TY - CHAP

T1 - Feynman path integral regularization using Fourier Integral Operator ζ-functions

AU - Hartung, Tobias

PY - 2018/4/28

Y1 - 2018/4/28

N2 - 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.

AB - 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.

KW - Feynman path integral

KW - Fourier Integral Operators

KW - Operator ζ-functions

UR - http://www.scopus.com/inward/record.url?scp=85046292869&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-75996-8_14

DO - 10.1007/978-3-319-75996-8_14

M3 - Chapter

AN - SCOPUS:85046292869

VL - 268

T3 - Operator Theory: Advances and Applications

SP - 261

EP - 289

BT - Operator Theory

PB - Springer International Publishing

ER -