Missing the point in noncommutative geometry.
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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
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
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.
Article, Noncommutative geometry, Emergent spacetime, Quantum field theory
External DOI: https://doi.org/10.1007/s11229-020-02998-1
This record's URL: https://www.repository.cam.ac.uk/handle/1810/330879