Local-time representation of path integrals

TL;DR

This paper introduces a local-time path-integral representation for one-dimensional quantum systems, offering an alternative to the Feynman-Kac formula, especially useful at temperature extremes, with applications to analyzing the Bloch density matrix.

Contribution

It presents a novel local-time path-integral formulation for quantum systems, generalizing existing methods and connecting with Sturm-Liouville and variational theories.

Findings

Provides a new local-time representation for the Bloch density matrix.

Demonstrates the method's effectiveness at high and low temperatures.

Connects the local-time approach with Sturm-Liouville and variational principles.

Abstract

We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high and low temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle are also discussed.