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Quantifying Intermembrane Distances with Serial Image Dilations
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Missing the point in noncommutative geometry.

Nick Huggett1, Fedele Lizzi2,3,4, Tushar Menon5

  • 1Department of Philosophy, University of Illinois at Chicago, Chicago, IL USA.

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|December 6, 2021
PubMed
Summary
This summary is machine-generated.

Points do not exist in noncommutative geometries. Researchers show that small regions lack formal and operational definitions, questioning the fundamental nature of points in these advanced geometric theories.

Keywords:
Emergent spacetimeNoncommutative geometryQuantum field theory

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Area of Science:

  • Theoretical Physics
  • Differential Geometry
  • Quantum Field Theory

Background:

  • Noncommutative geometries extend standard smooth geometries by incorporating a fundamental quantity with dimensions of area.
  • The concept of points and regions smaller than a fundamental scale is questioned within these generalized geometries.

Purpose of the Study:

  • To investigate the definability and operational existence of small regions and points in noncommutative geometries.
  • To explore the metaphysical implications of pointlessness in these geometries.
  • To understand how smooth manifold behavior can emerge as an approximation.

Main Methods:

  • Analysis within Connes' spectral triple approach to assess formal definability of regions.
  • Application of the scalar field Moyal-Weyl approach to evaluate operational definitions of regions.
  • Metaphysical and approximation-based investigations.

Main Results:

  • Arbitrarily small regions are not formally definable in the spectral triple approach.
  • Small regions lack an operational definition in the Moyal-Weyl approach.
  • The study concludes that points do not exist in noncommutative geometries.

Conclusions:

  • The fundamental nature of points is incompatible with noncommutative geometries.
  • Further investigation is needed into the metaphysical underpinnings and emergent properties of these geometries.
  • Smooth manifold appearance can be understood as an approximation to noncommutative geometry.