(Non-)unitarity of strictly and partially massless fermions on de Sitter space II: an explanation based on the group-theoretic properties of the spin-3/2 and spin-5/2 eigenmodes

Abstract

In our previous article [Letsios 2023 J. High Energ. Phys. JHEP05(2023)015], we showed that the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields, on $N$-dimensional ($N\ge 3$) de Sitter spacetime ($d{S}_N$) are non-unitary unless $N=4$. The (non-)unitarity was demonstrated by simply observing that there is a (mis-)match between the representation-theoretic labels that correspond to the Unitary Irreducible Representations (UIR's) of the de Sitter (dS) algebra spin$(N,1)$ and the ones corresponding to the space of eigenmodes of the field theories. In this paper, we provide a technical representation-theoretic explanation for this fact by studying the (non-)existence of positive-definite, dS invariant scalar products for the spin-3/2 and spin-5/2 strictly/partially massless eigenmodes on $d{S}_N$ ($N\ge 3$). Our basic tool is the examination of the action of spin$(N,1)$ generators on the space of eigenmodes, leading to the following findings. For odd $N$, any dS invariant scalar product is identically zero. For even $N>4$, any dS invariant scalar product must be indefinite. This gives rise to positive-norm and negative-norm eigenmodes that mix with each other under spin$(N,1)$ boosts. In the $N=4$ case, the positive-norm sector decouples from the negative-norm sector and each sector separately forms a UIR of spin$(4,1)$. Our analysis makes extensive use of the analytic continuation of tensor-spinor spherical harmonics on the $N$-sphere ($S^N$) to $d{S}_N$ and also introduces representation-theoretic techniques that are absent from the mathematical physics literature on half-odd-integer-spin fields on $d{S}_N$.

arXiv commentOnce upon a time, there was a very long paper. This paper has now been split into two parts. The first part is arXiv:2303.00420 which has been published in JHEP (doi: 10.1007/JHEP05(2023)015). The second part corresponds to the present article, arXiv:2206.09851, which has been published in the Journal of Physics A: Mathematical and Theoretical (doi: 10.1088/1751-8121/ad2c27)

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Cited by in corpus (5)

• de Sitter at all loops: the story of the Schwinger model
• New conformal-like symmetry of strictly massless fermions in four-dimensional de Sitter space
• Partially massless spin 2 and supersymmetry
• Cubic interactions for massless and partially massless spin-1 and spin-2 fields
• Unconventional conformal invariance of maximal depth partially massless fields on $d{S}_4$ and its relation to complex partially massless SUSY