Propagators of the Dirac fermions on spatially flat FLRW spacetimes

TL;DR

This paper develops a formalism for free Dirac fermions in spatially flat FLRW spacetimes, deriving propagators and solutions, and extends a recent integral representation to these geometries, including massless neutrinos.

Contribution

It introduces a new integral representation for Dirac fermion propagators applicable to any spatially flat FLRW spacetime, including solutions for massless neutrinos.

Findings

Integral representation applies to all spatially flat FLRW geometries.

Analytic solutions for massless neutrino propagators are derived.

The formalism recovers the Minkowski case with a change of variables.

Abstract

The general formalism of the free Dirac fermions on spatially flat $(1+3)$-dimensional Friedmann-Lema\^ itre-Robertson-Walker (FLRW) spacetimes is developed in momentum representation. The mode expansions in terms of the fundamental spinors satisfying the charge conjugation and normalization conditions are used for deriving the structure of the anti-commutator matrix-functions and, implicitly, of the retarded, advanced, and Feynman fermion propagators. The principal result is that the new type of integral representation we proposed recently in the de Sitter case can be applied to the Dirac fermions in any spatially flat FLRW geometry. Moreover, the Dirac equation of the left-handed massless fermions can be analytically solved finding a general spinor solution and deriving the integral representations of the neutrino propagators. It is shown that in the Minkowski flat spacetime our new integral representation is up to a change of variable just like the usual Fourier representation of the fermion propagators. The form of the Feynman propagator of the massive fermions on a spatially flat FLRW spacetime with a scale factor of Milne-type is also outlined.