Isometry generators in momentum representation of the Dirac theory on the de Sitter spacetime

TL;DR

This paper demonstrates that the covariant representation of the Dirac field in de Sitter spacetime decomposes into two unitary irreducible representations of the $Sp(2,2)$ group, linking them to principal series representations characterized by fermion mass and spin.

Contribution

It explicitly constructs the basis generators and Casimir operators for these representations, establishing their equivalence to principal series UIRs and analyzing associated conserved observables.

Findings

Representation decomposes into two UIRs of $Sp(2,2)$

Generators and Casimir operators are explicitly derived

Conserved observables linked to de Sitter isometries are characterized

Abstract

In this paper, it is shown that the covariant representation (CR) transforming the Dirac field under de Sitter isometries is equivalent to a direct sum of two unitary irreducible representations (UIRs) of the $Sp(2,2)$ group transforming alike the particle and antiparticle field operators in momentum representation. Their basis generators and Casimir operators are written down for the first time finding that these representations are equivalent to a UIR from the principal series whose canonical labels are determined by the fermion mass and spin. The properties of the conserved observables (i. e. one-particle operators) associated to the de Sitter isometries via Noether theorem and of the corresponding Pauli-Lubanski type operator are also pointed out.