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 (SN). 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 spinor functions that satisfy the eigenvalue equation of the Dirac operator on SN,
ψ=iλψ,
(1.1)
have been studied by Camporesi and Higuchi.1 More specifically, the eigenspinors on SN have been recursively constructed in terms of eigenspinors on SN−1 using separation of variables in geodesic polar coordinates, and their eigenvalues have been calculated. The line element for SN may be written as
dsN2=dθN2+sin2θNdsN12,
(1.2)
where θN is the geodesic distance from the North Pole and dsN12 is the line element of SN−1. Similarly, the line element of Sn (n = 2, 3, …, N − 1) can be expressed as
dsn2=dθn2+sin2θndsn12,
(1.3)
while ds12=dθ12.
The N-dimensional de Sitter space–time is the maximally symmetric solution of the vacuum Einstein field equations with positive cosmological constant Λ,2 
Rμν12gμνR+Λgμν=0.
(1.4)
The cosmological constant is given by
Λ=(N2)(N1)2R2,
(1.5)
where R is the de Sitter radius. Throughout this paper, we use units in which R=1.
The N-dimensional de Sitter space–time can also be obtained by an “analytic continuation” of SN. More specifically, by replacing
θNxπ/2it
(1.6)
in the SN metric (1.2), we find the line element for dSN with SN−1 spatial sections [see Eq. (2.2)],
ds2=dt2+cosh2tdsN12.
(1.7)
Motivated by the above, one can obtain the mode solutions to the Dirac equation on dSN,
ψMψ=0,
(1.8)
just by analytically continuing the eigenmodes of (1.1). The Dirac spinors obtained by analytic continuation can be used to describe spin-1/2 particles in de Sitter space–time, and they form a representation of Spin(N,1). The latter has to be unitary to ensure that negative probabilities will not arise. In order to study the unitarity of the representation, we are going to introduce a de Sitter invariant inner product among the analytically continued eigenspinors (see Sec. V). Note that this approach has been previously applied for the divergence-free and traceless tensor eigenfunctions of the Laplace–Beltrami operator on SN,3 where the restriction of unitarity gave rise to the forbidden mass range for the spin-2 field on dSN.

In this paper, our main aim is the identification of the mode functions for the free Dirac field on global dSN with SN−1 spatial sections. As a consistency check, we reproduce the expected form for the massless spinor Wightman function4 using the mode-sum method. We also use this Wightman function to find an analytic expression for the spinor parallel propagator. To our knowledge, such an expression is absent from the literature. Solutions of the free Dirac equation on de Sitter space–time with static charts may be found in Ref. 5, with moving charts in Refs. 6–8 and with open charts in Ref. 9.

The rest of this paper is organized as follows: In Sec. II, we discuss the global coordinate system that is relevant to the analytic continuation of SN and we review the geodesic structure of dSN. In Sec. III, we present the basics about Dirac spinors and Clifford algebras on dSN. In Sec. IV, we begin by reviewing the eigenspinors of the Dirac operator on SN following Ref. 1. Then, we obtain the mode solutions of the Dirac equation on dSN by analytically continuing the eigenmodes on SN, and we give a criterion for generalized positive frequency modes. We also construct spinors satisfying the Dirac equation with the sign of the mass term changed. These spinors are used in  Appendix A for an alternative construction of the negative frequency modes via charge conjugation. In Sec. V, we define a de Sitter invariant inner product among the analytically continued eigenmodes and we show that the associated norm is positive-definite (i.e., the representation is unitary). Using this norm, we normalize the analytically continued eigenspinors. Then, the transformation properties of the positive frequency solutions under Spin(N,1) are studied using the spinorial Lie derivative.10 It is shown that the positive frequency solution subspace is Spin(N,1) invariant (so is the corresponding vacuum). In Sec. VII, after presenting the negative frequency solutions of the Dirac equation, we perform the canonical quantization procedure for the free Dirac quantum field. Then, we review the coordinate independent construction of Dirac spinor Green’s functions on dSN following Ref. 4. We present a closed-form expression for the massless spinor Wightman two-point function obtained by the mode-sum method. This closed-form expression is in agreement with the construction given in Ref. 4. Then, we find an analytic expression for the spinor parallel propagator and we use it to obtain a closed-form expression for the massive Wightman two-point function in terms of intrinsic geometric objects. Our summary and concluding remarks are given in Sec. VIII.

There are six appendices. In  Appendix A, we construct the negative frequency solutions of the Dirac equation on dSN by charge conjugating our analytically continued eigenspinors. In  Appendix F, we compare the mode-sum method for the massive spinor Wightman function with the construction presented in Ref. 4 and we arrive at a closed-form conjecture for a series containing the Gauss hypergeometric function. The rest of the  Appendixes A–F concern technical details. Some minor details omitted in the main text are presented in  Appendixes B and  C. In  Appendix D, we present details about the mode-sum construction of the massless spinor Wightman function. In  Appendix E, we demonstrate that our analytic expression for the spinor parallel propagator satisfies the defining properties given in Ref. 4.

We use the mostly plus convention for the metric signature. When it comes to tensors, lowercase Greek indices refer to components with respect to the “coordinate basis,” while Latin ones refer to components with respect to the vielbein (i.e., orthonormal frame) basis. Spinor indices (when not suppressed) are denoted with capital Latin letters. For bitensors (or bispinors) that depend on two space–time points x, x′, unprimed indices refer to the tangent space at x, while primed ones refer to the tangent space at x′. Summation over repeated indices is understood throughout this paper.

The N-dimensional de Sitter space–time can be represented as a hyperboloid embedded in (N + 1)-dimensional Minkowski space. The de Sitter hyperboloid is described by
ηabXaXb=1,
(2.1)
where ηab = diag(−1, 1, 1, …, 1) (a, b = 0, 1, …, N) is the flat metric for the embedding space and X0, X1, …, XN are the standard Minkowski coordinates. The global coordinates used in this paper are given by
X0=X0t,θ=sinht,Xi=Xit,θ=coshtZi,i=1,,N,
(2.2)
where tR,θ=(θN1,θN2,,θ1) and the Zi’s are the spherical coordinates for SN−1 in N-dimensional Euclidean space
Z1=sinθN1sinθN2sinθ2sinθ1,Z2=sinθN1sinθN2sinθ2cosθ1,ZN1=sinθN1cosθN2,ZN=cosθN1,
(2.3)
where 0 ≤ θ1 < 2π and 0 ≤ θiπ (i ≠ 1). Using the coordinates (2.2), we obtain the line element (1.7) for dSN.
The non-zero Christoffel symbols for the coordinates (2.2) are
Γθiθjt=coshtsinhtg̃θiθj,Γθjtθi=tanhtg̃θjθi,Γθiθjθk=Γ̃θiθjθk,
(2.4)
where g̃θiθj,Γ̃θiθjθk are the metric tensor and the Christoffel symbols, respectively, on SN−1. The vielbein fields are given by
et0=1,eθii=1coshtẽθii,i=1,,N1,
(2.5)
where ẽθii are the vielbein fields on SN−1. The latter are given by
ẽθN1N1=1,ẽθjj=1sinθN1sinθN2sinθj+1,j=1,,N2.
(2.6)
The spin connection ωabc = ωa[bc] ≡ (ωabcωacb)/2 is given by
ωabc=eμaμeλb+Γμνλeνbeλc,
(2.7)
and its only non-zero components are
ωijk=ω̃ijkcosht,ωi0k=tanhtδik,i,j,k=1,,N1,
(2.8)
where ω̃ijk are the spin connection components on SN−1 and δij is the Kronecker delta symbol. (Note that the sign convention we use for the spin connection is the opposite of the one used in most supersymmetry texts.)

Geodesics on dSN are obtained by intersecting the hyperboloid (2.1) with two planes passing through the origin.11 Note that, contrary to the case of maximally symmetric Euclidean spaces (RN,SN,HN), on pseudo-Riemannian spaces, two points cannot always be connected by a geodesic.

Let x, x′ be two points on the de Sitter hyperboloid (2.1) and μ(x, x′) be the geodesic distance between them. Using the scalar product of the ambient space
Z(x,x)=ηabXa(x)Xb(x),
(2.9)
one can define the useful quantity,
z(x,x)=121+ηabXa(x)Xb(x).
(2.10)
If 1Z(x,x)<1 (i.e., z ∈ [0, 1)), the points x, x′ are spacelike separated (μR) and they can be connected by a spacelike geodesic. (The equality sign corresponds to antipodal points.) The geodesic distance is then defined by Z(x,x)=cos(μ(x,x)) or, equivalently,
z=cos2μ2.
(2.11)
If Z(x,x)<1 (i.e., z < 0), the points are spacelike separated, but there is no geodesic connecting them. However, the function μ(x, x′) can still be defined by Eq. (2.10) via analytic continuation.12 (Let x̄ be the antipodal point of x and let x′ be any point in the interior of the past or future light cone of x̄. Then, there is no geodesic connecting x and x′.12) If Z(x,x)=1 (i.e., z = 1), the geodesic distance is zero and the two points can be connected by a null geodesic (or they coincide). If Z(x,x)>1 (i.e., z > 1), the two points are timelike separated (μ=iκ,κR) and they can be connected by a timelike geodesic. The geodesic distance for timelike separation is given by
z=cos2μ2=cosh2κ2.
(2.12)
In the rest of this paper, we suppose that the points under consideration can be connected by a spacelike geodesic (unless otherwise stated). The corresponding results for the timelike case can be obtained just by replacing μ.
The unit tangent vectors at x and x′ to the geodesic connecting the two points are defined by
nκ(x,x)=κμ(x,x),nκ(x,x)=κμ(x,x),
(2.13)
respectively. Since dSN is a maximally symmetric space–time, the unit tangents satisfy12,
μnν=cotμ(gμνnμnν),
(2.14)
μnν=1sinμ(gμν+nμην),
(2.15)
κgμν=tanμ2(gκμnν+gκνnμ),
(2.16)
where gμν(x) is the metric tensor and gμν(x, x′) is the bivector of parallel transport. The latter is also known as the vector parallel propagator and it performs the parallel transport of a vector field Vν(x′) from x′ to x along the geodesic connecting these points,12,
Vμ(x)=gνμVν(x),
(2.17)
where Vμ(x) is the parallelly transported vector at x. (In this paper, by geodesic, we mean the shortest geodesic connecting the two points.) It is worth noting the relations12 
nμ=gμνnν,nμ=gμνnν,
(2.18)
gνμgλν=δλμ,gκμgνκ=δνμ.
(2.19)
Using the coordinates (2.2), we obtain the following expression for the geodesic distance:
cosμ(x,x)=sinhtsinht+coshtcoshtcosΩN1,
(2.20)
where
cosΩn=cosθncosθn+sinθnsinθncosΩn1
(2.21)
for n = 2, …, N − 1 and
cosΩ1=cos(θ1θ1).
(2.22)
Then, the coordinate basis components of the tangent vector nμ(x,x)=(nt(x,x),nθi(x,x)) (i = 1, …, N − 1) are given by
nt=1sinμ(coshtsinhtsinhtcoshtcosΩN1),
(2.23)
nθi=1sinμcoshtcoshtθi(cosΩN1),
(2.24)
where
θi(cosΩN1)=r=1N(i+1)sinθNrsinθNr(sinθicosθi+cosθisinθicosΩi1).
(2.25)
The components of nμ(x, x′) are given by analogous expressions with tt′, θiθi′. The vielbein basis components of the tangent vector at x, na(x,x)=eaμ(x)nμ(x,x) (a = 0, 1, …, N − 1), are given by
n0=nt,
(2.26)
nN1=coshtsinμsinθN1cosθN1+cosθN1sinθN1cosΩN2,
(2.27)
nb=coshtsinμr=1N(b+1)sinθNr(sinθbcosθb+cosθbsinθbcosΩb1),
(2.28)
with b = 1, …, N − 2, while the components of na(x,x)=eaμ(x)nμ(x,x) [a′ = 0′, 1′, …, (N − 1)′] can be obtained from Eqs. (2.26) and (2.28) with tt′, θaθa′. (Note that we define cos Ω0 ≡ 1.)
Dirac spinors are 2[N/2]-dimensional column vectors that appear naturally in Clifford algebra representations, where [N/2] = N/2 if N is even and [N/2] = (N − 1)/2 if N is odd. A Clifford algebra representation in (N − 1) + 1 dimensions is generated by N gamma matrices satisfying the anti-commutation relations,
{γa,γb}=2ηab1,a,b=0,1,,N1,
(3.1)
where 1 is the identity matrix and ηab is the inverse of the N-dimensional Minkowski metric ηab = diag(−1, +1, …, +1). We follow the inductive construction of Ref. 1 where gamma matrices in (N − 1) + 1 dimensions are expressed in terms of spacelike gamma matrices in (N − 1) dimensions (γ̃i) as follows:
  • For even N,
    γ0=i0110,γi=0iγ̃iiγ̃i0,i=1,,N1,
    (3.2)
    where the lower-dimensional gamma matrices satisfy the Euclidean Clifford algebra anti-commutation relations,
    {γ̃i,γ̃j}=2δij1,i,j=1,,N1.
    (3.3)
  • For odd N,
    γ0=i1001,γN1=0110,
    γj=γ̃j=0iγ̃̃jiγ̃̃j0,j=1,,N2.
    (3.4)
    The double-tilde is used to denote gamma matrices in N − 2 dimensions. For N = 1, the only (one-dimensional) gamma matrix is equal to 1.

Note that the gamma matrices we use here for dSN can be obtained by the Euclidean gamma matrices on SN used in Ref. 1 via the coordinate change (1.6). [Gamma matrices transform as vectors under coordinate transformations and it can be checked that all Euclidean γa’s remain the same under (1.6) apart from γN; the latter transforms into the timelike gamma matrix: γNN = γ0.]

Spinors transform under 2[N/2]-dimensional spinor representations of Spin(N − 1,1) [double cover of SO(N − 1,1)] as
ψ(x)S(Λ(x))ψ(x),
(3.5)
where S(Λ(x)) ∈ Spin(N − 1,1) is a spinorial matrix. The N(N − 1)/2 generators of Spin(N − 1, 1) are given by the commutators,
Σab=14[γa,γb]
(3.6)
=12γaγb12ηab,a,b=0,,N1,
(3.7)
and they satisfy the Spin(N − 1, 1) algebra commutation relations,
[Σab,Σcd]=ηbcΣadηacΣbd+ηadΣbcηbdΣac.
(3.8)
The covariant derivative for a spinor along the vielbein is
aψ=eaψ12ωabcΣbcψ,
(3.9)
where ea=eaμμ. The Dirac adjoint of a spinor is defined as
ψ̄iψγ0,
with the covariant derivative given by
aψ̄=eaψ̄+12ψ̄ωabcΣbc.
(3.10)
The covariant derivative of the gamma matrices is
aγk=eaγkωakcγc12ωabc[Σbc,γk]=0.
(3.11)
One can show the following properties of the gamma matrices given by Eqs. (3.2) and (3.4):
(γ0)T=γ0,(γr)T=(1)N2(1)r2γr,
(3.12)
(γ0)*=γ0,(γr)*=(1)N2(1)r2γr,
(3.13)
r = 1, …, N − 1, and
(γa)=γ0γaγ0,a=0,,N1,
(3.14)
where the asterisk denotes complex conjugation. Note that the timelike gamma matrix is anti-Hermitian, while the spacelike ones are Hermitian.

We first present the basic results from Ref. 1 regarding the eigenmodes of the Dirac operator on SN, and then, we perform analytic continuation for the two cases with N even and N odd.

Case 1: N even. The eigenvalue equation for the Dirac operator on SN is
ψ±nσ(s,s̃)=±in+N2ψ±nσ(s,s̃),
(4.1)
where n = 0, 1, … and = 0, …, n are the angular momentum quantum numbers on SN and SN−1, respectively. The index s indicates the two different spin projections (s = ±). The symbol σ stands for the angular momentum quantum numbers N−2N−3 ≥ ⋯ ≥ 21 ≥ 0 on the lower-dimensional spheres, while s̃ stands for the (N/2 − 1) spin projection indices sN−2, sN−4, …, s2 on the lower-dimensional spheres SN−2, SN−4, …, S2, respectively. (Note that there exists one spin projection index for each lower-dimensional sphere of even dimension.) For each value of n, we have a representation of Spin(N + 1) on the space of the eigenspinors ψ+nσ(s,s̃) (or ψnσ(s,s̃)) with dimension1 
dn=2[N/2](N+n1)!n!(N1)!.
(4.2)
The solutions of the eigenvalue equation for the Dirac operator on SN (1.1) are found by writing the spinor ψ in terms of “upper” (φ+) and “lower” (φ) components as follows:
ψφ+φ.
(4.3)
By substituting Eq. (4.3) into Eq. (1.1), one obtains two coupled differential equations for φ+, φ. By eliminating φ+ (or φ), one finds1,
θN+N12cotθN2+1sin2θÑ2±cosθNsin2θNĩφ±=λ2φ±,
(4.4)
where ̃ is the Dirac operator on SN−1. [Equation (4.4) is equivalent to 2ψ=λ2ψ.] Then, by separating variables, the normalized eigenspinors of |SN are found to be1,
ψ±nσ(,s̃)(θN,ΩN1)=cN(n)2ϕn(θN)χσ(s̃)(ΩN1)±iψn(θN)χσ(s̃)(ΩN1)
(4.5)
and
ψ±nσ(+,s̃)(θN,ΩN1)=cN(n)2iψM(θN)χ+σ(s̃)(ΩN1)±ϕM(θN)χ+σ(s̃)(ΩN1),
(4.6)
where ΩN−1SN−1 and the normalization factor is given by
|cN(n)|2=Γ(n+1)Γ(n+N+)2N2|Γ(N/2+n)|2.
(4.7)
The eigenspinors on SN−1, χ±σ(s̃)(ΩN1), satisfy the eigenvalue equation
̃χ±σ(s̃)=±i+N12χ±σ(s̃).
(4.8)
They are normalized by
SN1dΩN1χsσ(s̃)(ΩN1)χsσ(s̃)(ΩN1)=δssδδσσδs̃s̃,
(4.9)
while the eigenspinors on SN are normalized by
SNdΩNψ±nσ(s,s̃)(θN,ΩN1)ψ±nσ(s,s̃)(θN,ΩN1)=δssδnnδδσσδs̃s̃,
(4.10)
where all the ψ+ eigenspinors are orthogonal to all the ψ eigenspinors. The functions ϕnℓ(θN), ψnℓ(θN) are given in terms of the Gauss hypergeometric function by
ϕn(θN)=κϕ(N)(n)cosθN2+1sinθN2Fn+N+,n+;N/2+;sin2θN2
(4.11)
and
ψn(θN)=κϕ(N)(n)(n+N/2)N/2+cosθN2sinθN2+1Fn+N+,n+;N/2++1;sin2θN2,
(4.12)
where
κϕ(N)(n)=Γ(n+N/2)Γ(n+1)Γ(N/2+).
(4.13)
The condition n and the quantization of the eigenvalue of the Dirac operator λ2 = (n + N/2)2 (n = 0, 1, …) arise by requiring that the mode functions are not singular.1 The functions ϕnℓ, ψnℓ are related to each other by
ddθN+N12cotθN1sinθN+N12ϕn(θN)=n+N2ψn(θN),
(4.14)
ddθN+N12cotθN+1sinθN+N12ψn(θN)=+n+N2ϕn(θN).
(4.15)
As mentioned in the Introduction, we can obtain the Dirac spinors, which solve the Dirac equation (γaaM)ψ = 0 on dSN by analytically continuing the eigenmodes of the Dirac operator on SN. The eigenvalues on SN will be replaced by the spinor’s mass M. It is easy to check that under the replacement θNπ/2 − it, one finds |SN|dSN. Without loss of generality, we choose to analytically continue the eigenspinors ψ+ with the positive sign for the eigenvalue [see Eqs. (4.5) and (4.6)] by making the replacements
θNxπ/2it,niMN2.
(4.16)
The solutions of the Dirac equation on dSN are then
ψMσ(,s̃)(t,ΩN1)=cN(M)2ϕM(t)χσ(s̃)(ΩN1)iψM(t)χσ(s̃)(ΩN1)
(4.17)
and
ψMσ(+,s̃)(t,ΩN1)=cN(M)2iψM(t)χ+σ(s̃)(ΩN1)ϕM(t)χ+σ(s̃)(ΩN1),
(4.18)
where cN(Mℓ) is a normalization factor that will be determined later ( = 0, 1, …). [Alternatively, we can choose to analytically continue the eigenspinors ψ in order to obtain the solutions (4.17) and (4.18) of the Dirac equation. In this case, we need to make the replacements θNπ/2 − it, niMN/2 in Eqs. (4.5) and (4.6) instead of the replacements (4.16).] The un-normalized functions that describe the time dependence are
ϕM(t)=cosx2+1sinx2FN2++iM,N2+iM;N2+;sin2x2
(4.19)
and
ψM(t)=iMN/2+cosx2sinx2+1FN2++iM,N2+iM;N2++1;sin2x2,
(4.20)
where
cosx2=22cosht2+isinht2,
(4.21)
sinx2=22cosht2isinht2,
(4.22)
sin2x2=1isinht2.
(4.23)
It is clear from Eq. (4.20) that ψMℓ(t) vanishes in the massless limit. Note the analytically continued version of Eqs. (4.14) and (4.15),
ddt+N12tanht+icosht+N12ϕM(t)=+MψM(t),
(4.24)
ddt+N12tanhticosht+N12ψM(t)=MϕM(t).
(4.25)
Using the relation13 
F(a,b;c;z)=(1z)cabF(ca,cb;c;z),
(4.26)
we can rewrite the functions ϕMℓ, ψMℓ as
ϕM(t)=cosx2N+1sinx2FiM,iM;N2+;sin2x2
(4.27)
and
ψM(t)=iMN/2+cosx2N+2sinx2+1FiM+1,iM+1;N2++1;sin2x2.
(4.28)
The short wavelength limit ( ≫ 1) of these functions can be found, by noting that the hypergeometric functions here tend to 1 in this limit, as
ddtϕM(t)icoshtϕM(t),
(4.29)
ddtψM(t)icoshtψM(t).
(4.30)
We see that the time derivative of our mode solutions (4.17) and (4.18) reproduces locally the positive frequency behavior of flat space–time. Thus, our modes can serve as the analogs of the positive frequency modes, and we can use this criterion as well as de Sitter invariance (see Sec. V) in order to define a vacuum.
Note that by making the replacements (4.16) in the expressions for the spinors ψ with the negative sign for the eigenvalue on SN [see Eqs. (4.5) and (4.6)], we obtain the spinors
ψMσ(,s̃)(t,ΩN1)=ϕM(t)χσ(s̃)(ΩN1)iψM(t)χσ(s̃)(ΩN1)
(4.31)
and
ψMσ(+,s̃)(t,ΩN1)=iψM(t)χ+σ(s̃)(ΩN1)ϕM(t)χ+σ(s̃)(ΩN1),
(4.32)
which are not solutions of the Dirac equation (1.8) on dSN. However, these spinors satisfy the equation ψM=MψM and they serve as a tool in the construction of the negative frequency solutions of the Dirac equation (1.8) using charge conjugation in  Appendix A. (Note that the negative frequency solutions are obtained in two different ways: by separating variables in Sec. VI and via charge conjugation in  Appendix A.)
Case 2: N odd. For the construction of the eigenmodes of Eq. (1.1), it is convenient to consider the eigenvalue equation for the iterated Dirac operator 2ψ=λ2ψ. The latter may be written as follows:1,
θN+N12cotθN2+1sin2θÑ2cosθNsin2θNγÑψ=λ2ψ.
(4.33)
By separating variables, the spinor eigenfunctions of the Dirac operator on SN are found to be1,
ψ±nσ(s,s̃)(θN,ΩN1)=cN(n)2(ϕn(θN)χ̂σ(s,s̃)(ΩN1)±iψn(θN)χ̂+σ(s,s̃)(ΩN1)),
(4.34)
where
χ̂σ(s,s̃)=12(1+iγN)χσ(s,s̃)
(4.35)
and the eigenvalues are the same as in Eq. (4.1) [i.e., λ = ±(n + N/2), with n = 0, 1, …]. The spinors χ+σ(s,s̃) and χ̂+σ(s,s̃) are given by
γNχσ(s,s̃)=χ+σ(s,s̃)
(4.36)
and
χ̂+σ(s,s̃)=γNχ̂σ(s,s̃).
(4.37)
Here, s is the spin projection index on SN−1 and s̃ stands for the rest of the spin projection indices on the lower-dimensional spheres of even dimensions. The functions ϕnℓ, ψnℓ are given by Eqs. (4.11) and (4.12), while the spinors χ̂±σ(s,s̃)(ΩN1) are eigenfunctions of the Hermitian operator γÑ (which commutes with the iterated Dirac operator 2) satisfying1 
γÑχ̂±σ(s,s̃)=±+N12χ̂±σ(s,s̃).
(4.38)

As in the even-dimensional case, for each value of n, the eigenspinors ψ+nσ(s,s̃) (or ψnσ(s,s̃)) form a representation of Spin(N + 1) with dimension dn given by Eq. (4.2). (The dimension is half the dimension for the case with N even because there is no contribution from spin projections on SN.) Notice that on S1, the Dirac operator is just /∂θ1 and the eigenspinors are χ±1(θ1)=exp(±i(1+1/2)θ1) [the normalization constant is (2π)−1/2]. The eigenspinors (4.34) are normalized as in the case with N even, and the normalization factors are given again by Eq. (4.7).

We choose to analytically continue the ψ+ eigenmodes. By making the replacements (4.16) in the expression for the eigenspinors ψ+nσ(s,s̃)(θN,ΩN1) [Eq. (4.34)], we obtain the solutions of the Dirac equation on odd-dimensional dSN,
ψMσ(s,s̃)(t,ΩN1)=cN(M)2(ϕM(t)χ̂σ(s,s̃)(ΩN1)+iψM(t)χ̂+σ(s,s̃)(ΩN1)),
(4.39)
where the normalization factor will be determined later. The functions ϕMℓ(t), ψMℓ(t) are given again by Eqs. (4.19) and (4.20). Hence, the solutions (4.39) can be used as positive frequency modes.
As in the even-dimensional case, we can analytically continue the eigenspinors ψ to obtain
ψMσ(s,s̃)(t,ΩN1)=(ϕM(t)χ̂σ(s,s̃)(ΩN1)iψM(t)χ̂+σ(s,s̃)(ΩN1)),
(4.40)
which satisfy the Dirac equation (1.8) with M → −M.

For each value of M, the set of the analytically continued eigenspinors of the Dirac operator |SN forms a representation of the Lie algebra of Spin(N,1) [which is also a representation of the group Spin(N,1)]. If we want to use these mode functions to describe spin-1/2 particles on N-dimensional de Sitter space–time, the corresponding representation has to be unitary. Unitarity ensures that no negative probabilities will arise. A representation is unitary if there is a positive definite inner product that is preserved under the action of the group. In this section, we show that the representation formed by our analytically continued eigenspinors is unitary by introducing a Spin(N,1) invariant inner product among the solutions of the Dirac equation and by verifying the positive-definiteness of the associated norm for our positive frequency solutions. In addition, we calculate the normalization factors cN(Mℓ) and we show that the positive frequency modes transform among themselves under the action of a boost generator. In view of a mode expansion of the quantum Dirac field using our analytically continued modes, the transformation properties thus obtained imply that the corresponding vacuum is de Sitter invariant.

Let ψ and ψ′ be any two Dirac spinors on a globally hyperbolic spacetime (global hyperbolicity is assumed for later convenience). The Dirac inner product of ψ, ψ′ is then given by
(ψ,ψ)=iΣdΣn̂μψ̄γμψ,
(5.1)
where the integration is over any Cauchy surface Σ and n̂μ is the unit normal to the Cauchy surface with n̂0>0. Below, we use this inner product in order to show that our positive frequency modes on dSN have positive norm. (The Dirac inner product is also used in Sec. VI in order to normalize the negative frequency solutions and show that the positive and negative frequency solution subspaces are orthogonal to each other.)
Now, let ψ, ψ′ in Eq. (5.1) be positive frequency solutions of the Dirac equation on global dSN with same mass M [see Eqs. (4.17) and (4.18)]. Then, the Dirac inner product (5.1) is written as
(ψMσ(s,s̃),ψMσ(s,s̃))=idθgψ̄Mσ(s,s̃)γ0ψMσ(s,s̃)
(5.2)
=dθgψMσ(s,s̃)ψMσ(s,s̃),
(5.3)
where the integration is over the Cauchy surface Σ = SN−1 and dθ stands for 12N−1. The square root of the determinant of the de Sitter metric is
g=coshN1tsinN2θN1sinθ2=coshN1tg̃,
(5.4)
where g̃ is the determinant of the SN−1 metric. First, we show that the inner product (5.2) is both time independent and Spin(N,1) invariant. Let ψ(1), ψ(2) be two analytically continued eigenspinors, which satisfy the Dirac equation (1.8). The Dirac equation and Eq. (3.11) imply that the vector current
Jμ=iψ̄(1)γμψ(2)
(5.5)
is covariantly conserved. Hence, the inner product (5.2) is time independent. As for the invariance under Spin(N,1), we can show that the change in the inner product due to infinitesimal Spin(N,1) transformations vanishes (as in Ref. 3). Let ξμ be a Killing vector of dSN satisfying
μξν+νξμ=0.
(5.6)
The Lie derivative of Jμ with respect to the Killing vector ξμ (LξJμ) gives the change in Jμ under the corresponding transformation, that is,
δJμ=LξJμ=ξννJμJννξμ=ν(ξνJμJνξμ),
(5.7)
where we used the fact that both Jμ, ξμ are divergence free. Then, we find
δJ0=ν(ξνJ0Jνξ0)=1gθκg(ξθκJ0Jθκξ0),
(5.8)
where κ = 1, …, N − 1. By integrating Eq. (5.8) over SN−1, we find
δ(ψ(1),ψ(2))=dθgδJ0=0.
(5.9)
Below, we study the positive-definiteness of the norm associated with the inner product (5.2) for our positive frequency modes.
Case 1: N even. Substituting the analytically continued eigenspinors (4.17) [or (4.18)] into the inner product (5.2), we find
(ψMσ(s,s̃),ψMσ(s,s̃))=|cN(M)|22coshN1tϕM*(t)ϕM(t)+ψM*(t)ψM(t)δssδs̃s̃δδσσ,
(5.10)
where the positive-definiteness is obvious (i.e., the representation is unitary).
Using Eqs. (4.24) and (4.25), one finds
ddtcosh(N1)/2tϕM=icosh(N3)/2t+N12ϕM+Mcosh(N1)/2tψM
(5.11)
and
ddtcosh(N1)/2tψM=+icosh(N3)/2t+N12ψMMcosh(N1)/2tϕM,
(5.12)
respectively. Consequently,
coshN1tϕM*(t)ϕM(t)+ψM*(t)ψM(t)=K,
(5.13)
where K is a positive real constant (since the time derivative of the left-hand side vanishes). We can determine the value of K just by letting t = 0 in Eq. (5.13). The functions (4.19) and (4.20) for t = 0 are
ϕM(t=0)=2212Fδ,δ*,δ+δ*2;12
(5.14)
and
ψM(t=0)=i2MN+212Fδ,δ*,δ+δ*2+1;12,
(5.15)
respectively, where
δ=N2++iM.
Using the two formulas,14,15
Fa,b,a+b2;12=πΓa+b21Γ((a+1)/2)Γ(b/2)+1Γ((b+1)/2)Γ(a/2),
(5.16)
Fa,b,a+b2+1;12=2πabΓa+b2+11Γ((b+1)/2)1Γ(a/2)1Γ((a+1)/2)Γ(b/2),
(5.17)
we find
K=ϕM*(0)ϕM(0)+ψM*(0)ψM(0)=2N1|ΓN2+|2|ΓN2++iM|2,
(5.18)
where we also used the Legendre duplication formula,
Γ(2z)=1π22z1Γ(z)Γ(z+1/2).
(5.19)
Since (ψMσ(s,s̃),ψMσ(s,s̃))=|cN(M)|2K/2, it is straightforward to calculate the normalization factor as
|cN(M)|2=2(2N)|ΓN2++iM|2|ΓN2+|2.
(5.20)
Our positive frequency solutions are now normalized by
(ψMσ(s,s̃),ψMσ(s,s̃))=δssδs̃s̃δδσσ.
(5.21)
Case 2: N odd. Substituting the analytically continued eigenspinors (4.39) into the inner product (5.2), we obtain again Eq. (5.10). Thus, the Spin(N,1) representation is unitary (due to the positive-definiteness of the norm) and the normalization is again given by Eqs. (5.20) and (5.21).

In this section, we use the spinorial Lie derivative10 with respect to the Killing vector field ξ in order to study the Spin(N,1) transformations of the analytically continued modes of |SN generated by ξ. More specifically, we show that our positive frequency modes transform among themselves under the action of an infinitesimal boost in the θN−1 direction.

The coordinate expression for the spinorial Lie derivative of a spinor field ψ with respect to the Killing vector ξ is10,
Lξsψ=ξμμψ+14κξλγκγλψ.
(5.22)
(We use the superscript s to distinguish the spinorial Lie derivative from the usual Lie derivative.) We are interested in the transformation generated by the boost Killing vector,
ξ=cosθN1ttanhtsinθN1θN1.
(5.23)
After a straightforward calculation, we find
Lξsψ=ξμμψ+sinθN12coshtγN1γ0ψ
(5.24)
=cosθN1tψtanhtsinθN1θN1ψ+sinθN12coshtγN1γ0ψ.
(5.25)
The spinorial Lie derivative with respect to Killing vectors commutes with the Dirac operator.10 Hence, if ψ is an analytically continued eigenspinor of |SN, we can express Eq. (5.25) as a linear combination of other such eigenspinors. In order to proceed, it is useful to introduce the ladder operators for the functions ϕM(t),ψM(t),ϕ̃N2(θN1),ψ̃N2(θN1), sending the angular momentum quantum number to ± 1. [The functions ϕ̃N2,ψ̃N2 are given by Eqs. (4.11) and (4.12), respectively, with NN − 1, n and N−2.] The ladder operators are given by the following expressions:
Tϕ(+)=ddt+12tanhti2cosht,
(5.26)
Tψ(+)=ddt+12tanht+i2cosht,
(5.27)
Tϕ()=ddt++N32tanht+i2cosht,
(5.28)
Tψ()=ddt++N32tanhti2cosht,
(5.29)
T̃ϕ̃(+)=sinθN1ddθN1++N32cosθN1N2+N222+N2,
(5.30)
T̃ψ̃(+)=sinθN1ddθN1++N32cosθN1+N2+N222+N2,
(5.31)
T̃ϕ̃()=sinθN1ddθN1cosθN1+12+N2+N222+N22,
(5.32)
T̃ψ̃()=sinθN1ddθN1cosθN1+12N2+N222+N22.
(5.33)
The corresponding ladder relations are
Tf(+)fM(t)=k(+)fM+1(t),
(5.34)
Tf()fM(t)=k()fM1(t),
(5.35)
T̃f̃(+)f̃N2(θN1)=k̃(+)f̃+1N2(θN1),
(5.36)
T̃f̃()f̃N2(θN1)=k̃()f̃1N2(θN1),
(5.37)
where fMℓ(t) ∈ {ϕMℓ(t), ψMℓ(t)}, f̃N2(θN1){ϕ̃N2(θN1),ψ̃N2(θN1)}, and
k(+)=i(N/2+)2+M2N/2+,
(5.38)
k()=i(N/2+1),
(5.39)
k̃(+)=(+N1+N2)(N2+1)(+N/2),
(5.40)
k̃()=((N1)/2+1)((N1)/2+)(N2)/2+.
(5.41)
The ladder relations (5.34)–(5.37) can be proved using the raising and lowering operators for the parameters of the Gauss hypergeometric function given in  Appendix B. Below, we describe how to express the spinorial Lie derivative (5.25) of a mode solution ψMσ(s,s̃) as a linear combination of other solutions with the same M.
Case 1: N even (>2). Using Eq. (3.2), one finds
γN1γ0=γ̃N100γ̃N1.
(5.42)
Let ψ be the eigenspinor ψMN2σ̃(±,s̃), where σ̃ stands for quantum numbers other than , N−2. Since the partial derivatives in Eq. (5.25) refer only to the coordinates {t, θN−1}, we want to extract the t and θN−1 dependence from our analytically continued eigenspinors. By combining Eqs. (4.17), (4.18), and (4.34), we can express the spinors ψMN2σ̃(±,s̃)(t,ΩN1) in terms of eigenspinors on SN−2 (χ̃̂±N2σ̃(s)̃(ΩN2)) as follows:
ψMN2σ̃(,s̃)(t,ΩN1)=cN(M)2cN1(N2)2UMN2σ̃(s̃)(t,θN1,ΩN2)DMN2σ̃(s̃)(t,θN1,ΩN2),
(5.43)
ψMN2σ̃(+,s̃)(t,ΩN1)=cN(M)2cN1(N2)2D+MN2σ̃(s̃)(t,θN1,ΩN2)U+MN2σ̃(s̃)(t,θN1,ΩN2),
(5.44)
where
UMN2σ̃(s̃)(t,θN1,ΩN2)=ϕM(t)ϕ̃N2(θN1)χ̃̂N2σ̃(s̃)(ΩN2)iψ̃N2(θN1)χ̃̂+N2σ̃(s̃)(ΩN2)
(5.45)
and DMN2σ̃(s̃) is given by an analogous expression with ϕMℓ(t) → Mℓ(t). By substituting Eqs. (5.43) and (5.45) into the expression for the spinorial Lie derivative (5.25) and making use of Eqs. (5.34)–(5.41), we find after a lengthy calculation
LξsψMσ(,s̃)=RMN2(N)ψM+1σ(,s̃)+LMN2(N)ψM1σ(,s̃)+CMN2(N)ψMσ(±,s̃),
(5.46)
where the coefficients on the right-hand side are given by the following expressions:
RMN2(N)=cN(M)cN1(N2)cN(M+1)cN1(+1,N2)k(+)k̃(+)2+N12
(5.47)
=i2N2+2+M2N2+×(N2+1)(+N2+N1),
(5.48)
LMN2(N)=(1)×cN(M)cN1(N2)cN(M,1)cN1(1,N2)k()k̃()2+N12
(5.49)
=RM,1,N2(N)*,
(5.50)
and
CMN2(N)=iMN2+N222+N22+N2.
(5.51)
Notice that in the last term of the linear combination in Eq. (5.46), the spin projection sign is flipped. We have checked the validity of the above results by using the de Sitter invariance of the inner product (5.2). More specifically, we have verified that (LξsψM,ψM±1)+(ψM,LξsψM±1)=0. [Some details regarding the derivation of Eq. (5.46) can be found in  Appendix C along with the N = 2 case.] It is clear from Eq. (5.46) that our positive frequency solutions transform to other positive frequency solutions with the same M under the transformation generated by ξ. Based on this observation, we can conclude that the vacuum corresponding to these positive frequency modes is de Sitter invariant (see Refs. 16 and 17).
Case 2: N odd. Using Eq. (3.4), we find
γN1γ0=i0110.
(5.52)
As in the case with even N, it is convenient to express the analytically continued eigenspinors ψMN2σ̃(s,s̃)(t,ΩN1) [Eq. (4.39)] in terms of eigenspinors on SN−2 [χ̃±N2σ̃(s̃)(ΩN2)]. By combining Eqs. (4.35), (4.5), and (4.6), we can rewrite Eq. (4.39) as
ψMN2σ̃(,s̃)(t,θN1,ΩN2)=cN(M)2cN1(N2)212(1+i)ϕ̃N2[ϕM+iψM]χ̃N2σ̃(s̃)(1+i)iψ̃N2[ϕMiψM]χ̃N2σ̃(s̃)
(5.53)
and
ψMN2σ̃(+,s̃)(t,θN1,ΩN2)=cN(M)2cN1(N2)212(1+i)iψ̃N2[ϕM+iψM]χ̃+N2σ̃(s̃)(1+i)ϕ̃N2[ϕMiψM]χ̃+N2σ̃(s̃).
(5.54)
Working as in the case with N even, we find after a lengthy calculation
LξsψMσ(,s̃)=RMN2(N)ψM+1σ(,s̃)+LMN2(N)ψM1σ(,s̃)±CMN2(N)ψMσ(,s̃).
(5.55)
Notice that, unlike the even-dimensional case, the two spin projections do not mix with each other. As in the case with N even, we conclude that the vacuum is de Sitter invariant.

In this section, we follow the canonical quantization procedure and give the mode expansion for the free quantum Dirac field on N-dimensional de Sitter space–time with (N − 1)-sphere spatial sections using the analytically continued spinor modes of |SN. As mentioned earlier, our analytically continued eigenspinors can be used as the analogs of the flat space–time positive frequency modes. However, the latter are not the only solutions of the Dirac equation (1.8) on dSN. New solutions (i.e., the negative frequency modes) can be obtained by separating variables. Below, we present the negative frequency solutions before proceeding to the canonical quantization. (Note that the negative frequency solutions can also be obtained using charge conjugation, as demonstrated in  Appendix A.)

Case 1: N even. By making the replacements (4.16) in the expression for the iterated Dirac operator on SN (4.4), one finds
t+N12tanht21cosh2t̃2±sinhtcosh2t̃φ±=M2φ±.
(6.1)
Then, by separating variables (as in Ref. 1), one finds the negative frequency solutions,
VMσ(,s̃)(t,ΩN1)=cN(M)2ϕM*(t)χ+σ(s̃)(ΩN1)iψM*(t)χ+σ(s̃)(ΩN1),
(6.2)
VMσ(+,s̃)(t,ΩN1)=cN(M)2iψM*(t)χσ(s̃)(ΩN1)ϕM*(t)χσ(s̃)(ΩN1).
(6.3)
These are normalized using the inner product (5.2) as
(VMσ(s,s̃),VMσ(s,s̃))=δssδs̃s̃δδσσ
(6.4)
and they are orthogonal to the positive frequency solutions, i.e.,
(ψMσ(s,s̃),VMσ(s,s̃))=0.
(6.5)
As we can see, the negative frequency modes are given by the positive frequency solutions (4.17) and (4.18) by replacing the functions ϕMℓ(t), ψMℓ(t) with their complex conjugate functions and by exchanging χ±N−1) and χN−1). The time derivatives of the spinors (6.2) and (6.3) reproduce the flat space–time behavior in the large limit, i.e., the complex conjugate of Eqs. (4.29) and (4.30).
Case 2: N odd. Working as in the even-dimensional case, the negative frequency modes are found to be
VMσ(s,s̃)(t,ΩN1)=cN(M)2ϕM*(t)χ̂+σ(s,s̃)(ΩN1)+iψM*(t)χ̂σ(s,s̃)(ΩN1),
(6.6)
and they satisfy the conditions (6.4) and (6.5).
The Lagrangian density for a free spinor field Ψ is
L=gΨ̄γμμMΨ
(6.7)
=giΨA(γ0)BA(γμ)CB(μΨ)CMΨB,
(6.8)
where we have written out the spinor indices explicitly in the second line (A, B, C = 1, …, 2[N/2]). The corresponding equation of motion for Ψ is the Dirac equation (1.8). By the standard canonical quantization procedure, we find
{Ψ(t,θ)A,Ψ(t,θ)B}=1g(t,θ)δ(N1)(θθ)δBA,
(6.9)
{Ψ(t,θ)A,Ψ(t,θ)B}={Ψ(t,θ)A,Ψ(t,θ)B}=0.
(6.10)
The mode expansion for the free Dirac field is
Ψ(t,θ)=,σs,s̃aMσ(s,s̃)ψMσ(s,s̃)(t,θ)+bMσ(s,s̃)VMσ(s,s̃)(t,θ),
(6.11)
where we are summing over all angular momentum quantum numbers and over all the possible spin projections. (There are [N/2] spin projection indices in total.) Using the normalization conditions (5.21) and (6.4) and the orthogonality condition (6.5), we may express the annihilation operators, aMσ(s,s̃) and bMσ(s,s̃), as
aMσ(s,s̃)=(ψMσ(s,s̃)(t,θ),Ψ(t,θ))=dθgψMσ(s,s̃)(t,θ)Ψ(t,θ)
(6.12)
and
bMσ(s,s̃)=dθgΨ(t,θ)VMσ(s,s̃)(t,θ).
(6.13)
By combining Eqs. (6.12) and (6.13) with the anti-commutation relations (6.9) and (6.10), we obtain
{aMσ(s,s̃),aMσ(s,s̃)}=δssδs̃s̃δδσσ,
(6.14)
{bMσ(s,s̃),bMσ(s,s̃)}=δssδs̃s̃δδσσ,
(6.15)
while all the other anti-commutators are zero. The de Sitter invariant vacuum is defined by
aMσ(s,s̃)|0=bMσ(s,s̃)|0=0
(6.16)
for all ,σ,(s,s̃). Using the mode expansion of the Dirac field (6.11), we can obtain the mode-sum form for the Wightman two-point function,
W(t,θ),(t,θ)0|Ψ(t,θ)Ψ̄(t,θ)|0
(6.17)
=,σs,s̃ψMσ(s,s̃)(t,θ)ψ̄Mσ(s,s̃)(t,θ).
(6.18)
The high frequency behavior of our mode solutions (4.29) and (4.30) implies that we should adopt the − prescription (i.e., the time variable t should be understood to have an infinitesimal negative imaginary part: tt, ϵ > 0).

In this section, we first review the basics about the construction of Dirac spinor Green’s functions on dSN using intrinsic geometric objects following the work of Mück.4 (Mück gave the coordinate independent construction of the spinor Green’s function in terms of intrinsic geometric objects on maximally symmetric spaces of arbitrary dimensions using Dirac spinors. An analogous construction on four-dimensional maximally symmetric spaces using two-component spinors was first presented in Ref. 18.) Then, using the mode-sum method (6.18), we obtain a closed-form expression for the massless spinor Wightman two-point function on dSN that agrees with the construction presented in Ref. 4. Using this massless two-point function, we infer the analytic expression for the spinor parallel propagator and then obtain the massive spinor Wightman two-point function in a closed form.

Let |ψ⟩ be a state invariant under the action of the de Sitter group. Then, two-point functions [such as ψ|Ψ(x)Ψ̄(x)|ψ] define maximally symmetric bispinors.12 These bispinors can be expressed in terms of the following “preferred geometric objects”: the geodesic distance (2.11), the unit tangent vectors (2.13) to the geodesic with endpoints x, x′, and the bispinor of parallel transport Λ(x, x′), also known as the spinor parallel propagator.4,19,20 The spinor parallel propagator parallel transports a spinor ψ(x′) from x′ to x along the (shortest) geodesic joining these points, i.e.,
ψ(x)A=Λ(x,x)AAψ(x)A,
(7.1)
where ψ(x) is the parallelly transported spinor. The following relations can be used as the defining properties of the spinor parallel propagator for arbitrary the space–time dimension:4,
Λ(x,x)=[Λ(x,x)]1,
(7.2)
γν(x)=Λ(x,x)γμ(x)gμν(x,x)Λ(x,x),
(7.3)
nμμΛ(x,x)=0,
(7.4)
where the parallel transport equation (7.4) holds along the geodesic connecting x and x′. Equation (7.3) describes the parallel transport of gamma matrices. By contracting Eq. (7.3) with nν(x, x′) and using Eqs. (2.18) and (7.2), we find
[Λ(x,x)]1nΛ(x,x)=n,
(7.5)
where nγμ(x)nμ(x,x) and nγμ(x)nμ(x,x). Equation (7.5) conveniently describes the parallel transport property of n. In  Appendix E, we show that our result for the spinor parallel propagator [given by Eq. (7.31)] is consistent with the defining properties (7.2)–(7.5). On dSN, the covariant derivatives of Λ(x.x′) can be expressed as4 
μΛ(x,x)=12tanμ2(γμnnμ)Λ(x,x),
(7.6)
μΛ(x,x)=12tanμ2Λ(x,x)(γμnnμ).
(7.7)
Note that n2=1 and (n)2=1, where 1, 1′ are the identity spinor matrices at x and x′, respectively.
The massive case. The massive spinor Green’s function SM(x, x′) on dSN satisfies the inhomogeneous Dirac equation,
[MSM(x,x)]AA=δ(N)(xx)g(x)δAA.
(7.8)
The Green’s function SM(x, x′) can be expressed in terms of intrinsic geometric objects as follows:4,
SM(x,x)=(αM(μ)+βM(μ)n)Λ(x,x),
(7.9)
where αM(μ), βM(μ) are scalar functions of the geodesic distance. By requiring that SM(x, x′) in Eq. (7.9) satisfies Eq. (7.8), we find the following system of ordinary differential equations for αM(μ), βM(μ),
dαMdμN12tanμ2αMMβM=0,
(7.10)
dβMdμ+N12cotμ2βMMαM=δ(xx)g(x).
(7.11)
Using the variable z = cos2(μ/2) [see Eq. (2.10)], this system of equations is solved by4,
αM(z)=M|ΓN2+iM|2ΓN2+1(2π)N/22N/2zFN2iM,N2+iM;N2+1;z
(7.12)
and
βM(z)=1zMzddz+N12zαM(z).
(7.13)
Using Eqs. (7.12) and (B2), we can rewrite Eq. (7.13) as
βM(z)=|ΓN2+iM|2ΓN2+1(2π)N/22N/21zN2FN2iM,N2+iM;N2;z.
(7.14)
[Note that there is a misprint in the corresponding equation for βM(z)—Eq. (29)—in Ref. 4. Equation (7.14) of the present paper and Eq. (29) of Ref. 4 agree with each other after inserting a missing prefactor.] The proportionality constant for αM(μ) [hence for βM(μ)] has been determined by requiring that the singularity in Eq. (7.12) for μ → 0 has the same strength as the singularity of the flat space–time Green’s function.4 This ensures that the spinor Green’s function (7.9) has the desired short-distance behavior. [Note that since n,αM and βM are known, the only remaining step for obtaining an explicit expression for the two-point function (7.9) is to derive an analytic expression for the spinor parallel propagator.]
The massless case. Letting M = 0 in Eqs. (7.12) and (7.14), we find
α0(z)=0,
(7.15)
β0(z)=Γ(N/2)2N/2(2π)N/21(1z)(N1)/2,
(7.16)
where z = cos2(μ/2) and where we used Eq. (D30). These are just the solutions (with the appropriate singularity strength) of the decoupled system,
dα0dμN12tanμ2α0=0,
(7.17)
dβ0dμ+N12cotμ2β0=δ(xx)g(x).
(7.18)
The massless Green’s function is then given as follows:
S0(x,x)=β0(z)nΛ(x,x)
(7.19)
=Γ(N/2)2N/2(2π)N/2(1z)(N1)/2nΛ(x,x).
(7.20)
We find that the defining properties of Λ(x, x′) [Eqs. (7.2)–(7.4)] translate to the following properties for the massless Green’s function:
[S0(x,x)]1n=1β02nS0(x,x),
(7.21)
[S0(x,x)]1=1β02S0(x,x),
(7.22)
nμμ+N12cotμ2S0(x,x)=0.
(7.23)
Note that by combining Eqs. (7.21) and (7.22), one obtains
nS0(x,x)=S0(x,x)n,
(7.24)
which is equivalent to Eq. (7.5).

In the massive case, the mode-sum approach for the Wightman function (6.18) leads to complicated series involving products of hypergeometric functions and it seems that their corresponding closed-form expressions do not exist in the literature. Fortunately, the situation is simpler in the massless case, and we can obtain a closed-form expression for the Wightman two-point function. This directly results in the knowledge of the spinor parallel propagator Λ(x, x′) due to Eq. (7.19). The spinor parallel propagator Λ(x, x′), in turn, can be used to obtain an analytic expression for the massive spinor Wightman two-point function via Eq. (7.9).

Below, we present the closed-form expression we have obtained by the mode-sum method for the massless Wightman two-point function in agreement with Eq. (7.19). We present the details of the lengthy calculation in  Appendix D (as well as the result for the N = 2 case).

Case 1: N even (N > 2). By letting M = 0 in Eqs. (4.17) and (4.18), we obtain the massless positive frequency modes,
ψ0σ(,s̃)(t,ΩN1)=2(2N)/22ϕ0(t)χσ(s̃)(ΩN1)0,
(7.25)
ψ0σ(+,s̃)(t,ΩN1)=2(2N)/22ϕ0(t)0χ+σ(s̃)(ΩN1).
(7.26)
Now, the function describing the time dependence has the following form:
ϕ0(t)=tanx2cosx2N1,
(7.27)
where cos(x/2) is given in Eq. (4.21) and
tanx2=1isinhtcosht.
(7.28)
Exploiting the rotational symmetry of SN−1, we may let θN−1′ = θN−2′ = ⋯ = θ2′ = θ1′ = 0 in the mode-sum (6.18). After a long calculation, we obtain the following 2N/2-dimensional bispinorial matrix:
W0[(t,θ),(t,0)]=(β0(μ)n)|θ=0expλ(t,θN1,t)2γ0γN1j=2N1expθNj2γNj+1γNj,
(7.29)
where
n|θ=0=γ0n0[(t,θ),(t,0)]+γN1nN1[(t,θ),(t,0)]
(7.30)
[see Eqs. (2.26) and (2.27)]. By comparing Eq. (7.29) with Eq. (7.19), we find
Λ(t,θ),(t,0)=expλ(t,θN1,t)2γ0γN1j=2N1expθNj2γNj+1γNj.
(7.31)
The biscalar λ(t, θN−1, t′) is defined by the following relations:
coshλ2=w+n++wn2isin(μ/2)=w1n0+w2nN1sin(μ/2),sinhλ2=w+n+wn2isin(μ/2)=w1nN1+w2n0sin(μ/2),
(7.32)
where n0n0[(t, θ), (t′, 0)], nN−1nN−1[(t, θ), (t′, 0)] and
w1(t,θN1,t)=sinhtt2cosθN12,
(7.33)
w2(t,θN1,t)=cosht+t2sinθN12,
(7.34)
w±(t,θN1,t)i[w1(t,t,θN1)±w2(t,θN1,t)],
(7.35)
n±n0±nN1.
(7.36)
(This definition of λ is motivated naturally in the mode-sum construction of the massless Wightman function given in  Appendix D.) It is worth mentioning that the biscalar functions w+ and w satisfy w+w = sin2(μ/2), i.e., β0(μ)(w+w)(N1)/2 [see Eqs. (2.20) and (7.16)]. We have verified that Eq. (7.32) are consistent with the relation cosh2(λ/2) − sinh2(λ/2) = 1.

It is natural that the spinor parallel propagator (7.31) is given by a product of N − 1 matrices ∈ Spin(N − 1, 1); these correspond to one boost and N − 2 rotations (see  Appendix D).

As mentioned earlier, we do not follow the mode-sum method for the construction of the massive Wightman function. A closed-form expression for the latter can be found using our result for the spinor parallel propagator (7.31). To be specific, by substituting Eq. (7.31) into Eq. (7.9), one can straightforwardly obtain an analytic expression for the massive Wightman function [with x = (t, θ) and x′ = (t′, 0)] in terms of intrinsic geometric objects. In  Appendix F, we compare the mode-sum form of the massive Wightman function with timelike separated points, x = (t, 0) and x′ = (t′, 0), with the expression coming from Eq. (7.9) with μ = i(tt′). Based on this comparison, we make a conjecture for the closed-form expression of a series containing the Gauss hypergeometric function. Note that a closed-form expression for the spinor parallel propagator on anti-de Sitter space–time [along with the construction of the Feynman Green’s function for the Dirac field according to Eq. (7.9)] can be found in Ref. 20.

Case 2: N odd. The massless positive frequency solutions (4.39) are given by
ψ0σ(s,s̃)(t,ΩN1)=2(2N)/22ϕ0(t)χ̂σ(s,s̃)(ΩN1).
(7.37)
Working as in the even-dimensional case, we obtain again Eqs. (7.29) and (7.31) [where γ0 is given by Eq. (3.4)], and then, we can construct the massive two-point function using Eq. (7.9).

In this paper, we analytically continued the eigenspinors of the Dirac operator on SN to obtain solutions to the Dirac equation on dSN that serve as analogs of the positive frequency modes of flat space–time. Our generalized positive frequency solutions were used to define a vacuum for the free Dirac field. The negative frequency solutions were also constructed. The de Sitter invariance of the vacuum was demonstrated by showing that the positive frequency solutions transform among themselves under infinitesimal Spin(N,1) transformations.

In order to check the validity of our mode functions, the Wightman function for massless spinors was calculated using the mode-sum method and it was expressed in a form that is in agreement with the construction in terms of intrinsic geometric objects (μ,n,Λ) given in Ref. 4. An analytic expression for the spinor parallel propagator was found. This expression was tested using the defining properties of the spinor parallel propagator, as presented in Ref. 4 (see  Appendix E). Note that it has been checked that the spinor Green’s functions expressed in terms of μ,n,Λ have Minkowskian singularity strength in the limit μ → 0.4 Thus, the conditions for the unique vacuum21 are satisfied by the vacuum for the free massless Dirac field defined in this paper.

Although we did not obtain a closed-form expression for the massive spinor Wightman function by the mode-sum method using our analytically continued eigenspinors, we constructed it in terms of intrinsic geometric objects. Since the short-distance behavior has been checked in Ref. 4, the requirements for a preferred vacuum are again satisfied. The mode-sum method and the geometric construction of Ref. 4 should give the same result for the massive Wightman function. This observation leads to the series conjecture of  Appendix F.

The author is grateful to Atsushi Higuchi for guidance, encouragement, and useful discussions. He also thanks Wolfgang Mück for communications and the referee for useful comments. Subsection V A was part of the author’s M.Sc. thesis at Imperial College London. This work was supported by a studentship from the Department of Mathematics, University of York.

In this Appendix, we demonstrate how the negative frequency solutions given by Eqs. (6.2), (6.3), and (6.6) are constructed by charge conjugating our analytically continued eigenspinors. First, let us review charge conjugation for Dirac spinors on dSN and on spheres following Ref. 22. For convenience, our discussion will be based on the unitary matrices B± that relate the gamma matrices to their complex conjugate matrices by similarity transformations, i.e.,
(γa)*=B+γaB+1,(γa)*=BγaB1,
(A1)
and not in terms of the conventional charge conjugation matrices C± that relate γa to (γa)T. These two ways of defining charge conjugation are equivalent.22 From this point, we will refer to the matrices B± as the charge conjugation matrices. [We should note that the representation we use for the gamma matrices (3.2) and (3.4) is different from the one used in Ref. 22. Also, note that charge conjugation matrices are defined up to a phase factor and that γN ≡−0.]

1. Charge conjugation on N-dimensional de Sitter space–time and on spheres

For convenience, let us work in d = τ + s dimensions, with τ ∈ {0, 1} being the number of timelike dimensions and s being the number of spacelike dimensions.

For even d dimensions, there are both B+ and B. For d odd dimensions, we can use one of the matrices from the (d − 1)-dimensional case.22 (As it will be clear in the following, one needs to modify the charge conjugation matrix on dSd−1 before using it on dSd. This is not the case in Ref. 22 because a different representation for γa’s is used.) More specifically, on odd-dimensional spaces with Lorentzian (Euclidean) metric signature, there is only B+ (B) for [d/2] odd and only B (B+) for [d/2] even (see Refs. 22 and 23 for more details).

Let Ψ be a 2[d/2]-dimensional Dirac spinor transforming under Spin(s, τ). Its charge conjugated spinor is defined with either one of the following two ways:
ΨC+B+1Ψ*orΨCB1Ψ*.
(A2)
Suppose now that Ψ± is an eigenspinor of the Dirac operator with the eigenvalue κ(τ,s)±, i.e.,
(τ,s)Ψ±=κ(τ,s)±Ψ±,
(A3)
where (1,N1)|dSN is the Dirac operator on dSN with κ(1,N1)±±M and (0,N1)̃ is the Dirac operator on SN−1 with κ(0,N1)±±i(+(N1)/2). The charge conjugated counterparts of the eigenspinors of the Dirac operator are also eigenspinors. This can be understood as follows: taking the complex conjugate of Eq. (A3) and using Eqs. (A1) and (A2), we find
(τ,s)Ψ±C+=+(κ(τ,s)±)*Ψ±C+,
(A4)
(τ,s)Ψ±C=(κ(τ,s)±)*Ψ±C,
(A5)
where we also used (Σab)*=B±ΣabB±1. It is clear from Eqs. (A4) and (A5) that performing charge conjugation with B changes the sign of the mass term on dSN. Also, Eqs. (A4) and (A5) imply the following relations for the eigenspinors of the Dirac operator on Sn [with Ψ±=χ±nσ(s̃) and κ(0,n)±=±i(n+n/2)]:
(χ±nσ(s̃))Cχ±nσ(s̃),(χ±nσ(s̃))C+χnσ(s̃),
(A6)
where n is arbitrary, σ stands for angular momentum quantum numbers other than n, and s̃ represents the [n/2] spin projection indices that correspond to this eigenspinor. The label s̃ is no necessarily equal to s̃.

Below, we use the tilde notation for quantities defined on SN−1.

2. Negative frequency solutions for N even

Case 1: N/2 even. The charge conjugation matrices B±, satisfying Eq. (A1) on dSN, are given by the following products of gamma matrices:
B+=γ1r=1(N4)/4γ4rγ4r+1,
(A7)
B=γ0r=1N/4γ4r2γ4r1.
(A8)
On the odd-dimensional spatial part SN−1, there is only B̃ since [(N − 1)/2] is odd. This is given by
B̃=γ̃1r=1(N4)/4γ̃4rγ̃4r+1.
(A9)
For convenience, we choose to define charge conjugation using B+, which preserves the sign of the mass term in the Dirac equation. Using the representation (3.2) for the gamma matrices, we can express B+ as follows:
B+=0iB̃iB̃0.
(A10)
The charge conjugated counterparts of the positive frequency solutions ψMσ(,s̃) [Eq. (4.17)] can be constructed using Eqs. (A6) and (A10). Then, we have (omitting the normalization factors)
(ψMσ(,s̃)(t,ΩN1))C+=(i)iψM*(t)(χσ(s̃)(ΩN1))C̃ϕM*(t)(χσ(s̃)(ΩN1))C̃iψM*(t)χσ(s̃)(ΩN1)ϕM*(t)χσ(s̃)(ΩN1).
(A11)
After normalizing these modes, we find the negative frequency solutions (6.3). Similarly, starting from the positive frequency solutions ψMσ(+,s̃) [Eq. (4.18)], we find the negative frequency modes (6.2).
Case 2: N/2 odd. The charge conjugation matrices on dSN are given by
B+=γ0γ1r=1(N2)/4γ4rγ4r+1,
(A12)
B=1×r=1(N2)/4γ4r2γ4r1.
(A13)
Since [(N − 1)/2] is even, the only charge conjugation matrix on SN−1 is B̃+. The matrices B and B̃+ are related to each other as follows:
B=r=1(N2)/4γ̃4r2γ̃4r100γ̃4r2γ̃4r1
(A14)
=B̃+00B̃+.
(A15)

In order to construct the negative frequency solutions, it is convenient to use the charge conjugation matrix B that flips the sign of the mass term in the Dirac equation and the “negative mass” spinors ψMσ(s,s̃) [Eqs. (4.31) and (4.32)]. Then, by working as in the case with N/2 even, we obtain the negative frequency solutions (6.2) and (6.3) [with VMσ(,s̃)(ψMσ(,s̃))C and VMσ(+,s̃)(ψMσ(+,s̃))C].

3. Negative frequency solutions for N odd

Case 1: [N/2] even. The only charge conjugation matrix on dSN is B, which changes the sign of the mass term of the Dirac equation. It is given by
B=γ0r=1(N1)/4γ4r2γ4r1.
(A16)
Note that this is the matrix (A8) with NN − 1, where now, γ0 is given by Eq. (3.4). Then, Eq. (A16) may be expressed in terms of the charge conjugation matrix on SN−1 as
B=iγNB̃+=B̃+iγN.
(A17)
By performing charge conjugation for the spinors ψMσ(s̃N1) [Eq. (4.40)], we find
ψMσ(s̃N1)(t,ΩN1)C=iϕM*(t)γNχ̂σ(s̃N1)(ΩN1)C̃++iψM*(t)γNχ̂+σ(s̃N1)(ΩN1)C̃+,
(A18)
where s̃N1 represents the spin projection indices sN−1, sN−3, …, s4, s2 on the lower-dimensional spheres and the charge conjugated counterparts of the “hatted” spinors can be found using Eqs. (4.35)–(4.37) and Eq. (A6). More specifically, by introducing the proportionality constant c such that (χσ(s̃N1))C̃+=cχ+σ(s̃N1), we find
(χ̂±σ(s̃N1))C̃+=icχ̂±σ(s̃N1).
(A19)
By substituting this equation into Eq. (A18), we obtain the negative frequency solution (6.6).
Case 2: [N/2] odd. The only charge conjugation matrix on dSN is B+. This is given by
B+=γ0γ1r=1(N3)/4γ4rγ4r+1
(A20)
=γ0B̃=B̃γ0.
(A21)
As in the case with [N/2] even, we introduce the proportionality constant m such that (χσ(s̃N1))C̃=mχσ(s̃N1), and we find
(χ̂±σ(s̃N1))C̃=mχ̂±σ(s̃N1).
(A22)
Then, we use the matrix (A21) in order to find the charge conjugate of the spinors ψMσ(s̃N1) [Eq. (4.39)], and working as in the previous case, we obtain the negative frequency solution (6.6).
The Gauss hypergeometric function F(a, b; c; z) satisfies13,
ddzF(a,b;c;z)=abcF(a+1,b+1;c+1;z),
(B1)
zddz+c1F(a,b;c;z)=(c1)F(a,b;c1;z),
(B2)
zddz+aF(a,b;c;z)=aF(a+1,b;c;z).
(B3)
By combining Eq. (B3) with the even relation24 
(cb)F(a+1,b1;c;z)+(ba1)(1z)F(a+1,b;c;z)=(ca1)F(a,b;c;z),
(B4)
we find
a(bc)+a(b+a+1)z(b+a+1)z(1z)ddzF(a,b;c;z)=a(bc)F(a+1,b1;c;z).
(B5)
Using Eqs. (B1) and (B2), we can show the ladder relations (5.34) and (5.35), while using Eq. (B5), we can show the ladder relations (5.36) and (5.37).

1. Transformation properties for N > 2; some details for the derivation of Eq. (5.46)

Here, we present some details for the derivation of Eq. (5.46) that expresses the spinorial Lie derivative (5.25) of the analytically continued eigenspinors (4.17) and (4.18) as a linear combination of solutions of the Dirac equation. The case with odd N [i.e., Eq. (5.55)] can be proved similarly, and its derivation is not presented.

In order to obtain Eq. (5.46), it is useful to introduce the following relations (where θθN−1):
ξμμ(ϕM(t)ϕ̃N2(θ))+iϕM(t)2coshtsinθψ̃N2(θ)=12+N12Tϕ(+)×T̃ϕ̃(+)Tϕ()×T̃ϕ̃()ϕM(t)ϕ̃N2(θ)+MN2+N222+N2+N22ψM(t)ϕ̃N2(θ),
(C1)
ξμμ(ϕM(t)ψ̃N2(θ))iϕM(θ)2coshtsinθϕ̃N2(θ)=12+N12Tϕ(+)×T̃ψ̃(+)Tϕ()×T̃ψ̃()ϕM(t)ψ̃N2(θ)MN2+N222+N2+N22ψM(t)ψ̃N2(θ),
(C2)
ξμμ(ψM(t)ψ̃N2(θ))+iψM(t)2coshtsinθϕ̃N2(θ)=12+N12Tψ(+)×T̃ψ̃(+)Tψ()×T̃ψ̃()ψM(t)ψ̃N2(θ)+MN2+N222+N2+N22ϕM(t)ψ̃N2(θ),
(C3)
ξμμ(ψM(t)ϕ̃N2(θ))iψM(t)2coshtsinθψ̃N2(θ)=12+N12Tψ(+)×T̃ϕ̃(+)Tψ()×T̃ϕ̃()ψM(t)ϕ̃N2(θ)MN2+N222+N2+N22ϕM(t)ϕ̃N2(θ).
(C4)
We can prove relation (C1) as follows: We express ψ̃N2 on the left-hand side in terms of ϕ̃N2,dϕ̃N2/dθN1 using Eq. (4.14). As for the right-hand side, we expand Tϕ(±),T̃ϕ̃(±) using Eqs. (5.34)–(5.37), and then, we express ψMℓ in terms of ϕMℓ, Mℓ/dt using Eq. (4.24). Then, it is straightforward to show that the two sides are equal. Relations (C2)–(C4) can be proved in the same way.
Let us now derive Eq. (5.46) for the negative spin projection solution (the positive spin projection case can be treated in the same way). Substituting Eqs. (5.42) and (5.43) into Eq. (5.25), we find
LξsψMN2σ̃(,s̃)=C1C2ξμμUMN2σ̃(s̃)sinθ2coshtγ̃N1UMN2σ̃(s̃)ξμμDMN2σ̃(s̃)+sinθ2coshtγ̃N1DMN2σ̃(s̃),
(C5)
where C1cN(M)/2 and C2cN1(N2)/2. Then, using
γ̃N1χ̃̂±N2σ̃(s̃)(ΩN2)=χ̃̂N2σ̃(s̃)(ΩN2)
[see Eq. (4.37)] and Eq. (5.45), it is straightforward to find
LξsψMN2σ̃(,s̃)=C1C2×χ̃̂N2σ̃(s̃)ξμμ[ϕMϕ̃N2]+isinθ2coshtϕMψ̃N2iχ̃̂+N2σ̃(s̃)ξμμ[ϕMψ̃N2]isinθ2coshtϕMϕ̃N2iχ̃̂N2σ̃(s̃)ξμμ[ψMϕ̃N2]isinθ2coshtψMψ̃N2+χ̃̂+N2σ̃(s̃)ξμμ[ψMψ̃N2]+isinθ2coshtψMϕ̃N2.
(C6)
At this point, we can use relations (C1)–(C4) to find
LξsψMN2σ̃(,s̃)=C1C212+N12Tϕ(+)ϕMiTψ(+)ψMχ̃̂N2σ̃(s̃)T̃ϕ̃(+)ϕ̃N2iχ̃̂+N2σ̃(s̃)T̃ψ̃(+)ψ̃N212+N12Tϕ()ϕMiTψ()ψMχ̃̂N2σ̃(s̃)T̃ϕ̃()ϕ̃N2iχ̃̂+N2σ̃(s̃)T̃ψ̃()ψ̃N2iMN2+N222+N2+N22iψMϕMχ̃̂N2σ̃(s̃)ϕ̃N2+iχ̃̂+N2σ̃(s̃)ψ̃N2.
(C7)
Then, using Eqs. (5.43) and (5.45) as well as the ladder relations (5.34)–(5.37), we obtain Eq. (5.46).

2. Transformation properties for N = 2.

The massive positive frequency solutions (4.17) and (4.18) for N = 2 are given by
ψM()(t,φ)=c2(M)2πϕM(t)iψM(t)ei(+1/2)φ,
(C8)
ψM(+)(t,φ)=c2(M)2πiψM(t)ϕM(t)e+i(+1/2)φ,
(C9)
where 0 ≤ φθ1 < 2π and = 0, 1, …. By calculating the spinorial Lie derivative with respect to the boost Killing vector (5.23), we arrive again at Eq. (5.25), where ψM(±)/φ=±i(+12)ψM(±). By expressing cos φ and sin φ in terms of exp(±) and using the ladder operators (5.34) and (5.35) with N = 2, it is straightforward to find
LξsψM(±)=k(+)2c2(M)c2(M,+1)ψM+1(±)+k()2c2(M)c2(M,1)ψM1(±)
(C10)
=i2(+1iM)ψM+1(±)i2(+iM)ψM1(±).
(C11)
By using Eq. (C11), we have verified that (LξsψM,ψM±1)+(ψM,LξsψM±1)=0, in agreement with the de Sitter invariance of the inner product (5.2).

In this Appendix, we present the derivation of the massless Wightman two-point function using the mode-sum method (6.18) for even N. The derivation of the two-point function for N odd has many similarities with the even-dimensional case and therefore is just briefly discussed. The case with N = 2 is presented separately at the end.

Let us first introduce the notation and some useful relations used in the calculations. The functions (4.11) and (4.12) used in the recursive construction of the eigenspinors of the Dirac operator on SNr (Nr = 1, 2, …, N − 2) are denoted as
ϕ̃NrNr1(θNr)ϕ̃NrNr1(Nr),ψ̃NrNr1(θNr)ψ̃NiNr1(Nr),
(D1)
with ϕ̃00(Nr)=cos(θNr/2) and ψ̃00(Nr)=sin(θNr/2) [see Eqs. (D13) and (D14) below]. We let θNr = (θNr, θNr−1, …, θ1). The dimension of the Spin(N − 1,1) representation is denoted as D ≡ 2N/2. Also, let s̃N2 represent the spin projection indices (sN−2, sN−4, …, s4, s2), s̃N4 represent (sN−4, …, s4, s2), and so forth. Similarly, σNr represents the angular momentum quantum numbers (Nr, Nr−1, …, 2, 1). Note that for θN−1′ = θN−2′ = ⋯ = θ1′ = 0, we have
cosμ|θ=0=sinhtsinht+coshtcoshtcosθN1
(D2)
[see Eq. (2.20)], while the only non-zero (vielbein basis) components of the tangent vector na|θ′=0 [see Eqs. (2.26)–(2.28)] are given by
n0|θ=0=1sinμ(coshtsinhtsinhtcoshtcosθN1),
(D3)
nN1|θ=0=coshtsinμsinθN1=1coshtnθN1|θ=0.
(D4)
(For brevity, we will denote n0|θ′=0, nN−1|θ′=0 and n|θ=0 by n0, nN−1 and n respectively.) Also, notice that Spin(N − 1, 1) transformation matrices can be expressed as
exp(aΣ0j)=expa2γ0γj=1cosha2+γ0γjsinha2,
(D5)
exp(bΣkj)=expb2γkγj=1cosb2+γkγjsinb2
(D6)
(with kj and k, j = 1, 2, …, N − 1), where a and b are the transformation parameters. The corresponding generators are given by Eq. (3.6). Also, many of the following calculations involve the variables x = π/2 − it and x′ = π/2 − it′ [see Eq. (1.6)].
We can now start deriving the massless Wightman two-point function for N even. By expanding the summation over the spin projections (s = ±), Eq. (6.18) becomes
W0(t,θN1),(t,0)==0σN2s̃N2ψ0σN2(+,s̃N2)(t,θN1)ψ̄0σN2(+,s̃N2)(t,0)+ψ0σN2(,s̃N2)(t,θN1)ψ̄0σN2(,s̃N2)(t,0).
(D7)
Then, using Eqs. (7.25) and (7.26), we find
W0(t,θN1),(t,0)=cN(M=0)22=0σN2ϕ0(t)ϕ0*(t)×s̃N4sN20χσN2(sN2,s̃N4)(θN1)χσN2(sN2,s̃N4)(0)χ+σN2(sN2,s̃N4)(θN1)χ+σN2(sN2,s̃N4)(0)0,
(D8)
where χ±σN2(sN2,s̃N4) are the eigenspinors on SN−1. In order to proceed, we need to express the eigenspinors on SNr, with Nr being odd, in terms of eigenspinors on SNr−2. Therefore, using Eqs. (4.34), (4.5), and (4.6), we derive the following two recursive relations:
χ±NrσNr1(,s̃Nr3)(θNr)=cNr(NrNr1)2cNr1(Nr1Nr2)212×(1+i)ϕ̃Nr1Nr2(Nr1)[ϕ̃NrNr1(Nr)±iψ̃NrNr1(Nr)](1+i)ψ̃Nr1Nr2(Nr1)[ϕ̃NrNr1(Nr)iψ̃NrNr1(Nr)]χNr2,σNr3(s̃Nr3)(θNr2),
(D9)
χ±NrσNr1(+,s̃Nr3)(θNr)=cNr(NrNr1)2cNr1(Nr1Nr2)212×(1+i)ψ̃Nr1Nr2(Nr1)[ϕ̃NrNr1(Nr)±iψ̃NrNr1(Nr)](1+i)ϕ̃Nr1Nr2(Nr1)[ϕ̃NrNr1(Nr)iψ̃NrNr1(Nr)]χ+Nr2,σNr3(s̃Nr3)(θNr2)
(D10)
(for r odd and N − 3 ≥ r ≥ 1). Since ψ̃NrNr1(Nr)(0)=0 and ϕ̃NrNr1(Nr)(0) is non-zero only for Nr−1 = 0, it is clear from the recursive relations (D9) and (D10) that the only non-vanishing terms in Eq. (D8) are the ones with N−2 = N−3 = ⋯ = 2 = 1 = 0. Thus, only the summation over N−1 survives in the mode-sum. Substituting Eqs. (D9) and (D10) (with r = 1) into Eq. (D8), one obtains (after some calculations)
W0(t,θN1),(t,0)=cN(M=0)22=0ϕ0(t)ϕ0*(t)cN1(0)22cN2(00)22×ϕ̃0(N1)(0)*iϕ̃0(N1)(θN1)γ0+ψ̃0(N1)(θN1)γN1I2ϕ̃00(N2)ψ̃00(N2)ψ̃00(N2)ϕ̃00(N2)ID/4×I2s̃N4χ00(s̃N4)(θN3)χ00(s̃N4)(0)00χ+00(s̃N4)(θN3)χ+00(s̃N4)(0),
(D11)
where Id is the identity matrix of dimension d. Also, we are going to use the following results:
|cN1(0)|2=Γ(+1)Γ(+N1)2N3|Γ+N12|2,
(D12)
ϕ̃0(N1)(θN1)=κϕ(N1)(0)cosθN12F+N1,;N12;sin2θN12,
(D13)
ψ̃0(N1)(θN1)=κϕ(N1)(0)(+(N1)/2)(N1)/2sinθN12F+N1,;N+12;sin2θN12,
(D14)
where
κϕ(N1)(0)=Γ+N12!ΓN12
(D15)
[see Eqs. (4.7) and (4.11)–(4.13)].
We complete the derivation of the massless two-point function in three steps: (1) we calculate the proportionality constant [also we show that it agrees with the proportionality constant in Eq. (7.20), (2) we obtain a closed-form result for the infinite sum over and we determine the dependence on {t, θN−1, t′}, and (3) we obtain analytic expressions for the terms of the two-point function that depend only on the angular variables θN−2, θN−3, …, θ1. We call the latter the “angular part” of the two-point function and we denote it as follows:
W̃0(ΩN2)j=3N1cNj(00)22×I2ϕ̃00(N2)ψ̃00(N2)ψ̃00(N2)ϕ̃00(N2)ID/4×I2s̃N4χ00(s̃N4)(θN3)χ00(s̃N4)(0)00χ+00(s̃N4)(θN3)χ+00(s̃N4)(0).
(D16)
After completing these steps, it will be clear that the obtained two-point function is of the form (7.19) (i.e., it agrees with the construction presented in Ref. 4].

1. The proportionality constant

The proportionality constant for the massless two-point function arises from the normalization factors in Eq. (D11). [Note that apart from cN(M = 0) and cN−1(0), there are N − 2 additional normalization factors; one for each lower-dimensional sphere.] The overall contribution from the normalization factors is given by the following product:
cN(M=0)22cN1(0)2κϕ(N1)(0)2j=2N1cNj(00)22
(D17)
=12πcN(M=0)22cN1(0)2κϕ(N1)(0)2j=2N2cNj(00)22,
(D18)
where c1(00)1/π is the normalization factor for eigenspinors on S1, while the normalization factors for eigenspinors on higher-dimensional spheres are given by Eq. (4.7). Using Eqs. (D12) and (D15), we observe that
cN1(0)κϕ(N1)(0)2=cN1(00)2(N1)!,
(D19)
where (N − 1) = Γ(N − 1 + )/Γ(N − 1) is the Pochhamer symbol for the rising factorial. Using Eq. (D19), we may rewrite Eq. (D17) as
1π2N2N2j=1N2Γ(Nj)|ΓNj2|22Nj2×(N1)!=Γ(N/2)2N/2(2π)N/2×(N1)!,
(D20)
where we also used Eqs. (4.7) and (5.20) and the Legendre duplication formula (5.19). Equation (D20) clearly gives the desired form for the proportionality constant [see Eq. (7.20)]. The -dependence in Eq. (D20) will be discussed later (it will be used in the summation over ).

2. Obtaining a closed-form expression for the series

Using Eqs. (D11), (D13), (D14), and (D20), we collect all the -dependent terms of the two-point function. Then, the mode-sum expression (D11) can be written as
W0(t,θN1),(t,0)=Γ(N/2)2N/2(2π)N/2[iAγ0+BγN1]W̃0(ΩN2),
(D21)
where
A=cosθN12=0(N1)!ϕ0(t)ϕ0*(t)F+N1,;N12;sin2θN12,
(D22)
B=2sinθN12N1=0(N1)!+N12ϕ0(t)ϕ0*(t)F+N1,;N+12;sin2θN12.
(D23)
Using Eq. (7.27) for ϕ0(t),ϕ0*(t), we find
A=cos(θN1/2)(cos(x/2)cos(x/2)*)N1=0(N1)!ρ(t,t)F+N1,;N12;sin2θN12,
(D24)
B=2N1sin(θN1/2)(cos(x/2)cos(x/2)*)N1=0(N1)!+N12ρ(t,t)F+N1,;N+12;sin2θN12,
(D25)
where
ρ(t,t)tanx2tanx2*=(1isinht)(1+isinht)coshtcosht.
(D26)
Let us first find the infinite sum in A. By using the formula25 
k=0(a)kk!tkF(k,a+k;c;z)=(1t)aF(a2,a+12;c;4t(1t)2z),|t|<1,
(D27)
Eq. (D24) can be written as
A=cos(θN1/2)(cos(x/2)cos(x/2)*)N1(1ρ(t,t))N+1F(N12,N2;N12;4ρ(t,t)(1ρ(t,t))2sin2θN12)
(D28)
=cos(θN1/2)(cos(x/2)cos(x/2)*)N1(1ρ(t,t))N+1((1ρ(t,t))2)N/2(1ρ(t,t))2+4ρ(t,t)sin2θN12N/2,
(D29)
where we also used
F(a,b;b;z)=1(1z)a.
(D30)
[Note that since |ρ(t, t′)| = 1, the series in Eq. (D24) diverges. Therefore, we make the replacement tt with ϵ > 0 before applying (D27), and then, we let ϵ → 0.] By expressing x, x′ and ρ(t, t′) in terms of t and t′, we can write Eq. (D29) as
A=isinhtt2cosθN12sin2μ2N/2
(D31)
=iw1(t,θN1,t)sin2μ2N/2.
(D32)
[The biscalar function w1 is given in Eq. (7.33), while sin2(μ/2) can be found by Eq. (D2)].
Let us now find the infinite sum in B. We can rewrite Eq. (D25) as
B=2N1sin(θN1/2)(cos(x/2)cos(x/2)*)N1ρρ