Commuting Heisenberg operators as the quantum response problem: Time-normal averages in the truncated Wigner representation

TL;DR

This paper extends the truncated Wigner approximation to multitime averages of Heisenberg operators, introducing a path-integral approach and linking response properties to time-normal averages, demonstrated on the Bose-Hubbard model.

Contribution

It develops an (almost) exact phase-space path-integral method for multitime averages and connects response functions to time-normal averages within the truncated Wigner framework.

Findings

Introduces a path-integral approach for time-symmetric averages.

Links response properties to time-normal averages using Kubo's formula.

Demonstrates techniques on the Bose-Hubbard model.

Abstract

The applicability of the so-called truncated Wigner approximation (-W) is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. Firstly, what class of multitime averages the -W approximates, and, secondly, how to proceed if the average in question does not belong to this class. To answer the first question we develop an (in principle, exact) path-integral approach in phase-space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The -W equations emerge as an approximation within this path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages Kubo's renowned formula relating the linear response function to two-time commutators suffices. The -W is trivially generalised to the response properties of the system allowing one to calculate approximate time-normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.