F Fuzzy Topology of Phase Space and Gauge Fields S.Mayburov SSss - - PowerPoint PPT Presentation

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F Fuzzy Topology of Phase Space and Gauge Fields S.Mayburov SSss - - PowerPoint PPT Presentation

Feb 19, 2024 368 likes •577 views

F Fuzzy Topology of Phase Space and Gauge Fields S.Mayburov SSss J.Phys A 41 (2008) 164071 Motivations: Study of Mathematics foundations can be important for the construction of quantum space-time Axioms of Set theory and Topology are the

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F Fuzzy Topology of Phase Space and Gauge Fields

SSss

S.Mayburov

J.Phys A 41 (2008) 164071 Motivations: Study of Mathematics foundations can be important for the construction

  • f quantum space-time

Axioms of Set theory and Topology are the basis of any geometry Examples:

Discrete space-time (Snyder, 1947) Noncommutative geometry (Connes, 1991)

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S Sets, Topology and Geometry

Example: 1-dimensional Euclidian geometry is constructed on ordered set of elements X = { xl } ; xl - points

j i x

x , ∀

i j j i

x x

  • r

x x ≤ ≤ . .

PPartial ordered set – Poset P x :

j i

x x ≤

Beside ,Itit can be also xi ~ xj

ixi , xj are incomparable elements

X f xj

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Example: P = X U P f ; P f = { f l }

X

x xi

f1

DDx Ff1 ~ xl ,

Dx xl ∈ ∀

f j - fuzzy points, (Zeeman, 1968)

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Conclusions

1.1. Fuzzy topology is the most simple and natural formalism for introduction

  • f quantization into physical theory
  • 2. Shroedinger equation is obtained from simple assumptions
  • 3. Gauge invariance of fields corresponds to dynamics on fuzzy manifold
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Red Blue UV

photon energy