Quantized Field Theory of Fuzzy Space

Aleksey M Avdeenko

doi:10.4236/jamp.2021.910163

Abstract

The proposed work handles the concept of fuzzy space as a set of real numbers with a set finite membership function. Extending standard arithmetic operations through a fuzzy space, the fuzzy Green’s function is created here with an analysis of its behavior inside and outside the light cone. The fuzzy causality principle is generalized to field models. Also, this work demonstrates the ability to use fuzzy space to regularize divergences in quantum field theory. The passage to the limit to a system of interacting particles enables the obtaining of the dissipative projection operator, represented earlier. The Liouville equation is solved here by successive approximations in the range of times much larger than the typical scale of fuzziness, by assuming the interaction as a small parameter. As well, here was applied the standard diagram technique.

Highlights

IntroductionThe gravitational field quantizing leads to some problems: the strong nonlinearity of the gravity field leads to the impossibility of regularizing divergent diagrams, which number rises with increasing the order of the perturbation theory [1] [2] [3]
The general relativity theory describes the gravity field
The gravitational field quantizing leads to some problems: the strong nonlinearity of the gravity field leads to the impossibility of regularizing divergent diagrams, which number rises with increasing the order of the perturbation theory [1] [2] [3]

Summary

Introduction

The gravitational field quantizing leads to some problems: the strong nonlinearity of the gravity field leads to the impossibility of regularizing divergent diagrams, which number rises with increasing the order of the perturbation theory [1] [2] [3] It is only in recent times, the theory of non-abelian fields gauge fields [4] [5] has played a significant role in the understanding of weak and strong nuclear forces [6] and raised hope of using similar methods for quantizing gravity fields; there is no any significant progress observed up-to-date in this direction [1]-[9]. It is fundamental that using the fuzzy Green’s function, the problem of regularization of divergences is solved in nonlinear theories, including quantizing the gravity field

Green’s Function in a Fuzzy Space

The Fuzzy Causality Principle for a Field

Regularization of Divergences

Discussion of Results

Conclusions