(Non-)unitarity of strictly and partially massless fermions on de Sitter space

Abstract

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.

(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

“…The main aim of this paper is to provide a technical explanation for the results of our previous paper [1], and, in particular, of the main result:…”

“…This is a technical sequel to our previous paper [1], in which we constructed a 'field theoryrepresentation theory' dictionary for totally-symmetric spin-s = 3/2, 5/2 tensor-spinors on Ndimensional (N ⩾ 3) de Sitter (dS) spacetime (dS N ). Totally symmetric tensor-spinors, Ψ µ1...µr , of spin s ≡ r + 1/2 on dS N satisfy [2,3] (∇ / + M) Ψ µ1...µr = 0 (1.1) ∇ α Ψ αµ2...µr = 0, γ α Ψ αµ2...µr = 0, (…”

“…Main results of our previous paper. In our previous article [1], we constructed the spins = 3/2, 5/2 eigenmodes of equations (1.1) and (1.2) on global dS N (N ⩾ 3). Then, we investigated the (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 eigenmodes.…”

“…We found that for real values of M the representations are unitary. However, the main interesting result of [1] concerns the strictly and partially massless theories (i.e. the theories that enjoy a gauge symmetry).…”

“…[The analysis of the previous [1], as well as of the present, papers focuses only on the cases with r = 1 and r = 2. However, the 'field theory-representation theory' dictionary of our previous paper [1] suggests that our main result extends to all strictly/partially massless (totally symmetric) tensor-spinors of spin s ⩾ 7/2.] The quantity τ is known as the depth of the strictly/partially massless field (i.e.…”