Glauber-Sudarshan P-representations for fermions

TL;DR

This paper explores the fermionic counterpart of the Glauber-Sudarshan P-representation, deriving both bosonic and fermionic distributions and explaining their differences and relationships in single and multi-mode systems.

Contribution

It provides a derivation of fermionic P-representations and clarifies the existence of two distinct fermionic forms and their connection to bosonic cases.

Findings

Derived fermionic P-representations for single and multi-mode systems

Clarified the relationship between two fermionic forms

Compared fermionic and bosonic distributions

Abstract

The Glauber-Sudarshan P-representation is well-known within quantum optics, and is widely applied to problems involving photon statistics. Less familiar, perhaps, is its fermionic counterpart. We present a derivation of both the bosonic and fermionic distributions and, in doing so, demonstrate the reason for the existence of two distinct fermionic forms and the relationship between these. We consider both single mode systems and also multiparticle systems with many modes. For simplicity only one type of boson or fermion will be considered.