Canonical quantization of the covariant fields on de Sitter spacetimes

TL;DR

This paper develops a canonical quantization framework for covariant quantum fields on de Sitter spacetimes, establishing their equivalence with unitary irreducible representations of the de Sitter isometry group, especially for Dirac fields.

Contribution

It provides a detailed analysis of covariant quantum fields on de Sitter space, linking covariant representations with unitary irreducible ones and explicitly deriving their generators and Casimir operators.

Findings

Covariant representations are equivalent to principal series unitary irreducible representations.

Dirac fields' covariant representation decomposes into a sum of two irreducible representations.

Explicit forms of generators and Casimir operators are derived for these representations.

Abstract

The properties of the covariant quantum fields on de Sitter spacetimes are investigated focusing on the isometry generators and Casimir operators in order to establish the equivalence among the covariant representations and the unitary irreducible ones of the de Sitter isometry group. For the Dirac quantum field it is shown that the spinor covariant representation, transforming the Dirac field under de Sitter isometries, is equivalent with a direct sum of two unitary irreducible representations 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 finding that the covariant representations are equivalent with unitary irreducible ones from the principal series whose canonical weights are determined by the fermion mass and spin.