Grassmann phase-space methods for fermions: uncovering classical probability structure

TL;DR

This paper develops a method to interpret Grassmann phase-space representations of fermionic quantum systems as classical probabilities, enabling more accessible analysis and computation.

Contribution

It introduces a $c$-number probability interpretation for Grassmann phase-space methods, making fermionic quantum descriptions more computationally and conceptually accessible.

Findings

Defined a measure of size and proximity for Grassmann numbers

Introduced Grassmann derivatives based on infinitesimal variations

Presented a $c$-number interpretation of Grassmann equations

Abstract

The phase-space description of bosonic quantum systems has numerous applications in such fields as quantum optics, trapped ultracold atoms, and transport phenomena. Extension of this description to the case of fermionic systems leads to formal Grassmann phase-space quasiprobability distributions and master equations. The latter are usually considered as not possessing probabillistic interpretation and as not directly computationally accessible. Here, we describe how to construct $c$-number interpretations of Grassmann phase space representations and their master equations. As a specific example, the Grassmann $B$ representation is considered. We disscuss how to introduce $c$-number probability distributions on Grassmann algebra and how to integrate them. A measure of size and proximity is defined for Grassmann numbers, and the Grassmann derivatives are introduced which are based on infinitesimal variations of function arguments. An example of $c$-number interpretation of formal Grassmann equations is presented.