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

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.Pacs: 04.62.+v

Canonical quantization of the covariant fields on de Sitter space–times

“…For this reason, we concentrate on the Dirac field, investigating the relation among the CRs and UIRs of the Dirac theory in momentum rep. on the de Sitter background [20]. As mentioned, these CRs are induced by the linear reps. of the group Ĝ without to meet explicitly the linear reps. of the group S(M).…”

“…is written in terms of the Pauli-Lubanski operator of componentsW A . After performing the calculation on computer we find [20] W…”

“…In this report we would like to present this approach focusing on our principal results concerning the properties of the CRs and UIRs of the de Sitter isomety group, the CR-UIR equivalence and the role of the generators of these reps. in determining the principal invariants of the quantum field theory (QFT) on the de Sitter spacetime [18,20].…”

Time evolution of the free Dirac field in spatially flat FLRW space–times

“…Note that these functions transform under isometries alike according to a unitary rep. of the isometry group such that Eq. ( 33) can be seen as defining the equivalence between the covariant rep. ( 24) and the unitary one transforming the functions a and b [26,15,16].…”

“…In general relativity the Dirac field on Riemannian manifolds is less studied [1][2][3][4][5][6][7][8][9] since one prefers the scalar field that can be manipulated easier in the actual developments in astrophysics and cosmology. For this reason we devote this paper to the Dirac field on the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes paying a special attention to the framework we need for studying the analytical solutions of the Dirac equations that can be derived on such manifolds [10][11][12][13][14][15][16], including a new type of solution reported here for the first time.…”