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Missing the point in noncommutative geometry

Huggett, Nick and Lizzi, Fedele and Menon, Tushar (2020) Missing the point in noncommutative geometry. [Preprint]

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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.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Huggett, Nickhuggett@uic.edu0000-0002-5699-5479
Lizzi, Fedele
Menon, Tushartushar.menon@philosophy.ox.ac.uk
Additional Information: This paper partially overlaps http://philsci-archive.pitt.edu/15432/, which also contains an appendix for those wanting a summary of the mathematics involved.
Keywords: spacetime, geometry, quantum gravity, noncommutative geometry
Subjects: General Issues > Scientific Metaphysics
Specific Sciences > Physics > Fields and Particles
Specific Sciences > Physics > Quantum Gravity
Depositing User: Nick Huggett
Date Deposited: 25 Jun 2020 02:52
Last Modified: 25 Jun 2020 02:52
Item ID: 17373
Subjects: General Issues > Scientific Metaphysics
Specific Sciences > Physics > Fields and Particles
Specific Sciences > Physics > Quantum Gravity
Date: 23 June 2020
URI: http://philsci-archive.pitt.edu/id/eprint/17373

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