(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
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| Formato: | Preprint |
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2022
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| _version_ | 1866912127348899840 |
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| author | Letsios, Vasileios A. |
| author_facet | Letsios, Vasileios A. |
| contents | 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 \geq 3 $) de Sitter spacetime ($dS_{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 $dS_{N}$ ($N \geq 3$). 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 $dS_{N}$ and also introduces representation-theoretic techniques that are absent from the mathematical physics literature on half-odd-integer-spin fields on $dS_{N}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_09851 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | (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 Letsios, Vasileios A. High Energy Physics - Theory Mathematical Physics 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 \geq 3 $) de Sitter spacetime ($dS_{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 $dS_{N}$ ($N \geq 3$). 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 $dS_{N}$ and also introduces representation-theoretic techniques that are absent from the mathematical physics literature on half-odd-integer-spin fields on $dS_{N}$. |
| title | (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 |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2206.09851 |