Geometric Aspects of Gauge and Spacetime
Symmetries
Steffen Christian Martin Gielen
Trinity College
A dissertation submitted to the
University of Cambridge
for the degree of
Doctor of Philosophy
March 2011
2Declaration
This dissertation is my own work and contains nothing which is the outcome of work
done in collaboration with others, except as specified in the text (section 1.2). It is
not substantially the same as any other thesis that I have submitted or will submit
for a degree at this or any other university.
This thesis is based on the research presented in the following refereed publica-
tions:
(1) Gary W. Gibbons and Steffen Gielen. The Petrov and Kaigorodov-
Ozsva´th Solutions: Spacetime as a Group Manifold. Classical and Quantum
Gravity 25 (2008), 165009.
(2) Gary W. Gibbons and Steffen Gielen. Deformed General Relativity and
Torsion. Classical and Quantum Gravity 26 (2009), 135005.
(3) Gary W. Gibbons, Steffen Gielen, C. N. Pope, and Neil Turok.
Measures on Mixing Angles. Physical Review D79 (2009), 013009.
(4) Steffen Gielen. CP Violation Makes Left-Right Symmetric Extensions With
Non-Hermitian Mass Matrices Appear Unnatural. Physical Review D81 (2010),
076003.
(5) Steffen Gielen and Daniele Oriti. Classical general relativity as BF-
Plebanski theory with linear constraints. Classical and Quantum Gravity 27
(2010), 185017.
These papers appear in the text as references [65], [66], [67], [68], and [69] respec-
tively. The results of section 5 of (3) were also separately published in
(6) Gary W. Gibbons, Steffen Gielen, C. N. Pope, and Neil Turok.
Naturalness of CP Violation in the Standard Model. Physical Review Letters
102 (2009), 121802.
Steffen Gielen
Thursday, 3 March 2011
3Acknowledgments
My first and foremost thanks go to my supervisor, Prof. Gary Gibbons, for teaching
me so much about physics and mathematics, for many fruitful discussions and sug-
gestions, for his encouragement and for his support of all of my plans over the last
three years.
I have also greatly enjoyed working with, and learning from, Dr. Daniele Oriti,
Prof. C. N. Pope and Dr. Neil Turok on different projects that are part of this thesis.
Invitations by Dr. Turok and Dr. Oriti that allowed me to spend an intensive week
at Perimeter Institute in October 2008 and several months at AEI Golm in 2009/10
deserve special thanks.
I am grateful to Trinity College for all its support, both financial and otherwise,
over the course of the last four years, without which I would not have been able to
complete a PhD at Cambridge. My work has also been supported by an EPSRC
studentship.
I would like to thank my friends in Cambridge, who have hopefully improved
my understanding of physics and made my time more enjoyable, not least through
countless hours of ‘foosball’, and my friends in Germany who have kept in touch while
I was away for so long, and gave support in difficult times.
Going back a bit further in time, I would not have ended up as a PhD student in
Cambridge (and certainly not at Trinity) without the encouragement of Dipl.-Phys.
Igor Pikovski and the strong support of Profs. Ingo Peschel and Ju¨rgen Bosse. The
latter two together with Hermann Schulz (Hochschuldozent) were also influential in
shaping my interest in theoretical physics early on.
Last but not least, I would like to thank my aunt Odina and uncle Uwe for their
support, especially for the first year at Cambridge.
4Geometric Aspects of Gauge and Spacetime Symmetries
Steffen Christian Martin Gielen
Summary
We investigate several problems in relativity and particle physics where symmetries
play a central role; in all cases geometric properties of Lie groups and their quotients
are related to physical effects.
The first part is concerned with symmetries in gravity. We apply the theory of Lie
group deformations to isometry groups of exact solutions in general relativity, relating
the algebraic properties of these groups to physical properties of the spacetimes. We
then make group deformation local, generalising deformed special relativity (DSR)
by describing gravity as a gauge theory of the de Sitter group. We find that in our
construction Minkowski space has a connection with torsion; physical effects of torsion
seem to rule out the proposed framework as a viable theory. A third chapter discusses
a formulation of gravity as a topological BF theory with added linear constraints
that reduce the symmetries of the topological theory to those of general relativity.
We discretise our constructions and compare to a similar construction by Plebanski
which uses quadratic constraints.
In the second part we study CP violation in the electroweak sector of the standard
model and certain extensions of it. We quantify fine-tuning in the observed magnitude
of CP violation by determining a natural measure on the space of CKM matrices,
a double quotient of SU(3), introducing different possible choices and comparing
their predictions for CP violation. While one generically faces a fine-tuning problem,
in the standard model the problem is removed by a measure that incorporates the
observed quark masses, which suggests a close relation between a mass hierarchy and
suppression of CP violation. Going beyond the standard model by adding a left-right
symmetry spoils the result, leaving us to conclude that such additional symmetries
appear less natural.
CONTENTS 5
Contents
1 Introduction 7
1.1 Lie Groups and Homogeneous Spaces . . . . . . . . . . . . . . . . . . 11
1.2 Structure of this Thesis and Conventions . . . . . . . . . . . . . . . . 15
I Deformed and Constrained Symmetries in Gravity 17
2 Deformations of Spacetime Symmetries 18
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Deformation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 The Petrov and Kaigorodov-Ozsva´th Solutions . . . . . . . . . . . . . 25
2.4 The Kaigorodov Solution and Lobatchevski Plane Gravitational Waves 42
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Deformed General Relativity and Torsion 50
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Curved Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Deformed Special Relativity . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 A de Sitter Gauge Theory of Gravity . . . . . . . . . . . . . . . . . . 60
3.5 Cartan Connections on Homogeneous Bundles . . . . . . . . . . . . . 67
3.6 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.8 Gauge Invariance Broken? . . . . . . . . . . . . . . . . . . . . . . . . 77
3.9 Symmetric Affine Theory . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 General Relativity From a Constrained Topological Theory 80
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Actions for Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
CONTENTS 6
4.3 Constraining BF Theory . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4 Linear Constraints for BF-Plebanski Theory . . . . . . . . . . . . . . 90
4.5 Lagrangian and Hamiltonian Formulation . . . . . . . . . . . . . . . . 101
4.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 103
II Gauge Symmetries and CP Violation 106
5 SU(3) and its Quotients 107
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 The Homogeneous Space SU(3)/U(1)2 . . . . . . . . . . . . . . . . . 111
5.3 Metrics on U(1)2\SU(3)/U(1)2 . . . . . . . . . . . . . . . . . . . . . 112
5.4 Hermitian and Complex Matrices . . . . . . . . . . . . . . . . . . . . 114
6 Naturalness of CP Violation 120
6.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . 120
6.2 CP Violation in the Standard Model and Beyond . . . . . . . . . . . 124
6.3 Statistics on the Space of CKM Matrices . . . . . . . . . . . . . . . . 129
6.4 Statistics on the Space of Mass Matrices . . . . . . . . . . . . . . . . 136
6.5 Extension to Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Summary and Outlook 157
References 161
List of Figures 176
List of Tables 177
Index 178
Introduction 7
Chapter 1
Introduction
“Ich kann es nun einmal nicht lassen, in diesem Drama von
Mathematik und Physik – die sich im Dunkeln befruchten,
aber von Angesicht zu Angesicht so gerne einander
verkennen und verleugnen – die Rolle des (wie ich genugsam
erfuhr, oft unerwu¨nschten) Boten zu spielen.” [164]
Hermann Weyl
It is impossible to overemphasise the importance of symmetry as a fundamental
concept in both mathematics and theoretical physics.
In mathematics, the study of groups arose as an abstraction of “transformation
groups”, invertible maps of a given “point field” into itself [164]. Groups are the
mathematical realisation of the concept of symmetry. Felix Klein’s Erlangen Pro-
gramme [96] aimed to describe any geometry by its transformation group; as we will
detail shortly, the geometries considered in the Erlangen Programme are given by
coset spaces of the form G/H = {gH, g ∈ G}, where G is a continuous (Lie) group
and H a closed subgroup of G. Klein himself found the concept of groups most char-
acteristic of 19th century mathematics [164]. The study of continuous symmetries is,
in mathematical terms, the study of Lie groups and homogeneous spaces which nicely
combines aspects of group theory and differential geometry. Beyond the direct study
of groups, symmetry is an important concept in other parts of mathematics, such as
the study of differential equations.
In physics, the underlying symmetries of any theory are arguably its most fun-
damental property. As an example, the first physical theory in the modern sense,
Newtonian mechanics, fundamentally rests on assumptions about the symmetries of
Introduction 8
the physical world: A configuration of bodies in (gravitational) interaction will behave
in the same way if it is rotated as a whole (space is isotropic); a common motion of
all bodies at constant velocity is not observable (relativity of inertial frames). Con-
sequently, the laws of Newtonian mechanics are invariant under the action of the
semidirect product of the rotation group SO(3) and commutative Galileian boosts,
the Euclidean group E(3). When Einstein understood in 1905, in the wake of the
Michelson-Morley experiment [118], that it was necessary to modify the concept of
relativity of inertial frames, he did so by replacing the Euclidean group by the Lorentz
group SO(3, 1) as the fundamental symmetry group. We will see in chapter 2 that
this process can be mathematically viewed as a Lie group (or algebra) deformation.
The Galilean and Lorentz groups are examples of spacetime symmetries; their
primary action on space(time) itself induces an action on all fields according to their
tensorial structure. In general relativity, these global spacetime symmetries are made
local, although global symmetries also appear in many exact solutions of physical
interest, as we shall see in chapter 2. Both global and local spacetime symmetries
will play a role in the first part of this thesis.
More generally, the equations defining a given theory (or the action, in the usual
case that the equations of motion can be derived from an action principle) may also
be invariant under a group of transformations of the physical variables which is not
associated to a transformation of spacetime. A simple example in field theory is that
of a complex scalar field φ with equations of motion invariant under a global rotation
φ→ eiαφ. Promoting such a global symmetry to a local symmetry leads to a gauge
theory. We will discuss gauge symmetries in both gravity and particle physics in this
thesis.
Mathematically, the fundamental structures describing local symmetries are prin-
cipal fibre bundles over a given manifold which is interpreted as spacetime, where
the structure group becomes the “gauge group” of the theory, and a connection on
such a bundle is interpreted as a gauge potential. The remaining fields that make up
the physical content of the theory then live in associated vector and tensor bundles.
While this underlying structure suggests a very close relationship of all fundamental
interactions in Nature, there are important differences between Yang-Mills theory,
defined in terms of a principal bundle with compact gauge group, and general rela-
tivity, where in addition to the connection one has a frame field as a second structure
Introduction 9
which is crucial for the geometric interpretation of the theory.∗ In chapter 3, we will
study a formulation of general relativity as a Yang-Mills-like theory which tries to
exploit this analogy; the usual Lorentz algebra valued connection and the frame field
are combined into a connection on a bundle of the Poincare´ group (or its deformation,
the (anti-)de Sitter group). This study will elucidate some of the virtues as well as
shortcomings of such a formulation.
Symmetry is our most important guiding principle towards discovering the laws
of Nature. In particle physics, the electroweak model [72, 146, 161] unifying the
electromagnetic and weak interactions was discovered by postulating the fundamen-
tal symmetries and writing down the most general action invariant under them. A
crucial step in completing the present picture of particle physics was the proof by
’t Hooft and Veltman [156] that the mathematical structure of Yang-Mills theory is
compatible with the framework of quantum field theory; namely, Yang-Mills theory
is renormalisable. By far most attempts to go beyond the standard model of par-
ticle physics, such as supersymmetric theories and GUTs, are based on postulating
new symmetries. Part of our analysis of chapter 6 will be to discuss possible observ-
able consequences of such an existence of additional symmetries beyond those of the
electroweak model.
General relativity, however, has so far evaded a similar treatment in terms of
conventional field theory formulated on a background Minkowski spacetime, and by
now it seems unlikely that adding (super)symmetries will be sufficient for a consistent
quantum-mechanical treatment, although there has recently been increasing interest
in the possiblity that maximal (N = 8) supergravity in four dimensions is perturba-
tively finite [22]. String theory is one attempt to go beyond standard quantum field
theory by replacing the notion of point particles by extended objects; symmetries
(such as dualities) play a fundamental theory in string theory as well. Besides the
symmetries that will be important for a consistent fundamental theory of quantum
gravity, if we manage to construct (and understand) one, one may wonder whether
at low energy some effects of quantum gravity will be manifest in a deformation of
spacetime symmetries. This idea, which has been proposed on a purely phenomeno-
logical basis [4] as well as a consequence of developments in non-perturbative quantum
gravity [6], will be taken up in chapter 3.
∗A second difference is of course the non-compactness of the gauge group for Lorentzian signature,
but this seems a less important difference to Yang-Mills theory in the basic structure of the theory
– one often considers Riemannian signature as well.
Introduction 10
The notion of symmetries we have discussed so far refers to invariance of the
action or equations of motion under certain group actions. From a canonical view-
point, there is another important class of symmetries, namely those generated by
constraints. To obtain the canonical formulation of a given theory, one performs a
Legendre transformation, leading to a Hamiltonian
pi(x) ≡ δS
δq˙i(x)
, H ≡
∫
dd−1xH(x) ≡
∫
dd−1x
(
pi(x)q˙
i(x)− L(x)) , (1.0.1)
where S =
∫
dt
∫
dd−1xL(x) is the action defining the theory, and a symplectic struc-
ture on the phase space parametrised by (qi, pi),
{qi(x), pj(y)} = δij δd−1(x− y) , {qi(x), qj(y)} = {pi(x), pj(y)} = 0 . (1.0.2)
The Hamiltonian density H(x) typically splits into a “dynamical” part which is a
generic function of the pi and q
i, and a “constraint” part of the form λαfα(pi, q
i);
here λα are Lagrange multipliers, variables that appear in the original action whose
canonical momentum is identically zero and that are conventionally not treated as be-
longing to the phase space. The functions fα(pi, q
i) then act as constraints imposed
on the dynamical variables. A subset of these will be first class†, i.e. their Pois-
son brackets vanish up to a linear combination of constraints. The Poisson brackets
{fα, ·} of the first class constraints then determine vector fields on the phase space
whose integral curves are interpreted as describing physically equivalent configura-
tions. Although not necessarily associated with connections on fibre bundles, the
transformations on the phase space generated by these vector fields are, by their
physical interpretation, usually interpreted as “gauge transformations” as well. We
will see an instance of such a symmetry in the context of a topological field theory,
which is then constrained by a term that is added to the action, in chapter 4.
Although our discussions of chapter 4 will remain classical, the motivation for
the reformulation of general relativity that we will consider comes from the fact
that the quantisation of topological field theories is relatively well understood. One
might therefore hope that starting from a topological quantum theory which is then
constrained may provide a different route to quantum general relativity. This also
provides a bridge to the discussion of chapter 3 in that both consider possible reformu-
lations of general relativity at the classical level which are suggested by developments
in non-perturbative approaches to quantising gravity.
†We are here and in the following using the terminology introduced by Dirac [46].
Introduction 11
In all of our discussions, an important distinction is that between discrete and
continuous symmetries. The first systematic study of groups in physics was probably
that of discrete groups describing the symmetries of crystals [164]. We shall be mainly
interested in continuous symmetries which are described by Lie groups. The spacetime
and gauge symmetries described above are of this type. Their differentiable structure
means that differential geometry can be used to study them. In particular, we will
see that the group structure is encoded in a bracket operation on the tangent space
to the group at the identity element.
We will also encounter discrete symmetries, such as charge conjugation and parity
in particle physics which are represented by the finite group Z2. These will not be
studied mathematically, but their physical significance will be discussed.
In the spirit of Hermann Weyl’s self-image as a “messenger” between mathemat-
ics and physics, we will highlight different examples where a study of the differential
geometry of (spacetime or gauge) symmetry groups leads to important physical con-
sequences. Let us first introduce the most important aspects of the theory of Lie
groups and homogeneous spaces that will be important throughout this thesis.
1.1 Lie Groups and Homogeneous Spaces
The study of symmetries is the study of groups. If we consider continuous symme-
tries, the corresponding groups have an additional differentiable structure: They are
manifolds as well as groups‡. As simple examples one may think of the groups of
isometries of Rn,Cn or Hn, denoted by O(n), U(n) and Sp(n) respectively, or more
generally, of groups of matrices with continuous parameters which are submanifolds
of Rn×n,Cn×n or Hn×n. All examples of relevance in this thesis will be of this type.
Such groups are known as Lie groups, named after the Norwegian mathemati-
cian Sophus Lie (1842-1899) who initiated their systematic study in the context of
systems of differential equations. Lie himself seems to have referred to Lie groups as
“continuous groups” which describes appropriately what they are. The study of Lie
groups combines aspects of differential geometry with group theory.
Two natural actions of a Lie group G on itself are given by the left and right
‡One also requires the group multiplication map (x, y) 7→ x · y and the inversion map x 7→ x−1
to be differentiable.
Introduction 12
multiplication maps
L : G×G→ G , (h, g) 7→ hg , R : G×G→ G , (h, g) 7→ gh . (1.1.1)
By definition of a group these maps act simply transitively: Given a, g ∈ G there
is a unique element h such that L(h, g) = a and a unique element h′ such that
R(h′, g) = a. The pull-back and push-forward of these maps act on the tangent and
cotangent bundle of G.
Starting with a basis of the tangent space at the identity element e, one can use
the push-forward of left translation to define a basis of vector fields on G, known as
left-invariant vector fields. These define a global frame field on G and hence every
Lie group is parallelisable. (Right-invariant vector fields can be defined analogously
but often left invariance is preferred.) The tangent space at e, or alternatively the
set of left-invariant vector fields on G, is known as the Lie algebra g. It inherits a
Lie bracket [·, ·] from the commutator of vector fields. For matrix Lie groups, the
bracket is given by the commutator of matrices.
Lie’s important results are summarised in the following theorem [9]:
Theorem 1.1.1 (Lie)
(1) For any Lie algebra g there is a Lie group G (not necessarily unique) whose Lie
algebra is g.
(2) Let G be a Lie group with Lie algebra g. If H is a Lie subgroup of G with Lie
algebra h, then h is a subalgebra of g. Conversely, for each Lie subalgebra h of g,
there exists a unique connected Lie subgroup H of G which has h as its Lie algebra.
Furthermore, normal subgroups of G correspond to ideals in g.
(3) Let G1, G2 be Lie groups with corresponding Lie algebras g1, g2. Then if g1 and
g2 are isomorphic as Lie algebras, then G1 and G2 are locally isomorphic. If the Lie
groups G1, G2 are simply connected, then G1 is isomorphic to G2.
Loosely speaking, all of the structure of the groupG is encoded in its Lie algebra. If
one is not interested in the topological properties of G and considers only connected
and simply connected Lie groups, one can study the geometry of G by studying g
which is just a linear vector space together with a bracket structure.
For matrix Lie groups, it is usually most convenient to fix a basis of generators
Ma and to expand the Maurer-Cartan form in terms of this basis:
g−1 dg = σaMa (1.1.2)
Introduction 13
determines a basis of left-invariant one-forms σa. Similarly, right-invariant forms can
be obtained by computing dg g−1 which is clearly invariant under g 7→ gh. The basis
dual to {σa} is then a basis of left-invariant vector fields. By taking the exterior
derivative of both sides of (1.1.2) and using d(g−1) = −g−1 dg g−1 for matrices, one
can show that the forms σa satisfy the Maurer-Cartan relations
dσa = −1
2
Cb
a
c σ
b ∧ σc , (1.1.3)
where Ca
c
b are the structure constants of the Lie algebra defined by
[Ma,Mb] = Ca
c
bMc . (1.1.4)
One defines left-invariant tensors accordingly, e.g. left-invariant metrics as generated
by symmetrised combinations of σa: Any metric of the form
g = habσ
a ⊗ σb (1.1.5)
for constant symmetric hab is by definition left-invariant. One can phrase this by
saying that there is a one-to-one correspondence between scalar products on the Lie
algebra g (specified by hab) and left-invariant metrics on G [9]. These metrics are the
ones compatible with the left action on G on itself.
Lie groups are in some sense special cases of a more general class of manifolds
that will be important in the following: In many situations of physical interest, one
considers a manifold X which a Lie group G acts on transitively, but not necessarily
simply transitively. That is, for any x, x′ ∈ X, there is a (not necessarily unique)
g ∈ G with g · x = x′. (Again we take this to be a left action, the case of a right
action is analogous.) Then X is called a homogeneous space. For a given x ∈ X,
the subgroup Gx = {g ∈ G : g · x = x} leaving x invariant is called the stabiliser,
little group or isotropy subgroup of x. Then we have the following result [9]
Proposition 1.1.2 Let G × X → X be a transitive action of a Lie group G on a
manifold X, and let H = Gx be the isotropy subgroup of a point x. Then:
(a) The subgroup H is a closed subgroup of G.
(b) The natural map j : G/H → X given by j(gH) = g · x is a diffeomorphism. In
other words, the orbit G · x is diffeomorphic to G/H.
(c) The dimension of G/H is dimG− dimH.
Introduction 14
Since we assume transitivity of the group action, the orbit G · x is the whole of
X; one can then identify X with the coset space G/H.§ The choice of x in this
identification is arbitrary and corresponds to a choice of origin in X; different choices
for x lead to groups Gx related by conjugation. The natural left action of G on G/H
is just induced by the group multiplication law.
In many applications, an additional assumption is satisfied which simplifies the
construction of metrics on a homogeneous space: A given homogeneous space G/H
is called reductive if there is a splitting of Lie algebras of the form
g = h⊕ x (1.1.6)
invariant under the adjoint action of H. Then the subspace x of the Lie algebra g is
isomorphic [9] to the tangent space to G/H at H (or any other coset gH). It therefore
defines a basis of left-invariant vector fields on X, and one can take the dual basis
of a basis for x to construct left-invariant metrics on X. Reductive geometries will
be of importance in chapter 3 (with G = SO(d, 1) and H = SO(d − 1, 1)) and in
chapters 5 and 6 (with G = SU(3) and H = U(1)2). We will encounter an example
of a non-reductive geometry in section 2.4.
Homogeneous spaces are the simplest generalisations of (flat) Euclidean geom-
etry: In the 19th century, attempts to deduce Euclid’s fifth postulate, the parallel
postulate [50], from the other four¶, led to the discovery of elliptic and hyperbolic
geometries based on the sphere Sn ' SO(n + 1)/SO(n) and hyperbolic space Hn '
SO(n, 1)/SO(n), the natural counterparts of Euclidean space Rn ' E(n)/SO(n).
In Lorentzian geometry, the spaces of constant curvature, Minkowski, de Sitter and
anti-de Sitter space, are also homogeneous spaces. Many other known exact solutions
in general relativity are also homogeneous spaces; see [154, chapter 12].
Homogeneous spaces appear whenever there are (gauge or spacetime) symmetry
groups. Another example would be a Higgs potential spontaneously breaking a sym-
metry group G down to a subgroup H; then the space of vacua would be G/H.
§This result justifies dropping the distinction between the terms “coset space” and “homogeneous
space” as we will do in the following.
¶About 30 such “proofs” were shown to be unsatisfactory in the 1763 dissertation of A. Kaestner’s
student Georg Simon Klu¨gel entitled “Conatuum praecipuorum theoriam parallelarum demonstrandi
recensio” (Critique of the foremost attempts to prove the theory of parallels) [97, p.274].
Introduction 15
1.2 Structure of this Thesis and Conventions
The thesis consists of two main parts.
The first part explores symmetries in relativity which are deformed or constrained.
In chapter 2 we introduce deformation theory of Lie algebras and investigate its appli-
cation to global symmetries (isometries) of certain exact solutions to Einstein’s equa-
tions in four dimensions. We will see how the group-theoretic viewpoint of spacetimes
in terms of their isometry groups relates to physical properties such as causality vio-
lation. Section 2.3 contains calculations and results taken from [65] and are the result
of a collaboration with Gary Gibbons, except subsection 2.3.6 which is new. We omit
those parts of [65] which are of less relevance to the discussion of symmetry groups.
In chapter 3 we turn to a study of local symmetries in relativity, namely those of
the tangent space. We investigate how Cartan geometry, a generalisation of Rieman-
nian geometry where the flat tangent space is replaced by an arbitrary homogeneous
space, can be used to generalise the proposed deformed special relativity (DSR) to
a “deformed general relativity” in the same way that (pseudo-)Riemannian geome-
try describes general relativity as an extension of special relativity. We analyse the
physical predictions of a particular example supposedly describing Minkowski space,
and note that it involves a connection with torsion potentially leading to disastrous
predictions of charge nonconservation. This analysis serves as another example of the
interplay between an algebraic viewpoint on (infinitesimal) symmetries and physical
effects. The material presented in chapter 3 is mainly taken from [66] which again
resulted from collaboration with Gary Gibbons. In chapter 4 we study yet another
possible reformulation of classical general relativity as a topological BF theory with
constraints. Such a formulation has been considered before and is of relevance in
covariant approaches to quantising gravity, but we give a new version of it where the
constraints are linear, removing some of the ambiguity in their solutions. We see how
a very large symmetry on the phase space of the theory – the local equivalence of all
solutions to the equations of motion – can be constrained to give general relativity, a
theory with local degrees of freedom. The results detailed in chapter 4, published in
[69], were obtained in collaboration with Daniele Oriti.
The second part is concerned with gauge symmetries in particle physics and their
influence on CP violation in the electroweak sector. We aim to give estimates for
the likelihood of the observed magnitude of CP violation, measured by the Jarlskog
invariant J , by constructing a natural measure on the space of Cabibbo-Kobayashi-
Introduction 16
Maskawa (CKM) matrices. This space is a double quotient of SU(3) and does not
have a clearly preferred measure. After introducing different choices for the measure
in section 5, motivated first only by geometric and then also by physical considerations
– we will incorporate the observed quark masses into the analysis as well – in chapter
6 we make a detailed study of their physical predictions. We find that once the quark
masses are assumed, predictions for CP violation agree well with observation. We
then provide another example of how modified symmetries affect physical predictions
by assuming an extended left-right symmetry given by a second SU(2) gauge group.
This will influence our predictions on CP violation and lead to completely different
results. The analysis for the standard model was done in collaboration with Gary
Gibbons, Chris Pope and Neil Turok in [67]. Much of the calculations presented in
sections 6.3 and 6.4 are taken from this paper; I have omitted some parts of [67] where
my original contribution has been limited. The calculations in subsection 6.4.2 are
my own and were published in [68].
We close with a brief summary and outline possible directions for future research.
Our conventions for general relativity are those of Hawking & Ellis; the signature of
a Lorentzian metric is mainly positive; Greek letters are used for spacetime and Latin
letters for internal indices. The curvature of a connection is always F = dA+A ∧A
without any coupling constants as customary in some of the particle physics literature.
In discussions of CP violation in particle physics we follow the book of Jarlskog [91].
17
Part I
Deformed and Constrained
Symmetries in Gravity
Deformations of Spacetime Symmetries 18
Chapter 2
Deformations of Spacetime
Symmetries
2.1 Overview
The archetypal examples of groups describing spacetime symmetries in physics are the
Galilean and Lorentz groups of classical mechanics and special relativity. The crucial
insight of Einstein in 1905 was the necessity of replacing the Galilean group by the
Lorentz group in order to have an invariant speed of light c; as we will see shortly, this
process is also the archetypal example of a Lie algebra deformation. Historically,
the development of a mathematical theory of deformation and contraction of Lie
algebras was motivated by advances in theoretical physics, but it is at least conceivable
that the replacement of the Galilean group by the Lorentz group had been suggested
by mathematicians working on Lie group theory, instead of physicists like Einstein.
Since physical theories are to some extent always idealised approximations of a
much more complex Nature, the process of deformation of a symmetry group allows
one to make statements about the stability of a given theory. This stability point
of view was explored in detail in [116]. The passage from non-relativistic to special-
relativistic mechanics can be understood as a deformation of an “unstable” to a
“robust stable” theory [116]. The point here is that the zero commutators of Galilean
boosts constitute “fine-tuning” and any perturbation of the commutation relations
leads to noncommuting boosts. On the other hand, given noncommuting boosts, a
slight perturbation of the coefficient appearing in the commutator can be undone by
redefining the algebra generators, hence one has “robustness” in the theory.
Deformations of Spacetime Symmetries 19
In [116], the possible central extension of the Poisson algebra of the phase-space
coordinates of Hamiltonian classical mechanics, leading to quantum mechanics, was
interpreted as a similar instability of the original algebra. It was then argued that
although the resulting Heisenberg algebra admits a deformation, it can be undone by
a nonlinear redefinition of the position coordinate.
In this chapter, we shall focus on Lie algebra deformations, perturbations of the
structure constants which can be expanded in a power series. We are maintaining the
structure of a Lie algebra, i.e. of a vector space, and redefinitions of the generators
will always be linear transformations. One could consider more general kinds of de-
formations, where the Lie algebra structure is lost, either by allowing commutators of
two generators to be nonlinear in the generators or by allowing “structure functions”.
While the existence or non-existence of deformations of a given Lie algebra is a
mathematically well-posed question, the physical interpretation of its answer is less
clear in general. Both the generators and the deformation parameters of a given Lie
algebra may have different possible interpretations. For example, while the Poincare´
group admits a deformation to the de Sitter or anti de Sitter groups, it was argued in
[116] that the deformation parameter R, defining a length scale, may be taken to be
infinity if one is only interested in local kinematics, which of course is what happens
in general relativity. It is also clear that while linear transformations can formally be
applied to the generators in the Lie algebra, it may not appear physically meaningful
to take linear combinations of, for instance, translation and rotation generators in
the Poincare´ algebra. This point was stressed in [37], who also pointed out that the
interpretation of position operators appearing in a “kinematical algebra” acting on
single- or multi-particle quantum states is physically rather unclear, since position is
not additive in a way that momentum and angular momentum are.
The classic cases of the de Sitter group and its contractions were studied in detail in
[12], as we will review shortly. One could argue that the mathematical theory behind
deformations seems to have predictive power in terms of extensions or generalisations
of fundamental theories, but these are only a posteriori predictions [37]. There seem
to be two obvious classes of new applications for the concept of deformations of
spacetime symmetries:
• Situations where the relevant symmetry group is a subgroup of the Poincare´
group, or more generally a group of lower dimension than the Poincare´ group.
One example for the former is the proposed “very special relativity” [39], where
Deformations of Spacetime Symmetries 20
the fundamental symmetry group is taken to be ISIM(n− 2), consisting of the
maximal subgroup of the Lorentz group leaving a null direction invariant plus
translations. The possibility of deforming this group to obtain “general very
special relativity” was discussed in [62]; a physically acceptable one-parameter
deformation was found, similar to the case of the Poincare´ group which can be
deformed into the (anti-)de Sitter group. Interestingly enough, this deformed
group turns out to be the symmetry group of an asymptotically AdS space-
time playing a role in the construction of nonrelativistic hydrodynamics in the
context of AdS/CFT: The “Schro¨dinger spacetime” [81]
ds2 = r2
(−2 du dv − r2νdu2 + d~x2)+ dr2
r2
(2.1.1)
can be viewed as a left-invariant metric on a group manifold which is a subgroup
of the deformed group DISIMb(n− 2)[63, app. B].
On a less fundamental level, given a homogeneous space G/H with isometry
group G, e.g. as a given solution of general relativity, one may always ask what
deformations its isometry group admits, and try to construct spaces with the
deformed symmetry. The two examples we give in this chapter describe this
kind of situation. In principle, this could be used to facilitate the search for
new solutions of general relativity, although one is restricted to homogeneous
spaces, and curvature invariants are constant. One may still obtain spaces
with interesting causal properties, as exemplified by the BTZ black hole [152]
(obtained from anti-de Sitter space by periodic identifications), and by the
closed timelike curves that we will see occur in the Petrov and Kaigorodov-
Ozsva´th solutions.
• Alternatively, one may assume that the symmetry group is larger than the
Poincare´ group. In particular, the formalism can be straightforwardly extended
to supergroups [52]. It was found in [52] that the Killing superalgebra of the M2-
brane in eleven dimensions admits a deformation, which leads to the conjecture
that a solution with the deformed Killing superalgebra exists, possibly having
an interpretation as a perturbation of the M2-brane. Unfortunately, the search
for such a solution does not seem to have been successful∗.
Another example is Peter West’s conjecture of E11 as being the relevant sym-
metry group for M-theory [162], where one could look for possible deformations.
∗Jose´ Figueroa-O’Farrill, private communication (May 2009)
Deformations of Spacetime Symmetries 21
As outlined in section 1.1, the general setup for our considerations is a manifold X
which a group of transformations G acts on transitively, which may then be identified
with G/H where H is a subgroup of G.
When using deformations to look for interesting manifolds “close to” a given one,
one considers group deformations of the group G and then identifies the appropriate
stabiliser H in the deformed group. It simplifies the interpretation of the manifold
one obtains if this stabiliser can be viewed as a deformation of the stabiliser subgroup
of the original undeformed group, but in general this need not be the case [62].
By definition, the deformation theory reviewed here can only make statements
about Lie algebras. As explained in section 1.1, Lie’s theorems allow us to use the
terms “Lie algebra” and “Lie group” interchangeably, meaning by “Lie group” the
unique connected, simply connected group specified by a given algebra. In particular,
terms such as “Poincare´ group” or “de Sitter group” refer to connected components
of the identity.
We review the mathematical theory of Lie algebra deformations in section 2.2,
and then give two examples for pairs of solutions of general relativity related by such
deformations: The Petrov and Kaigorodov-Ozsva´th solutions are discussed in section
2.3. We shall see that they are both geodesically complete, and try to relate their
causal properties. We then turn to the Kaigorodov solution in section 2.4, and find it
can be deformed into a solution known as a “Lobatchevski plane gravitational wave”.
2.2 Deformation Theory
The theory of deformation of Lie algebras was reviewed in [104], where it was con-
nected to the perhaps more familiar operation of Lie algebra (Inonu-Wigner [88])
contraction. A Lie algebra g is deformed by redefining the Lie brackets as a power
series in a parameter t
ft(a, b) = [a, b] + tF1(a, b) + t
2F2(a, b) + . . . , a, b ∈ g , (2.2.1)
where the series is required to converge in some neighbourhood of the origin. The
Jacobi identity then leads to integrability conditions on the functions Fi at each order
in t: ∑
P(a,b,c)
∑
µ+ν=n
Fµ(Fν(a, b), c) + Fν(Fµ(a, b), c)
!
= 0 , (2.2.2)
where the sum is over all cyclic permutations and n = 0, 1, 2, . . .
Deformations of Spacetime Symmetries 22
Choosing n = 0 gives back the Jacobi identity for [·, ·], and at linear order (n = 1)
one gets ∑
P(a,b,c)
F1([a, b], c) + [F1(a, b), c]
!
= 0 . (2.2.3)
A deformation that only corresponds to a change of basis in the algebra g will be
regarded as trivial; these are of the form
f(a, b) = S[S−1a, S−1b] , S ∈ GL(n) , (2.2.4)
and expanding S in a power series around the identity gives the general form of trivial
deformations at each order in t.
It is perhaps more common to express these operations in terms of structure
constants Ca
c
b, as defined in (1.1.4) for a given basis. Denoting the structure constants
of the deformed algebra, with respect to the same basis, by Cˆa
c
b, the Jacobi identity
for the deformed algebra can be written as
Cˆd
e
[aCˆb
d
c] = 0 . (2.2.5)
One can then contemplate Rn
2(n−1)/2 (where n is the dimension of g) parametrised by
coordinates Cˆa
c
b antisymmetric in the lower indices, and the submanifold of R
n2(n−1)/2
defined by the Jacobi identity. A deformation of the given algebra g describes a smooth
curve in this submanifold which can be expanded in a power series in a parameter t:
Cˆa
c
b(t) = Ca
c
b + tAa
c
b + t
2Ba
c
b + . . . (2.2.6)
At linear order in t the Jacobi identity (2.2.5) holds if
Cd
e
[aAb
d
c] + Ad
e
[aCb
d
c] = 0 . (2.2.7)
Then a linear deformation only gives rises to a deformation if the requirement (2.2.5)
can be satisfied at each order in t.
Under a change of basis in g, expressed as a matrix S, the structure constants will
change according to
Cˆa
b
c(t) = S
b
eCd
e
f (S
−1)da(S
−1)f c . (2.2.8)
Expanding Sab(t) = δ
a
b+tM
a
b+. . ., this means that to first order a trivial deformation
can be written as
Aa
b
c =M
b
eCa
e
c − CebcM ea − CabeM ec . (2.2.9)
Deformations of Spacetime Symmetries 23
One can rephrase these conditions in the language of differential forms by choosing
a basis {λa} of left-invariant 1-forms for the original algebra satisfying the Maurer-
Cartan relations dλa = −1
2
Cb
a
cλ
b ∧ λc, and also defining Aa = 1
2
Ab
a
cλ
b ∧ λc to be
a vector-valued 2-form and Cab = Cc
a
bλ
c to be a matrix-valued 1-form [62]. Here
Cb
a
c are the structure constants of the original algebra and Ab
a
c is their infinitesimal
deformation. Then at linear order, the Jacobi identity is
DA := dA+ C ∧ A = 0 , (2.2.10)
and a deformation that can be written as A = DS for some S = Sabλ
b is regarded as
trivial. Hence non-trivial infinitesimal deformations is in one-to-one correspondence
with the cohomology group H2(g, g) [53]. At higher order one has to consider the
higher cohomology groups. In general, if a non-trivial deformation has been found
infinitesimally (i.e. at linear order), it does not necessarily extend to a deformation
satisfying the full set of conditions (2.2.2); obstructions to “integrability” of infinites-
imal deformations are given by the cohomology group H3(g, g) [53]. In practice, once
all infinitesimal deformations have been found, one can verify the full Jacobi identity,
and if it is violated, try to add higher order terms to the deformation.
Using cohomology theory, one can determine all general deformations of a given
algebra, as was done for the Galilean algebra, with and without central extension, in
[53]. We will focus on deformations at linear order in the following.
Applying this formalism, the authors of [12] classified all possible kinematical
groups in four dimensions, by which they meant classifying ten-dimensional Lie al-
gebras consisting of “translations”, “rotations”, and “inertial transformations”. This
splitting meant singling out three generators (“rotations”) and fixing their commuta-
tors with the other generators to assure isotropy of space. They further assumed that
parity and time-reversal (defined on the generators) leave the Lie brackets invariant,
and that the subgroup of inertial transformations is non-compact. The classification
one then obtains consists of eight types of Lie algebras, all of which may be obtained
by either contracting the de Sitter algebra so(4, 1) or anti-de Sitter algebra so(3, 2),
or alternatively as deformations of the Abelian algebra, as shown in figure 2.1. The
possible contractions correspond to different physical limits such as vanishing cosmo-
logical constant, infinite speed of light or zero speed of light, as explained in detail in
[12]. For instance, the Newton groups describe the symmetries of a non-relativistic
cosmological model with nonzero cosmological constant; this was discussed in detail
in [64].
Deformations of Spacetime Symmetries 24
(A)dS SO(3, 2), SO(4, 1)ffpara-Poincare´/inh. SO(4)
?
ff
?
Newton groupspara-Galilei




Poincare´Carroll
??
GalileiStatic (Abelian) ff
ff
Figure 2.1: Possible contractions and deformations of kinematical groups, according to
[12]. Arrows denote possible Inonu-Wigner [88] contractions.
Looking at figure 2.1, it could be argued that the introduction of noncommut-
ing translations, the most direct effect of a curved spacetime, is very natural from
the group deformation viewpoint as the de Sitter and anti-de Sitter groups arise as
deformations of the Poincare´ group.
For completeness, we mention that in order to obtain the algebra of quantum
mechanics from a deformation of that of classical mechanics, as discussed in [116],
one either deforms the Poisson algebra of functions on phase-space, replacing the
Poisson bracket by a Moyal bracket
{f, g}M = {f, g} − ~
2
4 · 3!
∑
i1,i2,i3
j1,j2,j3
ωi1j1ωi2j2ωi3j3∂i1i2i3(f)∂j1j2j3(g) + . . . , (2.2.11)
or alternatively considers the phase-space coordinates as elements of an Abelian Lie
algebra which is replaced by the Heisenberg algebra. In the first case one has an
infinite-dimensional algebra, in the second case one performs a central extension to-
gether with a deformation. The very illuminating discussion in [116] used purely
mathematical arguments as well as physical intuition to establish the stability of the
Heisenberg algebra, considered as a test for the “robustness” of quantum mechan-
ics. We stress again that this is a different notion of deformation, since one is now
considering non-linear transformations of the generators.
The maximal algebra one could consider in this discussion is the algebra of the
Poincare´ group, together with “spacetime coordinates” and an “identity” (a central
element) which appear as operators in quantum mechanics. In four dimensions, this
Deformations of Spacetime Symmetries 25
is a 15-dimensional group which is naturally deformed into so(6 − t, t), where the
original central element ceases to commute with the other generators in general. This
maximally extended case is most obviously fraught with interpretational difficulties,
as clearly illustrated in [37]. Formally, the deformation process naturally leads to non-
commuting spacetime coordinates [116]. This proposal goes back to Snyder [151],
whose algebra is however not the same as the one proposed in [116]. The idea of
“spacetime being non-commutative” is very interesting physically, however since it
is separate from the deformation theory of spacetime symmetries reviewed in this
chapter, it will be discussed in more detail in chapter 3.
2.3 The Petrov and Kaigorodov-Ozsva´th Solutions
Among the wide range of known exact solutions to Einstein’s field equations (the most
comprehensive source is [154]), there are some with interesting symmetry properties.
In this section we concentrate on the particularly simple case of a simply transitive
group of motions, such that the stabiliser is trivial and there is a one-to-one correspon-
dence between spacetime points and elements of the group of motions. Therefore the
spacetime is not only a manifold but also a Lie group. One example, and in fact the
only example among vacuum solutions without a cosmological constant, is provided
by the Petrov solution (2.3.1). An analogous example with negative cosmological
constant is provided by the Kaigorodov-Ozsva´th solution (2.3.10).
The Petrov solution has a physical interpretation as the exterior solution of an
infinite rigidly rotating cylinder. To gain an understanding of the physical proper-
ties of the Kaigorodov-Ozsva´th solution, we compute the stress-energy tensor of the
boundary theory in the context of the AdS/CFT correspondence from an expansion
near the conformal boundary. We then give a matrix representation of the isometry
group of the Petrov solution and its algebra and use it to construct left-invariant
one-forms on the group which could be used to compute left-invariant metrics.
We apply the theory of Lie algebra deformations to relate the Petrov solution to the
Kaigorodov-Ozsva´th solution, and discuss the physical interpretation of this result.
It is well known that the Petrov solution contains closed timelike curves (CTCs), so
an obvious question is whether the Kaigorodov-Ozsva´th solution exhibits similarly
exotic properties. Since in the context of possible causality violation by CTCs a
central question is whether these can be created by some process in a spacetime which
Deformations of Spacetime Symmetries 26
did not exhibit CTCs initially, we give a brief discussion of the possible appearance
of CTCs by spinning up a rotating cylinder. This elementary example supports
Hawking’s chronology protection conjecture which asserts that the appearance of
CTCs is forbidden by the laws of physics [77].
In a more detailed analysis of the physical properties of the Petrov and Kaigorodov-
Ozsva´th solutions, we show that they are geodesically complete. We discuss the global
causality properties of the two spaces; while the Petrov solution is totally vicious, due
to a theorem of Carter, a similar analysis for the Kaigorodov-Ozsva´th solution remains
inconclusive. The computation of the stress-energy tensor in AdS/CFT however sug-
gests causal pathologies in the Kaigorodov-Ozsva´th solution as well.
2.3.1 Vacuum Solutions With Simply-Transitive Groups of
Motions
Λ = 0 - The Petrov Solution
The Petrov solution is introduced in [154] in the following theorem: The only vac-
uum solution of Einstein’s equations admitting a simply-transitive four-dimensional
maximal group of motions is given by
k2ds2 = dr2 + e−2rdz2 + er(cos
√
3r(dφ2 − dt2)− 2 sin
√
3r dφ dt) , (2.3.1)
where k is an arbitrary constant, which shall be set equal to one, and we have rela-
belled the coordinates compared to [154]. The solution was first given in [134] and
also discussed in [43]. It describes a hyperbolic plane H2 (the (r, z)-plane) with a
timelike two-plane attached to each point.
The isometry group is generated by the Killing vector fields
T ≡ ∂
∂t
, Φ ≡ ∂
∂φ
, Z ≡ ∂
∂z
, R ≡ ∂
∂r
+z
∂
∂z
+
1
2
(
√
3t−φ) ∂
∂φ
− 1
2
(t+
√
3φ)
∂
∂t
, (2.3.2)
which satisfy the algebra
[R, T ] =
1
2
T −
√
3
2
Φ , [R,Φ] =
1
2
Φ +
√
3
2
T , [R,Z] = −Z . (2.3.3)
The isometry group contains three-dimensional subgroups of Bianchi types I and VIIh
acting on timelike hypersurfaces, and the solution (2.3.1) is Petrov type I [154]. The
first three Killing vectors obviously generate translations while the action of the one-
parameter subgroup generated by R on spacetime is given by the integral curves of
Deformations of Spacetime Symmetries 27
R, solutions of
dxµ(λ)
dλ
= Rµ(x(λ)) . (2.3.4)
These integral curves have the form
xµ(λ) =
(
r0 + λ , z0e
λ , φ0e
−λ
2 cos
√
3
2
λ+ t0e
−λ
2 sin
√
3
2
λ ,
−φ0e−λ2 sin
√
3
2
λ+ t0e
−λ
2 cos
√
3
2
λ
)
, (2.3.5)
where we label the coordinates by xµ = (r, z, φ, t).
The metric components gφφ and gtt become zero at certain values
† of r, but as
the determinant of the metric in (2.3.1) is always −1, it is possible to extend the
coordinates to infinite ranges and the coordinates (r, z, φ, t) define a global chart.
The manifold is also time-orientable, as the vector field tµ = (0, 0, sin
√
3
2
r, cos
√
3
2
r)
defines a global arrow of time, though this amounts to ∂
∂t
being future- as well as
past-directed at some points (as well as spacelike at others).
Bonnor [25] pointed out that the solution can be viewed as a special case of
the exterior part of a Lanczos-van Stockum solution [103, 155] describing an infinite
cylinder of rigidly rotating dust. Since this allows a physical interpretation of (2.3.1),
let us give the general solution for an infinite rigidly rotating dust cylinder, which in
Weyl-Papapetrou form is given by [157]
ds2 = H(ρ)(dρ2 + dz˜2) + L(ρ)dχ2 + 2M(ρ) dχ dτ − F (ρ)dτ 2, (2.3.6)
where H, L, M, F are functions of the radial variable ρ containing two parameters
a and R, interpreted as the angular velocity and radius of the cylinder respectively.
The high energy case aR > 1
2
, which contains closed timelike curves (CTCs), is of
interest here, and with the choices R =
√
e and aR = 1 the exterior solution is given
by
H(ρ) =
1
ρ2
, L(ρ) = −2
√
e
3
ρ sin
(√
3 log
ρ√
e
)
, (2.3.7)
M(ρ) =
2√
3
ρ sin
(
pi
3
+
√
3 log
ρ√
e
)
, F (ρ) =
2√
3e
ρ sin
(
pi
3
−
√
3 log
ρ√
e
)
,
where ρ is a radial coordinate in the exterior of the cylinder and so is restricted to
ρ ≥ √e , (2.3.8)
†these values of r can be shifted by a coordinate transformation φ→ αφ+βt, t→ −βφ+αt and
hence have no coordinate-independent significance
Deformations of Spacetime Symmetries 28
and χ is an angular coordinate and periodically identified with period 2pi; τ and z˜
are unconstrained. Applying the coordinate transformations
z˜ =
√
e z , ρ =
√
e er , χ =
1
4
√
3
√
2e
(√
2 +
√
3z +
√
2−
√
3t
)
, τ =
1
4
√
3
(z − t)
(2.3.9)
to the line element (2.3.6) indeed gives back (2.3.1). Hence if we adopt the interpre-
tation of (2.3.1) as describing the exterior of a spinning cylinder, we restrict the coor-
dinates to r ≥ 0 and identify
√
2 +
√
3z +
√
2−√3t with
√
2 +
√
3z +
√
2−√3t+
2pi 4
√
3
√
2e.
The general Lanczos-van Stockum solution has three linearly independent Killing
vectors ∂
∂χ
, ∂
∂τ
and ∂
∂z
but a fourth Killing vector is only present in the special case
aR = 1 because the algebraic invariants of the Riemann tensor are independent of ρ
just in this case [26].
Λ < 0 - The Kaigorodov-Ozsva´th Solution
A solution of the vacuum Einstein equations with negative cosmological constant
which has a simply-transitive four-dimensional group of motions was first given by
Kaigorodov [93] and rediscovered by Ozsva´th [125]. It has the line element
ds2 = − 3
Λ
dr2 + e−2r(dz2 + 2 dt dφ) + e4rdφ2 − 2
√
2er dz dφ . (2.3.10)
This solution was also given in [154]. It is Petrov type III and the metric asymptoti-
cally (as r → −∞) approaches that of anti-de Sitter space. Obvious Killing vectors
are
Z ≡ ∂
∂z
, Φ ≡ ∂
∂φ
, T ≡ ∂
∂t
, (2.3.11)
and the metric (2.3.10) has a further isometry
r → r + λ, z → eλz, φ→ e−2λφ, t→ e4λt (2.3.12)
which is generated by the fourth Killing vector R ≡ ∂
∂r
+ z ∂
∂z
− 2φ ∂
∂φ
+ 4t ∂
∂t
. The
Killing vector fields satisfy the algebra
[R,Z] = −Z , [R,Φ] = 2Φ , [R, T ] = −4T . (2.3.13)
No analogous solution for a positive cosmological constant exists [125, 154]. Because of
the similarity to (2.3.3) one would expect (2.3.13) to arise as a deformation of (2.3.3).
Physically both algebras describe the isometries of vacuum solutions of Einstein’s
Deformations of Spacetime Symmetries 29
equations, one with Λ < 0 and one with Λ = 0, and hence one might expect one of
them to arise as some limit of the other.
This spacetime is also time-orientable, as the vector field tµ = (0,− 1√
2
e3r,−1, 1)
defines a global arrow of time.
We can express (2.3.10) in coordinates corresponding to Poincare´ coordinates on
AdS (with Λ = −3)
ds2 =
dρ2 + dz2 + dφ2 − dt2
ρ2
− 2ρ dz(dt+ dφ) + 1
2
ρ4(dt+ dφ)2 . (2.3.14)
The limit ρ→ 0 in Poincare´ coordinates corresponds to the timelike boundary I
of anti-de Sitter space. After setting ρ˜ = ρ2 the line element is
ds2 =
dρ˜2
4ρ˜2
+
1
ρ˜
(
dz2 + dφ2 − dt2 − 2ρ˜3/2 dz(dt+ dφ) + 1
2
ρ˜3(dt+ dφ)2
)
, (2.3.15)
and (2.3.15) is an expansion of the form
ds2 =
dρ˜2
4ρ˜2
+
1
ρ˜
gµνdx
µdxν , gµν(x, ρ˜) = g
(0)
µν (x) + g
(2)
µν (x)ρ˜+ g
(3)
µν (x)ρ˜
3/2+ . . . (2.3.16)
as given in [80]. The coefficient g(3) = −dz ⊗ (dt + dφ) − (dt + dφ) ⊗ dz encodes
the stress energy tensor of the boundary dual theory in the context of the AdS/CFT
correspondence [15, 80]. It does not satisfy even the null energy condition on the three-
dimensional conformal boundary, since g
(3)
µν nµnν = −2 for the null vector n = ∂∂t + ∂∂z .
This presumably reflects causal pathologies of the bulk spacetime that we will try to
understand by more conventional means shortly.
A general analysis of stationary cylindrically symmetric Einstein spaces was done
in [109]. These authors assume the Lewis form of the metric, where a cross term dz dφ
would be absent. We did not find it possible to bring (2.3.10) to the Lewis form. The
theorem by Papapetrou [127] that any solution with two commuting Killing vectors
(one timelike, one spacelike with periodic orbits) can be written in the Lewis form,
only applies to solutions of the vacuum Einstein equations without cosmological term.
Hence one cannot make a connection with spaces of the Lewis form as was possible
for the Petrov solution.
2.3.2 Left-Invariant Forms
Since the action of the elements of the group manifold on itself has been given in
(2.3.5), we can write down a matrix representation of this group of motions, with a
Deformations of Spacetime Symmetries 30
general element given by
g =


1 0 0 0 r
0 er 0 0 z
0 0 e−
r
2 cos(
√
3
2
r) e−
r
2 sin(
√
3
2
r) φ
0 0 −e− r2 sin(
√
3
2
r) e−
r
2 cos(
√
3
2
r) t
0 0 0 0 1


. (2.3.17)
The group is generated by
Z =


0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0


, Φ =


0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0


, (2.3.18)
T =


0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0


, R =


0 0 0 0 1
0 1 0 0 0
0 0 −1
2
√
3
2
0
0 0 −
√
3
2
−1
2
0
0 0 0 0 0


, (2.3.19)
so that g = ezZ+φΦ+tT erR and (r, z, φ, t) are coordinates on the group. The generators
satisfy the algebra
[R, T ] = −1
2
T +
√
3
2
Φ , [R,Φ] = −1
2
Φ−
√
3
2
T , [R,Z] = Z . (2.3.20)
This differs from the Killing algebra (2.3.3) by the usual overall minus sign coming
from the fact that right-invariant vector fields generate left actions and vice versa.
The matrix representation (2.3.17) gives the group multiplication law
(r, z, φ, t) · (r′, z′, φ′, t′) = (r + r′, z + erz′, φ+ e− r2 (t′s+ φ′c), t+ e− r2 (t′c− φ′s)) ,
(2.3.21)
where s ≡ sin
√
3
2
r, c ≡ cos
√
3
2
r. The Maurer-Cartan form is
g−1dg = e−rR(Z dz + Φ dφ+ T dt)erR +Rdr ≡ Rλ1 + Z λ2 + Φλ3 + T λ4 , (2.3.22)
which gives the desired basis of left-invariant one-forms:
λ1 = dr , λ2 = e−r dz , λ3 = e
r
2
(
cos
(√
3
2
r
)
dφ− sin
(√
3
2
r
)
dt
)
,
λ4 = e
r
2
(
sin
(√
3
2
r
)
dφ+ cos
(√
3
2
r
)
dt
)
. (2.3.23)
Deformations of Spacetime Symmetries 31
We obtain a left-invariant metric on the group
ds2 = ηabλ
a ⊗ λb
= dr2 + e−2rdz2 + er
(
cos(
√
3r)(dφ2 − dt2)− 2 sin(
√
3r)dφ dt
)
(2.3.24)
with η = diag(1, 1, 1,−1), which is the same as (2.3.1) and shows that our chosen
coordinates agree with the initial Petrov coordinates. We see how to recover a metric
on a group manifold; note that one could obtain this metric by just starting from the
algebra (2.3.3).
2.3.3 Deformations of the Petrov Killing Algebra
We examine possible infinitesimal deformations of the four-dimensional Lie algebra.
UsingMathematica for the computations, equations (2.2.7) give the following con-
ditions on linear deformations:
0 = Az
r
φ +
√
3At
r
z = Az
r
t −
√
3Aφ
r
z = Aφ
r
t ;
0 = 2Aφ
r
r + Aφ
z
z +
√
3At
z
z = −
√
3Aφ
z
z + 2At
r
r + At
z
z = Aφ
z
t = Aφ
r
t ;
0 = Ar
r
z + 2Aφ
φ
z +
√
3Az
t
φ −
√
3At
φ
z =
√
3Ar
r
z + 2Az
φ
t −
√
3Az
t
t −
√
3Aφ
φ
z
=
√
3Ar
r
φ + At
φ
φ −
√
3Aφ
t
t − Atrr =
√
3Az
r
φ − Atrz ;
0 =
√
3Ar
r
z −
√
3Az
φ
φ − 2Aztφ −
√
3At
t
z = Ar
r
z −
√
3Az
φ
t + 2At
t
z +
√
3Aφ
t
z
= Ar
r
φ −
√
3Aφ
φ
t +
√
3At
r
r − Attφ = Azrφ +
√
3At
r
z . (2.3.25)
These constraints for Aa
c
b reduce the number of free parameters from 24 to twelve.
We list the most general deformation parameters satisfying (2.3.25) in a table:
Table 2.1: Infinitesimal deformations of the Petrov Killing algebra.
c = r c = z c = φ c = t
Ar
c
z 2C x1 x2 x3
Ar
c
φ −
√
3A− B x4 x5 x6
Ar
c
t −A+
√
3B x7 x8 x9
Az
c
φ 0 2B C
√
3C
Az
c
t 0 2A −
√
3C C
Aφ
c
t 0 0 −
√
3B − A B −√3A
Deformations of Spacetime Symmetries 32
The parameters x1, x2, . . . , x9, A,B and C can be arbitrary real constants. We need to
investigate which of these correspond to trivial deformations. The conditions (2.2.9)
mean that trivial deformations can be written as
Ar
r
z = −M rz , Arzz = −M rr , Arφz = −
3
2
Mφz+
√
3
2
M tz , Ar
t
z = −
√
3
2
Mφz−3
2
M tz .
(2.3.26)
The parameters C, x1, x2 and x3 correspond to trivial deformations and can be set to
zero by a change of basis. Furthermore,
Ar
r
φ =
1
2
M rφ +
√
3
2
M rt , Ar
r
t = −
√
3
2
M rφ +
1
2
M rt ,
Az
z
φ = −M rφ , Azzt = −M rt (2.3.27)
etc., so that we can set A = B = 0,
Ar
z
φ =
3
2
M zφ +
√
3
2
M zt , Ar
φ
φ = −
1
2
M rr +
√
3
2
Mφt +
√
3
2
M tφ ,
Ar
t
φ = −
√
3
2
M rr −
√
3
2
Mφφ +
√
3
2
M tt , (2.3.28)
so that we can set x4 = x5 = x6 = 0,
Ar
z
t =
3
2
M zt −
√
3
2
M zφ , Ar
φ
t =
√
3
2
M rr −
√
3
2
Mφφ +
√
3
2
M tt ,
Ar
t
t = −
1
2
M rr −
√
3
2
Mφt −
√
3
2
M tφ , (2.3.29)
so that we can set x7 = 0, but must treat x8 and x9 as nontrivial perturbations. After
a relabelling of these parameters, the modified Lie algebra is now
[R, T ] = −aT − bΦ , [R,Φ] = −1
2
Φ−
√
3
2
T , [R,Z] = Z . (2.3.30)
These relations satisfy the full Jacobi identity and so the linear deformation indeed
defines a deformation of the Lie algebra. We may modify the matrix representation
by setting
R =


0 0 0 0 1
0 1 0 0 0
0 0 −1
2
−b 0
0 0 −
√
3
2
−a 0
0 0 0 0 0


. (2.3.31)
Deformations of Spacetime Symmetries 33
In the case where a 6= 1
2
, one can always find a linear transformation of the basis
vectors Φ and T such that the algebra takes the more symmetric form
[R, T ] = a′T + b′Φ , [R,Φ] = a′Φ± b′T , [R,Z] = Z , (2.3.32)
with ± depending on the value of b in the original deformation. This means that
there are three distinct cases:
First case: Positive sign. Then a matrix representation of R is
R =


0 0 0 0 1
0 1 0 0 0
0 0 a′ b′ 0
0 0 b′ a′ 0
0 0 0 0 0


(2.3.33)
and a general group element looks like
g = ezZ+φΦ+tT erR =


1 0 0 0 r
0 er 0 0 z
0 0 ea
′r cosh(b′r) ea
′r sinh(b′r) φ
0 0 ea
′r sinh(b′r) ea
′r cosh(b′r) t
0 0 0 0 1


. (2.3.34)
The Maurer-Cartan form is
g−1dg = Z e−r dz + Φ e−ar(cosh(b′r) dφ− sinh(b′r) dt)
+T e−ar(cosh(b′r) dt− sinh(b′r) dφ) +Rdr (2.3.35)
and we can read off the left-invariant forms.
The special case a′ = 1, b′ = 3 gives the algebra of the Kaigorodov-Ozsva´th
solution (2.3.10), as can be seen by setting Φ′ = Φ − T and T ′ = Φ + T , which
amounts to
[R,Φ] = Φ + 3T , [R, T ] = 3Φ + T ⇒ [R,Φ′] = −2Φ′ , [R, T ′] = 4T ′ , (2.3.36)
which is just the sign-reversed version of (2.3.13). One can recover the metric (2.3.10)
by choosing the symmetric matrix
hab =


− 3
Λ
0 0 0
0 1 −√2 √2
0 −√2 −1 −1
0
√
2 −1 3

 (2.3.37)
Deformations of Spacetime Symmetries 34
and computing the left-invariant metric
habλ
a⊗λb = − 3
Λ
dr2+ e−2rdz2− 2e−2r(dφ2− dt2)+ e4r(dφ− dt)2− 2
√
2erdz(dφ− dt) ,
(2.3.38)
which after the coordinate transformations φ − t = φ′ and −φ − t = t′ reduces to
(2.3.10). Since the matrix (2.3.37) has one negative and three positive eigenvalues,
there exists a (vierbein) basis of left-invariant one-forms σa such that ηabσ
a⊗σb gives
the metric (2.3.10).
Second case: Negative sign.
R =


0 0 0 0 1
0 1 0 0 0
0 0 a′ b′ 0
0 0 −b′ a′ 0
0 0 0 0 0


, g = ezZ+φΦ+tT erR =


1 0 0 0 r
0 er 0 0 z
0 0 ea
′r cos(b′r) ea
′r sin(b′r) φ
0 0 −ea′r sin(b′r) ea′r cos(b′r) t
0 0 0 0 1


(2.3.39)
The Maurer-Cartan form is
g−1dg = Z e−rdz+Φ e−ar(cos(b′r)dφ−sin(b′r)dt)+T e−ar(cos(b′r)dt+sin(b′r)dφ)+Rdr
(2.3.40)
and a left-invariant metric will be given by
ds2 = ηabλ
a ⊗ λb = dr2 + e−2rdz2 + e−2a′r(cos(2b′r)(dφ2 − dt2)− 2 sin(2b′r)dφ dt) .
(2.3.41)
The original Petrov algebra is of course the special case a′ = −1
2
, b′ =
√
3
2
.
For the metric (2.3.41) the Ricci tensor has non-vanishing components
Rrr = −1− 2a′2 + 2b′2 , Rzz = −(1 + 2a′)e−2r ,
Rφφ = −(1 + 2a′)e−2a′r(a′ cos(2b′r) + b′ sin(2b′r)) ,
Rφt = (1 + 2a
′)e−2a
′r(a′ sin(2b′r)− b′ cos(2b′r)) ,
Rtt = (1 + 2a
′)e−2a
′r(a′ cos(2b′r) + b′ sin(2b′r)) . (2.3.42)
The manifold is an Einstein manifold only if a′ = −1
2
and b′ = ±
√
3
2
(Λ = 0, Petrov
solution) or a′ = 1 and b′ = 0 (Λ = −3, anti-de Sitter space). In the general case the
energy-momentum tensor defined by Tµν =
1
8piG
Gµν does not satisfy the weak energy
condition; without loss of generality assume sin(
√
3r) = 0 and cos(
√
3r) = 1 and
choose a timelike vector tµ = (0, 0, t3, t4) (t
2
4 ≥ t23), then
Gµνt
µtν = (1 + a′ + a′2 − b′2)(t23 − t24)− 2(1 + 2a′)b′t3t4 (2.3.43)
Deformations of Spacetime Symmetries 35
can be made arbitrarily negative by letting t3, t4 → ±∞ while keeping t24 − t23 small
and positive, unless a′ = −1
2
and b′2 ≥ 3
4
. In the case a′ = −1
2
and b′2 ≥ 3
4
, the
Einstein tensor can be written as
Gµν = −λgµν + 2λuµuν , λ ≡ b′2 − 3
4
≥ 0, uµ = (1, 0, 0, 0) , uµuµ = 1 . (2.3.44)
Note that uµ is spacelike. This tensor satisfies the dominant energy condition as
2trvr − tava ≥ 0 for any timelike and future-directed t, v. These statements are
independent of the choice of the arrow of time, i.e. hold for both t4 < 0 or t4 > 0.
Third case: a = 1
2
. Let us introduce a new parameter c, so that
[R, T ] = −1
2
T − 2√
3
c2Φ , [R,Φ] = −1
2
Φ−
√
3
2
T , [R,Z] = Z (2.3.45)
for positive b which gives
R =


0 0 0 0 1
0 1 0 0 0
0 0 −1
2
−2c2√
3
0
0 0 −
√
3
2
−1
2
0
0 0 0 0 0


(2.3.46)
and a general group element looks like
g = ezZ+φΦ+tT erR =


1 0 0 0 r
0 er 0 0 z
0 0 e−
r
2 cosh(cr) − 2c√
3
e−
r
2 sinh(cr) φ
0 0 −
√
3
2c
e−
r
2 sinh(cr) e−
r
2 cosh(cr) t
0 0 0 0 1


. (2.3.47)
The Maurer-Cartan form is
g−1dg = Rdr + Z e−r dz + Φ
(
e
r
2 cosh(cr) dφ+
2c√
3
e
r
2 sinh(cr) dt
)
+T
(
e
r
2 cosh(cr) dt+
√
3
2c
e
r
2 sinh(cr) dφ
)
. (2.3.48)
In the limit c→ 0 or b→ 0 this becomes
g−1dg = Rdr + Z e−r dz + Φ e
r
2 dφ+ T
(
e
r
2 dt+
√
3
2
e
r
2 r dφ
)
. (2.3.49)
Deformations of Spacetime Symmetries 36
The only remaining case, namely negative b, is
[R, T ] = −1
2
T +
2√
3
c2Φ , [R,Φ] = −1
2
Φ−
√
3
2
T , [R,Z] = −Z , (2.3.50)
which turns the hyperbolic into trigonometric functions.
If c 6= 0 one can rescale the coordinate t (for instance) and recover the same left-
invariant forms as before, hence this does not give anything new. In the case c = 0 a
left-invariant metric is given by
ds2 = dr2 + e−2rdz2 + er
((
1− 3
4
r2
)
dφ2 −
√
3r dφ dt− dt2
)
. (2.3.51)
The Einstein tensor for this metric can be written as
Gµν =
3
16
gµν +
3
8
(
diag
(−2, e−2r, 2er, 0))
µν
. (2.3.52)
This does not satisfy the weak energy condition as Gµνt
µtν = − 3
16
(3+ er) < 0 for the
timelike vector tµ = (1, 0, 0, 1).
Of course, for all three possible cases considered, one could take any constant
symmetric matrix with one negative and three positive eigenvalues instead of ηab to
construct a left-invariant metric; this gives a 10-parameter family (subject to con-
straints) of metrics whose Killing algebras contain one of the deformations of the
four-dimensional Petrov Killing algebra as a subalgebra. One of these parameters
can be absorbed into an overall scale, and any cross terms dr dz, dr dφ or dr dt can
be absorbed into a redefinition of z, φ and t. One is left with a 6-parameter family
of metrics, by far most of which will not appear to have a physical interpretation.
We have chosen to just give examples; more efficient methods of constructing general
metrics with a given isometry group are certainly known.
2.3.4 Spinning Cylinders
In the case of an infinite rigidly rotating dust cylinder one could imagine trying to
speed up this cylinder by shooting in particles with some angular momentum which
enter the interior region on causal curves and increase the angular velocity a, so as
to reach and surpass the critical value aR = 1
2
above which CTCs appear. We will
show that this is not possible.
The interior part of the general van Stockum solution
ds2 = H(ρ)(dρ2 + dz2) + L(ρ)dφ2 + 2M(ρ)dφ dt− F (ρ)dt2 , (2.3.53)
Deformations of Spacetime Symmetries 37
describing the region ρ < R is given by [157]
H = exp(−a2ρ2) , L = ρ2(1− a2ρ2) , M = aρ2 , F = 1 . (2.3.54)
As there are closed timelike curves for ρ > 1
a
we require aR ≤ 1. The exterior solution
is for a < 1
2
, from now on setting R = 1 for simplicity which is no loss of generality,
H = e−a
2
ρ−2a
2
, L =
ρ sinh(3+ θ)
2 sinh 2 cosh 
, M =
ρ sinh(+ θ)
sinh 2
, F =
ρ sinh(− θ)
sinh 
(2.3.55)
where θ(ρ) =
√
1− 4a2 log ρ and  = Artanh√1− 4a2. Note that always−FL−M2 =
−ρ2 and so the metric has the right signature for all ρ (this of course is also true as
a→ 1
2
). The point-particle Lagrangian is (a dot denotes differentiation with respect
to an affine parameter λ)
L = gµν x˙µx˙ν = H(ρ˙2 + z˙2) + Lφ˙2 + 2Mφ˙t˙− F t˙2 , (2.3.56)
and since the Lagrangian does not depend on z, φ and t there are three conserved
quantities associated with geodesics:
P ≡ Hz˙ , J ≡ Lφ˙+Mt˙ , E ≡ F t˙−Mφ˙ . (2.3.57)
The Lagrangian for timelike or null geodesics becomes
L = Hρ˙2 + P
2
H
+
1
ρ2
(
FJ2 − 2MEJ − LE2) = −µ2 ≤ 0 (2.3.58)
and we obtain the radial equation(
dρ
dλ
)2
=
L
Hρ2
(
−ρ
2µ2
L
− ρ
2P 2
HL
+ E2 +
2M
L
EJ − F
L
J2
)
=
L
Hρ2
(E − V +eff (ρ))(E − V −eff (ρ)) , (2.3.59)
where we have introduced an effective potential
V ±eff (ρ) =
M(ρ)
L(ρ)
J ± ρ
√
1
L(ρ)
(
µ2 +
P 2
H(ρ)
+
J2
L(ρ)
)
. (2.3.60)
This is well-defined for all ρ as H, L, M all remain positive for all ρ. A particle
falling in on a geodesic can enter the cylinder if
E > V +eff (1) =
a
1− a2J +
√
1
1− a2
(
µ2 + P 2ea2 +
J2
1− a2
)
≥ a+ 1
1− a2J =
1
1− aJ .
(2.3.61)
Deformations of Spacetime Symmetries 38
Any particle entering the cylinder on a geodesic must have J
E
< 1 − a. In the limit
a → 0 the conserved quantities J and E clearly describe angular momentum and
energy per mass. We can identify the ratio J
E
with the angular velocity of an infalling
particle at R = 1.
If we are considering accelerated observers, equation (2.3.58) still holds, but P, E
and J will no longer be conserved quantities. However, only the local values of these
quantities at R = 1 will decide about whether or not a particle will be able to enter
the interior region of the cylinder.
This means that the above considerations also hold for accelerated observers and
as any particle entering the cylinder must have J
E
< 1
2
for a = 1
2
, one cannot speed
up the cylinder beyond a = 1
2
using particles on timelike or null curves.
2.3.5 Geodesic Completeness
We ask whether the Petrov spacetime, with the radial coordinate r extended to take
arbitrary values, is geodesically complete, i.e. whether all timelike and null geodesics
can be extended to infinite values of the affine parameter. First we give an example
that this need not be possible on a group manifold: Remove the null hyperplane z = t
from Minkowski space and consider the half-space z > t, denoted byM−. It is clearly
geodesically incomplete. Null translations and boosts
(z, t)→ (z + c, t+ c) ; (t+ z, t− z)→
(
λ(t+ z),
1
λ
(t− z)
)
(2.3.62)
act on M−, and together with translations (x, y) → (x + a, y + b) they form a four-
dimensional group which acts simply-transitively on M−. For instance, the point
(x, y, z, t) = (0, 0, 1, 0) is, by a null translation and a successive boost, taken to
(0, 0, 1, 0)→ (0, 0, 1 + c, c)→
(
0, 0, λc+
1
2
− 1
2λ
, λc+
1
2
+
1
2λ
)
. (2.3.63)
There is a one-one correspondence between points in M− and group parameters
(a, b, λ, c), where λ > 0. The space M− can be identified with the group G × R2,
where G is the unique two-dimensional non-Abelian Lie group.
As a second example, introduced in a slightly different context in [76], consider
the dilatation group generated by translations and dilatations
xµ → xµ + cµ , xµ → ρxµ , (2.3.64)
Deformations of Spacetime Symmetries 39
where we denote the generators by Pa and D respectively. Parametrising the group
elements by‡ g = ex
aPaeλD, the Maurer-Cartan form is
g−1dg = e−λdxa Pa + dλD (2.3.65)
and hence a left-invariant metric is
ds2 = dλ2 + e−2ληµνdxµdxν =
1
ρ2
(dρ2 + ηµνdx
µdxν) , (2.3.66)
where ρ = eλ. This is the metric of anti-de Sitter space in five dimensions in Poincare´
coordinates, which is geodesically incomplete as these coordinates cover only a patch
of the full spacetime. Hence geodesic completeness is a non-trivial property of a group
manifold.
To show geodesic completeness of the Petrov solution, we need to show that no
geodesic reaches infinity for finite values of the affine parameter. Consider the La-
grangian
L = gµν x˙µx˙ν , (2.3.67)
which for the metric (2.3.1) is
L = r˙2 + e−2rz˙2 + er
(
cos
√
3r(φ˙2 − t˙2)− 2 φ˙ t˙ sin
√
3r
)
. (2.3.68)
Evidently, from the Euler-Lagrange equations, there are three conserved quantities
because the Lagrangian does not depend on z, φ or t explicitly, the conjugate momenta
P ≡ e−2rz˙ , j ≡ er(φ˙ cos
√
3r − t˙ sin
√
3r) , E ≡ er(φ˙ sin
√
3r + t˙ cos
√
3r) . (2.3.69)
The Lagrangian now takes the form
L = r˙2 + e2rP 2 + e−r
(
cos
√
3r(j2 − E2) + 2Ej sin
√
3r
)
, (2.3.70)
and since the Lagrangian is a conserved quantity in geodesic motion the equation(
dr
dλ
)2
= e−r(E2 cos
√
3r − 2Ej sin
√
3r − j2 cos
√
3r)− e2rP 2 − µ2 (2.3.71)
is satisfied by any geodesic, where µ2 is positive for timelike, zero for null and negative
for spacelike geodesics. For timelike geodesics (µ2 > 0), right-hand side of (2.3.71)
‡The coordinate λ only covers dilatations with ρ > 0, so that one may obtain geodesic complete-
ness by adding the disconnected component with ρ < 0. This suggests that geodesic completeness
of groups is a topological question that cannot be decided at the level of the Lie algebra.
Deformations of Spacetime Symmetries 40
becomes negative for large r, so that r is bounded and we may extend geodesics
infinitely. For null geodesics, r˙ is bounded§. So for any finite values of the affine
parameter, r˙ remains finite and so do z˙, φ˙ and t˙. Hence the Petrov spacetime is
geodesically complete. It will be incomplete if we interpret it as the exterior solution
of a rotating cylinder and cut off the region described by the Petrov solution at some
value of r.
By very similar arguments we can show geodesic completeness of the Kaigorodov-
Ozsva´th solution with line element (2.3.14). In this case, the Lagrangian is
L = 1
ρ2
(
ρ˙2 + z˙2 + φ˙2 − t˙2
)
− 2ρ z˙(t˙+ φ˙) + 1
2
ρ4(t˙+ φ˙)2 (2.3.72)
and the conserved quantities are
P ≡ 1
ρ2
z˙− ρ(t˙+ φ˙), j ≡ 1
ρ2
φ˙− ρz˙+ 1
2
ρ4(t˙+ φ˙), E ≡ 1
ρ2
t˙+ ρz˙− 1
2
ρ4(t˙+ φ˙) (2.3.73)
so that the radial equation is(
dρ
dλ
)2
= −ρ4(P 2 + j2 − E2)− 2ρ7P (j + E)− 1
2
ρ10(j + E)2 − ρ2µ2 . (2.3.74)
The right-hand side becomes negative if µ2 > 0 for both ρ→∞ and ρ→ 0, hence all
timelike geodesics are bounded in ρ. For µ = 0 the right-hand side is either constant
zero or becomes negative for large r, so null geodesics are bounded from above in ρ.
For very small ρ, ρ˙ goes to zero, so as before ρ˙ is bounded for null geodesics. This
shows geodesic completeness. Note that in the case of anti-de Sitter space (2.3.74)
would be (
dρ
dλ
)2
= −ρ4(P 2 + j2 − E2)− ρ2µ2 (2.3.75)
and depending on the magnitudes of E, j and P the right-hand side blows up as
ρ→∞ for some geodesics, which can reach ρ =∞ in finite affine parameter distance.
2.3.6 Causal Properties
It was observed by Tipler [157] that the Petrov solution contains CTCs. As shown
in [158], its causal behaviour is actually even more pathological: The Petrov solution
§If we allow r to take negative values, then for E 6= 0 or j 6= 0 the right-hand side becomes
oscillatory for large negative r, taking positive as well as negative values. Hence both timelike and
null geodesics are bounded from below in r. For E = j = 0 the right-hand side is either constant
zero or always negative.
Deformations of Spacetime Symmetries 41
is totally vicious (for a summary of causality conditions see [59]), i.e. for any two
points a and b one has a < b and b < a, where “<” is the usual causal relation “can
be joined by a timelike curve”. The proof of this statement proceeds by giving the
future and past light cones at the identity element of the group manifold, and the
observation that the adjoint action of elements of the form exp(λR) rotates the cone
around, such that for certain values of λ a future-directed timelike tangent vector is
mapped to a past-directed timelike tangent vector.
One might hope that a similar statement could be made for the Kaigorodov-
Ozsva´th solution, but this turns out not to be case, as is apparent from the deformed
group element (2.3.34) or the commutation relations (2.3.13). The action of R on the
plane generated by Φ and T is now just a scaling of the generators, which amounts
to an opening and closing of the light cones, but no rotation. The assumptions of
Theorem 2 in [158] are not satisfied.
Apart from the results in [158] which refer to homogeneous spacetimes, we may try
to apply a criterion for the causality properties of spacetimes with an Abelian isometry
group acting transitively on timelike surfaces which was given by Carter in his 1967
PhD thesis [34]. For the Kaigorodov-Ozsva´th solution, this Abelian isometry group
is generated by ∂
∂z
, ∂
∂φ
and ∂
∂t
which act on surfaces {r = constant}. Carter’s result is
that a sufficient condition for such a spacetime to be totally vicious is the nonexistence
of a one-form in the dual of the Abelian Killing algebra which is everywhere timelike
or null. In the present case, such a one-form is of the form ω = Adz+B dφ+C dt, with
A, B and C constants. Conversely, if there is such a one-form which is everywhere
timelike, the spacetime is virtuous, i.e. causally well-behaved. In the present case,
one finds that the norm of such an ω is
g˜(ω, ω) = e2r
(
A2 + 2
√
2AC e3r + C(2B + Ce6r)
)
, (2.3.76)
where g˜ is the contravariant version of (2.3.10). In order to make (2.3.76) non-positive
for all values of r, one has to choose A = C = 0 to be left with the null one-
form dφ. Put differently, the subgroup generated by ∂
∂z
and ∂
∂t
acts on null surfaces
{r = φ = constant}; were these spacelike it would follow that the Kaigorodov-Ozsva´th
solution is causal (cf. Proposition 9 in [33]).
Carter’s condition for causality violation was used by Tipler in [157]¶ to show the
pathological behaviour of the Petrov solution. Here the situation is less clear; we
¶whose wording is slightly unfortunate, since he states the relevant condition on ω as being
“everywhere timelike”
Deformations of Spacetime Symmetries 42
cannot show directly that in our example spacetimes related by Lie algebra defor-
mation have similar causal properties. The strongest indication we have for causal
pathologies of the Kaigorodov-Ozsva´th spacetime comes from the computation of the
boundary stress-energy tensor in AdS/CFT in (2.3.16).
2.4 The Kaigorodov Solution and Lobatchevski Plane
Gravitational Waves
As a different example of a homogeneous space which appears as an exact solution in
general relativity, we consider the Kaigorodov solution, which has a well-known inter-
pretation as the AdS analogue of homogeneous pp-waves on a Minkowski background
[139]. It has received recent interest in the context of string theory as the near-horizon
limit of an M2 brane [42, 129]. By using Lie algebra deformations we again try to
find a spacetime which has a physical interpretation “close to” the Kaigorodov solu-
tion. An obvious candidate would be a member of the family of “Lobatchevski plane
gravitational waves” discussed by Siklos in [149] as spacetimes admitting a Killing
spinor. The expectation that the particular solution considered by Siklos which has
a five-dimensional isometry group arises as a deformation of the Kaigorodov solution
will be confirmed. Since the isometry group is now five-dimensional and not four-
dimensional as in the previous section, one has to consider a quotient G/H. The
larger symmetry will place constraints on the general form of a left-invariant metric.
2.4.1 The Kaigorodov Solution
The Kaigorodov solution (of Petrov type N) [93] is another example of a homo-
geneous space-time which solves the vacuum Einstein equations with a cosmological
constant Λ < 0. The form given in [154] is
ds2 = −12
Λ
dz2 + 10k e2zdx2 + e−4zdy2 − 10u ezdz dx− 2ez du dx . (2.4.1)
The five-dimensional Killing algebra of (2.4.1) is spanned by
Y ≡ ∂
∂y
, X ≡ ∂
∂x
, U ≡ e−5z ∂
∂u
,
Z ≡ ∂
∂z
+ 2y
∂
∂y
− x ∂
∂x
, W ≡ ye−5z ∂
∂u
+ x
∂
∂y
. (2.4.2)
Deformations of Spacetime Symmetries 43
The stabiliser of the origin is the one-parameter subgroup generated by W , which we
call H to be consistent with previous notation. The transformations generated by Z
are
z → z + λ, y → ye2λ, x→ xe−λ , (2.4.3)
while W generates
y → y + λx, u→ u+ ye−5zλ+ 1
2
xe−5zλ2 . (2.4.4)
The Killing vectors have non-zero commutators
[W,Y ] = −U , [W,X] = −Y , [Z,U ] = −5U ,
[Z, Y ] = −2Y , [Z,X] = X , [W,Z] = 3W ; (2.4.5)
Y,X and U span a three-dimensional Abelian subalgebra. The generators of the
kinematical group will have the sign-reversed algebra
[W,Y ] = U , [W,X] = Y , [Z,U ] = 5U ,
[Z, Y ] = 2Y , [Z,X] = −X , [W,Z] = −3W . (2.4.6)
We denote the isometry group by K ‖; it has three-dimensional subgroups of Bianchi
types I, II and VIh. The spacetime is naturally represented as a coset space
M = K/H . (2.4.7)
Represented as matrices, the group K is generated by
X =


0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0


, Y =


0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0


, Z =


5 0 0 0 0
0 0 0 0 1
0 0 2 0 0
0 0 0 −1 0
0 0 0 0 0


,
U =


0 0 0 0 −1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0


, W =


0 0 −1 0 0
0 0 0 0 0
0 0 0 1 0
0 0 0 0 0
0 0 0 0 0


. (2.4.8)
‖The K stands for Kaigorodov or Killing, according to taste
Deformations of Spacetime Symmetries 44
Any element of the group can be parametrised by
g = exX+yY+ue
5zUezZewW , (2.4.9)
such that (x, y, z, u) are coordinates on the coset spaceM. The Maurer-Cartan form
ωMC ≡ g−1dg = eaλa , (2.4.10)
where ea are the group generators, gives a basis of left-invariant one-forms on K:
λ1 = ez dx , λ2 = e−2z dy − w ez dx , λ3 = dz ,
λ4 = du− w e−2z dy + 1
2
w2 ez dx+ 5u dz , λ5 = dw + 3w dz . (2.4.11)
As usual, one can now construct a left-invariant metric on the group by
ds2 = Gabλ
aλb , (2.4.12)
where Gab is constant and non-degenerate. Any such metric will depend on the
coordinate w on W , since the Lie algebra splitting
k = m⊕ h (2.4.13)
is not reductive: [W,Z] = 3W where one would require [h,m] ⊂ m. (Recall the
definition of a reductive geometry given in (1.1.6)).
The original Kaigorodov metric can be reconstructed by taking
ds2 = 10k(λ1)2 + (λ2)2 − 12
Λ
(λ3)2 − 2λ1λ4 + (λ5)2 (2.4.14)
= 10k e2zdx2 + e−4z dy2 − 12
Λ
dz2 − 2ez dx(du+ 5u dz) + (dw + 3w dz)2
and reducing to the coset by dropping the last term.
We summarise the most important physical properties of this solution, as discussed
in more detail in [139]: The Kaigorodov spacetime has a curvature singularity at
z = ∞, and z = −∞ represents null and spacelike infinity; the metric approaches
the AdS metric asymptotically as z → −∞. The patch covered by the coordinates
(x, y, z, u) can be extended by changing coordinates to q = e2z, and extending to
negative q. The enlarged spacetime splits into two disjoint regions q > 0 and q < 0
which cannot mutually communicate. Then X = ∂
∂x
is a timelike Killing vector for
q < 0, and the enlarged spacetime is stationary in this region. This means that
gravitational radiation can exist in a stationary spacetime.
Deformations of Spacetime Symmetries 45
2.4.2 Deformations of the Lie Algebra
In order to find infinitesimal deformations of the Killing algebra (2.4.6), we again
consider the Jacobi identity at linear order (2.2.7), again using Mathematica for
explicit computations. Imposing these constraints on Aa
c
b, one is left with 20 free
parameters, as shown in table 2.2.
Table 2.2: Infinitesimal deformations of the Kaigorodov Killing algebra.
c = x c = y c = z c = u c = w
Ax
c
y (2+ η)/6 (2δ + 5τ)/3 0 −γ/4 0
Ax
c
z −ω + 2P 3α + β −(δ + τ)/3 c2 γ
Ax
c
u σ/3 η/2 0 5/3(δ + τ) 0
Ax
c
w (c9 − 2c8 − 5c7)/5 c1 (2+ η)/6 α δ
Ay
c
z (6c9 − 7c8 − 15c7)/5 −ω + P (2+ η)/3 β τ
Ay
c
u 0 −2σ/3 0 (5+ 4η)/3 0
Ay
c
w m c7 σ/3 c5 ε
Az
c
u −6m c8 −5σ/3 ω η
Az
c
w c3 c4 (3c9 − c8)/5 c6 P
Au
c
w 0 −m 0 c9 σ
Again it turns out that almost all of the possible deformations should be considered
trivial (i.e. correspond to a change of basis in the Lie algebra). Only a one-parameter
deformation, which can be given by ω, cannot be obtained by a basis transformation.
This infinitesimal deformation satisfies the full Jacobi identity, and we obtain the
new algebra
[W,Y ] = U , [W,X] = Y , [Z,U ] = (5 + α)U ,
[Z, Y ] = (2 + α)Y , [Z,X] = (α− 1)X , [W,Z] = −3W . (2.4.15)
One would perhaps have expected that at least one possible deformation would be to
deform the commutation relations of Z to arbitrary numbers:
[Z,U ] = aU , [Z, Y ] = bY , [Z,X] = cX , [W,Z] = dW . (2.4.16)
The one-parameter deformation we have obtained is the most general deformation of
this type: By a rescaling of Z one can always keep one of the four numbers a, b, c, d
Deformations of Spacetime Symmetries 46
fixed. Then the Jacobi identity puts two constraints on changing the coefficients in
the three other commutators, leaving a single deformation parameter.
Put differently, one is free to replace the deformed algebra (2.4.15) by any one-
parameter deformation of the action of Z which satisfies the Jacobi identity, e.g.
[W,Y ] = U , [W,X] = Y , [Z,U ] = 5U ,
[Z, Y ] = (2 + α)Y , [Z,X] = (2α− 1)X , [W,Z] = (α− 3)W . (2.4.17)
We will use this last form in the following. In the matrix representation we only have
to modify the matrix for Z:
Zα =


5 0 0 0 0
0 0 0 0 1
0 0 2 + α 0 0
0 0 0 2α− 1 0
0 0 0 0 0


. (2.4.18)
2.4.3 Construction of the Manifold
We are trying to construct a metric on the quotient space
M′ = K′/H , (2.4.19)
where the stabiliser H is again generated by W . We use the same parametrisation as
before:
g = exX+yY+ue
5zUezZαewW (2.4.20)
By computing the Maurer-Cartan form one finds the left-invariant forms are now
λ1 = e(1−2α)z dx , λ2 = e−(2+α)z dy − w e(1−2α)z dx , λ3 = dz , (2.4.21)
λ4 = du− w e−(2+α)z dy + 1
2
w2 e(1−2α)z dx+ 5u dz , λ5 = dw + (3− α)w dz .
Again, it is not possible to find a left-invariant metric
ds2 = Gabλ
aλb (2.4.22)
which is independent of w. We can, however write any such metric in the form
ds2 = G55(dm+ Aµdx
µ)2 + G˜µνdx
µdxν , (2.4.23)
Deformations of Spacetime Symmetries 47
where xµ = (x, y, z, u), and demand that G˜µν , in order to be a metric on the coset
space, be independent of w. This reduces the number of free parameters in G from
15 to nine, and the coset metric is (setting G55, which determines the overall scale,
to −1)
ds2 = p3e
z(2−4α) dx2 + p1e−2z(2+α) dy2 + p2dz2 − 2ez(1−2α) dx(p1du+ (5 p1 u− p4)dz) ,
(2.4.24)
where we have defined
p1 := G
2
25−G22G55 , p2 = G235−G33G55 , p3 = G215−G11G55 , p4 = G15G35−G13G55 .
(2.4.25)
It is clear that by shifting u by a constant one can always achieve that p4 = 0. This
means the most general metric on the deformed coset space is
ds2 = p3e
z(2−4α) dx2 + p1e−2z(2+α) dy2 + p2dz2 − 2p1 ez(1−2α) dx(du+ 5u dz) . (2.4.26)
One discovers that the determinant of (2.4.26) is
det g = −e−2(1+3α)zp31p2 , (2.4.27)
and one finds that for Lorentzian signature (−+++) we must have p1 > 0 and p2 > 0.
One can verify that the metric, as it should be, has five Killing vectors
Y ≡ ∂
∂y
, X ≡ ∂
∂x
, U ≡ e−5z ∂
∂u
,
Zα ≡ ∂
∂z
+ (2 + α)y
∂
∂y
+ (2α− 1)x ∂
∂x
, W ≡ ye−5z ∂
∂u
+ x
∂
∂y
, (2.4.28)
which satisfy the algebra
[W,Y ] = −U , [W,X] = −Y , [Zα, U ] = −5U ,
[Zα, Y ] = −(2 + α)Y , [Zα, X] = (1− 2α)X , [W,Zα] = (3− α)W . (2.4.29)
One finds that the Ricci tensor of (2.4.26) is, in the coordinate basis,
Rµν = −3(2 + α)
2
p2
gµν − 5e
(2−4α)zp3α(α− 3)
p2
δµ
xδν
x . (2.4.30)
There is a cosmological term with
Λ = −3(2 + α)
2
p2
, (2.4.31)
Deformations of Spacetime Symmetries 48
consistent with p2 = −12/Λ for the original Kaigorodov solution, and a traceless
second term which describes null dust. In an orthonormal frame, given by
E0 = du+ ez(1−2α)
(
− p3
2p1
+
25p1u
2
2p2
+
p1
2
)
dx ,
E1 = du+ ez(1−2α)
(
− p3
2p1
+
25p1u
2
2p2
− p1
2
)
dx ,
E2 =
√
p2 dz − 5p1√
p2
u ez(1−2α)dx ,
E3 =
√
p1e
−z(2+α)dy , (2.4.32)
the nonvanishing components of T ab are
T 00 = T 01 = T 10 = T 11 =
5p3α(3− α)
p2p21
. (2.4.33)
The number (2.4.33) is chosen to be positive so that T ab describes an electromagnetic
field; then depending on the value of α the sign of p3 has to be chosen appropriately.
Using deformations of the isometry group of the Kaigorodov solution we have
constructed a spacetime which belongs to a more general class of “Lobatchevski plane
gravitational waves” discussed in [149]. To bring the metric (2.4.26) into the standard
form we apply the coordinate transformation
q =
√
p2
2 + α
e(2+α)z , y˜ =
√
p1y , u˜ = −p1e5zu . (2.4.34)
In these coordinates the metric becomes
ds2 =
p2
(2 + α)2q2
[
dq2 + dy˜2 + 2dx du˜+H(q)dx2
]
, (2.4.35)
with
H(q) = p3p
α−3
2+α
2 ((2 + α)q)
6−2α
2+α . (2.4.36)
In the notation of [154], we have the metric form,
ds2 =
3
|Λ|q2
(
dq2 + dy˜2 + 2dx du˜± q2kdx2) (2.4.37)
with k = 3−α
2+α
and the sign corresponding to the sign of p3. The condition for positive
energy given in [154] is equivalent to
p3(3− α)α > 0 , (2.4.38)
which is consistent with the remarks below (2.4.33). The case k = −1, corresponding
to Defrise’s pure radiation solution [44] with enhanced symmetry, is the only value
Deformations of Spacetime Symmetries 49
of k which does not correspond to any choice of α. Metrics of the form (2.4.37) are
conformal to pp-waves and a more detailed physical interpretation is given in [149].
Similar to the previous case of the Petrov solution, using a deformation of the
Killing algebra of a given spacetime we have constructed a solution with similar
physical properties, namely in this case a pp-wave on an AdS background which now
describes gravitational radiation as well as a null Maxwell field.
2.5 Summary
We have seen how the mathematical theory of Lie algebra deformations can relate dif-
ferent physical situations, in this case exact solutions of general relativity. The Petrov
and Kaigorodov-Ozsva´th solutions, the unique vacuum solutions without and with
cosmological constant with simply-transitive four-dimensional isometry groups, were
shown to be related by Lie algebra deformations; they were shown to be both geodesi-
cally complete. While the Petrov solution displays pathological causal behaviour as
first observed by Tipler [157] (for the more general class of Lanczos-van Stockum
solutions), we were not able to determine the causal properties of the Kaigorodov-
Ozsva´th solution. There is however strong indication that the latter shows causal
pathologies since we found an unphysical energy-momentum tensor on the boundary
in the context of AdS/CFT. We have to leave a more concrete study of the relation
of these two properties to further work. Another interesting question that has been
left open is whether the Kaigorodov-Ozsva´th solution can be understood as a certain
limit of an exterior solution of a matter distribution, perhaps representing a rotating
cylinder as well.∗∗ More generally its physical interpretation which is so far largely
unknown deserves further study.
In a second calculation, we analysed deformations of the Killing algebra of the
Kaigorodov spacetime and saw that one is naturally led to a solution including an
electromagnetic field. Although we have not constructed new exact solutions here, we
have shown how this more algebraic approach to exact solutions of general relativity
can give new insights on physical properties.
∗∗A class of rotating cylinder solutions with negative cosmological constant has been studied in
[27], who discuss the appearance of CTCs as well, but leave out certain limiting null cases. Our
study strongly suggests that the Kaigorodov-Ozsva´th solution, if related to the results of [27], would
be a limiting case not considered in [27].
Deformed General Relativity and Torsion 50
Chapter 3
Deformed General Relativity and
Torsion
“Seit die Mathematiker u¨ber die Relativita¨tstheorie
hergefallen sind, verstehe ich sie selbst nicht mehr.” [54]
Albert Einstein
3.1 Introduction
In the previous chapter, we have discussed exact solutions of general relativity which
are homogeneous spaces G/H. Since these are special cases with more symmetry
than one would expect in a generic physical situation, it is natural to investigate a
more general geometric description in which the symmetry of a given homogeneous
space is only present infinitesimally.
The passage from special to general relativity can be understood as making the
symmetries of a homogeneous space local: General relativity is formulated in terms
of (pseudo-)Riemannian geometry, which makes the symmetries of Euclidean space,
which can be thought of as a homogeneous space E(p, q)/SO(p, q) ∗, local, and can
thus be formulated on arbitrary manifolds. Physically, this is the incorporation of
Einstein’s equivalence principle: At at a given point, any spacetime looks to an inertial
observer like Minkowski space. In the vier-/vielbein formalism of general relativity
(originally due to Cartan as well as the Cartan connections we will use in this chapter),
one chooses a basis of the tangent space at each point, and thus a “moving frame” of
∗For p spacelike and q timelike dimensions. We will set p = d− 1 and q = 1 in the following.
Deformed General Relativity and Torsion 51
vector fields, such that the metric in this frame is constant and diagonal. The frame
field then encodes the metric information of the manifold.
In this chapter, we shall investigate a generalised formulation of general relativity
in terms of Cartan geometry which incorporates the symmetries of an arbitrary
given homogeneous space G/H on an infinitesimal level, thus providing an extension
of both Felix Klein’s Erlangen Programme [96] and (pseudo-)Riemannian geometry,
as summarised in the following picture that we essentially take from [148]:
Euclidean
Geometry -
curvature 6= 0 (pseudo-)Riemannian
Geometry
? ?
generalise
transformation group
generalise
tangent geometry
Klein
Geometry -
curvature 6= 0 Cartan
Geometry
Figure 3.1: Cartan geometry extends both Klein geometry and (pseudo-)Riemannian ge-
ometry.
In general relativity, if an orthonormal frame has been found, one can still use
general local Lorentz transformations to obtain a different frame with the same prop-
erties, and so in this formulation general relativity appears as a gauge theory. The
description of general relativity is in terms of an h-valued Ehresmann connection
(with H = SO(d − 1, 1), the Lorentz group), together with a soldering form (the
frame field) determining how the tangent spaces at different points are “soldered”
to the spacetime manifold. The gauge group H acts on the tangent space which is
viewed as a vector space Rd−1,1.
We will use a similar gauge-theoretic formulation, but of a more general character:
Instead of viewing Minkowski space as Rd−1,1, we will view it as a homogeneous space
E(d − 1, 1)/SO(d − 1, 1). A description in terms of Cartan geometry will allow us
to deform general relativity by replacing E(d− 1, 1) by its deformation, the de Sitter
group † SO(d, 1). We effectively try to construct a theory of gravity built on tangent
de Sitter spaces which replace the usual (co-)tangent spaces in general relativity. As
†The anti-de Sitter group SO(d− 1, 2) could be used equally well, and in fact is used in [153].
Deformed General Relativity and Torsion 52
we saw in the previous chapter, from the viewpoint of Lie group deformations this
process is a very natural extension of the standard framework. In addition to the
speed of light c, a second scale κ with dimensions of momentum is introduced on
a fundamental level. It is tempting to speculate that this scale could be related
to quantum gravity, although the basic idea of “deforming” the kinematic algebra
has existed at least since Heisenberg, who thought in terms of atomic rather than
quantum-gravitational scales.
The motivation for investigating this possible generalisation of the usual for-
mulation of Einstein-Cartan theory, or general relativity in first order formulation,
comes from the recently proposed framework of deformed, or doubly, special rela-
tivity (DSR), where one effectively proposes a momentum space that is acted on
by the de Sitter, not the Poincare´ group, rather in the spirit of Snyder’s proposal
[151]. It seems natural to try to construct a general geometric framework which in-
corporates this symmetry on the infinitesimal level. We will use a formulation of
Einstein-Cartan theory given by Stelle and West, and interpret it in the concrete ex-
ample of Minkowski space. By doing this construction, we hope to shed light on the
physical interpretation of DSR, which seems rather unclear at present. In particular,
while the noncommuting translations appearing in the de Sitter group are commonly
interpreted as noncommuting “spacetime coordinates”, and one passes to a noncom-
mutative geometry description to account for this, our discussion will stay within
conventional differential geometry. We will see that the description of Minkowski
space in this theory includes a connection with torsion instead of a fundamentally
noncommutative structure.
In order to confront this mathematical framework with physical reality, we note
that such a connection with torsion may lead to the usual ambiguities in minimal
coupling. We give estimates concerning observable effects and note that observable
violations of charge conservation induced by torsion should happen on a time scale
of 103 s, which seems to rule out these modifications as a serious theory. Our con-
siderations show, however, that the noncommutativity of translations in the Snyder
algebra need not correspond to noncommutative spacetime in the usual sense.
To set the scene, we start with the Einstein-Hilbert action in four dimensions,
without cosmological constant, written in first order (Palatini) form:
SEH =
1
16piG
∫
abcd
(
ea ∧ eb ∧Rcd) , (3.1.1)
where Rcd is the curvature two-form of an so(3, 1)-valued connection one-form ωab.
Deformed General Relativity and Torsion 53
One discovers that variation with respect to the vierbein field ea gives the usual
vacuum field equation Rica = 0 (where Rica is the Ricci one-form), while variation
with respect to ωab gives the condition of vanishing torsion. Thus, this formulation
has the advantage over the second order form that the condition of a torsion-free
connection does not have to be put in by hand. On the other hand, if one now adds
a matter action to the Einstein-Hilbert action, both equations now potentially have
a non-zero right-hand side. While the functional derivative of the matter action with
respect to the vierbein determines an energy-momentum tensor, one will also have
nonvanishing torsion if
δSmat
δωab
6= 0. (3.1.2)
This is the case for matter with internal spin, such as spinor fields. This generali-
sation of general relativity which incorporates the possibility of torsion is known as
Einstein-Cartan theory. Since torsion does not propagate, i.e. the field equation
contains the full torsion two-form instead of contractions of it (as for curvature in Ein-
stein’s equations for d > 3), it seems difficult to rule out such a theory experimentally,
since one would only measure torsion inside a matter distribution with spin.
The basic observation is now the following: Combine the connection ωab and the
vierbein ea into a single connection, taking values in the algebra of the Poincare´ group,
A =


ωab
1
L
ei
0 0

 , (3.1.3)
where L is a length introduced to get dimensions right, but which is completely
arbitrary at this stage. Then consider the Yang-Mills-like action (with now d = 4)
S = − L
2
32piG
∫
abcd
(
F ab ∧ F cd) = − L2
32piG
∫
d4x
1
4
abcd
µνρτF abµνF
cd
ρτ , (3.1.4)
involving the curvature F ab of the connection Aab (Latin indices here run from 1 to 4,
and so one projects F ab to its so(3, 1) part in this action). Writing out F ab in terms
of the curvature Rab of ωab and the vierbein field e
i, one discovers that the action
(3.1.4) contains a topological Gauss-Bonnet term plus the Einstein-Hilbert action
(3.1.1). Thus, the field equations of Einstein-Cartan theory are recovered.
The idea that general relativity may be interpreted as a gauge theory with the
Poincare´ group as gauge group is of course far from new. In the usual textbook
Deformed General Relativity and Torsion 54
discussions, the significance of a gauge theory is expressed as the promotion of a
global symmetry in a given theory to a local one by the introduction of an additional
field which appears in gauge-covariant derivatives. This viewpoint was probably first
elaborated in detail by Utiyama [159], who showed that the promotion of a global
to a local symmetry directly leads to a gauge field, valued in the Lie algebra of the
symmetry group under consideration, with the usual transformation properties, and
thus geometrically to the well known formulation in terms of connections on principal
bundles (see e.g. [120]). Utiyama also considered the Lorentz group as an example
and was led to general relativity, but only by making several assumptions, such as
the existence of a Riemannian metric and a local Lorentz (vierbein) frame and the
symmetricity of the affine connection in its lower indices, without further motivation.
It was then realised by Kibble [94] that the more natural gauge group to take is
the Poincare´ group, so that the vierbein appears naturally as part of the connection.
Kibble showed that one is led to general relativity in the first order formalism, i.e.
to Einstein-Cartan theory, and so to a connection with torsion when matter with
internal spin, such as fermionic matter, is present. The appearance of torsion when
fermions are coupled to gravity had earlier been noted by Weyl [165].‡
The result of MacDowell and Mansouri [110] is essentially that by considering a
connection taking values in the algebra so(4, 1) of the de Sitter group, one obtains a
cosmological term in addition to the Einstein-Hilbert action (3.1.1). (MacDowell and
Mansouri also extended the calculations to supergravity, which we will not discuss
further.) We will see that the result is not just a mathematical trick, but has a
geometric interpretation in terms of a Cartan connection.
The rest of this chapter is organised as follows: We discuss the idea of a curved mo-
mentum space in section 3.2 and give a brief introduction into the ideas of deformed
(doubly) special relativity (DSR) most relevant to the following discussion in sec-
tion 3.3. In section 3.4 we outline how Einstein-Cartan theory can be formulated
as a gauge theory of gravity with the de Sitter group SO(d, 1) as gauge group; this
theory includes a gauge field that plays a crucial role in what follows. In this section
we essentially re-derive the results of Stelle and West, using a different set of coordi-
nates which we find more closely related to the DSR literature. This section uses the
‡Rather amusingly, Weyl wondered in [165] whether the “gauge transformations” of electromag-
netism should not be called “phase transformations” instead – the term “gauge transformation” had
of course been introduced by Weyl himself [163] in the context of local conformal transformations
(changes in “gauge”).
Deformed General Relativity and Torsion 55
language of gauge theory familiar to physicists; we then give a more mathematical
account of the geometry of Cartan connections in section 3.5.
We then return to a concrete example: To justify our claim that this geometric
framework can be used to generalise the ideas of DSR, we show in section 3.6 how,
if spacetime is taken to be Minkowski space, the simplest non-trivial choice of zero
section leads to a connection with torsion, providing a geometric interpretation for
the noncommuting “coordinates” appearing in the Snyder algebra. We close with a
discussion of our results and their possible physical implications, which show that the
theory, at least in its given form, is not physically viable. We conclude that there
may be different physical interpretations of algebraic commutation relations such as
those used in DSR.
Since the two most obvious extensions of general relativity are admitting either
connections with torsion or non-metric connections, we briefly discuss the theory of
a torsion-free non-metric connection, known as symmetric affine theory, in a final
section. We will see that it does not fit as well into a description by Cartan geometry
as the case highlighted in this chapter.
3.2 Curved Momentum Space
It is commonly assumed that quantum gravity sets a fundamental length scale, the
Planck scale [137], which can not be resolved by any physical experiment. Different
approaches to quantum gravity, such as string theory or loop quantum gravity, in-
corporate such a scale. This leads to the idea that some kind of “space discreteness”
should be apparent even in a low-energy “effective” theory.
The idea of putting quantum mechanics on a discrete lattice§ seems to have been
first considered by Heisenberg in the spring of 1930 [130], in an attempt to remove the
divergence in the electron self-energy. Because the absence of continuous spacetime
symmetries leads to violations of energy and momentum conservation, this approach
was not pursued further, but later in the same year Heisenberg considered modifying
the commutation relations involving position operators instead [130].
A fundamental length scale is absent in special relativity, where two observers
will in general not agree on lengths or energies they measure. Hence the usual ideas
of Poincare´ invariance must be modified in some way. Snyder observed [151] that
§with spacing equal to the Compton wavelength of the proton, lc ≈ 1.3 fm
Deformed General Relativity and Torsion 56
this could be done by deforming the Poincare´ algebra into the de Sitter algebra, i.e.
considering the isometry group of a (momentum) space of constant curvature. As we
have seen in chapter 2, if one maintains the structure of a Lie algebra and considers
deformations of the Poincare´ algebra, replacing it by the de Sitter (or anti-de Sitter)
algebra is the unique way of implementing a modified kinematic framework [12].
A d-dimensional de Sitter momentum space with curvature radius κ is defined as
the submanifold of a (d+ 1)-dimensional flat space with metric signature (d, 1) by
(P 1)2 + (P 2)2 + . . .+ (P d−1)2 − (P d)2 + (P d+1)2 = κ2 , (3.2.1)
where κ has dimensions of mass. Its isometry group is generated by the algebra
[Mab,Mcd] = ηacMbd + ηbdMac − ηbcMad − ηadMbc ,
[Xa,Mbc] = ηacXb − ηabXc , [Xa, Xb] = 1
κ2
Mab . (3.2.2)
Here Mab correspond to a Lorentz subalgebra of the de Sitter algebra, while Xa ≡
1
κ
Md+1,a are interpreted as (noncommuting) translations. These translations are then
interpreted as corresponding to coordinates on spacetime; Snyder thought of operators
acting on a Hilbert space. Since the operators X1, . . . , Xd−1 correspond to compact
rotations in the (d + 1)-dimensional space, their spectrum is discrete. In this way,
one obtains “quantised spacetime”, while maintaining Lorentz covariance.
It should be stressed at this point that we are interested in the full isometry group
acting on momentum space, i.e. the Poincare´ group in the case of a Minkowski space.
Clearly, an equation such as (3.2.1) defines a scale κ already in the (d+1)-dimensional
Minkowski space; this scale would however not be invariant under translations of this
space. In other words, there is no natural length scale in Minkowski space, whereas
there is one in de Sitter space, namely the scale κ defining the manifold. There has
been some confusion in the literature (see e.g. [147]) about the way in which Lorentz
invariance is deformed in the Snyder algebra and the more recent ideas of DSR that
we will review shortly. It should be clear from the commutation relations (3.2.2) that
the Lorentz subgroup generated by Mab is not deformed in any way.
One can give explicit expressions for the algebra elements by choosing coordinates
on de Sitter space (3.2.1). The choice made by Snyder is taking Beltrami coordinates
p1 = κ
P 1
P d+1
, p2 = κ
P 2
P d+1
, . . . , pd = κ
P d
P d+1
, (3.2.3)
whence one has (P d+1)2 = κ4/(κ2 + ηabp
apb) to satisfy (3.2.1), and ηabp
apb ≥ −κ2,
corresponding to an apparent maximal mass if pa were interpreted as Cartesian co-
Deformed General Relativity and Torsion 57
ordinates on a Minkowski momentum space. (Up to this point one could in principle
have chosen anti-de Sitter instead of de Sitter space. Then this inequality becomes
ηabp
apb ≤ κ2, which perhaps seems less motivated physically.) A necessary sign choice
means that these coordinates only cover half of de Sitter space. In these coordinates,
the translation generators
Xa =
1
κ
(
P d+1
∂
∂P a
− Pa ∂
∂P d+1
)
=
∂
∂pa
+
1
κ2
pap
b ∂
∂pb
(3.2.4)
generate “displacements” in de Sitter space. (In this notation, indices are raised and
lowered with ηab, the d-dimensional Minkowski metric, so that pa = ηabp
b.)
The motivation behind these ideas was to cure the infinities of quantum field the-
ory, which evidently arise from allowing arbitrary high momenta (or short distances).
In a somewhat similar spirit, Gol’fand suggested [73] to define quantum field theory
on a momentum space of constant curvature, using Beltrami coordinates as momen-
tum variables. This makes the volume of the corresponding Riemannian space finite
and so presumably leads to convergent loop integrals in the Euclideanised theory. The
consequences for standard quantum field theory were further explored in [92, 74].
Gol’fand only assumed that κ m for all elementary particles; thinking of quan-
tum gravity, one would perhaps identify κ with the Planck scale, whereas the original
authors seem to have thought of the Fermi (i.e. weak interaction) scale.
The induced metric on de Sitter space in terms of the coordinates pa is
gnr =
κ2
κ2 + p · p
(
ηnr − pnpr
κ2 + p · p
)
, (3.2.5)
where p · p ≡ ηcdpcpd. The metric (3.2.5) becomes singular when p · p → −κ2, and
negative definite when extended to what Gol’fand calls the exterior region p ·p < −κ2.
In four dimensions,
det g = −κ10(κ2 + p · p)−5 , (3.2.6)
and the volume element is d4p κ5(κ2 + p · p)−5/2.
In Gol’fand’s approach (assuming d = 4 of course), the standard Feynman rules
were modified by replacing the addition of momenta p and k at a vertex by
(p(+)k)a =
κ
κ2 − p · k
(
pa
√
κ2 + k · k + ka
(
κ− p · k
κ+
√
κ2 + k · k
))
, (3.2.7)
which corresponds to a translation by k of the vector p. (Again p · k ≡ ηabpakb, etc.)
It was also noted that spinors now transform under “displacements” as well, which is
Deformed General Relativity and Torsion 58
made more explicit in [92] and [74]. As is well known, five-dimensional Dirac spinors
still have four components and the matrix γ5 appears in the Dirac Lagrangian, hence
there is no chirality. This alone seems to imply that the original Gol’fand proposal
cannot be used for an appropriate model of the known particles.
Gol’fand’s approach is very different from more recent approaches to quantum field
theory on noncommutative spaces (see e.g. [48]) in that the field theory is defined on
a momentum space which is curved, but neither position nor momentum space are
noncommutative in the usual sense.
3.3 Deformed Special Relativity
The idea that the classical picture of Minkowski spacetime should be modified at
small length scales or high energies was re-investigated in more recent times, moti-
vated by the apparent existence of particles in ultra high energy cosmic rays whose
energies could not be explained within special relativity [5]. The proposed framework
of deformed special relativity (DSR) [4] modifies the Poincare´ algebra, introducing
an energy scale κ into the theory, in addition to the speed of light c. This leads
to a quantum (κ-)deformation of the Poincare´ algebra [113], with the parameter κ
associated with the newly introduced scale.
It was soon realised [100] that this deformed algebra is the algebra of the isometry
group of de Sitter space, and that the symmetries of DSR could hence be obtained by
identifying momentum space with de Sitter space, identifying Xa as the generators of
translations on this space. The constructions of DSR thus appear to be a resurrection
of Snyder’s and Gol’fand’s ideas. We take this observation as the defining property of
DSR, and will seek to describe a framework in which momentum space, or rather the
(co-)tangent space in general relativity, is replaced by an “internal” de Sitter space.
We will see that this can best be done using Cartan geometry.
When discussing DSR as a modification of special relativity, we take the view
that special relativity is defined as a kinematic framework with preferred inertial
systems, related to one another by (proper) Lorentz transformations. That is, one
has a flat spacetime on which there exist certain preferred coordinate systems, those
in which the metric is diagonal with entries ±1. From this point of view, the choice
of coordinates on the internal de Sitter space plays quite an important role if one is
looking for a “deformation” of special relativity including an energy scale κ. Such
Deformed General Relativity and Torsion 59
a deformation can only arise if the chosen coordinate system reduces to Cartesian
coordinates on Minkowski space as κ → ∞. The choice of coordinates is obviously
not unique.
The generators of the algebra will take different explicit forms when different
coordinate systems (on four-dimensional de Sitter space) are chosen. In [100] “natural
coordinates” are defined by, in the notation of section 3.2, ¶
g = exp
[
pI(MI4 +XI)
]
exp
[
p4X4
]O , (3.3.1)
where O = (0, 0, 0, 0, κ) is taken to be the origin of de Sitter space in five-dimensional
Minkowski space, and MI5 and M45 correspond to translations in space and time.
The coordinates one obtains are related to the five-dimensional coordinates by
P I = pIe
p4
κ , P 4 = κ sinh
(
p4
κ
)
+
~p2
2κ
e
p4
κ , P 5 = κ cosh
(
p4
κ
)
− ~p
2
2κ
e
p4
κ . (3.3.2)
Again, these cover only half of de Sitter space where P 4 + P 5 > 0. The metric in
these “flat” coordinates is
ds2 = −(dp4)2 + e 2p4κ δIJdpI dpJ . (3.3.3)
Slices of constant p4 are flat; to an observer using these coordinates the spacetime
appears as expanding exponentially. An illuminating discussion of different coordinate
systems and kinematics on de Sitter space is given in [29].
The Magueijo-Smolin model [112] corresponds to the following choice of coordi-
nates:
p1 = κ
P 1
P 5 − P 4 , p
2 = κ
P 2
P 5 − P 4 , p
3 = κ
P 3
P 5 − P 4 , p
4 = κ
P 4
P 5 − P 4 , (3.3.4)
The generators of boosts in de Sitter space take the form
KI ≡ pI ∂
∂p4
+ p4
∂
∂pI
+
1
κ
pIpJ
∂
∂pJ
, (3.3.5)
and translations (not considered by the authors of [112]) would take the form
XI =
p4 + κ
κ
∂
∂pI
+
1
κ2
pIp
b ∂
∂pb
, X4 =
1
κ
pb
∂
∂pb
+
p4 + κ
κ
∂
∂p4
. (3.3.6)
¶Capital Latin indices such as I and J used in this section only run over spatial coordinates (from
1 to 3).
Deformed General Relativity and Torsion 60
This choice of coordinates is somewhat peculiar as p4 takes a special role, as is also
apparent from the modified dispersion relations presented in [112]. The quantity
||p||2 = ηabp
apb
(1 + 1
κ
p4)2
(3.3.7)
is invariant under boosts and rotations in de Sitter space, as would ηabp
apb be in
Beltrami coordinates.
Each DSR model corresponds to a choice of coordinates on de Sitter space, such
that all expressions reproduce the expressions for special-relativistic Minkowski co-
ordinates as κ → ∞. What Magueijo and Smolin call a “U map” is essentially
a coordinate transformation from Beltrami coordinates to a different set of coordi-
nates, which becomes the identity as κ→∞. In the remaining sections we shall use
Beltrami coordinates. Note that this means we always have p · p ≥ −κ2.
3.4 A de Sitter Gauge Theory of Gravity
The most direct implementation of the ideas discussed so far into a framework describ-
ing more general spacetimes is replacing the cotangent (or tangent) bundle usually
taken as phase space by a general symplectic manifold {P , ω}, which can be locally
viewed as a product U ×D of a subset U ⊂M of spacetimeM with de Sitter space
D. We want to retain the differentiable structure of a manifold, which we do not
assume to be present in a full theory of quantum gravity. We also assume that the
structure of momentum space is fixed and in particular does not depend on matter
fields, as suggested in [119].
If phase space is described as such a manifold, with a choice of origin in the
“tangent” de Sitter space at each point, the appropriate mathematical language is
that of fibre bundles. The theory of connections on fibre bundles of this type, called
homogeneous bundles in [51], was developed by E´lie Cartan (e.g. in [32]). Adopting
this framework means there is now an so(d, 1) connection, instead of an so(d − 1, 1)
connection, defining parallel transport on spacetime.
It was noted by MacDowell and Mansouri [110] that gravity with a cosmological
term in four dimensions could be obtained from a theory of such an so(d, 1) connection
by projecting it onto its so(d− 1, 1) part in the action. A more elaborate description
in terms of Einstein-Cartan theory was then given by Stelle and West [153]. Their
analysis included the gauge field needed to identify the fibres at different spacetime
Deformed General Relativity and Torsion 61
points, which will be crucial for the interpretation of the theory. The mathematical
side of MacDowell-Mansouri gravity as a theory of a Cartan connection is nicely
illustrated in [166]; we follow this article as well as the more computationally based
presentation of [153], who use the language of non-linear realisations. An overview
over the mathematics of Cartan connections is given in [148].
For clarity we first describe the framework in a language more common to physi-
cists; a more mathematical account of Cartan connections on homogeneous bundles
is given in the following section 3.5.
The usual description of general relativity as a gauge theory of the Lorentz group,
known as vier-/vielbein formalism, was reviewed at the beginning of this chapter.
Since the tangent bundle is in our description replaced by a homogeneous bundle
with a curved “tangent” space, one has to effectively use a “double vielbein” formal-
ism, in which spacetime vectors are mapped to vectors in the tangent space to the
internal (curved) space by a soldering form (vielbein). The picture we have in mind
is that of a de Sitter space rolled along the manifold. One then needs to introduce
a new field which specifies the point of tangency, expressed in a given coordinate
system on the internal space, at each spacetime point. We denote it by pa(x). This
corresponds mathematically to a necessary choice of zero section (see below), and
physically to a gauge field. Picking a point of tangency at each spacetime point
breaks the gauge group SO(d, 1) down to the Lorentz subgroup SO(d− 1, 1) leaving
the point of tangency invariant.
In more general terms [167], one has a principal bundle over a base manifold M
with gauge group G. Locally, a connection on this bundle can be viewed as a one-
form taking values in the Lie algebra g. One then considers the projection of the
connection to its g/h part and demands that at any point in the manifold, this be an
isomorphism between the tangent space at this point and the algebra g/h. In other
words, the g/h part eµ
i, written as a matrix, has to have full rank. This of course
means that the dimension of g/h must be equal to the dimension of the manifold.
The gauge field pa(x) now comes in because the projection of the connection to its
g/h part is not canonical; for a given subgroup H ⊂ G viewed as the stabiliser of a
particular point in G/H, one could equally well take a conjugate group gHg−1 ' H
stabilising a different point in G/H. This will become more explicit shortly when we
consider the MacDowell-Mansouri action.
The procedure does not depend on the choice of gauge group G, and would be
Deformed General Relativity and Torsion 62
equally possible if the gauge group were chosen to be the Poincare´ group E(d− 1, 1);
such a formulation for gravity, including copuling to matter, is indeed discussed in
[75]. In that case, the homogeneous space G/H would be Minkowski space, viewed as
an affine space. As the value of the gauge field pa(x) at a particular spacetime point
picks an origin in this affine space, it may then be identified with the vector space
R
d−1,1: At a given point in Minkowski space, the tangent space to this point can
be identified with Minkowski space itself. Although all calculations we will present
go through in the case of E(d − 1, 1) as gauge group (resulting in different formulae
of course), the formalism may seem a bit redundant since there is no real necessity
for the second vielbein field. In the more general case of a homogeneous space G/H
which acts as a “tangent space” instead of a flat vector space, it is however necessary.
We consider a theory with gauge group SO(d, 1), so that the connection A takes
values in the Lie algebra so(d, 1). It can be split as (again introducing a length l on
dimensional grounds)
A =


ωab
1
l
ei
−1
l
ei 0

 , (3.4.1)
so that ωab acts as the usual so(d − 1, 1)-valued connection of general relativity and
ei as a vielbein one-form. We have simultaneously unified the usual connection and
the vielbein, and replaced the (flat) tangent space by a curved “internal” space, such
that the de Sitter group and not the Poincare´ group now appears as a gauge group.
(Lorentz) indices on ωab and e
i are raised and lowered using ηab.
A gauge transformation, i.e. a local transformation g(x) taking values in the de
Sitter group, can be split as g(x) = s(x)Λ(x), where s(x) changes the zero section,
i.e. changes the local identification of points of tangency at each spacetime point, and
Λ(x) is a local Lorentz transformation in the vielbein formalism of general relativity
which does not mix the ωab and e
i parts of the connection. In this notation, the
connection transforms under a gauge transformation as
A(x)→ A′(x) = Λ−1(x)s−1(x)A(x)s(x)Λ(x)+Λ−1(x)s−1(x)ds(x)Λ(x)+Λ−1(x)dΛ(x) .
(3.4.2)
One can use this equation to relate the connection A0 corresponding to the triv-
ial zero section, where the point of tangency is the origin of the internal space at
each spacetime point, pa(x) ≡ (0, 0, 0, 0), to a connection corresponding to any given
Deformed General Relativity and Torsion 63
zero section. The physical significance of this is the following. Assume we have
fixed pa(x) ≡ (0, 0, 0, 0). Then an action can be defined from the curvature of the
connection A (here R is the curvature of the so(d− 1, 1) part of A),
F = dA+ A ∧ A =


Rab − 1l2 (ea ∧ eb) 1l T i ≡ 1l (dei + ωij ∧ ej)
−1
l
Ti 0

 . (3.4.3)
In four dimensions, the MacDowell-Mansouri action [110, 166] is
S = − 3
32piGΛ
∫
abcd
(
F ab ∧ F cd) = − 3
32piGΛ
∫
d4x
1
4
abcd
µνρτF abµνF
cd
ρτ , (3.4.4)
where the Latin indices run from 1 to 4, and so one projects F to its so(3, 1) part in
this action.
Apart from a topological Gauss-Bonnet term, the action (3.4.4) is equivalent to
the Einstein-Hilbert action with a cosmological term
S =
3
16piGΛ
1
l2
∫
abcd
(
ea ∧ eb ∧Rcd − 1
2l2
ea ∧ eb ∧ ec ∧ ed
)
, (3.4.5)
where we have to identify
Λ =
3
l2
(3.4.6)
as the cosmological constant.‖
In order to define the projection of F in the action (3.4.4), one has used a splitting
so(d, 1) ' so(d, 1)/so(d− 1, 1)⊕ so(d− 1, 1) , (3.4.7)
which depends on the gauge field since the subgroup SO(d − 1, 1) leaving a given
point in de Sitter space invariant depends on the choice of this point.
When the action (3.4.4) is coupled to matter, the so(d, 1)/so(d−1, 1) part ea of the
connection appears in a volume element in the matter Lagrangian. By varying the
action one obtains the field equations of Einstein-Cartan theory with a cosmological
constant Λ = 3/l2. The length scale l, which is so far arbitrary, can be chosen
to reproduce the Λ of the observed universe, which means it must be chosen to be
very large (the “cosmological constant problem”). By the field equations, one can
‖Note that in the formulation in terms of the Poincare´ group, with connection (3.1.3), no cosmo-
logical constant appears and the length scale L remains undetermined.
Deformed General Relativity and Torsion 64
determine for a given matter distribution a connection A0 consisting of an so(d−1, 1)
connection (ωab)0 and a vielbein e
i
0.
The MacDowell-Mansouri action reproducing Einstein-Cartan theory with a cos-
mological constant includes a gauge choice. We can hence view it as the gauge-fixed
version of a more general theory. Since (3.4.2) determines how the connection trans-
forms under a gauge transformation, we can generalise a given solution of Einstein-
Cartan theory to an arbitrary gauge choice. The extension of the theory to arbitrary
configurations of the gauge field, and hence arbitrary choices of tangency points of the
internal space to spacetime, is what we call Einstein-Cartan-Stelle-West theory.
Any solution of Einstein-Cartan theory, in particular any (torsion-free) solution of
general relativity, gives rise to more general solutions of Einstein-Cartan-Stelle-West
theory via (3.4.2). We will later see that one can construct an so(d− 1, 1) connection
with torsion from a torsion-free one.
In (3.4.2), s(x) takes values in the de Sitter group, more precisely in the subgroup
generated by “translations” which leaves no point of de Sitter space invariant. The
correspondence between Beltrami coordinates pa(x) on de Sitter space and such group
elements is given explicitly by
s(p(x)) = exp
[
pi(x)√−p(x) · p(x) Artanh
(√−p(x) · p(x)
κ
)
κXi
]
. (3.4.8)
Then the group element s(p(x)) maps (0, 0, 0, 0) to (p1(x), p2(x), p3(x), p4(x)) in Bel-
trami coordinates. A different choice of coordinates in the internal de Sitter space
would correspond to a different parametrisation of the elements of the subgroup of
translations of the de Sitter group.
Inserting (3.4.8) into (3.4.2) and setting Λ(x) ≡ e, we obtain
ωab(p(x)) =
paeb0
lκγ(p)
+
(
1− 1
γ(p)
)
padpb + ωca0 p
bpc
p · p +
1
2
ωab0 − (a↔ b) ,
ei(p(x)) =
lκ
p · p+ κ2
(
pi
pcdp
c
p · p (1− γ(p)) + dp
iγ(p) + (ωib)0p
bγ(p)
)
+piea0pa
1 + κ
2
p·p(1− γ(p))
p · p+ κ2 +
ei0
γ(p)
, (3.4.9)
where
γ(p) ≡
√
p · p+ κ2
κ2
= 1 +
p · p
2κ2
+ . . . (3.4.10)
Because p ·p ≥ −κ2 in Beltrami coordinates, the square root is always real. In the
limit p · p→ 0, our parametrisation is the same as that used in [153], and we recover
Deformed General Relativity and Torsion 65
their results
ωab(p(x)) =
(
1
2
ωab0 +
1
lκ
paeb0 +
1
2κ2
(
padpb + ωca0 p
bpc
))− (a↔ b) ,
ei(p(x)) = ei0 +
l
κ
(
− 1
2κ2
pipcdp
c + dpi + ωib0 pb
)
+
1
2κ2
piea0pa . (3.4.11)
Near p = 0, we have
ωab(p(x)) = ωab0 +O
(p
κ
)
, ei(p(x)) = ei0 +
l
κ
dpi +O
(p
κ
)
. (3.4.12)
As mentioned above, the so(d, 1)/so(d − 1, 1) part of the connection A acts as a
vielbein and maps vectors in the tangent space at a point x in spacetime to vectors
in the tangent space at p(x) in the internal de Sitter space, given in components
with respect to an orthonormal basis at p(x). In order to give their components in
the coordinate-induced basis { ∂
∂pa
}, we need another vielbein, which can be obtained
from (3.4.9) by setting ω0 = e0 = 0 (corresponding to spacetime being de Sitter space
with cosmological constant Λ) and pa(x) = κ
l
xa, as in [153]. We obtain
ln
a(p(x)) = κ2
δn
a(p · p)γ(p)− papn(γ(p)− 1)
(p · p)(p · p+ κ2) , (3.4.13)
where n is a coordinate index in the internal space and a denotes a Lorentz index, as
before. This vielbein is of course independent of the underlying spacetime.
Parallel transport can be defined for the so(d, 1) connection using the notion of
development, which generalises the usual covariant derivative. One introduces a de-
velopment operator [153]
D = d− 1
2
ωabMab − (e · V ) , (3.4.14)
where the second term is the usual infinitesimal relative rotation of tangent spaces at
different spacetime points, and the last term compensates for the change of point of
tangency and hence generates maps from the tangent space at one point of de Sitter
space to the tangent space at a different point of de Sitter space. Again one should
think of an internal space rolled along spacetime [166].
In components, in our conventions we have
(ωabMab)
c
d = −2ωcd , (3.4.15)
and the combination eaVa acts on Lorentz indices as an element of so(d − 1, 1), rep-
resenting the map from one tangent space to another in the respective bases. We use
Deformed General Relativity and Torsion 66
the result obtained by [153] using the techniques of non-linear realisations, namely
that when expressed as an so(d− 1, 1) matrix,
l(e · V ) = κ s(p)−1(eaXa)s(p)− s(p)−1 [s(p+ δp)− s(p)] , (3.4.16)
where s(p) is defined according to (3.4.8) and δp is determined from the equation
[s(p+ δp)]a5 =
[
(1 + ebXbκ)s(p)
]a
5
(3.4.17)
where only terms linear in ea are kept in δp. An explicit calculation shows that
δpa =
pa
κ
(ηbce
bpc) + eaκ , (3.4.18)
and hence near p = 0, we have δpa = κea, as expected. We find that (e · V ) has
components
(e · V )bc =
κ(ebpc − ecpb)(1− γ(p))
l(p · p) . (3.4.19)
One then has a notion of holonomy, mapping closed loops in spacetime into the
internal space by development. In particular, if one develops the field p(x) describing
the point of tangency around an infinitesimal closed loop at x0, the developed value
will in general differ from the original value at x0 [153]:
∆pa(x0) ∝ Tµνi(x0)lai(p(x0))
∮
xµdxν , (3.4.20)
where lai(p(x)) is the inverse of the vielbein (3.4.13) and Tµν
i are the components of
the torsion tensor T = de + ω ∧ e. This is because by specifying pa(x), one locally
identifies the “internal” de Sitter spaces, thinking of a tangent de Sitter space rolled
along the manifold. In a generic situation, where the manifold M is not identical to
the internal space, such an identification is, strictly speaking, only possible at a point.
To understand the geometry, one may picture a sphere being rolled along a plane, as
in figure 3.2. In general, when the sphere is rolled around and returns to its initial
position on the plane, its point of tangency will not be the original one. This is just
the situation for the Lorentzian analogues, Minkowski space and de Sitter space, as
we will discuss shortly.
The central result we will try to justify in the following is that, starting from
Minkowski spacetime, if we assume the internal de Sitter space is rolled along Minkowski
space in a non-trivial way, we obtain a connection with torsion. In our interpretation,
this is the only way that “coordinates” can act as translations on momentum space,
as one normally assumes when associating the Snyder algebra with a noncommutative
spacetime.
Deformed General Relativity and Torsion 67








































&%
'$
Figure 3.2: When the curved internal space is rolled along Minkowski space, a path in
spacetime corresponds to a path in the internal space. Because of the curvature of the
internal space, a closed path in Minkowski space does not correspond to a closed path in
the internal space, which is manifest as torsion. (We have drawn Riemannian spaces, a
plane and a sphere, for clarity of presentation.)
3.5 Cartan Connections on Homogeneous Bundles
This more mathematical introduction into Cartan connections on homogeneous bun-
dles relies mainly on [166], but mentions some additional points which are of impor-
tance to our discussion of Einstein-Cartan-Stelle-West theory.
The tangent bundle of a manifold needs to be replaced by a fibre bundle whose
fibres are homogeneous spaces D ≡ SO(d, 1)/SO(d − 1, 1). This can be achieved
by starting with a principal bundle P (M, SO(d, 1)), and considering the associated
bundle P = E(M,D, SO(d, 1), P ) = P ×SO(d,1) D (taken as phase space); it can be
identified with P/SO(d− 1, 1) by the map
ν : P → P/SO(d− 1, 1), [u, a · SO(d− 1, 1)] 7→ ua · SO(d− 1, 1). (3.5.1)
Then the structure group SO(d, 1) is reducible to SO(d − 1, 1) if the associated
bundle P admits a cross section σ :M→ P [99]; furthermore, there is a one-to-one
correspondence between reductions of the structure group and cross sections. This
cross section, called a zero section in [135], corresponds to a choice of origin in the
momentum space attached to each point. In physicist’s terms, the de Sitter group is
spontaneously broken down to the Lorentz group by the choice of points of tangency
in the tangent de Sitter spaces at each spacetime point.
The bundle reduction depends on the choice of zero section, or rather, its local
representation in coordinates as a function M ⊃ U → U × D. This is because the
Deformed General Relativity and Torsion 68
embedding of SO(d−1, 1) into SO(d, 1) is not canonical, as the stabilisers of different
points in D are isomorphic but related by conjugation. In other words, the mappings
appearing in the exact sequence
0→ SO(d− 1, 1)→ SO(d, 1)→ SO(d, 1)/SO(d− 1, 1)→ 0 (3.5.2)
are not canonically chosen (cf. the discussion for the affine group in [135]).
It is of course possible to choose canonical coordinates such that the function
representing the zero section is just x 7→ (x, [e]) ≡ (x, SO(d − 1, 1)) ∈ M × D.
However, in general we want to locally identify the fibres at nearby base space points,
adopting the viewpoint that there is a single tangent D space which is “rolled along”
the manifold. Then we need to retain the general coordinate freedom. (This point
is missing in the discussion of [166].) An exact identification is only possible when
the connection is flat. Let us assume that coordinates on P have been fixed, and
that it is the zero section, and hence the identification of the fibres, that is varied∗∗.
After a choice of zero section, there is still a local gauge freedom corresponding to the
stabiliser SO(d− 1, 1). We express a given section as s(x), where σ(x) = (x, s(x)) ∈
M×D in our coordinates. The section that corresponds to s0(x) ≡ [e] will be called
“trivial”.
An so(d, 1)-valued Ehresmann connection A in P is in general not reducible to an
so(d−1, 1)-valued connection in the reduced SO(d−1, 1) bundle PR(M, SO(d−1, 1)).
It can, however, be pulled back using the inclusion
ιx : SO(d− 1, 1)→ SO(d, 1), Λ 7→ s(x)Λs(x)−1 (3.5.3)
to a Cartan connection AC on the reduced bundle
††. Of course reducing the connec-
tion to an so(d − 1, 1)-valued connection and pulling it back to a Cartan connection
are very different operations, since in the latter case one wants the so(d, 1)/so(d−1, 1)
part of the pulled-back connection to act as a soldering form, so in particular to be
non-singular. We obtain a bundle sequence (cf. [166])
∗∗one is free to choose an “active” or “passive” viewpoint here
††We assume here that the necessary condition kerA ∩ (ιx)∗(TPR(M, SO(d − 1, 1))) = {0} (see
[148]) is satisfied.
Deformed General Relativity and Torsion 69
PR(M, SO(d− 1, 1)) - P (M, SO(d, 1)) - P/ιx(SO(d− 1, 1)) ' P
ν−1x
@
@
@
@
@
@R ?





	
M
The reduced bundle PR(M, SO(d− 1, 1)) is mapped into P (M, SO(d, 1)) by
p 7→ [p, e] = {(pΛ−1, s(x)Λs(x)−1)|Λ ∈ SO(d− 1, 1)}
∈ PR(M, SO(d− 1, 1))×ιx(SO(d−1,1)) SO(d, 1) . (3.5.4)
The connection one-form A on M, induced by the connection A on P , depends on
a choice of section τ : M→ P (M, SO(d, 1)). If the zero section σ is fixed, one can
choose an arbitrary (local) section τR :M→ PR(M, SO(d−1, 1)) to obtain a section
τ ; in local coordinates,
σ(x) = (x, s(x)) , τR(x) = (x,Λ(x))
−→ τ(x) = (x, s(x)Λ(x)s(x)−1 · s(x)) = (x, s(x)Λ(x)) . (3.5.5)
For practical computations, it is often useful to first consider the trivial section. The
induced connection corresponding to this section, denoted by A0(x), is related to the
connection for a general section by
A(τ(x)) = Λ−1(x)s−1(x)A0(x)s(x)Λ(x) + Λ−1(x)s−1(x)ds(x)Λ(x) + Λ−1(x)dΛ(x) .
(3.5.6)
Once the zero section s(x) has been fixed, there is still the freedom of SO(d − 1, 1)
transformations, corresponding to different choices of Λ(x) in (3.5.6). These are the
standard local Lorentz transformations in the vielbein formalism of general relativity.
The choice of zero section induces a local splitting of the so(d, 1) connection,
according to
so(d, 1) ' so(d, 1)/so(d− 1, 1)⊕ so(d− 1, 1) . (3.5.7)
This splitting is invariant under the adjoint action of SO(d− 1, 1), thus the different
parts of the connection will not mix under SO(d − 1, 1) transformations. We are
considering a reductive geometry (cf. (1.1.6)); recall that the existence of such a
splitting
g = x⊕ h (3.5.8)
Deformed General Relativity and Torsion 70
invariant under the adjoint action of H, implies that we may identify g/h with the
subspace x of g. In the present case, this is of course the subspace spanned by the
generators Md+1,a of SO(d, 1). For the de Sitter group, just as for the Poincare´ and
anti de Sitter groups, one can say more: The Lie algebra so(d, 1) is equipped with a
Z2 grading; the subspaces in (3.5.8) satisfy
[h, h] ⊂ h , [h, x] ⊂ x , [x, x] ⊂ h (3.5.9)
and the homogeneous space G/H is a symmetric space [167].
The fact that de Sitter space viewed as a homogeneous space is a symmetric space
is directly responsible for the splitting of the curvature F of the Cartan connection
A into curvature and torsion of the so(d − 1, 1) connection ω that we have observed
above [167]. A general Cartan connection A = ω + e, where ω takes values in h and
e takes values in x, has curvature
F = R +
1
2
[e, e] + dωe , (3.5.10)
where [e, e] denotes taking the wedge product on the form part and the commutator
on the Lie algebra part, i.e.
[e, e]iµν = Cj
i
k
(
ej ∧ ek)
µν
= Cj
i
k
(
ejµe
k
ν − ekµejν
)
(3.5.11)
if Cj
i
k are the structure constants as in (1.1.4). The middle term generically takes
values in h and x, so that one has a splitting
Fh = R +
1
2
[e, e]h , Fx =
1
2
[e, e]x + dωe . (3.5.12)
For a symmetric space, however, [e, e]x vanishes [167], and we obtain the result used
in the construction of the MacDowell-Mansouri action (3.4.4).
Because we assume A to be a Cartan connection, the so(d, 1)/so(d−1, 1) part acts
as a soldering form, corresponding to the standard vielbein of general relativity; in
particular, eµ
i is an invertible matrix. The soldering form maps vectors in the tangent
space TxM at a point x in spacetime to vectors in the tangent space Tp(x)D at p(x)
in the internal de Sitter space, given in components with respect to an orthonormal
basis at p(x). The vielbein that maps between the components of a vector in the
orthonormal basis and the coordinate-induced basis is given in (3.4.13).
Deformed General Relativity and Torsion 71
3.6 Synthesis
The notion of development along curves in spacetime is central to the interpretation
of Einstein-Cartan-Stelle-West theory, because it allows “spacetime coordinates” to
act as translations in the internal de Sitter space. The situation described by DSR,
where noncommuting translations on a curved momentum space are interpreted as
noncommuting spacetime coordinates, here corresponds to a Minkowski spacetime
with an internal de Sitter space rolled along this Minkowski space. The gauge field
pa(x) specifies the points of tangency of the internal space at each spacetime point,
and we have chosen Beltrami coordinates on de Sitter space which look like Cartesian
coordinates on Minkowski space near the “origin” of de Sitter space. Since the internal
space has a natural scale κ and we needed to introduce a natural scale l in spacetime,
we choose the gauge field to be
pa(x) =
κ
l
xa (3.6.1)
in a vicinity of the origin of spacetime which is now taken to be Minkowski space,
where xa are the standard Minkowski coordinates such that the connection vanishes
in general relativity. In general a closed path in spacetime will not correspond to a
closed path traced out on the internal space, hence such an identification is only local
and, strictly speaking, only valid the origin of Minkowski spacetime. On dimensional
grounds, the effects of torsion scale as x
l
or p
κ
. For (3.6.1) to be well-defined, we must
guarantee that x · x ≥ −l2, so l should be large in Planck units. We will comment on
the significance of the scale l at the end of this section.
It should perhaps be emphasised that the gauge field pa(x) does not represent
physical momentum, but determines the point of tangency of the internal space we
have introduced which is to some extent arbitrary. Tangent vectors to the original
spacetime can be mapped to tangent vectors to the internal space via the vielbein. The
physical interpretation of motion in an internal “momentum” space which is related
to motion in spacetime seems obscure, but if coordinates are to act as translations
in the internal space, the two must be connected in some way. In this sense, we are
constructing the minimal non-trivial gauge field which leads to observable effects, and
an alternative interpretation of noncommuting generators Xa in the Snyder algebra.
In our interpretation, different points in the internal de Sitter space do not repre-
sent different values for physical four-momentum. Hence we avoid problems with the
physical interpretation of DSR, such as the “spectator problem” of noncommutative
momentum addition and the “soccer ball problem” of how to describe extended ob-
Deformed General Relativity and Torsion 72
jects, given that any momentum would appear bounded by the relation p·p ≥ −κ2. In
our framework, tangent vectors representing a particle’s (or extended body’s) velocity
remain vectors and as such live in an unbounded space with commutative addition.
As explained before, we can use equations (3.4.9) to obtain the connection com-
ponents ω and e corresponding to this choice of our gauge field; we set ω0 = 0 and
(eµ
a)0 = δµ
a and substitute (3.6.1) to get
ωµ
ab =
(
xaδµ
b − xbδµa
) x · x+ l2(γ(x)− 1)
l2(x · x)γ(x) , (3.6.2)
eµ
i =
1
(x · x)(x · x+ l2)
(
xixµ(x · x− 2l2(γ(x)− 1)) + δµi(x · x)2l2γ(x)
)
and
∂νeµ
i − ∂µeνi =
(
xνδµ
i − xµδνi
) l2(2l2(γ(x)− 1)− 3(x · x))− (x · x)2
(x · x)(x · x+ l2)2 ,
ων
ibeµb − ωµibeνb = (2l
2 + x · x) (x · x+ l2(γ(x)− 1))
(x · x)(x · x+ l2)l2γ(x)
(
xνδµ
i − xµδνi
)
, (3.6.3)
where now
γ(x) ≡
√
x · x+ l2
l2
, (3.6.4)
which gives a non-zero torsion
Tµν
i =
(
xνδµ
i − xµδνi
) 1
l2
√
x·x
l2
+ 1
. (3.6.5)
Interestingly enough, for the choice of zero section (3.6.1) the scale κ drops out of all
expressions. Expressed in coordinates on the internal space, one has
Tµν
i =
(
pνδµ
i − pµδνi
) 1
lκ
√
p·p
κ2
+ 1
. (3.6.6)
The quantity Tµν
i will be multiplied by an infinitesimal closed loop
∮
xµdxν to give
the difference in the value p(x) caused by development along this loop. In momentum
coordinates, this is equal to l
κ
∮
pµdpν , and the effect of going around the developed
curve in the internal space is (near x = 0 or p = 0) proportional to κ−2, just as was
suggested by (3.2.2).
Expressing Minkowski space in the usual coordinates, together with the (local)
identification pa(x) = κ
l
xa, in this framework gives a connection with torsion. De-
veloping a closed curve in spacetime in the internal space will give a curve that does
not close in general, which is the effect of noncommuting translations in the internal
space.
Deformed General Relativity and Torsion 73
The reader may wonder how the “deformation” of the Minkowski solution de-
scribed here is manifest in a metric. We can define a metric by the usual expression
gµν = e
a
µe
b
νηab . (3.6.7)
This metric would not determine the connection, but could be used to define distances
in the spacetime in the usual way. Then, from (3.6.2), we get
gµν = ηµν
4
1 + x·x
l2
+ xµxν
(x · x)
((x · x) + l2)2 . (3.6.8)
It should be stressed that the connection on spacetime is not the Levi-Civita con-
nection of this metric, as we are working in a first order formulation where metric
and connection are independent. There is a factor of 4 because of a term in (3.4.12)
which does not necessarily go to zero as p → 0. With the identification (3.6.1), the
soldering form always gets a contribution
eµ
i(x) = (eµ
i)0 + δµ
i +O
(x
l
)
. (3.6.9)
The limit κ → ∞ is now identified with the limit l → ∞, in which we recover the
(rescaled) Minkowski metric.
In deriving the expressions (3.6.2) we started with Minkowski space, which clearly
solves the field equations of the Einstein-Cartan theory for an energy-momentum
tensor cancelling the cosmological constant term, and vanishing internal spin. In
changing the zero section, we then performed a SO(d, 1) gauge transformation, under
which the curvature F transformed as
F (s(x)) = s−1(x)F (x)s(x). (3.6.10)
This is a general SO(d, 1) rotation which mixes up the so(d−1, 1) and so(d, 1)/so(d−
1, 1) parts of the connection and the curvature. Hence, the resulting connection will
no longer solve the original field equations, but the field equations for an energy-
momentum tensor which has also undergone a SO(d, 1) transformation. This mixes
up the energy-momentum and internal spin parts, combining them into an element
of the Lie algebra so(d, 1), the interpretation of which seems obscure at least.
A comment is in order with regard to physical units. In addition to the energy
scale κ, which is perhaps naturally identified with the Planck scale, the identification
of lengths with momenta, necessary in the framework presented here, requires the
choice of a unit of length l which is not necessarily connected to the scale κ. It may
Deformed General Relativity and Torsion 74
well be that it is instead the cosmological constant which sets this length scale, leading
to an astronomical scale instead of a sub-atomic one. And indeed, some more recent
approaches to quantum gravity (e.g. [58]) use the product GΛ as a dimensionless
parameter in a perturbative expansion. A fixed positive Λ also seems to be required in
non-perturbative approaches to quantum gravity [6]. Then the cosmological constant
may play the role of a fundamental parameter in quantum gravity.
3.7 Summary and Discussion
It has been argued that the algebra of DSR describes the symmetries of a semiclas-
sical limit of (a generic theory of) quantum gravity (see e.g. [6]). If this claim is
taken seriously, one has to give an interpretation of the noncommuting translations
appearing in the algebra, and usually they are supposed to represent a spacetime
with a fundamentally noncommutative structure [111]. Alternatively, one may view
the apparent noncommutativity as an artefact of the finite resolution of lengths [70].
However, there are fundamental difficulties in associating these noncommuting op-
erators directly with coordinates on spacetime, as position is not additive in a way
that momentum and angular momentum are [37]. Furthermore, as also pointed out
in [37], a proposed noncommutativity of spacetime of the form (3.2.2), proportional
to angular momentum or boost generators, and hence vanishing at a given “origin”,
seems deeply at odds with any idea of (even Galilean) relativity. This would also
be an obvious criticism of the framework presented in this chapter, when taken as a
theory that is supposed to describe the real world.
What we have shown here, is that using the framework of Einstein-Cartan-Stelle-
West theory, one reaches a different conclusion from the usual one: The noncommu-
tativity of translations on a momentum space of constant curvature is interpreted as
torsion of a connection that solves the equations of Einstein-Cartan theory with a
modified energy-momentum tensor that mixes with the spin tensor. If one takes this
seriously, one is led to conclude that there is an effect of torsion induced by quantum
gravity, whose effects would however only become measurable over distances compa-
rable to l, a length scale presumably associated with the cosmological constant.
No such effect appears in de Sitter space with an appropriate cosmological con-
stant, or indeed any vacuum solution of the theory. Vacuum solutions are then just
described by the Poincare´ algebra, and hence undeformed special relativity. It is curi-
Deformed General Relativity and Torsion 75
ous that the curvature in the internal “momentum” space is reflected in torsion when
Minkowski spacetime is considered as spacetime, which would however seem rather
unnatural as it is not a vacuum solution any more. This result does however seem
to be the most natural interpretation of the DSR algebra in a differential-geometric
framework capable of describing more general spacetimes.
Any non-zero energy-momentum tensor, however, will lead to a connection having
torsion. In theories such as Einstein-Cartan theory, this leads to well-known problems
when trying to couple the gravitational field to Maxwell fields, for instance, as there
is no unambiguous procedure of minimal coupling. This is because the statement
that the exterior derivative is independent of the choice of connection,
(dA)µν ∝ ∂[µAν] = ∇[µAν] , (3.7.1)
is true precisely when torsion vanishes. Using an so(d − 1, 1) connection, this is
apparent from
d(eiAi) = ∇(eiAi) = Ai∇ei − ei ∧∇Ai = −ei ∧∇Ai + AiT i (3.7.2)
where ∇ei = dei+ωij ∧ ej etc. One has two different candidates for the field strength
F , namely ei ∧ ∇Ai and d(eiAi), with possibly observable differences between these
choices, although it could be argued that F = dA is the only meaningful choice
because it preserves gauge invariance [21].
In the framework of Einstein-Cartan-Stelle-West theory, gauge fields should be
coupled to gravity via development, i.e. replacing F = dA by F = DA. We compute
from (3.4.9) and (3.4.19) that development can be expressed in terms of ω0 and e0 by
D = d+ ω − (e · V ) = d+ ω0 + 2(p⊗A e0)κ(γ(p)− 1)
l(p · p) =: d+ ωeff , (3.7.3)
where ⊗A is an antisymmetrised tensor product, (U ⊗A V )ab ≡ U [aV b] ≡ 12(UaV b −
U bV a). Parallel transport is effectively described by the connection ωeff , whose torsion
is in general non-zero. One can give an explicit formula for the torsion which is
however rather complicated and does not seem to give much insight; to linear order
in pi, one has
T i =
l
κ
(Rib)0p
b − 1
2lκ
ei0 ∧ (e0jpj) +
1
2κ2
(
piej0 ∧ dpj − pjei0 ∧ dpj
)
+O(p2). (3.7.4)
If we assume a universal relation of internal momenta and spacetime lengths of the
form p ∼ κ
l
x, the second and third terms seem to give contributions of order x/l2. The
Deformed General Relativity and Torsion 76
first term is proportional to the local curvature of ω0, R
i
b = dω
i
b+ω
i
j∧ωjb, contracted
with xb. Note that it is the Riemann tensor, not the Ricci tensor, that appears, so
that propagating degrees of freedom of the gravitational field are included. This first
term should in realistic situations, even in vacuum, give the dominant contribution.
Assuming that minimal coupling is achieved through the development operator D,
or equivalently by using the effective connection which has torsion, one would couple
vector or matter fields (using Dψ for spinors) to torsion, breaking gauge invariance.
Such an effect of course leads to the absence of charge conservation, and this should
be experimentally observable in the presence of a non-trivial gravitational field, i.e.
in regions where spacetime is not exactly de Sitter. Let us recall that in standard
tensor calculus one uses the identity
[∇µ,∇ν ]Mλρ = RµνλσMσρ +RµνρσMσλ − Tµσν∇σMλρ (3.7.5)
which gives for an antisymmetric Mλρ when contracted
[∇λ,∇ρ]Mλρ = −gµλgνρTµσν∇σMλρ, (3.7.6)
to establish that the right-hand side of Maxwell’s equation ∇λFλρ = 4piJρ satisfies a
continuity equation in the absence of torsion. With torsion present, one has then for
any region R ∫
∂R
d3x
√
h nλJλ =
1
4pi
∫
R
d4x
√
g
(−gµλgνρTµσν∇σFλρ) . (3.7.7)
Effects become important when the size of the region R is comparable to the length
scale of torsion.
As an example consider the Schwarzschild solution, which has Kretschmann scalar
RabcdR
abcd ∼ r
2
S
r6
, (3.7.8)
so roughly Rabcd ∼ rSr−3. Assuming that the origin of the x coordinate system
corresponds to the centre of the Earth, we would, on the surface of the Earth, measure
a torsion of order rSR
−2, where R is the radius of the Earth. With rS ∼ 10−2 m and
R2 ∼ 1013m2, the length scale for effects of torsion would be about 1011 m. The
other two contributions, given that l ∼ 1026 m, would be much smaller. Although
this crude estimate suggests that effects will be very small, even tiny violations of
charge conservation should have been observed experimentally. For a discussion of
experimental tests of charge conservation and possible extensions of Maxwell theory
Deformed General Relativity and Torsion 77
in Minkowski space, see [102]. Processes such as electron decay on a length scale of
1011 m, or a time scale of 103 s, can clearly be ruled out.
The example presented here shows that the correct physical interpretation of
purely algebraic relations, such as the commutators of the Snyder algebra, may not
be the seemingly obvious one. We conclude that the physical motivation for assuming
spacetime is “noncommutative” may not be as clear as often assumed.
3.8 Gauge Invariance Broken?
When discussing the issue of possible consequences of broken gauge invariance, we
must bear several points in mind.
The idea that an asymmetry between the proton and electron charges could have
interesting astrophysical consequences goes back to Lyttleton and Bondi [108], who
argued that a charge difference, and hence a net charge of the hydrogen atom, of
10−18 elementary charges, might explain the observed expansion of the universe by
electrostatic repulsion. This idea was proposed in connection with Hoyle’s ideas
of a universe in a steady state, which required continuous production of material
via a “creation field” [85], and a modification of Maxwell’s equations was proposed
to accommodate charge nonconservation. From Hoyle’s perspective, however, the
steady state model was incompatible with expansion of the universe by electrostatic
repulsion, and should lead to electrostatic attraction instead [86].
There seems to be no need for the electron and proton charges to be of equal
magnitude to maintain gauge invariance. However, if the universe as a whole is not
neutral, but it is homogeneous, gauge invariance must be broken. Hence the two
issues are closely related. Modern laboratory experiments [115] give a bound of 10−21
elementary charges on the difference of electron and proton charge; astrophysical
considerations give bounds of 10−26 elementary charges using the isotropy of the
cosmic microwave background [30], or 10−29 elementary charges by considering cosmic
rays [124]. Recently, an interesting proposal to measure net charges of atoms and
neutrons, sensitive to 10−28 elementary charges, was put forward [8].
From a theoretical viewpoint, if gauge invariance is broken, it is natural to assume
a nonvanishing photon mass. One then considers Einstein-Proca theory, an outline
of which can be found in [79]. The photon may also be charged. Here, experimental
bounds on the charge are 10−29 elementary charges using pulsars [141], and possibly
Deformed General Relativity and Torsion 78
10−35 elementary charges from CMB isotropy [30].
Experimental bounds on violations of gauge invariance in electrodynamics are very
tight, and hence any theory predicting torsion which is coupled to electromagnetism
faces severe problems when confronted by experiment. In the framework of Einstein-
Cartan-Stelle-West theory, it is possible to maintain gauge invariance by choosing
F = dA, but using the development operator is the most natural choice.
3.9 Symmetric Affine Theory
If Einstein-Cartan theory is considered as the extension of general relativity which
allows for torsion, there is an analogous extension which allows for a non-metric con-
nection. This theory can be formulated in terms of a torsion-free gl(n,R) connection
and is known as symmetric affine theory. It is equivalent to standard general
relativity with a massive vector field, known as (nonlinear) Einstein-Proca theory
[79].
One could attempt to embed this theory into a theory of a connection taking
values in the algebra of the affine group a(n,R)‡‡,
A =


ωab
1
l
ei
0 0

 , (3.9.1)
where now ωab is not constrained by ω
ab = −ωba. Geometrically, this means that the
connection does not preserve the lengths of vectors under parallel transport.
The corresponding curvature of A (R is the curvature of the gl(n,R) part),
F = dA+ A ∧ A =


Rab
1
l
T i ≡ 1
l
(dei + ωij ∧ ej)
0 0

 , (3.9.2)
would then be constrained by demanding that T i ≡ 0. This seems rather unnatural
from the perspective of Cartan geometry. Furthermore, the length scale l is now
completely arbitrary as it does not appear in the gl(n,R) part of the curvature any
more, just as for a connection taking values in the Poincare´ algebra.
‡‡For a comprehensive review of general theories of this type, see [78].
Deformed General Relativity and Torsion 79
One proceeds by considering Lagrangians that only depend on the Ricci tensor,
which is a one-form obtained by contracting the components of the Riemann curva-
ture, written in the basis of one-forms given by the vielbein ei:
Rica = Riciae
i, Ricia = Rji
j
a, (3.9.3)
where the curvature two-form is
Rab =
1
2
Rij
a
be
i ∧ ej. (3.9.4)
One then splits the Ricci tensor into symmetric and antisymmetric part, symmetrising
over a component (with respect to the given basis) index and a gl(n,R) index. The
antisymmetric part can be interpreted as a spacetime two-form
iea
(
Rab ∧ eb
)
, (3.9.5)
where iea is interior multiplication with the vector ea, defined by being dual to the
one-forms eb:
eb(ea) = δ
b
a. (3.9.6)
No such construction is possible for the symmetric part, which is normally more
relevant in concrete constructions. The splitting itself seems depend on the choice of
basis.
General Relativity From a Constrained Topological Theory 80
Chapter 4
General Relativity From a
Constrained Topological Theory
“What, then, is time? If no one ask me, I know; if I wish to
explain to him who asks, I know not.” [11]
St. Augustine
4.1 Overview
In this chapter, we shall investigate another possible formulation of general relativity
in four dimensions, in terms of a constrained topological theory. Here the term
“topological” means that the theory has a very large symmetry: All of its solutions
are locally gauge equivalent, and there are no local degrees of freedom. Globally,
not all solutions are necessarily gauge equivalent if there are topological obstructions.
Starting from such a theory, we will use constraints that restrict the possible gauge
transformations and hence allow for a less trivial theory, namely one with local degrees
of freedom that are those of general relativity.
The topological theory we consider is usually known as BF theory, deriving its
name from the action
S =
∫
Bab ∧ Fab[A] , (4.1.1)
where F is the curvature of a G-connection A and B is a g-valued two-form (in our
case G will be a rotation group). The equations of motion are
F ab = 0 , ∇Bab = 0 , (4.1.2)
General Relativity From a Constrained Topological Theory 81
where ∇ is a covariant exterior derivative involving the connection A. All flat con-
nections are locally gauge equivalent, and the (local) equivalence of all solutions to
the second equation can be seen [13] by noting that the transformation
Bab 7→ Bab +∇ηab , (4.1.3)
where ηab is a g-valued one-form, is a symmetry of the action (4.1.1). This transfor-
mation is generated by the equation F ab = 0, in a canonical sense, and should be
viewed as a generalised gauge transformation. It is a symmetry on the phase space
of the theory. Locally, any solution to ∇Bab = 0 is of the form ∇ηab, so any solution
is gauge equivalent to Bab = 0 in this generalised sense.
The motivation for considering BF theory is that general relativity in three di-
mensions is just of this form: The first order formulation can be written as
S =
∫
abc
(
ea ∧Rbc[ω]) , (4.1.4)
and hence general relativity in three dimensions is topological. Since the quantisation
of this system is relatively well understood (see [168] for Witten’s formulation as
a Chern-Simons theory, and the book [31] for an overview over different routes to
quantisation), and seems to have many conceptual aspects such as diffeomorphism and
Lorentz invariance in common with the higher-dimensional case, one might hope to
gain understanding of higher-dimensional general relativity from studying BF theory.
On the other hand, there is of course a fundamental difference between a system with
an infinite number of (local) degrees of freedom and one with only (usually a finite
number of) global degrees of freedom, which makes it rather unclear whether this
approach will be fruitful.
It has been shown [56] that in any number of dimensions d ≥ 4 general relativity
can be formulated as a constrained BF theory, where one adds to the BF action
constraints quadratic in the field Bab, restricting it to be a wedge product of one-
forms ea that can be interpreted as a frame field. In four dimensions, where one has
to enforce
Bab = abcde
c ∧ ed (4.1.5)
to reproduce the first order GR action (3.1.1), this formulation (in slightly different
complex form) was first studied by Plebanski [138]. This is also the situation most
studied in the spin foam approach to quantum gravity, where one tries to quantise
general relativity by quantising a BF-type theory and then implementing constraints.
General Relativity From a Constrained Topological Theory 82
In this chapter, we shall investigate a formulation which is in the spirit of the
Plebanski formulation, but involving only linear constraints, of the type used re-
cently as a new idea in the spin foam approach to quantum gravity. We identify
both the continuum version of the linear simplicity constraints used in the quantum
discrete context and a linear version of the quadratic volume constraints that are nec-
essary to complete the reduction from the topological theory to gravity. We illustrate
and discuss also the discrete counterpart of the same continuum linear constraints.
Moreover, we show under which additional conditions the discrete volume constraints
follow from the simplicity constraints, thus playing the role of secondary constraints.
In the general context of the thesis, we see how a very large symmetry present
in the original action (4.1.1) can be constrained appropriately so as to give a theory
with rather different physics – one with local degrees of freedom. This symmetry acts
neither on spacetime nor on fibre bundles, but on a symplectic manifold, namely the
phase space of a dynamical system.
4.2 Actions for Gravity
It should not be surprising that the equations of general relativity can be derived
from several different action principles, leading to equivalent classical theories (in
the case of pure gravity, at least). The statement that given equations of motion
arise from different actions is the functional equivalent of saying that there are many
functions with the same stationary points. Apart from constructing different actions
as functionals of the same variables, one can also express the same physical content
in terms of different variables (by performing a Legendre transformation, or other-
wise), leading to even more possibilities for putting the same physics into a different
mathematical form.
This introduces several possible sources of ambiguity into quantisation; Feynman’s
path integral approach to quantum mechanics is based on the fact that off-shell con-
figurations, i.e. those that do not solve the equations of motion and are not stationary
points of the action, contribute to transition amplitudes as well. Hence different func-
tionals of the same variables, while giving the same equations of motion, may give
different quantum theories. For formulations in terms of different variables it is even
less clear whether the resulting quantum theories are related or not. One may see this
as a problem or as a virtue; when a classical theory such as general relativity seems
General Relativity From a Constrained Topological Theory 83
to resist conventional quantisation techniques, one may hope that one of the possi-
ble classical reformulations offers more promise for quantisation. The most obvious
example of this is Ashtekar new variables [10] as a possible new route to quantis-
ing general relativity in four dimensions. In three dimensions, a number of consistent
quantisation schemes for general relativity are known, leading to (at least apparently)
inequivalent theories [31].
But already classically, in the case of general relativity equivalent formulations
for pure gravity can give non-equivalent theories when coupled to matter. We en-
countered an instance of this in chapter 3 when discussing Einstein-Cartan theory;
in contrast to conventional general relativity, where torsion is set to zero by fiat, in
Einstein-Cartan theory matter can act as a source of torsion. This is a non-equivalence
on the level of equations of motion which can therefore be tested more easily by ex-
periment. In this chapter we shall consider a reformulation of pure general relativity
only.
Hilbert-
Palatini
L(eµa , ωabµ )
-
δS
δω
= 0⇒ ω = ω(e)
Einstein-
Hilbert
L(eµa)
6
?
Legendre
transform
ADM
Hamiltonian
H(eia, pibj)
Hilbert-Palatini
Hamiltonian
H(ωabi , piiab)
6
?
Legendre
transform
-
solve 2nd class
constraints
BF-Plebanski
(real form)
L(ωabµ , Babµν ,Ξ)
6
?
δS
δΞ
= 0 ⇒ B = B(e)
-
δS
δB
= 0, δS
δΞ
= 0⇒
B = B(ω),Ξ = Ξ(ω)
SO(3, 1)→ SO(3,C) CDJ
Lagrangian
L(η, ωabµ )
Ashtekar
Hamiltonian
H(Eai, Aai)-
complex canonical
transform
@
@
@ILegendre
transform
Figure 4.1: Lagrangians and Hamiltonians for general relativity in four dimensions. (Es-
sentially taken from [131].)
In figure 4.1, essentially taken from the review [131], we show a few possible action
principles for general relativity and their interrelations.∗ The most commonly used
∗Figure 4.1 is by no means meant to be a complete display of all known actions for general
General Relativity From a Constrained Topological Theory 84
ones are the Einstein-Hilbert action [82], the Palatini first order formulation, and its
modification proposed by Holst [83]. This last one is of special interest because it is
the classical, covariant starting point for the canonical quantisation leading to loop
quantum gravity [145]. Even if not the only possible useful one [150], a particularly
popular action in covariant approaches to quantising gravity [122], like the spin foam
[133] and group field theory approach [121], is the formulation as a constrained BF
(or Plebanski [138, 101]) theory. Here one starts from topological BF theory [84] in
four spacetime dimensions, and adds suitable constraints on the two-form B variables
of the theory such that, on solutions of these constraints, the action reduces to the
Palatini or Holst action for general relativity. We will summarise the idea behind this
formulation in the following. In the original Plebanski formulation the constraints
on the B variables are quadratic, and so are the discrete constraints that are then
implemented in the spin foam models based on a simplicial discretisation. On the
other hand, the most recent developments in the spin foam and group field theory
approach to quantum gravity are based on a linear set of discrete constraints, which
can be shown to be slightly stronger, in the restrictions they impose on the original
BF configurations, than some of the original discrete quadratic constraints. Once
more, we will detail this construction in the following.
Here we investigate whether a formulation in terms of linear constraints is also
possible in the classical continuum theory, and what it implies. We will see that the
replacement of the so-called diagonal and cross-simplicity constraints with linear con-
straints at the continuum level is relatively straightforward, after one has introduced
new variables na forming a basis of three-forms at each point. One then needs addi-
tional constraints corresponding to the volume constraints. We will see that one can
linearise these constraints too. We then give a discrete version of these linear volume
constraints (thinking of spacetime which is discretised in a triangulation), which bears
a striking resemblance to the so-called “edge simplicity” constraints of [47]. We note
that only certain linear combinations of the volume constraints one would naively
write down are necessary to constrain the bivectors ΣAB(4) sufficiently. Similarly to
the quadratic case, we will also see that when linear diagonal and off-diagonal con-
straints hold everywhere in a 4-simplex, and one also imposes “closure” constraints
on both bivectors Σab(4) (associated to triangles) and normals na(,) (associated to
relativity; for instance, an interesting variant of the (complex) Plebanski formulation has recently
been explored by Eyo Eyo Ita III (Ph.D. thesis, University of Cambridge (2010)).
General Relativity From a Constrained Topological Theory 85
tetrahedra), the sufficient set of linear combinations of the linear volume constraints
follows. This additional four-dimensional closure constraint on the normal vectors
has, to the best of our knowledge, not been considered or implemented as an ad-
ditional condition in the spin foam literature yet, although it does appear in some
first order formulation of Regge calculus [35], and it plays also a role in the discrete
analysis of [40].
4.3 Constraining BF Theory
Let us briefly review what is known at the classical continuum and discrete level,
concerning the Plebanski formulation of classical gravity. We limit our considerations
to the covariant, Lagrangian context, and to a very small subset of the available
results, those which have been already of direct relevance for quantum gravity model
building, especially in the spin foam context. For recent results in the canonical
Hamiltonian setting, both continuum and discrete, see [28, 47, 160].
In this chapter we consider the Einstein-Hilbert-Palatini-Holst [82, 83] Lagrangian
(without cosmological constant), a modification of the action (3.1.1) used earlier,
SEHPH =
1
8piG
∫
R×R
(
1
2
abcd e
a ∧ eb ∧Rcd[ω] + 1
γ
ea ∧ eb ∧Rab[ω]
)
, (4.3.1)
where spacetime is assumed to be of the form R × R so that a (3+1) splitting can
be performed, ωab is a G-connection one-form (the gauge group G is SO(3, 1) or
SO(4), or an appropriate cover), Rab its curvature, and ea is an R4-valued one-form
representing an orthonormal frame. The term involving γ, known as the Holst term,
is not relevant classically; it is, up to a total derivative†, proportional to T a ∧ Ta,
and hence does not modify the classical equation of motion T a = 0. When (4.3.1) is
coupled to matter, the Holst term leads to a rescaling of the coupling of the matter
spin density to torsion.
Though not relevant classically, the Holst term is of fundamental importance in
loop quantum gravity (LQG), where γ is known as the Barbero-Immirzi parame-
ter, and more generally for any canonical formulation of gravity, as it modifies the
symplectic structure of the theory.
If one introduces a g-valued two-form,
Bab =
1
8piG
(
1
2
abcd e
c ∧ ed + 1
γ
ea ∧ eb
)
, (4.3.2)
†For a discussion of the total derivative term see [117].
General Relativity From a Constrained Topological Theory 86
then the action (4.3.1) becomes‡
S =
∫
Bab ∧Rab[ω] + λαCα[B] , (4.3.3)
i.e. it takes the form of a topological BF theory with additional constraints Cα which
enforce that Bab is indeed of the form (4.3.2), and that are enforced by means of
Lagrange multipliers λα.
As said, BF theory without constraints is topological. Its equations of motion
imply that ωab is flat and the covariant exterior derivative of Bab vanishes. Having no
local degrees of freedom, the quantisation of such a theory is therefore rather simple
and quite well understood. Inspired by this classical formulation, the main issue when
trying to construct a quantum theory related to quantum gravity, in four dimensions,
is then the correct implementation of appropriate constraints that lead to (4.3.2) for
some set of one-forms ea, either at the level of quantum states or in a path integral
formulation. Indeed, the bulk of the work in the spin foam approach [49, 57, 105, 133]
(as well as in the group field theory formalism [16, 121, 123]), in recent years, has
been devoted to this task. These constraints are also the subject of this chapter.
To simplify the following calculations, we introduce another two-form field Σab,
Σab ≡ 1
1− sγ2
(
Bab − γ
2
abcdB
cd
)
, (4.3.4)
where s is the spacetime signature, s = −1 for G = SO(3, 1) and s = +1 for
G = SO(4) (and we assume γ2 6= s)§. This is a linear redefinition which simplifies
the constraint (4.3.2),
Σab =
1
8piγG
ea ∧ eb , (4.3.5)
but leads to more terms in the action. The translation of all calculations from one
set of variables to the other is usually straightforward.
The traditional way to enforce the restriction (4.3.5), the one matching the origi-
nal classical Plebanski formulation of gravity, was to add quadratic simplicity con-
straints¶ to the action [45, 143],
abcdΣ
ab
µνΣ
cd
ρσ = V µνρσ , (4.3.6)
‡We use indices from the beginning of the Greek alphabet such as α to denote abstract indices
not associated with certain transformation groups.
§For uniformity of the discussion, we shall in the following talk about “time” and use the label
0 even when the gauge group is SO(4) and the signature Riemannian.
¶“Simplicity” because a two-form that can be written as a wedge product of one-forms is called
simple.
General Relativity From a Constrained Topological Theory 87
where V can be expressed in terms of Σab by contracting (4.3.6) with µνρσ, to give:
V = s
24
µνρσabcdΣ
ab
µνΣ
cd
ρσ. This is itself a reformulation of the original Plebanski con-
straint, which would read:
µνρσΣabµνΣ
cd
ρσ = V 
abcd , (4.3.7)
and is equivalent to the first under assumption that V 6= 0 everywhere. The version
(4.3.6) has the advantage of permitting a much simpler discretisation and thus a
more straightforward implementation within the spin foam formalism. Under the
same assumption V 6= 0, there are the following four classes of solutions to (4.3.6):
either Σab = ±ea ∧ eb or Σab = ±1
2
abcdE
c ∧ Ed (4.3.8)
for some set of one-forms Ea or ea (in the following, we reserve ea for one-forms
satisfying the first relation, viewed as encoding the metric; since we consider pure
gravity, the factor 8piγG can obviously be introduced by rescaling). One would like
to select only the first class of solutions Σab = +ea ∧ eb, which, when substituted in
the BF action, gives the Holst action (4.3.1). Classically, this is not a severe problem.
As shown in [143], non-degenerate initial data of a solution of the form Σab = +ea∧eb
generically remain within the same branch of solutions. The situation in the quantum
theory, where one necessarily has contributions from all branches, is less clear.
More troublesome, if V = 0, the field Σab does not permit a straightforward
geometric interpretation at all. Since in the region of the phase space where V = 0,
the theory is less constrained, and hence has more degrees of freedom, these non-
geometric configurations should be expected to be dominating in a path integral
[143], unless measure factors are such that this is avoided.
Spin foam models are usually defined in a piecewise flat context, and spin foam
amplitudes are defined for given simplicial complexes [133]. Therefore one is interested
in identifying a discrete version of the above constraints that could be imposed at
the level of each complex. The version (4.3.7) of the simplicity constraints admits
only a rather involved discrete counterpart [45] and, upon quantisation, leads to the
Reisenberger model [45, 142], which has so far received only limited attention.
The discrete analogue of the constraints (4.3.6) led instead [45, 143] to the con-
struction of the Barrett-Crane model [17], in the case in which the Barbero-Immirzi
parameter is excluded from the original action (γ → ∞). The construction is ini-
tially limited to a single 4-simplex, the convex hull of five points in R4 (R1,3, in the
Lorentzian case) with the topology of a 4-ball, whose boundary is triangulated by the
General Relativity From a Constrained Topological Theory 88
five tetrahedra identified by the five independent subsets of four such points, while
subsets of three points identify the four triangles belonging to each of these five tetra-
hedra, each of the triangles being shared by a pair of tetrahedra (see figure 4.2). One
p1 p2
B
B
B
B
B
B











Q
Q
Q
Q
Q
p5 p3
p4









B
B
B
B
B
B
B
B
B








Z
Z
Z
Z
Z
Z
Z
Z
Figure 4.2: A 4-simplex is generated by five points.
then associates a Lie algebra element (bivector) Σab4 ∈ so(4) ' ∧2R4 (similarly in
the Lorentzian case) to each triangle 4 in a given triangulation by integrating the
two-form Σab over 4. The task is then to constrain appropriately these Lie algebra
variables (or their quantum counterpart) following the continuum treatment.
It is useful to split the set of continuum equations (4.3.6) into two sets. Out of
the 21 equations (4.3.6), one first identifies and imposes those 18 which have zero on
the right-hand side (the “diagonal” and “off-diagonal” simplicity constraints),
abcdΣ
ab
µνΣ
cd
µν = abcdΣ
ab
µνΣ
cd
µρ = 0 ∀µ, ν, ρ (no sum over µ, ν) . (4.3.9)
This corresponds to the case if one or two of the indices of the two fields Σ coincide.
At the discrete level, this translates into two triangles on which the same fields are
discretised which either coincide or at least share a single edge, and thus belong to
the same tetrahedron. Thus all bivectors Σab4 are required to satisfy
abcdΣ
ab
4Σ
cd
4 = 0 (diagonal simplicity constraint)
and
abcdΣ
ab
4 Σ
cd
4′ = 0 for all 4,4′ sharing an edge (cross-simplicity constraint).
These two sets of equations can be imposed at the level of each tetrahedron in the
4-simplex.
The remaining three equations (the “volume” constraints) are equivalent to the
requirement that:
abcdΣ
ab
01Σ
cd
23 = −abcdΣab02Σcd13 = abcdΣab03Σcd12 ∝ V (Σ) , (4.3.10)
General Relativity From a Constrained Topological Theory 89
and can be imposed at the discrete level as the requirement that, for each 4-simplex:
abcdΣ
ab
4 Σ
cd
4′ = V for all 4,4′ not sharing an edge (volume constraints)
(4.3.11)
where V is defined by the above equation, and is interpreted, on the solutions of the
constraints, as the volume of the 4-simplex.
An additional condition on the bivectors is usually considered, namely the “clo-
sure” constraint, which states that the sum of four bivectors corresponding to the
faces of one tetrahedron is zero: ∑
4⊂,
Σab4 = 0 . (4.3.12)
This constraint can be understood in two ways. One can either view it as the condition
that the triangles described by the variables Σab4 close to form a tetrahedron [14], or
as a consequence of the equations of motion. In a topologically trivial region such
as the interior of a tetrahedron, a flat connection can be set to zero by a gauge
transformation. Then using Stokes’ theorem, the integral over the equation dΣab =
0 can be written as
∫
,
Σab = 0, which is the closure constraint. The canonical
counterpart of this condition is the so-called Gauss constraint, which generates local
gauge (rotation) transformations and is to be imposed on the quantum states of the
theory.
The same picture appears in three spacetime dimensions, where there are no sim-
plicity constraints and one directly deals with a su(2) connection one-form ωa and an
su(2)-valued (using su(2) ' R3) one-form ea. Here the equation dea = 0 is integrated
over a (spacetime) triangle to give a closure constraint. The vectors associated to
the edges of the triangle add up to zero, and thus have a consistent geometric in-
terpretation as edge vectors in R3. In this sense, an n-form with vanishing exterior
derivative and appropriate internal indices can be given a geometric interpretation
as describing n-simplices closing up to form an (n + 1)-simplex. We shall encounter
another instance of this statement later on.
The closure constraint, being linear in the Σ’s and local in each tetrahedron, is
obviously easier to impose at the discrete level, and in the quantum theory, than the
volume constraints. Thus it is a useful fact that it can indeed be imposed instead
of them. More precisely, it can be shown [105] that the volume constraints in each
4-simplex are implied if one has enforced the diagonal and cross-diagonal simplicity
constraints, plus the closure conditions everywhere, i.e. in all the tetrahedra of the
General Relativity From a Constrained Topological Theory 90
4-simplex (in general, i.e. for non-degenerate 4-simplices, involving tetrahedra be-
longing to different “time slices”). From a canonical perspective, this observation is
usually phrased as an interpretation of the volume constraints as “secondary con-
straints” required to guarantee conservation of the other constraints (including the
Gauss (closure) constraint) under time evolution.
After a period of investigations, several potentially worrying issues have been put
forward regarding the Barrett-Crane model [1, 19] (for a more recent analysis of
the geometry of the Barrett-Crane model, see [16]), and have given impetus to the
development of alternative spin foam models [49, 57]. These models are known to have
nice semiclassical properties [18], and, importantly, generalise the spin foam setting
to include the Barbero-Immirzi parameter at the quantum level (for an early attempt,
see [107]), and thanks to this allow for a more direct contact with the canonical loop
quantum gravity. Their study is still somewhat preliminary, but the above properties
make them promising candidates for a quantum theory related to gravity. One of
the central features of the new models is the replacement of the quadratic simplicity
constraints (4.3.9) by linear constraints of the form
na(,)Σ
ab(4) = 0 ∀4 ⊂ , , (4.3.13)
where na is the normal associated to the tetrahedron , and 4 is any of the faces of
,.
It can be shown that these are lightly stronger than the discrete diagonal and off-
diagonal quadratic simplicity constraints, and remove some of the discrete ambiguity
in the solution for Σab: out of the classes of solutions (4.3.8), one can restrict to (the
discrete version of) Σab = ±ea∧eb only. For a geometric analysis of these conditions in
the discrete setting, see [40, 49, 57], and for a proof that the same discrete conditions
can also lead to the Barrett-Crane model, see [16].
4.4 Linear Constraints for BF-Plebanski Theory
The purpose of this chapter is to investigate whether a formulation in terms of linear
constraints is also possible in the classical continuum theory, and what it implies.
Let us work backwards, at first. Assume that the two-form field Σab is of the form
Σab = Ea ∧ Eb, and that the “frame field” (not necessarily associated with a metric)
Ea is non-degenerate, i.e. that the matrix (Eaµ) is invertible. It follows that
EρaΣ
ab
µν = δ
ρ
µE
b
ν − δρνEbµ . (4.4.1)
General Relativity From a Constrained Topological Theory 91
In order to make a connection to the discrete setting it is more convenient to work
with exterior powers of the cotangent bundle only (n-forms can be integrated over n-
dimensional submanifolds). Hence we multiply (4.4.1) by ρστυ and insert the relation
ρστυE
ρ
a = (detE
µ
a ) adefE
d
σE
e
τE
f
υ , which is true for invertible matrices, obtaining
adefE
d
σE
e
τE
f
υΣ
ab
µν = (detE
a
µ)
(
µστυE
b
ν − νστυEbµ
)
. (4.4.2)
One can define the three-form naστυ ≡ na[στυ] ≡ adefEdσEeτEfυ , so that (4.4.2) take
the form
naστυΣ
ab
µν = (detE
a
µ)
(
µστυE
b
ν − νστυEbµ
)
. (4.4.3)
naστυ can be interpreted as a 3D volume form for the submanifold parametrised by
(xσ, xτ , xυ) embedded in 4D spacetime, whose internal index gives the normal to this
submanifold. If Ea are a basis of one-forms at each spacetime point, then na are a
basis of three-forms at each spacetime point, and so one can choose to work either
with one or the other set of variables. Clearly Ea can be reconstructed from na:
1
6
ρστυnaστυ = s(detE
a
µ)E
ρ
a = s
3
√
det
(
1
6
νστυnbστυ
)
Eρa . (4.4.4)
This means that the set of variables na(x) define a co-tetrad frame at any point of
the spacetime manifold (for the discrete analogue of the above, see [40]).
4.4.1 Linearised Diagonal and Off-Diagonal Constraints
So far we have just rewritten the equation we want to obtain for Σab. Let us now
consider the implications of imposing (4.4.3) as constraints, where we restrict to those
with zero right-hand side, i.e. those for which {µ, ν} ⊂ {σ, τ, υ}. These are half of the
equations (4.4.3). This will identify the continuum analogue of the linear simplicity
constraints.
Claim 4.4.1 For a basis na of three-forms, the general solution to
naστυΣ
ab
µν = 0 ∀{µ, ν} ⊂ {σ, τ, υ} (4.4.5)
is
Σabµν = GµνE
[a
µ E
b]
ν , (4.4.6)
where Eaµ is defined in terms of naστυ as in (4.4.4), and so in particular non-degenerate,
and Gµν = Gνµ and Gµµ = 0. (Obviously, as the variables Σ
ab and Ea, the “coeffi-
cients” Gµν are spacetime dependent.)
General Relativity From a Constrained Topological Theory 92
Proof. First note that we can rewrite (4.4.5) as
adefE
d
σE
e
τE
f
υΣ
ab
µν = 0 (4.4.7)
with Eaµ defined by (4.4.4). Then E
a
µ by assumption defines a basis in the cotangent
space, so that
Σabµν = G
χξ
µνE
a
χE
b
ξ (4.4.8)
for some coefficients Gχξµν with G
χξ
µν ≡ G[χξ][µν]. Substituting this into (4.4.7), we get
0
!
= adefE
a
χE
d
σE
e
τE
f
υG
χξ
µνE
b
ξ = χστυ det(E
a
µ)E
b
ξG
χξ
µν , (4.4.9)
and since det(Eaµ) 6= 0 and Ebξ form a basis of (the internal) R4, this implies that
χστυG
χξ
µν = 0 ∀{µ, ν} ⊂ {σ, τ, υ} . (4.4.10)
It follows that Gχξµν = 0 unless {χ, ξ} = {µ, ν} and so Gχξµν ≡ δ[χµ δξ]ν Gµν .

By a linear redefinition eaµ = λµE
a
µ one might try to set some of the Gµν to a given
value (usually ±1, but one might prefer ± 1
8piγG
), but it is clear that one needs two
conditions on the Gµν for this to be possible.
In the discrete context, one sets na(,) = (1, 0, 0, 0) for each , by a gauge trans-
formation.‖ One could use some of the gauge freedom here to restrict the form
of na: This amounts to finding a convenient parametrisation for the coset space
GL(4)/SO(3, 1). Let us make the (usual) assumption that the normal to hypersur-
faces {t = constant} is indeed timelike. Then one can use the boost part of SO(3, 1)
to set na123 = (C, 0, 0, 0). The remaining SO(3) subgroup can then be used to make
the (3× 3) matrix ni0στ , where i ∈ {1, 2, 3}, upper diagonal, so that one has the form
naστυ ∼


∗ ∗ ∗ ∗
0 ∗ ∗ ∗
0 0 ∗ ∗
0 0 0 ∗

 . (4.4.11)
Clearly, when this form of naστυ is assumed, integrating the three-form na over a
region where t is constant gives a vector in R4 that only has a time component. Its
magnitude specifies the three-dimensional volume of such a region.
‖This seems to involve an implicit assumption, namely that there is a non-degenerate normal to
each tetrahedron, as well.
General Relativity From a Constrained Topological Theory 93
4.4.2 Linearised Volume Constraints
As in the quadratic case, further constraints, in addition to the linear simplicity
constraints (4.4.5), are needed to complete the identification Σab = ±ea ∧ eb.
First of all, one can show the following.
Claim 4.4.2 Under the assumption that all Gµν are non-zero, the necessary and
sufficient conditions for the existence of a linear redefinition eaµ = λµE
a
µ, such that
either Σab = cea ∧ eb or Σab = −cea ∧ eb, where c is a given positive number, are
G12G03 = G01G23 = G13G02( 6= 0) . (4.4.12)
Proof. Set c = 1; the extension to arbitrary c amounts to a further rescaling by√
c. Then the required redefinition is possible if and only if there exist λ0, . . . , λ3,
such that either Gµν = λµλν for all µ 6= ν or Gµν = −λµλν for all µ 6= ν. Clearly
(4.4.12) are necessary. They are also sufficient: Take
λ1 =
√∣∣∣∣G12G13G23
∣∣∣∣, λ2 = sgn
(
G12G13
G23
)
G12
λ1
, λ3 = sgn
(
G12G13
G23
)
G13
λ1
, (4.4.13)
which solves the equations for G12, G13 and G23 with sgn
(
G12G13
G23
)
specifying the
overall sign. The remaining three equations for G01, G02 and G03 are then solved by
the two relations (4.4.12) and
λ0 = sgn
(
G12G13
G23
)
G01
λ1
. (4.4.14)

The assumption Gµν 6= 0 is necessary: One solution to (4.4.12) is G12 = G23 =
G13 = 0 with the other Gµν non-zero, which cannot be expressed as Gµν = ±λµλν .
Further constraints, in addition to the linear simplicity constraints (4.4.5), are
needed to complete the identification Σab = ±ea ∧ eb. One possibility is to use the
quadratic volume constraints (4.4.12). Take the three volume constraints (4.3.10),
abcdΣ
ab
01Σ
cd
23 = −abcdΣab02Σcd13 = abcdΣab03Σcd12 , (4.4.15)
and substitute the solution Σabµν = GµνE
[a
µ E
b]
ν of (4.4.5). This gives precisely (4.4.12).
The non-degeneracy assumption needed for (4.4.12) is then the usual one, namely
V 6= 0 in (4.3.6).
General Relativity From a Constrained Topological Theory 94
This shows that imposing the linear version of the diagonal and off-diagonal sim-
plicity constraints (4.4.5) together with the quadratic volume constraints (4.3.10) and
a non-degeneracy assumption on Σab implies that
Σab = ±cea ∧ eb (4.4.16)
for some set of one-forms ea, where c > 0 can be chosen at will. Thus, linearising the
diagonal and off-diagonal simplicity constraints means that two of the four types of
solutions for Σab are removed, but on the other hand one needs to introduce a basis
of three-forms na at each spacetime point, which is put in as an additional variable.
One also still has to assume V 6= 0.
There is also generically no evolution of initial data with V 6= 0 into a degenerate
Σab with V = 0 and a non-geometric interpretation (this is part of the discussion
of [143]). The geometry of the spacetime manifold is specified by ea and not by Ea
which is only used to determine normals in the constraints.
Alternatively, one might prefer to use a linear version of the volume constraints
as well. Consider the original equation (4.4.3)
naστυΣ
ab
µν = (detE
a
µ)
(
µστυE
b
ν − νστυEbµ
)
, (4.4.17)
which was equivalent to Σab = Ea ∧ Eb for a basis of one-forms Ea. So far we only
considered one half of these equations, namely those with {µ, ν} ⊂ {σ, τ, υ}. The
other half have the form
naντυΣ
ab
µν = (detE
a
µ)µντυE
b
ν , no sum over ν , (4.4.18)
with µντυ 6= 0. One way to read these equations is as the requirement on the left-hand
side to be totally antisymmetric in (µ, τ, υ):
naντυΣ
ab
µν = naνυµΣ
ab
τν = naνµτΣ
ab
υν , no sum over ν . (4.4.19)
We could again try to turn the argument around and impose (4.4.19) as constraints
on a g-valued two-form Σab together with the linear simplicity constraints (4.4.5).
Substituting the solution Σabµν = GµνE
[a
µ E
b]
ν of the linear simplicity constraints into
(4.4.19) gives (after diving by a non-zero factor 1
2
det(Eaµ))
µντυGµνE
b
ν = τνυµGτνE
b
ν = υνµτGυνE
b
ν . (4.4.20)
General Relativity From a Constrained Topological Theory 95
For µντυ 6= 0 this would imply Gµν = Gτν = Gυν . By Claim 4.4.2, imposing (4.4.19)
for one fixed ν, say ν = 0, is generically not sufficient: If we know that G01 = G02 =
G03 6= 0, we still have the condition
G12 = G13 = G23 , (4.4.21)
so that one needs more conditions of the form (4.4.19). These will then imply that
all Gµν are equal, Gµν = c
′ for some c′ that could be positive, negative, or zero. One
can absorb |c′| by an overall redefinition, so that one has
Σab = ±c ea ∧ eb , (4.4.22)
for any chosen c, as before. Note that here it is possible, if c′ = 0 at a point, that
all ea are zero this point and so Σab = 0 as well. While this is a very degenerate
geometry, it is still a geometry.
While the conditions (4.4.19), imposed for all values of ν, are therefore sufficient
to complete the identification of the two-form field Σab as ±cea∧eb, note that (4.4.19)
is a massively redundant set of constraints: In order to obtain at most five relations
on the coefficents Gµν (two relations (4.4.12) if all Gµν are nonzero), we are imposing
eight vector equations! We have not exploited the fact that (4.4.20) is a multiple of
one of the vectors Ebν , which are by assumption linearly independent. We could add
several of the equations (4.4.19) for different ν, instead of considering all equations
for different ν separately. Let us try to impose
∑
ν
∑
{µ,υ}6∈{ν,τ}
naντυΣ
ab
µν = 0, τ ∈ {0, 1, 2, 3} fixed. (4.4.23)
Again substituting the solution Σabµν = GµνE
[a
µ E
b]
ν of the linear simplicity constraints
into (4.4.23), we obtain
1
2
det(Eaµ)
∑
ν
∑
{µ,υ}6∈{ν,τ}
µντυGµνE
b
ν = 0, τ fixed, (4.4.24)
which implies, by linear independence of the Ebν , that indeed Gµν = Gυν for all
τ 6∈ {µ, ν, υ}. It is then sufficient to impose the constraint (4.4.23) for three different
choices of τ , say τ = 0, 1, 2, so that we only need three vector equations instead of
eight.
By absorbing the constant Gµν = c
′ (all Gµν are equal) we rescale all Ea by the
same factor to obtain the variables ea that will have the physical intepretation of frame
General Relativity From a Constrained Topological Theory 96
fields encoding the metric geometry of spacetime. While in the case of quadratic vol-
ume constraints the one-forms Ea, or alternatively the three-forms na, only specified
the normals to submanifolds {xµ = constant}, for linear volume constraints they can
be directly interpreted, up to a position-dependent normalisation, as specifying an
orthonormal basis in the cotangent space.
Note that this implies that one can assume a convenient normalisation for the
one-forms Ea. Instead of just assuming non-degeneracy det(Eaµ) 6= 0, one could fix
det(Eaµ) = 1. This is no restriction of the physical content of the theory as the E
a, for
both linear and quadratic volume constraints, only have a geometric interpretation
after rescaling. One could then interpret Eaµ as a map into SL(4,R). For linear volume
constraints, the relation between the normalised one-forms Ea and the variables ea
that are interpreted as frame fields is a single function on spacetime which may be
viewed as a “gauge” in the sense of Weyl [163].
In contrast to the case of the quadratic volume constraint, no non-degeneracy
assumption on the two-form Σab is needed to enforce simplicity. One might get
Σab = 0 in some region as a solution to the constraints, in which case the action for
this region will be zero. This is analogous to a metric with vanishing determinant
in general relativity and, in contrast to the requirement V 6= 0 outlined above, not
an additional issue. Notice, however, that one still has to assume that the tetrad
field Ea and, equivalently, the co-tetrad field na are non-degenerate, in order for the
simplicity and volume constraints to imply (4.4.16). Failing this, one gets solutions
of the constraints that admit no proper geometric interpretation.
In the end, writing the action for BF theory in terms of Σab,
S =
∫
Bab ∧Rab =
∫
Σab ∧Rab + γ
2
abcdΣ
cd ∧Rab , (4.4.25)
we substitute (4.4.16) into this action, which gives (setting c = 1
8piγG
)
S =
1
8piG
∫
R×R
σ(x)
(
1
2
abcde
a ∧ eb ∧Rcd + 1
γ
ea ∧ eb ∧Rab
)
. (4.4.26)
One is left with a field σ(x) that can take the values ±1, but in the classical theory
one may again argue that if σ = 1 everywhere on an initial hypersurface, there will
be no evolution into σ = −1. What we obtain is first order general relativity where
one uses (det e) = ±| det e| instead of | det e| as a volume element in the action. If σ
is continuous as classical fields usually are assumed to be, this differs from the action
with | det e| by an overall sign at most.
General Relativity From a Constrained Topological Theory 97
To summarise, we have identified both a linear version of the quadratic simplicity
constraints and a linear version of the (quadratic) volume constraints in the con-
tinuum, which can be used to reduce topological BF theory to 4D gravity in the
continuum. We have found also that both linear versions are slightly stronger (i.e.
more restrictive) than the corresponding quadratic constraints, so that the resulting
constrained theory is likely to be closer to gravity at the quantum level than the
one in which quadratic constraints are implemented. We now discuss the discrete
counterpart of the constraints found above.
4.4.3 Discrete Linear Constraints and their Relations
The discrete analogue of (4.4.5) is just the linear constraint used in [49, 57], as desired:
na(,)Σ
ab(4) = 0 ∀4 ⊂ , . (4.4.27)
One could write down also a discrete version of (4.4.19), obtained in the natural
way, demanding that within the same 4-simplex
na(,)Σ
ab(4′) = na(,′)Σab(4′′) (4.4.28)
whenever 4′ 6⊂ , and 4′′ 6⊂ ,′ and the edge shared by 4′ and , is the same as that
shared by ,′ and 4′′.
In the following we adopt the notation of [105], where the tetrahedra in a given
4-simplex are labelled by A,B,C,D,E, so that triangles are represented by AB,AC,
etc., and edges by combinations ABC,ABD, etc. The orientation of the triangles
and tetrahedra in (4.4.28) is then fixed by the signs of the permutations of the letters,
na(,A)Σ
ab(4BC) = −na(,B)Σab(4AC) = na(,C)Σab(4AB), etc. (4.4.29)
In analogy to the continuum case, it will be sufficient to impose, instead of the
full set of conditions (4.4.29), certain linear combinations of (4.4.29) to complete the
geometric interpretation of the bivectors Σab(4). The discrete analogue of the three
continuum equations (4.4.23), where the index e was kept fixed, is to pick one of
the tetrahedra and add those six equations out of (4.4.29) which involve triangles
belonging to this tetrahedron. Starting with A, we impose the constraint
∑
{i,j}63A
na(,i)Σ
ab(4Aj) = 0, (4.4.30)
General Relativity From a Constrained Topological Theory 98
and the equivalent conditions for the tetrahedra B to E, thereby needing to satisfy
only five instead of 20 volume constraints.
The above discrete formulation of the linearised volume constraints resembles
strongly the edge simplicity constraints studied, in a canonical setting, in [47], and it
imposes indeed the same restriction on the discrete data. However, it does not match
exactly any of the various expressions given for these edge simplicity constraints in
[47]. The correspondence between the two, therefore, deserves to be studied in more
detail, given also that edge simplicity constraints have been shown to be crucial for
the kinematical phase space of BF theory (and of loop gravity) to reduce to that of
discrete gravity, in accordance with what we find here in a covariant setting.
In spin foam models such as [49, 57], as we mentioned earlier, only the diagonal
and off-diagonal simplicity constraints, but no quadratic volume constraints (4.3.10)
are imposed. This is because in the discrete setting, one can use the closure con-
straint (4.3.12), imposed in all the tetrahedra in a 4-simplex, to relate the (quadratic)
simplicity constraints to the volume constraints, so that if the former are imposed
everywhere the latter follows. Since the quadratic simplicity constraints follow from
the linear ones, as can be easily checked, this argument is still valid if one uses linear
simplicity constraints.
One might hope that the sufficient set of linear volume constraints (4.4.30) would
also follow from the linear simplicity constraints and the closure constraints. This
is almost the case, but not quite. In fact, one more constraint should be added to
simplicity and closure imposed in the five tetrahedra in the 4-simplex. This is a “4D
closure” constraint of the form
na(,A) + na(,B) + na(,C) + na(,D) + na(,E) = 0 , (4.4.31)
where ,i are the (appropriately oriented) tetrahedra of a given 4-simplex.
Just as for the usual closure constraint (4.3.12), there are two ways to understand
why such a constraint must be imposed. Recall that if one demands the triangles
described by discrete variables Σab4 close to form a tetrahedron, they have to satisfy
(4.3.12). Alternatively, one can start with the continuum field equation ∇(ω)[µ Σabνρ] = 0,
where ∇(ω) is the covariant derivative for the connection ωab, set the (flat) connection
to zero by a gauge transformation, and integrate this over an infinitesimal 3-ball
(whose triangulation is a tetrahedron).
The new constraint (4.4.31) seems to be the analogous statement that tetrahedra
close up to form a 4-simplex. By Hodge duality ∧1R4 ' ∧3R4 and any internal cov-
General Relativity From a Constrained Topological Theory 99
ector na(,) can be mapped to a three-form; unlike for two-forms, any three-form can
be written as E1 ∧ E2 ∧ E3 for some eα. Demanding that the tetrahedra described
by these three-forms form a closed surface is then (4.4.31). Thus the simplicial geo-
metric reasoning goes through also for this new constraint. In terms of the equations
of motion of the theory, on the other hand, the only argument for the need of this
constraint is the following. If ∇(ω)[µ Σabνρ] = 0 and we assume that Σab = ±ea ∧ eb, then
it follows that ∇(ω)[µ naνρσ] = 0 as well. Integrating this equation (with the connection
again set to zero) over a 4-ball (which can be thought of as our 4-simplex) whose
boundary is a 3-sphere, triangulated by tetrahedra, then leads to (4.4.31). We then
however have to assume simplicity of Σab. A more direct derivation of (4.4.31) from
the equations of motion would be desirable.
A B
B
B
B
B
B
B











Q
Q
Q
Q
Q
E C
D









B
B
B
B
B
B
B
B
B








Z
Z
Z
Z
Z
Z
Z
Z
Figure 4.3: Dual picture of a 4-simplex; points represent tetrahedra, lines represent trian-
gles shared by adjacent tetrahedra.
The role of this constraint, anyway, is the following. Consider the closure con-
straint
Σab(4AB) + Σab(4AC) + Σab(4AD) + Σab(4AE) = 0. (4.4.32)
Contracting with na(,B) gives, using the linear simplicity constraint na(,B)Σ
ab(4AB) =
0,
na(,B)Σ
ab(4AC) + na(,B)Σab(4AD) + na(,B)Σab(4AE) = 0. (4.4.33)
Alternatively, one may start with the 4d closure constraint and contract with ΣAB(4AB)
to get, again using the linear simplicity constraints,
na(,C)Σ
ab(4AB) + na(,D)Σab(4AB) + na(,E)Σab(4AB) = 0. (4.4.34)
In total one obtains 20 + 10 = 30 equations of this kind that can be used to express
some of the combinations na(,)Σ
ab(4) in terms of others. Substituting the result-
ing expressions into the five discrete volume constraints (4.4.30) one finds that the
General Relativity From a Constrained Topological Theory 100
equations (4.4.30) indeed follow from the relations (4.4.33) and (4.4.34). We have
seen in the continuum that the summed constraints (4.4.23) are sufficient to identify
ΣAB = ±eA∧eB, and hence we find that in the discrete case the situation is analogous
to the case of quadratic constraints in that a sufficient set of volume constraints can
be viewed as secondary.
To see more clearly what happens in both our construction and in the case of
quadratic constraints analyzed in [105], note that in our linear case one could use the
3d and 4d closure constraints to express the variables na(,E) and Σ
ab(4AE),Σab(4BE),
Σab(4CE),Σab(4DE) in terms of the others. Taking the linear simplicity constraints
into account, one is then left with twelve independent combinations na(,)Σ
ab(4),
just as in the continuum. In the continuum, we saw that one can impose the three
additional constraints (4.4.23) on the twelve contractions naµτυΣ
ab
µν to complete the
identification Σab = ±ea ∧ eb. In the discrete case, one has the following three addi-
tional conditions coming from linear cross-simplicity constraints:
0 = na(,E)Σ
ab(4AE)
= na(,B)Σ
ab(4AC) + na(,B)Σab(4AD) + na(,C)Σab(4AB)
+na(,C)Σ
ab(4AD) + na(,D)Σab(4AB) + na(,D)Σab(4AC) (4.4.35)
and similar ones coming from na(,E)Σ
ab(4BE) = 0 and na(,E)Σab(4CE) = 0. These
are precisely the analogue of the continuum constraints (4.4.23).
Similarly, in the case of quadratic simplicity constraints, one can use 3d closure
to eliminate Σab(4AE),Σab(4BE),Σab(4CE),Σab(4DE). Then one observes that ad-
ditional quadratic cross-simplicity constraints give expressions such as
0 = abcdΣ
ab(4AE)Σcd(4BE)
= abcdΣ
ab(4AC)Σcd(4BD) + abcdΣab(4AD)Σcd(4BC) (4.4.36)
which are equivalent to the desired (two) volume constraints.
All of this is an exercise in solving a system of linear equations for which there
might be a more simple and elegant description, but the upshot is the following.
The sufficient set of linear volume constraints (4.4.23) does indeed follow from the
linear simplicity constraints and the closure constraints, once one also imposes a
four-dimensional closure constraint on the normals to tetrahedra that seems very
natural in light of their geometric interpretation. Just as in the formulation in terms
of quadratic simplicity constraints [105], the volume constraints can be viewed as
General Relativity From a Constrained Topological Theory 101
secondary constraints that imply conservation of the simplicity constraints in time,
or put differently, the volume constraints follow if the simplicity constraints hold
everywhere. Once more this strenghtens the relationship between the discrete linear
volume constraints we have identified and the edge simplicity constraints of [47].
4.5 Lagrangian and Hamiltonian Formulation
Let us briefly outline the Lagrangian formulation of 4D gravity resulting from our
linear constraints added to BF theory. One adds the linear simplicity and volume
constraints to the action of BF theory using Lagrange multipliers:
S =
∫
d4x
(
1
4
µνρσΣabµνRabρσ[ω] +
γ
8
µνρσabcdΣ
ab
µνR
cd
ρσ[ω] + Ξ
µνστυ
b naστυΣ
ab
µν
)
,
(4.5.1)
where the Lagrange multiplier field Ξµνστυb satisfies Ξ
µνστυ
b ≡ Ξ[µν][στυ]b , and
µτυΞ
µνντυ
b = 0 (no sum over ν) . (4.5.2)
Indeed, varying with respect to Ξb then gives back the constraints
naστυΣ
ab
µν =

0, {µ, ν} ⊂ {σ, τ, υ},µτυf bν , ν = σ (for some f bν) . (4.5.3)
Note that the second line corresponds to the set of constraints (4.4.19) and not to
the summed version (4.4.23), and that it is clearly sufficient for the geometric inter-
pretation of Σab. The field equation from varying with respect to the connection ω is
the usual
∇(ω)[µ Σabνρ] = 0 , (4.5.4)
where ∇ is the covariant derivative for the connection ωab. The remaining equations
involve the Lagrange multipliers, as would be expected:
1
4
µνρσRabρσ[ω] +
γ
8
µνρσabcdR
cd
ρσ[ω] + Ξ
µνστυ
[b na]στυ = 0 , Ξ
µνστυ
b Σ
ab
µν = 0 . (4.5.5)
We have seen that the constraints imply that Σab = ±ea ∧ eb, and when substituting
this back into the action one will recover general relativity, modulo the possible sign
ambiguity we have already discussed.
We leave a complete Hamiltonian analysis of this theory to future work. However,
we note a feature of the theory that follows directly from the use of linear constraints,
and from the introduction of the additional variables na.
General Relativity From a Constrained Topological Theory 102
As in unconstrained BF theory the initial dynamical variables will be the spatial
part of the connection ωabk and its conjugate momentum P
k
ab ≡ 12ijkΣabij. We also
saw that the equation of motion ∇(ω)[µ Σabνρ] = 0 is unaffected by the constraints. Hence
there will be Gauss constraints of the form
Gcd ≡ ∂iP cdi + ωceiP edi + ωdeiP cei (4.5.6)
on the canonical momenta. Their role is to generate G gauge transformations.
Looking at the action (4.5.1), one would already require that (4.5.6) should be
modified to generate gauge transformations on the normals naστυ; (4.5.1) is only
invariant under gauge transformations if the three-forms naστυ are transformed. The
need for such a modification is also seen if one computes Poisson brackets between
the linear simplicity and Gauss constraints. Define “smeared” constraints
C[Ξ] :=
∫
Ξij,στυb naστυijkP
abk , G[Λ] :=
∫
ΛcdGcd . (4.5.7)
One then finds that
{C[Ξ],G[Λ]} = −
∫
δC[Ξ]
δP abk
δG[Λ]
δωabk
= −
∫
Ξij,στυB naστυijk
[
ΛadP bd
k − ΛbdP adk
]
= −C[Λ · Ξ]−
∫
Ξij,στυB Λ
adnaστυijkP
b
d
k
, (4.5.8)
where (Λ ·Ξ)ij,στυd = ΛdbΞij,στυb . The first term alone would imply that G[Λ] generates
gauge transformations, but the second term is an unwanted extra piece. For G to be
a generator of gauge transformations, it must be first class (i.e. commute with other
constraints up to linear combinations of constraints). We can remedy this by adding
the variables naµνρ to the phase space, together with their conjugate momenta pi
aµνρ.
Now we can define a new Gauss constraint
G ′cd ≡ Gcd − n[cµνρpid]µνρ . (4.5.9)
Then, computing the Poisson brackets of the new Gauss constraint with C[Ξ], one
finds
{C[Ξ],G ′[Λ]} = {C[Ξ],G[Λ]} −
∫
Ξij,στυB Λ
cancστυijkP
abk = −C[Λ · Ξ] , (4.5.10)
as desired. We have however increased the number of phase space variables at each
point by 32.
General Relativity From a Constrained Topological Theory 103
A similar reformulation of the Gauss constraint, leading to a relaxation of the
gauge invariance properties of spin network states, has been already suggested by the
Hamiltonian analysis of the Plebanski theory [28], and it has been advocated in the
loop quantum gravity context in [3, 106] as well as spin foam and group field theory
context [2, 16, 123].
4.6 Summary and Outlook
We have investigated a formulation of classical BF-Plebanski theory where the con-
straint Σab = ±ea ∧ eb, needed to reproduce general relativity in four dimensions,
starting from topological BF theory, is imposed through constraints linear in the
bivector field Σab. The discrete counterpart of a part of these linear constraints (the
“simplicity constraints”), in fact, has proven very useful in the spin foam approach
to quantum gravity [49, 57].
The corresponding continuum constraints have been easily identified, and can
indeed be used to replace the quadratic “diagonal” and “off-diagonal” parts of the
simplicity constraints appearing in the Plebanski formulation. As in the discrete case,
one needed to introduce a new set of variables na which are assumed to form a basis
of three-forms at each point of spacetime, and are slightly stronger than the quadratic
constraints: they eliminate two of the four sectors of solutions that are present for
quadratic constraints.
In the second part of the analysis we found that the quadratic volume constraints
of the Plebanski formulation, needed to complete the identification Σab = ±ea∧eb, can
also be replaced by linear constraints, which again are stronger than their quadratic
analogues. They do not require an additional non-degeneracy assumption on Σab.
However, a non-degeneracy assumption on the three-forms na is still necessary, and
only when this is imposed one can hope to eliminate all “non-geometric” degenerate
configurations for Σab, which are feared to dominate the quantum theory in the case
of quadratic volume constraints. Also, while for quadratic volume constraints the
variables na merely specify normals to submanifolds {xµ = constant} and hence can
be independently rescaled arbitrarily at each point, for linear volume constraints
they directly specify, up to an overall rescaling, the frame field encoding the metric
geometry, i.e. an orthonormal basis in the cotangent space at each spacetime point.
We have then analysed the discrete (simplicial) translation of the linear constraints
General Relativity From a Constrained Topological Theory 104
we identified. In the context of spin foams, the quadratic volume constraints follow
from imposing the (quadratic) diagonal and off-diagonal simplicity constraints every-
where together with closure constraints on the discrete variables Σab(4). We have
shown a similar property for the linear volume constraints. If (linear) diagonal and
off-diagonal simplicity constraints and closure constraints for both bivector variables
Σab(4) and normals na(,) are imposed everywhere, a sufficient set of linear volume
constraints follows. This means that “non-geometric” bivector configurations can-
not appear if the additional closure constraint on the normals holds, and the same
normals are assumed to be non-degenerate.
We have not performed a complete Hamiltonian analysis of the resulting linear
constrained BF action for gravity, but only noted that the use of linear simplicity and
volume constraints immediately requires a modification of the usual Gauss constraint
to generate a transformation of the normal 3-form variables na alongside that of
the Σ’s; a similar relaxation of the Gauss constraint, which translates at the spin
foam and discrete gravity level into a closure constraint for simplices, and in the
canonical quantum gravity context into a generalisation of spin network states, has
been suggested on more than one occasion in the literature [2, 3, 16, 106], even if
its proper implementation at the quantum level has not been yet developed. On
the classical level, therefore, a full Hamiltonian analysis of the constraints would be
highly desirable. This would involve adding momenta for the components Σab0i , which
are Lagrange multipliers in unconstrained BF theory, as well as those for the normals
na we have introduced, so that all variables can transform nontrivially under G gauge
transformations generated by a modified Gauss constraint, as shown.
Still at the classical level, but with obvious implications for the quantisation,
one aspect of our construction that deserves further work is the relation between the
discretised linear volume constraints we have found and the edge simplicity constraints
used in [47]. As noted, the two sets of constraints appear to be very similar, and their
role in the classical theory is the same, in particular, they remove (partly) the non-
geometric configurations from the configuration space (or phase space) of the theory
and appear as “secondary” in the sense specified above. So it natural to conjecture
that one is simply a reformulation of the other. The implications for the quantum
theory are not only due to the dominant role that non-geometric configurations may
play in the quantum theory, if not removed, but also in the fact that one discrete
formulation of these constraints can actually be simpler to implement in a spin foam
General Relativity From a Constrained Topological Theory 105
context than the other.
The possible use of our findings in the spin foam and group field theory con-
text, and more generally in any quantisation based on the formulation of gravity
as a constrained BF theory, are in fact most interesting. In particular, it seems to
be important to explore how a closure constraint on normals could be implemented
into existing spin foam models, given that we found it to be necessary for the full
imposition of the geometric constraints on the variables of topological BF. A conve-
nient setting to do so could be the GFT formulation of [16], since there the simplicial
geometry and the contact with classical actions is brought to the forefront.
106
Part II
Gauge Symmetries and CP
Violation
SU(3) and its Quotients 107
Chapter 5
SU(3) and its Quotients
5.1 Introduction
The Lie group that will play a central role in this second part of the thesis is the special
unitary group SU(3). In the standard model of particle physics, this group plays two
important roles: Firstly as the (colour) gauge group of quantum chromodynamics
(QCD), the version of Yang-Mills theory describing the strong interactions, secondly
as the group of flavour symmetry which relates different generations of quarks, fa-
mously introduced in Gell-Mann’s “Eightfold Way” [60]. We shall in the following
focus on the electroweak sector of the standard model [72, 146, 161], where the gauge
group is SU(2) × U(1)/Z2 ' U(2) [132], and possible extensions of it, where one
adds another SU(2) group acting on “right-handed” fields. We are interested in CP
violating processes which can change quark flavour, which are again described by an
element of SU(3), the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Ambiguities
in the definition of the bases with respect to which this matrix is defined mean that
all matrices obtained from a given element of SU(3) by left or right multiplication
by an element of the maximal Abelian subgroup U(1)2 should be regarded as equiva-
lent to the original matrix. Hence the space of CKM matrices is the double quotient
U(1)2\SU(3)/U(1)2. In chapter 6 we will make statements about statistics of the
Jarlskog invariant J , a measure of the magnitude of CP violation, which assume a
choice of measure on this quotient space. In this chapter we study the geometry of
SU(3) and its quotients to motivate different possible choices of measure.
We will also discuss measures on the space of Hermitian and complex 3×3 matrices
whose construction relies on the knowledge of measures on quotients of SU(3), and
SU(3) and its Quotients 108
which become important if one considers the space of quark mass matrices.
Let us first introduce coordinates on SU(3) that we find convenient. The Lie
algebra su(3) is generated by (i times) the Gell-Mann matrices
λ1 =


0 1 0
1 0 0
0 0 0

 , λ2 =


0 −i 0
i 0 0
0 0 0

 , λ4 =


0 0 1
0 0 0
1 0 0

 ,
λ5 =


0 0 −i
0 0 0
i 0 0

 , λ6 =


0 0 0
0 0 1
0 1 0

 , λ7 =


0 0 0
0 0 −i
0 i 0

 ,
λ3 =


1 0 0
0 −1 0
0 0 0

 , λ8 = 1√3


1 0 0
0 1 0
0 0 −2

 . (5.1.1)
To make the passage from SU(3) to the single or double quotients straightforward,
we note that any element of SU(3) can be written as
U = AL V AR , (5.1.2)
where
AL = e
i
2
(3p−q)λ3+ i
√
3
2
(p+q)λ8 , AR = e
itλ3+i
√
3rλ8 , (5.1.3)
and
V = eixλ7 e−iwλ3 eiyλ5 eiwλ3 eizλ2 , (5.1.4)
where the ranges of the Euler angles x, y, z and the complex phases w, p, q, r, t are
0 ≤ x , y , z ≤ pi
2
, 0 ≤ w , p , q , r , t < 2pi . (5.1.5)
Explicitly, the matrix V takes the form
V =


cos y cos z cos y sin z e−iw sin y
− cos x sin z − eiw sin x sin y cos z cos x cos z − eiw sin x sin y sin z sin x cos y
sin x sin z − eiw cos x sin y cos z − sin x cos z − eiw cosx sin y sin z cos x cos y

 ,
(5.1.6)
whereas the diagonal matrices AL and AR are
AL = diag(e
2ip, ei(q−p), e−i(p+q)) , AR = diag(ei(r+t), ei(r−t), e−2ir) . (5.1.7)
It should be clear the orbits of the left action of the subgroup U(1)2 are parametrised
by (p, q), while the orbits of the right action are parametrised by (r, t). Therefore one
SU(3) and its Quotients 109
may use (x, y, z, w) to parametrise the double quotient U(1)2\SU(3)/U(1)2. Indeed,
(5.1.6) is commonly used in the literature as a parametrisation of the CKM matrix.
We may now obtain left-invariant forms on SU(3) by computing the Maurer-
Cartan form
U−1dU = iλa σa . (5.1.8)
The resulting expressions are somewhat involved, and we use the usual complex nota-
tion to slightly shorten the expressions (taking real and imaginary parts one recovers
the σi which are all real),
σ1 + iσ2 = ie
2it ω − e2it sin y cos(2z)(sinw dx+ cosw sin(2x) dq) + 1
4
e2it sin(2z) Ω ,
σ3 =
1
4
cos(2z) Ω + dt+ sin y sin(2z) (sinw dx+ cosw sin(2x) dq) ,
σ4 + iσ5 = ie
i(3r+t+w) cos z dy − cos y sin z(iei(3r+t) dx+ ei(3r+t) sin(2x) dq)
+ei(3r+t+w) cos y sin y cos z (3 dp− dw + cos(2x) dq) ,
σ6 + iσ7 = ie
i(3r−t+w) sin z dy + cos y cos z(iei(3r−t) dx+ ei(3r−t) sin(2x) dq)
+ei(3r−t+w) cos y sin y sin z(3 dp− dw + cos(2x) dq) , (5.1.9)
σ8 =
√
3
4
[−dp+ 4 dr + dw + 2 dq cos(2x) cos2 y + (3 dp− dw) cos(2y)] ,
where we defined the 1-forms
ω = dz + dx cosw sin y − 2 dq cos x sinw sin x sin y ,
Ω = 3 dp+ dw + dq cos(2x)(cos(2y)− 3) + (3 dp− dw) cos(2y) . (5.1.10)
A general left-invariant metric on the group will have the form
ds2 = gabσaσb (5.1.11)
for some constant symmetric matrix gab. Depending on the choice of gab, the resulting
metric will have a certain subgroup of SU(3) as isometries acting by right multipli-
cation on SU(3). The possible cases are detailed in [41]. The most symmetric case is
the bi-invariant metric, invariant under SU(3)L × SU(3)R, which results from taking
gab proportional to δab.
Since we will be mainly interested in the measure, we note that for any choice of
gab the (left-invariant) volume form is, up to an (irrelevant) overall constant,
µ
SU(3)
l.inv. ∝ σ1 ∧ σ2 ∧ σ3 ∧ σ4 ∧ σ5 ∧ σ6 ∧ σ7 ∧ σ8 (5.1.12)
=
3
√
3
2
sin(2x) sin y cos3 y sin(2z) dx ∧ dy ∧ dz ∧ dw ∧ dp ∧ dq ∧ dr ∧ dt ,
SU(3) and its Quotients 110
which is independent of the complex phases w, p, q, r and t. Clearly left invariance
fixes the volume element up to an overall constant: Pick a group element g (say the
identity), then the space of 8-forms at g is one-dimensional. The volume form at any
other group element is then determined to be the push-forward of the volume form
at g under left translation. The volume form (5.1.12) will be the central quantity in
many of the following calculations.
One could of course have started demanding right-invariance, computing the right-
invariant forms from
dU U−1 = iλa τa . (5.1.13)
If the left- and right-invariant measures are identical the group is called unimodular.
Compact and semisimple groups are unimodular. Hence we can use left and right
invariance as interchangeable conditions on a natural measure on SU(3).
If G is compact and semisimple, (the negative of) the Killing form is a bi-invariant
Riemannian metric which is also an Einstein metric [9]. It may therefore be regarded
as the most natural choice of metric on such a group. For a (complex) matrix Lie
group this metric is defined by
ds2 = Tr(dg dg†) . (5.1.14)
It can also be viewed as the induced metric on the group G, interpreted as a sub-
manifold of Cn×n with flat metric. (The same considerations are of course true for a
group of real matrices.)
We will use this metric as the most natural choice later on. It does give the same
measure as any left-invariant metric, as we have noted. For completeness, we give its
form explicitly for SU(3):
ds2 =
1
2
Tr
(
dU dU †
)
= δabσaσb
= 3 dp2 + dq2 + 3 dr2 + dt2 +
3
2
(3 cos(2y)− 1) dp dr
+3 cos2 y (cos(2z) dp dt+ cos(2x) dq dr)
+
1
2
[cos(2x) cos(2z) (cos(2y)− 3) + 4 sin(2x) sin(2z) sin y cosw] dq dt
− sin2 y (3 dp− 3 dr + cos(2x) dq − cos(2z) dt)dw
+2 sin y sinw (sin(2z) dt dx− sin(2x) dq dz)
+dx2 + dy2 + dz2 + sin2 y dw2 + 2 sin y cosw dx dz . (5.1.15)
SU(3) and its Quotients 111
5.2 The Homogeneous Space SU(3)/U(1)2
The coset space SU(3)/U(1)2 is a reductive geometry as explained in section 1.1:
The splitting of su(3) into an Abelian subalgebra u(1)2 and the orthogonal subspace
is invariant under the adjoint action of SU(3). We may hence identify an appropriate
subset of generators of SU(3) and use the corresponding one-forms to define metrics
on the homogeneous space SU(3)/U(1)2.
The one-forms σ3 and σ8 correspond to the Gell-Mann matrices generating the
Abelian subgroup U(1)2. We can construct left-invariant metrics on SU(3)/U(1)2
from the remaining six left-invariant forms σ1, σ2, σ4, . . . , σ7. The left group action of
SU(3) on the homogeneous space again guarantees the existence of a unique (up to
a constant) left-invariant volume form, and we find that this unique volume form is
µ
SU(3)/U(1)2
l.inv. ∝ σ1 ∧ σ2 ∧ σ4 ∧ σ5 ∧ σ6 ∧ σ7 (5.2.1)
=
3
√
3
2
sin(2x) sin y cos3 y sin(2z) dx ∧ dy ∧ dz ∧ dw ∧ dp ∧ dq ,
where we note that, due to the SU(3) decomposition (5.1.2), we may parametrise
the quotient SU(3)/U(1)2 by the SU(3) coordinates (x, y, z, w, p, q). This is of course
the same volume form as the one obtained by starting with the left-invariant volume
form on SU(3) and integrating over the coordinates r and t, or by just considering a
slice {r = constant, t = constant} in SU(3) with the measure (5.1.12). (We shall see
in the next section that this need not be true if one reduces to the double quotient
U(1)2\SU(3)/U(1)2.)
There is an additional possible source of non-uniqueness of the measure, namely
a possible ambiguity of the definition of the same manifold as different homogeneous
spaces X = G/H = G′/H ′. In practice, this does not seem to lead to different results
for the measure.
It should be noted that the homogeneous space considered here, where the sub-
group is the maximal Abelian subgroup, the “maximal torus”, of a compact group, is
what is known as a flag manifold. Such a manifold is naturally a Ka¨hler manifold. For
any G-invariant complex structure on a flag manifold, there is a unique G-invariant
Ka¨hler-Einstein metric [9]. This metric may then be regarded as the most natural
choice on such a homogeneous space. An explicit construction for such metrics on
spaces SU(M)/U(1)M−1, where U(1)M−1 is the maximal torus of SU(M), is given in
[136]. The case M = 6 might be of interest for neutrino mixing; here the analogue of
the CKM matrix is the Maki-Nakagawa-Sakata matrix [114], usually taken to be
SU(3) and its Quotients 112
an element of the single quotient U(1)2\SU(3). In the see-saw mechanism one adds
very heavy right-handed neutrinos, and the most general mixing matrix would be an
element of U(1)5\SU(6). The construction of [136] would be the starting point of an
analysis of this quotient, which is beyond the scope of this thesis.
5.3 Metrics on U(1)2\SU(3)/U(1)2
The double quotient U(1)2\SU(3)/U(1)2 does not have a natural SU(3) group action,
because the left and right actions of SU(3) on itself do not commute with the Abelian
subgroup U(1)2 and hence do not induce a group action on the double quotient:
If g and g′ are in the same equivalence class, [g] = [g′], then g = hg′h′ for some
h, h′ ∈ U(1)2. For a group action to be defined on the double quotient, one would
need [j · g] = [j · g′], j · g = k(j · g′)k′ = k(j · h−1gh′−1)k′ for some k, k′ ∈ U(1)2. This
can only be done for those j that commute with h−1.
It is therefore less straightforward than in the previous cases to find a geometrically
motivated measure. The simplest way of reducing the left-invariant measure on the
group SU(3) or the homogeneous space SU(3)/U(1)2 to the double quotient would
be to just integrate (5.2.1) over the coordinates p and q, which is trivial since the
volume form (5.2.1) is independent of p and q. One would therefore essentially use
the same measure for the double quotient.
Alternatively, one might start from a metric on either the group or the homoge-
neous space, reduce it in some way to the double quotient, and then determine the
associated volume form, but this introduces two possible sources of ambiguity; the
choice of metric and the choice of method of reduction. We will see that indeed the
measure on the double quotient is vastly non-unique.
For example, one may be tempted to identify the double quotient with the sub-
manifold p = q = r = t = 0 of SU(3), take a symmetric metric on SU(3), say the
bi-invariant metric (5.1.15), and set p = q = r = t = 0. This approach was indeed
followed by Ozsva´th and Schu¨cking [126], and leads to an appealingly simple form of
the metric
ds2 = dx2 + dy2 + dz2 + 2 sin y cosw dx dz + sin2 y dw2, (5.3.1)
but seems not to be justified geometrically: It corresponds to a gauge choice, a choice
of section in the orbits of U(1)2 × U(1)2 over the double quotient, and introduces
SU(3) and its Quotients 113
symmetries (three commuting Killing vectors) into the metric which should not be
present. It seems much better motivated geometrically to do a Kaluza-Klein reduction
of a metric, i.e. to project it orthogonally to the orbits of U(1)2. One writes a given
metric on SU(3) (say) as
ds2 = hij(x) (dy
i + Aiµ(x)dx
µ)(dyj + Ajν(x)dx
ν) + g˜µν(x)dx
µ dxν , (5.3.2)
and identifies g˜µν as a metric on the quotient space parametrised by (x
µ).
To understand what is going wrong in (5.3.1), we compare with the simpler case
of SU(2). The bi-invariant metric on SU(2) is
ds2 = (dψ + cos θ dφ)2 + dθ2 + sin2 θ dφ2 , (5.3.3)
where ∂
∂φ
generates the left action of U(1) and ∂
∂ψ
generates the right action of U(1).
Projecting the metric orthogonally to the orbits of right translations, a` la Kaluza-
Klein, gives the round metric
ds2 = dθ2 + sin2 θ dφ2 (5.3.4)
on SU(2)/U(1) = S2. By contrast, simply setting dψ = 0 (the analogue of the
construction of Ozsva´th and Schu¨cking) instead gives the flat metric
ds2 = dθ2 + dφ2 . (5.3.5)
The round metric (5.3.4) is invariant under SO(3). The flat metric (5.3.5) appears
to be invariant under the Euclidean group, with ∂
∂θ
and ∂
∂φ
having the appearance of
translations, but these are only local symmetries since φ is a periodic coordinate and
θ lies in an interval.
The example of SU(2) also illustrates the difference between taking the left-
invariant measure on the homogeneous space and a measure obtained by Kaluza-Klein
reduction of a metric to a biquotient. The biquotient U(1)\SU(2)/U(1) = U(1)\S2 is
just an interval. Its metric becomes, after performing another Kaluza-Klein reduction
of (5.3.4),
ds2 = dθ2 . (5.3.6)
The measure would be dθ, and not dθ sin θ as obtained by integrating a function
f(θ) over the coordinate φ. It is apparent from this simple example that there are
inequivalent ways of calculating integrals of a function on a right quotient that is
invariant under the left group action; namely, one can either reduce the metric to
SU(3) and its Quotients 114
obtain a measure on the double quotient or take the measure on the single quotient
and integrate out the left phases.
Performing a Kaluza-Klein reduction of the bi-invariant metric (5.1.15) we find
a rather complicated metric on the double quotient, whose determinant gives the
measure
√
det g˜ ∝ sin 2x sin 2z sin y cos2 y ·
(
(sin2 2x+ sin2 2z) sin2 y
+
1
8
(5 cos 2y − 3) sin2 2x sin2 2z + 1
2
sin 4x sin 4z sin3 y cosw
+
1
8
(3 cos 2y − 5) sin2 2x sin2 2z sin2 y cos2w
)− 1
2
. (5.3.7)
One might think that, starting from a more general (“squashed”) left-invariant
metric on the homogeneous space SU(3)/U(1)2, there might be some which after
Kaluza-Klein reduction give rise to a simpler expression for the measure on the biquo-
tient, but we have only found more complicated ones.
5.4 Hermitian and Complex Matrices
In chapter 6 we will require measures on the space of either all Hermitian or all
complex 3× 3 matrices which will appear as the space of quark mass matrices in the
standard model of particle physics or certain extensions of it. These will involve the
left- (or right-)invariant measure on the homogeneous space SU(3)/U(1)2, as we will
see shortly. Let us start with the space of 3× 3 Hermitian matrices. It has a natural
group action by U(3) by conjugation, corresponding to a change of basis. We would
like to determine a metric invariant under this action.
Any Hermitian matrix can be diagonalised, i.e. written in terms of a real diagonal
and a unitary matrix:
MH = U
†DU , (5.4.1)
It should be clear that U is only defined up to left multiplication by elements of
U(1)2; U should be regarded as an element of the homogeneous space U(1)2\SU(3) '
SU(3)/U(1)2.
The most symmetric measure on the space of Hermitian matrices is induced by
the bi-invariant metric
ds2 = Tr(dMH dMH) = Tr (dD dD) + 2Tr
((
dU U †D
)2 − (dU U †)2D2) (5.4.2)
SU(3) and its Quotients 115
which is clearly invariant under the action of U(3) by conjugation. Define right-
invariant one-forms τa on SU(3) by
dU U † = iλa τa , (5.4.3)
this becomes∗ [with D ≡ diag(D1, D2, D3)]
ds2 = Tr (dD · dD)− 2τaτbTr (λa[D, λb]D)
= dD21 + dD
2
2 + dD
2
3 + 2
{
(D1 −D2)2(τ 21 + τ 22 ) + (D1 −D3)2(τ 24 + τ 25 )
+(D2 −D3)2(τ 26 + τ 27 )
}
. (5.4.4)
The corresponding volume form is
(D1−D2)2(D1−D3)2(D2−D3)2 dD1∧dD2∧dD3∧ τ1∧ τ2∧ τ4∧ τ5∧ τ6∧ τ7 . (5.4.5)
We have explained above that any left- or right-invariant measure on the coset
U(1)2\SU(3) must be given by (5.2.1) and hence there is no need to compute the
τa explicitly. We obtain a Riemannian measure
DMH := (D1 −D2)2(D1 −D3)2(D2 −D3)2 sin(2x) cos3 y sin y sin(2z) dD1 dD2 dD3
× dx dy dz dw dr dt (5.4.6)
on the space of Hermitian 3×3 matrices. Note that we use coordinates (x, y, z, w, r, t)
on U(1)2\SU(3) according to the decomposition (5.1.2). The measure (5.4.6) favours
strongly non-coinciding eigenvalues.
From (5.4.1), it is apparent that each Hermitian matrix with three distinct eigen-
values is associated with six different elements of R3 × U(1)2\SU(3), related by the
action of the discrete group S3:
MH = U
†DU = (U †P−1)PDP−1(PU) =: U˜ †D˜U˜ , P ∈ S3 , (5.4.7)
where S3 is the symmetric group of degree 3 (the dihedral group of order 6, sometimes
denoted by D3 or D6) which permutes the canonical basis vectors of R
3. The set of
matrices with coinciding eigenvalues has zero measure and hence can be ignored in
the present discussion.
Thus we need to consider the space R3× (U(1)2×S3)\SU(3) instead†, restricting
the coordinates on the quotient U(1)2\SU(3) to an appropriate range to pick one of
∗Compare with the corresponding result for real matrices given in [71]
†Note that the space of Hermitian matrices is not topologically identified with this space; we
have discarded a subset of Hermitian matrices which has zero measure for our construction.
SU(3) and its Quotients 116
the six matrices related by the S3 action. We can use the fact that the S3 action
permutes the rows of an SU(3) matrix to demand that the elements of the third
column (cf. (5.1.6); note that absolute values will not depend on rephasing, i.e. right
multiplication by an element of U(1)2) satisfy the relation
| sin y| ≤ | sin x cos y| ≤ | cos x cos y| , (5.4.8)
which restricts the coordinates x and y to
0 ≤ y ≤ arctan(sin x) , 0 ≤ x ≤ pi
4
. (5.4.9)
Using the right-invariant measure (5.2.1) on U(1)2\SU(3), we see that this region
has precisely one-sixth of the total volume:
pi/2∫
0
dz
pi/4∫
0
dx
arctan(sinx)∫
0
dy sin 2x cos3 y sin y sin 2z
pi/2∫
0
dz
pi/2∫
0
dx
pi/2∫
0
dy sin 2x cos3 y sin y sin 2z
=
1
6
. (5.4.10)
Notice that while the total volume of (U(1)2 × S3)\SU(3) will be finite, the in-
tegrals over the R3 part will diverge; the eigenvalues of arbitrary Hermitian matrices
can of course be arbitrarily large. One would have to introduce some kind of cutoff
to make integrals over the space of Hermitian matrices convergent, as we will do later
on.
The space of 3 × 3 complex matrices can be discussed along similar lines. Here,
one has a left and right action by independent elements of U(3). We should therefore
determine a metric invariant under both of these group actions. An arbitrary complex
matrix can be represented as
Mc = U
†
LDUR , (5.4.11)
where UL, UR are unitary and D = diag(D1, D2, D3) is real diagonal. UL and UR are
only defined up to simultaneous left multiplication by a diagonal unitary matrix
UL → AUL , UR → AUR , A ∈ U(1)3 (5.4.12)
which so that one has the correct number of real parameters, only 18 instead of 21.
Since here the signs of the elements Di can be changed by a different choice of UL
and UR, and one can permute the Di as for Hermitian matrices, there are additional
discrete ambiguities given by elements of the group S3 × Z32. We will see shortly
SU(3) and its Quotients 117
that the measure, similar to the measure for Hermitian matrices, vanishes whenever
elements of D coincide up to sign, so we can restrict to matrices with D21 6= D22 6=
D23 6= D21. Hence we can identify this subspace of the space of 3× 3 complex matrices
with R3 × (U(1)3 × S3 × Z32)\(U(3)× U(3)) ' R3+ × (U(1)3 × S3)\(U(3)× U(3)).
The metric we will use to determine a measure is
ds2 = Tr(dMc dM
†
c ) , (5.4.13)
which clearly is invariant under M → OMO′ for O,O′ ∈ U(3). To evaluate this, use
dMc = −U †L dUL U †LDUR + U †L dDUR + U †LDdUR ,
dM †c = −U †R dUR U †RDUL + U †R dDUL + U †RDdUL (5.4.14)
and [D, dD] = 0 to obtain
Tr(dMc dM
†
c ) = Tr(dD dD)− Tr(D2(dUL U †L)2)− Tr(D2(dUR U †R)2)
+2Tr(DdUL U
†
LDdUR U
†
R) . (5.4.15)
We introduce right-invariant one-forms
dULU
†
L = iλa τ
a
L , dURU
†
R = iλb τ
b
R , (5.4.16)
where λ1, . . . , λ8 are the Gell-Mann matrices, and
λ9 =
√
2
3


1 0 0
0 1 0
0 0 1

 , (5.4.17)
so that iλa are a basis for the Lie algebra u(3). Then
Tr(dMc dM
†
c ) = Tr(dD dD)+Tr(D
2 λ(aλb))(τ
a
Lτ
b
L+τ
a
Rτ
b
R)−Tr(DλaDλb)(τaLτ bR+τaRτ bL) .
(5.4.18)
SU(3) and its Quotients 118
The only nonvanishing traces in (5.4.18) are
Tr(D2λ1λ1) = Tr(D
2λ2λ2) = Tr(D
2λ3λ3) = Tr(Dλ3Dλ3) = D
2
1 +D
2
2 ,
Tr(D2λ(3λ8)) = Tr(Dλ(3Dλ8)) =
1√
3
(D21 −D22) ,
Tr(D2λ(3λ9)) = Tr(Dλ(3Dλ9)) =
√
2
3
(D21 −D22) ,
Tr(D2λ(8λ9)) = Tr(Dλ(8Dλ9)) =
√
2
3
(D21 +D
2
2 − 2D23) ,
Tr(D2λ4λ4) = Tr(D
2λ5λ5) = D
2
1 +D
2
3 ,
Tr(D2λ6λ6) = Tr(D
2λ7λ7) = D
2
2 +D
2
3 ,
Tr(D2λ8λ8) = Tr(Dλ8Dλ8) =
1
3
(D21 +D
2
2 + 4D
2
3) ,
Tr(D2λ9λ9) = Tr(Dλ9Dλ9) =
2
3
(D21 +D
2
2 +D
2
3) ,
Tr(Dλ1Dλ1) = Tr(Dλ2Dλ2) = 2D1D2 ,
Tr(D2λ4λ4) = Tr(D
2λ5λ5) = 2D1D3 ,
Tr(D2λ6λ6) = Tr(D
2λ7λ7) = 2D2D3 . (5.4.19)
The metric can be written in the form
ds2 = dD21 + dD
2
2 + dD
2
3 +
1
2
(D1 −D2)2(τ 1L + τ 1R)2 +
1
2
(D1 +D2)
2(τ 1L − τ 1R)2
+
1
2
[
(D1 −D2)2(τ 2L + τ 2R)2 + (D1 +D2)2(τ 2L − τ 2R)2
+(D1 −D3)2(τ 4L + τ 4R)2 + (D1 +D3)2(τ 4L − τ 4R)2
]
+
1
2
[
(D1 −D3)2(τ 5L + τ 5R)2 + (D1 +D3)2(τ 5L − τ 5R)2
+(D2 −D3)2(τ 6L + τ 6R)2 + (D2 +D3)2(τ 6L − τ 6R)2
]
+
1
2
(D2 −D3)2(τ 7L + τ 7R)2 +
1
2
(D2 +D3)
2(τ 7L − τ 7R)2
+2D21
(
1√
2
(τ 3L − τ 3R) +
1√
6
(τ 8L − τ 8R) +
1√
3
(τ 9L − τ 9R)
)2
+2D22
(
− 1√
2
(τ 3L − τ 3R) +
1√
6
(τ 8L − τ 8R) +
1√
3
(τ 9L − τ 9R)
)2
+2D23
(√
2
3
(τ 8L − τ 8R) +
1√
3
(τ 9L − τ 9R)
)2
. (5.4.20)
The fact that the metric only depends on τ 3L− τ 3R etc., and not on τ 3L+ τ 3R etc., again
reflects the U(1)3 that has to be factored out. One finds that the volume form is
SU(3) and its Quotients 119
proportional to
(D21−D22)2(D21−D23)2(D22−D23)2|D1D2D3| dD1∧dD2∧dD3∧τ 1L∧τ 1R∧. . .∧(τ 8L−τ 8R)∧(τ 9L−τ 9R) .
(5.4.21)
Note that this expression only depends on the absolute values of Di, as expected.
Since the range of the Di is infinite, integration over these coordinates will give an
infinity, as was the case for Hermitian matrices. Note the main difference to the
measure (5.4.6) in the different powers of Di.
Now assume that all quantities we are interested in are independent of the pa-
rameters on UR, which will be the case in the discussion of chapter 6. Then we
can integrate over these coordinates, obtaining a constant which is irrelevant in the
averaging process.
We are then left with integrating over the space of possible matrices UL, the coset
U(1)3\U(3) = U(1)2\SU(3), and the volume form is proportional to
(D21−D22)2(D21−D23)2(D22−D23)2|D1D2D3| dD1∧dD2∧dD3∧τ 1L∧τ 2L∧τ 4L∧τ 5L∧τ 6L∧τ 7L .
(5.4.22)
In terms of the coordinates on U(1)2\SU(3) introduced in (5.1.2), the wedge product
of right-invariant forms gives the usual measure on SU(3), so that we finally get
DMc =
∏
i<j
(D2i−D2j )2|D1D2D3| sin 2x cos3 y sin y sin 2z dD1 dD2 dD3 dx dy dz dw dr dt.
(5.4.23)
The discrete S3 symmetry can be taken into account just as in the previous case:
We integrate only over one sixth of the homogeneous space U(1)2\SU(3), correspond-
ing to
0 ≤ y ≤ arctan(sin x) , 0 ≤ x ≤ pi
4
. (5.4.24)
This restriction amounts to removing unitary matrices that permute the elements of
D and hence to fixing an ordering.
Naturalness of CP Violation 120
Chapter 6
Naturalness of CP Violation
“Well, I don’t like to get involved in these philosophical
issues very much.”
Murray Gell-Mann (attributed)
6.1 Introduction and Motivation
In this chapter we will apply the mathematical theory of measures on quotients of
SU(3), outlined in the previous chapter, to the space of CKM matrices in the elec-
troweak sector of the standard model in order to make statements about naturalness
of the observed magnitude of CP violation in electroweak processes. We will see that
the observed value is rather small when compared to the maximal possible magni-
tude and aim to reach a conclusion if this should be viewed as a fine-tuning problem.
The task of making mathematically precise statements is simplified by the fact that
SU(3) is compact, and so normalisation of measures on SU(3) or a quotient of it will
normally not be problematic.
In modern theoretical physics one often tries to make statements about “natu-
ralness” or “fine-tuning” of the observed values of fundamental parameters, where
fine-tuning of a parameter is interpreted as an indication for incompleteness of the
theory. Popular examples of fine-tuning problems include the quark mass hierarchy
and the cosmological constant problem in particle physics. Since statements about
naturalness are fundamentally of a statistical character, to make them mathemati-
cally precise one has to assume a well-motivated probability distribution on the pa-
rameter space relevant for the theory. A fine-tuning problem then indicates that the
Naturalness of CP Violation 121
probability distribution one has used should be modified by introducing new physical
considerations. As an example, as long as observation was consistent with a vanishing
cosmological constant, it seemed reasonable to assume that a postulated symmetry
would constrain it to vanish. With more recent observations indicating that it must
be taken to be very small and positive, there seems to be an issue of fine-tuning∗. We
note that much of the motivation to extend the standard model of particle physics
is driven by such considerations, that it is by no means necessary to contemplate a
“Multiverse” where all possible values of a given parameter are actually realised. We
also need not consider anthropic arguments, which for the problem at hand would not
give a satisfactory explanation of very weak CP violation; strong CP violation might
be anthropically preferred since baryogenesis necessarily requires CP violation, and
the CP violation in the electroweak sector does not seem sufficient [87].
Of course we can only ever observe and make measurements in a single Universe,
and if a parameter takes a value that appears unlikely maybe this just means that
an unlikely possibility is realised in our Universe. One is merely doing statistics. But
one should recall Bayes’ theorem [20]
P (A|B) = P (B|A)P (A)
P (B)
=
P (B|A)P (A)∑
Ai
P (B|Ai)P (Ai) , (6.1.1)
where one can take A as a hypothesis and B as an observation. In order to calculate
the probability that a hypothesis follows from a given observation, one either needs a
priori probabilities for both A and B and the conditional probability of B assuming A
(which should be calculable), or a priori probabilities for a complete set of hypotheses
{Ai}, together with all conditional probabilities of B given these hypotheses. These
a priori probabilities are referred to as priors. In the second case the sum also has
to be made explicit, and if it is an integral one needs to specify a measure.
In cosmology, the role of priors is particularly important as there is no possibility
to confirm observations in different experimental situations, and P (B) does not seem
accessible. Here A would be a statement about our Universe. While it may be
appropriate to consider P (A) as independent ofA (Laplace’s Principle of Indifference),
a measure on the set of all universes has to be determined [61].
The situation considered in this chapter is similar, as “experiments” would involve
different universes with different values of the fundamental physical parameters. For
instance, one could try to use anthropic arguments, taking B to be the observation
∗For an alternative interesting but presumably non-mainstream viewpoint, see [23]
Naturalness of CP Violation 122
that human beings have evolved in our Universe, to claim that the values of (in this
case) CP violating CKM matrix parameters must take values close to the observed
values (hypothesis A). The conditional probability P (B|A) could in principle be
calculated using baryogenesis etc., but then one still needs a measure on the space
of possible values that A can take. Again, one is forced to pick a measure on the
parameter space. We shall therefore investigate predictions for the magnitude of
CP violation, measured by the Jarlskog invariant J , comparing different choices for
the measure on the parameter space that we regard as natural. The observation by
Kobayashi and Maskawa [98] that CP violation is only possible for at least three
quark families has led to a Nobel prize, but the issue of possible fine-tuning in the
magnitude of CP violation is much less understood.
It is true that, if one is talking about parameters in quantum field theory, they
should really not be regarded as constants, but have an evolution with energy scale
given by renormalisation group equations. However, since fine-tuning means a dis-
crepancy of several orders of magnitude, it may well be that a fine-tuning problem
is present at all energy scales. This is true for the quark mass hierarchy [170], and
also for the case of CP violation: Recent numerical studies [36] indicate that J2 does
not run strongly with energy scale, but that the value at extremely high energies
(∼ 1015GeV) is merely about twice the value at low energies. It is then meaningful
to talk about “naturalness” of the value of such a parameter.
Since any statements one tries to make depend very directly on the choice of mea-
sure, it is helpful if geometric considerations allow for a natural choice of probability
distribution. We saw in the previous chapter that, if the parameter space is a homo-
geneous space G/H for G a compact Lie group and H a closed subgroup, the natural
requirement on the measure determining the probability distribution is invariance
under the left action of G, which leads to a unique measure (up to normalisation).
Probabilities for a given function on G/H to take certain values are then well-defined.
We also saw that if, as in the case of the space of CKM matrices, one has a double
quotient H\G/H, the problem of determining a natural probability distribution is
more involved than for a homogeneous space G/H. We have discussed several possible
choices that we compare here. We will find that while there is no clearly preferred
choice of measure, there always seems to be fine-tuning in the observed value for J ,
unless additional input is used.
In a second part we shall take a different approach, taking the observed values for
Naturalness of CP Violation 123
the quark masses into account by considering not the space of CKM matrices, but the
space of mass matrices, as the fundamental parameter space. This is motivated by
the observation that the mass matrices are directly linked to the Yukawa couplings
and the Higgs vacuum expectation value, whereas the CKM matrix is only a derived
quantity. The observed values for the quark masses will be taken as given, and a
probability distribution be constructed that can reproduce these values. The choice
made for this distribution will be as simple as possible in the following sense: The
natural group action on the space of Hermitian mass matrices is the action of the
unitary group U(3) by conjugation, M → UMU †. There is essentially a unique
measure invariant under this action. This measure is then modified by introducing
the simplest possible function that would allow for a modification of the expectation
values for quark masses fitting observation. No further assumptions will be needed.
We will then find that this simple choice for the measure gives an expectation value
for J that is remarkably close to the observed value. Hence we will conclude that
once one assumes the quark masses as given, one does not face an additional fine-
tuning problem with J . This statement, while not new, is hence made precise using
a geometrically motivated measure on the parameter space.
In the standard model one can restrict, without loss of generality, to Hermitian
mass matrices, which is why we construct a measure on the space of 3× 3 Hermitian
matrices. However, in left-right symmetric extensions of the standard model, such as
Pati-Salam, such an assumption can no longer be made and the mass matrices have
to be regarded as arbitrary complex matrices†. This presumably has an effect on the
statements one makes about naturalness of J ; a similarly well-motivated measure on
the space of all 3 × 3 complex matrices might lead to very different results. We will
therefore in a further calculation redo the same analysis for general complex matrices.
We will again use the most symmetric measure on the space of mass matrices, here
the space of general complex matrices, and modify it in the simplest possible way to
incorporate the observed quark mass hierarchy. While this is a choice that could of
course be made very differently, it is a simple choice that uses as few assumptions
as possible, and that has worked very well for the case of Hermitian mass matrices.
What we will find is that the resulting probability distribution is different, and the
result is different too: The observed value for J now appears to be unnaturally large,
since CP violation should be more heavily suppressed by the quark mass hierarchy.
†We thank Ben Allanach for pointing this out.
Naturalness of CP Violation 124
One faces a fine-tuning problem, and needs additional assumptions to modify the
measure appropriately.
We should point out that the only real input we use from the physical theory
(standard model or a left-right symmetric extension of it) is, apart from the very def-
inition of CP -violating parameters, how the theory restricts the type of mass matrices
that appear. In particular, for any extension of the standard model (such as left-right
symmetric models where parity is the left-right symmetry‡) that also has Hermitian
mass matrices we would not see any modification in the results. Nevertheless, our
calculations provide another example of how symmetries, in this case the presence or
absence of a left-right symmetry, influence physical predictions of a given theory.
The structure of the remaining part of the chapter is as follows: In section 6.2,
we review CP violation in the electroweak sector and detail why mass matrices may
be assumed to be Hermitian in the standard model, but not if one has an extended
left-right symmetry. In section 6.3, we compute expectation values for the Jarlskog
invariant J using several of the measures on the space of CKM matrices that we have
introduced. We also perform a more detailed analysis of how unlikely the observed
value for J appears in these distributions. In section 6.4 we focus on the space of
mass matrices instead, detailing how the observed values for the quark masses can be
incorporated into a measure and how this completely changes predictions about likely
values for J . We briefly mention how our analysis relates to the case of neutrinos and
close with a summary and discussion of our findings.
6.2 CP Violation in the Standard Model and Beyond
We summarise how CP violation arises, first in the standard model and then in
the more general case of left-right symmetric extensions, essentially following [91].
In the standard model, the quark fields appear as left-handed SU(2) doublets and
right-handed SU(2) singlets:(
qjL
q′jL
)
, qjR , q
′
jR , j = 1, 2, . . . , N . (6.2.1)
Here N is the number of quark families, which is arbitrary in the standard model,
and normally taken to be three. The fields are written in a flavour basis which can be
considered unphysical, since flavour eigenstates do not correspond to mass eigenstates.
‡We thank an anonymous referee for Phys. Rev. D for clarification on this point.
Naturalness of CP Violation 125
The coupling of the Higgs doublet H to quarks is given by
LHiggs =
N∑
j,k=1
(
Yjk(q, q′)jLHCqkR + Y ′jk(q, q′)jLHq
′
kR + h.c.
)
, (6.2.2)
where the Higgs doublet H and its C-conjugate HC can be written as
H =
1√
2
(
φ1 + iφ2
φ0 + iφ3
)
, HC =
1√
2
(
φ0 − iφ3
−φ1 + iφ2
)
, (6.2.3)
and Yjk and Y
′
jk are complex (Yukawa) couplings. H transforms under SU(2) as
a doublet and the term LHiggs is invariant under SU(2). This symmetry is then
spontaneously broken by the Higgs potential V (H) which gives rise to a vacuum
expectation value v for φ0. The remaining components φj are ‘eaten’ by the W and
Z bosons, which become massive, and the only remaining terms involve φ′0 = φ0 − v,
giving masses to the quark fields:
LHiggs SSB−→ −
N∑
j,k=1
(
mjkqjLqkR +m
′
jkq
′
jLq
′
kR + h.c.
)(
1 +
1
v
φ′0
)
. (6.2.4)
The new mass matrices m and m′ are (up to a factor of
√
2) given by the original
Yukawa couplings, multiplied by v. Thus, it seems appropriate to regard either the
set of Yukawa couplings together with the Higgs vacuum expectation value or the
collection of elements of m and m′ as fundamental parameters of the theory.
This part of the Lagrangian is formally CP invariant if and only ifm andm′, which
are so far arbitrary complex matrices, are real. Since this condition is not satisfied in
Nature, one has formal CP violation. However, as remarked before, the Lagrangian
has been written in the unphysical flavour basis. One can, for general m and m′, pass
to a different basis, namely the basis of mass eigenstates, by diagonalising the mass
matrices with unitary matrices:
m = U †L∆UR , m
′ = (U ′L)
†∆′U ′R , (6.2.5)
thus the basis of mass eigenstates is related to the previously considered basis by
qLphys = ULqL , qRphys = URqR , etc. (6.2.6)
It is always possible to choose the unitary matrices so that ∆ and ∆′ are real, and
in this new basis this part of the Lagrangian is invariant under C and P separately,
Naturalness of CP Violation 126
and hence also under CP . However, the electroweak Lagrangian also contains charged
current terms mixing up- and down-type quarks, coupled to the W boson fields via
(we are now using the basis of mass eigenstates)
XC := (W
1
µ − iW 2µ)Jµc + h.c. , Jµc := (u, c, t)LγµV


dL
sL
bL

 , (6.2.7)
where V := UL(U
′
L)
† is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. In the
basis of mass eigenstates, this term XC is not invariant under CP unless V is real.
Since we consider the mass eigenstates as physical, CP is violated through these
charged current terms.
An important observation made in [55] is that one can redefine the right-handed
quark bases by arbitrary unitary transformations,
UR → OUR , U ′R → O′U ′R , (6.2.8)
obtaining a new basis which is to be regarded as equally physical. This is due to
the absence of charged current terms involving right-handed quarks, since they are
singlets under SU(2). It is therefore no loss of generality to set UR = UL and U
′
R = U
′
L
in (6.2.5), and to assume that m and m′ are Hermitian.
A natural way to extend the standard model is to assume the existence of a
second SU(2) symmetry which acts on the right-handed quarks, as in the Pati-Salam
model [128]. In such extensions, one adds a term (6.2.7) for right-handed quarks to
the Lagrangian. This has the important consequence that a general transformation
(6.2.8) for arbitrary unitary transformations O and O′ can no longer be regarded as
giving an equivalent quark basis, since it modifies this new charged current term. The
mass matrices cannot in general be taken to be Hermitian, but are arbitrary complex
matrices.
There are now also two possibly CP violating terms, and two CKM matrices. We
focus on V = UL(U
′
L)
† which involves the processes that are actually observed [89]
and disregard VR = UR(U
′
R)
† in the following.
Mathematically, V is an element of SU(N), but since the phases of the quark
fields are arbitrary (even for non-Hermitian mass matrices), V is only defined up
to left or right multiplication by a diagonal element of SU(N), i.e. an element of
the maximal torus U(1)N−1. The space of CKM matrices is therefore the double
quotient U(1)N−1\SU(N)/U(1)N−1, characterised by (N − 1)2 parameters, out of
Naturalness of CP Violation 127
which 1
2
N(N − 1) may be taken to be real (Euler) angles and the remaining 1
2
(N −
1)(N − 2) appear as complex phases. It follows that the matrix V can be taken to
be real for N = 2, and so in the formalism explained here one needs at least three
quark families to have a possibility of CP violation. Kobayashi and Maskawa were
awarded the 2008 Nobel Prize for using this observation to predict the existence of a
third quark family [98]. We set N = 3 in what follows.
The mathematical theory of observable measures of CP violation in the standard
model was developed by Jarlskog [91]. She showed [89] that all necessary and sufficient
conditions for CP violation can be summarised as the following condition on the
commutator of the Hermitian matrices m,m′:
detC := det (−i [m,m′]) 6= 0 . (6.2.9)
One finds that explicitly
detC = −2J(mt−mc)(mc−mu)(mu−mt)(mb−ms)(ms−md)(md−mb) , (6.2.10)
where J := Im(V11V22V
∗
12V
∗
21) is the Jarlskog invariant which is invariant under
left or right multiplication of V by a diagonal matrix, i.e. an element of U(1)2.
The geometrical interpretation of the quantity J is given by the so-called unitarity
triangles. These express the requirement on V to be unitary, so that for instance
(V V †)12 = V11V ∗21 + V12V
∗
22 + V13V
∗
23 = 0 . (6.2.11)
In the complex plane, the three complex numbers that sum to zero form the sides of
a triangle. The absolute value |J | is twice the area of this triangle. Since there are
different unitarity triangles corresponding to different elements of V V †, all with the
same area, there are several ways of expressing J in terms of the elements of V . A
general formula is given by [89]
J
∑
γ,l
αβγjkl = Im(VαjVβkV
∗
αkV
∗
βj) . (6.2.12)
The quantities describing the CKM matrix which are invariant under rephasing of
the quark fields are J and the absolute values |Vαj|.
In the general case where m and m′ are not assumed to be Hermitian, the corre-
sponding quantity would be
detC := det
(−i [mm†,m′m′†]) (6.2.13)
= −2J(m2t −m2c)(m2c −m2u)(m2u −m2t )(m2b −m2s)(m2s −m2d)(m2d −m2b) ,
Naturalness of CP Violation 128
which of course leads to the same conditions on the mass matrices as before (given
that we took all masses to be positive before). In the literature, the use of C is
perhaps more common than the use of C, and one may well argue that this second
measure of CP violation should be considered more fundamental as its value does
not depend on the arbitrary signs of the mass terms in the Lagrangian.
Recall from chapter 5 the parametrisation of the CKM matrix (5.1.6)
V =


cos y cos z cos y sin z e−iw sin y
− cos x sin z − eiw sin x sin y cos z cos x cos z − eiw sin x sin y sin z sin x cos y
sin x sin z − eiw cos x sin y cos z − sin x cos z − eiw cosx sin y sin z cos x cos y

 ,
(6.2.14)
where the ranges of the Euler angles x, y, z and the complex phase w are
0 ≤ x, y, z ≤ pi
2
, 0 ≤ w < 2pi . (6.2.15)
In this parametrisation, the Jarlskog invariant J is given by
J =
1
4
sin(2x) sin(2z) sin y cos2 y sinw . (6.2.16)
It appears that the observed value for J is very small, since the maximal value
would be 1
6
√
3
≈ 0.1, whereas in Nature [7]
J = 3.05+0.19−0.20 × 10−5 . (6.2.17)
This discrepancy in 3.5 orders of magnitude is the fine-tuning problem we are inves-
tigating in this chapter.
In a general discussion where the values of the quark masses are not fixed, J is
not an appropriate measure of CP violation, since even with nonvanishing J one
could have CP conservation if, for example, the up and charm quark masses were
coinciding. It was suggested in [90] to use an appropriately normalised form of detC,
namely
aCP = 3
√
6
detC
(TrC2)3/2
(6.2.18)
for three quark families as the unique basis independent measure of CP violation.
This is a dimensionless number which takes values between −1 and +1, and is again
observed to be very close to zero. When written out in terms of the CKM matrix
parameters and quark masses, it is a rather complicated expression which is therefore
not extremely useful in practical computations. In the present analysis, we assume
the quark masses as known and regard J as the measure of CP violation.
Naturalness of CP Violation 129
6.3 Statistics on the Space of CKM Matrices
Let us compute expectation values for J for different choices of measure on the double
quotient U(1)2\SU(3)/U(1)2, parametrised by coordinates (x, y, z, w). Obviously,
given a metric g, the expectation value of any function f(x, y, z, w) is
〈f〉 =
∫
dx dy dz dw
√
det g f(x, y, z, w)∫
dx dy dz dw
√
det g
. (6.3.1)
As we mentioned in chapter 5, one possible viewpoint of a given function on
U(1)2\SU(3)/U(1)2 is to see it as function on the homogeneous space SU(3)/U(1)2
which is independent of the coordinates parametrising the left U(1)2 (p and q in
our conventions). Since we also saw that there is a unique left-invariant measure on
the homogeneous space, this gives a unique definition of expectation values of such
functions.
We computed this left-invariant measure on SU(3)/U(1)2 to be
√
g = N sin(2x) sin(2z) sin y cos3 y (6.3.2)
for some constant N .
Note that there is no need to pass from the double quotient to the homogeneous
space if one regards the former as the fundamental domain of definition of a given
function. One might equally well regard the left-invariance on the homogeneous space
that has led to a unique measure as a spurious symmetry requirement.
Remembering that the CKM matrix is defined as V = U(U ′)† in terms of two uni-
tary matrices U and U ′ (denoted by UL and U ′L for the general case of non-Hermitian
mass matrices above), one might think about defining the Jarlskog invariant J in
terms of coordinates on the Cartesian product of SU(3) × SU(3), the spaces of U
and U ′.§ The natural measure on SU(3)× SU(3) is, in terms of left-invariant forms
σi and σ
′
i,
µ = N σ1∧σ2∧σ3∧σ4∧σ5∧σ6∧σ7∧σ8∧σ′1∧σ′2∧σ′3∧σ′4∧σ′5∧σ′6∧σ′7∧σ′8 . (6.3.3)
Since it is only V = UU ′† that enters into the CP violating parameters, one could
consider U and V as independent variables, i.e. write U ′ = V †U for some matrix V .
§Ambiguities in the definition of U and U ′ mean that they should be regarded as living in quotient
spaces such as U(1)2\SU(3), but since our left-invariant measures do not depend on the left and
right U(1)2 parameters such subtleties may be ignored.
Naturalness of CP Violation 130
Then the Maurer-Cartan form on the second SU(3) is
iλaσ
′
a ≡ U ′†dU ′ = U †dU − U †(dV V †)U , (6.3.4)
which gives σ′a = σa − habτb, where τb are right-invariant forms on SU(3) in terms
of V coordinates and hab only depends on the U coordinates. The measure (6.3.3),
expressed in terms of V and U coordinates, is thus a product of a function of the
U coordinates and the natural measure in V coordinates (left- and right-invariant
forms on SU(3) give the same measure). Integration over the U coordinates then
just gives an irrelevant constant, and one is left with the measure (6.3.2) on the
space of V matrices. This justifies the use of (6.3.2) instead of the more complicated
constructions obtained by reducing to the double quotient, and we will regard (6.3.2)
as the most natural choice of measure on the parameter space.
For the measure (6.3.2) the evaluation of the necessary integrals is very simple
and we find that all odd powers of J average to zero, and
〈J2〉 = 1
720
≈ 1.389× 10−3 , 〈J4〉 = 1
201600
≈ 4.960× 10−5 . (6.3.5)
Thus we find that ∆J for the Jarlskog invariant is given by
∆J =
1
12
√
5
≈ 0.0373 , (6.3.6)
which is about three orders of magnitude larger than the experimental value (6.2.17).
We argued in section 5.3 that a geometrically motivated measure on the double
quotient space arises from Kaluza-Klein reduction of a left-invariant metric on SU(3)
or SU(3)/U(1)2, and obtained the measure (5.3.7) from the bi-invariant metric on
SU(3). It is interesting how expectation values for powers of J compare to the
previous calculations if this measure is used. Unfortunately, the expression for the
measure is too complicated to allow us to perform the integrations analytically. Using
numerical integration, we find that
〈J2〉 ≈ 1.1161× 10−3 , 〈J4〉 ≈ 3.750× 10−6 , (6.3.7)
with the odd powers of J again averaging to zero. Thus we find
∆J ≈
√
〈J2〉 ≈ 0.03341 , (6.3.8)
which is very close to the previous result.
Naturalness of CP Violation 131
Naively, one might have thought that since J is independent of all the U(1) phases,
the results would be the same whether one averaged over the space U(1)2\SU(3)/U(1)2,
or else the flag manifold SU(3)/U(1)2. Of course we know that this is not correct,
since, as we have seen, the measure for the biquotient takes a different form than
the measure one would obtain from just integrating over the left U(1)2 parameters.
Nevertheless, this does not seem to have a great impact on expectation values for J .
We briefly mentioned in chapter 5 that one could construct more general mea-
sures on the double quotient by Kaluza-Klein reduction of more general left-invariant
metrics on SU(3)/U(1)2. While this makes the expression for the measure even more
complicated, the effect on expectation values of J seems somewhat limited, unless
one takes extreme values for the “squashing parameters”. The general conclusion,
namely that there appears to be a fine-tuning problem, is unchanged.¶
Given that we argued that the measure used by Ozsva´th and Schu¨cking (5.3.1) was
not appropriate from a geometric viewpoint, we should compute expectation values
for this measure also. We find that again 〈J〉 = 0 and
〈J2〉 = 35× 2−16 ≈ 5.341 × 10−4 , 〈J4〉 = 27027× 2−34 ≈ 1.573 × 10−6 , (6.3.9)
and that the standard deviation is
∆J2 =
√
〈J4〉 − 〈J2〉2 =
√
22127× 2−17 ≈ 1.135 × 10−3 , (6.3.10)
and
∆J =
√
〈J2〉 =
√
35
256
≈ 0.02311 . (6.3.11)
Again, the results are rather similar to the previous cases.
Assuming a uniform distribution over the angles, and hence treating the double
coset as a flat four-dimensional manifold so that the measure is simply
√
det g = 1,
would give
〈J〉 = 0, 〈J2〉 = 1
2048
≈ 4.883× 10−4 , 〈J4〉 = 189× 2−27 ≈ 1.408× 10−6 .
(6.3.12)
Hence, this simplest possible choice gives
∆J ≈ 0.02210 . (6.3.13)
¶These calculations, mainly carried out by Chris Pope, are not included in detail in this thesis,
but can be found in [67].
Naturalness of CP Violation 132
We have to conclude that, although we argued that some choices of measure on
the space of CKM matrices are better motivated geometrically than others, if one
is only interested in predictions about the magnitude of J there seems to be little
difference between all of the choices we have examined. In all cases one would expect
the observed value of J to be at least of order 10−2, which leaves three orders of
magnitude between prediction and observation.
6.3.1 Fine-tuning of J
In the previous subsection we saw that different measures on the space of mixing
angles all seem to lead to expectation values for J which are about three orders of
magnitude larger than the observed value. The value for J that we observe hence
appears to be finely tuned. In this section we shall do a closer, mainly numerical,
analysis of the fine-tuning involved. We compare results obtained by taking the
SU(3)-invariant (descending from the left-invariant measure on SU(3)/U(1)2) and
Kaluza-Klein (obtained from Kaluza-Klein reduction of the bi-invariant metric on
SU(3)) measures, which seem natural from a geometric perspective, with a uniform
distribution which is just the simplest possible choice.
The observed value for the Jarlskog invariant J is
J ≈ 10−4.51 ≈ e−10.39 . (6.3.14)
In order to obtain a probability distribution for J we have used Mathematica to
numerically compute integrals of the form∫
dx dy dz dw
√
det g θ(a− |J |) θ(|J | − b) ≡ P (b ≤ |J | ≤ a) ·
∫
dx dy dz dw
√
det g
(6.3.15)
using Monte Carlo methods. The SU(3)-invariant measure and the Kaluza-Klein
measure disfavour small values of J more strongly than a uniform distribution would.
For example, we obtain
Pflag(|J | ≤ 10−4) ≈ 0.25% , PKK(|J | ≤ 10−4) ≈ 0.44% . (6.3.16)
Taking a uniform distribution
√
det g ≡ 1, we get
Punif(|J | ≤ 10−4) ≈ 7% . (6.3.17)
In figures 6.1 to 6.3, we have plotted the probability distribution of |J | logarithmically,
together with a fit to a power law p(|J |) ∝ |J |λ assumed to be valid for small |J |.
Naturalness of CP Violation 133
Figure 6.1: Probability distribution for log |J | using the SU(3)-invariant flag measure.
The degree of fine-tuning required to reproduce a very small J is considerably higher
if one uses the measure induced by a SU(3)-invariant flag metric or the Kaluza-Klein
metric, maybe contrary to what one might expect. Values of J close to its maximal
value of 1
6
√
3
≈ 0.0962 are disfavoured in both cases. This motivated the use of a
logarithmic scale for |J |.
Figure 6.2: Probability distribution for log |J | using the Kaluza-Klein measure.
In all three cases the numerical results for small |J | are well approximated by a power
law of the form p(|J |) = α · |J |λ for the probability density of |J |. The logarithmic
graphs show p(log |J |) ∝ |J |λ+1. For the SU(3)-invariant flag measure (Fig. 6.1), the
best fit to the data in the region below |J | = 10−2.3 or log |J | = −5.3 is
λflag = −0.042(±0.006) , αflag = 18.1(±0.7) ; (6.3.18)
Naturalness of CP Violation 134
for the Kaluza-Klein measure (Fig. 6.2) we fitted the data in the region below |J | =
10−2.7 or log |J | = −6.2 and obtained
λKK = −0.097(±0.008) , αKK = 18.9(±1.0) ; (6.3.19)
finally for the uniform measure (Fig. 6.3), the best fit to the data in the region below
|J | = 10−3.4 or log |J | = −7.8 is
λunif = −0.500(±0.005) , αunif = 3.51(±0.15) . (6.3.20)
Figure 6.3: Probability distribution for log |J | using a uniform distribution.
6.3.2 Wolfenstein Parametrisation
A different parametrisation of the CKMmatrix which is frequently used in the particle
physics literature was introduced by Wolfenstein and is based on the experimentally
observed hierarchy
y  x z  1 (6.3.21)
in the mixing angles. One rewrites [169]
sin z = λ , sin x = Aλ2 , sin ye−iw = Aλ3(ρ− iη) (6.3.22)
and treats λ as a small parameter while A, ρ, and η are supposed to be parameters
of order unity. In the modern literature one also frequently uses ρ¯, η¯ instead of ρ and
Naturalness of CP Violation 135
η because then the combination ρ¯+ iη¯ is independent of the phase convention in the
CKM matrix [7]. These parameters are defined by
ρ =
√
1− A2λ4
1− λ2
ρ¯− A2λ4(ρ¯2 + η¯2)
(1− A2λ4ρ¯)2 + A4λ8η¯2 ,
η =
√
1− A2λ4
1− λ2
η¯
(1− A2λ4ρ¯)2 + A4λ8η¯2 . (6.3.23)
The experimental values for λ,A, ρ¯, η¯ are [7]‖
λ = 0.2272± 0.0010, A = 0.818+0.007−0.017, ρ¯ = 0.221+0.064−0.028, η¯ = 0.340+0.017−0.045 . (6.3.24)
One viewpoint on the Wolfenstein parametrisation is that it is adapted to the values
for the CKM matrix entries that we observe and has no deeper significance; but often
the viewpoint is expressed that this parametrisation expresses some kind of “natural
hierarchy” in the mixing angles coming from physics beyond the standard model (see
e.g. [24]). Treating the other parameters as “naturally of order unity” reduces our
calculations to a one-dimensional problem as everything is only expanded in terms of
λ. We find that the SU(3)-invariant measure on the flag manifold is now, to leading
order in λ, ∣∣∣∣∂(x, y, z, w)∂(λ,A, ρ¯, η¯)
∣∣∣∣√g ∝ A3λ11 (1 + λ2 +O(λ4)) , (6.3.25)
and the Jarlskog invariant J is
J =
A2η¯λ6(1− A2λ4) (1− λ2 − 2A2ρ¯λ4 − A2(η¯2 + (ρ¯− 2)ρ¯)λ6 + A4(η¯2 + ρ¯2)λ8)
(1− λ2) (1− 2A2ρ¯λ4 + A4(η¯2 + ρ¯2)λ8)2
= A2η¯λ6 +O(λ10) . (6.3.26)
Inverting this expression to leading order gives the probability distribution for J
p(J) ∝ J
Aη¯2
(
1 +
(
J
A2η¯
)1/3
+O(J2/3)
)
, (6.3.27)
which is incompatible with the numerical results. Trying to improve this approximate
result by letting A, ρ¯ and η¯ take all possible values leads to inconsistencies since the
expansion in powers of J contains poles of arbitrary order in A. From our present
viewpoint, where no mechanism for fixing these parameters close to one is known,
the Wolfenstein parametrisation seems rather misleading when discussing geometric
probability.
‖Note that only even powers of λ appear in all expansions, so that it is λ2 ≈ 0.05 which is the
small parameter.
Naturalness of CP Violation 136
6.4 Statistics on the Space of Mass Matrices
In the previous sections we have focussed on U(1)2\SU(3)/U(1)2, the space of CKM
matrices, as the space of CP violating parameters. Since SU(3) is compact, this
space has finite volume for a natural measure. But the CKM matrix is derived from
the quark mass matrices, which could be viewed as more fundamental and more
directly determined by physics beyond the standard model. In this section, we try
to obtain statistics of the Jarlskog invariant J from a random distribution on the
space of mass matrices; in the standard model, this is the space of 3 × 3 Hermitian
matrices, whereas in certain extensions with left-right symmetry one has to take all
3× 3 complex matrices into account.
In this section, as suggested in [89], we define a dimensionless mass matrix M by
M = m/Λ, where m is the quark mass matrix in the notation of section 6.2, and Λ
is a scale which may be chosen for convenience. A natural choice would be Λ = mt
for the up-type or Λ′ = mb for the down-type quarks, but since we in principle allow
arbitrary values for the quark masses we leave Λ arbitrary.
We now need to pick a measure on the space of mass matrices in order to do
statistics. Let us summarise the main results of section 5.4: For a given Hermitian
matrix M , one defines a unitary matrix U and a real diagonal matrix D by
MH = U
†DU , (6.4.1)
and the natural measure, invariant under conjugation by U(3), on the space of such
matrices was found to be
DMH := (D1 −D2)2(D1 −D3)2(D2 −D3)2 sin(2x) cos3 y sin y sin(2z)
×dD1 dD2 dD3 dx dy dz dw dr dt . (6.4.2)
We noted that to avoid overcounting one should integrate over R3 times a quotient
of SU(3), more precisely (U(1)2 × S3)\SU(3).
Let us contrast this with the space of all complex matrices; here we used the
representation
Mc = U
†
LDUR , (6.4.3)
and identified the relevant space one has to integrate over as R3+×(U(1)3×S3)\(U(3)×
U(3)). We then argued that if we are only interested in functions that depend on UL,
but not on UR, such as the Jarlskog invariant J , we can integrate over the space of
Naturalness of CP Violation 137
UR. Then the resulting measure on R
3
+ × (U(1)2 × S3)\SU(3) is
DMc :=
∏
i<j
(D2i −D2j )2|D1D2D3| sin(2x) cos3 y sin y sin(2z)
×dD1 dD2 dD3 dx dy dz dw dr dt , (6.4.4)
which obviously differs from (6.4.2) by different powers of the real numbers Di that
had the role of eigenvalues for Hermitian matrices. Since the measure (6.4.4) is
invariant under Di → −Di, we will integrate over all of R3 for simplicity. This only
leads to an extra factor 8 which drops out in expectation values.
It is clear that in both cases the integrals over R3 with the given measures will
diverge. We could introduce a cutoff for the quark masses, but then any expectation
values for quark masses would strongly contradict observation, as there is no way to
explain the observed mass hierarchy.
To both make the integrals converge and introduce a possibility for introducing the
quark masses as additional parameters into our distribution, we introduce a function
decaying sufficiently fast for large |Di| into the measure. A natural and simple choice
is a Gaussian.
Since there are actually two integrations over the space of mass matrices, corre-
sponding to two mass matrices M and M ′ for the up-type and down-type quarks, we
will use the measure∗∗
DM DM ′ exp(−Tr(MM †A)− Tr(M ′(M ′)†A′))
= DM DM ′ exp(−Tr(U †D2UA)− Tr((U ′)†(D′)2U ′A′)) , (6.4.5)
which for complex matrices is still invariant under the two right U(3) actions on the
complex matrices M and M ′, but the invariance under the left U(3) actions is broken
to the subgroups commuting with A and A′ respectively. We will assume A and A′
to be Hermitian with non-negative eigenvalues, satisfying [A,A′] = 0 and use the
redefinitions U → UW , U ′ → U ′W , which leave J invariant, to diagonalise A and A′.
Hence we can assume A and A′ to be diagonal with positive entries in the following,
and the subgroup commuting with A and A′ is U(1)3. For Hermitian matrices, (6.4.5)
becomes
DMHDM
′
H exp(−Tr(M2HA)) exp(−Tr((M ′H)2A)) , (6.4.6)
∗∗DM is to replaced byDMH andDMc for Hermitian and complex matrices respectively, and from
now on we write U instead of UL in the case of complex matrices as well, since UR has disappeared
from our calculations.
Naturalness of CP Violation 138
where the Gaussian breaks the symmetry of the measure from invariance under
(U(3) × U(3)) to invariance under (U(1)3 × U(1)3), again the subgroup commuting
with A and A′.
We will find that this symmetry breaking is necessary to obtain a distribution
that can reproduce the observed quark masses. Our proposal is to fit the diagonal
matrices A and A′ to the observed quark masses and to use the resulting probability
distribution for statistics of J .
An expectation value of a quantity such as J2 is then given by††
〈J2〉 = N
∫
DM DM ′ e−Tr(MM
†A)−Tr(M ′M ′†A′) J2(M,M ′) (6.4.7)
= N
∫
R6
dD dD′
∫
((U(1)2×S3)\SU(3))2
DU DU ′ e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) J2 .
Here DU and DU ′ are the measures on (U(1)2 × S3)\SU(3), given by (5.1.12),
DU := sin(2x) sin y cos3 y sin(2z) , (6.4.8)
and
dD :=

(D1 −D2)
2(D1 −D3)2(D2 −D3)2 dD1 dD2 dD3 , M Hermitian ,
(D21 −D22)2(D21 −D23)2(D22 −D23)2|D1D2D3| dD1 dD2 dD3 , M complex ,
(6.4.9)
and the normalisation factor N is defined by
1
N
:=
∫
R6
dD dD′
∫
((U(1)2×S3)\SU(3))2
DU DU ′ e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) . (6.4.10)
From J = Im
(
V11 V22 V
∗
12 V
∗
21
)
and V = UU ′†, we have
J(U,U ′) =
3∑
a,b,c,d=1
Im
(
U1aU2bU
∗
1cU
∗
2dU
′∗
1aU
′∗
2bU
′
2cU
′
1d
)
. (6.4.11)
At this point, it is perhaps instructive to note that choosing A proportional to the
identity would split the integral (6.4.7) into a product of an integral over R6, i.e.
the quark masses, which just gives a constant, and an integral of J2 over ((U(1)2 ×
S3)\SU(3))2. Since all even powers of J are invariant under the S3 action on U and
U ′, this can be replaced by an integral over (U(1)2\SU(3))2 if averages are concerned.
By the arguments presented in section 6.3, a change of coordinates reduces this to a
††In all cases considered, all odd powers of J again have expectation value zero.
Naturalness of CP Violation 139
single integration over a copy of SU(3)/U(1)2, and one recovers the results of section
6.3 for expectation values of powers of J . The introduction of more general diagonal
matrices A and A′ means that the maximal invariance of the measure DM DM ′ is
broken down to the action of a subgroup.
It should be clear from (6.4.7) that multiplying A (or A′) by a constant is the
same as rescaling the eigenvalues Di (or D
′
i) and so amounts to a rescaling of Λ (or
Λ′). We can therefore, without any loss of generality, choose
A =


1 0 0
0 1/µ2c 0
0 0 1/µ2u

 , A′ =


1 0 0
0 1/µ2s 0
0 0 1/µ2d

 , (6.4.12)
where µc, µu, µs, and µd are dimensionless parameters that we are free to choose so
as to reproduce the observed quark masses as expectation values. (In the case of an
exponential exp(−Tr(D2A)), these would of course be equal to the respective quark
masses, expressed in units where Λ = mt and Λ
′ = mb.) Because of experimental un-
certainties in the up and quark masses, one can modify this distribution to reproduce
different values for these masses.
6.4.1 Hermitian Mass Matrices
We shall now present the result of evaluating the integral (6.4.7), first for Hermitian
and then for complex mass matrices, and therefore for the two different choices of
measure (6.4.9). We will have to use various approximations which we try to justify in
the following, and the final result is obtained independently by numerical integration
and by an analytical approximation, where we find relatively good agreement between
the two.
The need for approximations in the integral (6.4.7) is immediate since the expres-
sion for J in terms of coordinates on ((U(1)2×S3)\SU(3))2 is too complicated to be
given explicitly. However, since
Tr(D2UAU †) =
∑
a
D2a
∑
c
Ac|Uac|2 =:
∑
a
D2aξa , Tr((D
′)2U ′A′U ′†) =:
∑
a
(D′a)
2ξ′a
(6.4.13)
with
ξ1 = A1 cos
2 y cos2 z + A2 cos
2 y sin2 z + A3 sin
2 y ,
ξ′1 = A
′
1 cos
2 y′ cos2 z′ + A′2 cos
2 y′ sin2 z′ + A′3 sin
2 y′ , (6.4.14)
Naturalness of CP Violation 140
and we assume A3  1 and A′3  1, the integrand is negligibly small unless y ≈ 0 and
y′ ≈ 0. We use this to approximate the integrals over y and y′ (recall the restriction
of the coordinates on U(1)2\SU(3) given by (5.4.9) which restricts from U(1)2\SU(3)
to (U(1)2 × S3)\SU(3)):
arctan(sinx)∫
0
dy
arctan(sinx′)∫
0
dy′ cos3 y sin y cos3 y′ sin y′e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) J2(U,U ′)
≈
∞∫
0
dy
∞∫
0
dy′ y y′ e−A3y
2−A′
3
(y′)2 ·
(
e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) J2(U,U ′)
) ∣∣
y=y′=0
≈ 1
4A3A′3
(
e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) J2(U,U ′)
) ∣∣
y=y′=0 . (6.4.15)
It turns out that this is independent of w and w′. Constant prefactors such as
1/4A3A
′
3 appearing in both numerator and denominator can be dropped, and so we
have
〈J2〉 ≈
∫
R6
dDdD′
∫
d4xd4x′ s2xs2zs2x′s2z′
(
e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) J2(U,U ′)
) ∣∣
y=y′=0∫
R6
dDdD′
∫
d4xd4x′ s2xs2zs2x′s2z′
(
e−Tr(D2UAU†)−Tr((D′)2U ′A′U ′†)
) ∣∣
y=y′=0
,
(6.4.16)
where s2x = sin 2x etc. and
∫
d4x ≡
pi/4∫
0
dx
pi/2∫
0
dz
2pi∫
0
dr
2pi∫
0
dt (6.4.17)
and similarly for
∫
d4x′.
Now we can integrate over both copies of R3 in (6.4.16), using
∞∫
−∞
dD1
∞∫
−∞
dD2
∞∫
−∞
dD3 (D1 −D2)2(D1 −D3)2(D2 −D3)2e−ξ1D21−ξ2D22−ξ3D23
=
3pi3/2
8ξ
5/2
1 ξ
5/2
2 ξ
5/2
3
(
ξ21(ξ2 + ξ3) + ξ
2
2(ξ1 + ξ3) + ξ
2
3(ξ2 + ξ1)− 2ξ1ξ2ξ3
)
. (6.4.18)
The explicit expression for J at y = y′ = 0 is
J(U,U ′)
∣∣
y=y′=0 =
1
4
s2xs2x′
{
c2z′s
3
zsz′ sin(3rˆ + tˆ) + c
3
zcz′s
2
z′ sin(3rˆ − tˆ) (6.4.19)
−c2zszsz′(c2z′
[
sin(3rˆ + tˆ) + sin(3rˆ − 3tˆ)]− s2z′ sin(3rˆ + tˆ))
+czcz′s
2
z(c
2
z′ sin(3rˆ − tˆ)− s2z′
[
sin(3rˆ + 3tˆ) + sin(3rˆ − tˆ)])}
Naturalness of CP Violation 141
where sx = sin x, cz′ = cos z
′, etc., rˆ = r− r′, and tˆ = t− t′. Integrating (6.4.19) over
r, r′, t, and t′ indeed gives zero, which is why we choose to use J2.
We need to determine the parameters appearing in the matrices A and A′ in
(6.4.12) by fitting them to the expectation values for quark masses that we want to
reproduce. We first observe that expectation values for squared mass matrices take
the relatively simple form
〈D21〉 ≈
∫
R3
dDD21
pi/4∫
0
dx
pi/2∫
0
dz sin 2x sin 2z
(
e−Tr(D
2UAU†)
) ∣∣
y=0
∫
R3
dD
pi/4∫
0
dx
pi/2∫
0
dz sin 2x sin 2z
(
e−Tr(D2UAU†)
) ∣∣
y=0
. (6.4.20)
The denominator of (6.4.20) is explicitly
ID :=
pi/4∫
0
dx
pi/2∫
0
dz sin 2x sin 2z
3pi3/2
8ξ
5/2
1 ξ
5/2
2 ξ
5/2
3
×
(
ξ21(ξ2 + ξ3) + ξ
2
2(ξ1 + ξ3) + ξ
2
3(ξ2 + ξ1)− 2ξ1ξ2ξ3
)
, (6.4.21)
where within our approximation the quantities ξa are taken at y = 0,
ξ1 = A1 cos
2 z + A2 sin
2 z , ξ2 = A1 cos
2 x sin2 z + A2 cos
2 x cos2 z + A3 sin
2 x ,
ξ3 = A1 sin
2 x sin2 z + A2 sin
2 x cos2 z + A3 cos
2 x , (6.4.22)
with A3  A2  A1. We notice that all ξa are nonzero for all values of x and z.
Furthermore, the integral is dominated by very small x and z (we cannot have x = pi
2
),
and we can approximate ID well by only keeping the terms of leading order in x and
z in the trigonometric functions, and
ξ21(ξ2 + ξ3) + ξ
2
2(ξ1 + ξ3) + ξ
2
3(ξ2 + ξ1)− 2ξ1ξ2ξ3 ≈ A33x2 + A23A2 , (6.4.23)
which are the leading terms (as we shall see, the first of these is effectively also of
Naturalness of CP Violation 142
order A23A2):
ID ≈ 3pi
3/2
8
pi/4∫
0
dx
pi/2∫
0
dz 4xz (A33x
2 + A23A2)(A1 + A2z
2)−5/2(A2 + A3x2)−5/2A
−5/2
3
≈ 3pi
3/2
8
∞∫
0
dX
∞∫
0
dZ (A33X + A
2
3A2)(A1 + A2Z)
−5/2(A2 + A3X)−5/2A
−5/2
3
=
3pi3/2
8
∞∫
0
dX(A33X + A
2
3A2)(A2 + A3X)
−5/2A−5/23 ·
2
3A
3/2
1 A2
=
3pi3/2
8
2
3A
3/2
1 A2

2
3
A
−1/2
2 A
−3/2
3 +
∞∫
0
dX
2
3A
1/2
3
(A2 + A3X)
−3/2


=
pi3/2
4A
3/2
1 A2
(
2
3
A
−1/2
2 A
−3/2
3 +
4
3A
3/2
3 A
1/2
2
)
=
pi3/2
2A
3/2
1 A
3/2
2 A
3/2
3
. (6.4.24)
Similarly, we find for the numerator of (6.4.20)
ID〈D21〉 ≈
15pi3/2
16
∞∫
0
dX
∞∫
0
dZ (A33X + A
2
3A2)(A1 + A2Z)
−7/2(A2 + A3X)−5/2A
−5/2
3
=
15pi3/2
16
∞∫
0
dX (A33X + A
2
3A2)(A2 + A3X)
−5/2A−5/23 ·
2
5A2A
5/2
1
=
3pi3/2
4A
5/2
1 A
3/2
2 A
3/2
3
, (6.4.25)
hence
〈D21〉 ≈
3
2A1
. (6.4.26)
Redoing the same calculation for D2 and D3 gives
〈D22〉 ≈
1
2A2
, 〈D23〉 ≈
1
2A3
. (6.4.27)
There is a relative factor of 3 which has to be taken into account when determining
A and A′.
Because of the dependence of masses on the energy scale in quantum field theory,
described by the renormalisation group, there is some ambiguity in what is meant by
the “quark masses” we want to reproduce. Following [144], for example, we take all
the quark masses evolved to the scale of the Z boson mass. These are given in [170]:
(mu,mc,mt) = (1.27
+0.50
−0.42 MeV, 0.619± 0.084 GeV, 171.7± 3.0 GeV) ;
(md,ms,mb) = (2.90
+1.24
−1.19 MeV, 55
+16
−15 MeV, 2.89± 0.09 GeV) . (6.4.28)
Naturalness of CP Violation 143
We use the central values
(mu,mc,mt) := (1.27 MeV, 0.619 GeV, 171.7 GeV) ;
(md,ms,mb) := (2.9 MeV, 55 MeV, 2.89 GeV) . (6.4.29)
The mass scales Λ and Λ′ that were so far arbitrary are fixed by setting 〈D21〉 =
(mt/Λ)
2 and 〈(D′1)2〉 = (mb/Λ′)2. By comparing the results obtained by numerical
integration with the values we want to reproduce, we can then fix the parameters
µc, µu, µs and µd.
In the case of the positively charged top, charm and up quarks, which exhibit
a more extreme quark mass hierarchy, we find that numerical calculations (using
Mathematica) reproduce the results we have obtained analytically very well (see
Table 6.1). For the negatively charged quarks, we find numerically that we have to
use relative factors different from 3 to reproduce the observed masses. Comparing
the numerical results with (6.4.29), we fix the parameters appearing in A and A′ to
µ2c = 3
(
mc
mt
)2
≈ 3.90× 10−5 , µ2u = 3
(
mu
mt
)2
≈ 1.64× 10−10 ,
µ2s =
3
2
(
ms
mb
)2
≈ 5.43× 10−4 , µ2d =
12
5
(
md
mb
)2
≈ 2.42× 10−6 . (6.4.30)
As a brief side remark, we see that the dominant contributions to these integrals
come from the regions
y ≈
√
1
A3
, y′ ≈
√
1
A′3
, x ≈
√
A2
A3
, z ≈
√
A1
A2
, x′ ≈
√
A′2
A′3
, z′ ≈
√
A′1
A′2
,
(6.4.31)
and these values are all small compared to one. This observation allows us to give
rough estimates for magnitudes of individual elements of the CKM matrix.
In the standard convention the ordering of the quark families is (u, c, t) and not
(t, c, u) as used in (6.4.12), which means that in our parametrisation,
|(UU ′†)13| = |Vtd| , |(UU ′†)12| = |Vts| , |(UU ′†)23| = |Vcd| . (6.4.32)
Since all of the numbers in (6.4.31) are small, we only keep leading terms in the angles
on U and U ′:
|(UU ′†)13| ≈ |x′(z′ − z)− eiw′y′ + . . . | ≈ x′z′ ≈ µd ≈ 2× 10−3 ,
|(UU ′†)12| ≈ z′ ≈ µs ≈ 0.02 , |(UU ′†)23| ≈ x′ ≈ µd
µs
≈ 0.07 . (6.4.33)
Naturalness of CP Violation 144
Experimental values are [7]
|Vtd| = (8.14+0.32−0.64)× 10−3 , |Vts| = (41.61+0.12−0.78)× 10−3 , |Vcd| = 0.2271+0.0010−0.0010 .
(6.4.34)
Our rough estimates reproduce the right ordering of the three parameters and are
accurate to factors of order a few. A more careful analysis would involve computing
expectation values for these parameters in the distribution we have assumed.
We return to the task of computing the expectation value of J2. In order to obtain
an analytical expression, we use the fact that the main contribution to the integral
(6.4.16) will come from small z to only take the term in (6.4.19) that is nonzero at
z = 0. Averaging over r, t, r′,and t′ gives a factor of 1/2, as one might have expected,
and therefore we use
J2small z :=
1
2
sin2 x cos2 x sin2 x′ cos2 x′ cos2 z′ sin4 z′ (6.4.35)
for our calculations. The numerator of (6.4.16) is the product (using again that only
small z contributes)
9pi3
32
×
pi/2∫
0
dz
2z
(A1 + A2z2)5/2
×
pi/2∫
0
dz′
sin 2z′ cos2 z′ sin4 z′
(A1 cos2 z′ + A2 sin2 z′)5/2
(6.4.36)
×
pi/4∫
0
dx
sin 2x sin2 x cos2 x(A33 cos
2 x sin2 x+ A23A2(cos
6 x+ 2 cos2 x sin2 x+ sin6 x))
(A2 cos2 x+ A3 sin
2 x)5/2(A3 cos2 x+ A2 sin
2 x)5/2
times an integral identical to the second line of (6.4.36), except that A2 and A3
are replaced by A′2 and A
′
3. The first two factors are
2
3A
3/2
1
A2
and 4
3
√
A′
2
(
√
A′
1
+
√
A′
2
)4
,
respectively; for the second line of (6.4.36) we change variables to X = cos2 x to
obtain
1∫
1/2
dX
X(1−X)(A33X(1−X) + A23A2(X2 −X + 1))
(A2X + A3(1−X))5/2(A3X + A2(1−X))5/2 ≈
1
A23
(
arctan
(
1
2
√
A3
A2
)
− 2
√
A2
A3
)
,
(6.4.37)
Naturalness of CP Violation 145
where we are dropping corrections of order A2
A3
. Putting everything together, we obtain
〈J2small z〉 ≈
(A′1)
3/2A′2
√
A2√
A3A′3(
√
A′1 +
√
A′2)4
(
arctan
√
A3
4A2
−
√
4A2
A3
)
×
(
arctan
√
A′3
4A′2
−
√
4A′2
A′3
)
(6.4.38)
=
4√
15
mumdmb
mcm2s
(
1 +
√
2
3
mb
ms
)4
(
arctan
mc
2mu
− 2mu
mc
)
×
(
arctan
√
5
32
ms
md
−
√
32
5
md
ms
)
,
where the numerical factors appearing in the last line come from the different factors
chosen in (6.4.30). Note that the top quark mass does not appear in this approximate
result.
We also use numerical integration to check the validity of our analytical approxi-
mations. In these calculations we use both the simplified expression J2small z and the
expression for J given in (6.4.19). We find that for the first quantity, the numerically
evaluated expectation value 〈J2small z〉 is about 7/6 of (6.4.38), and the numerical result
for 〈J2〉 (taken at y = y′ = 0) is
〈J2〉 ≈ 5.28× 10−9 , (6.4.39)
which gives
∆J =
√
〈J2〉 ≈ 7.27× 10−5 (6.4.40)
which is much closer to the observed value than any of the previously obtained results.
Assuming a Gaussian distribution for J which is peaked at zero, the probability of
finding a small J , in the sense of Sec. 6.3.1, is now
Pmass(|J | ≤ 10−4) ≈ 83% , (6.4.41)
whereas the probability of finding a J which is even smaller than the observed value
is
Pmass(|J | ≤ 3× 10−5) ≈ 32% . (6.4.42)
The observed value for J can no longer be viewed as being finely tuned if the distri-
bution used in our calculations is assumed.
Naturalness of CP Violation 146
Table 6.1: Analytical and numerical results for integrals of interest.
Quantity ID (over x, z) ID〈D21〉 ID〈D22〉 ID〈D23〉
Analytical result 1.43× 10−21 2.14× 10−21 2.78× 10−26 1.17× 10−31
Numerical result 1.43× 10−21 2.14× 10−21 2.78× 10−26 1.18× 10−31
Quantity I ′D (over x
′, z′) I ′D〈(D′1)2〉 I ′D〈(D′2)2〉 I ′D〈(D′3)2〉
Analytical result 1.32× 10−13 1.99× 10−13 3.60× 10−17 1.60× 10−19
Numerical result 1.32× 10−13 1.98× 10−13 7.28× 10−17 1.99× 10−19
Quantity I˜D (over x, z, x
′, z′) 〈J2small z〉 〈J2〉
Analytical result 1.89× 10−34 3.22× 10−9 —
Numerical result 1.88× 10−34 3.73× 10−9 5.28× 10−9
To test the sensitivity of our results to changes in the parameters, we take values
at the upper or lower limit in (6.4.28) and try to find the highest and lowest values
for 〈J2〉. We find that setting
(mu,mc,mt) := (0.85 MeV, 0.535 GeV, 174.7 GeV) ;
(md,ms,mb) := (1.71 MeV, 40 MeV, 2.98 GeV) (6.4.43)
gives
〈J2〉 ≈ 1.86× 10−9 (6.4.44)
and
∆J =
√
〈J2〉 ≈ 4.31× 10−5 , (6.4.45)
whereas setting
(mu,mc,mt) := (1.77 MeV, 0.535 GeV, 168.7 GeV) ;
(md,ms,mb) := (4.14 MeV, 71 MeV, 2.8 GeV) (6.4.46)
gives
〈J2〉 ≈ 1.52× 10−8 (6.4.47)
and
∆J =
√
〈J2〉 ≈ 1.23× 10−4 . (6.4.48)
While the result for ∆J depends on the choice of parameters, we see that the general
conclusion, which is a comparison of orders of magnitude, is unchanged by reasonable
Naturalness of CP Violation 147
modifications of the parameters (i.e. values for the quark masses). Even the greatest
possible value for ∆J is significantly lower than any of the values obtained in previous
sections.
In this subsection, we have established that assuming the observed hierarchy in
quark masses in a Gaussian distribution over the space of mass matrices gives expec-
tation values for J2 which are small enough to regard the observed value as “natural”
and not finely tuned. This statistical observation seems to open up the possibility
that the same mechanism that is responsible for the apparently unlikely hierarchy in
quark masses might also explain why the observed value for |J | is so small.
A more detailed analysis including a probability density for |J | for this distribution
is left to future work, since the numerical methods used here do not give sufficiently
accurate results.
6.4.2 Complex Mass Matrices
The corresponding calculation for general complex mass matrices differs from the
one for Hermitian mass matrices detailed in the previous subsection by a different
measure over R6, the space of possible quark masses. Hence after the integration
over R6 has been performed, the resulting integrands will be different functions of
the quantities ξa introduced in the previous section. The steps of calculating an
expectation value for J2 will be the same as in the previous subsection, and we will
use similar analytical approximations as well. It suffices therefore to highlight those
steps where the resulting formulae differ from the previous ones.
In the previous subsection we concluded that a natural measure on the space
of Hermitian mass matrices that involves the observed quark masses reproduces the
observed magnitude of CP violation very well. The quark mass hierarchy had the
effect of significantly lowering the expectation value of J2. Looking at the different
measures (6.4.9), we would expect this effect much stronger if general complex mass
matrices are considered. This expectation will be confirmed: There now appears to
be fine-tuning in the opposite way, i.e. one would expect CP violation to be much
smaller than is observed.
Using the same approximation of small y and y′ as previously, we are again left
Naturalness of CP Violation 148
with computing an integral of the form
〈J2〉 ≈
∫
R6
dDdD′
∫
d4xd4x′ s2xs2zs2x′s2z′
(
e−Tr(D
2UAU†)−Tr((D′)2U ′A′U ′†) J2(U,U ′)
) ∣∣
y=y′=0∫
R6
dDdD′
∫
d4xd4x′ s2xs2zs2x′s2z′
(
e−Tr(D2UAU†)−Tr((D′)2U ′A′U ′†)
) ∣∣
y=y′=0
,
(6.4.49)
where ∫
d4x ≡
pi/4∫
0
dx
pi/2∫
0
dz
2pi∫
0
dr
2pi∫
0
dt (6.4.50)
and similarly for
∫
d4x′. The symbol dD of course now has a different meaning. Again
we can integrate over both copies of R3 in (6.4.49), using
fξ1ξ2ξ3 :=
∞∫
−∞
dD1 dD2 dD3 (D
2
1 −D22)2(D21 −D23)2(D22 −D23)2|D1D2D3|e−ξ1D
2
1
−ξ2D22−ξ3D23
=
24
ξ51ξ
5
2ξ
5
3
(
2(ξ21ξ
2
2 + ξ
2
1ξ
2
3 + ξ
2
2ξ
2
3)(ξ
2
1 + ξ
2
2 + ξ
2
3 − ξ1ξ2 − ξ1ξ3 − ξ2ξ3) (6.4.51)
−3ξ1ξ2ξ3(ξ31 + ξ32 + ξ33 − ξ22ξ3 − ξ23ξ1 − ξ21ξ2 − ξ23ξ2 − ξ21ξ3 − ξ22ξ1)− 8ξ21ξ22ξ23
)
.
To fix the parameters appearing in the matrices A and A′, we again use expectation
values for squared quark masses:
〈D21〉 ≈
∫
R3
dDD21
pi/4∫
0
dx
pi/2∫
0
dz sin 2x sin 2z
(
e−Tr(D
2UAU†)
) ∣∣
y=0
∫
R3
dD
pi/4∫
0
dx
pi/2∫
0
dz sin 2x sin 2z
(
e−Tr(D2UAU†)
) ∣∣
y=0
. (6.4.52)
The denominator of (6.4.52) is explicitly
ID :=
pi/4∫
0
dx
pi/2∫
0
dz sin 2x sin 2z fξ1ξ2ξ3 , (6.4.53)
where ξ1, ξ2, ξ3 at y = 0 were defined in (6.4.22). Again, we can approximate ID well
by only keeping the terms of leading order in x and z in the trigonometric functions,
and approximating fξ1ξ2ξ3 by the leading term
fξ1ξ2ξ3 ≈
24
ξ51ξ
5
2ξ
5
3
× 2ξ43ξ22 ≈
48
(A1 + A2z2)5(A2 + A3x2)3A3
, (6.4.54)
Naturalness of CP Violation 149
which leads to
ID ≈ 48
A3
pi/4∫
0
dx
pi/2∫
0
dz 4xz (A1 + A2z
2)−5(A2 + A3x2)−3
≈ 48
A3
∞∫
0
dX
∞∫
0
dZ (A1 + A2Z)
−5(A2 + A3X)−3
=
48
A3
· 1
4A2A41
· 1
2A3A22
=
6
A41A
3
2A
2
3
. (6.4.55)
The result is very well reproduced by numerical calculations. Similarly,
ID〈D21〉 ≈
240
A3
∞∫
0
dX
∞∫
0
dZ (A1 + A2Z)
−6(A2 + A3X)−3
=
240
A3
· 1
5A2A51
· 1
2A3A22
=
24
A51A
3
2A
2
3
, (6.4.56)
hence
〈D21〉 ≈
4
A1
. (6.4.57)
Redoing the same calculation for D2 and D3 gives
〈D22〉 ≈
2
A2
, 〈D23〉 ≈
1
A3
. (6.4.58)
The relative factors, which are now slightly different from the ones obtained in
(6.4.27), have to be taken into account when determining A and A′.
Recall that we used the following central values for the quark masses:
(mu,mc,mt) := (1.27 MeV, 0.619 GeV, 171.7 GeV)
(md,ms,mb) := (2.9 MeV, 55 MeV, 2.89 GeV) . (6.4.59)
We now proceed as before, using numerical calculations to check our analytical
approximations. As before, the analytical approximations are better reproduced for
the positively charged top, charm and up quarks, which exhibit a more extreme quark
mass hierarchy, than for the negatively charged quarks. We use the numerical results
to fix the parameters in A and A′ to
µ2c = 2
(
mc
mt
)2
≈ 2.60× 10−5 , µ2u = 4
(
mu
mt
)2
≈ 2.19× 10−10 ,
µ2s =
(
ms
mb
)2
≈ 3.62× 10−4 , µ2d = 4
(
md
mb
)2
≈ 4.03× 10−6 . (6.4.60)
Naturalness of CP Violation 150
In order to obtain an analytical expression for expectation value of J2, we again
take the approximation of small z
J2small z :=
1
2
sin2 x cos2 x sin2 x′ cos2 x′ cos2 z′ sin4 z′ (6.4.61)
for our calculations. Within this approximation for J , still taking fξ1ξ2ξ3 ≈ 48ξ−51 ξ−32 ξ−13 ,
the numerator of (6.4.49) is the product (using again that only small z contributes)
1152×
pi/2∫
0
dz
2z
(A1 + A2z2)5/2
×
pi/2∫
0
dz′
sin 2z′ cos2 z′ sin4 z′
(A1 cos2 z′ + A2 sin2 z′)5
×
pi/4∫
0
dx
sin 2x sin2 x cos2 x
(A2 cos2 x+ A3 sin
2 x)3(A3 cos2 x+ A2 sin
2 x)
(6.4.62)
times the integral in the second line with all quantities replaced by “primed” ones.
The first two factors are 1
4A4
1
A2
and 1
12(A′
1
)2(A′
2
)3
, respectively; for the other two (which
have the same form) we change variables to X = cos2 x to obtain
1∫
1/2
dX
X(1−X)
(A2X + A3(1−X))3(A3X + A2(1−X)) =
1
2A33A2
(
1− 2A2
A3
+O
(
(A2/A3)
2)) ;
(6.4.63)
putting everything together, we obtain
〈J2small z〉 ≈
1
6
A2(A
′
1)
2
A3A′2A
′
3
=
4
3
m2sm
2
um
2
d
m4bm
2
c
, (6.4.64)
where the numerical factors appearing in the last line come from the different fac-
tors chosen in (6.4.60). Just as before, the top quark mass does not appear in this
approximation. This compares with the scaling behaviour obtained in the previous
section,
〈J2small z〉 ∼
m2smumd
m3bmc
. (6.4.65)
For numerical calculations we use both the simplified expression J2small z and the
expression for J given in (6.4.19). We find that for the first quantity, the numerically
evaluated expectation value, 〈J2small z〉 ≈ 1.89 × 10−15, is about 94% of (6.4.64), and
the numerical result for 〈J2〉 (taken at y = y′ = 0) is
〈J2〉 ≈ 2.07× 10−15 , (6.4.66)
which gives
∆J =
√
〈J2〉 ≈ 4.55× 10−8 (6.4.67)
Naturalness of CP Violation 151
which is now almost three orders of magnitude smaller than the observed value.
Assuming a Gaussian distribution peaked at zero, we now get
P (|J | ≤ 10−7) ≈ 97% , (6.4.68)
When the measure presented here is used, there seems to be extreme fine-tuning in
J , but now we would say that one observes unnaturally large CP violation! This
result may look surprising, given that the maximal value for J is around 0.1 and the
observed value just 3 × 10−5, but it shows how strongly the quark mass hierarchy
suppresses large values of J in our distribution.
6.5 Extension to Neutrinos
In the standard model, neutrinos are taken to be massless. There are no Yukawa
couplings to the Higgs field, and no possibility of CP violation. However, since the
phenomenon of neutrino oscillations has been confirmed experimentally, neutrinos
must be assumed to have mass. Curiously, a mixing matrix for leptons, now referred
to as the Maki-Nakagawa-Sakata (MNS) matrix, was already proposed in 1962
[114]. It appears in a charged current that has the same form as (6.2.7) for baryons,
with massive leptons taking the place of up-type quarks and neutrinos taking the place
of down-type quarks. With compelling evidence for neutrino oscillations [7] and hence
massive neutrinos, such an interaction is now assumed to exist. By diagonalising the
leptonic mass terms one will again encounter a mixing matrix M [95].
The mathematical problem is therefore the same as before; the only outstanding
issue is the ambiguity in the definition of this matrix due to possible rephasing of the
lepton fields. There are two different possible types of mass terms for fermions. In
the most commonly used notation, fermion fields are given by complex 4-component
(Dirac) spinors, consisting of two 2-component (Weyl) spinors:
ψ =
(
ψ1
ψ2
)
, ψ¯ := ψ†γ0 = (ψ†2 ψ
†
1). (6.5.1)
Normally, a mass term contains ψ and its Dirac conjugate ψ¯, thus coupling ψ1 and
ψ2:
Lmass = mψ¯ψ = m(ψ†2ψ1 + ψ†1ψ2). (6.5.2)
In this case, the four complex components of ψ represent a particle together with its
antiparticle, which have the same mass but are otherwise distinct. It is however also
Naturalness of CP Violation 152
possible to have Majorana mass terms, which are of the form
LMajorana = m1ψ†1ψ1 +m2ψ†2ψ2. (6.5.3)
Here m1 and m2 are not necessarily equal. In the case m1 6= m2, the 4-spinor
represents two different particles, which are each equal to their own antiparticle. It
is then appropriate to take real and imaginary parts and rewrite the theory in terms
of two real 4-component (Majorana) spinor fields.‡‡
These fields, being real, have no rephasing freedom, and the matrix relating the
flavour and mass eigenbases for neutrinos is in SO(3). Therefore, for an MNS matrix
which couples three Majorana neutrinos to the massive leptons, known to be Dirac
spinors, there is only the freedom of rephasing on one side, and this matrix would be
an element of the single quotient SU(3)/U(1)2. Of course, all intermediate cases are
also conceivable, where some neutrinos have Majorana and others have Dirac masses.
The most general mixing matrix is hence of the form U(1)k\SU(N)/U(1)N−1, for
some k between 0 and N − 1.
There are at present no strong constraints on the magnitude of CP violation from
observation. The results presented in this chapter may possibly put constraints on it,
if there is some experimental information about neutrino masses, or conversely make
near-coincident neutrino masses appear more or less likely, if the magnitude of CP
violation is known.
6.5.1 Neutrino Mixing Matrix
If we assume that the neutrinos are Majorana spinors, the lepton mixing matrix be-
longs to U(1)2\SU(3), since only phasing of the lepton charge eigenstates (νe, νµ, ντ ),
but not the mass eigenstates (ν1, ν2, ν3) is possible. The former are defined by

νe
νµ
ντ

 = M


ν1
ν2
ν3

 . (6.5.4)
In other words, |να〉 is the neutrino state created in the decay W+ → l+α +ν, where lα
is an antilepton of flavour α = e, µ, τ [7]. Such a state is in general a superposition of
‡‡There is of course no need for a theory formulated in terms of Majorana spinors to have an even
number of fields, since the Dirac spinor should not be regarded as fundamental. The presentation
only started from the Dirac spinor field because it is most commonly encountered in the literature.
Naturalness of CP Violation 153
mass eigenstates which will have different propagation amplitudes, leading to nonzero
probabilities for neutrino oscillation να → νβ.
One conventionally fixes the phases so that M takes the form
M =


1 0 0
0 c23 s23
0 −s23 c23




c13 0 s13e
−iδ
0 1 0
−s13eiδ 0 c13




c12 s12 0
−s12 c12 0
0 0 1




eiα1/2 0 0
0 eiα2/2 0
0 0 1

 ,
(6.5.5)
where the three angles θ12, θ13, and θ23 lie in the first quadrant.
The Jarlskog invariant for the neutrino mixing matrix is defined as in (6.2.16) but
with (x, y, z, w) now parametrising the MNS matrix M . Note, in particular, that it
is independent of the phases α1 and α2.
Experimentally, parameters of the neutrino mixing matrix are not completely
known. According to [7],
sin2(2θ12) = 0.87± 0.03 , 0.92 < sin2(2θ23) ≤ 1 , sin2(2θ13) < 0.19 , (6.5.6)
and there is no experimental information about the Dirac angle δ. Thus, we can cer-
tainly deduce that there is an upper bound on the Jarlskog invariant for the neutrino
mixing matrix, given by
|J | < 0.049 . (6.5.7)
For six different neutrino mass eigenstates, as in the see-saw mechanism, which
are assumed to all couple to charged leptons, a general mixing matrix would be an
element of U(1)5\SU(6), since one would diagonalise a 6× 6 Hermitian matrix. The
present viewpoint is however that such additional neutrino fields would be “sterile”
and not couple to either W or Z bosons [7].
6.5.2 Statistics of J
We have seen that the parameter space for the neutrino mixing matrix is the six-
dimensional single quotient U(1)2\SU(3), and that the Jarlskog invariant for the
neutrino mixing matrix takes the same form (6.2.16) as it does for the CKM matrix.
Therefore, all results obtained in section 6.3 apply equally to the case of neutrinos.
For completeness, we quote the results obtained in section 6.3 for a right-invariant
measure on the homogeneous space U(1)2\SU(3),
〈J2〉 = 1
720
≈ 1.389× 10−3 , 〈J4〉 = 1
201600
≈ 4.960× 10−5 , (6.5.8)
Naturalness of CP Violation 154
and
∆J =
1
12
√
5
≈ 0.0373 . (6.5.9)
This can be compared with the experimental bound given in (6.5.7).
One should also repeat the calculations of section 6.4, assuming particular values
for the neutrino masses (and including the known masses for the charged leptons).
A strong hierarchy in the neutrino masses would then presumably again lead to
“naturally” small CP violation from the corresponding mixing matrix. Alternatively,
an experimental observation of small CP violation for neutrinos would perhaps be
an indication of a mass hierarchy in neutrinos. At present, neither the magnitude
of CP violation nor any values of neutrino masses have been measured sufficiently
accurately to allow predictions.
6.6 Summary and Outlook
In this chapter, we analysed the problem of finding a natural measure on a space of
coupling constants, which in our case was the space of CKM matrices, the double
quotient U(1)2\SU(3)/U(1)2. We saw that the measure on this double quotient is
nonunique, and we analysed several possible choices of measure on the double quo-
tient. One class of measures was given by squashed Kaluza-Klein measures, induced
by a Kaluza-Klein reduction of a left-invariant metric on the flag manifold. Alterna-
tively, one could take the unique measure on SU(3)/U(1)2 and simply integrate over
the left angles. The measure used by Ozsva´th and Schu¨cking seemed not to be very
well motivated from a geometric perspective.
When calculating expectation values for J , we found that all of the measures we
considered led to rather similar results. In each case, the observed value was about
three orders of magnitude below what one would normally expect; the observed value
appears to be finely tuned. The same applied to the Ozsva´th-Schu¨cking measure,
an extremely squashed Kaluza-Klein measure, or a flat measure, which is just the
simplest choice and not justified geometrically.
In section 6.4, we adopted the different viewpoint that the CKM matrix should not
be viewed as separate from the quark masses, but that it is really the mass matrices
which are “chosen” by a yet unknown physical mechanism. We took the observed
values for the quark masses as an input and chose the simplest distribution which
was able to reproduce these observed values, while inducing a different measure on
Naturalness of CP Violation 155
the space of CKM matrices. While this is a choice we made, and all results depend
on this choice, our measure is the combination of a maximally symmetric measure,
invariant under a redefinition of a Hermitian matrix by conjugation by a unitary
matrix, and a Gaussian incorporating the observed values of the quark masses. We
would have to make additional rather strong assumptions to motivate a different
choice of measure that would differ appreciably from this simple construction. It may
well be that such assumptions are justified by the underlying mechanism determining
the mass matrices, but we do not know of such a mechanism yet. Assuming such a
distribution, together with the assumption that the mass matrices can be taken as
Hermitian as is possible in the standard model, we found that the observed value of
J now appears very natural and not finely tuned at all. In this statistical approach,
regarding the Yukawa couplings determining the mass matrices as randomly chosen
seems more appropriate than separating quark masses and mixing angles.
In order to test the dependence of this nice result on the assumption of Hermitian
mass matrices, we then repeated the calculation under the assumption that there is
a left-right symmetry which implies that the quark mass matrices can not in general
be taken to be Hermitian. We constructed the analogous probability distribution on
the space of 3× 3 complex matrices, again fitting four free parameters to expectation
values for quark masses that can reproduce the observed values. We saw that using
such a probability distribution the conclusion one reaches is rather different: One
would now expect J to be about three orders of magnitude smaller than the observed
value. Hence, there is a fine-tuning problem; without further assumptions, a fun-
damental theory leading to a left-right symmetric electroweak sector at low energy
should generically be expected to reproduce very weak CP violation. Invoking the
principle of Occam’s razor, “entia non sunt multiplicanda praeter neces-
sitatem,” we would like to conclude that, only looking at possible explanations for
the magnitude of CP violation in the electroweak sector, the standard model should
be preferred to left-right symmetric extensions such as Pati-Salam: In the latter one
needs additional assumptions on the fundamental parameters that resolve the issue
of observing “unnaturally large” CP violation, that are not necessary in the standard
model, or any extension of it that allows a restriction to Hermitian mass matrices.
In the general context of this thesis, these calculations provide an example of gauge
symmetries, described by one or two copies of SU(2), whose presence or absence has a
direct effect on measurable quantities, although we have followed a purely statistical
Naturalness of CP Violation 156
approach rather than proposing dynamical mechanisms that could lead to certain
values for these quantities.
Although we have tried to argue that our results are independent of renormalisa-
tion group flow since the relevant quantities do not run strongly with energy scale,
there is another more subtle issue: The low-energy limit of a left-right symmetric ex-
tension with non-Hermitian mass matrices would still be the standard model, where
mass matrices can be assumed to be Hermitian, leading us back to the measure we
considered previously. It would be desirable to incorporate this dependency of the
assumptions one has used to construct the measure on energy scale into the analy-
sis, namely to use a measure which depends on energy scale also. A starting point
would be a quantification of “non-Hermiticity” that could then flow from zero at low
energies to some non-zero value at high energies. At present these ideas are however
somewhat vague, so that we will have to leave them to exploration in future work.
We have argued that our analysis also applies to the case of massive neutrinos,
where the predictions will conceivably be tested by future experiments. We have seen
that for Majorana spinors, the Maki-Nakagawa-Sakata matrix [114] which appears is
naturally an element of the single quotient U(1)2\SU(3). Since the right phases do
not play any role in neutrino oscillations and the relevant J is independent of these
phases, the calculations are identical to the ones presented here, although with the
appropriate values of the µ parameters appearing in A and A′. Based on our results,
one would expect a strong hierarchy in neutrino masses to lead to suppressed CP
violation.
In the seesaw mechanism one adds very heavy right-handed neutrinos, and the
most general mixing matrix (making the additional, at present experimentally unjus-
tified, assumption that the heavy neutrinos couple to charged leptons as well) would
be an element of U(1)5\SU(6). This is naturally a Ka¨hler manifold, and the measure
induced by the Ka¨hler metric can be obtained from the analysis in [136]. We leave a
detailed treatment of this case, following our approach here, to future work.
Finally, one could analyse the effects of a fourth generation of quarks on CP viola-
tion by repeating the calculations for 4× 4 Hermitian matrices. If this generalisation
spoils the agreement with the observed J , one might obtain interesting lower bounds
on the masses of a hypothetical extra generation of quarks.
Summary and Outlook 157
Summary and Outlook
“La recherche de la ve´rite´ doit eˆtre le but de notre activite´;
c’est la seule fin qui soit digne d’elle.” [140]
Henri Poincare´
In this thesis we have investigated several examples within general relativity and
particle physics where symmetries, and more precisely the geometry of symmetry
groups and homogeneous spaces, played a fundamental role. In each of these cases, the
study of geometric aspects of the underlying symmetries led to physical predictions,
providing illuminating examples for the interplay between mathematical structures
and the physical world that modern theoretical physics is based on.
The first part consisting of chapters 2 to 4 was concerned with deformed and
constrained symmetries in relativity.
In chapter 2, we studied exact solutions of general relativity in four dimensions
from a group-theoretic viewpoint, focussing on their global symmetries (isometries).
The isometry groups of these spacetimes act simply-transitively; the spacetimes are
homogeneous spaces G/H, where H was either trivial or one-dimensional. We applied
the theory of deformations of Lie groups and algebras to these isometry groups,
finding that some solutions are related by deformation of their isometry groups, and
that these algebraic properties are partly related to physical properties; we have at
least found indications that the Kaigorodov-Ozsva´th solution contains closed timelike
curves (CTCs) just as the Petrov solution which it is a deformation of. This work,
which focussed on a few concrete relatively straightforward examples, leaves much
scope for extension. First, one could go to higher dimensions, since in four dimensions
all homogeneous solutions are more or less known. Second, one could investigate what
the generic relationship is between algebraic properties (related to isometry groups)
and physical properties (such as causal behaviour); one might be able to relate a
Summary and Outlook 158
classification of group deformations to a classification of physical properties of certain
classes of exact solutions. We leave this to further work.
In chapter 3, we considered a formulation of general relativity as a gauge theory
of the de Sitter group, shifting our focus to local symmetries. This formulation, an
extension of the MacDowell-Mansouri formulation given by Stelle and West, replaces
the perhaps more usual Poincare´ group by the de Sitter group, and provides another
example of group deformation. While the viewpoint on general relativity as some kind
of gauge theory is not new, we argued how Einstein-Cartan-Stelle-West theory may
provide a “deformation” of general relativity analogous to the framework of deformed
special relativity (DSR), and focussed on Minkowski space as an explicit example,
thereby uncovering close relations between two apparently disconnected branches of
the general relativity literature. We showed how the apparent “noncommutativity of
spacetime” given by noncommuting translations on the internal (“momentum”) de
Sitter space is manifest in torsion of the connection. We gave estimates for the torsion
one would get in the simplest situation, and discussed worrying physical consequences
if electromagnetic fields couple to torsion, such as breaking of charge conservation.
While this means that the framework we discussed is presumably not viable as a
physical theory, it provided an interesting alternative interpretation of DSR.
Our study was an application of Cartan geometry which is a generalisation of
(pseudo-)Riemannian geometry in that the “tangent” space is replaced by an arbitrary
homogeneous space G/H. It would be worth formulating other examples of such a
situation in physics in this mathematical language. One example is the traditional
formulation of loop quantum gravity (LQG) where the so(3, 1)-valued connection is
reduced to an so(3)-valued connection by partial gauge fixing. A recent study [38]
seems to generalise this gauge choice by essentially working in a Cartan geometric
construction, but the precise relation is rather unclear to us.
In chapter 4 we studied yet another reformulation of general relativity, as a topo-
logical BF theory with constraints. Instead of deforming a symmetry, we were con-
straining it; the large symmetry on the phase space of the topological theory which
implies the (local) equivalence of all solutions to the equations of motion could be
constrained to be only the gauge symmetry of general relativity in four dimensions,
where not all solutions are locally equivalent. Such a formulation, which is popular in
covariant approaches to quantising gravity, has so far mainly been given in terms of
quadratic constraints, which suffer from ambiguities in their solution whose relevance
Summary and Outlook 159
in a quantum theory is not entirely clear. It is therefore an improvement if, as we
have shown, an alternative formulation in terms of linear constraints can be found.
This formulation seems closely related to discrete constraints on representations that
are used in spin foam models, but requires in its corresponding classical formulation
the introduction of a new “4D closure” constraint that has not been studied very
much so far. An obvious extension of our work would be to study the effects of the
presence of such a new constraint on spin foam models for quantum gravity. Linear
constraints, whose solutions are formally given by projections, are also more closely
related to the group field theory approach to quantum gravity, where constraints are
usually enforced as projections [121].
The second part consisting of chapters 5 and 6 was a geometric study of gauge
symmetries in the electroweak model and their role in CP violation.
In chapter 5 we introduced a class of metrics on the group SU(3), the homogeneous
space SU(3)/U(1)2 and the double quotient space U(1)2\SU(3)/U(1)2. While for the
first two cases the SU(3) left group action provided a clear criterion (left-invariance)
for preferred classes of metrics, the situation for the double quotient was less clear
and several possible choices were introduced. We also computed bi-invariant metrics
(with respect to the natural group actions) on the space of Hermitian and on the
space of complex 3× 3 matrices.
In chapter 6 we saw how CP violation in the electroweak sector is determined by
the Jarlskog invariant J which can be viewed as a function on U(1)2\SU(3)/U(1)2,
the space of Cabibbo-Kobayashi-Maskawa (CKM) matrices. We tried to give a math-
ematically precise answer to the question whether the observed magnitude of J , which
appears to be rather small, should be considered natural or fine-tuned. Since finding
this answer is equivalent to specifying a natural measure on the double quotient space,
we analysed several possible choices of the measure induced by metrics presented in
the previous chapter. While it seems not entirely clear which of these metrics should
be preferred, we found that they all suggest fine-tuning in the magnitude of J . We
then reformulated the problem by considering the space of quark mass matrices in-
stead, using a measure which incorporates the observed quark masses and modifies
the induced distribution on the space of CKM matrices. The effect of this modified
distribution was the following: If the mass matrices can be taken to be Hermitian
as in the standard model, we found that the input of observed quark masses is suf-
Summary and Outlook 160
ficient to reproduce the observed magnitude of CP violation to a remarkably high
accuracy. If on the other hand one takes general complex mass matrices, one finds
an “inverse” fine-tuning problem; the magnitude of J should be expected to be much
smaller than is observed. This distinction is physically relevant since in models with
extended left-right symmetry, such as the Pati-Salam model, one has to consider
general complex mass matrices. Our results, which rely on assuming the simplest
and geometrically best motivated probability distributions, seem therefore to suggest
that the standard model should be preferred by Occam’s razor: One does not need
additional fine-tuning assumptions to explain the magnitude of CP violation. Hence,
these calculations provide another example of how the modification of symmetries
(the absence or presence of a left-right symmetry) leads to different physical (in this
case statistical) predictions.
Similar calculations could be performed for the case of four quark generations,
where the mass of a fourth quark would have an effect on the probability distribu-
tions we have considered. Another important extension of our results is to the case
of neutrinos, where the formalism for CP violation is rather similar and the mathe-
matics essentially the same. Our results would suggest that a large mass hierarchy
in neutrinos (together with the known hierarchy in lepton masses) makes highly sup-
pressed CP violation likely, whereas on the other hand large CP violation might
suggest that near-coincident neutrino masses appear more likely. Such predictions
can obviously be tested in experiments such as those at LHC.
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LIST OF FIGURES 176
List of Figures
2.1 Possible contractions and deformations of kinematical groups. . . . . 24
3.1 Cartan geometry extends both Klein geometry and (pseudo-)Riemannian geometry. 51
3.2 Rolling an internal de Sitter space along a closed path in Minkowski space. 67
4.1 Lagrangians and Hamiltonians for general relativity in four dimensions. 83
4.2 A 4-simplex is generated by five points. . . . . . . . . . . . . . . . . . . 88
4.3 Dual picture of a 4-simplex. . . . . . . . . . . . . . . . . . . . . . . . 99
6.1 Probability distribution for log |J | using the SU(3)-invariant flag measure. 133
6.2 Probability distribution for log |J | using the Kaluza-Klein measure. . . . . 133
6.3 Probability distribution for log |J | using a uniform distribution. . . . . . . 134
LIST OF TABLES 177
List of Tables
2.1 Infinitesimal deformations of the Petrov Killing algebra. . . . . . . . . . . 31
2.2 Infinitesimal deformations of the Kaigorodov Killing algebra. . . . . . . . 45
6.1 Analytical and numerical results for integrals of interest. . . . . . . . . . 146
INDEX 178
Index
Z2 grading, 70
4-simplex, 84
AdS/CFT, 20, 29, 49
anti-de Sitter space, 39, 40
Ashtekar new variables, 83
Bayes’ theorem, 121
BF theory, 80
bi-invariant metric
for complex matrices, 118
for Hermitian matrices, 115
on SU(3), 110
Bianchi classification, 26, 43
Cartan geometry, 51
chirality, 58
CKM matrix, 126, 128
closed timelike curves, 20, 27, 40
cohomology, 23
constraints, 10, 84, 86
first class, 10, 102
Gauss, 89, 102
simplicity, 86
cosmic microwave background (CMB), 78
cosmological constant problem, 63, 121
creation field, 77
cylinder, rotating, 27, 36
de Sitter space, 56, 59, 70
deformed special relativity (DSR), 58
development, 65, 71, 75
dihedral group S3, 115
dilatation, 38
discrete symmetries, 11
dominant energy condition, 35
double quotient, 107, 154
effective potential, 37
Einstein-Cartan theory, 53
Einstein-Cartan-Stelle-West theory, 64
Einstein-Hilbert action, 52, 63, 81, 85
Einstein-Proca theory, 77
electroweak model, 9, 107
equivalence principle, 50
Erlangen Programme, 7, 51
Euclid’s fifth postulate, 14
Euclidean group, 8
Euler angles, 108, 128
Euler-Lagrange equations, 39
exact sequence, 68
experiment, 77, 153
fine-tuning, 120, 124
flag manifold, 111, 131
gauge, 96
gauge theory, 8, 54
gauge transformation, 54, 62, 92
Gell-Mann matrices, 108
geodesic completeness, 38
INDEX 179
gravitational radiation, 44, 48
Higgs, 125
homogeneous bundles, 60
homogeneous space, 13, 129
Jacobi identity, 21, 45
Jarlskog invariant, 122, 127, 128
Kaigorodov solution, 42
Kaigorodov-Ozsva´th solution, 28
Kaluza-Klein reduction, 113, 114, 130
Killing form, 110
Killing vector, 26, 28, 42, 47
kinematical group, 23, 56
left-invariant metric, 13, 31, 34, 36, 39,
44, 46, 109
left-invariant one-form, 30
left-invariant vector field, 12
Lewis form, 29
LHC, 160
Lie algebra, 12, 54, 62
Lie bracket, 12
Lie group, 11, 122
Lie group/algebra deformation, 8, 19, 21,
31, 45, 56
Lie’s theorems, 12
M-theory, 20
MacDowell-Mansouri formulation, 53, 63
mass hierarchy, 123, 154
Maurer-Cartan form, 12, 30, 33–35, 39,
44, 46, 109, 130
Maurer-Cartan relations, 13, 23
minimal coupling, 75
MNS matrix, 151
Moyal bracket, 24
Occam’s razor, 155
orthonormal frame, 48, 51, 65, 81, 96
Pati-Salam model, 123, 126, 155
Petrov classification, 26, 28, 42
Petrov solution, 26
physical units, 73
point particle, 37
pp-wave, 42
priors, 121
quantum chromodynamics, 107
quantum gravity, 9, 55, 60, 74, 81, 85
loop, 84
spin foam approach, 81
quark mass matrix, 125
reductive geometry, 14, 44, 69, 111
renormalisation, 9, 122, 142, 156
Schro¨dinger spacetime, 20
Schwarzschild, 76
Snyder’s non-commuting coordinates, 25,
52, 56, 66
special relativity, 58
spinor, 151
Majorana, 152
stability of theories, 18
Stokes’ theorem, 89
string theory, 9, 42, 55
structure constants, 13, 22, 70
supersymmetry, 9
symmetric affine theory, 78
symmetric space, 70
symmetry breaking, 138
INDEX 180
symplectic structure, 10
time-orientable, 27, 29
topological field theory, 10, 80
torsion, 52, 66, 71, 72, 75
totally vicious, 40
transformation group, 7
unimodular, 110
unitarity triangle, 127
weak energy condition, 34, 36
Witten, 81
Wolfenstein parametrisation, 134
zero section, 67