The (partially) massless spin-3/2 and spin-5/2 fields in de Sitter spacetime as unitary and non-unitary representations of the de Sitter algebra

Abstract

The divergence-free and gamma-traceless vector-spinor eigenfunctions, as well as the divergence-free and gamma-traceless rank-2 symmetric tensor-spinor eigenfunctions, of the Dirac operator on the N -sphere (S N ) are written down explicitly for N ≥ 3. The spin-3/2 and spin-5/2 eigenmodes of the Dirac operator with arbitrary imaginary mass parameter on N -dimensional (N ≥ 3) de Sitter spacetime (dS N ) are obtained by analytic continuation. Their transformation properties under the de Sitter algebra spin(N, 1) are studied. For N odd, we show that there is no de Sitter (dS) invariant scalar product for these eigenmodes. For N even, although dS invariant scalar products exist, positive-definiteness of the norm occurs only for the strictly and partially massless theories in N = 4 dimensions. For N = 4, the way in which the eigenmodes form unitary strictly and partially massless representations of spin(4, 1) is emphasised. The analysis presented in this paper reveals previously unknown features of the gauge-invariant theories with spin 3/2 and 5/2 on dS N (N ≥ 3): the strictly massless spin-3/2 field theory, as well as the strictly and partially massless spin-5/2 field theories, are unitary only for N = 4. In particular, a unitary theory for the gravitino field on dS N does not exist unless N = 4.

Particles of a de Sitter Universe

“…On the other hand, the massless spin− 3 2 field is the gravitino field, known to be a fermionic gauge field in supersymmetry literature. From this perspective the results of [35] supported by representation theory of the de Sitter group, connect supersymmetry with four spacetime dimensions in the presence of the cosmological constant. We leave it to future studies to extend our list of late-time operators to include cases of nonzero spin.…”

“…Following [13], in four dimensions the exceptional series category is considered equivalent to a discrete series category and we recognize α M N and β M N to belong to discrete series representations, which also agrees with [32]. Other examples in this category are the nonzero spin fields in Higher Spin theory on de Sitter [34] and the spin− 3 2 and 5 2 fermions [35]. Another example of identifying discrete series representations comes from modified gravity literature [36], where the tensor degrees f freedom at a perturbative level are recognized to belong to discrete series representations of the de Sitter group.…”

“…In our discussion of late-time operators, we paid particular attention to their normalization. Normalization of states that can arise from fields on de Sitter have also been a guiding property in obtaining ranges of masses for gauge fields [88] or dimensionality for fermions [35]. Reference [88] studies allowed mass ranges for scalar, vector and spin two fields on de Sitter by considering equal-time commutations in Poincaré patch and analysing the norm of the states these fields give rise to.…”

“…As a result, for the mass M in terms of the value of the cosmological constant Λ, the range 0 < M 2 < 2 3 Λ is forbidden for spin−2 as it gives rise to negative norm states. Reference [35] analysis covers spin− 3 2 and spin− 5 2 in terms of the unitary irreducible representations of Spin(d + 1, 1), and the double cover of de Sitter group in general dimensions using analytic continuation methods from the sphere to de Sitter. To be more precise, it is the strictly massless spin− 3 2 , strictly and partially massless spin− 5 2 fields that are studied.…”