We present the dictionary between the one-particle Hilbert spaces of totally symmetric tensor-spinor fields of spin s = 3/2, 5/2 with any mass parameter on D-dimensional (D ≥ 3) de Sitter space (dSD) and Unitary Irreducible Representations (UIR’s) of the de Sitter algebra spin(D, 1). Our approach is based on expressing the eigenmodes on global dSD in terms of eigenmodes of the Dirac operator on the (D − 1)-sphere, which provides a natural way to identify the corresponding representations with known UIR’s under the decomposition spin(D, 1) ⊃ spin(D). Remarkably, we find that four- dimensional de Sitter space plays a distinguished role in the case of the gauge-invariant theories. In particular, the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on dSD, are not unitary unless D = 4.

The mode solutions of the Dirac equation on N -dimensional de Sitter space-time (dSN ) with (N − 1)-sphere spatial sections are obtained by analytically continuing the spinor eigenfunctions of the Dirac operator on the N -sphere (S N ). The analogs of flat space-time positive frequency modes are identified and a vacuum is defined. The transformation properties of the mode solutions under the de Sitter group double cover (Spin(N ,1)) are studied. We reproduce the expression for the massless spinor Wightman two-point function in closed form using the mode-sum method. By using this closed-form expression and taking advantage of the maximal symmetry of dSN we find an analytic expression for the spinor parallel propagator. The latter is used to construct the massive Wightman two-point function in closed form.

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.

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