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

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 ($D \geq 3$) de Sitter space ($dS_{D}$) and Unitary Irreducible Representations (UIR's) of the de Sitter algebra spin$(D,1)$. Our approach is based on expressing the eigenmodes on global $dS_{D}$ 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$(D,1)$ $\supset$ 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 $dS_{D}$, are not unitary unless $D=4$.


Introduction
1.1 Strictly and partially massless field theories in de Sitter space 1.2 Eigenmodes, 'field theory-representation theory' dictionary and purpose of this paper 1.3 Main result for strictly and partially massless theories of spin s = 3/2, 5/2 1. 4  The de Sitter spacetime, apart from its relevance to inflationary cosmology, is also thought to be a good model for the asymptotic future of our Universe, as suggested by current experimental evidence in favor of a positive cosmological constant [1][2][3].The D-dimensional de Sitter spacetime (dS D ) is the maximally symmetric solution of the vacuum Einstein field equations with positive cosmological constant Λ [4] where g µν is the metric tensor, R µν is the Ricci tensor and R is the Ricci scalar.Throughout this paper we use units in which the cosmological constant is i.e. the de Sitter radius is one.Unlike Minkowskian field theories, possible field theories of spin s on dS D are not restricted to the two usual cases of massive and strictly massless theories, where for D = 4 the former has 2s + 1 propagating degrees of freedom (DoF), while the latter has only 2 helicity DoF (±s) due to the gauge invariance of the theory [5].On dS D there also exist intermediate gauge-invariant theories for s ≥ 2, known as partially massless1 theories [6][7][8][9][10].For a given spin s ≥ 1, there exists one strictly massless theory and [s] − 1 different partially massless theories, where [s] = s if the spin s is an integer and [s] = s − 1/2 if s is a half-odd integer.Partial masslessness was first observed for the spin-2 field by Deser and Nepomechie [11,12] and for higher integer-spin fields by Higuchi [14].Partially massless theories with various spins have been discussed further in a series of papers by Deser and Waldron [6][7][8][9][10]15].Note that this paragraph, as well as the rest of the paper, refers only to totally symmetric tensor and tensor-spinor fields.Mixed-symmetry tensor fields on dS D -for which strict and partial masslessness also occur -have been discussed in Ref. [17].
Each strictly or partially massless theory of spin s is conveniently labeled by a distinct value of the 'depth' τ = 1, 2, ..., [s] (where the value τ = 1 corresponds to strict masslessness) and in 4 dimensions there are 2τ propagating helicities, namely: (±s, ±(s − 1), ..., ±(s − τ + 1)) [6,8,9].For given spin s and depth τ, each of these gauge-invariant theories corresponds to a distinct tuning of the mass parameter to the cosmological constant Λ [6,8,9,14,15].Higuchi classified the tunings of the mass parameter for all strictly and partially massless theories with arbitrary integer spin by studying the group-theoretic properties of the eigenmodes of the Laplace-Beltrami operator on dS D [14,16].Deser and Waldron gave an analogous classification for arbitrary integer and half-odd-integer spins by using group representation methods based on the de Sitter/CFT correspondence [15].

Eigenmodes, 'field theory-representation theory' dictionary and purpose of this paper
Unitarity of field theories is very important for physical problems since it ensures the positivity of probabilities.A sufficient condition for field-theoretic unitarity on dS D is that of the unitarity of the underlying representation of the de Sitter (dS) algebra, spin(D, 1).
Particles in a D-dimensional dS universe correspond to Unitary Irreducible Representations (UIR's) of spin(D, 1).Representation-theoretic insight from eigenmodes.The interplay between free field theory on dS D and representation theory of spin(D, 1) manifests beautifully itself in the solution space -consisting of eigenmodes -of the corresponding field equation. 2Let us briefly discuss Higuchi's work [14,16] in order to demonstrate the great amount of representationtheoretic knowledge that we can obtain for a free field theory on dS D by studying its eigenmodes.In particular, in Refs.[14,16] Higuchi studied the group-theoretic properties of totally symmetric tensor eigenmodes of the Laplace-Beltrami operator on dS D (D ≥ 3).
In these works, he showed that the phenomenon of partial masslessness exists for all totally symmetric tensor fields of spin s ≥ 2 on dS D by detecting pure gauge modes (these eigenmodes indicate the gauge invariance of the theory).Also, by calculating the norm of the physical strictly/partially massless eigenmodes using a dS invariant scalar product, he showed that all strictly and partially massless theories with arbitrary integer spin s are unitary for all D ≥ 3.Moreover, he showed that for all integer spins there exist mass (parameter) ranges where the eigenmodes have negative norm -i.e. the corresponding spin(D, 1) representations are non-unitary.The unitary strictly/partially massless theories appear at special tunings of the mass parameter corresponding to the boundaries of the 'forbidden' mass ranges -see Deser and Waldron's works for a detailed analysis and a physical insight into these 'forbidden' ranges [6][7][8][9].Last, Higuchi's group-theoretic analysis of the eigenmodes showed that there is a lower bound for the mass parameter of integer-spin fields, below which the fields can only be non-unitary 3 .This bound is known as the 'Higuchi bound' in the modern literature -see, e.g Ref. [13,28].'Field theory-representation theory' dictionary and a gap in the literature.The basis elements of spin(D, 1) correspond to the (D + 1)D/2 Killing vectors of dS D and they act on eigenmodes in terms of Lie derivatives (or spinorial generalizations thereof [26,37]).The (spinorial) Lie derivatives with respect to Killing vectors commute with the field equation of the free theory [26,37] and the solution space is identified with the representation space of a -often irreducible -representation of spin(D, 1) [14,16].What we would like to know is whether this representation, which is formed by eigenmodes, is unitary.Fortunately, all UIR's of spin(D, 1) have been classified by Ottoson and Schwarz [19,20] (see also Refs.[30][31][32]).Thus, as field theorists, we would like to construct a dictionary between the known UIR's of spin(D, 1) and eigenmode spaces (i.e.one-particle Hilbert spaces) of free field theories on dS D .Such a dictionary was first constructed by Higuchi [16] for to-tally symmetric integer-spin fields 4 and was later extended to mixed-symmetry integer-spin fields by Basile, Bekaert and Boulanger [17].However, a detailed study of the dictionary for tensor-spinor fields for arbitrary D is absent from the literature5 .Main aim.It is the purpose of the present article to construct the dictionary between one-particle Hilbert spaces (consisting of eigenmodes) and UIR's of spin(D, 1) for the vectorspinor (i.e.spin-3/2) field and symmetric rank-2 tensor-spinor (i.e.spin-5/2) field on dS D .
1.3 Main result for strictly and partially massless theories of spin s = 3/2, 5/2 The dictionary between one-particle Hilbert spaces of unitary spin-s = 3/2, 5/2 field theories on dS D and UIR's of spin(D, 1) will be given in Section 7 (for both massive and strictly/partially massless fields).However, here we would like to draw attention to our remarkable main result concerning the strictly and partially massless theories: • Main result: The strictly massless spin-3/2 field (gravitino field) and the strictly and partially massless spin-5/2 fields on dS D (D ≥ 3) are not unitary unless D = 4.
(The case with D = 2 is not discussed in the present article.)As we will see later, our analysis for the spin-3/2 and spin-5/2 cases suggests that our main result should hold for all strictly and partially massless fields with half-odd-integer spin s ≥ 3/2.According to our main result, four-dimensional dS space plays a distinguished role in the unitarity of the strictly massless spin-3/2 field and the strictly and partially massless spin-5/2 fields.This is an example of a remarkable and previously unknown feature of dS field theory that has no known field-theoretic counterparts in anti-de Sitter and Minkowski spacetimes.As will become clear, the significance of four-dimensional dS space is related to the representation theory of spin(D, 1), where the latter allows (totally symmetric) fermionic strictly/partially massless UIR's only for D = 4 (corresponding to a direct sum of spin(4, 1) UIR's in the Discrete Series -see Section 7).Also, although it might be a mere mathematical coincidence, it is interesting that the dimensionality that plays a special representationtheoretic role happens to correspond to the number of the observed macroscopic dimensions of our Universe.

Strategy
Our strategy in order to construct the dictionary between spin(D, 1) UIR's and spin-s = 3/2, 5/2 one-particle Hilbert spaces on dS D is based on constructing the dS eigenmodes using the method of separation of variables [18,21,22].More specifically, we are going to express the spin-3/2 and spin-5/2 eigenmodes on global dS D in terms of tensor-spinor eigenmodes of the Dirac operator on S D−1 .This will help us determine the spin(D) content of the spin(D, 1) representations formed by the eigenmodes on dS D -by spin(D) content we mean the irreducible representations of spin(D) that appear in a spin(D, 1) representation under the decomposition spin(D, 1) ⊃ spin(D) [19,20].We will also obtain the values of the spin(D, 1) quadratic Casimir corresponding to the eigenmodes on dS D .Once we have determined both the quadratic Casimir and the spin(D) content for the representations formed by the dS eigenmodes, we will be able to construct the dictionary between oneparticle Hilbert spaces and UIR's of spin(D, 1) by using the known classification of UIR's [19,20] under the decomposition spin(D, 1) ⊃ spin(D).We also provide the dictionary for the spin-1/2 field (as the group-theoretic properties of the spin-1/2 eigenmodes on global dS D have been already studied by the author [22]), while our analysis also allows us to propose a dictionary for totally symmetric tensor-spinors of any spin s ≥ 3/2.
As for our main result concerning the strictly/partially massless theories of spin s = 3/2, 5/2, we will show that for D = 4 there is a mismatch between the values of the quadratic Casimir for the strictly/partially massless eigenmodes and the values corresponding to the UIR's of spin(D, 1) and/or another mismatch between the representation labels of the eigenmodes and the allowed labels in spin(D, 1) UIR's.(The spin(D, 1) representation labels we use in this paper specify a spin(D, 1) representation under the decomposition spin(D, 1) ⊃ spin(D) [14,16,19,20] and their role is similar to the role played by the highest weights in spin(D + 1) representations -see Section 3.) In other words, we will demonstrate that there are no UIR's of spin(D, 1) that correspond to the strictly massless spin-3/2 field and to the strictly and partially massless spin-5/2 fields on dS D for D = 4.However, for D = 4, both the quadratic Casimir and the representation labels of the strictly/partially massless theories correspond to the Discrete Series UIR's of spin(4, 1).
An alternative technical explanation.A technical explanation of all the results reported in this paper can be given by studying the (non-)existence of positive-definite dS invariant scalar products for the spin-3/2 and spin-5/2 eigenmodes on dS D .Such an analysis has been carried out in detail by the author and will be presented in a separate article [40,41], in which the author has extended Higuchi's methods [14,16] to the case of spin-3/2 and spin-5/2 eigenmodes on dS D (D ≥ 3).In particular, in Refs.[40,41] the author has proved the following results for the strictly/partially eigenmodes of spin s = 3/2, 5/2 on dS D (D ≥ 3): • For odd D all dS invariant scalar products are identically zero.
• For even D > 4 all dS invariant scalar products are indefinite giving always rise to positive-norm and negative-norm eigenmodes that mix with each other under spin(D, 1) boosts.
• The D = 4 case is special as the positive-norm sector decouples from the negativenorm sector.Then, both sectors can be viewed as positive-norm sectors and each sector independently forms a spin(4, 1) UIR in the Discrete Series.
Although we have not performed such a technical analysis for the eigenmodes with halfodd-integer spin s ≥ 7/2, the analysis of our present paper suggests that our main result extends to all strictly and partially massless fields with half-odd-integer spin s ≥ 7/2 on dS D .

Outline of the paper, notation and conventions
The rest of the paper is organised as follows.In Section 2, we begin by presenting the basics about tensor-spinor fields on dS D (gamma matrices, vielbein fields, spin connection, and the spinorial generalisation of the Lie derivative) and, then, we specialise to the global slicing of dS D .In Section 3, we review the classification of the spin(D, 1) UIR's under the decomposition spin(D, 1) ⊃ spin(D) given originally in Refs.[19,20].In Section 4, we begin by discussing the totally symmetric tensor-spinor eigenmodes of the Dirac operator on S D−1 that are also gamma-traceless and divergence-free, as well as the way they form representations of spin(D) (Subsection 4.1).Then, using the aforementioned eigenmodes on S D−1 , we present the construction of the TT eigenmodes of the spin-3/2 field on dS D for both even D ≥ 4 (Subsecton 4.2) and odd D ≥ 3 (Subsection 4.3), in order to illustrate the method of separation of variables for tensor-spinor fields.The spin(D) content of the spin(D, 1) representations formed by the spin-3/2 eigenmodes is also identified and the main results are tabulated in Tables 1 and 2. In Subsection 4.4, we present our basic results concerning the TT eigenmodes for the spin-5/2 field on dS D (D ≥ 3).In Section 5, we obtain the quadratic Casimir for the spin(D, 1) representation formed by eigenmodes with half-odd-integer spin s ≥ 1/2 on dS D by using "analytic continuation" techniques that relate dS D to S D .In Section 6, after identifying the pure gauge and physical modes of our strictly/partially massless theories (Subsection 6.1), we prove the main result of this paper, i.e. the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on dS D , are not unitary unless D = 4 (Subsection 6.2).In order to achieve this, we take advantage of both the spin(D) content and the quadratic Casimir corresponding to our physical modes on dS D and then we show that they do not agree with any UIR of spin(D, 1) unless D = 4.In Section 7, we present our dictionary between spin(D, 1) UIR's and (totally symmetric) tensor-spinor fields with arbitrary mass parameters on dS D (D ≥ 3).Although in the main part of the present paper we discuss the spin-3/2 and spin-5/2 fields, our analysis allows us to propose a dictionary for all (totally symmetric tensor-)spinor fields with spin s ≥ 1/2.
Notation and conventions.We use the term 'tensor-spinor field of rank r' in order to refer to a r th -rank tensor where each one of its tensor components is a spinor.Other authors prefer the name spinor-tensors for these objects -see, e.g., Ref. [18].We use the mostly plus metric sign convention for dS D .Lowercase Greek tensor indices refer to components with respect to the 'coordinate basis' on dS D .Coordinate basis tensor indices on S D−1 are denoted as μ1 , μ2 , ... .Lowercase Latin tensor indices are 'flattened', i.e. they refer to components with respect to the vielbein basis (the indices a, b, c, d, f run from 0 to D − 1, while the indices i, j, k run from 1 to D − 1).Summation over repeated indices is understood.We denote the symmetrisation of a pair of indices as A (µν) ≡ (A µν + A νµ )/2 and the anti-symmetrisation as A [µν] ≡ (A µν −A νµ )/2.Spinor indices are always suppressed throughout this paper.The rank of tensor-spinors on dS D is denoted as r, while the rank of tensor-spinors on S D−1 as r.The complex conjugate of the complex number z is z * .
2 Background material concerning tensor-spinors on dS D Fermionic fields with arbitrary half-odd-integer spin s ≡ r + 1/2 and mass parameter M on dS D can be described by totally symmetric tensor-spinors Ψ µ 1 ...µr satisfying the onshell conditions [6,15]: where / ∇ = γ ν ∇ ν is the Dirac operator.From now on, we will refer to the divergence-free and gamma-tracelessness conditions in eq. ( 2.2) as the TT conditions.
The half-odd-integer-spin theories described by eqs.(2.1) and (2.2) become gaugeinvariant (i.e.strictly/partially massless) for the following imaginary values of the mass parameter M = i M [15]: Real values of M -including M = 0 -correspond to non-gaugeinvariant theories.

Gamma matrices, vielbein fields, spin connection and Lie-Lorentz derivative on dS D
The 2 [D/2] -dimensional6 gamma matrices γ a (with 'flattened' indices a = 0, 1, ..., D − 1) satisfy the anti-commutation relations where 1 is the spinorial identity matrix and η ab = diag(−1, 1, ..., 1).The vielbein fields e a = e µ a ∂ µ , determining an orthonormal frame, satisfy where the co-vielbein fields e a = e µ a dx µ define the dual coframe.The gamma matrices with coordinate basis indices are defined using the vielbein fields as γ µ (x) ≡ e µ a (x)γ a .The covariant derivative for a vector-spinor field is where ω νbc = ω ν[bc] = e ν a ω abc is the spin connection, Γ λ νµ are the Christoffel symbols and γ bc = γ [b γ c] .The covariant derivatives for higher-spin tensor-spinors are given by straightforward generalisations of eq.(2.6).It is easy to check that the gamma matrices are covariantly constant, as According to our sign convention, we have7 For each value of the mass parameter M in eq.(2.1), the set of TT eigenmodes Ψ µ 1 ...µr forms a representation of the de Sitter algebra spin(D, 1), which -as we will see belowmay be unitary or non-unitary depending on both M and the dimension D. The Killing vectors generating spin(D, 1) act on tensor-spinors in terms of the spinorial generalisation of the Lie derivative [26,37] -also known as Lie-Lorentz derivative -as: where ξ µ is any dS Killing vector -i.e.∇ (µ ξ ν) = 0.The Lie-Lorentz derivative satisfies [26] L ξ e a µ = 0, (2.9a) as well as and hence L ξ commutes with the Dirac operator.Moreover, the Lie-Lorentz derivative preserves the Lie bracket between any two vectors ξ µ , X µ ∈ spin(D, 1) as As for the representation of our gamma matrices on dS D , we choose the following: • For D even: the 2 D/2 -dimensional gamma matrices are (j = 1, ..., D − 1) where the 2 (D−2)/2 -dimensional gamma matrices γ j generate a Euclidean Clifford algebra in D − 1 dimensions, as One can construct the extra gamma matrix γ D+1 which is given by the product γ D+1 ≡ γ 1 γ 2 ...γ D−1 γ 0 , where is a phase factor.The matrix γ D+1 anti-commutes with each of the γ a 's in eq.(2.12).We choose the phase factor such that

.14)
For D = 4 this is the familiar matrix γ 5 .

Specialising to global coordinates
In order to obtain explicit expressions for the TT eigenmodes of the field equation (2.1), we will choose to work with the global slicing of dS D .In global coordinates the line element is (t ∈ R) where dΩ 2 D−1 is the line element of S D−1 .The line element of S m can be parameterised as ) The non-zero Christoffel symbols on global dS D are where gθ i θ j and Γθ k θ i θ j are the metric tensor and the Christoffel symbols, respectively, on S D−1 .We choose the following expressions for the vielbein fields on dS D : where ẽθ i i are the vielbein fields on S D−1 .The non-zero components of the spin connection on dS D are given by where ωijk are the spin connection components on S D−1 .
3 Classification of the UIR's of spin(D, 1) Here we review the classification of the spin(D, 1) UIR's by Ottoson [19] and Schwarz [20].These authors have classified the UIR's of spin(D, 1) under the decomposition spin(D, 1) ⊃ spin(D) -in the present paper spin(D) denotes the Lie algebra of SO(D).Under this decomposition, an irreducible representation of spin(D) appears at most once in a UIR of spin(D, 1) [36].The case with D = 2p and the case with D = 2p + 1, where p is a positive integer, are studied separately.Below we will adopt the notation for the labels of UIR's that were used by Higuchi in Ref. [16].However, we will use the names of the UIR's that are used in the modern literature [17,33,34].

Representations of spin(D).
Let us review the basics concerning spin(D) representations.As is well-known, a representation of spin(2p) or spin(2p + 1) is specified by the highest weight of the representation [24,25], denoted here as where The labels f j (j = 1, ..., p) in eqs.(3.2) and (3.3) are all integers or all half-odd integers.
Note.In the present paper, following Schwarz [20] and Ottoson [19], for even D the value F 0 = −(D − 1)/2 is not included in the Principal Series UIR's, but it is included in the Discrete Series UIR's instead.For odd D, the value F 0 = −(D − 1)/2 is included in the Principal Series UIR's in the present paper.However, in Ref. [17] the value F 0 = −(D−1)/2 (corresponding to the weight ∆ c = (D − 1)/2) is included in the Principal Series UIR's for arbitrary D. The present note is important for reasons of clarity, as we are going to show that the spin-3/2 and spin-5/2 fields on even-dimensional dS D with mass parameter M = 0 have F 0 = −(D − 1)/2 and they correspond to the Discrete Series UIR's in our paper (i.e.Principal Series in Ref. [17]) -see Section 7.
• In Subsections 4.2 and 4.3, we present the construction of spin-3/2 TT eigenmodes on dS D in order to illustrate the method of separation of variables for tensor-spinor fields.Some basic results are tabulated in Tables 1 and 2.
• In Subsection 4.4, we summarise our main results concerning the spin-5/2 TT eigenmodes on dS D .The spectrum of the Dirac operator acting on tensor-spinor eigenmodes on spheres, as well as the representations of spin(D) formed by the eigenmodes, have been discussed in Refs.[18,21,29,39] (see also references therein).
For both even D [eqs.(4.4) and (4.5)] and odd D [eq.(4. 3)], if the aforementioned irreducible representations of spin(D) are contained in a spin(D, 1) representation, then the allowed values for the angular momentum quantum number might not just be = r, r + 1, ...; might have to satisfy extra conditions because of the branching rules (3.7) and (3.17).This will become clear in the next Subsection as will have to satisfy ≥ r, where r is the rank of the tensor-spinor eigenmodes on dS D .The Dirac equation (2.1) is expressed as (j = 1, 2, ..., D − 1), where we have made use of eqs.(2.6), (2.12) and (2.18)-(2.20),while γ θ j = e k θ j γ k .There are two different ways in which we can separate variables for the TT vector-spinor Ψ µ (t, θ D−1 ) giving rise to two different types of eigenmodes: the type-I modes and the type-II modes.These two different types of eigenmodes correspond to spin(D) representations with different spin.In particular, the spin(D) content that is relevant to type-I modes corresponds to the spinor representation 2 ) with = 1, 2, ... . 14The spin(D) content that is relevant to type-II modes corresponds to the vector-spinor representation 2 ) with = 1, 2, .... Type-I modes.Let us denote the type-I modes with spin(D) content given by f ± 0 = ( + 1 2 , 1 2 , ..., 1 2 , ± 1 2 ) as Ψ (M ; r=0, ± ; m) µ (t, θ D−1 ), where the label m has the same meaning as in Subsection 4.1.We start with the case of f − 0 = ( + 1 2 , 1 2 , ..., 1 2 , − 1 2 ), i.e. with the type-I modes Ψ (M ; r=0, − ; m) µ (t, θ D−1 ).As in Refs.[18,21,22], we separate variables for the t-component by expressing it in terms of upper and lower spinor components, as   M (t) -the superscript '(1)' in these functions has been used for later convenience.By substituting eq.(4.8) into the Dirac equation (4.6), we can eliminate the lower component in eq.(4.8).We find in this manner the second order equation for Φ M , (4.9) 14 Under the decomposition spin(D, 1) ⊃ spin(D), the branching rules (3.7) give rise to the restriction ≥ 1.One can also arrive at this restriction on by requiring the regularity of type-I eigenmodes, as we will discuss below.See Refs.[40,41] for more details concerning the explicit form of the eigenmodes.
where the differential operator D (1) is a special case of the following family of differential operators: where we have defined with cos x = i sinh t and sin x = cosh t.For later convenience, instead of just solving the eigenvalue equation (4.9), we can solve the more general equation for arbitrary integer a.The solution is given by where F (A, B; C; z) is the Gauss hypergeometric function [23], while M is given by eq.(4.13) with a = 1.
In order to determine the lower component in eq.(4.8), we substitute eq.(4.8) into the Dirac equation (4.6) and we straightforwardly find the relations Then, substituting eq.(4.13) (with a = 1) into eq.(4.16) and using well-known properties of the hypergeometric function [23], we find (4.17) For later convenience, let us note that Ψ (1) M (t) corresponds to a special case (i.e. the case with a = 1) of the following functions: 2 ).We find and, thus, the regularity of type-I eigenmodes gives rise to the restriction ≥ 1.However, here we will not present explicit expressions for Ψ (M ; r=0, ± ; m) θ j (j = 1, ..., D − 1) as they are lengthy and they are not needed for our analysis.The interested reader can find the explicit expressions in Refs.[40,41].Type-II modes.Let us denote the type-II modes with spin(D) content given by f can be determined by applying the method of separation of variables as in the case of the type-I modes.However, now we have to express Ψ (M ; r=1, ± ; m) θ j (t, θ D−1 ) in terms of TT eigenvector-spinors on S D−1 , instead of eigenspinors on S D−1 .By applying the method of separation of variables to the Dirac equation (4.7), we find Summary.Some basic results concerning the spin-3/2 eigenmodes for even D ≥ 4 are tabulated in Table 1.

Separating variables for spin-3/2 eigenmodes on dS D for odd D ≥ 3
The Dirac equation (2.1) is expressed as (j = 1, 2, ..., D − 1), where the gamma matrices are now given by eq.(2.15).As in the evendimensional case, we have two different types of eigenmodes depending on their spin(D) content.

Spin-5/2 eigenmodes on dS D
In the case of rank-2 totally symmetric tensor-spinors Ψ µν -which satisfy eqs.(2.1) and (2.2) with r = 2 on dS D -the method of separation of variables can be applied in a way analogous to the case of TT vector-spinors.Depending on the spin(D) content of the spin-5/2 dS eigenmode we can distinguish three types of modes: type-I, type-II and type-III modes (the last two exist for D > 3).Here we will just summarise some basic results for the TT spin-5/2 eigenmodes on dS D .Below we use the same notation for the labels of the eigenmodes as in the spin-3/2 case, while we refer again to the spin(D) content of the eigenmodes using the highest weights f ± r for even D [eqs.(4.4) and (4.5)] and f r for odd D [eq. ( 4.3)].
5 Quadratic Casimir for spin-3/2 and spin-5/2 eigenmodes on dS D In order to find the values of the spin(D, 1) quadratic Casimir corresponding to the representation formed by our spin-3/2 and spin-5/2 eigenmodes we will use the "analytic continuation" techniques that have been already used in Refs.[14,22].More specifically, we will use the fact that dS D can be obtained by an "analytic continuation" of S D .The line element of S D can be written as where 0 ≤ θ D ≤ π.By replacing the angle θ D in dΩ 2 D as: (t ∈ R) we find the line element (2.16) for global dS D (x(t) coincides with the 'useful' variable that we have already introduced in eq.(4.11)).Quadratic Casimir for tensor-spinor eigenmodes on S D .Motivated by the aforementioned observation, we can obtain the field equations (2.1) and (2.2) for spin-(r + 1/2) fields on dS D by analytically continuing the equations for totally symmetric TT tensorspinors of rank r on S D : where ψ ±µ 1 ...µr is a tensor-spinor on S D , while n is the angular momentum quantum number on S D .Equations (5.3) and (5.4) are essentially the D-dimensional counterparts of eqs.(4.1) and (4.2), while now n on S D plays the role of on S D−1 .As we discussed in Subsection 4.1, the spin(D + 1) representations formed by tensor-spinor eigenmodes of the Dirac operator on S D are known [29].Using eqs.(3.4) and (3.5), the spin(D + 1) quadratic Casimir corresponding to the eigenmodes ψ ±µ 1 ...µr on S D is readily found to be for all D ≥ 3, while in the second line we used that ∇ µ ∇ µ acts on ψ ±µ 1 ...µr as Analytic continuation to dS D .Without loss of generality, we can choose to analytically continue the eigentensor-spinors with either one of the two signs for the eigenvalue in eq. ( 5.3), since each of the two sets of modes, {ψ +µ 1 ...µr } and {ψ −µ 1 ...µr }, forms independently a unitary representation of spin(D + 1) labelled by n (see Subsection 4.1).Here we choose to analytically continue the eigentensor-spinors ψ −µ 1 ...µr .We perform analytic continuation by making the following replacements in eqs.(5.3) and (5.4)15 : and we obtain eqs.(2.1) and (2.2), respectively, for tensor-spinors Ψ µ 1 ...µr with mass parameter M on dS D .Recall that the values of interest for M are: M ∈ R (corresponding to massive fermions of spin s ≥ 3/2), as well as the purely imaginary values of M corresponding to the strictly/partially massless tunings (2.3).The prescription for obtaining the explicit form of dS eigenmodes by analytically continuing eigenmodes on S D can be found in Refs.[14,22,40,41].Quadratic Casimir for tensor-spinor eigenmodes on dS D .With the use of the replacements (5.6), we analytically continue the quadratic Casimir on S D [Eq.(5.5)], and we find the value of the quadratic Casimir on dS D : (with s = r + 1/2), which holds for all D ≥ 3 and for all totally symmetric TT tensor-spinor eigenmodes with spin s ≥ 1/2 and mass parameter M on dS D .Specialising to the spin-3/2 TT eigenmodes we find while for the spin-5/2 TT eigenmodes we find (5.9) Type of eigenmode Notation spin(D) content Type-I Ψ (M ; r=0,± ; m) µ 2 ), = 1, 2, ... } r=0,1 separately form Discrete Series UIR's of spin(D, 1).For M = ±i(D −2)/2 (strictly massless tuning) the type-I modes become pure gauge modes, while the type-II modes are the physical modes forming a non-unitary representation for D = 4 and a direct sum of two 'chiral' UIR's in the Discrete Series for D = 4 -see Section 7. All these results have been also explained by studying the group-theoretic properties of the eigenmodes in Refs.[40,41].

Type of eigenmode
Notation spin(D) content Type-I Ψ (M ; r=0, ; m) µ 2 ), = 1, 2, ... Type-II (for D > 3) Ψ (M ; r=1, ; m) µ 2 ), = 1, 2, ... Apart from the values of the quadratic Casimir (6.1)-( 6.3), we also know the spin(D) content of the spin(D, 1) representations formed by our dS eigenmodes -see Tables 1 and 2, as well as Subsections 4.2-4.4.Keeping these results in mind, we can use the classification of the UIR's in Section 3 in order to readily deduce the (non-)unitarity of the representations formed by our strictly/partially massless eigenmodes on dS D .First, let us identify which types of dS eigenmodes correspond to pure gauge modes and which to physical modes in the strictly/partially massless theories.By 'physical modes' we mean the eigenmodes that form the strictly/partially massless representation of spin(D, 1) and that correspond to the (non-gauge) propagating degrees of freedom of the theory.(If the representation formed by the eigenmodes is non-unitary, then the name 'physical modes' could be misleading as the theory is, of course, unphysical due to the appearance of negative probabilities.)The pure gauge modes describe pure gauge degrees of freedom of the theory.If a dS invariant scalar product exists, then the pure gauge modes have zero norm and they are orthogonal to all physical modes [14,40,41].The generators of spin(D, 1) act in terms of the Lie-Lorentz derivative (2.8) on equivalence classes of physical modes with equivalence relation given by: "For any two physical modes Ψ where for convenience we have omitted all quantum number labels from Ψ for some TT vector-spinor gauge functions λ ±µ (t, θ D−1 ) with / ∇λ ±µ = ∓i D + 2 2 λ ±µ (6.7) where the spinor gauge functions ϕ ± (t, θ D−1 ) satisfy Explicit expressions on global dS D for the eigenmodes corresponding to the gauge functions in eqs.(6.4), (6.6) and (6.9) can be found in Refs.[40,41].
Remark 6.1.On dS 3 , both spin-3/2 and spin-5/2 theories with arbitrary mass parameters have only type-I modes.Thus, specialising to the strictly/partially massless theories on dS 3 , we conclude that all eigenmodes for these theories are pure gauge modes.
The validity of Remarks 6.1-6.3 for the spin-3/2 and spin-5/2 fields has been demonstrated in this paper, as well as in Refs.[40,41].However, we expect that these remarks also hold for all strictly/partially massless fields with half-odd-integer spins s ≥ 3/2.This expectation is also motivated by the well-studied case of totally symmetric tensors [14].

Studying the (non-)unitarity of the strictly/partially massless theories
with spin s = 3/2, 5/2 Our 'tools' in order to demonstrate that the unitarity of the strictly/partially massless fields of spin s = 3/2, 5/2 occurs only for D = 4 are: on the one hand the values of the quadratic Casimir [eqs.(6.1)-(6. 3)] and the spin(D) content of the physical modes [see Tables 1 and  2 and Remarks 6.1-6.3]and, on the other hand, the classification of the UIR's in Section 3.
Although the readers can readily convince themselves about the non-unitarity for D = 4 (given our aforementioned tools), we will present here a detailed discussion concerning the strictly massless spin-3/2 field.The cases of the strictly and partially massless spin-5/2 fields can then be treated in the same manner and, therefore, we will not present their details here.

Summary and discussions
In the present paper, we demonstrated that four-dimensional dS space plays a distinguished role in the unitarity of the strictly and partially massless (symmetric) tensor-spinor fields of spin s = 3/2, 5/2.In particular, the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields on dS D , are not unitary unless D = 4.The explanation relies on the representation theory of spin(D, 1), where the latter does not allow strictly/partially massless UIR's for (symmetric) tensor-spinors unless D = 4.This is a remarkable feature of dS field theory, while it is also very interesting that the dimensionality that plays a special representation-theoretic role matches the dimensionality of our physical Universe.We also expect that this result should hold for all totally symmetric tensorspinors with spin s ≥ 7/2, while this expectation of ours is justified by the classification of the spin(D, 1) UIR's.A technical explanation of our results in terms of the (non-)existence of positive-definite dS scalar products for the spin-3/2 and spin-5/2 eigenmodes has been given in Refs.[40,41].modes (4.20) form the Discrete Series UIR D − (− 1 2 , 3 2 ) with the positive-definite scalar product (8.6), while the physical modes (4.21) form the Discrete Series UIR D + (− 1 2 , 3 2 ) with positive-definite scalar product given by the negative of eq.(8.6).The pure gauge modes (6.4) have zero norm with respect to the scalar product (8.6) and they are orthogonal to all physical modes.For more details concerning the eigenmodes see Refs.[40,41].
The fermionic strictly and partially massless tunings (2.3) were found in Ref. [15], but the non-unitarity of the corresponding theories for D = 4 could not be revealed with the methods used in this reference.

Strategy 1 . 5 2 7 3 8 1 Introduction 1 . 1
Outline of the paper, notation and conventions 2 Background material concerning tensor-spinors on dS D 2.1 Gamma matrices, vielbein fields, spin connection and Lie-Lorentz derivative on dS D 2.2 Specialising to global coordinates 3 Classification of the UIR's of spin(D, 1) 4 Spin-3/2 and spin-5/2 eigenmodes on dS D 4.1 Tensor-spinor eigenmodes of the Dirac operator on S D−1 and representations of spin(D) 4.2 Separating variables for spin-3/2 eigenmodes on dS D for even D ≥ 4 4.3 Separating variables for spin-3/2 eigenmodes on dS D for odd D ≥ 3 4.4 Spin-5/2 eigenmodes on dS D 5 Quadratic Casimir for spin-3/2 and spin-5/2 eigenmodes on dS D 6 Strictly and partially massless representations: non-unitarity for D = 4 and unitarity for D = 4 6.1 Pure gauge modes and physical modes 6.2 Studying the (non-)unitarity of the strictly/partially massless theories with spin s = 3/2, 5/Dictionary between (symmetric) tensor-spinor fields on dS D and UIR's of spin(D, 1) for D ≥ Summary and discussions A The only totally symmetric TT tensor-spinor eigenmodes of the Dirac operator that exist on S 2 are the spinor eigenmodes Strictly and partially massless field theories in de Sitter space

M
(t) = 0 where the differential operator D(a) is given by eq.(4.10) with x replaced by π − x.Thus, we have now also determined the lower component of Ψ (M ; r=0, − ; m) t in eq.(4.8).Now, by following the same procedure as the one described above, we can separate variables for the type-I modes Ψ (M ; r=0, + ; m) µ (t, θ D−1 ) corresponding to the spin(D) highest weight f

Table 2 .
Spin-3/2 TT eigenmodes with mass parameter M on dS D (odd D ≥ 3).Type-II modes exist for D > 3.For real M , type-I and type-II modes together form a Principal Series UIR of spin(D, 1) for D > 3.For real M and D = 3, type-I modes form a Principal Series UIR of spin(3,1).For M = ±i(D − 2)/2 (strictly massless tuning) and D > 3, the type-I modes become pure gauge modes, while the type-II modes are the physical modes forming a non-unitary strictly massless representation.At the strictly massless tuning M = ±i/2 for D = 3, the type-I modes are again pure gauge modes and they form a non-unitary representation of spin(3, 1) -see Section 7. All these results have been also explained by studying the group-theoretic properties of the eigenmodes in Refs.[40,41].6Strictly and partially massless representations: non-unitarity for D = 4 and unitarity for D = 4Here we will obtain the main result of this paper: the strictly massless spin-3/2 field, as well as the strictly and partially massless spin-5/2 fields, on dS D (D ≥ 3) cannot be unitary unless D = 4.Note that we already know the values of the quadratic Casimir [eqs.(5.8) and (5.9)] for the representations formed by our dS eigenmodes for any mass parameter M .By specialising to the strictly/partially massless tunings (2.3), we find C

. 8 )(
The gauge functions for type-I modes are different from the gauge functions for type-II modes -for more details see Refs.[40,41].)Pure gauge and physical modes for partially massless spin-5/2 field.The mass parameter for the partially massless spin-5/2 field is given by M = ±i(D − 2)/2 [this is found by letting r = 2 and τ = 2 in eq.(2.3)].The type-I modes are the pure gauge modes of the theory.Both type-II and type-III modes are physical modes that form the (partially massless) representation of spin(D, 1).For M = ±i(D −2)/2 all type-I modes are expressed in a pure gauge form as: