Path-integral quantization of Galilean Fermi fields

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. First, we review the five-dimensional approach to the Galilean Dirac equation, which leads to the Lévy-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 λ(Ψ¯Ψ)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.