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Missing the point in noncommutative geometry.

Authors :
Huggett, Nick
Lizzi, Fedele
Menon, Tushar
Source :
Synthese; Dec2021, Vol. 199 Issue 1/2, p4695-4728, 34p
Publication Year :
2021

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. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00397857
Volume :
199
Issue :
1/2
Database :
Complementary Index
Journal :
Synthese
Publication Type :
Academic Journal
Accession number :
153650982
Full Text :
https://doi.org/10.1007/s11229-020-02998-1