Missing the point in noncommutative geometry.
Publication Date
2021Journal Title
Synthese
ISSN
0039-7857
Publisher
Springer Science and Business Media LLC
Volume
199
Issue
1-2
Pages
4695-4728
Language
en
Type
Article
This Version
VoR
Metadata
Show full item recordCitation
Huggett, N., Lizzi, F., & Menon, T. (2021). Missing the point in noncommutative geometry.. Synthese, 199 (1-2), 4695-4728. https://doi.org/10.1007/s11229-020-02998-1
Description
Funder: John Templeton Foundation; doi: http://dx.doi.org/10.13039/100000925
Funder: American Council of Learned Societies; doi: http://dx.doi.org/10.13039/100000962
Funder: Institute of Philosophy, University of London
Abstract
Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale-and ultimately the concept of a point-makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes' spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal-Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.
Keywords
Article, Noncommutative geometry, Emergent spacetime, Quantum field theory
Sponsorship
MINECO (MDM-2014-0369)
Identifiers
s11229-020-02998-1, 2998
External DOI: https://doi.org/10.1007/s11229-020-02998-1
This record's URL: https://www.repository.cam.ac.uk/handle/1810/330879
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
Statistics
Total file downloads (since January 2020). For more information on metrics see the
IRUS guide.