Wigner Functions on a Lattice

TL;DR

This paper demonstrates that Wigner functions can be defined on one-dimensional lattices with any number of sites, providing a general solution form and discussing their quantum expectation values.

Contribution

It corrects previous misconceptions by showing Wigner functions exist for all lattice sizes and introduces a heuristic method and general solutions for their form.

Findings

Wigner functions exist on lattices with any number of sites.

Multiple solutions satisfy the Wigner function conditions.

Quantum expectation values can be expressed using these Wigner functions.

Abstract

The Wigner functions on the one dimensional lattice are studied. Contrary to the previous claim in literature, Wigner functions exist on the lattice with any number of sites, whether it is even or odd. There are infinitely many solutions satisfying the conditions which reasonable Wigner functions should respect. After presenting a heuristic method to obtain Wigner functions, we give the general form of the solutions. Quantum mechanical expectation values in terms of Wigner functions are also discussed.