On Bargmann Representations of Wigner Function

TL;DR

This paper derives a new Bargmann integral representation of the Wigner function using coherent states, offering a potentially useful approach for numerical computations and exploring geometric relations.

Contribution

It introduces a non-integral quadratic form representation of the Wigner function based on Bargmann functions, simplifying numerical calculations.

Findings

Derived a straightforward Bargmann integral representation of Wigner function

Presented a non-integral quadratic form involving a self-adjoint matrix and a recursive vector

Discussed geometric relations between Wigner function and coherent state basis uncertainty

Abstract

By using the localized character of canonical coherent states, we give a straightforward derivation of the Bargmann integral representation of Wigner function (W). A non-integral representation is presented in terms of a quadratic form V*FV, where F is a self-adjoint matrix whose entries are tabulated functions and V is a vector depending in a simple recursive way on the derivatives of the Bargmann function. Such a representation may be of use in numerical computations. We discuss a relation involving the geometry of Wigner function and the spacial uncertainty of the coherent state basis we use to represent it.