Star-exponential for Fermi systems and the Feynman-Kac formula

TL;DR

This paper develops a fermionic star-exponential formalism within deformation quantization, deriving a Feynman-Kac formula for Fermi systems, and validates it on simple oscillators as an alternative computational method.

Contribution

It introduces a fermionic star-exponential formalism and a Feynman-Kac formula within deformation quantization, extending bosonic methods to fermionic systems.

Findings

Derived a closed-form fermionic star-exponential expression.

Established a fermionic Feynman-Kac formula for phase space calculations.

Validated the approach on harmonic and driven Fermi oscillators.

Abstract

Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.