The mode solutions of the Dirac equation on *N*-dimensional de Sitter space–time (*dS*_{N}) with (*N* − 1)-sphere spatial sections are obtained by analytically continuing the spinor eigenfunctions of the Dirac operator on the *N*-sphere (*S*^{N}). The analogs of flat space–time positive frequency modes are identified, and a vacuum is defined. The transformation properties of the mode solutions under the de Sitter group double cover [Spin(*N*,1)] are studied. We reproduce the expression for the massless spinor Wightman two-point function in closed form using the mode-sum method. By using this closed-form expression and taking advantage of the maximal symmetry of *dS*_{N}, 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.

## I. INTRODUCTION

*S*

^{N},

^{1}More specifically, the eigenspinors on

*S*

^{N}have been recursively constructed in terms of eigenspinors on

*S*

^{N−1}using separation of variables in geodesic polar coordinates, and their eigenvalues have been calculated. The line element for

*S*

^{N}may be written as

*θ*

_{N}is the geodesic distance from the North Pole and $dsN\u221212$ is the line element of

*S*

^{N−1}. Similarly, the line element of

*S*

^{n}(

*n*= 2, 3, …,

*N*− 1) can be expressed as

*N*-dimensional de Sitter space–time is the maximally symmetric solution of the vacuum Einstein field equations with positive cosmological constant Λ,

^{2}

*N*-dimensional de Sitter space–time can also be obtained by an “analytic continuation” of

*S*

^{N}. More specifically, by replacing

*S*

^{N}metric (1.2), we find the line element for

*dS*

_{N}with

*S*

^{N−1}spatial sections [see Eq. (2.2)],

*dS*

_{N},

*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

*S*

^{N},

^{3}where the restriction of unitarity gave rise to the forbidden mass range for the spin-2 field on

*dS*

_{N}.

In this paper, our main aim is the identification of the mode functions for the free Dirac field on global *dS*_{N} with *S*^{N−1} spatial sections. As a consistency check, we reproduce the expected form for the massless spinor Wightman function^{4} 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 *S*^{N} and we review the geodesic structure of *dS*_{N}. In Sec. III, we present the basics about Dirac spinors and Clifford algebras on *dS*_{N}. In Sec. IV, we begin by reviewing the eigenspinors of the Dirac operator on *S*^{N} following Ref. 1. Then, we obtain the mode solutions of the Dirac equation on *dS*_{N} by analytically continuing the eigenmodes on *S*^{N}, 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 *dS*_{N} 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 *dS*_{N} 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.

## II. GEOMETRY OF *N*-DIMENSIONAL DE SITTER SPACE–TIME

### A. Coordinate system, Christoffel symbols, and spin connection

*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

*η*

_{ab}= diag(−1, 1, 1, …, 1) (

*a*,

*b*= 0, 1, …,

*N*) is the flat metric for the embedding space and

*X*

^{0},

*X*

^{1}, …,

*X*

^{N}are the standard Minkowski coordinates. The global coordinates used in this paper are given by

*Z*

^{i}’s are the spherical coordinates for

*S*

^{N−1}in

*N*-dimensional Euclidean space

*θ*

_{1}< 2

*π*and 0 ≤

*θ*

_{i}≤

*π*(

*i*≠ 1). Using the coordinates (2.2), we obtain the line element (1.7) for

*dS*

_{N}.

*S*

^{N−1}. The vielbein fields are given by

*S*

^{N−1}. The latter are given by

*ω*

_{abc}=

*ω*

_{a[bc]}≡ (

*ω*

_{abc}−

*ω*

_{acb})/2 is given by

*S*

^{N−1}and

*δ*

_{ij}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.)

### B. Geodesics on *dS*_{N}

Geodesics on *dS*_{N} 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.

*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*∈ [0, 1)), the points

*x*,

*x*′ are spacelike separated ($\mu \u2208R$) and they can be connected by a spacelike geodesic. (The equality sign corresponds to antipodal points.) The geodesic distance is then defined by $Z(x,x\u2032)=cos(\mu (x,x\u2032))$ or, equivalently,

*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 $x\u0304$ be the antipodal point of

*x*and let

*x*′ be any point in the interior of the past or future light cone of $x\u0304$. Then, there is no geodesic connecting

*x*and

*x*′.

^{12}) If $Z(x,x\u2032)=1$ (i.e.,

*z*= 1), the geodesic distance is zero and the two points can be connected by a null geodesic (or they coincide). If $Z(x,x\u2032)>1$ (i.e.,

*z*> 1), the two points are timelike separated ($\mu =i\kappa ,\kappa \u2208R$) and they can be connected by a timelike geodesic. The geodesic distance for timelike separation is given by

*μ*→

*iκ*.

*x*and

*x*′ to the geodesic connecting the two points are defined by

*dS*

_{N}is a maximally symmetric space–time, the unit tangents satisfy

^{12}

^{,}

*g*

_{μν}(

*x*) is the metric tensor and

*g*

_{μν′}(

*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

*V*

^{ν′}(

*x*′) from

*x*′ to

*x*along the geodesic connecting these points,

^{12}

^{,}

*x*. (In this paper, by geodesic, we mean the shortest geodesic connecting the two points.) It is worth noting the relations

^{12}

*n*= 2, …,

*N*− 1 and

*i*= 1, …,

*N*− 1) are given by

*n*

_{μ′}(

*x*,

*x*′) are given by analogous expressions with

*t*↔

*t*′,

*θ*

_{i}↔

*θ*

_{i}′. The vielbein basis components of the tangent vector at

*x*, $na(x,x\u2032)=ea\mu (x)n\mu (x,x\u2032)$ (

*a*= 0, 1, …,

*N*− 1), are given by

*b*= 1, …,

*N*− 2, while the components of $na\u2032(x,x\u2032)=ea\u2032\mu \u2032(x\u2032)n\mu \u2032(x,x\u2032)$ [

*a*′ = 0′, 1′, …, (

*N*− 1)′] can be obtained from Eqs. (2.26) and (2.28) with

*t*↔

*t*′,

*θ*

_{a}↔

*θ*

_{a}′. (Note that we define cos Ω

_{0}≡ 1.)

## III. DIRAC SPINORS AND CLIFFORD ALGEBRA ON *N*-DIMENSIONAL DE SITTER SPACE-TIME

^{[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,

**1**is the identity matrix and

*η*

^{ab}is the inverse of the

*N*-dimensional Minkowski metric

*η*

_{ab}= 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 ($\gamma \u0303i$) as follows:

- For even
*N*,where the lower-dimensional gamma matrices satisfy the Euclidean Clifford algebra anti-commutation relations,(3.2)$\gamma 0=i0110,\gamma i=0i\gamma \u0303i\u2212i\gamma \u0303i0,i=1,\u2026,N\u22121,$(3.3)${\gamma \u0303i,\gamma \u0303j}=2\delta ij1,i,j=1,\u2026,N\u22121.$ - For odd
*N*,$\gamma 0=i100\u22121,\gamma N\u22121=0110,$The double-tilde is used to denote gamma matrices in(3.4)$\gamma j=\gamma \u0303j=0i\gamma \u0303\u0303j\u2212i\gamma \u0303\u0303j0,j=1,\u2026,N\u22122.$*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 *dS*_{N} can be obtained by the Euclidean gamma matrices on *S*^{N} 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 *γ*^{a}’s remain the same under (1.6) apart from *γ*^{N}; the latter transforms into the timelike gamma matrix: *γ*^{N} → *iγ*^{N} = *γ*^{0}.]

^{[N/2]}-dimensional spinor representations of Spin(

*N*− 1,1) [double cover of SO(

*N*− 1,1)] as

*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,

*N*− 1, 1) algebra commutation relations,

*r*= 1, …,

*N*− 1, and

## IV. SOLUTIONS OF THE DIRAC EQUATION ON *N*-DIMENSIONAL DE SITTER SPACE–TIME

We first present the basic results from Ref. 1 regarding the eigenmodes of the Dirac operator on *S*^{N}, 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

*S*

^{N}is

*n*= 0, 1, … and

*ℓ*= 0, …,

*n*are the angular momentum quantum numbers on

*S*

^{N}and

*S*

^{N−1}, respectively. The index

*s*indicates the two different spin projections (

*s*= ±). The symbol

*σ*stands for the angular momentum quantum numbers

*ℓ*

_{N−2}≥

*ℓ*

_{N−3}≥ ⋯ ≥

*ℓ*

_{2}≥

*ℓ*

_{1}≥ 0 on the lower-dimensional spheres, while $s\u0303$ stands for the (

*N*/2 − 1) spin projection indices

*s*

_{N−2},

*s*

_{N−4}, …,

*s*

_{2}on the lower-dimensional spheres

*S*

^{N−2},

*S*

^{N−4}, …,

*S*

^{2}, 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 $\psi +n\u2113\sigma (s,s\u0303)$ (or $\psi \u2212n\u2113\sigma (s,s\u0303)$) with dimension

^{1}

*S*

^{N}(1.1) are found by writing the spinor

*ψ*in terms of “upper” (

*φ*

_{+}) and “lower” (

*φ*

_{−}) components as follows:

*φ*

_{+},

*φ*

_{−}. By eliminating

*φ*

_{+}(or

*φ*

_{−}), one finds

^{1}

^{,}

*S*

^{N−1}. [Equation (4.4) is equivalent to $\u22072\u2061\psi =\u2212\lambda 2\psi $.] Then, by separating variables, the normalized eigenspinors of $\u2207|SN$ are found to be

^{1}

^{,}

_{N−1}∈

*S*

^{N−1}and the normalization factor is given by

*S*

^{N−1}, $\chi \xb1\u2113\sigma (s\u0303)(\Omega N\u22121)$, satisfy the eigenvalue equation

*S*

^{N}are normalized by

*ψ*

_{+}eigenspinors are orthogonal to all the

*ψ*

_{−}eigenspinors. The functions

*ϕ*

_{nℓ}(

*θ*

_{N}),

*ψ*

_{nℓ}(

*θ*

_{N}) are given in terms of the Gauss hypergeometric function by

*n*≥

*ℓ*and the quantization of the eigenvalue of the Dirac operator

*λ*

^{2}= (

*n*+

*N*/2)

^{2}(

*n*= 0, 1, …) arise by requiring that the mode functions are not singular.

^{1}The functions

*ϕ*

_{nℓ},

*ψ*

_{nℓ}are related to each other by

*γ*

^{a}∇

_{a}−

*M*)

*ψ*= 0 on

*dS*

_{N}by analytically continuing the eigenmodes of the Dirac operator on

*S*

^{N}. The eigenvalues on

*S*

^{N}will be replaced by the spinor’s mass

*M*. It is easy to check that under the replacement

*θ*

_{N}→

*π*/2 −

*it*, one finds $\u2207|SN\u2192\u2207|dSN$. 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

*dS*

_{N}are then

*c*

_{N}(

*Mℓ*) is a normalization factor that will be determined later (

*ℓ*= 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

*θ*

_{N}→

*π*/2 −

*it*,

*n*→

*iM*−

*N*/2 in Eqs. (4.5) and (4.6) instead of the replacements (4.16).] The un-normalized functions that describe the time dependence are

*ψ*

_{Mℓ}(

*t*) vanishes in the massless limit. Note the analytically continued version of Eqs. (4.14) and (4.15),

^{13}

*ϕ*

_{Mℓ},

*ψ*

_{Mℓ}as

*ℓ*≫ 1) of these functions can be found, by noting that the hypergeometric functions here tend to 1 in this limit, as

*ψ*

_{−}with the negative sign for the eigenvalue on

*S*

^{N}[see Eqs. (4.5) and (4.6)], we obtain the spinors

*dS*

_{N}. However, these spinors satisfy the equation $\u2207\psi \u2212M=\u2212M\psi \u2212M$ 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 $\u22072\u2061\psi =\u2212\lambda 2\psi $. The latter may be written as follows:

^{1}

^{,}

*S*

^{N}are found to be

^{1}

^{,}

*λ*= ±(

*n*+

*N*/2), with

*n*= 0, 1, …]. The spinors $\chi +\u2113\sigma (s,s\u0303)$ and $\chi \u0302+\u2113\sigma (s,s\u0303)$ are given by

*s*is the spin projection index on

*S*

^{N−1}and $s\u0303$ stands for the rest of the spin projection indices on the lower-dimensional spheres of even dimensions. The functions

*ϕ*

_{nℓ},

*ψ*

_{nℓ}are given by Eqs. (4.11) and (4.12), while the spinors $\chi \u0302\xb1\u2113\sigma (s,s\u0303)(\Omega N\u22121)$ are eigenfunctions of the Hermitian operator $\gamma N\u2207\u0303$ (which commutes with the iterated Dirac operator $\u22072$) satisfying

^{1}

As in the even-dimensional case, for each value of *n*, the eigenspinors $\psi +n\u2113\sigma (s,s\u0303)$ (or $\psi \u2212n\u2113\sigma (s,s\u0303)$) form a representation of Spin(*N* + 1) with dimension *d*_{n} 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 *S*^{N}.) Notice that on *S*^{1}, the Dirac operator is just *∂*/*∂θ*_{1} and the eigenspinors are $\chi \xb1\u21131(\theta 1)=exp(\xb1i(\u21131+1/2)\theta 1)$ [the normalization constant is (2*π*)^{−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).

*ψ*

_{+}eigenmodes. By making the replacements (4.16) in the expression for the eigenspinors $\psi +n\u2113\sigma (s,s\u0303)(\theta N,\Omega N\u22121)$ [Eq. (4.34)], we obtain the solutions of the Dirac equation on odd-dimensional

*dS*

_{N},

*ϕ*

_{Mℓ}(

*t*),

*ψ*

_{Mℓ}(

*t*) are given again by Eqs. (4.19) and (4.20). Hence, the solutions (4.39) can be used as positive frequency modes.

*ψ*

_{−}to obtain

*M*→ −

*M*.

## V. NORMALIZATION FACTORS AND TRANSFORMATION PROPERTIES UNDER SPIN(*N*,1) OF THE ANALYTICALLY CONTINUED EIGENSPINORS OF THE DIRAC OPERATOR ON THE *N*-SPHERE

For each value of *M*, the set of the analytically continued eigenspinors of the Dirac operator $\u2207|SN$ 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 *c*_{N}(*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.

### A. Unitarity of the Spin(*N*,1) representation and normalization factors

*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

*dS*

_{N}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.)

*ψ*,

*ψ*′ in Eq. (5.1) be positive frequency solutions of the Dirac equation on global

*dS*

_{N}with same mass

*M*[see Eqs. (4.17) and (4.18)]. Then, the Dirac inner product (5.1) is written as

*S*

^{N−1}and

*d*

**stands for**

*θ**dθ*

_{1}

*dθ*

_{2}…

*dθ*

_{N−1}. The square root of the determinant of the de Sitter metric is

*S*

^{N−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

*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

*ξ*

^{μ}be a Killing vector of

*dS*

_{N}satisfying

*J*

^{μ}with respect to the Killing vector

*ξ*

^{μ}($L\xi J\mu $) gives the change in

*J*

^{μ}under the corresponding transformation, that is,

*J*

^{μ},

*ξ*

^{μ}are divergence free. Then, we find

*κ*= 1, …,

*N*− 1. By integrating Eq. (5.8) over

*S*

^{N−1}, we find

**Case 1:**

*N***even**. Substituting the analytically continued eigenspinors (4.17) [or (4.18)] into the inner product (5.2), we find

*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

^{14,15}

**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).

### B. Transformation properties of the positive frequency solutions under Spin(*N*,1)

In this section, we use the spinorial Lie derivative^{10} with respect to the Killing vector field *ξ* in order to study the Spin(*N*,1) transformations of the analytically continued modes of $\u2207|SN$ generated by *ξ*. More specifically, we show that our positive frequency modes transform among themselves under the action of an infinitesimal boost in the *θ*_{N−1} direction.

*ψ*with respect to the Killing vector

*ξ*is

^{10}

^{,}

*s*to distinguish the spinorial Lie derivative from the usual Lie derivative.) We are interested in the transformation generated by the boost Killing vector,

^{10}Hence, if

*ψ*is an analytically continued eigenspinor of $\u2207|SN$, 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 $\varphi M\u2113(t),\psi M\u2113(t),\varphi \u0303\u2113\u2113N\u22122(\theta N\u22121),\psi \u0303\u2113\u2113N\u22122(\theta N\u22121)$, sending the angular momentum quantum number

*ℓ*to

*ℓ*± 1. [The functions $\varphi \u0303\u2113\u2113N\u22122,\psi \u0303\u2113\u2113N\u22122$ are given by Eqs. (4.11) and (4.12), respectively, with

*N*→

*N*− 1,

*n*→

*ℓ*and

*ℓ*→

*ℓ*

_{N−2}.] The ladder operators are given by the following expressions:

*f*

_{Mℓ}(

*t*) ∈ {

*ϕ*

_{Mℓ}(

*t*),

*ψ*

_{Mℓ}(

*t*)}, $f\u0303\u2113\u2113N\u22122(\theta N\u22121)\u2208{\varphi \u0303\u2113\u2113N\u22122(\theta N\u22121),\psi \u0303\u2113\u2113N\u22122(\theta N\u22121)}$, and

*M*.

**Case 1:**

*N***even**

**(>2)**. Using Eq. (3.2), one finds

*ψ*be the eigenspinor $\psi M\u2113\u2113N\u22122\sigma \u0303(\xb1,s\u0303)$, where $\sigma \u0303$ stands for quantum numbers other than

*ℓ*,

*ℓ*

_{N−2}. Since the partial derivatives in Eq. (5.25) refer only to the coordinates {

*t*,

*θ*

_{N−1}}, we want to extract the

*t*and

*θ*

_{N−1}dependence from our analytically continued eigenspinors. By combining Eqs. (4.17), (4.18), and (4.34), we can express the spinors $\psi M\u2113\u2113N\u22122\sigma \u0303(\xb1,s\u0303)(t,\Omega N\u22121)$ in terms of eigenspinors on

*S*

^{N−2}($\chi \u0303\u0302\xb1\u2113N\u22122\sigma \u0303(s)\u0303(\Omega N\u22122)$) as follows:

*ϕ*

_{Mℓ}(

*t*) →

*iψ*

_{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

*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

*N*, it is convenient to express the analytically continued eigenspinors $\psi M\u2113\u2113N\u22122\sigma \u0303(s,s\u0303)(t,\Omega N\u22121)$ [Eq. (4.39)] in terms of eigenspinors on

*S*

^{N−2}[$\chi \u0303\xb1\u2113N\u22122\sigma \u0303(s\u0303)(\Omega N\u22122)$]. By combining Eqs. (4.35), (4.5), and (4.6), we can rewrite Eq. (4.39) as

*N*even, we find after a lengthy calculation

*N*even, we conclude that the vacuum is de Sitter invariant.

## VI. CANONICAL QUANTIZATION

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 $\u2207|SN$. 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 *dS*_{N}. 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.)

### A. Negative frequency solutions

**Case 1:**

*N***even**. By making the replacements (4.16) in the expression for the iterated Dirac operator on

*S*

^{N}(4.4), one finds

*ϕ*

_{Mℓ}(

*t*),

*ψ*

_{Mℓ}(

*t*) with their complex conjugate functions and by exchanging

*χ*

_{±}(Ω

_{N−1}) and

*χ*

_{∓}(Ω

_{N−1}). The time derivatives of the spinors (6.2) and (6.3) reproduce the flat space–time behavior in the large

*ℓ*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

### B. Canonical quantization

*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

*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, $aM\u2113\sigma (s,s\u0303)$ and $bM\u2113\sigma (s,s\u0303)$, as

*iϵ*prescription (i.e., the time variable

*t*should be understood to have an infinitesimal negative imaginary part:

*t*→

*t*−

*iϵ*,

*ϵ*> 0).

## VII. THE WIGHTMAN TWO-POINT FUNCTION

In this section, we first review the basics about the construction of Dirac spinor Green’s functions on *dS*_{N} 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 *dS*_{N} 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.

### A. The spinor parallel propagator on *dS*_{N}

*ψ*⟩ be a state invariant under the action of the de Sitter group. Then, two-point functions [such as $\u3008\psi |\Psi (x)\Psi \u0304(x\u2032)|\psi \u3009$] 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*) 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}

^{,}

*x*and

*x*′. Equation (7.3) describes the parallel transport of gamma matrices. By contracting Eq. (7.3) with

*n*

_{ν′}(

*x*,

*x*′) and using Eqs. (2.18) and (7.2), we find

*dS*

_{N}, the covariant derivatives of Λ(

*x*.

*x*′) can be expressed as

^{4}

**1**,

**1**′ are the identity spinor matrices at

*x*and

*x*′, respectively.

### B. Constructing spinor Green’s function on *dS*_{N} using intrinsic geometric objects

**The massive case**. The massive spinor Green’s function

*S*

_{M}(

*x*,

*x*′) on

*dS*

_{N}satisfies the inhomogeneous Dirac equation,

*S*

_{M}(

*x*,

*x*′) can be expressed in terms of intrinsic geometric objects as follows:

^{4}

^{,}

*α*

_{M}(

*μ*),

*β*

_{M}(

*μ*) are scalar functions of the geodesic distance. By requiring that

*S*

_{M}(

*x*,

*x*′) in Eq. (7.9) satisfies Eq. (7.8), we find the following system of ordinary differential equations for

*α*

_{M}(

*μ*),

*β*

_{M}(

*μ*),

*z*= cos

^{2}(

*μ*/2) [see Eq. (2.10)], this system of equations is solved by

^{4}

^{,}

*β*

_{M}(

*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

*α*

_{M}(

*μ*) [hence for

*β*

_{M}(

*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 $n,\alpha M$ and

*β*

_{M}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

*z*= cos

^{2}(

*μ*/2) and where we used Eq. (D30). These are just the solutions (with the appropriate singularity strength) of the decoupled system,

*x*,

*x*′) [Eqs. (7.2)–(7.4)] translate to the following properties for the massless Green’s function:

### C. Analytic expressions for the massless and massive Wightman two-point function and the spinor parallel propagator

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**

**. By letting**

*(N > 2)**M*= 0 in Eqs. (4.17) and (4.18), we obtain the massless positive frequency modes,

*x*/2) is given in Eq. (4.21) and

*S*

^{N−1}, we may let

*θ*

_{N−1}′ =

*θ*

_{N−2}′ = ⋯ =

*θ*

_{2}′ =

*θ*

_{1}′ = 0 in the mode-sum (6.18). After a long calculation, we obtain the following 2

^{N/2}-dimensional bispinorial matrix:

*λ*(

*t*,

*θ*

_{N−1},

*t*′) is defined by the following relations:

*n*

_{0}≡

*n*

_{0}[(

*t*,

**), (**

*θ**t*′,

**0**)],

*n*

_{N−1}≡

*n*

_{N−1}[(

*t*,

**), (**

*θ**t*′,

**0**)] and

*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*

_{−}= sin

^{2}(

*μ*/2), i.e., $\beta 0(\mu )\u221d(w+w\u2212)\u2212(N\u22121)/2$ [see Eqs. (2.20) and (7.16)]. We have verified that Eq. (7.32) are consistent with the relation cosh

^{2}(

*λ*/2) − sinh

^{2}(

*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*(

*t*−

*t*′). 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

*γ*

^{0}is given by Eq. (3.4)], and then, we can construct the massive two-point function using Eq. (7.9).

## VIII. SUMMARY AND CONCLUSIONS

In this paper, we analytically continued the eigenspinors of the Dirac operator on *S*^{N} to obtain solutions to the Dirac equation on *dS*_{N} 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 ($\mu ,n,\Lambda $) 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 $\mu ,n,\Lambda $ have Minkowskian singularity strength in the limit *μ* → 0.^{4} Thus, the conditions for the unique vacuum^{21} 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.

## ACKNOWLEDGMENTS

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.

### APPENDIX A: CHARGE CONJUGATION AND NEGATIVE FREQUENCY SOLUTIONS

*dS*

_{N}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.,

*C*

_{±}that relate

*γ*

^{a}to $(\gamma a)T$. 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

*γ*

^{N}≡−

*iγ*

^{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 *dS*_{d−1} before using it on *dS*_{d}. This is not the case in Ref. 22 because a different representation for *γ*^{a}’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).

^{[d/2]}-dimensional Dirac spinor transforming under Spin(

*s*,

*τ*). Its charge conjugated spinor is defined with either one of the following two ways:

_{±}is an eigenspinor of the Dirac operator with the eigenvalue $\kappa (\tau ,s)\xb1$, i.e.,

*dS*

_{N}with $\kappa (1,N\u22121)\xb1\u2261\xb1M$ and $\u2207(0,N\u22121)\u2261\u2207\u0303$ is the Dirac operator on

*S*

^{N−1}with $\kappa (0,N\u22121)\xb1\u2261\xb1i(\u2113+(N\u22121)/2)$. 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

*B*

_{−}changes the sign of the mass term on

*dS*

_{N}. Also, Eqs. (A4) and (A5) imply the following relations for the eigenspinors of the Dirac operator on

*S*

^{n}[with $\Psi \xb1=\chi \xb1\u2113n\sigma (s\u0303)$ and $\kappa (0,n)\xb1=\xb1i(\u2113n+n/2)$]:

*n*is arbitrary,

*σ*stands for angular momentum quantum numbers other than

*ℓ*

_{n}, and $s\u0303$ represents the [

*n*/2] spin projection indices that correspond to this eigenspinor. The label $s\u0303\u2032$ is no necessarily equal to $s\u0303$.

Below, we use the tilde notation for quantities defined on *S*^{N−1}.

#### 2. Negative frequency solutions for *N* even

**Case 1:**

*N*/2**even**. The charge conjugation matrices

*B*

_{±}, satisfying Eq. (A1) on

*dS*

_{N}, are given by the following products of gamma matrices:

*S*

^{N−1}, there is only $B\u0303\u2212$ since [(

*N*− 1)/2] is odd. This is given by

*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:

**Case 2:**

*N*/2**odd**. The charge conjugation matrices on

*dS*

_{N}are given by

*N*− 1)/2] is even, the only charge conjugation matrix on

*S*

^{N−1}is $B\u0303+$. The matrices

*B*

_{−}and $B\u0303+$ are related to each other as follows:

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 $\psi \u2212M\u2113\sigma (s,s\u0303)$ [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 $VM\u2113\sigma (\u2212,s\u0303\u2032)\u2261(\psi \u2212M\u2113\sigma (\u2212,s\u0303))C\u2212$ and $VM\u2113\sigma (+,s\u0303\u2032)\u2261(\psi \u2212M\u2113\sigma (+,s\u0303))C\u2212$].

#### 3. Negative frequency solutions for *N* odd

**Case 1:**

**[N/2]**

**even**. The only charge conjugation matrix on

*dS*

_{N}is

*B*

_{−}, which changes the sign of the mass term of the Dirac equation. It is given by

*N*→

*N*− 1, where now,

*γ*

^{0}is given by Eq. (3.4). Then, Eq. (A16) may be expressed in terms of the charge conjugation matrix on

*S*

^{N−1}as

*s*

_{N−1},

*s*

_{N−3}, …,

*s*

_{4},

*s*

_{2}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 $(\chi \u2212\u2113\sigma (s\u0303N\u22121))C\u0303+=c\chi +\u2113\sigma (s\u0303N\u22121\u2032)$, we find

**Case 2:**

**[**

*N*/2]**odd**. The only charge conjugation matrix on

*dS*

_{N}is

*B*

_{+}. This is given by

*N*/2] even, we introduce the proportionality constant

*m*such that $(\chi \u2212\u2113\sigma (s\u0303N\u22121))C\u0303\u2212=m\chi \u2212\u2113\sigma (s\u0303N\u22121\u2032)$, and we find

### APPENDIX B: SOME RAISING AND LOWERING OPERATORS FOR THE PARAMETERS OF THE GAUSS HYPERGEOMETRIC FUNCTION

*F*(

*a*,

*b*;

*c*;

*z*) satisfies

^{13}

^{,}

^{24}

### APPENDIX C: TRANSFORMATION PROPERTIES OF THE POSITIVE FREQUENCY SOLUTIONS UNDER SPIN(*N*,1)

#### 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.

*θ*≡

*θ*

_{N−1}):

*ψ*

_{Mℓ}in terms of

*ϕ*

_{Mℓ},

*dϕ*

_{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.

#### 2. Transformation properties for *N* = 2.

*N*= 2 are given by

*φ*≡

*θ*

_{1}< 2

*π*and

*ℓ*= 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 $\u2202\psi M\u2113(\xb1)/\u2202\phi =\xb1i(\u2113+12)\psi M\u2113(\xb1)$. By expressing cos

*φ*and sin

*φ*in terms of exp(±

*iφ*) and using the ladder operators (5.34) and (5.35) with

*N*= 2, it is straightforward to find

### APPENDIX D: DERIVATION OF THE MASSLESS WIGHTMAN TWO-POINT FUNCTION USING THE MODE-SUM METHOD

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.

*S*

^{N−r}(

*N*−

*r*= 1, 2, …,

*N*− 2) are denoted as

*θ*_{N−r}= (

*θ*

_{N−r},

*θ*

_{N−r−1}, …,

*θ*

_{1}). The dimension of the Spin(

*N*− 1,1) representation is denoted as

*D*≡ 2

^{N/2}. Also, let $s\u0303N\u22122$ represent the spin projection indices (

*s*

_{N−2},

*s*

_{N−4}, …,

*s*

_{4},

*s*

_{2}), $s\u0303N\u22124$ represent (

*s*

_{N−4}, …,

*s*

_{4},

*s*

_{2}), and so forth. Similarly,

*σ*

_{N−r}represents the angular momentum quantum numbers (

*ℓ*

_{N−r},

*ℓ*

_{N−r−1}, …,

*ℓ*

_{2},

*ℓ*

_{1}). Note that for

*θ*

_{N−1}′ =

*θ*

_{N−2}′ = ⋯ =

*θ*

_{1}′ = 0, we have

*n*

_{a}|

_{θ′=0}[see Eqs. (2.26)–(2.28)] are given by

*n*

_{0}|

_{θ′=0},

*n*

_{N−1}|

_{θ′=0}and $n|\theta \u2032=0$ by

*n*

_{0},

*n*

_{N−1}and $n$ respectively.) Also, notice that Spin(

*N*− 1, 1) transformation matrices can be expressed as

*k*≠

*j*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)].

*N*even. By expanding the summation over the spin projections (

*s*= ±), Eq. (6.18) becomes

*S*

^{N−1}. In order to proceed, we need to express the eigenspinors on

*S*

^{N−r}, with

*N*−

*r*being odd, in terms of eigenspinors on

*S*

^{N−r−2}. Therefore, using Eqs. (4.34), (4.5), and (4.6), we derive the following two recursive relations:

*r*odd and

*N*− 3 ≥

*r*≥ 1). Since $\psi \u0303\u2113N\u2212r\u2113N\u2212r\u22121(N\u2212r)(0)=0$ and $\varphi \u0303\u2113N\u2212r\u2113N\u2212r\u22121(N\u2212r)(0)$ is non-zero only for

*ℓ*

_{N−r−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

*ℓ*

_{N−2}=

*ℓ*

_{N−3}= ⋯ =

*ℓ*

_{2}=

*ℓ*

_{1}= 0. Thus, only the summation over

*ℓ*

_{N−1}≡

*ℓ*survives in the mode-sum. Substituting Eqs. (D9) and (D10) (with

*r*= 1) into Eq. (D8), one obtains (after some calculations)

*d*. Also, we are going to use the following results:

*ℓ*and we determine the dependence on {

*t*,

*θ*

_{N−1},

*t*′}, and (3) we obtain analytic expressions for the terms of the two-point function that depend only on the angular variables

*θ*

_{N−2},

*θ*

_{N−3}, …,

*θ*

_{1}. We call the latter the “angular part” of the two-point function and we denote it as follows:

#### 1. The proportionality constant

*c*

_{N}(

*M*= 0) and

*c*

_{N−1}(

*ℓ*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:

*S*

^{1}, while the normalization factors for eigenspinors on higher-dimensional spheres are given by Eq. (4.7). Using Eqs. (D12) and (D15), we observe that

*N*− 1)

_{ℓ}= Γ(

*N*− 1 +

*ℓ*)/Γ(

*N*− 1) is the Pochhamer symbol for the rising factorial. Using Eq. (D19), we may rewrite Eq. (D17) as

*ℓ*-dependence in Eq. (D20) will be discussed later (it will be used in the summation over

*ℓ*).

#### 2. Obtaining a closed-form expression for the series

*ℓ*-dependent terms of the two-point function. Then, the mode-sum expression (D11) can be written as

*A*. By using the formula

^{25}

*ρ*(

*t*,

*t*′)| = 1, the series in Eq. (D24) diverges. Therefore, we make the replacement

*t*→

*t*−

*iϵ*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

*w*

_{1}is given in Eq. (7.33), while sin

^{2}(

*μ*/2) can be found by Eq. (D2)].

*B*. We can rewrite Eq. (D25) as