(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

Abstract In our previous article (Letsios 2023 J. High Energy 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 ⩾ 3 ) de Sitter (dS) 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 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 ⩾ 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 .