Spectral functions and zeta functions in hyperbolic …calvino.polito.it/~camporesi/JMP94.pdfSpectral functions and zeta functions in hyperbolic spaces Roberto Camporesia) Theoretical

Spectral functions and zeta functions in hyperbolic spacesRoberto Camporesi and Atsushi Higuchi Citation: J. Math. Phys. 35, 4217 (1994); doi: 10.1063/1.530850 View online: http://dx.doi.org/10.1063/1.530850 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v35/i8 Published by the American Institute of Physics. Related ArticlesTopology optimization of viscoelastic rectifiers Appl. Phys. Lett. 100, 234102 (2012) A characterization of causal automorphisms by wave equations J. Math. Phys. 53, 032507 (2012) Ray congruences that generate conformal foliations J. Math. Phys. 53, 032504 (2012) Curvature in sub-Riemannian geometry J. Math. Phys. 53, 023513 (2012) Geometric spectral inversion for singular potentials J. Math. Phys. 52, 112102 (2011) Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Spectral functions and zeta functions in hyperbolic spaces

Roberto Camporesia) Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta T6G 231, Canada

Atsushi Higuchib) Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637

(Received 9 February 1994; accepted for publication 18 February 1994)

The spectral function (also known as the Plancherel measure), which gives the spectral distribution of the eigenvalues of the Laplace-Beltrami operator, is calculated for a field of arbitrary integer spin (i.e., for a symmetric traceless and divergence-free tensor field) on the N-dimensional real hyperbolic space (HN). In odd dimensions the spectral function p(h) is analytic in the complex X plane, while in even dimensions it is a meromorphic function with simple poles on the imaginary axis, as in the scalar case. For N even a simple relation between the residues of p(X) at these poles and the (discrete) degeneracies of the Laplacian on the N sphere (SN) is established. A similar relation between p(X) at discrete imaginary values of X and the degeneracies on SN is found to hold for N odd. These relations are generalizations of known results for the scalar field. The zeta functions for fields of integer spin on HN are written down. Then a relation between the integer- spin zeta functions on HN and SN is obtained. Applications of the zeta functions presented here to quantum field theory of integer spin in anti-de Sitter space-time are pointed out.

I. INTRODUCTION

In the Euclidean approach to quantum field theory in curved space-time,’ the Wick rotation, i.e., the rotation of the time variable from the real axis to the imaginary axis, is performed in order to make the path integral well defined. If the space-time admits a Euclidean section, the metric becomes a positive-definite (Riemannian) metric, and the one-loop functional determinant can be calculated with the l-function method.2’3 It is well known that the N sphere (SN> is the Euclidean section for N-dimensional de Sitter space-time. One-loop calculations have been performed using the well-known spectrum of the Laplacian on SN.4*5

It has been pointed out6+7 that one can similarly compute one-loop quantities for scalars and spinors in anti-de Sitter space-time by using the 5 function on the real hyperbolic space HN, which is the appropriate Euclidean section in this case. In general for a noncompact Riemannian symmetric space of rank one [The rank of a symmetric space M is the maximal dimension of a flat, totally geodesic submanifold of M, or, equivalently, the dimension of the (commutative) algebra of invariant differential operators on M, known as Casimir or Laplace operators.8*9], such as HN=SO,(N,l)lSO(N), the 5 function takes the form

(1.1)

‘kresent address: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abmzzi 24, 10129 Torino, Italy. “Present address: Department of Physics, University of Berne, Sidlerstrasse 5, CH-3012 Beme, Switzerland.

0022-2488/94/35(8)/4217/30/$6.00 J. Math. Phys. 35 (8), August 1994 0 1994 American Institute of Physics 4217


4218 Ft. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces

where the real parameter A labels the continuous spectrum of the Laplacian and the wx are the corresponding eigenvalues. The spectral function p(A) plays the same role as the discrete degeneracies d, of the Laplacian in the case of a compact Euclidean section, where the 5 function is given as

I(z)=5 2. n=O n

The knowledge of ,u(X) is therefore essential in the construction of the noncompact 5 function. In the case of scalar fields the function p(X) is known in the mathematical literature as the

Pluncherel measure.9 The explicit form of p(X) is available in this case for any Riemannian symmetric space of the noncompact type (with negative curvature), and is given by

where the function C(A), known as the Harish-Chandra function, is given in terms of a product over the positive roots of the symmetric space [see, e.g., Eq. (5.38) of Ref. IO]. [If the rank of the symmetric space is r (sl), the spectrum label A is a vector with r components, A&Y where Y=R’ is a Cartan subspace of the symmetric space.“] The function C(A) is related to the asymptotic form at infinity of the spherical functions A(X), the eigenfunctions of the radial Laplacian (see, e.g., Ref. 11). However, little is known about ,u(A) in the case of fields of nonzero spin.

Recently the concept of spherical functions has been generalized to spinors on HN, and ,X(A) has been calculated in that case.” In this article the spectral function p(A), whose precise definition will be given in Sec. II, will be calculated for fields of arbitrary integer spin s on HN. The fields of integer spin s are defined here to be the symmetric, transverse, and traceless (SIT) tensor fields of rank s. This definition is a natural generalization of spin-s fields in four dimensions. (Note here that there are other tensor fields that form inequivalent unitary representations of SO,(N,l), such as p-forms, if the dimension N is larger than 4.)

Generally speaking, p(A) can be obtained from the eigenmodes of the Laplace-Beltrami operator by imposing the &function normalization of the continuous spectrum. [It will be seen in Sec. II that the spectrum of the Laplacian acting on rank-s SIT tensors over HN is purely continuous except for N=2. In this case there are square-integrable STT eigentensors which contribute a discrete part to the spectrum and are related to the discrete series of SO,(Z, I).] In Ref. 13 the symmetric tensor spherical harmonics (STSH) on the N sphere, defined as the SIT eigentensors of the Laplace-Beltrami operator, have been explicitly constructed recursively, by working in geodesic polar coordinates. Now, it is well known that HN and SN are related by analytic continuation in the geodesic distance, (+ -+ ia. (Alternatively the metrics on HN and SN are related by a 4 ia if the radius a is not normalized to 1.) In this article we analytically continue the STSH’s given in Ref. 13 to obtain the eigenmodes of the Laplace-Beltrami operator on HN. Then we compute their normalization factors to find the spectral function p(X).

The spectral function obtained in this way has the following properties. For N odd p(A) is an analytic function and in fact it reduces to a polynomial in the variable A*. For N even ,u(A) can be continued to a meromorphic function in the complex A plane, with simple poles on the imaginary axis. Comparing Eqs. (1.1) and (1.2), one naturally expects some relation between p(A) and the degeneracies d, of the Laplacian on SN acting on a spin-s field. Such a relation is indeed dem- onstrated here and takes the following form:

d 2 -iuG(n+p+s)) - do

p(i(p+s)) , n=O,l,...,

J. Math. Phys., Vol. 35, No. 8, August 1994


R. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces 4219

where p=(N- I)/2 and where it is understood that for N even the right-hand side means the ratio of the residues of p(A) at the poles A,, = i(n + p + s) and A,. This is a generalization of the results obtained for scalar fields in Refs. 10 and 11. A similar result has been obtained also for spinor fields.‘* It is natural to conjecture that a relation analogous to Eq. (1.4) holds for arbitrary fields on any pair of compact and noncompact Riemannian symmetric spaces that are “dual” to each other (see Ref. 8 for the definition of dual symmetric spaces). The general proof of this statement in the scalar case has been obtained only recently.i4

Note that the Plancherel measure p(A) can also be determined by specializing the Plancherel formula for the Lorentz group G=SOo(N,l) (given, e.g., by Hirai”) to the (vector valued) right SO(N)-covariant functions on G that define SIT tensor fields on HN. This group-theoretic approach is described in Ref. 16 in the case of vector fields, and in Ref. 17 in the case of arbitrary vector bundles over HN.

The rest of the article is organized as follows. In Sec. II we compute the spectral function p(A) for the field of arbitrary integer spin s. In Sec. III we demonstrate the relation (1.4). We also sketch an alternative proof of Eq. (1.4) which does not require the explicit formulas for the degeneracies on SN or the spectral function on HN. In Sec. IV we examine the analytic properties of the 5 function (1.1). The results, in agreement with general theory,18 show that the zeta function l(z) extends to a meromorphic function in the complex z plane with simple poles at z= N/Z,N/2 - 1 ,..., --c4 for N odd and z=NlZ,N/Z- l,..., 1 for N even, and with so-called “trivial” zeros at the negative integers in the odd-dimensional case. In Sec. V we demonstrate a relation between the 5 functions on SN and HN. The relation (1.4) between the residues of p(A) and the degeneracies d, on SN (for N even) suggests that a relation between the spin-s 5 functions on HN and SN may be obtained by deforming the contour of integration of Eq. (1.1) in the complex A plane. By exploring this idea we obtain a contour representation of the spin-s 5 function on the even-dimensional sphere which yields its analytic continuation in the complex z plane. A variation of this idea is then used to find similar results in the odd-dimensional case. In Sec. VI we conclude by pointing out two possible applications of the results of this article. The first is the calculation of the one-loop effective potentials and stress-energy tensors for higher-spin fields in anti-de Sitter space-time. The second is the computation of the one-loop Casimir energy in static space-times with topology RX HNlr, where HNIT is a compact hyperbolic space with l? being a discrete subgroup of SOo(N,l). In the Appendix the 5 functions for symmetric traceless fields, without the requirement that they be divergenceless, are studied to provide some insight into the relation between the spin-s 5 functions on HN and SN obtained in Sec. V.

Il. THE SPECTRAL FUNCTION FOR INTEGER SPIN

In this section we derive an explicit formula for the spectral function p(A) of rank-s SIT eigentensors of the Laplace-Beltrami operator. We start by motivating and giving the precise definition of the spectral function p(A). Consider the heat kernel of the operator -V”V,+K (where K is a fixed constant) acting on SIT tensors of rank s on HN (Na3) of radius 1, defined by the following equations:

(2.1)

K P”1”‘Ps y”“yqx,x’,o)= s/l,...Irs~““~~(x,x’), (2.2)

where the differential operator VT’, acts on X. The heat kernel K~,...~sVI”‘V~(~,~‘,t) is a symmetric bitensor, i.e., a tensor at the point x with indices ,ui,...,,~~ and at the point x’ with indices v, ,...,vs . It is required to be traceless and divergenceless as a tensor at x and at x’. The 6function in Eq. (2.2) is the STI S function which extracts the transverse-traceless (TT) part of a symmetric tensor. That is,




I dNx’~~~,...B,Y,“‘Y,(X,XI)Ty,...~~(x’)=the TT part of TP,...r,(x) (2.3)

for any smooth symmetric tensor TY,...YQ (x’) with compact support. Here dNxJg(x) is the volume element of HN. (Note that the support of the TT part of T,, . . .

Now, let the SIT tensors Lr,lr!.,l(x) satisfy Ps(x) is noncompact in general.)

-VaV,fi~,‘f!.,s=(A2+p2+s)i~,:!.,l, (2.4)

where u is the discrete label for distinguishing eigentensors with the same eigenvalue and where p=(N- 1)/2. As we shall see, the label A* is continuous and positive. Let the eigentensors fi?!. be normalized as fil ps

where

(jpu’,p W)s/ dNx~~(AU)*.r;cX’U’)(x)=s,,,s(A-A’), (2.5)

jp)* .I;cA’u’,(x),~cAu)P,~~~~~(x)*~~,~~~~~(x). (2.6)

(The asterisk indicates complex conjugation throughout this article.) The heat kernel can be expressed in terms of fir,!!.,(x) as

I

m

KI*,-.PLs ~I”‘~S(X,X’,J)=~ d), I;~,~~~p,(X)~(~~)~,~~~YI(XI)*e-(~z+~2+~+K)~~ (2.7)

u 0

We define the l function for SIT tensor fields in analogy with the scalar case6 as

dt tZ-kp ,... pSp,“‘*s(x,x,t). (2.8)

Homogeneity of the hyperbolic space HN implies [‘H’(z,~) = <‘N’(z,O), where 0 denotes the “origin” of HN (which may be chosen arbitrarily). We shall denote eH)(z,x) simply by eH)(z) for this reason. By using the mode expansion (2.7) we find

$X)(z)=CN g(s) I

- dA AA>

aN- 1 0 (A2+b2)2 ’

where b* = p* + s + K, and the spectral function p(A) is defined by

/L(h)=$$ c jpu)* . r;CW(()), u

with

27?* nN-lGr(N,2) ?

2N-2 cNc-

9.r ’

(2.9)

(2.10)

(2.11)

(2.12)




g(s) = (Zs+N-3)(s+N-4)!

(N-3)!s! . (2.13)

The spin factor g(s) is the number of independent solutions with a given momentum (k’,k* ,...,kN) in flat N-dimensional Euclidean space. [For N=3, g(O)=1 and g(s)=2 for ~21.1 The factor RN-t is the volume of SN-t. With this normalization the spectral function ,x(A) approaches the flat space scalar spectral function

,rrAN- * pA(A)=[2N-*r(N/2)]f (2.14)

for X%1. To compute the spectral function p(A) we need to discuss the eigentensors in detail. We start

with the discussion of the scalar eigenfunctions. We write the line element of the hyperbolic space HN (N32) as

dsi= dy* + sinh’ y dli- 1, (2.15)

where dlk- , is the line element of SN- ‘. This is related to the negative of the line element of SN

ds$= dx* + sin* x dli- , (2.16)

by x=iy. Consequently, there is a close connection between the scalar spherical harmonics on SN and the eigenfunctions of the Laplacian on HN. On SN the Laplace eigenvalue equation is

d2 2 +(N-l)cot$ +

ii JX ax 22-y <P= -L(L+N- l)Ca, L=O,l,Z ,..., (2.17)

where 6 is the Laplacian on SN- ’ . The solutions are given by

cp ~kP(sin x> -(N-2)/2p-I-(N-2)/2

L+(N-2)/2 tcos x)&o, (2.18)

where P;:(x) is the associated Legendre function of the first kind” (see, e.g., Ref. 13). The functions Y,, are the orthonormal scalar spherical harmonics on SN-’ satisfying

i&= -l(l+N-Z)Y,,. (2.19)

The label (+ is for distinguishing eigenfunctions with the same eigenvalues labeled by L and 1. By using the definition of associated Legendre functions in terms of hypergeometric functions F(cr,P;y;x) and the transformation formula (9.131.1), p. 1043 of Ref. 19, we find that one can let

@ ~~cT=Qdx)ho, (2.20)

where

QLl(x)=(sin ,y)‘F L+l+N- I,,-L;l+g ;sin* $ .

Notice that we have normalized the function QLl(,y) as

(2.21)

(2.22) lim (sin x)-‘QLl(x) = 1. x-0



4222 R. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces

The “angular momentum” L is restricted to be larger than or equal to 1 by the requirement that QL1~) be nonsingular at X=T. Hence, as is well known, l=O, l,..., L.

The eigenfunctions of the Laplacian satisfying

q (b=-(A*+p*)c$ (2.23)

on HN [where p=(N- 1)/Z] can immediately be obtained from Eqs. (2.20) and (2.21) by letting x=iy and

L= -p+iA. (2.24)

Thus, the solutions of Eq. (2.23) are

+Alo= 4*r(Y 1 Yl, 9 (2.25)

where l=O,l,...,a and

qkl(y)=(i sinh y)‘F iA+p+l,-iA+p+l;l+t ;-sinh* i i

. (2.26)

The behavior of qxl(y) for large y can be obtained by using the transformation formula (9.132), p. 1043 of Ref. 19. We have

where

qAl(y)“i’[c~(A)e(-P+iA)Y+(A H -A)], (2.27)

2N+'-*r(z+N/2)r(iA) "(')= J;; I-'(iA+(N- l)/Z+Z) .

(2.28)

It is clear that the volume integral of I&]* will be badly divergent if A is imaginary. Hence the complete set fo; square-integrable functions on HN is given by Ahla with positive A.

We define &la to be the normalized eigenfunctions proportional to c&, i.e.,

~A,CT=NA,~Al*~ (2.29)

(4Ad A,I,o,)=S(A-A’)SrtrS~~, , (2.30)

where the inner product is defined by Eq. (2.5). As is well known in the general theory of Sturm-Liouville equations, the normalization

integral of c#+~~ is determined by its large-y behavior. Thus, we find

NAl=c~*IcI(A)I-’ (2.3 1)

up to a phase factor, where cN is defined by Eq. (2.12). In particular, we have A - 2 lim (sinh Y)-*~~~~~~~*=cNIc~(A)I-*~Y~~~ . (2.32)

y-0

Using the fact that &,.j Ylal * is a constant over SN- ’ , we find

c liJ=~, * (2.33)



Ft. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces 4223

where G(I) is the degeneracy of the eigenvalue -1( 2 + N- 2) of the Laplacian h, which is given by

G(l)= (21+N-2)(Z+N-3)!

Z!(N-2)! * (2.34)

[Notice that the spin factor g(s) in Eq. (2.13) is equal to the degeneracy of the eigenvalue -s(s+N-3) of the Laplacian on SNm2.] Then we have the following result:

Proposition 2.1: The normalized scalar eigenfinctions &,.la on fl with eigenvalues -X2-p’ of q and -I(l+N-2) of & satisfy

ytmo C (siti y)-“‘l&.,12=~ G(OIcr(A)Im2* fY (2.35)

An immediate consequence of this formula is the following well-known result (see, e.g., Ref. 6): Corollary 2.2: The spectral function &h) for the scalar field on fl (Na2) is given by

P”(x)=[2’-2r;N/2)]2 I’(ih+(N- 1)/2) 2

r(iX) *

Proof From the definition (2.10) of the spectral function, we find

(2.36)

(2.37)

The corollary immediately follows from Eq. (2.28) with l=O. Q.E.D. Now we move on to the SIT eigentensors of the Laplace-Beltrami operator of rank s. As in

the case of the scalar field, one can use the known results for SN and simply let x=iy . For this reason we first review some results in Ref. 13 which are relevant here. We specialize to NZ=3.

The SIT eigentensors HP,.. ,Ps on SN satisfy

q HP,-YS =[-L(L+N-l)+s]HP,...~~ (2.38)

VaH,p’l...ps-l=O, (2.39)

gaSHapp,.-CLS-2=0, (2.40)

where g+ is the metric tensor on SN and gas is its inverse. The “angular momentum” L( 2s) is an integer.20 It is convenient to introduce the following resealing:

H xx...x=(sin x)-~F, H,...,i ,... it=(sin X)2k-SFi ,... ikv (2.41)

where the indices i i ,. . .,i, refer to angular coordinates on SN- ‘. Equations (2.38)-(2.40) are equivalent to the following set of equations:

d2 3 +(N-1)COt Xd +!$ Fi,...it+2k

- I

cot x -

ax G V(iIFi2...ik)

cot2 x +k(k- 1) -e 77~i,i2Fi3...ik)=-L(L+N- l)Fi,...i,. (2.42)



4224 FL Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces

V’Fji,...i,-,=- A +(N+k-3)COt X Fi,...ik-,p dX 1

1 d1Fjli,...ike2= -e Fi,...ikm2 7 (2.44)

where vij is the metric on the unit SN- ’ covariant derivative on SN-’

and $ is its inverse. The differential operator qk is the and the indices are raised and lowered by 77ii. It is interesting that

these equations do not depend on the spin s. For a given STI eigentensor E’$y!Y,!m on SN- * satisfying

~l~!~~=[-I(Z+N-2)+m]Hlmf~~,

with Iam and normalized by

(2.45)

dfiNml I;l!l~~~*~(ml’u’);l”‘im= 8 II lrn

,& 11 ud (2.46)

we define the symmetric tensor ?$:jYfzin as the unique traceless tensor of rank m fn that is a -

. ._ linear combination of the tensors

V(ili2 * * * ?7i,,- ,i2tVi2k+,’ “Vinfi!~~~!..i In+” ) y

with the coefficient of V(i, * * .~i,iT?j~~~~..i,+,) being 1. For example,

jyw;‘= fipy) 1 lx’

?(I ;mfo) = qi $m’u) ~l”‘rt,.l+l I ‘2”‘Jm+ ,) ’

pyy =qi ~i*~!m~3 )+ (l-m)(Z+m+N-2) ‘I’ ‘?lZ+2 1 ‘3”“i7!+2 N+2m-1 9(i,i,fil~f5~+2)

These tensors satisfy

6~:;r?~~~,=A(nrnl)~~~~~~~~,

\?kf+vnly) ki,...r m+n-I

= - C(nml)fi:.Y.:~~~J,,

(2.47)

(2.48)

(2.49)

(2.50)

(2.5 1)

(2.52)

where

A(nml)=n(2m+n+N-4)-l(l+N-2)+m+n, (2.53)

n(N+2m+n-4) C(nml)=(m+n)(N+2m+2n-5) (l-m-n+l)(l+N+m+n-3). (2.54)

For the case with N=3, n = 1, and m =0, Eq. (2.54) becomes indefinite. The correct formula is given by letting N --+ 3+~ and taking the limit E -+ 0. This procedure works for all the formulas that become indefinite for N=3 in this section. The formulas (2.53)-(2.54) can be proved by induction starting with

A(Oml)= -Z(l+N-2)+m, (2.55)



R. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces

C(OmZ)=O. (2.56)

Equation (2.52) can be used to evaluate the inner product. Lemma 2.3: The inner product of fly;f’.‘:zyn is given by

4225

X(Z+N+m+k-3). 1 (2.57)

Proof: One can assume that m ’ ==rn without loss of generality. Since ?$~.$~~. is traceless, we have

(~~;mlo),~T(n+m-m’;m’l’u’) - )- 1 dfiNml ii;(n;mlu)ii”‘i,+~*~j,. . .~i,im-m,ljjnm:~~~!+l...im+n.

(2.58)

By integrating by parts and using Eq. (2.52) we find

(fwmra),~ n+m~m’~m’l’u’))=C(nm~)C(~-l,m~)...C(m’-m

+ 1 ;ml) I

daN-, ~(m’-m:mlrr)*.I;l(m’l’u’). (2.59)

Since keel!:“,” is traceless and divergenceless, the right-hand side of this equation is zero unless m’=m. If m’=m, then

($wW, (2.60)

due to the orthonormality of fi$y!:k. The (2.60).

n we obtain the lemma by substituting Eq. (2.54) in Eq.

Since the inner product is positive definite, ‘i:.?‘!f,“@, Q.E.D.

is zero if and only if its norm vanishes. Hence we have the following corollary of this lemma: Corollary 2.4: The traceless tensor fi:.?,fEi” is nonzero if and only if m -tnsl.

Proof: The norm of q:jllf”) f?l+lJ to Lemma 2.3.

is nonvanishing for m + n s 1 and is zero for m + n > 1 according Q.E.D.

This corollary implies that the tensors (2.47) are linearly independent if and only if m +n~l. For m+n>l the tensor Q,i,4iz...Qi n I?~~‘~!..i,+.J is a linear combination of tensors (2.47) with k # 0. For example,

(2.61)

fjjw~L~i~j~lo+ qzijI;l,=o. (2.62)

Thus, as is well known, the vectors @’ ‘cl are Killing vectors and vifrO are conformal Killing vectors on SN- ’ .

Now, one can solve Eqs. (2.42)-(2&l) as follows. First let




F@+) F(Lm[u) z.z Fip’“) = . . . = F~f,‘$‘)-, = 0. I m (2.63)

Then the solution to Eq. (2.42) with k = m is

F?‘+) = QLr( x)$Tfy? ‘,“‘lm ‘RI ’ (2.64)

where Q&) is defined by Eq. (2.21). Then, postulate that FiF.?iz+, (1 snss- m) can be written as

F~~~.‘P_)+~=c~“;~“‘)(x)~:~~~~~~-c(;;~”~)(x) qi,j2f$;.-;;;;‘+ *** + (- l)k~~‘Lm’)(X)

x 77(iliz”‘* ?li,,-,i,, ~:;+:kill’tK1.)+...+Eit...i,+,, (2.65)

where

I C-1) “‘2C$$m1)(X) 7j(i,i2’ .* 77j,-,infi~~~~!..i,+,) (n even)

Ei,...i,+,= (- 1)‘“-“‘2~I~;L’;;:),(~)77(, .** 7/i,_*l,_,~i,HI~~~~..i,+,) (n odd). (2’66) 11’2

From Eq. (2.64) we find by definition

c~~‘~““)( ,Y) = Q&Y). (2.67)

The other coefficient functions c~;~~‘)(x) are uniquely determined by the divergence and trace conditions (2.43) and (2.44). These conditions lead to the following recursion formulas:

cp;LmO(X) = (m+n)!

2kk!(m+n-2k)! b, N+2(m+n-;k+a- l)- 1 Cg-2k’Lm1)(X) ’ 1 sin2k x

n(N+2m+n-4)

(m+n)(N+2m+2n-5) (l-m-n+ 1)(l+N+m+n-3)c~;Lm’)(x)

d - +(N+m+n-3)cot x c~-‘~~~‘)(x)-(~+~~+~~~~)~~~~ x c~-~;~~‘)(x) dx I

(2.69)

for n 22. The last term on the right-hand side of Eq. (2.69) is omitted for n = 1. [We have corrected a misprint in the equation corresponding to Eq. (2.68) in Ref. 13.1 The formula (2.52) with (2.54) is essential in deriving Eq. (2.69).

The postulate (2.65) does not work if Z<m + n because of Corollary 2.4. Hence, one needs to require that 13 s for a solution to be found in this manner because n = 0, 1,. . . ,s - m. In fact it is known that there are no solutions of Eqs. (2.42)-(2&t) with Eqs. (2.63) and (2.64) unless Las.

In summary, for each STSH on SN- ‘, film!:: (m = 0, 1 ,. . . ,s for N>4, and m =O,l for N=3 as will be shown later in this section) with sGlGL, there is a solution of Eqs. (2.42)-(2&t) which satisfies Eqs. (2.63) and (2.64).

As an example, we consider the case with s =2. The solutions to Eqs. (2.38)-(2.40) with label m =2 are given by H($f’O) = Hs210) = 0 and

(2.70)



R. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces

The solutions with m = 1 are given by HFirU) =O and

H$““)= QLz(x)~~““‘, (2.7 1)

HpW= 2 sin2 x

CJ (I- l)(f+N- 1) d +(N- 1)cot x dyy 1 Qrr(x)~&‘“‘. (2.72)

Finally, for m =0 we have

H(““‘)= sinm2 xQ,J x) ?, xx L7, (2.73)

Hpw = Xl x 1 Q&)f%, , (2.74)

c, (2.75)

where c~~‘~~‘)(x) is given by

N-2 N-l (I- l)(l+N- 1)c~z~Lo”(~)=~(,+~~2) & +(N- l)cot x 1

d 1 1 Q,,(x) - +W-2M x Qdx)-- dx N-l sx

(2.76)

The above analysis for eigentensors on SN can be applied to those on HN (N>3) simply by letting x=iy and L = - p+ ik. Thus, the eigentensors hc,‘?!:J on HN are obtained from those on SN by analytic continuation as

,+A+) = ik( sinh y)2k-‘fi~.?.‘~) , Y’“YI,“‘Zk (2.77)

where l=s,s+ l,...,~ and

~A.~zP)=F!~m.zP)~,_-p+ix x=iy. *, “Lk ‘,” ‘k

We write the normalized STT eigentensors fi::mf.$ as

(Ada) ij$!‘p,‘,= ck,INX~hpl...p . s

(2.78)

(2.79)

The scalar normalization factors NA, are given by Eq. (2.31). From the normalization condition (2.5) we have

(h 3 (Xmzo) h(x’m’z’a’))=~~Aml~-2[NAz~-2~(~-~’)~,,r~z~~6,~~.

To evaluate this inner product and find CA& we first note that

(2.80)

(2.8 1)

where




J,~h~,“.,‘,~s*V,h(A’ ’ ’ ’ m I c )/.q...~~- (

v h(hd”h (1 P,“‘ll* 1

h(k’m’l’U’)P,.-,t$

(This equation follows from the eigenvalue equation for h::?.‘$ .) Thus we have

(2.82)

=( sitiN- y] dfiNe,[ h(Am'c)*. g h(A'"'l"J')-$ h(Ania)*.h(A'm"'"'l]]y=*.

(2.83)

It is clear that this integral vanishes if (mla) # (m’1’c’). From Eqs. (2.27), (2.68), and (2.69), we find that Ic~‘~~‘) (iy) I 2 goes to zero like eVCN- ’ +4k)y for large y. Hence if k> 1

Ic~;Lm’)(iy)l~lc6”‘Lml)(iy)l, for y*l. (2.84)

This implies that

/+A?‘?) =ik( sinh y) y” YZ,‘“‘k 2k-sCbk-m;Lml)(iy)TI:-~~mlu). (2.85)

Therefore we have

h(“mlF)* h(hmla)y”‘yil”‘ik,(sinh y)2k-2SIC~k-m;Lm~)(iy)l2~k-m;mlo)*~k-m;m~u)i~~~~ik Y Yll”“k 1,” lk

(2.86)

for large y . Thus the contribution from the component hi,. . .i, (Am’u) dominates the right-hand side of Eq. (2.83) for large A. Now, notice that

c~“‘Lm’)(iy)=qkl(y)~[cl(~)e(-p+iA)y+(~ ‘4 -A)] (2.87)

[see Eq. (2.27)]. From this formula and Eq. (2.69) one can readily find the large-y behavior of cgiLm’)( iy) as

c~~LmL)(iy)=i’[K”‘(X)cl(X)eOY+(X ++ -X)], (2.88)

where

N+2m+2k-5 iX+(N-3)/2+m+k- 1 N+2m+k-4 < I-m-k+l)(l+N+m+k-3) 1 . (2.89)

Thus we have

(A2-Ar2)JoA dy sinhN-’ y/ dQN-, h(xmlu)* h(A’m’/‘rr’)

~2~N+3X~KS~m~‘(X)c,(X)~2(~~~m~m~u~,~s~m~m[o~)sin[(X-~‘)h]S,,~S~~,S,,~

(2.90)

for IX-h’l=Sl. By using

lim sin[(X-~'Ml A-A' =7rlqX-A')

h+m (2.9 1)




and the formulas for KS-m*‘(X), (~s-m;m’a),~~-m;mla)) [i.e., Eqs. (2.89) and (2.57)] and the relation (2.3 1) for NA,, we obtain

N+2m+k-4 (Z-m-k+ l)(Z+N+m+k-3) = h’+[(N-3)/2+m+k- 112 . (2.92)

In particular, for m = 0 and E = s, we have

s!(N+2s-3)! ‘CAo”‘2=(N+s-3)(N-4)! k=, fi {(N+2k-5)(X2+[(N-3)/2+k- 112)}-‘. (2.93)

Next we establish the following lemma: Lemma 2.5: At the origin y =0 the normalized STT eigentensors satisfy

2 ~(Amru)*.~(hmlo)(o)=S(s)C Ih~~.“g’(O)l”, do *

where

S(s) = N(N+2).-.(N+2s-2)

s!

Proof: We analyze the left-hand side of Eq. (2.94) in an orthonormal basis. Define

K PI “‘/A*:Y,~~~VS =z dU

i~l~.!$s*(o)i~lr!;~ (0).

(2.94)

(2.95)

(2.96)

Let {er,ef , . . . ,ef;) be an orthonormal basis at y =O. Define

PI @Ls vi vs K, ,... a,Eb ,... b,=Kp,l...p.v ,... Yseal...ea,ebl...eb,. (2.97)

By the invariance property of this quantity under rotations that fix the origin y =O, we find

KI...II:I ...ll=K, ,... p., ,... ,,fe -PLl...~~s~q...ps (2.98)

where

P=cos ffe(;+ sin aeg . (2.99)

Comparison of the terms of order CJ? for small cr leads to

0=-sK,...,,:,...,,+s2K ,... 12:1...*2+- sts;l) (K ,... II:1 . ..2 2fK1...22 :1...11>. (2.100)

Using the fact that Kp, . . .~‘s :y . . ys is traceless in the first and second sets of indices separately and also using the isotropy of H”I, we have

1 K,...,,,,...,,=K,...,,:,...,,= --

N-l Ki . ..*1.1...11. (2.101)




Substituting this in Eq. (2.100), we find

1 K ,... 12:1...12=; (2.102)

Then

c K ,... rc :,..., c=K ,... r1:r...r,+(N-1)K,...r2:r . . . . 2=N+2;-1) KI...II:I...II- c

(2.103)

By letting s -+ s - 1 and then KFl. ..ps-, :“r.. .ys-, --f X, K, ,... ILI-lc:yl. ..y3- ,c in this formula, we find

T c Kl...lbc:l...lbc= N+ftl-2) c K ,..., IC:l...l,C c c

=[N+~(s-~>I[N+~(s-~)I K1 ,11,1 . ,,1 (s- 1)s . . . . . . . (2.104)

Continuing in this manner, we obtain

This is equivalent to

c -c K,l....s., ,... =,=S(s)K ,..., :I...I. =I CI

(2.105)

c R(Amr~)*.~(Amlu)(o)=s(~)C Ij$$)(o)l2 (2.106) do mla

according to the definition (2.96). The lemma follows from the fact that h$‘.YY(0) vanishes unless m =0 and l=s [see Bqs. (2.63), (2.64) with m =0, (2.41) and (2.21)]. Q.E.D.

Now, we have all the ingredients necessary for establishing the following result: Theorem 2.6: The spectral function dejned in Eq. (2.10) for the spin-sfield on Z? (Nz=~) is

given by

ol; equivalently

N odd,

(2.107)

(2.108)

~[A~+(s+(N-3)/2)~] (N-912

,4X)= [2N-21-‘(N/2)]2 A tanh(7rA) n (A2+j2), N even, (2.109) j= 112

where for N=3 and N=4 the products are omitted. Proof: Using the definition (2.10) and Lemma 2.5, we have



FL Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces 4231

Ah)=$) S(s)2 Iijlt!y(0)12. (T

.

(2.110)

From Eq. (2.79) we find

lq?!y(0)12= lc,,,1~1N,,1~1h~~~~~0~12= ICXOs121~~s121L12~ (2.111)

where Eqs. (2.41), (2.64), and (2.21) have been used to obtain the second equality. Hence

m=$) w421~As12 2 7 (2.112)

where Eq. (2.57) has been used. Finally, the theorem follows from Eqs. (2.12), (2.13), (2.95), (2.93), (2.34), and (2.31) with (2.28). Q.E.D.

Next, let us treat the case N=2. For s=l the divergenceless eigenvectors of the Laplace- Beltrami operator are given by

vy’= EpvVV@~l, (2.113)

where I+ is the invariant two-form on H2. The functions Qkr satisfy

0*x,= -(i-x2+ $IQJ. (2.114)

It is straightforward to compute the spectral function /A(X) using the formula (2.10) [with g (1) = 11. We find that it is the same as that for the scalar field. Hence we have the following result:

Theorem 2.7: The spectral functions on Hz for spin 0 and I are both given by

p(X)=rrX tanh rrX. (2.115)

It is interesting that for iX = l/2 there are square-integrable eigenvectors. They are given by

A(‘)=V P Q, /.L (r/2)l* 1=-+1,+2 ,..., +m, (2.116)

@ci,2jl=$ ( c:;y+ l)‘5~@. (2.117)

There are no other discrete values of A which allow square-integrable eigenvectors. There are no SIT eigentensors of the Laplace-Beltrami operator for continuous values of h

for ~32. This fact can be proved in the same way as in the case of S2 (see, e.g., Ref. 13). Let h P]“‘P’1 be transverse and traceless with ~22. Define pP1...PLs-l by

VICLh’l~l...~Ls-,~EI*~~,...CLs-,. (2.118)

By taking the trace of this equation over p and ,u,, we find P~,...~~-, = 0 because the tensor h cI]“‘P, is both traceless and divergenceless. Then

2V”V~,lh,,,2...PLs= -(V”V,+s)h,,...,s=O. (2.119)

There are square-integrable STT solutions to this equation. They are given by




I

II2 (i si& y)l'lwk

de (cash y+ l)lV e , (2.120)

where I= ts,+(s+ l),...+a. Let us verify that there are no imaginary values of A which allow square-integrable SIT

eigentensors for Nz=~. The tensor h(Xm’o) /*, . . .cLs would be square integrable for imaginary A only if P’(A)c[(A)=O in Eq. (2.88) with Im A>O. From Eq. (2.89) we find that the coefficient K”‘(X) vanishes only if N=3, m=O, and X=0. However, the coefficient cl(X) given in Eq. (2.28) has a pole at X=0. Thus, the product K”‘(A)cl(A) is nonzero even for this case. Therefore, any square- integrable SIT tensor on HN (Ns3) can be expanded in terms of the eigentensors in the continuous spectrum.

Ill. A RELATION BETWEEN ,x AND THE DEGENERACIES ON SN

For N odd the spectral function calculated in the previous section is a polynomial in X2. For N even it can be continued to a meromorphic function in the complex A plane, with simple poles on the imaginary axis at

A=ki(p-l),tip ,..., +i(p+s-2), (3.1)

+i(p+s),+i(p+s+l) ,..., fiw, (3.2)

where p=(N- 1)/2. [For s =0 we only have the “tower” of poles (3.2).] Consider now the degeneracies d it’ of the eigenvalues -L(L + N - 1) + s of the Laplace-

Beltrami operator V”v, acting on STSH’s of rank s on SN. Proposition 3.1: The degeneracies &’ are given by

di;)=g(s) (L-s+ l)(L+s+N-2)(2L+N- l)(L+N-3)!

(L+ l)!(N- l)! , (3.3)

where L=s,s+I,... and g(s) is the spin-factor (2.13). (The index L is related to n in Eq. (1.2) by L =n +s and d there means dN)

[It is poss;ble to find diy)ni”e’ using the general formula given by Weyl (see, e.g., Ref. 21). Here we give an elementary proof by induction on N and s.]

Proofi For N=3 and 4, Eq. (3.3) gives

d$)=(L+ 1)2 ,

di;)=2[(L+ 1)2-s2], (s> 1) ,

dE’=;(2s+ l)(L+ $)[(L+ ;)2-(s+ $)‘I.

(3.4)

These agree with the known formulas. Now, suppose that dit-‘) is correctly given by Eq. (3.3). The branching rule for SO(N+l)XO(N) (see, e.g., Ref. 13) implies

d& i dj;-‘). l=s k=O

(3.5)

Note that this formula uniquely determines d i‘) Hence, in order to prove the proposition one only . needs to show that Eq. (3.3) satisfies this formula. First consider the case of s=O. Since d&!)(= 1) is correctly given by Eq. (3.3), Eq. (3.5) for s=O is equivalent to




d&@, o=d~$--‘), (3.6)

which can readily be shown to be satisfied by Eq. (3.3). Next, suppose that di;’ is given correctly by Eq. (3.3) for k~s - 1. Then the branching rule

(3.5) is satisfied if

[d~~)-dlfv-)l,,]-[d~~~-l-d~~,,s-l]=c~~-l) (3.7)

(N) for all L(as). [We define d,-l,,- -0, which is consistent with Eq. (3.3).] This equation can readily be shown to hold. Hence, by induction, Eq. (3.5) holds for all N and s. Q.E.D.

We find the following relation between the degeneracies di:’ on SN and the spin-s spectral function p(A) on HN, which is generalization of the known relation for s =0:

Theorem 3.2: The spectral function &A) on HN satisfies

Ai(L + p)) d$ p(i(s+p)) =p 9 L=s,s+l,...,

ss (3.8)

where for N even the lefi-hand side means the ratio of the residues of &A) at the given points. Proof: Using Bq. (2.107) for p(A) we see that the factor A2+(s+p- 1)2 gives, at A=i(L+p),

the first two terms in the numerator of Eq. (3.3). The remaining (L-dependent) terms in Eq. (3.3) arise from calculating the ratio of I? functions in Eq. (2.107) at A=i(L + p) and A=i(s + p), and substituting on the left-hand side of Eq. (3.8). Q.E.D.

Since the proof of Theorem 3.2 requires the explicit formulas of the degeneracy dit’ and the spectral function p(A) which were derived separately, it cannot be generalized easily to other pairs of symmetric spaces “dual” to each other. In the rest of this section we shall present a sketch of a proof for N>4 which may admit a generalization. We let~dLs-d~~) and &(A)=p(A).

Fist define the normalized scalar spherical harmonics aLIU by

&Lla= NE)@ LlU

and

I da, &&&‘,y= &~‘&‘c&‘,

where da, is the volume element of SN. Then write the normalized STSH I!??,?.$ as

$h,l,d = ,~s&,&~)H(Lml”) ccl’ ll.s Pl”‘Ps *

Then the degeneracy dLs can be written as

dLs= I

daN c ~(Lmlu)*.I;T(Lmlu)(X)=nNC $Lm[d*.$Lmlo)(o), mlu da

(3.9)

(3.10)

(3.11)

(3.12)

where CI,v is the volume of SN. The second equality holds because the integrand is x independent. By the same argument as that which led to Eq. (2.112) we find

dLs=s S(s)G(s)lc~sd,121N~~)12, (3.13)

where S(s) and G(s) are given by Eqs. (2.95) and (2.34), respectively. Hence

J. Math. Phys., Vol. 3.5, No. 8, August 1994


4234 FL Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces

d b’d?12 2 =SWWl&l’ 1~(s)12 .

LO

Similarly, from Eq. (2.112) we find

mu,@) h12 PO(A) =WG(sho,12 p----g.

(3.14)

(3.15)

Now, we shall sketch an alternative proof of the following theorem, which is obvious if we use the explicit formulas for d,, and p$(A):

Theorem 3.3: The degeneracy dL, on SN and the spectral function &A) on HN satisfy

cLsW + PI) dL, =- LcoW+ p)) dLo ’

Proof We shall show that

N$ 2 =(-1Y fl

I I

(3.16)

(3.17)

and

Ihzll~=i(L+p)= (- 1 Yw2~12. (3.18)

From these equations and Eqs. (3.14) and (3.15) the theorem will immediately follow. Recall that the un-normalized eigenfunctions (DLIC on SN and Ala on HN satisfy by definition

(@ @Lr,‘o’ Liar )=IN~~)I-2~LL,Sll,S~~, , (3.19)

(AlCTP4 X,l,cT,)=INhll-2~(A-A’)Sll,S~~, . (3.20)

Since these inner products are invariant under SO(N+ 1) and SO(N,l) transformations, respectively, we have

(~x~Ll~,~L’~‘~‘)+(~Llrr,~~~L’~‘o’)=o, (3.21)

(~~~Xlot~X’l’a’)+(~*lcrr~~~*‘l’a’)=O, (3.22)

where Zx and Zr are the Lie derivatives with respect to the Killing vectors XP on SN and YP on HN, respectively. Now, let YP be the following Killing vector on HN:

Ypd,=cos 6 -&coth y sin 8 & ,

where the line element of HN is written as

dsi=dy2+sinh2 y(de2+sin2 B dli-,), (3.24)

with dli-, being the line element of SNm2. By evaluating the Lie derivative Zr&o (see Ref. 13) we find that the label 1 changes by one and the label CT remains unchanged if they are chosen appropriately. Hence

~~~,,,=c+(Ala)~h,l+~,o+~-(Al~)~~.1-1.u. (3.25)




By letting I’=1 - 1 in Eq. (3.22) and using Eqs. (3.20) and (3.25), we find

NAl I I 2 c+(U- 134 - =...-

Nx,l- 1 c-(hla)* *

Next, let the Killing vector XP on SN be

P”d,=cos 8 $-cot x sin e -& ,

(3.26)

(3.27)

where the line element of SN is the one obtained from Eq. (3.24) by letting y = - ix and changing sign. Note that X”= - iYp under this substitution. Hence we have from Eq. (3.25)

where the identification L = - p + iX has been used. By a similar argument as that which led to Bq. (3.26), we find

Ng) I I c+(i(L+p),l- 1,~) NAl

I I

2

c = c-(i(L+p>Jd* =- NA,l-l A=i(L+p)’ (3.29)

where Eq. (3.26) has been used for the second equality. This implies @. (3.1’8. Next by an explicit calculation we find

~yh’~“,~p’ =a(Xml~)h(A’“-‘.‘~)+c+(~mz~)h(X’f.’C’l-””) PI PLS Pl “CL* fil “CL* . ‘F s

+ b(Xmlcr)h;;17f.+,;*1u). (3.30)

(See Ref. 13 for the values of the coefficients a, ct , an d b.) By using Eq. (2.79) in the formula

(3.31)

we find

CA& I I 2 a(h,m+ 1,Za) - =- CA,m+ I,1 b(Xmla)* ’

(3.32)

By considering the Lie derivative Z’J-Z~,?.!~~ on SN and using Eq. (3.11) and applying the above procedure for SN, we find

&!l I I ’ a(i(L+p),m+ l,Zg) CAmI

I I

2

Ez = b(i(L+p),mZu)* =- ~

3 1 CLm+l,l A=i(L+p) (3.33)

This implies I%& (3.18) because CL’\= cksl= 1. Now, by dividing Eq. (3.16) by its special case of L=s we have

Q.E.D.

/-d(L+p)) /-di(s+p)) dLs 40 =-- ,+(i(s+p)) di(L+p)) 4, dLO .

(3.34)

It has been shown”.” that




I*&@ + p>) 40 i-dip) 2i.Y ’

Hence

di(L+ PI) dL0 ZZ- ,di(s+p)) 40 ’

(3.35)

(3.36)

It is clear that Eqs. (3.34) and (3.36) imply Theorem 3.2.

IV. THE c FUNCTION

Let the radius of HN be a. (We require N>3 in this section.) Then the 5 function (2.9) becomes

c$H)(Z)=U2Z-Ng(S)bN I

m ,dA)dA o (X2+b2)z ’

where

cN

bN=&+,

,.&N/2)+ 1

(4.1)

The constant b in Eq. (4.1) depends on the spin s and on the mass m of the field. For example, for the minimally coupled scalar fields we have b2 = p2 + u2m2. For the vector (spin-l) field theory the Hodge-deRham operator (da+ Sd) acting on transverse vectors corresponds to the massless wave operator. Since its eigenvalues are [X2+(p- 1)2], we find b2= (p- 1)2+a2m2 for the Rroca field with mass m. In general the wave operator is (-V’*V,+clu2+m2), where c is a given constant. The integral in Eq. (4.1) converges for Re z > N/2 and is defined by analytic continuation for the other values of z.

Let N be odd and define the numbers CX$, by

[x2+ ( s+32]‘y-y (k’+j’)-y,; 4p. (4.3)

(For N=3 the product is to be omitted and we have a$;= 1 and cyt{=s2.) Assuming that Re z >N/2, we can perform the integration in Eq. (4.1) by using Eq. (3.251.2) of Ref. 19. The result is

g(s)b1-2Z (N- 1)12

pqz) = c ($) b2k r(k+ 1/2)T(z-k- l/2)

(4~r)~‘~I’(N/2) k=O ak,N Uz) (4.4)

The 5 function exhibits “trivial” zeros at z =O,- l,-2,... It is meromorphic in the complex z plane with simple poles at

(4.5)

in agreement with general theory.18 For N even define the numbers ;B’ksk by




(X2+ (s+ y)‘i(;$ (h,+j+‘z; p&A2k. (4.6)

[For N=4 the product is omitted and we have &k= 1, PO,4 (‘) = (s + 02.] Using the identity

2 tanh(d)=l- 2vA+1

e (4.7)

we obtain

go) (N-2)/2

5’H’(z)=(4,)N’2r(N/2) k=O c Pi%

b2k+2-2z w+ 1)~(z-k- 1) r(z)

-4 (4.8)

The last term in this expression is analytic in z. The first term carries only a finite number of simple poles at

NN z=y , y- l,...,l (4.9)

again in agreement with Ref. 18.

V. THE RELATION BETWEEN 5”‘(z) AND 5’w(z)

The close relationship between the degeneracies on SN and the spectral function on HN (Theorem 3.2) suggests that there is a simple relation between the spin-s 5 functions on these two spaces. In this section we obtain this relation by means of complex contours. The technique used here is similar to that of Dowke?2 who obtained the analytic continuation of the 5 function for the minimally coupled massless scalar field on SN by writing the sum (1.2) in contour form and then deforming the contour in the complex X plane. We use here a slightly different method which leads to an analogous result for spin s in a somewhat simpler way. We continue to require Na3.

Consider a wave operator of the form -VT,+ [ cI(N(N- 1 ))]R where c is a fixed constant. The Ricci scalar R is +N(N- l)lu2 for SN and -N(N- l)/u2 for HN. (We shall comment on the case with a nongeometrical mass term later in this section.) The eigenvalues of this operator acting on the STSH’s of rank s on SN are

q=[L(L+N- 1)-s+c]Iu2=[(L+p)2-b2]lu2, L=s,s+l,..., (5.1)

b2=p2+s-c. (5.2)

We shall assume b2>0, i.e., c<p2+s. The eigenvalues of the same operator acting on the corresponding field on HN may be obtained from o, by replacing L by A-p and a by iu, i.e.,

o,=(X2+b2)/u2. (5.3)

For example, the field equation [ - VPV, + clu2]h,,...,s = 0 has solutions of the form h “‘P = ~(cL,4ycc,) (2 theS - sign) for

in de Sitter space-time (for the + sign) and in anti-de Sitter space-time

c=s-(s-2)(s+N-3) (5.4)




r R+iR 1 .

. i W-4

. L .

-R r r R

FIG. 1. The contour of integration r in Fq. (5.6).

(see Ref. 13). Therefore the value of b2 for the massless field with s 3 1 for the Euclidean sections SN and HN is

b2=p2+(s-2)(s+N-3)=(p+s-2)2. (5.5)

Hereafter the radius a will be normalized to the value 1. It is necessary to distinguish between even and odd N, since the spectral functions have different analytic properties in the two cases. The final formulas, however, will be the same.

A. N even

Consider for Re z > N/2 the integral

I= P ,4A)dA

r (-X2-b2)z (5.6)

over the contour I? shown in Fig. 1. We define the phase by letting (-X2- b2)z= 1 -X2- b21z on the segment [ib, + iw]. The integrand has two branch points at A= tib. The cuts are chosen to run from ib to ib--03, and from -ib to -ib+m. For simplicity we assume

p+s-2<b<p+s (5.7)

so that the branch points lie between the two sets of poles (3.1) and (3.2) of ,u(A) (this restriction will be removed later). The contour r is a rectangle of vertices (-R,R,R+iR,- R+iR) de- formed around the cut at ib to the contour y consisting of a small semicircle of radius e=q - b centered at ib and of the segments [iq- R, iq],[i(2b-q), i(2b-q) -RI. The point iR lies between consecutive poles, i.e.,

p+s+n<R<p+s+n+l. (5.8)

The integrand in I is analytic inside the contour r except for the simple poles on the imaginary axis. By applying the residue theorem we obtain




?l+s /-4X)

s-l

I=*riLTs res (-A2-b2)z A=i(p+L) +Pz2*iC res

P(A)

G2+b2Jz X=i(p+k- 1). (5.9)

k=O

Consider now the limit R,n --t ~0. As long as Re z>N/2 the integrals over the sides of IY other than the real line and y tend to zero in this m-nit. To see this define f(A) from I%+ (2.109) according to

p(A)=?T tanh(TTA)f(A). (5.10)

Now the function tanh(nL) is bounded over the sides of r and f(h) is a polynomial in X of order N- 1. Therefore the leading terms of the integrals over the sides behave like l/R2 Re z-N and approach zero if Re z> N/2. Combining this with Theorem 3.2, we obtain as R + w

2 I

m /-4A)dh

0 (A2t.b2Y

+

=2~i e-i”* res P(x)IX=i(s+p) m

d c dLs

ss L=s W+d2-b21Z +‘i f(ib+k- 1))

k=O P2--b+k- U21Z (5.11)

The first sum on the right-hand side is just the spin-s 5 function on the N sphere

c$~‘(z) = 5 dL&, L=s

(5.12)

where the eigenvalues oL and the degeneracies d,, are given by Eqs. (5.1) and (3.3), respectively. On the other hand the first integral on the left-hand side of Eq. (5.11) is proportional to e”‘(z) given in Eq. (4.1). A simple calculation using Eqs. (2.109) and (3.3) gives

d 3-s

res Ah)lX=i(s+p) =i7r(- 1Y2g(.s)bNfiZN, (5.13)

where g(s) and b, are given in Eqs. (2.13) and (4.2), and a,., is the volume of the N sphere [see Eq. (2.11)]. Using Eqs. (4.1), (5.12), and (5.13) we can rewrite Eq. (5.11) in the following way:

p)(z) = ,ir(z-(N/2))~,5(H)(z) + e’dz-W/2))g(s)bN,RN

/-dAMA .‘s f(i(p+k- 1)) (X2+b2)z-T~k=0 [b2-(p+~-1)21z .

To simplify this expression further we use Eq. (5.10) and the identity

in the integral over y, and show that the term “- 1” gives no contribution, i.e.,

I f(A)dA y cA2+b2jz =O, Re z$ .

(5.14)

(5.15)

(5.16)




In order to see this consider for Re z>N/2 the integral of f(A)I(A’+ b’)’ over the contour F. Since f(A) is odd [compare Eqs. (5.10) and (2.109)], the integral over the real line vanishes and the integrals over the sides of the rectangle again approach zero as R -+ a. Since the integrand is analytic and has no poles inside the contour I?, the integral over the remaining part y must vanish. This establishes Eq. (5.16).

We thus obtain the following form of f”‘(z), valid for all z:

p)(z) = eirr(z-(N/2))~nN5(H)(z) + ei~(z-(N/2))~g(s)b~~~

X f(A)dA ‘-l f(i(p+k- 1))

(e-2T”+1)(A2+b2)z-ik~o [b2-(p+k-1)2]z * (5.17)

Furthermore, one can show by using Proposition 3.1 that

-i7rg(s)bNRN(- 1)N’2f(i(p+k- l))=ds-l,k, (5.18)

where d, - 1 ,k is the degeneracy of the “angular momentum” L = s - 1 for spin k. Hence we have

5(S)(z)=e'"(Z-(N'2))S1N pqz) +

s-l

+C ds-l,k

k=O HP+k-1)2--21z' (5.19)

where [(pfk- 1)2-b2]Z=e-iAZ [b2- (pfk- 1)2]Z. This equation can also be applied in the scalar case (s =0) if the last term is omitted, in agreement with the fact that there are no poles of the form (3.1) in this case.

Since the last two terms in Eq. (5.17) are analytic, we see that e”(z) carries the same poles and residues as tH)(z) in Eq. (4.8) [ see Eq. (4.9)]. The finite parts of these 5 functions are of course different.

Equation (5.19) is well suited for analytic continuation to negative values of z. If Re z< 1, it is possible to obtain a simpler expression for the integral over ‘y. Consider the integral over the semicircle around the point ib. By letting A - ib = Eeie we see that as E + 0 this integral behaves as c1 -Re z. Thus, if Re z< 1, the contour y can be contracted to run along the edges of the cut and there is no contribution from the semicircle. By using the phases

(A-ib)z=r e z ioz, rze2Tiz above ( 8= 2 T)

rz below (19=0)

we obtain

I ftA)dA I ib-m

Cl?0 y te-2Tx+ l)or2+b2)z=*k-i"Z sin(m) f(A)dA ib (e-2rrh+ 1)(A2+b2)’

f(ib--y)dy [e2”(y-i6)+1][(y-ib)2+b2]z ’

(5.20)

(5.21)

Re z<l. (5.22)




Thus the integral along y in Eq. (5.17) vanishes for z =0, - 1, -2,... . However, the summation term in Eq. (5.19) does not. Hence the 5 functions on SN and HN are not proportional to each other by the volume factor !& (and a phase) for these values.

This should be contrasted with the scalar case, where the last term in Eq. (5.19) is absent and the proportionality between the 5 functions at zero and negative integers leads to

an (sNL+l)nafN),

where a,, are the (unintegrated) coefficients of the asymptotic (small time) expansion of the scalar heat kernel. [Since SN and HN are homogeneous, the coincidence limits an=an(x,x) are constants independent of x. The integrated coefficients on SN are simply u~=~,Lz, .] Equation (5.23) follows from the relations

NN n=y, 2 - + l,..., (5.24)

anKres 5(z)lZ=N/2-n, N

n=o,l,... y- 1 (5.25)

(valid for N even). It has also been proved independently in Ref. 10. In the higher-spin case the relations (5.24) and (5.25) between the coefficients of the SIT heat

kernel and the SIT 5 function are still valid for both SN and HN. However, since

C$s’(-p) # (- l)N’2+pf&%,{(H)(-P), p=0,1,2,..., (5.26)

the relation (5.23) between the SIT heat kernel coefficients on SN and HN must break down for spin s 2 1. This is possible because the SIT 8 function in Eq. (2.2) is nonlocal. Since this nonlo- cality is due to the divergence condition, Eq. (5.23) must hold if one imposes only the trace condition on the field. Hence, for a symmetric traceless field with no divergence condition, this mismatch between c”(z) and cH’(z) at zero and negative integers should disappear. In the Appendix it will be shown assuming a reasonable conjecture that this is indeed the case.

6. N odd

For N odd p(A) is analytic [see Eq. (2.108)]. T o use the residue theorem we need to introduce a fictitious spectral function ,G(A) defined by

,$A)-coth(~A),u(A). (5.27)

We then define f(A)=p(A)Im. We immediately see that the function ,G(A) has simple poles given again by Eqs. (3.1) and (3.2) and is otherwise analytic in the A plane. By calculating the residues at the poles in Eq. (3.2) we find

res iiW+p)) dL, res j-qi(s+p)) =d,, 9 L=s,s+ I,... . (5.28)

This is analogous to Theorem 3.2 for N even. Fist we treat the case with Ns5. Consider the integral of ,&(A)/( - X2-b*)’ over the same

contour l? of Fig. 1 [with R satisfying Eq. (5:S)]. Notice that ,G(A) is an odd function [since p(A) is even], and therefore the integral over the real line is now zero. By applying the residue theorem we obtain in the limit R --+ ~0

J. Math. Phys., Vol. 3.5, No. 8, August 1994



p’(z) = dsseiPZ

res ~(A)Ix=i(s+P) (5.29)

A simple calculation gives

d ss res riili~~+~)

=(-1)P7Tg(S)b&.,. (5.30)

We now use the identity

coth(rrA)= - 1 -e-2;L- 1 (5.3 1)

in the integral over yin Eq. (5.29). The contribution of the term “- 1” can be related to eH’(z) in the following way. Consider for Re z>N/2 the integral over lY of the function f(A)I(A2+ b2)z. Since f(A) is even and the integrand is analytic inside the contour l? we find

I f(A)dA y (A2+b2)z = -* I

- f(A)dA o (A2+b2)2 *

(5.32)

Using Eqs. (5.30), (5.31), and (5.32) in Eq. (5.29) we obtain

$‘)(z) = - i( - 1) peirzfiN[(H)(z)+ 7Teirz( - l)Pg(s)b,,,ti,.,

f(A)dA (e-2”x-1)(A2+b2)z k=O .

(5.33)

Again for Re z<l the integral over ycan be contracted to yield a result similar to Eqs. (5.21) and (5.22). The function $‘)(z) has the same poles and residues as cH)(z) in Eq. (4.4) [see Eq. (4.5)]. As in the even-dimensional case, we find

-(- 1)PTg(.3)bNfl,f(i(p+k- l))=d,+,. (5.34)

Hence

f(A)dA (1 -e-2wx)(A2+b2)z ]

+c ds- 1.k

k=O [CP+~- 02-b21z ' (5.35)

The finite sum of residues (the third term) will again contribute to f”‘(z) at z=O,- l,-2,... In particular, although the 5 function for the massive scalar field on SN, N odd, vanishes at zero and negative integers, the 5 function for the symmetric traceless divergenceless fields of spin s > 1 on SN, N odd, does not vanish at these values.

For N=3 we obtain Eq. (5.33) except that the first term in the sum over k must be taken with a coefficient l/2. That is, it must be --$(0)be2” rather than -f(0)be2’. The reason is that the “lowest” pole at A=i(p- 1) in Eq. (3.1) coincides with zero for N=3 (p= 1). This implies that the contour I of Fig. 1 must be modified by adding a small indentation around A=O. The contribution of this pole acquires consequently a factor l/2. However, the right-hand side of Eq. (5.34) needs an extra factor of 2 for k=O. As a result, Eq. (5.35) remains valid for N=3 as well.




One can write Eqs. (5.19) and (5.35) as

s(s)(z)=eilr(z-(N’2))~N p’(z)+ 7.&&(S) I f(A)dA

y [l+(-1)Ne~z*A](A2+b2)z]

s-l

+C ds- l,k

k=O UP+k-V2-b21Z

where

X’+(S+(N-~)/~)~ (N-5)/2

f(A)=A [2N-2r(lv/2)]2 n (X+-S).

k=-(N-5)/2

(5.36)

(5.37)

We also observe that the restriction (5.7) on b can be easily removed. If ib is not one of the poles of p(h)@(h)), the contour lY is modified in an obvious way and we obtain the same results. A slight modification is required in the case where ib coincides with one of the poles. For example, in the “massless” case we have b = p + s - 2 for s 31 [see Eq. (5.5)]. If, as in this case, ib is one of the poles given in Rq. (3.1), then Eq. (5.36) is still valid if we drop the corresponding term in the sum over k. If ib is one of the poles in Eq. (3.2) we have a zero mode on SN. This situation arises, for example, for the massless minimally coupled scalar field (s=O and b=p). [For higher spin this can happen only if the constant c in Eq. (5.1) is negative.] in this case Eq. (5.36) and related formulas are still valid if $“(z) is defined to be the 5 function with the zero mode subtracted off. In either case ib is both a pole and a branch point and the integral over the semicircle of radius E around ib behaves as eeRez as E -+ 0. Therefore, Eqs. (5.21) and (5.22) (and the corresponding ones for N odd) hold only for Re z<O.

Finally we make a few comments on the massive case. Suppose that the wave operator contains a (nongeometrical) mass term ITZ~. This term does not change sign when going from SN to HN unlike the term b2/a2 [see Eqs. (5.1) and (5.3)]. If we start with the integral (5.6) with denominator (-A2-b2-a2m2)z we obtain Eq. (5.11) with b2 -+ (b2+a2m2). The first sum on the right-hand side of ECq. (5.11) will be the spin-s 5 function on the N sphere with imaginary mass, i.e., $S)(z,im), where

m l’S’(z,m)= 2 dL,,[(L+p)2-b2+a2m2]-z.

L=s (5.38)

Therefore the right-hand side of Eq. (5.19) (with b2 --+ b2+a2m2) gives now $s’(z>], --+ im. In order to obtain the 5 function with real mass on the sphere a further analytic continuation m + im is required. Under this change the integral term in GM’(z), Eq. (4.8), will contain (A2+b2-a2m2)z in the denominator, and will converge for all values of z provided a2m2<b2. Therefore if a2m2<b2, $S’(z) is given by the right-hand side of Eq. (5.17) with b2 + ( b2 - a2m2) and with the contour y going around the cut at i d$?&?. [We can start with the integral (5.6) with denominator (-X2- b2+a2m2) to obtain the same result.] The case of u2m22 b2 is more complicated and will not be discussed here. Similar considerations can be repeated for N odd.

VI. CONCLUSIONS

It is natural to conjecture that Theorem 3.2 can be generalized to any field defined on a pair of “dual” Riemannian symmetric spaces. In the case of scalars the generalization of this theorem to any such pair was stated, without proof, in Refs. 23,11,10. The general proof for scalars, based on




the work of Vretare,” has been obtained by Helgason.14 The case of Dirac spinors and of anti- symmetric tensor fields on HN has been considered in Refs. 12,17. The general proof of a relation analogous to Eq. (3.8) for arbitrary vector bundles on G/K- UIK can be obtained by using the Plancherel formula for L2( G) .

A possible application of the results obtained in this article is to quantum field theory in anti-de Sitter space-time. As observed in the introduction, the functional determinant of the operator -V”V,+K may be defined from the corresponding 5 function on the Euclidean section H4 according to3

det[-V”V,+K]=exp[-l’(O)]. (6.1)

Thus one can carry out for anti-de Sitter space-time the one-loop calculation of the effective potentials and stress-energy tensors in the 5 function method (see Refs. 6,7,25). The corresponding calculation for de Sitter space-time, using S4 as the Euclidean section, was considered by Allen.4

Another application is to the computation of the one-loop Casimir energy for fields defined on static space-times with topology R X HNIT, where F is a discrete subgroup of SO&J, 1) acting freely and properly discontinuously on H*, such that H*lr is a compact space of constant negative curvature. This problem has been discussed recently in the case of scalar fields in Refs. 26 and 27 (N=2), (Ref. 28) (N=3), and (Ref. 29) (N arbitrary).

The case N=3 may be relevant for cosmology, where in the open Friedmann-Robertson- Walker models (k= - 1) the topology of the spacelike three-surfaces is not known a priori, and is not determined by Einstein’s equations.30

By making use of the Selberg trace formula one obtains the 5 function31’29

Pr(z) = f-k,pqz) + 5’(z), (6.2)

where flF is the volume of the fundamental domain of HNIT. The “topological term” c(z) may be written as a sum over the multiple closed geodesics of H*/r. The results obtained in this article allow one to calculate the first term in Eq. (6.2) for fields of integer spin.

ACKNOWLEDGMENTS

The authors would like to thank S. Helgason for helpful comments. This work was supported in part by the National Science Foundation under Grant No. PHY 89-18388 and by the Natural Sciences and Engineering Research Council of Canada.

APPENDIX: TRACELESS 5 FUNCTIONS ON SN AND HN

Consider the 5 function for the wave operator -VT,--p2-s + b2 acting on the symmetric traceless tensor field hp,. . .cls on HN without the divergence condition Vahall,...C1,-, = 0. (Thus, for s = 1 no condition is imposed on h, .) In this appendix we compare this [ function with the corresponding one on SN. Since the traceless S function is a local bitensor, the asymptotic expan- sions of the heat kernels on SN and HN must satisfy Eq. (5.23). We shall show that Eq. (5.23) is indeed satisfied by these 5 functions assuming a reasonable conjecture.

Let r~,~!.~~‘“’ (k=O,l,..., s - 1) be the unique traceless tensors on HN constructed from the transverse-traceless modes hcXm’@ “t”;“f in the same way as the unique traceless tensor ?$?‘!8@, was constructed from the STSH HiI .: on SN- ‘. Note that the divergence formula (2.52) in this case implies that $,~!.‘~~lQ) are nonzeri for all A (>O). From Eq. (2.51) with N -+ N+l, we find

vav &Syk;~W= _ a PI . P’s [A2+p2+s+(~-k)(k+s+N-3)]tjsl?;,k1u). (Al)



Ft. Camporesi and A. Higuchi: Spectral functions in hyperbolic spaces 4245

We use the conjecture that the spectral function for f&Y(j;i:‘@ , defined in the same way as that for h(AklU)

PI”‘&’ is pk(A) without proof. (We do not know a rigorous prOOf of this COUjeCtUre, but it is intuitively clear if one considers the spectral function as an appropriate limit of the degeneracies on HN with an infrared cutoff at large y. We have also checked this statement for s = 1 using the explicit formulas.) Then, the 5 function iiH’(z) for the operator -VT,--p2-s + b2 for the symmetric traceless tensor field is

‘i$?z> = 5 !$%>b2 + b: , 642) k=O

where

b;=b2+(s-k)(k+s+N-3). (A3)

One can similarly define the 5 function ~!$)(z) for the corresponding operator and field on SN. In this case we have

s-l

@‘(&2 + b:- c dpk p=k C(P+d2-b:lZ (A4)

The subtraction term is present because there are no (nonzero) symmetric traceless tensors r(;,Y.!‘~~‘“) constructed from the STSH HP,!!?:, if k<LGs - 1 (see Corollary 2.4).

Next we note the identity

(A5)

Hence Es. (A4) can be written as

&%db2 -+ b;- i ~~k+~~;;;-b7z p=ktl P

646)

Then using Eq. (5.36) we find

i(S)(z) = eiw(z-(N/2))fl .fk(A)dA s N

I

i:H’(Z>+ WbNi t?(k) ’ CA71 k=O I y [1+(-l)

i’.‘,-hr”](A2+@ I

where fk(A) is given by letting s + k in Eq. (5.37). Thus, the contribution due to the third term in Eq. (5.36) and that from the subtraction term in Eq. (A4) cancel each other, and there is no mismatch between ii’)(z) and iiH’(z) at z= -n (n=O,l,...). Hence, Eq. (5.23) follows from Eq. (5.24) for the traceless fields that are not required to be divergenceless.

’ R. M. Wald, Commun. Math. Phys. 70, 221 (1979). ‘J. S. Dowker and R. Critchley, Phys. Rev. D 13, 3224 (1976). 3S. Hawking, Commun. Math. Phys. 55, 133 (1977). 4B. Allen, Nucl. Phys. B 226, 228 (1983). ‘E. S. F&kin and A. A. Tseytlin, Nucl. Phys. 234, 472 (1984). ‘R. Camporesi, Phys. Rev. D 43, 3958 (1991). ‘R. Camporesi and A. Higuchi, Phys. Rev. D 45, 3591 (1992). ‘S. Helgason, Diffeerential Geometry, Lie Groups, and Symmetric Spaces (Academic, New York, 1978). 9S. Helgason, Groups and Geometric Analysis (Academic, New York, 1984).

‘OR. Camporesi, Phys. Rep. 196, 1 (1990).




‘IS. Helgason, Asterisque, hors serie, 151 (1985). “R. Camporesi, Commun. Math. Phys. 148, 283 (1992). 13A. Higuchi, J. Math. Phys. 28, 1553 (1987). I4 S. Helgason (private communication). “T. Hirai, Proc. Jpn. Acad. A 42, 323 (1966). 16E. Badertscher and H. M. Reimann, Math. 2. 202, 431 (1989). 17R. Camporesi and A. Higuchi, The Plancherel measure for p-forms in real hyperbolic spaces, University of Alberta and

University of Chicago preprint, October 1992 (to appear in J. Geometry Phys.). ‘*S Minakshisundaram and A. Pleijel, Can. J. Math. 1, 320 (1949) “L’S Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, review ed. (Academic, New York, 1980). z”M. A. Rubin and C. R. Ordoiiez, J. Math. Phys. 25,2888 (1984). ” P. G. Wyboume, Classical Groups (Wiley, New York, 1974). “J. S. Dowker, Class. Quantum Gravit. 1, 359 (1984). 23L. Vretare, Math. Stand. 39, 343 (1976). “L. Vretare, Math. Stand. 41, 99 (1977). “R. Carnporesi and A. Higuchi, Phys. Rev. D 47 (1993). s6G. Cognola, L. Vanzo, and S. Zerbini, J. Math. Phys. 33 222 (1992). 27A. A. Bytsenko and Yu. P. Goncharov, Class. Quantum Gravit. 8, 2269 (1991). **Yu. P. Goncharov and A. A. Bytsenko, Class. Quantum Gravit. 8, L211 (1991). 2gA. A. Bytsenko and S. Zerbini, Class. Quantum Gravit. 9, 1365 (1992). 30G. F. R. Ellis, Gen Relativ. Gravit. 2, 7 (1971). 3’B. Randol, Trans. Am. Math. Sot. 205, 241 (1975).



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