Feynman path integral regularization using Fourier Integral Operator ζ-functions

Tobias Hartung*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

2 Citations (Scopus)

Abstract

We will have a closer look at a regularized path integral definition based on Fourier Integral Operator ς-functions and the generalized Kontsevich-Vishik trace, as well as physical examples. Using Feynman's path integral formulation of quantum mechanics, it is possible to formally write partition functions and expectations of observables in terms of operator traces. More precisely, Let U be the wave propagator (a Fourier Integral Operator of order 0) and Ω an observable (a pseudo-differential operator), then the expectation 〈Ω〉 can formally be expressed as (Formula presented.). Unfortunately, the operators U and UΩ are not of trace-class in general. Hence, “regularizing the path integral” can be understood as “defining these traces.” In particular, the traces should extend the classical trace on trace-class operators. We therefore consider the generalized Kontsevich-Vishik trace (i.e., Fourier Integral Operator ς-functions) since its restriction to pseudo- differential operators (obtained through Wick rotations if they are possible) is the unique extension of the classical trace. Applying the construction of the generalized Kontsevich-Vishik trace yields a new definition of the Feynman path integral whose predictions coincide with a number of well-known physical examples.

Original languageEnglish
Title of host publicationOperator Theory
Subtitle of host publicationAdvances and Applications
PublisherSpringer International Publishing
Pages261-289
Number of pages29
Volume268
DOIs
Publication statusE-pub ahead of print - 28 Apr 2018

Publication series

NameOperator Theory: Advances and Applications
Volume268
ISSN (Print)0255-0156
ISSN (Electronic)2296-4878

Keywords

  • Feynman path integral
  • Fourier Integral Operators
  • Operator ζ-functions

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