they are perfect can- didates for being the temporal parts of classical particles (with matching locations and momenta) in that limit. The remaining wrinkle is the case where q1 = q2 =: q. In this case we can use one “individuation criterion” defined, as in Equation (8.19), from the interval ∆() := (q − , q + ). There is no ambiguity which branch-bound particle is picked out in this case: both are. Therefore we may additionally individuate using momentum. Provided that the assembly wavefunction is centred at a point that attributes different momenta of the two particles, then we can run a similar individuation campaign to that in the previous two paragraphs, will equal success. If, on the other hand, the assembly’s wavefunction is centred at a point which attributes the same location and momentum to both particles, then no two pro- 232 jectors will serve to individuate one without also picking out the other. But this is no objection to identifying the indiscernible branch-bound particles with their classical counterparts; for their shared location and momentum is definite in the classical limit, and in the classical case too the two particle-stages are indiscernible. (Whether such a situation is metaphysically possible is not a question the varietist has to answer.) Thus the varietist recovers classical particle-stages in the ~ → 0 limit. Unlike the factorist’s particles, in this limit the varietist’s particles may be attributed a definite position and momentum. But we can do more. One crucial characteristic of classical particles is that they have definite tra- jectories. To recover genuine classical particles (and not just particle-stages) in the classical limit, the varietist must provide uniquely natural trans-temporal in- dividuation criteria for her particles. But along the way to the classical limit, uniquely natural criteria exist. We need only make our original intervals ∆ () 1 and ∆ () 2 time-dependent. So define ∆ () 1 (t) := (q1(t) − , q1(t) + ) and ∆()2 (t) := (q2(t)− , q2(t) + ), where (q1(t), q2(t)) is the location of the system point in the RCS at time t, according to the classical dynamics. Then for any  > 0, at each time t, so long as q1(t) < q2(t), we succeed in individuating two branch-bound particles somewhere along the way to the classical limit (and thereafter). Trans-temporal individuals may then be constructed out of these individuated branch-bound particles with the requirement that any two branch-bound particles that exist at “near” times compose the same trans-temporal individual only if their wavefunctions significantly overlap. (This requirement can be made precise.) Again, collisions create a wrinkle: for if two branch-bound particles at a time them- selves have significantly overlapping wavefunctions, then it will be indeterminate which ought to belong to which trans-temporal individual. But again, this is no objection to the varietist’s claim to have recovered per- sisting classical particles in the ~ → 0 limit. For collisions provide as much of a problem for the trans-temporal individuation of classical particles. Thus it is enough for the varietist’s case to have recovered a unique pair of trans-temporal individuals for any segment of history for which the two individuals do not collide. 233 Varietist particles in QFT’s limit of conserved particle number Cross-theoretic ontological identifications are far more straightforward between quantum mechanics and QFT, since the state space of any quantum mechanical theory is a subspace of the state space for some quantum field theory. In QFT, the field’s state is restricted to one such subspace (in the sense that these subspaces are selected) when the dynamics preserve particle number. In this case the varietist’s proposal is simple: branch-bound particles are Fock space quanta. We have already seen, in Section 8.1.3, that varietism is very natu- rally extended to states with variable particle number. However, it is a further— and perhaps more surprising fact—that varietist particles have been familiar to us all along, albeit in a different theory. Recall what we know about Fock space quanta. We can glean all we need to know about a quantum from the operators with which it is associated—these are the creation and annihilation operators and the operators constructed from them. Starting from the unique vacuum state |0〉, it is familiar (cf. e.g. Maggiore (2005, p. 84)) that the mathematical state a†k,s|0〉 (8.20) represents (up to a multiplicative factor) a physical state in which there is one quantum, whose state comprises a definite momentum with wavevector k and internal state (spin, polarisation, etc., if any) s. Let us represent this state by |k, s〉. Then, by combining typical formalisms from QFT and elementary quantum mechanics, we may write a†k,s|0〉 ∝ |k, s〉. (8.21) Operations on (8.21) with more creation operators yield multiple-particle states with an ever-increasing number of particles. For example, if (k, s) 6= (l, r), then a†l,ra † k,s|0〉 ∝ 1√ 2 (|k, s〉 ⊗ |l, r〉 ± |l, r〉 ⊗ |k, s〉) . (8.22) where ‘±’ is positive for bosons and negative for fermions. The space of all states generated from finite operations on the vacuum state |0〉 by the creation operators 234 a†p,r is the Fock space for the quantum field. The operators a†p,r and ap,r, where p ∈ R and r is an internal state index, are the creation and annihilation operators for momentum quanta, which (by definition) satisfy the (anti-) commutation relations [ ap,r, a † q,s ] ± = (2pi) 3δ(3)(p− q)δr,s, (8.23) where the subscript ‘±’ indicates an anti-commutator or commutator, according to whether the quanta are fermions or bosons, respectively. And every annihilation operator annihilated the vacuum, i.e. for all k, s: ak,s|0〉 = 0. (8.24) If we now define a family of operators Nk,s := a † k,sak,s, (8.25) then from the (anti-) commutation relations (8.23), [ Np,r, a † q,s ] ± = (2pi) 3δ(3)(p− q)δr,sa†q,s. (8.26) This, combined with the vacuum condition (8.24), entails that the integral N(Ω, S) = ∫ Ω d3k (2pi)3 ∑ s∈S Nk,s (8.27) has a natural number spectrum on Fock space. This fact, combined with the identifications (8.21) and (8.22), entails that the operator N(Ω, S) is the quantity that counts the number of quanta whose momenta lie in Ω and whose internal states lie in S. But N(Ω, S) also counts the number of varietist particles that are maximally specific a` la states each of whose momentum lies in Ω, and internal state lies in S. To see this we only need to write N(Ω, S) out explicitly in a way that is more familiar from the point of view of elementary quantum mechanics, and our 235 previous discussions in Sections 7.1.2 and 7.2.1. We have N(Ω, S) = ∞⊕ n=1 [ n−1∑ k=1 ( k−1⊗ 1⊗ E(Ω, S)⊗ N−k⊗ 1 )] (8.28) where E(Ω, S) is the single-particle projector defined by E(Ω, S)ψs(k) := { ψs(k), k ∈ Ω, s ∈ S 0 otherwise, (8.29) and 1 is the identity on the single-particle Hilbert space. Thus each summand (in square brackets) of the right-hand side of (8.28) is an operator that acts on the appropriately (anti-) symmetrized n-particle Hilbert space. It may be checked that the operator on the right-hand side of Equation (8.28) is the unique operator that satisfies the requirements placed on it by the conditions from (8.21) to (8.27). Note that each n-particle summand of N(Ω, S) is of precisely the same form as a single-system quantity associated with a qualitatively individuated system, where E(Ω, S) is the criterion of individuation; cf. Equation (7.55) in Section 7.2.3 (where α ≡ (Ω, S)). In this case the single-system quantity in question is the identity, which yields a quantity which we interpret as counting the average number of systems maximally specific a` la states which lie in (Ω, S). But these maximally specific systems are, for the varietist, the branch-bound particles. Thus N(Ω, S) counts both the number of Fock space quanta, and, in each n- particle subspace, the number of varietist branch-bound particles, which are in states circumscribed by (Ω, S). Moreover, this number is that same for quanta as for branch-bound particles. The correct response is clear: the Fock space quanta are the varietist’s branch-bound particles. To end this Subsection, we note two significant consequences of the identifica- tion of Fock space quanta with the varietist’s branch-bound particles. 1. First, the lessons of anti-factorism already recommend something like a QFT ontology, even without having to consider superpositions of variable quantum number, difficulties in localization or the Unruh effect. 236 2. Second, any criticism facing varietism equally faces any interpretation of QFT that takes particles as the basic entities. So, in particular: the basis arbitrariness problem is a problem for a particle ontology in QFT as much as for varietism. This concludes our survey of the desiderata for varietist’s proposed target con- cept of particle. I have argued that varietist particles satisfy these desiderata well enough, despite possible problems facing trans-temporal individuation. But varietism’s viability relies on there being a satisfactory solution to the basis arbi- trariness problem, which I am yet to address. I will come to the problem soon, but first allow me a brief digression about discernibility. 8.2.6 Are varietist particles discernible? The varietist’s particles are composed of branch-bound particles (or are identical to them), and no two discernible objects may be composed of utterly discernible parts; so if any two particles are discernible, then they will be discernible by their parts. So our question comes to: Are branch-bound particles discernible? Branch-bound particles are “individuated” by single-particle states. Fermionic assemblies are characterized, due to the total anti-symmetrization of their available states, by Pauli exclusion: that is, any single-particle state is occupied at most once. Therefore, any two distinct fermionic branch-bound particles occupy differ- ent states, and are therefore discernible. Moreover, they are always absolutely dis- cernible (cf. Section 2.3), since the occupation of a particular single-particle state corresponds to a monadic property.15 This vindicates Weyl’s (1928, p. 241) claim, that ‘the Leibnizian principle of coincidentia indiscernibilium holds in quantum mechanics [for fermions],’ and his naming the exclusion principle as the ‘Leibniz- Pauli exclusion principle’ (Weyl (1949, p. 247)). In contrast, factorist fermions (and bosons and paraparticles) are always absolutely indiscernible (cf. Section 6.3), though they may be weakly discerned. If bosonic or paraparticle assemblies are in states where no single-particle state 15If we take a single-particle state as an intrinsic property of a branch-bound particle, then fermionic branch-bound particles are not just absolutely, but intrinsically discernible. 237 is occupied more than once, then any two branch-bound particles for those as- semblies are absolutely discernible too. However, bosons and paraparticles are not subject to Pauli exclusion, so single-particle states may be multiply occupied. Thus there are states for bosonic or paraparticle assemblies in which two bosons or two paraparticles may be indiscernible by their single-particle states. Does this make bosonic and paraparticle branch-bound particles utterly indis- cernible or just absolutely indiscernible? Recall (Section 6.3) that factorist par- ticles may be discerned, even though they occupy the same single-particle states, since they may be weakly discerned by some physical symmetric and irreflexive relation. Can these results for factorist particles be carried over for the varietist’s branch-bound particles? They cannot. Recall that the weak discernment of factorist particles relies on taking advantage of anti-correlations in the assembly’s state (cf. Section 6.3.5). Even a state with no anti-correlations in some single-particle basis may be ex- pressed in some new basis in which anti-correlations are guaranteed to arise. So the discernment of factorist particles relies on these particles surviving the basis change. The varietist’s branch bound particles do not survive single-particle basis changes. This is because, unlike factorist particles, they are individuated by their single- particle states. So if one changes the single-particle states in which the assembly’s state is expressed, then one changes the branch-bound particles about which one is talking. (Or at least, this is so for any reasonable attempt to solve the ba- sis arbitrariness problem; cf. Section 8.3.) Factorist particles are individuated by their factor Hilbert space labels, and these are of course preserved between ba- sis changes. Varietist branch-bound particles just don’t have anything similar to preserve them. Thus branch-bound bosons and fermions are utterly indiscernible in states of the assembly in which the same single-particle state is multiply occupied. Recall too that the varietist is anti-haecceitistic about her particles (cf. Section 8.1.1). Therefore the varietist endorses the metaphysical thesis which in Part I we labelled ‘QII’. If the varietist wishes to countenance only fermions, then any of the anti- haecceitistic theses, from SPII to QII, are consistent. 238 To conclude this Section, I will make a brief observation about how an ad- herence to varietism affects a well-established claim in the discernibility literature (this literature is more thoroughly discussed in Section 6.3). The claim is that a certain argument for anti-haecceitism about particles from quantum statistics, perhaps first criticised by Redhead (1987, p. 12) and French and Redhead (1988, pp. 235-8), does not work. The argument for anti-haecceitism runs as follows. Consider two two-state quantum systems. Choose the single- system orthobasis {|H〉, |T 〉} (“head” and “tails”). Then the tensor product Hilbert space for the two-system assembly is spanned by the four states |H〉 ⊗ |H〉; |T 〉 ⊗ |T 〉; |H〉 ⊗ |T 〉; |T 〉 ⊗ |H〉. (8.30) But if the systems are bosons, then the observed statistics seem to show that only three distinct possibilities exist—namely: two heads, two tails, or one head and one tail—since an equal probability of 1 3 is given to each. If the systems are fermions, then only one possibility exists: one head and one tail. These statistics can be explained (so the argument goes) if the last two states in (8.30) are not genuinely physically distinct. Yet the two states differ only haecceitistically. Therefore, there are no haecceitistic differences. The error in this argument, as pointed out by Redhead and French, is that the wrong basis states are appealed to. Under IP, the four pure states are |H〉⊗ |H〉; |T 〉⊗ |T 〉; 1√ 2 (|H〉 ⊗ |T 〉+ |T 〉 ⊗ |H〉) ; 1√ 2 (|H〉 ⊗ |T 〉 − |T 〉 ⊗ |H〉) . (8.31) Now the last two of these states are certainly distinct: one is bosonic; the other is fermionic. What explains the statistics is that, if the assembly is bosonic, then only the first three states are dynamically accessible; and if it is fermionic, then only the last state is accessible. The switch to varietist particles does not contradict this wisdom. For the vari- etist, as for anyone, the last two states in (8.31) are undeniably distinguishable— indeed they are qualitatively distinguishable, since an assembly’s symmetry type is an eigenvalue of a symmetric quantity. But nevertheless, varietism revives the 239 spirit of the argument for anti-haecceitism from statistics. The rehabilitated ar- gument runs as follows. Recall from Section 8.1.1 that the quantum formalism compels anti-haecceitism about varietist particles. That is, the varietist describes, for example, the state 1√ 2 (|H〉 ⊗ |T 〉+ |T 〉 ⊗ |H〉) as one in which there is one boson maximally specific a` la |H〉 and one boson maximally specific a` la |T 〉, and there is no further question which boson is which. But the quantum formalism may be incomplete. So suppose it is, and suppose there is the further question: Which boson is which? Then there would have to be two physical states corresponding to the mathematical state 1√ 2 (|H〉 ⊗ |T 〉+ |T 〉 ⊗ |H〉), related by a permutation of haecceities among the states a` la which they are max- imally specific. This permutation is not represented by the operation |φ〉 ⊗ |ψ〉 7→ |ψ〉⊗|φ〉, since varietist particles are not represented by factor Hilbert space labels. The permutation cannot be represented in the quantum formalism at all, since we supposed that the quantum formalism is incomplete in precisely this way. On the other hand, on the assumption of haecceitistic differences, there would still be only one physical state corresponding to each of the product states |H〉 ⊗ |H〉 and |T 〉 ⊗ |T 〉. Thus, if haecceitism were true of varietist particles, one would expect the physical states (plural!) represented by 1√ 2 (|H〉 ⊗ |T 〉+ |T 〉 ⊗ |H〉) to have twice the statistical weight assigned to |H〉 ⊗ |H〉 or |T 〉 ⊗ |T 〉 alone. Yet, in fact, they all have the same statistical weight, namely 1 3 . The best explanation of this is that our original assumption, namely that it made sense to ask which boson is which, beyond the distribution of single-system states, was incorrect. But this just is the assumption of haecceitistic differences. Thus, the quantum formalism is not incomplete, and anti-haecceitism is true of varietist particles. 8.3 A preferred basis problem for varietism I have argued that varietism satisfies, to a sufficient degree, the desiderata for any target concept of particle. But varietism’s viability depends on solving the problem first pointed out in Stage C of Section 7.1.2—the basis arbitrariness problem. This 240 Section argues for pessimism about varietism’s prospects for solving this problem. Because of this, I will be led, in Chapter 9, to advocate a rival target concept of particle—emergentism. The structure of this section is as follows. First, in Section 8.3.1, I will outline again the problem to be solved, and introduce five proposals, in increasing order of feasibility, to overcome it. Each of these five proposals will then be assessed, in Sections 8.3.2 to 8.3.6. 8.3.1 The problem of basis arbitrariness In Section 8.1.3, the problem of basis arbitrariness was discussed in particular for non-GM-entangled states for assemblies of fermions and paraparticles. The problem is that there are continuum-many single-particle bases in which such states are manifestly non-GM-entangled, so for each non-GM-entangled state it is under-determined which branch-bound particles the varietist is to say compose the assembly in that state. From Stage C of Section 7.1.2 we know that, for an assembly of N fermions, this arbitrariness is parameterized by the (N − 1)!2N−1- real-dimensional manifold ( CPN−1 × CPN−2 × · · · × CP1) /SN ; (8.32) i.e., that points in this manifold correspond one-to-one to a choice of a single- particle basis in which a given non-GM-entangled state of the fermionic assembly is manifestly non-GM-entangled. However, the problem—that the branch-bound particles out of which (the va- rietist will say) the assembly is composed are under-determined—is not limited to non-GM-entangled states of fermions and paraparticles. We also saw in Section 8.1.3 that it is under-determined, at least for some states, which non-GM-entangled “branches” the varietist should say superpose to yield a given GM-entangled state, even for bosonic assemblies. Again, this boils down to an under-determination of the single-particle basis which dictates which branch-bound particles the varietist says compose the assembly in that state. 241 Although in Section 8.1.3, I made this point by giving an example of a GM- entangled state which takes the same form for many choices of a single-particle basis, it may even be argued that the varietist’s problem plagues any state what- soever: for we can always re-express the same state using a different single-particle basis. I did not take this strict line, since it is a familiar fact—in classical me- chanics too—that a state space may be co-ordinatized in a multitude of different ways. This in no way compromises the claim that certain co-ordinatizations are nevertheless privileged in virtue of aligning with natural ontological divisions, so long as such natural divisions exist.16 The problem for the varietist is that there are plenty of states for which, given that natural divisions do exist, several “co- ordinatizations” (i.e., several single-particle bases) appear to have equal claim to align with them. It must also be emphasized that, given the clean meshing between the vari- etist’s branch-bound particles in elementary quantum mechanics and the quanta of QFT (cf. Section 8.2.5), these basis arbitrariness problems face those who seek a particle ontology of QFT. And, of course, this is all notwithstanding the additional problems facing a particle interpretation found there, such as the Unruh effect. In the following Sections, we investigate five potential responses to the basis ambiguity problem. The responses are not always exclusive: one may be imple- mented for some states, and others for other states, strategically. Nor do we claim that these responses are exhaustive. Perhaps a decent proposal exists that we have overlooked, in which case varietism would be saved. But we know of no such proposal. By way of introduction, we list the five responses here. They may be categorised into two groups: the responses which attempt to overcome basis arbitrariness by finding, for each state, a uniquely privileged basis; and the responses which 16This general way of looking at the matter is illustrated by what is now the consensus re- garding the empirical content of diffeomorphism covariance in general relativity, in the light of Kretschmann’s objection to Einstein. Despite the undeniable fact that the spacetimes in New- tonian mechanics and special relativity may too be expressed in arbitrary co-ordinates, general relativity is distinguished by the fact that no natural (in the sense of aligning with the inertial structure) co-ordinate system may be specified independently of a specification of the distribu- tion of mass-energy. For a better discussion of this response to the Kretschmann objection, see e.g. Misner, Thorne and Wheeler (1973, §17.7) and Brown (2006, pp. 154-6, 178-81). 242 attempt to overcome the arbitrariness by somehow accommodating all natural bases at once, without privileging any one over the other. The first two responses fall under the former category; the second three responses fall under the latter. The responses are written as claims for vividness. 1. One size fits all. There is a uniquely natural single-particle basis for each state of the assembly. It is the same basis for every state. That is, there is a categorically privileged single-particle basis. 2. Complicate. In realistic cases, there is more than one degree of freedom under consideration. These extra degrees of freedom provide the extra structure needed to determine a uniquely natural single-particle basis, for each state. 3. Coalesce. All of the rival single-particle bases may be reconciled by reify- ing all of the corresponding branch-bound particles. But in each non-GM- entangled branch, each particle associated with one single-particle basis is identical to some particle associated with any other single-particle basis. 4. Multiply. All of the rival single-particle bases may be reconciled by reifying all of the corresponding branch-bound particles. The particles in all bases are all distinct one from another. 5. Overlap. All of the rival single-particle bases may be reconciled by reifying all of the corresponding branch-bound particles. But particles associated with different bases are not wholly distinct. In fact particles in different bases overlap in such a way that for each non-GM-entangled branch, the sum of all branch-bound particles in one single-particle basis are jointly identical to the sum of all branch-bound particles in any other single-particle basis. 8.3.2 The ‘One size fits all’ response The One size fits all response is undeniably simple; but its simplicity issues from its rigidity, which is also the source of its drawbacks. It is clear that One size fits all easily and satisfactorily solves the basis ambiguity problem for non-GM-entangled fermions and paraparticles, whenever one of the rival decompositions belongs to 243 the categorically privileged basis. But if none of the rival decompositions belong to this basis, One size fits all comes into conflict with Sections 8.1.1 and 8.1.3’s prescription for deciding of which branch-bound particles the assembly in any given state is composed. For example, consider the two-fermion state 1√ 2 (|a〉 ⊗ |b〉 − |b〉 ⊗ |a〉) , (8.33) where |a〉 and |b〉 are both states with compact support centred at locations a and b, respectively, and 〈a|b〉 = 0. According to (V2) in Section 8.1.3, the possible pairs of branch-bound particles that could be said compose the assembly in this state are those for which (8.33) is manifestly non-GM-entangled. These pairs are maximally specific a` la 1√ 1 + |z|2 (|a〉+ z|b〉) and 1√ 1 + |z|2 (|b〉 − z ∗|a〉) , (8.34) where |z| 6 1, and 0 6 arg(z) < pi if |z| = 1.17 If for one value of z (and it will be for at most one value) the single-particle states in (8.34) belong to the categorically privileged basis under One size fits all, then the proponent of One size fits all may say that the assembly is composed of the pair corresponding to that value of z. So far, so good. But what if none of the rival pairs have states in the privileged basis? (What if, in our example, the privileged basis is position, or momentum?) What ought the proponent of One size fits all say then? A categorically privileged basis is categorically privileged, whether the state is manifestly non-GM-entangled in that basis or not. Therefore the proponent of One size fits all rejects the recommendations to the varietist I made in Sections 8.1.1 and 8.1.2 for all non-GM-entangled states, except those that happen to be mainfestly non-GM-entangled in the categorically privileged basis. According to this proposal, the objects of the quantum ontology are all branch-bound particles associated with the same privileged basis (or else they are composed from these 17This restriction is a convenient way to identify antipodal points on the Riemann sphere; cf. Stage C of Section 7.1.2. 244 branch-bound particles). Thus a state that is non-GM-entangled, but not mani- festly so in the privileged basis, is treated like any GM-entangled state: namely, as a superposition of “branches” of suitable branch-bound particles. The One size fits all response is clearly ad hoc: there is no reason to coun- tenance a categorically privileged single-particle basis, except that it solves the basis ambiguity problem. The proposed basis is not empirically accessible, and it conflicts with the principle that non-GM-entangled states are composed of un- superposed branch-bound particles. And, of course, there is no uniquely natural suggestion for what the privileged basis would be. (The two proposals that are perhaps the most intuitive, namely position and momentum, create particular trouble, since eigenstates for position or momentum do not exist in the standard Hilbert spaces.) Therefore we turn to the next proposal. 8.3.3 The ‘Complicate’ response The next proposal seeks to single out a preferred single-particle basis in a way that better reflects the physics. It takes advantage of the fact that, in realistic scenarios, particles have more than just the locational degree of freedom. Single- particle Hilbert spaces for all actual particles include an internal spin space, and possibly other degrees of freedom, such as flavour and colour. To simplify the discussion, let us ignore these other degrees of freedom and consider only location and spin. The utility of the spin degree of freedom in solving the basis arbitrariness prob- lem lies in the extra structure it provides in breaking the under-determination of single-particle bases. For, just as two “distinguishable” particles may be entangled, and two “indistinguishable” particles may be GM-entangled, so the individual de- grees of freedom associated with a single particle may be entangled. Entanglement between the degrees of freedom of a single system is exactly like entanglement be- tween distinct distinguishable systems: the state is entangled iff it is non-separable. The proponent of Complicate stipulates that branch-bound particles only pos- sess states in which distinct degrees of freedom are not entangled; i.e. they may be attributed both a pure spatial state and a pure spin state. The advantage of this 245 proposal over One size fits all, if it succeeds, is that it would be using the physi- cal phenomenon of entanglement to determine a uniquely preferred single-particle basis, so it could not be accused of being ad hoc. Apart from that, Complicate shares two important features with One size fits all. First, it breaks the under-determination of bases when one of the rivals is non-entangled in its degrees of freedom. For example, in the state 1√ 2 (|L, ↑〉 ⊗ |R, ↓〉 − |R, ↓〉 ⊗ |L, ↑〉), (8.35) the single-particle basis {|L, ↑〉, |R, ↓〉} is uniquely preferred over its rivals, such as { 1√ 2 (|L, ↑〉+ |R, ↓〉), 1√ 2 (|L, ↑〉 − |R, ↓〉)}.18 Second, however, the requirement that the single-particle basis be non-entangled may conflict with the principle that non-GM-entangled states are composed of un- superposed particles. Define an uncomplicated particle as a branch-bound particle whose state is non-entangled in its separate degrees of freedom; i.e. a branch-bound particle that may be ascribed a pure state in every degree of freedom. Then there are non-GM-entangled states whose maximally specific particles are not uncom- plicated. (Call not uncomplicated particles complicated.) For example, the single-particle states |φ〉 := α|L, ↑〉+ β|R, ↓〉; |χ〉 := β∗|L, ↑〉 − α∗|R, ↓〉 (8.36) (where |α|2 + |β|2 = 1) exhibit entanglement between the spatial and spin degrees of freedom so long as 0 < |α|, |β| < 1; so any particle that is maximally specific a` la |φ〉 or |χ〉 is complicated. Now consider the non-GM-entangled (bosonic) state |ψ〉 := 1√ 2 (|φ〉 ⊗ |χ〉+ |χ〉 ⊗ |φ〉) (8.37) ≡ √ 2αβ∗|L, ↑〉 ⊗ |L, ↑〉 − √ 2α∗β|R, ↓〉 ⊗ |R, ↓〉 + (|β|2 − |α|2) 1√ 2 (|L, ↑〉〉 ⊗ |R, ↓〉+ |R, ↓〉 ⊗ |L, ↑〉) . (8.38) 18Ghirardi et al (2002, p. 86) say of these rivals that they are ‘of no practical interest’. But I am trying to solve an ontological, not a practical, problem. 246 This state is a superposition of non-GM-entangled branches whose constituent particles are uncomplicated. Thus Complicate, like One size fits all, must suspend the recommendations to the varietist that I made in Sections 8.1.1 and 8.1.2 when the non-GM-entangled state is of the wrong type—in this case when the constituent particles could not be uncomplicated. Here we pick up where our discussion in Section 5.2.1, which applied to the “single-particle” Hilbert space, left off. The idea, under this proposal, is that non-entanglement of separate degrees of freedom trumps non-GM-entanglement of systems, so even a “single-particle” Hilbert space may contain states for which (it may be better to say) there is more than a single particle. There are two main objections to the Complicate response. The first, which is less serious, is that the demand that branch-bound particles be uncomplicated appears to conflict with the fact that complicated “single-particle” states may be used to qualitatively individuate particles (cf. Section 7.2.1). This objection is easily overcome by pointing out that it is already accepted that an individuation criterion may well succeed in picking out more than one branch-bound particle. True, the difference for the proponent of Complicate is that the criterion in question is associated with a one-dimensional projector, but to claim that branch-bound particles ought to be associated with one-dimensional projectors is simply to assert what was to be proved. The second, more serious, objection facing Complicate is that it will not work for every state, since states exist which continue to suffer a basis arbitrariness even when entanglement between degrees of freedom is taken into account. One such state is the ground state for the two electrons in a Helium atom: |φ1s〉1 ⊗ |φ1s〉2 ⊗ (|↑〉1 ⊗ |↓〉2 − |↓〉1 ⊗ |↑〉2) , (8.39) in which the states for each degree of freedom for both particles factorize. (Factor Hilbert space labels serve only to associate the right spin state with the right spatial wavefunction.) Here the demand that the constituent branch-bound particles be 247 uncomplicated helps not one bit in narrowing down the options.19 So the varietist must either appeal to one of the other four responses to mop up these cases, or else reject Complicate altogether. In fact, Complicate will play an important role later, in the discussion of emergentism in Section 9. 8.3.4 The ‘Coalesce’ response An alternative strategy in solving the basis arbitrariness problem is to somehow accommodate all of the single-particle bases in which the assembly’s state may be expressed. The first response that adopts this strategy is Coalesce, which stipulates that branch-bound particles that are maximally specific a` la states from one basis are identical to branch-bound particles that are maximally specific a` la states in other bases. Thus branch-bound particles are not only maximally specific a` la one state, but many. For example, for the singlet state 1√ 2 (|↑〉 ⊗ |↓〉 − |↓〉 ⊗ |↑〉) = 1√ 2 (|→〉 ⊗ |←〉 − |←〉 ⊗ |→〉) (8.40) the proponent of Coalesce claims that the branch-bound spin-up particle is iden- tical to either the branch-bound spin-left particle or the branch-bound spin-right particle, and that the branch-bound spin-down particle is identical to whichever of these is not identical to the spin-up particle. This may be generalised for any particle number N and any symmetry type. Thus, according to the proponent of Coalesce, a non-GM-entangled assembly is composed of N branch-bound particles which are in fact maximally specific a` la continuum-many single-particle states. GM-entangled assemblies, as per the discussion in Section 8.1.3, are then superpositions of collections of such particles. Now the objection. Let us ask: In the state (8.40), is the spin-up particle identical to the spin-right or the spin-left particle? That is, is the state composed 19This attempt to avoid the basis arbitrariness problem—and the example of the Helium ground state as a counterexample—has also recently been discussed by Bigaj in his lecture to the CLMPS 2011. 248 of a spin-up-spin-left particle and a spin-down-spin-right particle, or a spin-up- spin-right particle and a spin-down-spin-left particle? The proponent of Coalesce must allow both possibilities, so far as probabilities are concerned, since quantum mechanics tells us that p(← |↑) = p(→ |↑) = 1 2 , etc. Thus the proponent of Coalesce is forced to claim that the state (8.40)—and, with it, the entire quantum formalism—is incomplete, since it does not give us complete information about the constituents of the assembly. Rather, the state (8.40) must be interpreted as representing a statistical ensemble of collections of particles of both kinds. Indeed, since there are continuum-many bases in which the state is manifestly non-GM-entangled, the state must represent, for the proponent of Coalesce, a statistical ensemble of continuum-many kinds of particle collections. It would be enough to reject Coalesce that, at the outset (Chapter 1), we dis- avowed any interpretation that entails that the quantum formalism is incomplete. But even despite this Coalesce could not possibly work. The problems facing any attempt to interpret quantum probabilities as epistemic are well known; and it can be seen that Coalesce runs afoul the Kochen and Specker (1967) no-go theorem for non-contextual hidden-variable theories. The types of theories addressed by the Kochen-Specker theorem seek to at- tribute a definite true/false value to every ray in Hilbert space such that the quantum probabilities may be interpreted as arising from statistical ensembles of states corresponding to such attributions. This problem is equivalent to colouring the entire unit sphere in the Hilbert space in black and white so that, for any fam- ily of perpendicular points, all but one are painted black. As is now well known, this cannot be done for dimensions of three or more. Let us now consider a non-GM-entangled N -particle state in which no single- particle state is occupied more than once. For this state, the mathematical problem facing the proponent of Coalesce is to attribute one of N particle labels to every ray in the N -dimensional subspace spanned by the component single-particle states. Again, this must be done in a way that is consistent with the quantum probabilities arising from averages over statistical ensembles of assembly states corresponding to such attributions. This problem is equivalent to colouring the entire unit sphere in this subspace with N colours so that, for any family of perpendicular points, 249 all get painted one colour each. But if we label just one of the colours ‘white’ and the remaining N − 1 ‘black’ (consider them as shades of black, as it were), then it is clear that this problem can be solved only if Kochen and Specker’s problem can be solved—which it cannot, for N > 3. The only case for which Coalesce escapes the no-go theorem is N = 2. But this should come as no consolation, since we were seeking a general solution to the basis arbitrariness problem. We must therefore look elsewhere.20 8.3.5 The ‘Multiply’ response We could not solve the basis arbitrariness problem by stipulating that branch- bound particles from different bases are identical, so the natural next suggestion is to try the opposite: i.e. to say that any two branch-bound particles from different bases are distinct. Aside from the ontological extravangance of this response, one immediately wonders why it is that the branch-bound particles in different bases are always correlated in the same way, to accord with the quantum probabili- ties. (Why, for example, is there always one spin-left and one spin-right fermion whenever there is one spin-up and one spin-down?) In short, it seems we have a multitude of necessary connections between distinct existences. Now Hume’s dictum, that there are no such connections,21 has recently been subjected to some serious criticism.22 It may be argued that necessary connections between distinct existences are perfectly in order, so long as the objects in question are related in the right way, despite being distinct. Being related in the ‘right way’ might include: one of the objects composing the other (if one believed that composition was not identity); or being related by a difference in logical form, like object to universal. 20It is perhaps surprising to note that Coalesce does not run afoul of Bell’s Theorem (1964)— despite (8.40) being the very state that Bell used in his proof. The reason is that Bell’s Theorem assumes that the full algebra of quantities on C4 is available; whereas the varietist imposes the Indistinguishability Postulate, which greatly restricts the available algebra. In particular, joint probabilities for outcomes for different directions of spin cannot meaningfully be calculated, since the individuation criteria for the two particles would not be orthogonal. 21Hume (1740, Book I, Part III, §VI) wrote, ‘There is no object, which implies the existence of any other if we consider these objects in themselves.’ 22MacBride (1999), Cameron (2008), Wilson (2008). 250 Now all these cases provide interesting challenges to Hume’s dictum. They may even enforce a limitation on its application. But the current case is clearly not of this type: the branch-bound particles are all taken to be objects, and they are all taken to be wholly distinct. Why should it be that they always appear together as they do? Another response is available to the proponent of Multiply, and it was suggested by Lewis (1992): the connection between the particles may not be absolutely necessary, but only physically necessary. The claim of Multiply would then be that, somewhere in the full expanses of logical space, the branch-bound particles appear without particles from other bases, and it is only in the QM-worlds (the worlds that obey quantum mechanics) that they, due to physical necessity, appear together. One objection to this attempt to escape the Humean problem is that the branch-bound particles simply don’t exist in non-QM-worlds. This claim will be congenial to those who consider theoretical terms to be implicitly defined by the theory in which they are embedded (e.g. Carnap (1966), Lewis (1970b), Sneed (1971)). Entities that are so defined simply don’t exist in worlds in which the theory is not true. However, this clever response fails because, as we have seen in Section 8.2.5, ontological identifications may be made between classical and quan- tum mechanics. So the proponent of Multiply must accept that branch-bound particles can be found in non-QM-worlds. However, there is a more serious objection arising from considerations of the classical limit, which suggest that a wrong move has been made. As we have seen, in the classical limit, branch-bound particles at precise locations are identified with branch-bound particles with precise momenta: these “particles” are precisely the temporal stages of the familiar classical particles. Yet how can two groups of wholly distinct objects suddenly become identical in the classical limit? Multiply entails a sudden change where we need a continuous transition. This, together with worries about ontological extravagance and Hume’s dictum, points to a natural solution. 251 8.3.6 The ‘Overlap’ response The solution, and the best hope at solving the basis arbitrariness problem, is to retain the absolutely necessary connections between branch-bound particles from different bases, but deny that the particles are wholly distinct. This overcomes the Humean problem, since the particles now fall outside the domain of application of Hume’s dictum. It quells concerns about ontological extravagance, since particles from different bases may compose the same assembly, and are therefore jointly identical. Finally, it gets the classical limit right: we may say that branch-bound particles that are maximally specific a` la location and those that are maximally specific a` la momentum go from partial overlap to total coincidence in the classical limit. To illustrate the Overlap response, consider again the singlet state 1√ 2 (|↑〉 ⊗ |↓〉 − |↓〉 ⊗ |↑〉) = 1√ 2 (|→〉 ⊗ |←〉 − |←〉 ⊗ |→〉) . (8.41) The advocate of Overlap agrees with Coalesce, and against Multiply, that the as- sembly is in the same state no matter with what single-particle basis it is described. But Overlap disagrees, both with Coalesce that the constituent branch-bound particles are identical, and with Multiply that they are wholly distinct. Rather, they partly overlap, so that e.g. the spin-up particle is itself composed of parts of the spin-left and spin-right particles, and such that the sum of the spin-up and spin-down particles is exactly identical to the sum of the spin-left and spin-right particles (and every pair corresponding to every other decomposition). Similarly, the sphere has continuum-many decompositions into hemispheres; yet despite this, there remains just one sphere because hemispheres from different decompositions party overlap. The move made here is somewhat like the move made in Section 8.1.3, where the varietist accepted the overlap between less-than-maximally specific particles. Here, as there, we may endorse multiple but seemingly conflicting descriptions of the same state by declaring them equivalent. Each description has simply decomposed the same object in diverse ways. Thus the proponent of Overlap seeks to totally endorse (V2) in Section 8.1.3 with the reassuring addition that it 252 does not matter in which single-particle basis the assembly’s state is described. Following on from the calculations in Equation (8.10) and Table 8.1 in Section 8.1.3, we might even propose the squared inner product of two states as a measure of overlap for two branch-bound particles in those states. Since, e.g. |〈↑ |←〉|2 = |〈↑ |→〉|2 = 1 2 , we might say that the spin-up particle is composed of exactly half of the spin-left and spin-right particles. This measure has the desirable property that, given any single-particle state, its degree of overlap with all of the single-particle states in any chosen basis sums to unity. Mereological overlap has been marshalled in a variety of other areas to solve similar problems. Armstrong (1978, pp. 120-4) declares resembling universals to be part-identical as a way of accounting for their resemblance without having to posit higher-order universals, which would lead to an infinite regress. Lewis (1993) attempts to solve Unger (1980) and Geach’s (1980) ‘Problem of the Many’— namely, that any (putatively!) single macroscopic object may be identified with a surfeit of distinct microphysical sharpenings with clearly demarcated boundaries— by pointing out that the various sharpenings are all ‘almost identical’, and claiming that ‘almost identity’ is good enough for macroscopic objects, for most purposes. So: has mereological overlap come to the rescue again? There is an important difference between, on the one hand, the uses of overlap in Section 8.1.3, by Lewis (1993), and perhaps even by Armstrong (1978); and on the other hand, the use of overlap in this case. The difference is that, in all the former cases it is clear what the parts are whose shared composition entails the overlap. Overlap is part-identity, and part-identity is identity of parts. So any claim of overlap must be supported by a specification of the parts that are shared between the putatively overlapping entities. In the case of Section 8.1.3, the parts of the overlapping particles are the maximally specific, branch-bound particles in some basis.23 In the case of Lewis (1993), the parts of the overlapping sharpenings are 23Pleasingly, the claim that the less-than-maximally specific particles overlap at certain branch- bound particles is reflected perfectly in the geometry of Hilbert space, in which multi-dimensional subspaces associated with the the less-than-maximally specific particles overlap precisely at the corresponding one-dimensional subspaces that are associated with branch-bound particles. 253 the mereological sums of microphysical constituents, na¨ıvely conceived. In the case of Armstrong (1978), the parts of the overlapping universals are the simpler universals of which the original universals are logical constructions. What are the parts of the putatively overlapping branch-bound particles from different bases? Branch-bound particles cannot have other particles as parts, since they are already maximally specific. What else is there? One remaining suggestion is that the parts are the parts of wavefunctions that define the branch-bound particles. But this suggestion has the wrong results, since there are many even functions f(x) that overlap some odd function g(x); yet their inner product, as defined by∫∞ −∞ f ∗(x)g(x) dx, is always zero, if it exists. From here there appear to be only two routes ahead. Either we reject Overlap, which, since it is the final hope to save varietism, entails a rejection of varietism; or else we attempt to make sense of the overlap between branch-bound particles from different bases as something other than mereological. The problem with pursuing the latter route is that it is only mereology that, by being a part of logic,24 can overcome the worries of ontological extravagance, the violation of Hume’s dictum, and the discontinuity in the classical limit, all of which faced Multiply. Therefore, I reluctantly take the former route, and reject varietism. However, its merits were undeniable, so we might hope to find a nearby alternative that can overcome varietism’s fatal defect. With this, I turn to the final proposal for the target concept of particle, emergentism. 24Of course, it is controversial that mereology is logic. But even if mereology is not logic, it is hard to see that any other notion of composition (if any exists) can be claimed to be logical. These matters, of course, deserve a far more thorough study than there is space for here. 254 Chapter 9 Emergentism: winner in a poor field? In this short, final Chapter, I present an anti-factorist alternative to varietism. Its main feature is that is gives up on the desideratum that particles always compose the assembly. That is, it denies the strong version of compositionality (cf. Section 5.1.4); i.e., the claim that the assembly is a “derived” or “secondary” object, in the sense of being identical to the mereological sum of its constituent particles. And it also denies the weak version of compositionality; i.e., the claim that the properties of the assembly supervene on the properties and relations of the particles. For it claims that there are states in which the assembly exists but the particles do not. Instead, the assembly is taken as “fundamental” or “primary”, in the specific sense that each state in the assembly Hilbert space is construed as representing an ascription of properties to the assembly taken as a whole. Therefore, particles, if they exist, are construed as “derived” or “secondary” entities, in the sense of being emergent properties of some of the assembly’s states. For this reason, the proposal is called emergentism. This will be a somewhat disappointing de´nouement : I reluctantly plump for emergentism for want of a better alternative. In Section 9.1, emergentism is more precisely defined. I will argue that its most attractive version—what I call assembly realism—may usefully be conceived as a version of a field ontology found in the philosophy of QFT, albeit a version 255 restricted to the limit of conserved total particle number. Section 9.2 concludes the Chapter and the dissertation. After briefly recalling the main results of previous Chapters, I address the question whether particles under emergentism really are particles worthy of the name, especially given that some form of compositionality (cf. Section 5.1.4) is denied of them. I conclude that this question has no clear cut answer, but that some version of emergentism is the best proposal available for what particles are in quantum mechanics. 9.1 Emergentism defined The problem of basis arbitrariness facing varietism, discussed in Section 8.3, may be summarised in the following way. Prima facie, there is a single object—namely, the assembly—whose properties are represented by the states in Hilbert space. The varietist (and the factorist) wishes to consider this object to be complex, i.e. composed of simpler objects—namely, particles, as functionally defined by the desiderata in Section 5.1. However, several (indeed, continuum-many) putatively natural decompositions of the assembly exist, where the naturalness of decompo- sitions is governed by considerations of some version of entanglement between the constituent systems, whose own states must be meaningfully ascribable under the imposition of the Indistinguishability Postulate. The existence of several decompositions is problematic, because it seems impos- sible to interpret the particles issuing from each decomposition as part-identical, which would have permitted the interpretation that they are alternative decom- positions of the same assembly (cf. Section 8.3.6). Yet unless we can make sense of rival decompositions amounting to the same thing, we are not licensed to claim what is prima facie obvious: that there is a single entity, the assembly, being decomposed. Emergentism arises out of two possible retreats in the face of this problem. I define emergentism as the disjunction of the two, because they concur about one significant feature of particles: namely, that particles exist as (higher order) properties of some other object or objects. 256 The first retreat, which I discuss in Section 9.1.1, is to deny the original as- sumption above: namely, that the assembly is a single object, of which properties may be predicated. On this view, the assembly—and, with it, particles—are the (higher-order) properties of some other objects, perhaps spatial regions. I call this position mode realism. However, as I will argue, this position fares no better than the varietist’s One size fits all response (cf. Section 8.3.2), since it unjustifiably privileges one single-system basis over another. The second retreat, which I discuss in Section 9.1.2, is to accept the seemingly undeniable assumption that the assembly is a single object, but to deny that it may be categorically decomposed into simpler objects. On this view, particles are (higher-order) properties of the assembly, but they exist only for certain states of the assembly—in short, those for which basis arbitrariness can be overcome. I call this position assembly realism. I will argue that, though prima facie strange, assembly realism meshes best with the ontology of quantum field theory. 9.1.1 Mode realism The idea that, in elementary quantum mechanics, one might treat the modes as the basic objects is not unfamiliar (cf. e.g. Saunders 2006). But it is more familiar in the philosophy of quantum field theory. This is easily understandable, once one considers the formal properties of the Fock space. To illuminate these properties, consider a Fock space formed from a finite- dimensional Hilbert space. (So, strictly speaking, I am not considering a quantum field, which has infinitely many degrees of freedom. Nor am I considering a Fock space generated from a particle’s Hilbert space; since a particle must have location (cf. Section 5.1.2), and so has an infinite-dimensional Hilbert space.) For example, the Fock space for fermions whose single-system Hilbert space is Cd is F(Cd) = C⊕ Cd ⊕A(Cd ⊗ Cd)⊕A(Cd ⊗ Cd ⊗ Cd)⊕ . . . = C⊕ Cd ⊕ C 12d(d−1) ⊕ C 16d(d−1)(d−2) ⊕ . . . = ∞⊕ N=0 C( N d C) (9.1) 257 where Nd C := N ! d!(N−d)! . The Nth summand in (9.1) corresponds to the Hilbert space for an assembly of N fermions. But this tensor sum may be re-expressed as follows: F(Cd) = ∞⊕ N=0 C( N d C) = C ∑∞ N=0(Nd C) = C2d = d⊗ C2 . (9.2) So a Fock space for d-level fermions is equivalent to the Hilbert space for d dis- tinguishable 2-level quantum systems. These systems correspond one-to-one to the rays of the d-dimensional single-fermion Hilbert space, in some basis. In that basis, each ray represents the possession by a fermion of a particular eigenvalue (associated with that ray) for some single-fermion quantity whose eigenbasis is the basis in question. Thus, each of the d Hilbert spaces C2 is taken to represent, as it were, the space of possibilities associated with an eigenvalue. These possibilities are given by the possible number of fermions which may possess that eigenvalue. And since fermions are subject to Pauli exclusion, there are only two possibilities for each eigenvalue: 0 or 1; thus the Hilbert space for each is C2. Since the Hilbert space on the right-hand side of Equation (9.2) is a tensor product Hilbert space, it is tempting to think of it as having a natural decom- position into its factor Hilbert spaces. After all, recall (Section 6.1) that I en- dorse factorism for distinguishable systems, i.e. systems whose joint Hilbert space has a tensor product structure. Thus the temptation is to reify the eigenvalues mentioned in the previous paragraph. The states in Fock space F(Cd) are then construed, not as representing property attributions to a variable (indeed, possibly unsharp) number of systems; rather, they are construed as representing property attributions—more specifically, occupation numbers (which may be unsharp)—to a fixed number of these reified eigenvalues. Modes are simply these reified eigen- values. If we now return to elementary quantum mechanics, the mode ontology be- comes rather messy. This is because, in elementary quantum mechanics, the as- sembly’s Hilbert space is a single summand of the full Fock space in (9.1), and the breaking up of the Fock space into these summands is not particularly natural from the point of view of its decomposition into factor Hilbert spaces for each mode. Indeed, the demand that the “total system number” (the operator N(Ω, S), from 258 Section 8.2.5, where now (Ω, S) encompasses all single-system states) be conserved means that the states of the modes must be strictly correlated so as to ensure that the sum of all occupation numbers is a constant integer. But this is no argument against mode realism: after all, elementary quantum mechanics is the conserved “total system number” limit of quantum field theory. The correlation between the modes that ensures a constant “total system number” need not be interpreted as anything more than a dynamical phenomenon, whose contingent occurrence underpins the practical convenience of using elementary quantum mechanics over a more complicated theory that uses Fock space. Nor can it be objected against mode realism that its ontology is too weird. For, although the phrase ‘reified eigenvalue’ might cause more than a little hesi- tation, reified eigenvalues are actually much more familiar than it may seem. A compelling, but troublesome,1 example is location. The modes associated with the position quantity Q are nothing but locations ; i.e., roughly speaking, spatial points. Thus one particularly vivid instance of mode realism would be what Wallace and Timpson (2010) have recently presented as spacetime state realism. According to this view, any state in the assembly’s Hilbert space is interpreted as represent- ing the attribution of properties and relations (encoded in density operators) to spatial (or spacetime) regions. A typical state containing a single particle is then a (typically entangled) state of the various spatial regions in which the sum of all occupation numbers is 1 (Wallace and Timpson (2010, p. 711)). More generally, particles are construed as certain patterns of “excitation” (related to an increase of 1 in an occupation number) in the joint state of the various location. That is: particles are properties of the properties and relations between locations. I will not go into the details of spacetime state realism any further. Instead, I turn to an objection against mode realism (and hence spacetime state realism) that I cannot see a good response to. The objection is that mode realism, like varietism, suffers from its own basis arbitrariness problem. 1Location is troublesome because Hilbert spaces do not contain eigenstates for location. Nev- ertheless, location does have a spectral decomposition into a family projectors, all of which do act on the particle Hilbert space. 259 For consider again the right-hand side of Equation (9.2). Now, while it is true that the Fock space may be decomposed into d 2-level Hilbert spaces; there are, in fact, continuum-many ways of doing this. Specifically: a choice of a complete orthobasis in the single-system Hilbert space Cd determines a unique decomposi- tion of the Fock space into d “single-mode” Hilbert spaces, such that each vector in the chosen single-system basis corresponds to one of the d single-mode Hilbert spaces. But there are continuum-many complete orthobases in Cd. For example, let us consider a Fock space for 2-level fermions. For the sake of vividness, suppose the degree of freedom in question is spin for a spin-1 2 system. Then the Fock space is C4. A choice of modes now takes the form of a choice of direction of spin. A change of spin direction then induces a transformation between the single-mode Hilbert spaces. Writing |n〉ξ for the state in which the mode ξ is occupied n times, |1〉↑ ⊗ |0〉↓ = |↑〉 = 1√ 2 (|→〉+ |←〉) = 1√ 2 (|1〉← ⊗ |0〉→ + |0〉← ⊗ |1〉→) . (9.3) So a state that is non-entangled for one choice of modes is entangled for another choice of modes (cf. Zanardi (2001, p. 1)). This adds a new twist to the old basis arbitrariness problem of Section 8.3; since GM-entanglement, unlike the entangle- ment between modes shown here, is a basis-independent matter. One might now attempt to redeploy one or more of the five responses to the basis arbitrariness problem discussed in Section 8.3. However, Complicate—which relies on many degrees of freedom—could not possibly work, since modes have occupation number as their only degree of freedom. Apart from that, the results are the same as for varietism: i.e., the responses all fail to solve the problem. Incidentally, I note that Wallace and Timpson (2010) effectively opt for the One size fits all response (Cf. Section 8.3.2), since they favour the position representa- tion. Their argument is based on the intelligibility that the ‘spatial arena’ affords (Wallace and Timpson (2010, p. 724)). But it is not clear (to me, at least) why the momentum representation is any less intelligible; and the existence of more than one natural decomposition into modes—whether there are continuum-many or just two—is enough to be problematic for the mode realist. However, I should 260 emphasise, in defence of Wallace and Timpson, that they present spacetime state realism, not as the single best ontology for quantum mechanics, but rather as a rival to a particular interpretation they wish to criticise, namely wave-function realism (Wallace and Timpson (2010, pp. 702-6)). This concludes my discussion of mode realism. Since it runs afoul of its own basis arbitrariness problem, I turn to the remaining option, assembly realism. 9.1.2 The assembly is the object I claim that, like mode realism, assembly realism has a more familiar counterpart in quantum field theory. Specifically, my claim is that assembly realism is the conserved “total particle number” limit of field realism, the view that the quantum field is the basic object. Is it correct to liken an assembly to a quantum field? If a ‘field’ is defined as having infinite degrees of freedom (Huggett and Weingard, (1994b, p. 295)), then we may think that an assembly is unequivocally not a field. But why can we not identify the assembly with the quantum field under the condition of conserved “total particle number”? Just as modes appear to have strange correlations in the quantum-mechanical limit—which may be interpreted as a dynamically-induced phenomenon—couldn’t the assembly’s limited degrees of freedom be seen too as nothing but a dynamical restriction? An objection to this that can be dismissed out of hand is that the assembly is the mereological sum of its particles, and so could not have more degrees of freedom than are allowed by those of the particles. We can dismiss this objection because it assumes what the assembly realist has already denied: namely, that the assembly is the mereological sum of more basic objects. Assembly realism means treating the assembly as basic, so it is still up for grabs whether or not its degrees of freedom are constrained dynamically or by some stronger form of necessity. A second objection is that the identification of the assembly with a dynamically restricted quantum field could not be supported by a consideration of quantum mechanics alone, since the very idea of a quantum field is external to quantum 261 mechanics. But this objection does too much; for we want to be able to appeal to a theory’s “neighbours” when interpreting it (cf. our desideratum of inter-theoretic applicability in Section 5.1.5). And it must be noted that a positive case for the identification of the assembly with a dynamically restricted quantum field is, of course, ontological continuity in the QFT limit. Furthermore, it is not surprising that the metaphysics issuing from an interpre- tation of a formalism should point to possibilities, intelligible within that meta- physics, that the formalism neglects to represent. We saw an example of this very phenomenon in Section 8.1.3, where I claimed that superpositions of variable particle number are easily accommodated by the varietist. Treating the quantum field as the basic, or “fundamental” object in quantum field theory, is a doctrine with many adherents.2 For example Wald (1994, p. 46): ‘quantum field theory is the quantum theory of a field, not a theory of “particles”.’ And Clifton and Halvorson (2001, pp. 459): ‘quantum field theory is “fundamen- tally” a theory of a field, not particles . . . this view makes room for the reality of quanta, but only as a kind of epiphenomenon of the field associated with certain functions of the field operators.’ Also Malament (1996, p. 1): . . . in the attempt to reconcile quantum mechanics with relativity the- ory . . . one is driven to a field theory; all talk about particles has to be understood, at least in principle, as talk about the properties of, and interactions among, quantized fields. (And cf. also Huggett and Weingard (1994a, 1994b, 1996).) I emphasise that all the above quotes express support for a field ontology as a response to distinctly quantum-field-theoretic phenomena: namely, the Unruh effect (in the case of Wald and Clifton and Halvorson) and failures of localizability (in the case of Malament). Yet we have been pushed to a similar position—we may call it the quantum mechanical limit of field realism—not directly because of any quantum-field-theoretic phenomena, but for want of a better alternative. 2Though, of course, opinion is not unanimous. Notable critics of the field ontology are Teller (1995, Ch. 3) and Baker (2009). 262 Of course, the idea that the assembly has no decomposition makes the very use of the term ‘assembly’ Pickwickian, but that is no objection. A far more serious objection is that the claim that quantum assemblies have no proper parts is simply incredible. We have an understanding of single- and multi-particle states, or at least we thought we did. What has become of particles? 9.1.3 Regaining particles under assembly realism One of the major advantages of assembly realism is that one need not be committed to the idea that particles exist in every state. I combine this fact with what was a promising response to varietism’s basis arbitrariness problem—namely Complicate (cf. Section 8.3.3)—to give an account of the emergence of particles under assembly realism. Recall that, for the proponent of Complicate, particles are ‘uncomplicated’ max- imally specific systems. That is, they are maximally specific systems for which their separate degrees of freedom are not entangled. The idea is to use the re- quirement of non-entanglement of separate degrees of freedom to overcome the under-determination of single-particle bases. This response works for many states, but fails to overcome the under-determination for all states of the assembly. But assembly realists, unlike varietists, are not required to recover particles for all states of the assembly. Thus the assembly realist is at liberty to stipulate that, whenever particles exist, they are the unique, uncomplicated, maximally specific systems; and that in all other states of the assembly, there simply aren’t any particles. An enormous advantage of this proposal is that it manages to incorporate the varietist’s ‘branch-bound’ particles, whose merits I discussed in Section 8.2, without succumbing to the basis arbitrariness problem. Thus we may carry over, mutatis mutandis, almost all of the considerations from Section 8.2, which applied to varietist particles, to the assembly realist’s emergent particles. One important such result is the recovery of classical particles in the classical limit. For, in the classical limit, the varietist’s particles are not only non-GM- 263 entangled (cf. Section 8.2.5); they are also non-entangled in their separate degrees of freedom; that is, they are uncomplicated, as Complicate requires. This follows from the fact that, in the usual study of the ~ → 0 limit, we consider a series of ever-narrowing Gaussians centred at a single point in the (reduced) configuration space (Landsman (2007, §5)). Of course, it must be emphasised that, even in the classical limit of assembly realism, particles do not become objects, i.e. the subjects of predication. Rather, they become localized “spikes” in the assembly’s density function (like those dis- cussed by Redhead (1987, pp. 10-11)). Besides, particles exist in all classical limit states—even for assembly realism—since in the classical limit the basis arbitrari- ness problem vanishes (cf. the end of Section 8.3.5), and so there is a unique collection of uncomplicated maximally specific systems. Furthermore, in classical limit states of the assembly, the assembly state (begin non-GM-entangled) is de- termined by—i.e. it supervenes on—the properties of these “spikes”. So, in the classical limit, assembly realism even manages to recover weak compositionality. However, two issues remain for assembly realism. The first issue is that, by denying that the assembly is composed of constituent particles, assembly realism contradicts one of our desiderata for the concept of particle, namely composition- ality (cf. Section 5.1.4). I deal with this issue in the next, and final, Section. The second issue is that the claim that particles exist in some states of the assembly and not others seems to make them ontologically redundant. For the assembly exists in all states, and its properties suffice to make true or false any statement made in the quantum formalism. What work are the particles doing? We met this issue in Section 8.1.3, in the discussion of ‘cagey’ varietism. I make the same reply here: the particles are not idle objects, but ontological free- riders, once the assembly and its properties are taken into consideration. For the particles are certain features of the assembly’s state; they are not additions to the field ontology. 264 9.2 Conclusion: Is emergentism good enough? In this Section, I will consider whether emergentism—in the specific form of as- sembly realism—really does provide a good enough concept of particle. But first allow me to recapitulate the main results of this dissertation. 1. Following Quine (e.g. 1976), instances of discernment may be separated into four kinds: internal, external, relative and weak. The disjunction of internal and external discernibility—absolute discernibility—is related to the notion of individuality: specifically, an individual is an object that is absolutely dis- cernible from every other object in the domain. Discernibility is linked to the existence of symmetries on the domain of quantification. For any structure: if a permutation pi leaves invariant the indiscernibility equivalence classes, then pi is a symmetry (but not necessarily vice versa); and if pi is a symmetry, then it leaves invariant the absolute indiscernibility equivalence classes (but not necessarily vice versa; cf. Theorem 1, Section 2.4). Furthermore, for any finite structure: any two objects are absolutely indiscernible iff there is some symmetry that maps one to the other (Theorem 2, Section 2.4). 2. According to a weak version of the identity of indiscernibles (WPII) and a new metaphysical thesis called QII, individuality is conceptually distinct from identity; for (according to those two theses) there may be non-individual objects, whereas every object is self-identical. And according to QII and haecceitism, identity is distinct from indiscernibility; for (according to these theses) there may be utterly indiscernible, yet distinct objects. (Cf. Chapter 3.) 3. Following Haslanger (2006), a term may be associated with three concepts: its avowed concept (the concept we take it to connote); its operative concept (the concept which our use of the term reveals); and its target concept (what would be a good thing to mean by the term). The project of explication, or conceptual reform, for a given term may be characterised as bringing that term’s avowed and operative concepts in line with its target concept. (cf. Section 4.1.) 265 This framework may be adapted for the purposes of conceptual reform in physical theories in the following way. Interpreting a physical theory is a matter of laying down a representation relation between the theory’s formal- ism and physical items (including, perhaps, non-actual physical items). Thus the formalism is afforded physical content by referring to mathematical items that, in turn, represent physical items. A precise concept may be identified with an intension; i.e., roughly speaking, a function from possible physical worlds to extensions, which are physical items, or sets of them. Thus lay- ing down a representation relation associates concepts with elements of the theory’s formalism. Conceptual reform is thereby linked to finding the best representation relation between the mathematics and the physics. (Cf. Sec- tion 4.2.) A theory’s formalism provides its own standard of naturalness with which to identify the target concept for a given term. But this claim does not commit one to a Lewisian (1983) ontology of perfectly natural properties and relations. The identification of a target concept for a term is constrained, not only by naturalness, but also by that term’s operative concept. For otherwise the target concept could not count as an explication of that term. (Cf. Section 4.1.2.) 4. The operative concept of particle, applied generally, has five main strands. (1) A particle is a physical item. (2) A particle has location; so, inspired by Wigner (1939), its state space ought to provide a basis for a representation of the spacetime symmetry group. (3) A particle persists over time, but momentary particle-stages may exist even if there are no uniquely natural trans-temporal particles. (4) Particles compose assemblies, in the weak sense that an assembly’s properties supervene on the properties and relations pos- sessed by its particles; or perhaps in the stronger sense that an assembly is a mereological sum of its particles. (5) Particles are trans-theoretic entities; so particles from “neighbouring” theories ought to coincide in the limit that the mathematics of one theory tends to another. (cf. Section 5.1.) These strands of meaning are consistent with identifying several particles in certain states of what is usually called a ‘single-particle’ Hilbert space. This 266 occurs whenever the Hilbert space accommodates internal, as well as spatial, degrees of freedom. (Cf. Section 5.2.) 5. The “local” operative concept of particle, as applied in quantum mechanics, associates particles with factor Hilbert spaces; this conflicts with the general operative concept. The view that the “local” operative concept is also the target concept of particle is called factorism. Factorism is distinct from haecceitism, since factorism is purely a thesis about what particles are, while haecceitism is a thesis about whether permutations of particles generate a physical difference. (Cf. Section 6.2.) If the Indistinguishability Postulate (IP) is imposed, then factorist particles of the same species are all absolutely indiscernible one from another, and are therefore all non-individuals. However, they may still be weakly discerned, by appealing to multi-particle quantities which register anti-correlations be- tween singe-particle states. (Cf. Section 6.3.) Factorism is false for so-called “indistinguishable” particles—i.e. particles for which IP is imposed—since it makes an error analogous to reifying the average taxpayer. This is shown by their non-individuality, which entails that they cannot be picked out in language or in thought, and they cannot be associated with definite spatio-temporal trajectories in the classical limit. (Cf. Section 6.4.) 6. Anti-factorism prompts a new look at the notions of entanglement, assem- bly decomposition, and system individuation. The definition of entangle- ment proposed by Ghirardi et al (2002)—what I call GM-entanglement—is congenial to the anti-factorist. According to their account, an assembly’s state is non-GM-entangled iff it is obtained by (anti-) symmetrizing a prod- uct state. If an assembly is non-GM-entangled, its state supervenes on the single-particle states of which it is constructed, together with that assembly’s symmetry type (boson or fermion). (Cf. Section 7.1.) So-called “indistinguishable” systems may in fact be individuated (that is, picked out in language) using single-system projectors—what I call qualita- tive individuation. The Hilbert space for an assembly of “indistinguishable” 267 systems cannot be naturally decomposed in toto; but subspaces of it can be naturally decomposed. The resulting constituents are precisely the qualita- tively individuated systems. The failure of each qualitatively individuated system to exist in all states in the assembly’s Hilbert space, and the flexibil- ity in choosing individuation criteria, point to a quantum version of Lewis’s (1968) counterpart theory for such systems. (Cf. Section 7.2.) 7. A promising anti-factorist proposal for the target concept of particle is va- rietism, the “average” of whose particles may be identified with the fac- torist’s particles. The basic objects of the varietist ontology are branch- bound particles: particles which possess pure states and which compose non-GM-entangled branches of any state of the assembly. The qualitatively individuated systems of Chapter 7 may be identified with certain natural mereological sums of branch-bound particles. (Cf. Section 8.1.) Varietist particles satisfy the five strands of meaning of the operative con- cept of particle, laid out in Chapter 5. In the classical limit, branch-bound particles may be identified with classical particle-stages, and branch-bound particles may be identified with the quanta in QFT. Branch-bound fermions are always absolutely discernible, but branch-bound bosons and paraparti- cles may be utterly indiscernible. That is: the weak discernibility results established for factorist particles in Chapter 6 cannot be carried over to varietist branch-bound particles. (Cf. Section 8.2.) Varietism fails because it cannot escape a basis ambiguity problem. That is, for fermionic and paraparticle states, it is under-determined which branch- bound particles compose the assembly in a given non-GM-entangled state. The most promising attempt to escape the problem—to declare that branch- bound particles from different bases are part-identical—fails, since no ac- count can be given of what these parts are supposed to be. (Cf. Section 8.3.) 8. Thus we are led to our second and last anti-factorist proposal for the target concept of particle, emergentism. On this view, particles do not compose the assembly, even in the weak sense. Indeed, particles are not even objects, 268 according to emergentism. This proposal bifurcates: either the modes are treated as the basic objects (mode realism), or else the entire assembly is treated as the basic object (assembly realism). Mode realism runs afoul of its own basis arbitrariness problem, since it is under-determined which modes ought to be reified. Assembly realism suffers no such problem, and has on- tological continuity with a popular way of interpreting quantum field theory. According to assembly realism, particles are like the varietist’s branch-bound particles, with the added restriction that they are non-entangled in their separate degrees of freedom; except that they are construed as higher-order properties of the assembly. Thus the assembly realist’s particles obey most of the strands of meaning of the operative concept of particle; with the notable exception that they can no longer be said to compose the assembly. So assembly realism provides the best hope for an explication of the concept of particle in quantum mechanics. But is it good enough? That is: is the target concept proposed by the assembly realist sufficiently similar to the operative con- cept of particle, discussed in Chapter 5, to warrant counting it as an explication of ‘particle’? A mark against is that, even though assembly realism manages to recover weak compositionality in the classical limit, the stationary states of electrons in molecular orbitals do not belong to the classical limit. And in these states there is no unique collection of uncomplicated, maximally specific systems. (Recall the example of the ground state of Helium, at the end of Section 8.3.3.) Thus assembly realism is committed to the claim that, in realistic cases, Austin’s ‘moderate-sized specimens of dry goods’ are not composed of particles. A mark in favour is that, like the varietist’s particles, the assembly realist’s particles satisfies many of the operative concept’s strands of meaning. Specifi- cally: they are undeniably physical; they may be attributed a location (at least as a degree of freedom); and they enjoy ontological continuity with both quan- tum field theory and classical mechanics. They also satisfy some (at least Weyl’s (1928)) intuitions regarding discernibility. That is: fermions, whenever they exist, are always absolutely discernible; while bosons and paraparticles may be utterly indiscernible. 269 Another consideration in favour, of course, is that no viable alternative seems to exist. The factorist cannot offer particles that are anything other than non- individuals; and the varietist cannot tell us which branch-bound particles to de- compose non-GM-entangled states into. Assembly realism, even if it is not good enough, is the best we can get. I submit that we have here a case of semantic indecision (what I called in (v) of Section 4.1.2 a hard case). Therefore all I can do is present the unhappy facts: and it has been the aim of this dissertation to establish those facts. Whether we now decide to apply the term ‘particle’ to the assembly realists’ offerings is, as Lewis (e.g. 1995) might have said, a purely political matter. 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