The Phase Space Formulation of Time-Symmetric Quantum Mechanics, I: the Wigner Formalism

TL;DR

This paper reformulates time-symmetric quantum mechanics within the Wigner formalism, providing explicit formulas for state reconstruction and highlighting interference effects between pre- and post-selected states.

Contribution

It introduces a phase space approach to time-symmetric quantum mechanics using the Wigner formalism, generalizing state reconstruction to arbitrary observables.

Findings

Explicit formulas for state reconstruction in time-symmetric quantum mechanics

Identification of interference effects between pre- and post-selected states

Extension of Wigner formalism to time-symmetric quantum frameworks

Abstract

Time-symmetric quantum mechanics can be described in the usual Weyl--Wigner--Moyal formalism (WWM) by using the properties of the Wigner distribution, and its generalization, the cross-Wigner distribution. The use of the latter makes clear a strongly oscillating interference between the pre- and post-selected states. This approach allows us to give explicit formulas for the state reconstruction problem, thus generalizing known results to the case of arbitrary observables. In a forthcoming paper we will extend these results to other quantization schemes.