Path-integral quantization of Galilean Fermi fields

TL;DR

This paper develops a path-integral quantization framework for Galilean fermionic fields using a five-dimensional covariant approach, deriving Green's functions and analyzing interactions in non-relativistic quantum systems.

Contribution

It introduces a five-dimensional Lorentz-like covariant method for quantizing Galilean fermions and derives Green's functions for interacting theories, extending the non-relativistic field theory toolkit.

Findings

Compatible with existing non-relativistic many-body results

Provides compact expressions using extended (4+1) manifold diagrams

Derives Green's functions for quartic self-interacting potentials

Abstract

The Galilei-covariant fermionic field theories are quantized by using the path-integral method and five-dimensional Lorentz-like covariant expressions of non-relativistic field equations. Firstly, we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Levy-Leblond equations, and define the Galilean generating functional and Green's functions for positive- and negative-energy/mass solutions. Then, as an example of interactions, we consider the quartic self-interacting potential ${\lambda} (\bar{\Psi} {\Psi})^2$, and we derive expressions for the 2- and 4-point Green's functions. Our results are compatible with those found in the literature on non-relativistic many-body systems. The extended manifold allows for compact expressions of the contributions in $(3+1)$ space-time. This is particularly apparent when we represent the results with diagrams in the extended $(4+1)$ manifold, since they usually encompass more diagrams in Galilean $(3+1)$ space-time.