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

Abstract

The applicability of the so-called truncated Wigner approximation $(\ensuremath{-}W)$ is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the $\ensuremath{-}W$ approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (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 $\ensuremath{-}W$ equations emerge as an approximation within these 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 $\ensuremath{-}W$ is directly generalized 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.