Moyal Brackets, Star Products and the Generalised Wigner Function

TL;DR

This paper explores the generalization of the Wigner distribution function within the Wigner-Weyl-Moyal framework, linking it to non-local constructions and solutions of a generalized Moyal-Nahm equation, with implications for quantum and classical probability theories.

Contribution

It introduces a novel generalization of the Wigner function as a bilinear convolution of fermionic objects, connecting it to a BPS generalized Moyal-Nahm equation.

Findings

Wigner functions can be constructed as bilinear convolutions of fermionic objects.

The generalized Wigner functions solve a BPS Moyal-Nahm equation.

The approach emphasizes non-locality in quantum phase space representations.

Abstract

The Wigner-Weyl- Moyal approach to Quantum Mechanics is recalled, and similarities to classical probability theory emphasised. The Wigner distribution function is generalised and viewed as a construction of a bosonic object, a target space co-ordinate, for example, in terms of a bilinear convolution of two fermionic objects, e.g. a quark antiquark pair. This construction is essentially non-local, generalising the idea of a local current. Such Wigner functions are shown to solve a BPS generalised Moyal-Nahm equation.