Path integrals and deformation quantization:the fermionic case

TL;DR

This thesis develops a rigorous phase-space method for fermionic systems, enabling calculation of the star-exponential and ground state energies, and validates it on harmonic oscillators, advancing deformation quantization techniques.

Contribution

It introduces a formalism based on Grassmann variables and coherent states to compute the fermionic star-exponential and derive a fermionic Feynman-Kac formula, extending bosonic methods.

Findings

Derived a closed-form fermionic star-exponential expression.

Established a fermionic Feynman-Kac formula for ground state energy.

Validated the method on harmonic oscillators, showing the naive approach as a weak-coupling limit.

Abstract

This thesis addresses a fundamental problem in deformation quantization: the difficulty of calculating the star-exponential, the symbol of the evolution operator, due to convergence issues. Inspired by the formalism that connects the star-exponential with the quantum propagator for bosonic systems, this work develops the analogous extension for the fermionic case. A rigorous method, based on Grassmann variables and coherent states, is constructed to obtain a closed-form expression for the fermionic star-exponential from its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, allowing for the calculation of the ground state energy directly in phase space. Finally, the method is validated by successfully applying it to the simple and driven harmonic oscillators, where it is demonstrated that a simplified ("naive") approach (with an ad-hoc "remediation") is a valid weak-coupling limit of the rigorous ("meticulous") formalism, thereby providing a new and powerful computational tool for the study of fermionic systems.