Propagators of the Dirac fermions on spatially flat FLRW space–times

Abstract

The general formalism of the free Dirac fermions on spatially flat (1 + 3)-dimensional Friedmann-Lemaître-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.Pacs: 04.62.+v

Kinematics in spatially flat FLRW space-time

“…The second example we presented here for the first time is the kinematics of a new manifold we considered recently in quantum theory [17,18]. This is a spatially flat FLRW space-time with a Milne type scale factor produced by gravitational sources proportional with t −2 .…”

“…Let us finish with an example of manifold whose kinematics was never studied. This is the spatially flat FLRW manifold M with the Milne type scale factor a(t) = ω M t defined on the domain t ∈ (0, ∞), whose constant (frequency) ω M is introduced from dimensional reasons [17,18]. Then we may write the line element in the physical co-moving frame {t, x} as…”

“…Here the constant ω M is an useful free parameter representing the expansion speed of M. We remind the reader that in the case of the genuine Milne universe (of negative space curvature but globally flat) one must set ω M = 1 for eliminating the gravitational sources [2]. In contrast, our space-time M is produced by isotropic gravitational sources, i. e. the density ρ and pressure p, evolving in time as [17] ρ = 3 8πG…”

First order QED processes in a spatially flat FLRW space-time with a Milne-type scale factor

“…The fundamental spinors are solutions of the free Dirac equation whether the modulation functions u ± p (t) and v ± p (t) satisfy the first order differential equations 14) in the chart with the proper time. The solutions of these systems must satisfy the charge-conjugation symmetry [19], 15) and the normalization conditions…”

Integral representation of the scalar propagators on the de Sitter expanding universe

“…Therefore, we must look for another type of integral representation able to take over the effects of the Heaviside functions. Recently we succeeded to find a new integral representation of the propagators of the Dirac field on the de Sitter [20] or any spatially flat FLRW [21] spacetimes which is different from the usual Fourier integrals allowed in special relativity. Here we would like to continue this study applying the same method to the massive and charged Klein-Gordon fields, minimally coupled to the gravity of the de Sitter expanding universe, writing down for the first time the new integral representation of their propagators.…”