harmonic analysis and propagators on ...calvino.polito.it/~camporesi/physrep1990.pdfr. camporesi,...

PHYSICS REPORTS(Review Sectionof PhysicsLetters) 196. Nos. 1 & 2 (1990) 1—134. North-Holland

HARMONIC ANALYSIS AND PROPAGATORS ONHOMOGENEOUS SPACES

RobertoCAMPORESIDepartmentof Physicsand Astronomy,University of Maryland, College Park, MD 20742, (ISA

Editor: D.N. Schramm ReceivedMarch 1990

Contents:

1. Introduction 3 8.2. Fractional derivatives in one dimension: the2. The heat kernel and the Schwinger—DeWittexpansion 7 Riemann—Liouville integral 483. Example: the Einstein universe II 8.3. The heat kernelOfl S~in terms of fractional deriva-4. The eigenfunctionexpansionon a homogeneousspace 14 tives 5))

4.1. Spectral geometryof Riemannianmanifolds 14 8.4. The general rank-onecase 514.2. The heat kernel on a compacthomogeneousspace 16 8.5. Fractional representationof the Jacobipolynomials4.3. Example:S 19 and the spectrum 54

5. Freemotion on symmetricspacesandquantum integrahle 8.6. The noncompactcase and the Weyl fractional in-systems 20 tegral 575.1. The maximal torus and the lattice 21 9. Exactnessof the WKB approximationon the split-rank5.2. The radial Laplacian 23 symmetric spaces 585.3. The duality spectrum—geodesics 24 10. The partition function and thedimensionsof thespherical5.4. The intertwining operatormethod 26 representations 625.5. The noncompactcase 27 111.1. The Plancherel measureand the spectrum of a5.6. Complete integrability of the one-dimensionalquan- compactsymmetricspace 62

tum system 30 1(1.2. Examples:rank-oneSS, normal real form SS. and6. The heatequationon a torus and the Poissonsummation Lie groups 65

formula 32 1(1.3, Contour representationof the partition function 667. The group manifold case: theequivalenceof the eigen- 11. The heat kernel coefficients and the zetafunction in the

function expansionandthe sum over classical paths 37 rank-one case 707.1. The radial Laplacianon Lie groups and the inter- 11.1. Recursionrelations for the partition function 70

twining operator 37 11.2. The heat kernel coefficients 727.2. The spectrumand the “sum over classicalpaths” 41 11.3. Scalarzeta functions for arbitrarycoupling 767.3. Multiply connectedLie groupsand the phaseof the 11.4. Spinor Zeta functionson S’~ 82

indirect geodesics 43 12. Finite-temperaturequantum field theory in higher dimen-7.4. The noncompactsymmetricspaceGIL’ 45 dons 84

8. The finite propagatoron spheresand rank-onesymmetric 12.1. The scalar case 84spaces:fractional derivatives 46 12.2. The spinor case 888.1. The spherecase 46 12.3. The effective potential on M’ x V 9))

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HARMONIC ANALYSIS ANDPROPAGATORS ON

HOMOGENEOUS SPACES

Roberto CAMPORESI

Departmentof Physicsand Astronomy,University of Maryland, College Park, MD 20742, USA

iiNORTH-HOLLAND

R. Camporesi,Harmonicanalysisand propagatorson homogeneousspaces 3

12.4. The high-temperatureexpansion 93 Appendix B. Harmonicanalysison homogeneousspaces 11512.5. Explicit zero-temperaturecalculations 94 B.1. Homogeneousvector bundles and induced repre-12.6. Self-consistentEinsteinfield equations 99 Sentation 116

AppendixA. Geometryof cosetspaces 101 B.2. Frobeniusreciprocity 120Al. Killing vectors and thevielbein frame 101 B.3. Peter—Weyltheorem 123A.2. Invariant connectionson homogeneousspacesand B.4. Laplacianson G/H and theirspectrum 125

their curvature 107 B.5. Examples:scalarandvectorharmonicson GIH andA.3. Geodesicson G/H and the Van Vleck—Morette then-sphere 127

determinant 112 References 130

Abstract:The techniquesof harmonicanalysison homogeneousspacesarereviewed,andappliedto the theoryof propagators.Thespectralgeometryof

homogeneousand,in particular,of symmetricspacesis considered,with explicit calculationsof theheatkerneland thezetafunction.Severaltopicsrelevantto physical applicationsare discussed,including the Schwinger—DeWittexpansion,the exactnessof the WKB approximationin curvedspaces,theconnectionbetweenfree motion on symmetricspacesandquantumintegrablesystems,andfinite-temperaturequantumfield theoriesinhigher dimensions.The paper containssome new resultsof both mathematicaland physical interest; e.g., explicit formulas for the scalardegeneraciesof theLaplacianon acompactsymmetricspace,exactforms of thezetafunctioli on thesymmetricspacesof rankone,extensionof thefinite-temperatureformalism to spinor fields in higher-dimensionalstatic spacetimes,and Casimir energycalculationsin evendimensions.

1. Introduction

The theory of propagatorsin curvedspacetimesstartedin the fifties with the work of MoretteandDeWitt on the path integral approach[108, 38], and in the sixties with the developmentof theSchwinger—DeWittpropertime formalism and the heatkernel expansion.

Following the work of Schwinger,DeWitt obtainedan integral representationof the propagatorG(x, x’) of a scalarfield, in termsof a kernelK(x, x’, s) satisfyinga Schrödinger-likeor heatequationon the spacetimeM. Formally this kernel can be consideredas the transitionamplitudefor a freeparticle on M propagatingfrom x to x’in a fictitious propertime interval s. In view of this quantummechanicalanalogyit is often referredto as the “propagator”.

In 1949,the mathematiciansMinakshisundaramand Pleijel studiedthe solutionof the heatequationon a Riemannianmanifold, by using an asymptoticexpansionin the limit of small time interval andpoint separation.They were able to derive, from this expansion,the analytic propertiesof the zetafunction on the manifold (e.g., locations of the poles, residuesat thesepoles, existenceof “trivial”zeros,etc.).

Generalizingtheir work to the pseudo-Riemannianspacetimecase,DeWitt obtainedan asymptoticexpansionof the kernel K(x, x’, s) in powersof s, and of the propagatorG(x, x’) in termsof Hankelfunctions. Hereafter, the asymptotic expansionof K(x, x’, s) will be referred to as the MPSDexpansion.

As alreadyobserved,this expansiongives a good approximationof curvedspacetimepropagatorsonly whenthe pointsx andx’ arecloseto eachother andfor smallpropertime parameters. Therefore,it is very useful in isolating the (ultraviolet) divergent terms in quantumfield theorieson a curvedbackgroundand in giving a simple derivationof the existenceof conformal anomalies[19, 30, 1181.

Minakshisundaramand Pleijel gaveas early as in 1949 a generalformula for recursivelycalculatingthe heat kernel coefficients, {a~},in termsof an integral over geodesics.The actual details for anarbitrary spacetimeare rathercomplicated.Expandingthe Laplacian and the Van Vleck—Morettedeterminantin Riemannnormal coordinates(RNC), the perturbativeexpressionof the first few a~hasbeenobtainedin powersof the curvatureand its covariantderivatives [17, 30, 31, 39, 681.

4 R. Camporesi, Harmonic analysis and propagasors on homogeneous spaces

Momentumspacetechniquesin a RNC patchallow one to transferto curved spacesthe usualflatspace Feynmanrules, as exemplified by Bunch and Parker’s treatment [22] of A~”theory on anarbitrarycurvedspacetime (seealsoref. [821).However, all of thesetechniquesare local in natureandare not sensitive to the global structureof spacetime.

There are problemsthat may need a different approach,for example interacting fields, particleproductionand phasetransitionsin cosmologicalmodels, and symmetry breakingdue to changesintopology. In some of theseprocessesit is the infrared (rather than the ultraviolet) behaviourthat isimportant, and knowledge of the propagatorfor well separatedpoints is required. as well asconsiderationof global boundaryconditionsand topology.

Exact forms of the propagatorareobtainablefor fields in spacetimeswith high symmetry.andknowneigenvaluesfor the spacepart (e.g., a homogeneousmanifold) andsolutionsto the wave equationforthe time part. Examplesof treatablemodelsare the Einstein andTaubuniverses[50,32, 33, 81, 1311.de Sitter space [25, 48, 49], and some solutions of Robertson—Walkercosmologies[19]. NonlocalRiemannnormal coordinatesand group theoreticalmethodshave been developedfor the study ofhomogeneoustype IX cosmologicalmodels [83].

The purposeof this paper is to present, from a generalpoint of view as well as with specificexamples,the techniquesof geometricanalysison homogeneousspacesandseveralresultsin the theoryof propagatorson manifolds with symmetries.

The emphasisthroughout the paper is on exact results. For this reasonwe have restrictedourattentionto the very specialclassof homogeneousmanifolds,whosespectralpropertiesareknownfromthe representationtheory of Lie groups. The most explicit results will be obtainedfor symmetrichomogeneousspaces.

For example,in section10 thespectrumof the Laplacianon acompactsymmetricspacewill be givenin termsof the root vectorsandtheir multiplicities. In section11 we shall derivea closedexpressionforthe zetafunction on symmetricspacesof rank one.

Physicalapplicationsof spectralanalysison Riemannianmanifoldsarewell known. The zetafunctiontechniqueis probably the mostefficient way of regularizingpathintegralsin curvedspacetimes[49, 51,66, 73]. As an example,the one-loopeffective potentialfor finite-temperaturescalar andspinor fieldtheories on an arbitrary ultrastatic spacetimewill be calculatedin section 12. using zeta’functionregularization.

The resultsin sections10 and11 will then beusedto computethe(zero-temperature)Casimirenergyin Kaluza—Klein spacetimesof the form M4 x G/H, where GIH is a compact rank-one symmetricspace.This part will be a generalizationof previouswork in Kaluza—Klein theorieswith internalspacesequalto spheres[26, 47, 8].

An importantapplicationof the theory of symmetricspacesis to integrablesystems.Freemotion onsymmetric spacesis closely related to one-dimensionalsystemsthat are completely integrable,bothclassicallyand quantummechanically[115, 116]. The Hamiltonian of the quantumsystemassociatedwith a given symmetricspace,and the radial part of the Laplace—Beltramioperatoron the space,areconnectedby a simple transformation.The completeintegrabilityof the systemis then a consequenceof the known result [78] that the algebraof invariant differential operatorson a symmetric spaceiscommutative.This part will be reviewedin section 5.

A generalinvestigationof the heatkernelon manifoldswith symmetriesis importantbecauseit mayshednew light on the natureof the WKB approximationandof the path integral in curvedspaces.Aquestion that will be addressedin this paperis whetherthe WKB approximationof the heat kernel,representedby the MPSD expansion,may, in somecases,give exact, rather than just asymptotic,

R. Camporesi,Harmonicanalysisandpropagatorson homogeneousspaces 5

results.This questionarisesfrom theremarkablepropertyof Lie groups[45,56, 127] thatthe Gaussian(or leading WKB) approximationto the free particlepropagatoris exact there.

More precisely,the propagatorof a free particleon a Lie group, as well as the quantummechanicalpropagatorin a numberof well known problems(e.g., free particle on Euclideanspace,harmonicoscillator, Lagrangiansthat contain termsup to quadraticorder), are given by the squareroot of theVan Vleck—Morettedeterminantmultiplied by theexponentialof theclassicalaction and,in general,byan additional overall phase[128].

This form of the propagatoris properly referredto as the Gaussianapproximation,as it arisesin thepath integralcontext by expandingthe actionto secondorderandperforminga Gaussianintegration.However,we emphasizethat unlike the analogousquantummechanicalsystems,the resultsfor freemotion on Lie groupshavenot beenobtainedfrom the pathintegral,but either by solving the “radial”Schrödingerequation,or by usingthe spectrumof the Laplacian. As Lie groupsare nontrivial curvedmanifolds,one needsto explain how the higher orderterms(higher than quadratic)do (or rather,donot) contributeto the pathintegral.

Moreover,the equivalencebetweenthe (stationarystate)eigenfunctionexpansionandthe “sum overclassicalpaths”on a Lie groupG hasappeared,in the physicalliterature,only in thecaseof G = SU(n)[45].The completeequivalenceof the two approachesfor anarbitrary (semisimple)compactLie groupwill be proved in section7.

Whenwe movefrom Lie groupsto generalhomogeneousspaces,the Gaussianapproximationis notexactanymore.Still, it is a remarkablethat thereexist manifoldswherethe full WKB approximationisexact.The only knownexamplesarespaceswherethe MPSDexpansionof K(x, x’, s) in powersof thepropertime s terminates,up to a phase,and therebyconvergesfor any value of s andis exact. (Strictlyspeakingthis will betrue only if the spaceis noncompact;in the compactcasewe will needto add thecontributionsfrom the “indirect paths” as well, in order to get a periodic propagator.)

Examplesof this, besidesLie groups,are given by odd spheres[3], and moregenerallyby a specialclass of symmetricspacesknown as “split-rank” [76], which include Lie groupsand odd spheresasparticularcases.The exactnessof the WKB approximationwill be demonstratedin section8 for the oddspheres,and in section9 for the generalcase.

We noticeherethat if the semiclassicalMPSD expansionterminates(up to a phase),but not to itsfirst coefficient, the leadingterm of this expansion(which is just the Gaussianone, see ref. [108])willobviously not coincide with the full WKB expression.Therefore,while the full WKB approximationwill be exact, the Gaussianapproximation to the path integral will not be. This is preciselywhathappenson a split-rank symmetricspace,whenit is not a Lie group.

It is fair to saythat the “sum overclassicalpaths”is exacton all thesespaces,meaningthat theexactpropagatorfor a free particleis the sum of termsF( y) e’5~”1,whereS(y) is the actionon the classicalpath(geodesic)y, andF(y) is a functionalof the path.Again, while it is suggestivethat this form couldbe derivedfrom the path integral,this hasnot beendone as yet.

Of course the form of the propagatoras a “sum over classicalpaths” is a very restrictive one.Nevertheless,if the homogeneousspaceis symmetric, a simple “geometric representation”of thequantummechanicalpropagatorof afree particlecan be derivedin termsof paths,ratherthanin termsof aneigenfunctionexpansion.Remarkably,this geometricform of the propagator(heatkernel) canbeused to determinethe spectrum of the Laplacianwithout having to solve the eigenvalueproblemdirectly.

For example,using this approachon a rank-one symmetric space(e.g., a sphere), an integralrepresentationof the sphericaleigenfunctionsof the Laplacian(that are just Jacobipolynomials)will be

6 R. Camporesi, Harmonic analysis and propagators on homogeneous spaces

derived in section 8 in terms of fractional operators.This duality between the spectrum and thegeodesicscan probably be investigated on nonsymmetricspacesas well. However this point istechnically moredifficult, and will not be consideredhere.

We nowgive a brief outline of this paper.In section2 somebasicfactsaboutthe Schwinger—DeWittexpansionin curvedspaceswill be reviewed,emphasizingthe fact that whenthis expansionterminatesit is exact.The simple but importantexampleof the Einsteinuniverseis consideredin section3, wherethe Feynmanpropagatorof a scalarfield is calculated.

In section4, after reviewing the main resultsin zetafunction theory due to MinakshisundaramandPleijel, we shall derive the eigenfunctionexpansionof the heat kernel on an arbitrary (compact)homogeneousspace.Using the spectrum,we shall prove that the heatkernelon GIH can be obtainedfrom the (Gaussian)heatkernel on G by integratingout the isotropy subgroupH.

In section5 wereviewthe relationbetweenfree motion on symmetricspacesand integrablesystems.The geometricapproachto the calculation of the propagator,mentionedearlier,will be illustratedbywriting Schrödinger’sequationin “radial” coordinateson a maximal torus andusing the conceptof anintertwiningoperator.The heatequationand the duality spectrum—geodesicson a torus areconsideredin section6.

In section7 we extendDowker’s SU(n) proof of the equivalenceof the “sum over classicalpaths”andthe eigenfunctionexpansionof the propagatorto any compact,semisimpleLie group.The resultonthe exactnessof the WKB approximationwill be derivedby solving the radialheatequationandalsobyusingthe spectrum.The caseof a multiply connectedLie groupis discussedin detail, andwe showthatthe nontrivial phase associatedwith the indirect geodesicscoincides with the Gaussianphaseexp(—iirM7/2), whereM~is the total numberof conjugatepoints alongthe multiple geodesicy [128].

In section8 wederive the finite heat kernelon the rank-onesymmetricspacesin termsof fractionalderivatives. The result for the compactspaceshas appearedonly recently [4]. We shall treatboth thecompactand noncompactcasesby, usinganalyticcontinuation,andwill alsoreviewthe basicpropertiesof fractional operatorsin one dimension.

The compactsymmetricspacesof split-rank typeareconsideredin section9, wherethe exactnessofthe WKB approximationon thesespacesis discussed.The constructionof the intertwiningoperatorinthe rank-twocase(basedon ref. [4]) is given. With the appropriateanalyticcontinuation,this operatorgives the inverseAbel transform on the correspondingdual, noncompactsymmetricspaces— a newmathematicalresult.

In section10 westudy the spectralgeometryof an arbitrarysymmetricspace.We first show that thedimensionsof the spherical representationsare the residuesat the singular points of the Plancherelmeasure,andthencalculatethe scalardegeneraciesof the Laplacianin the compactcasein termsof theroot vectorsandthe multiplicities. As far as we know, the explicit formulaobtainedhereis new. It is

also simpler than the Weyl dimensionformula, sincethis involves the rootsystemof the full group G,while our formula involvesonly the rootsof GIH. A contourrepresentationof the partition function isalso obtained.

In section11, after reviewingthe resultsof Cahn and Wolf [23] on the heat kernelcoefficientsandextendingthem to the noncompactcase,we calculatethe scalarzetafunction (for arbitrarycoupling) onthe compact,rank-onesymmetricspacesin termsof a seriesof Riemannzetafunctions.The spinorzetafunction on odd and evenspheresis also calculated.

The formalism of finite-temperaturefield theoriesin arbitrarydimensionsis reviewedin section12,wherethe resultsobtainedby Dowker [47]in the scalarcasearegeneralizedto the spinorcaseas well.Casimir energycalculations for scalarand spinor fields in Kaluza—Klein spacetimesare considered.Consistencywith the quantum-correctedEinstein equationsis also discussed.

R. Camporesi,Harmonicanalysis andpropagatorson homogeneousspaces 7

In appendicesA andB we give aself-containedintroductionto the geometryof homogeneousspacesand to the techniquesof harmonic analysis, using group theoretic methods. We demonstratetheformula for the spectrumof the Laplacianof the H-connectiongiven by Strathdee[133]in applicationto Kaluza—Klein theories.

2. The heat kernel and the Schwinger—DeWitt expansion

Considerthe FeynmanpropagatorG(x,x’) of a massive, minimally coupledscalar field on anN-dimensionalspacetimeM with signature(—1, 1,. . . , 1)~

(L1~— m2)G(x,x’) = ô(x, x’). (2.1)

Here LI is the Laplace—Beltramioperatorand ~(x,x’) is the invariant deltafunction

(2.2)

wheredx’ = \/Ig(x’)~d”~’x’ is the invariant measure.In the Schwinger—DeWittrepresentationone has[39, 40, 19]

+~

—jam2G(x~x)—ij K(x,x,s)e ds, (2.3)

0

whereIm(m2)<0 is understood.The kernelK(x, x’, s) satisfiesa Schrödinger-likeor heatequationonM,

(i85 + E111)K(x,x’, s) = 0, (2.4)

with the boundarycondition

urn K(x, x’, s)= 6(x, x’). (2.5)

Equivalentlythe kernelK(x, x’, s) = 0(s)K(x, x’, s) (0 is the stepfunction) satisfiestheinhomogeneousheatequation

(ia5 + LI~)K(x,x’, s) = i~(s)6(x,x’) . (2.6)

The parameters is calledthe Schwingerpropertime [60, 111, 59, 129] andK the heatkernel. FormallyK is the amplitudefor a particle propagatingfrom x to x’ during the fictitious propertime intervals.However it is clear from (2.4) that s has dimensionsof length squaredand thereforecannot beinterpretedas a real time.

The quantumfield-theoryproblemof finding the propagatorof a scalarfield on acurvedbackground

•~Our conventionsfor thecurvature tensorare~ = a~w,, — ~ +‘‘‘, where are theconnectioncoefficients,and Ri,,, = ~

R= R.

8 R. C’arnporesi, Harmonic analysis and propagators on homogeneous spaces

M hasbeenreformulatedinto an equivalent“quantum mechanical”problem,solving the “evolution”equationin the Schwingerpropertime. Fromthe path integralpoint of view, K is given by a quantummechanicalpath integralfor a free particle propagatingon the manifold M.

On flat Euclideanspacethe heat kernel hasthe known form [29]

N ‘2 ir~ 4.,Kitai(X. x , s) = (4ins) e . (2.7)

whereN is the dimensionof the spaceandu = u(x,x’) is the geodesic(line segment)distancebetweenthe two points. The idea behind the Schwinger—DeWittexpansionis that the heat kernel should belocally representableas a “perturbation”of the flat one,sincelocally themanifold is like flat space.TheSchwinger—DeWitt proper time seriesis a (generally) asymptotic expansionof K in powersof theproper time

I :2 , 1~rI’4., \“ , .K(x,x,s) = (4irts) ~i (x, x ) e ~ a,,(x,x )(is) , (2.8)n = S

where, as in (2.7), a- is the geodesicdistance betweenx and x’ and ~ is the invariant VanVleck—Morette (VVM) determinant[108,42, 128], the biscalardefinedas

~(x,xs)=_[g(x)g(xr)]t2det(~2/2). (2.9)

The ansatz(2.8) was first consideredby the mathematiciansMinakshisundaramandPleijel [105,1041 inorder to solve the heatequationon a Riemannianmanifold in the asymptoticlimit of small time intervaland point separation.Later on, DeWitt [391,following the work of Schwinger[129],generalizedit tothe pseudo-Riemannian,spacetimecase. We shall therefore use the abbreviationMPSD for theasymptoticexpansion(2.8).

The heatkernel coefficientsa, can be calculatedin termsof geometricalquantitieson the manifoldM. The first stepis to introduceRiemannnormalcoordinates(RNC) centeredon oneof the two points,sayx’. The equationof the geodesicx(t) joining x’ to x with x(0) = x’ andx(1) = x is that of a straightline, x~’(t)= tx~,andxa = df(t) 7th’ arethe RNC of the point x. The geodesicdistancesquaredbetweenx and x’ is given by a-2 = ‘q~x”x0 x”x,,, and the inverse of the VVM determinantreducesto thesquareroot of the determinantof the metric tensor[118, 191

(2.10)

Therefore~ gives the invariant measurein RNC

f ~(x)dx = f (~oExp)~t(x)dNx (2.11)V=Exp”(U)

whereU is asmall (convex, normal)neighborhoodcontainingx andx’, V its imagein the tangentspaceT~and d”x = III dx”.

When acting on a scalar field c/i which is only a function of the geodesicdistancefrom x’, the


Laplacianbecomes

1 a2~ /älnii~t N—1\ acIl

LI~~~Ii(O’)oe a~[\I~jg”0a~(u)] = + + ) -~—- , (2.12)

wherewe used the RNC relations

= (x~Io’)0ff’~I, g”0x0 = = , X”t3,, = a~li.~, 9/m3a-.

If we now substitute(2.8) into (2.4) and use (2.12) we get the following recursionrelation for thecoefficientsa~[39, 17]:

~ a-t3,,a0=0. (2.13)

Fromthe last equationwe seethat a0can be an arbitraryfunction of the angularcoordinatesaroundx’.Choosinga0 = 1 we can solve (2.13) with the result

a~(x,x’) = a-”J [a-(t)] 1[112U(~t12a~

1)](x(t),x’) du(t), (2.14)

wheretheintegral is alongthe (supposed)uniquegeodesicconnectingx to x’ with parametera-(t) equalto the geodesicdistance,o-(t) = o(x(t),x’), ~s a-(0) = 0, a-(1) = a-.

With this solution (2.8) remainsan asymptoticexpansion,valid only for smalls andx closeto x’. Ithas thereforebeenuseful in dealing with renormalizationof ultraviolet divergences,e.g., in atheory on the curvedbackgroundM, as in the point-splittinganddimensionalregularisationtechniques[30, 40, 22, 191. For example, the trace anomalyof the stress—energytensorin four dimensionsisdirectly related to the coefficient a2 in the MPSD expansion[118, 19]. By (2.14) the heat kernelcoefficientson an arbitrarybackgroundcanbecalculatedperturbativelyin Riemannnormalcoordinates[17, 30, 39, 68, 97] in termsof the curvaturetensorand its covariantderivatives.

The main concernof this paperis to describethe heat kernelwhen the symmetriesof spacetimeallow one to explicitly solve the propagatorequation. Techniquesof geometricanalysis on homo-geneousspaces[78] can thenbe used,for examplein applicationsto cosmologicalspacetimes[81, 25,48, 50] andto multidimensionalunified theories[126,37, 34, 8, 52]. Thespectralpropertiesof theheatkernel on a homogeneousspacewill be discussedlater.

We shall considerhereaspecific examplein which the MPSD expansionis not just asymptotic,butexact,andgives the sameresultas the global approach.First of all, we notethat a nice propertyof theheat kernel is that it factorizeson a productspace

KMXM(z, s) = KM(xt, s)KM (x2, s) , (2.15)

wherez = (x1, x2) labelsthepointsin theproductspaceM1 x M2. Applicationsof this in sections3 and12 will be to static spacetimeswith topology Rx M and to Kaluza—Klein theorieson a backgroundM~X GIH, whereM~is n-dimensionalMinkowski spaceandGIH a compacthomogeneousspace.Forexample,the heat kernel for ordinary (zero temperature)scalar field theory on a static background

It) R. Camporesi. Harmonic analysis and propagators on homogeneous spaces

R x G/H is

—1.2 _)(,_~)2~4~K(t, t, y, y ,s) = (—4iris) e K~/~(y,y ,s). (2.16)

A finite temperature(T = /3 - I) field theory on the samebackgroundcan be convenientlyrepresentedasordinary zero-temperaturefield theory on the background S’ x GIH, upon a Wick rotation toimaginarytime. This meansthat the finite-temperatureheatkernelK~satisfiesthe sameequationas thezero-temperatureone, with the boundaryconditionthat K~is periodicin imaginarytime, with period /3.ThereforeK~can be expressedas a sumof zero-temperatureGreen’sfunctions[50, 51, 19] (seealsosection 12),

K0(t, t’, y, y’, s) = ~ K(t t’ + in/3, y, y’, s). (2.17)

On compactspacesan additional complication is that there will be an infinite numberof geodesicsconnectingtwo givenpoints, the shortestonebeingthe “direct path” andthe otherscircling aroundthemanifold at leastone time. A periodic kernel is thenbuilt up by summingoverthe “indirect geodesics”as well. For the S’ factorin the previousexamplethiscorrespondsto n = 0 andn ~ 0 respectively.It isa generalfact that only the direct path term gives rise to the divergencesof the theory: the indirectgeodesicsalwaysgive a finite contributionto the propagatorandthe stress—energytensor [50,40, 19].

It is conceivablethat on spaceswith a high degreeof symmetrythe MPSD expansionmay give exactresults. When this happenswe shall say that the WKB approximationrepresentedby the propertimeexpansion(2.8) is exact. Flat spaceis, in a sense,the simplestexampleof this, but spaceswith thisproperty neednot be flat. An exampleis given by the Lie groupmanifolds andwas first investigatedinrefs. [56,57, 127, 44, 45]. This casewill be reviewedin detail in section7, which includesadiscussionofthe multiply connectedLie groupsand of the noncompact,“dual” symmetricspaces.

Within the MPSD formalism the heat kernel on a group manifold can be derived as follows.Considera compactsemisimpleLie groupG. It is a well knownresult (seesection7) that if the spaceisequippedwith the usual Killing—Cartan metric the quantity ~1I2 in (2.14) is an eigenfunctionof theLaplacianwith eigenvalueR16, whereR is the curvaturescalarof the bi-invariant metric and equalsN14, where N is the dimensionof the group. Thereforefrom (2.14) we get

a~= (lIn!)(R16)” , (2.18)

and the MPSD expansionbecomes

KG = (4iiis)’2~~1”2exp(io-2/4s+ isR/6) . (2.19)

This is just the “direct contribution” of the exact heatkernel andexhibitsthe structureof a Gaussianapproximation.The term “Gaussian” originates from the stationary phaseapproximation to thepath-integral.According to this, if we expandthe action to secondorder in the correctionsto theclassical solution and do the resulting Gaussianintegration, we get the product of LI’ / 2 times theexponentialof the classicalaction (modulo a phasefactor) [108,42, 10, 128, 93].

On groupsthe semiclassicalWKB expansion(2.8) convergesfor any value of the propertime andequals(2.19). Therefore the Gaussianand WKB approximationscoincide and are exact (modulo the“indirect contributions”in the compactcase).In a sense,Lie groupsbehavevery similarly to flat space


as far as propagationof a free field is concerned.We might expectthat thesepropertiesof the heatkernelon a groupremainvalid in otherspaceswith highsymmetry,like spheres.After all, spheresare“more symmetric” than Lie groups, the isometry group of the round n-spherebeing n(n + 1)12-dimensional,while on an n-dimensionalLie groupit is only 2n-dimensional.

Howeverexplicit calculationrevealsthat the heatkernelexpansion(2.8) on a spaceas simple as thetwo-sphereis only asymptotic (sections 8 and 11), suggestingthat the exactnessof the WKBapproximationmight be related to other (possibly topological) properties,rather than just to thesymmetriesof the manifold; see also section9.

The simplestpossibility for (2.8) to be exact is whena phaseexp(isp2)(p being a constant)can befactoredout of the sumso that the remainingseriesterminates

a~(is)~= e~2~ 6~(is)~. (2.20)

The Lie groupresult (2.19) is aparticularcaseof this, with p2 = R16, ii~= â’~= 1, anda~’~= 0 for n � 1;i.e., the seriesterminatesto the first coefficientupon factoringexp(isR/6).Thehighestcoefficient c~’~in(2.20) satisfies

= p2(4112â’1) , (2.21)

i.e., multiplied by 41/2 it is aneigenfunctionof the Laplacianwith eigenvalue~2 To seethis,we noticethat with the new ansatz(2.20) the recursion(2.13) would become

+ (n + 1)ä’~~1= 4_h/2(D — p2)(4”2tT~). (2.22)

The solutionto this equationis given by the sameformula(2.14)with the replacementD—~(LI — p2).Requiring Tjei = 0 in (2.22)gives (2.21). Clearly (2.21) is a generalizationof thegrouppropertystatedearlier that 41/2 is an eigenfunctionof the Laplacian.

We shall seein section9 that spaceswith the property(2.20) arethe Riemanniansymmetricspaceswith evenmultiplicities for the root vectors,knownas “split-rank”. Theyincludethe Lie groupsandtheodd-spheresas particular cases,andthey all behavein a similar way from a topologicalpoint of view.The MPSD expansionterminatesat a coefficient ã,. I’ â’~,andthe phaseexp(isp2)is different from theGaussianphaseexp(isRI6),exceptwhenthe spaceis a group. For exampleon odd-spheresS”, withN> 3, the factorp2IR equalsthe conformalcouplinginN + 1 dimensions,the reasonbeingthat spheresare conformallyflat. Therefore,on a split-ranksymmetric spacewhich is not a Lie group the WKBapproximation(2.8) to the free-particlepropagatoris exact,but doesnot coincide with the Gaussianone [2—4].

Before turning to the harmonicexpansionof the heat kernel on a generalhomogeneousspaceweconsider,in the nextsection,the importantexampleof the Einstein universe,with spatial sectionequalto a three-sphere.

3. Example: the Einsteinuniverse

The Einsteinuniverseis the staticspacetimeR x S3. Sincethe three-sphereis the groupmanifold ofSU(2), we can usethe exactnessof the WKB approximationto calculatethe Feynmanpropagator.In


view of the productstructure,we havefor the heat kernel

— 1:2 ~ )2/4~

KFflSICIfl(X, x , s) = (—4lTls) e K~s(y,y , s) . (3.1)

Becauseof the homogeneityof ~3 SO(4)/SO(3) [SU(2)x SU(2)]/SU(2),we can fix the point y’ atthe origin Y() (the north pole) of the three-sphere.A furtheraction by the isotropygroup SO(3) leavesy() fixed and rotatesy on a two-sphere.Since the propagatoris invariant under this action, the kernelK53(y, y0, s) can only be afunction of thegeodesicdistancer = r(y, y0) betweenthe two points. If a isthe radiusof S

3, the equationfor Kçs is

(ia, + LR)Kss(0,s) = i6(s)8~s(0). (3.2)

where 0 = na and LR is the radial part of the Laplacian on S~,given by (2.12).

1 1 a2 /2 alnLI~\ a 1L =—I—-~+l—+ I—I. (3.3)a~La0 \0 ao I aoi

The inverseVVM determinanton the three-sphereis given by

LI’(O, 0) = [(sin0)1012. (3.4)

This can be seenby usingRiemannnormal coordinatesbasedat y0, which coincidewith the canonical

coordinateson SU(2). Using the well knownform of the SU(2)metric in canonicalcoordinates[29]and(2.10), we obtain (3.4) (seealso appendixA). Using this in (3.3) we get for the radial Laplacian

LR(0)=(1Ia2)(a~+2cotO8A) , (3.5)

wherea~ aiao. On the N-spherewe havea similar formula, with a factorN — 1 in place of 2 in thefirst orderderivative. Sincethe three-sphereis a Lie group,we expectthat the squarerootof the VVMdeterminant,41/2(0) = 0/sin0, will be an eigenfunctionof LR with eigenvalueR/6. This can be easilycheckedby rewriting LR in the form

LR(0)=(1Ia2)[(1/sin0)a~osin0+1], (3.6)

and rememberingthat, in the usual normalization,the curvaturescalarof S’~is N(N — 1)/a2, implyingR16= 1/a2 on ~ It is then straightforwardto solve for the heat kernel in (3.2). Omitting an overallstepfunction 0(s) we obtain [45, 50]

Kcs(0,s) = (4~is)3’2 e5~2 ~ 0 ~ exp[ia2(0 + 2~n)/4s]. (3.7)

The sum in this formula is over the geodesicsconnectingthe given point to the “north pole”, andproducesthe requiredperiodicity of the propagator.The n = 0 term is the “direct path” contribution,given by (2.19), and each term in the sum hasthe Gaussianform. The term (0 + 2irn)/sin 0 is the


squarerootof the VVM determinant,and 0 + 2irn the geodesicdistance(in units of a) on the multiplegeodesiclabelledby n.

Of course,this is a consequenceof the boundaryconditionsrequired by the compactnessof thespace.We can considerthe noncompactspacewhich is “dual” to S3, in the senseof symmetricspacetheory.This is justH3, the three-dimensionalhyperbolicspace.Analytically, the radial Laplacianon H3can be obtainedby letting 0—~ixand a—÷iain (3.5) and (3.6),

LR(x) = -~ ~ +2 coth(x)8~]= ~ (sin~(x)a~°sinh(x) — i). (3.8)

The curvatureis now just the oppositeof that on S3,namelyR’ = —6/a2,andtheheatkernel is given by

—3/2 . ja2x2/4s—js/a2K11s(x,s)(4ins) (x/slnhx)e . (3.9)

Notice that if a is normalized to 1 (so that it disappears from all the formulas) the noncompactpropagatorcan be obtained from the compact one by the replacements0—+ix, s—~—s and bymultiplying by an overall phase(_1)3/2.

Obviouslywe only havethe direct path term here,since thereis only one geodesicconnectingthetwo pointsandno sum over n is required.Thereforethe statementmadein ref. [50]that on H

3 onehas“imaginary geodesics” seemsincorrect. In order to get KH5 from K

53 we must use the abovereplacementswhile droppingthe sum over n and keepingonly the direct path contribution.

The regularisationprocedureused in ref. [50] on S3 (in which onesimply dropsthe infinity coming

from the n = 0 term in the calculation of the vacuumstresstensor),would then give strictly zero ifappliedto the hyerboliccase.

We can finally substitute (3.7) in (3.1), and then in (2.3), to get the Feynmanpropagatorof amassive,minimally coupled scalarfield on the Einstein universe

___________ (2) 2 —2 1/2 , 2 2 2 1/2

1 / 2 -2 0+2irn H1 ((m —a ) [(t—t ) —a (0+2irn) ] )G(t — t, 0) = — V m — a ~=_= sin 0 [(t — i”)

2 — a2(0 + 2~n)2]”2

(3.10)

whereH~2~is a Hankelfunction of thesecondkind of order1. For aconformallycoupledscalarfield weget the same expressionwith (m2 — a2)~”2—~m,in agreementwith ref. [50]. On the hyperbolicspacetimeR x H3 we get

1 2 —2 X H~((m+ a2)’’2[(t — tP)2 — a2x2]112)GRXH3(t — t , x) = — ~—= \/m + a sinhx [(t — i”)2 — a2x2]U2 . (3.11)

The results obtainedin this sectionfor S3 and H3 are particular casesof a moreelaboratetheory,whereS3 is replacedby an arbitrary symmetricspace.The geometricalapproachfollowed herewill bediscussedlater in the more general case. Of coursewe could havealso used the (stationarystate)eigenfunctionexpansionon the three-sphereto get the sameresult. This “duality” betweenthe “sumover paths” and the eigenfunctionrepresentationof the propagatorholds on an arbitrary symmetricspaceas well, andwill be discussedin sections5—8. We shall considerfirst the eigenfunctionexpansionon a homogeneousspacein group theoreticterms.

14 R. Camporesi. Harmonic analysis and propagators on homogenesus spaces

4. The eigenfunction expansion on a homogeneousspace

In this sectionwe calculatethe eigenfunctionrepresentationof the quantummechanicalpropagator(heat kernel) of a free scalar particle on a homogeneous space. The particular case of Lie groups andspheres has been considered, e.g., in refs. [56, 127, 45, 99, 3].

As already discussedin section 1. the motivation for studying the heat kernel on a homogeneousspaceis two-fold. Firstly one hopesto get a better understandingof the WKB approximationand thepath-integral formalism in curved spaces.On the other hand the spectral geometry of (compact)homogeneousspacescan be easily investigatedfrom the representationtheory of compactLie groups.This makes homogeneousspacessuitable to quantum (one-loop) calculations in homogeneouscos-mological models[25.48, 50. 131, 19] as well as in higher-dimensional(Kaluza—Klein) theories[26,87,70, 52].

Another importantaspectof free motion on homogeneousandsymmetricspaces(to be discussedinsection5) is that it is closely relatedto quantumintegrablesystemsin 1 + 1 dimensions[115,116].

We shall first reviewsomebasicfactsabout thespectrumof the Laplacianandthe zetafunction on aRiemannian manifold and then derive the finite heat kernel on an arbitrary, compact homogeneousspace.The example of the two-spherewill be discussedat the end.

4.1. Spectralgeometryof Riemannianmanifolds

Considera compact,boundaryless,N-dimensionalRiemannianmanifold M and let L denotetheLaplace—Beltramioperatoractingon scalarfields, i.e., in local coordinates

L = (1 /Vg)a1[vgg’a1]. (4.1)

The spectrumof L is discreteandwe can choosean orthonormalbasisof eigenfunctionsof L. {~,},inthe space of square integrable functions L

2(M) satisfying

(4.2)

~ dx = ~ (4.3)

where dx = sf~d’~x is the Riemannian measure and a starmeanscomplexconjugation.The eigenvaluesA. can be orderedaccordingto

= A55 < A1 A2 .. (44)

wherethe “equals” accountfor the multiplicity m1of a givenA1. We haveassumedthat M is connectedso that m0 = 1, since the eigenfunctionswith A = 0 are just the constantson a compact,connectedmanifold.

The heat kernel K is the fundamentalsolution of the heat equationon M

(—a/at + Lc)K(x, x’, t) = 0, (4.5)


with the initial condition

lim K(x, x’, t) = 6(x, x’), (4.6)

whereô(x, x’) is the invariant Diracdeltafunction on M. We shall alsoconsiderthe Green’sfunction Kto the Schrödingeroperator(ia5 + L~)in the Schwingerpropertime s, and usethe name“heat kernel”for K as well. The relation betweenK and K is just a “Wick rotation” t—~is,

K(x, x’, s)= K(x, x’, is) . (4.7)

It is straightforwardto showthat the deltafunction andthe heatkernelhavethe following eigenfunc-tion expansionon M

(4.8)

K(x, x’, s) = e1~~ô(x,x’) = ~ 41(x)~(x’)e’~’ . (4.9)

K is symmetricin x ~ x’, and in the asymptoticlimit of smalls and x closeto x’ the MPSD expansionconsideredin section2 gives agood approximationof K. Thepartition functionZ(s) is the integraloverM of the coincidencelimit K(x,x,s) of the heatkernel. Usingthe orthonormalityof the eigenfunctionswe find

Z(s) =J K(x, x, s)dx = ~ ~ (4.10)

A Mellin transformof K and Z gives the c-kerneland the C(zeta)-function

~=—~ J[K(x,x5,s)_V~1]s~1ds, (4.11)

~(z)= To f [Z(s) — 1Js~ds =J ~(x,x, z)dx, (4.12)

whereVM is the volume of M. Using (4.9) and (4.10) we obtain the eigenfunctionexpansions

~(x, x’, z) = ~ ~1(x)~(xf)A7z, (4.13)f+0

~(z)=~ AZ. (4.14)1+0

Notice that the zeroeigenvalueis omittedfrom the definition of thezetafunction. The series(4.13)and(4.14) convergefor Rez> N12 andfor the othervaluesof z they aredefinedby analyticcontinuation.The analyticstructureof thesekernelscan be determinedby usingthe asymptoticexpansion(2.8) for

16 R. (‘amporesi. Harmonic analysis and propagators on homogeneous spaces’

K, and the correspondingone for Z

Z(s)=(4iris)~’ ~ u,,(is)’, (4.15)if, -- (I

wherethe u,, are the integralsover M of the coincidencelimit of the heat kernel coefficients

u, a,(x, x) dx, (4.16)

and are Riemannianinvariants. The standardstatements[102, 105. 130. 73, 84, 85] are that the zetafunction ~(z) can be analytically continuedto a meromorphicfunction with simple poles at

z0=~N—p, p=O.I,2,... forNodd:(4.17)

z,,=~N—p. 0~p~N—l forNeven,

and residuesu1,/[(4ir) F(3N—p)j. Moreover for N odd

~(0)=—1, ~(—p)0, pl,2 (4.18)

while for N even

~(0)= (4~)” u,~,.,— I. ~(—p)= (—1)”p!(4~’~’2u\.

2,f/,.p = 1. 2 (4.19)

However, if we are interestedin the finite heat kernel or in the exact expansionof the c-functionwemust use the eigenfunction representation. The problem is that given a manifold with nontrivialtopology the spectrum and the eigenfunctions of the Laplacian are difficult to calculate. As we haveseenin section 2, this situationcan be improved by putting symmetrieson the manifold. If a groupGacts on M a numberof invariancepropertiesof the heat kernel can he derived which give furtherinformation and suggestnew geometricalapproachesto the solution of the propagatorequation.

4.2. Theheatkernel on a compacthomogeneousspace

Let M be a compacthomogeneousspaceG/H. The geometryof cosetspacesis consideredin detailin appendixA and theharmonicexpansionof arbitraryfields on G/H is given in appendixB, wherethespectrum of the Laplacian is also calculated in termsof the group theoretic, secondorder Casimirinvariants on G and H. The orthonormalized eigenfunctions of the Laplacian can be identified withsuitable matrix elementsof finite dimensional,irreducible, unitary representations(“irreps”) of thesymmetry group G.

In the caseof L actingon scalarfields the usefulirreps arethe so called sphericalrepresentationsofG with respectto the isotropy subgroupH. Let VA denotethe representationspace,and U

5(g) theoperators,of a spherical irrep of G. Then thereare vectors ~) in V

5 that are left fixed by all theelementsh in H, i.e.,

= . (4.20)

R. Camporesi,Harmonic analysisandpropagatorson homogeneousspaces 17

In other words, the spherical representationsof G are those that contain the singlet of H whenrestrictedto H. A scalarfield on GIH can be expandedin termsof the matrix elementsof theseirreps.Given any orthonormalbasis { I ~t} ~. in V5, we considerthe functions

= (d5IV)v2( U5(o~(x)’~’t)~I), (4.21)

where V is the volume of G/H and a-(x) E G is an elementof the cosetdefined by x EG/H. (Theembeddingfunction a- is describedin appendixA.) The index ~ in (4.21)can takevaluesfrom 1 to ~the dimensionof the subspaceof V

5 spannedby the sphericalvectors ~),on whichH actstrivially. Forexample,if G/H is a symmetricspace,4~= 1 for any sphericalirreps A [78].

An invariant metric is inducedon G/H by a metric on G invariant underthe adjoint representation(e.g., by the Killing—Cartan metric if G is semisimple,see appendixA). If L denotesthe Laplace—Betrami operatorof this metric on G/H we have

L~1= —[C(A)/a2]~

1, 04.22)

whereC(A) is the Casimirnumberof the irrep A, i.e., the valuetakenby the operator(—~~ TJ5TJ5),

wherethe T~arethe (antihermitian)generatorsof U5. Thequantity a hasdimensionsof lengthandsetsthe scaleof the space.The functions {4~} areorthonormal

I \~~‘*f \ —j (1J51~x)(,I)51.~x)ux—u55.u11.u

G/H

andwe concludethat the spectrumof the Laplacianon a homogeneousspaceis given by

{A~,~} = {C(A)1a2, 4~}, (4.24)

with A running over all the (inequivalent)sphericalrepresentationsof G. The multiplicity m5 of the

eigenvalueC(A) is justd5~5,whered5 is thedimensionof the representationand~ indicateshow manytimes the singletof H is containedin A. For example,if GIH is symmetric,~ = 1 and the multiplicityreducesto the dimensionfactor. [Weare not including herethe additionaldegeneracycomingfrom thefact that differentA’s can havethe sameC(A); this is correctly accountedfor in eq. (4.24)whenA variesover all the inequivalentrepresentations.]A similar analysiscan be carriedout for tensorand spinorfields on G/H (seeappendixB).

By eq. (4.9) we can write down the eigenfunctionexpansionof the heat kernel and the partitionfunction on G/H. Using the fact that the U

5 are unitary matrices,we obtain

KG/H(x,x’,s) ~ ~d5~U~(u(x)’a-(x’))~) e5c~

2, (4.25)

Z(s) = ~ d5~5e~~”

2. (4.26)

Fromthe invarianceof the Laplacianunderthe actionof G, or directly from (4.25), onecanshowthatthe heat kernel is a two-point invariant function, i.e.,

K(gx, gx’, s) = K(x, x’, s), (4.27)


for anyg E G. Choosingg = o-(x)’ gives

K(x, f, s) = K(x55, a-(x) ‘x’, s) , (4.28)

where x0 is the origin of the homogeneousspace,i.e., the elementof M whose isotropy group is H, orequivalently the identity coset of G/H, x0 = H. Equation (4.28) is a statementof translationalinvariance of the heat kernel. We can fix one point at the origin and simply computethe kernel forpropagationfrom x55 to x,

K(x,s)usK(x55,x, s)= ~ d5 (x)e’~51’~, (4.29)

wherethe functions ,(x) = ~ U5(o(x))~~ are called sphericalfunctions. Using (4.29), or directlyfrom (4.27) with x = x

0 andg = h E H, we seethat the heat kernelK(x, s) is a zonalfunction on G/H,i.e., it is invariant underthe action of the isotropy group H,

K(hx, s)= K(x,s), (4.30)

for anyh E H. In otherwords,oncethe first point hasbeenfixed at the origin we havethe freedomto“rotate” the secondpoint aboutx55 by the elementsof H. The propagatoris invariant underthis actionandwe can saythat K(x,s) (like anyzonalfunction) is rotationallyinvariant;i.e., it is only afunction oftheorbits Hx of H. This fact will be the startingpoint in the nextsectionfor a geometricconstructionofthe heatkernel on symmetricspaces[41.

Another important propertyof K on an arbitraryhomogeneousspaceis that it can be obtainedbyintegratingthe heat kernel of the symmetry group G over the isotropy subgroup[12, 46, 48, 84],

KG,.H(x~,x, s)= KG(e, gh, s) dh, (4.31)

where x = gx0. This can be proved in the following way using the eigenfunctionexpansion.We firstspecialize(4.29) to the casewhere GIH is the group manifold G itself; i.e., we chooseH = {e}, theidentity of G. The index ~ is then a full representationindex and runs from 1 to the dimensiond5.Thereforethe sumover ~ in (4.29) gives

~ A4~5(g)=x5(g)usTrace(U(g)), (4.32)

and (4.29) becomesan expansionover the charactersof G

K~(g,s) KG(e, g, s) = ~ d5~5(g) ~ (4.33)

The sumnow is over all the irreducible representationsof G, eventhe nonsphericalones.The sameformulacan bederivedby consideringG as the nontrivial homogeneousspace[GL x GR]/Gdlag, wherethe two G-factors act on G by left and right multiplications and Gdjag {(g, g), g E G} C G x G


(appendixB). The invarianceproperty (4.30) is then

K~(gg’g’~,s) = K0(g’, s), (4.34)

i.e., the zonalfunctionson G arethe class functions,invariant underconjugation.It is well knownthatthe charactersx5 form a basis for the class functionson G, so expansion(4.33) is well understood.

If we nowconsiderthe integraloverH in the right handsideof (4.31), anduse(4.33), wesee thatwe only get a nonzero result for those A which contain the singlet of H, i.e., the sphericalrepresentations.By normalizing the volumes so that VG = VHVG/H we then obtain precisely theexpansion(4.29).

4.3. Example:S2

The result (4.31) can be very useful in computingKG/H, since the heat kernel on any group G isknown exactly, both in terms of the eigenfunctions,by (4.33), and as a “sum over classicalpaths”(geodesics).(Theequivalenceof thesetwo forms for anarbitrarygroupwill be demonstratedin section7.) As an example,the propagatorfor a free particleon the two-spherecan be obtainedfrom the heatkernel on the groupsSO(3) or SU(2) by writing S2 as SO(3)/SO(2),or as SU(2)/U(1), and thenintegrating out the isotropy subgroupSO(2) U(1). The heat kernel on theseLie groupshasbeengiven, e.g., by Schulmanin termsof a sum over paths[127]

js/4 +

Kso(3)(0,s) = (4~is)

312~ (1) 2 sin(0/2) exp[(i/4s)(0 + 2irn)2], (4.35)

js/4 ~= 0+4KsU(

2)(0, s) = (4iris)312 2 sin(012) exp[(i/4s)(0 + 4lTn)2], (4.36)

where 0 is the length of the geodesicfrom the origin to the given point and coincideswith themagnitudeof the total angleof rotation.Both (4.35) and(4.36)can be obtainedfrom the eigenfunctionexpansion(4.33) using the standardgroup manifold normalization C(j) = 1(1 + 1) for the CasimirnumberandVsu(

2) = 16ir2 = 2Vs

0(3) for thevolumes.The characterseriesis identified as aderivativeofa Jacobitheta-function[~2 for SO(3) and 03 for SU(2)], andthe inversionformula for thesefunctions(see section 6) gives the above results. The SU(2) propagatorcoincides, upon a change in thenormalization,with the propagator(3.7) on the three-sphere.The relationbetweenthe two kernels(4.35) and (4.36) is

Kso(3)(0,s)= KsU(2)(0, s) + KsU(2)(21T — 0, s) , (4.37)

correspondingto the isomorphismSO(3) SU(2)/{1, —1}. SO(3) is doubly connectedandthepoints0and

21T — 0 of SU(2) areidentified in the quotient.TopologicallySU(2) is like S3,whereasSO(3)is thethree-dimensionalreal projectivespaceP3(R),which is isomorphicto S3 modulo the identification ofantipodalpoints [41]. (Seesection7 for the generalizationof this to a compact,multiply connectedLiegroup.)

If we now apply (4.31) to the SO(3) propagator (4.35),weget the heatkernelon the two-sphere[48]


K52(0,s) = V~e’~4~ ~, f d~(~ + 2~n)exp[(i/4s)(~+ 2~n)] (4.38)

(43r1s) -~ (cos0 — cos ~)

with the structureof a “continuousimagesum” of S0(3)propagators.Of course(4.38) is equivalenttothe eigenfunctionrepresentationin termsof the Legendrepolynomials (section8)

K52(0, s) = ~ (21 + 1)P1(cos0) e”

111~. (4.39)1=0

The sameanswer(4.38) is obtainedby applying(4.31) to the SU(2) propagator.The groupSU(2)hasintegerand half-integerrepresentations,and the latter cannotappearin the harmonicexpansionof ascalarfield on S2. In otherwords, the half-integerrepresentationsarenot spherical,becausetheynevercontain the singlet of U(1). The quantity KsU(

2) in (4.36) can be written as

KsU(2) = ~(K~ + K_), (4.40)

5,4 -.-~

K+(0, s)= (4~is)3~2~ 2sin(0/2) exp[(i/4s)(0 + 2~n)21. (4.41)

K_ coincideswith KS013) and correspondsto summingover the integer representations,while K

correspondsto the half-integers. K_ is the periodic propagatoron SO(3) whereas K~satisfies

antiperiodicboundaryconditions.Whenwe apply (4.31) to integrateover the onedimensionaltorus ofSU(2) (of length

41T), the & kernelgives zero and K_ reproduces(4.38).In turn, we can write down the heat kernel on the two-dimensionalhyperbolic spaceH2, the

noncompactsymmetricspacedual to the two-sphere,

_______ yeIY~4ifdyKH2(x, s) = . 3/2 I 1/2 (4.42)

(4iris) (coshy—coshx)

As in the caseof S3 and H3, this can be obtainedfrom (4.38) by replacing0—*ix, s—~—s, and bykeepingonly the “direct path” term in the sumover n. Furthermore,the rangeof integrationmust beextendedto ~.

The appearanceof the integral operatorin (4.38) and (4.42) is commonto the evenspheresandtothe evenprojectivespaces,and spoils the exactnessof the WKB approximationon thesespaces.Insection8 we shall interpretit as a fractionaloperator,andwill calculatetheheatkernelon the rank-onesymmetricspacesin termsof fractional derivatives.

5. Freemotion on symmetric spacesand quantum integrable systems

Theresultsin the previoussectioncan beconsiderablyimprovedin the casethat G/H is asymmetricspace.It appearsthat a geometricconstructionof the heat kernel is possibleon thesespaces,basedonthe notions of maximal torus, radial Laplacian, and intertwining operators.The duality spectrum—geodesics,that we havepreviouslynoticed in particularexamples,is a generalfeatureof thesespaces.


This doesnot meanthat the WKB approximationis exact,in general,but only that thereexists aduality transformationthat takes one from the eigenfunction representationto a sum over thecharacteristiclattice of the space,the lattice that gives the periodicityof the closedgeodesicsin themaximal torus. This form will be called the “sum over paths” representationof the propagator.Whetheror not this representationcan be consistentlyderivedfrom the pathintegral is not clearyet.Evenin the simplecaseof a Lie group, a pathintegral derivationof the knownresulton the exactnessof the Gaussianapproximationis still lacking (for a relateddiscussionsee refs. [99, 128, 93]).

The compactcasewill be consideredfirst, but it turns out that the compactand the noncompactsymmetric spaces(SS) can be dealt with together,using the concept of “duality”, and Harish—Chandra’s theory of spherical functions [72]. As we shall see, the heat kernel and the sphericalfunctionson “dual” symmetric spacesof the compactand noncompacttypes are relatedby analyticcontinuation.Recursionrelationsandasymptoticexpansionsof the heatkernelcan be easilyextendedfrom oneto the other typeof SS, and the spectralpropertiesof the compactspaces(e.g.,eigenvaluesanddegeneraciesof the Laplacian,analyticstructureof thezetafunction, etc.) canbe deducedfrom theGindikin-Karpelevicproductformula for the Plancherelmeasure[69, 78, 64] (seesections10 and 11).

The quantumproblemof a free particleon a symmetricspaceis of interestalso becauseit is closelyrelatedto one-dimensionalintegrablesystems.For a detaileddiscussionof this connectionwe refer torefs. [115, 116]. The prototypeof thesemodelsis the problemof n particlesin one spacedimension,interactingwith potentialsof the form v(a(q1— q.)) [a is aparameterwith dimension(length)’], wherethe functionsv are of threebasic types,

v(x) =(sinx)2 , v(x) =(sinhx)2 , v(x) x2 . (5.1)

These potentialsare associatedwith the compact,noncompact,and flat (zero-curvature)symmetricspaces,respectively.The key point in provingthe completequantumintegrability of thesesystemsis toshow that the Hamiltonianis simply relatedto the radial Laplacianon the correspondingSS,sothat theeigenfunctionscan be expressedin termsof the sphericalfunctions.A proofof the integrability will begiven atthe endof this section.We shall first developthenecessaryinformationaboutthe geometryofsymmetricspaces,including a discussionof the radial Laplacianand the lattice.

5.1. The maximaltorus and the lattice

Examplesof symmetricspacesareordinaryEuclideanspace,Lie groups,spheres,real andcomplexprojective spaces,and the correspondingnoncompacthyperbolic spaces.Let us recall that a homo-geneousspaceis symmetric if the covariantderivative of the Riemanntensorvanishes,i.e., thecurvaturetensoris parallel. There are two other equivalentdefinitions. One is that the geodesicsymmetry:Exp(X)—* Exp(—X) is an isometry,no matter on which point one choosesto define theexponentialmapping. The other definition involves some notions from the theory of Lie algebras.Considerthe direct sum decomposition(seeappendixA)

c~=~CEl~U (5.2)

of the Lie algebraof the group G, into the Lie algebra~Wof H and the complementarysubspace(ar)’ (with respectto the Killing metric if G is semisimple,or to anybi-invariantmetricy on G, if

it is not). A symmetricspaceis definedby the commutations


(5.3)

~ (5.4)

The first commutationalonedefinesreductivehomogeneousspaces,which include compactG/H aswell as noncompactG/H with G semisimple,or with H compact,and all symmetric homogeneousspaces.The secondcommutationcharacterizesSS. The subspace~Ucan be identified with the tangentspaceT~5at the origin, andthe restrictionof the metric y to .A~inducesa G-invariantmetric on G/H[113].In theappendices,the curvatureof this metric andthespectrumof its Laplacianarecalculatedinthe more general“reductive” case.

We haveseen,in section4, that the symmetriesof M G/H can be usedto simplify the problemofcalculatingtheheatkernel. Given two endpointsx andx’, oneof them can be translatedto the origin byan actionof the groupG. Undera furtheraction by an elementin theisotropy subgroupH, theorigin isheld fixed while the otherendpointis rotatedabout it on an orbit of H.

As the heat kernel is invariant under this action (it is a zonalfunction), it can only dependon thecoordinatesalong a subspaceorthogonalto the orbits of H. If G/H is symmetric thesesubspacesaremaximaltori [21, 78].

A maximal torus is a maximal, totally geodesic,flat submanifoldpassingthroughthe origin. If ~ is amaximal abeliansubspaceof At, then T = Exp 1ET is a maximal torus (Exp is the exponentialmappingbasedat the origin). Vice versa,any maximaltorus is the image,throughExp, of amaximalcommutingset of generatorsin the tangentspace. Its dimension1 is called the rank of the SS, and equals thenumberof independentCasimir operators(seelater).

The maximal tori areall conjugateunderthe actionof H, andintersectthe orbits of H orthogonally.Furthermore,any orbit of H intersectssomefixed maximal torus in a finite numberof points. Thesepoints aremappedinto eachotherby the actionof a finite group,knownas the Weylgroup. This groupis generatedby the reflectionsin the hyperplanesof ~?Jdefinedby the root vectorsa of the SS.

It follows that the secondendpointin the propagatorcan alwaysbe translatedinto a fixed maximaltorus T, andthat the heat kernel is only a function of separationin the maximal torus,andis invariantunderthe Weyl group W. If h is a vector in the Lie algebrai~Tof T, its coordinates(h’)~1in a (usually)orthonormal basis are the canonical coordinates of the point Exp(h)E T. The heat kernelK(x0,Exp(h),s) will be denotedK(h, s).

It is clear that K mustalsobe invariant undertranslationsin the lattice [~ of the SS,definedas the setof vectors2lTn in ~ suchthat h andh + 217-n representthe samepoint on the torus. In otherwords,17~is the kernel of the exponentialmap (restrictedto .9),

= {2irn E ~ Exp(2irn)= x0}, (5.5)

and the torus T is just the quotient ~/17~.The closedgeodesicsthrough the origin havethe formx(t) = Exp(2i~tn),with tE [0, 1]. Thus, the heat kernel satisfies

K(h+2i~n,s)=K(h,s), V2i~nEf5. (5.6)

An obviousway to implementthis periodicity is to build K in terms of a sum over the lattice I7~,

K(h,s)= ~ KdP(h+2lTn,s), (5.7)21TnE[~


whereKdP is the “direct path”, n = 0 term. Forexample,in the Gaussianapproximationthe heatkernelon a symmetric spaceis [93]

KGaussjafl(h,s) = (4e.)N/2 ~ H (i~~))) exp[i(h +2~n)2I4s] (5.8)

whereR is the curvaturescalar,N is the dimensionof GIH, andwe usedthefollowing expressionof theinverseVan Vleck—Morette determinanton a SS (seeappendixA)

4~(h) 4’(x~,Exp(h)) = fl (sin(ah))rn~ (5.9)

The productsin (5.8) and (5.9) areover the positiveroots of the SS,and the positive integersm~arethe multiplicities of the roots.For example,on S3 weonly haveone(positive) rootwith multiplicity 2,and (5.8) reproduces(3.7), sincethe Gaussianapproximationon S3 is exact.

5.2. The radial Laplacian

The direct-pathheat kernelcan be determinedby solving the “radial” heatequationon the torus.GivenadifferentialoperatorD on GIH, we can restrict it to the maximal torus to get its radial part DR.This is definedby [78]

= DR(~T), (5.10)

where~ is any zonalfunction on GIH (seesection4). The radialpart of the Laplace—BeltramioperatorL is given by the following Weyl-invariant,differential operatoron ~ [78]

LR=a2+~macot(a•h)a•O, (5.11)

where a2 = ~ a.a, (with a, ~I~h’) is the ordinary (flat) Laplacianon T and the sum is over thepositiveroots (a . D being the derivativein the direction of the roota). For example,on the N-spherethereis only one positive root with multiplicity N — 1 and (5.11) gives

LR = a~+ [(N— 1)/a] cot(r/a)a~, (5.12)

where r is the geodesicdistance,anda is the radiusof the sphere.The radial Laplaciancan be equivalentlywritten as

LR—(1IJ)a2oJ—(a2J)/J=(1/J2)aIJ2aI, (5.13)

whereJ2 is the “density” function,

J2(h)= [I (sin(a . h))m~ . (5.14)

The radial Laplacian on the symmetric space of the noncompact type dual to GIH (see later) is

24 R. Camporesi. Harmonic analysis and propagators on homogeneous spaces

obtainedfrom (5.11) by letting h—~ih,and changingsign. The result is that one merely replacesthetrigonometricfunctionswith the correspondinghyperbolicones,as we alreadysaw in the caseof ~

Another importantexampleis given by the compactsemisimpleLie groups.Thesearethe symmetricspacesof thecompacttype, with multiplicity equalto 2 for eachrootvector. It will be shown,in section7, that the functionJ is thenan eigenfunctionof a2 with eigenvalue—R/6, whereR is the curvaturescalar.Thus, the radial Laplacianon a compactLie group is [13, 45]

LR=(1/J)02oJ+R/6, (5.15)

where

J(h) = 112sin(a. h/2) (5.16)

in the “group manifold” normalization.Returningto thegeneralcase,the heatkernelis the Green’sfunction of the operator(ia

5+ LR), i.e.,

satisfies

(ia5 + LR)K(h,s) = i~(s)~R(h). (5.17)

Here ~R is the invariant radial deltafunction on the maximal torus,definedby

J 6R( (h)VJ(h) d1h = ~(0), (5.18)

where V’ is the volume factor from the coordinatesorthogonalto the torus. We see that the densityfunction J2 works as an invariant measureon the torus,andin fact it hasa simplegeometricalmeaning:the quantity V’J2(h) is just thevolume of the orbit of Hpassingthroughthe point Exp(h). Theintegralin (5.18) is over somefundamentaldomainof the Weyl groupin ~, i.e., a Weyl alcovein the compactcase, and a Weyl chamber in the noncompactone. (The Weyl chambersare the inner parts ofpolyhedralanglesandthe Weyl alcovesaresymplexes,see for examplethe appendicesin the papersbyOlshanetskiand Perelomov[115,116].)

5.3. The duality spectrum—geodesics

A formal solution to (5.17) is the eigenfunctionexpansion

K(h, s) = ~ d5~5(h) ~ (5.19)

G/H A

whereçb5(h) = K U5(e”)~)arethe sphericalfunctions,and the sum is over the sphericalrepresenta-

tions of G. The sphericalvector ~ can be normalizedso that (~I~)= 1 and the sphericalfunctionsare the Weyl-invarianteigenfunctionsof the radial Laplaciannormalizedto ~~(0)= 1.

It is a standardresult, in the theory of symmetric spaces[74, 78, 116], that the algebraof thedifferential operatorson G/H that are invariant under the action of G (known as Casimir or Laplaceoperators)is commutative.Thereare 1 algebraicallyindependentgeneratorsD,,. . . ,D, (1 is the rank),whosedegreesarecanonicallydeterminedby the root systemof the SS. A similar statementholdsfor


the algebraof the radial partsof the Laplaceoperators.The ordersof the generatorscoincidewith theso called invariants of the Weyl group. The lowest order generator is always the Laplace—Beltramioperator.

Since the Laplace operatorscommute,they havecommoneigenfunctions.The sphericalfunctionsare the zonal (H-invariant) eigenfunctions,while an arbitrary eigenfunctionis given by the matrixelement(~jU5oojI),see section4.

Assumingthat G/H is simply connected,weshall now showthat thereis a nice duality betweenthespectrumof the Laplacianand the geodesicsof the SS. It is proved,e.g. in ref. [76], that if G/H issimply connectedthe lattice F~,given in eq. (5.5), coincideswith the lattice F of the integer linearcombinationsof the normalizedsimple roots,

F={21T~n5â1, njinteger}, (5.20)

where{a1} arethe simple roots,and Ic = a~/~a5~2if 2a

1is not a root, andIc = a~/2~a5~2if 2a1 is a root.

On the other hand, Cartan hasshown that each sphericalrepresentationis parametrizedby itshighestweight vector A and that thesevectorslive in .~‘ (like the roots), and satisfy the followingintegrality condition: for eachpositive root a the numberA~a/~a~

2is a non-negativeinteger[78, 84].Therefore,the highestsphericalweightscan be written as linear combinationsA = ~ n

5~r1,wheren, arenon-negativeintegers,and {i~,} are the fundamentalsphericalweights,definedby

(5.21)

In other words, {&,} and {ir~} are dual basesin iT. The Casimirnumberin (5.19) can be written intermsof the highestweightA as [62,78]

C(A)=(A+p)2—p2A(A+2p), (5.22)

wherep is half the sum of the positive root vectors,countedwith their multiplicities,

p=~~maa. (5.23)

The vectorp is not necessarilya weightvector; however,it can be shownthat 2p is alwaysa sphericalweight.The lattice A of sphericalweightsis given by all possiblelinear combinationsof thefundamentalweights,with integercoefficients.

It is clear from their definitions and from (5.21) that F/2ir and A are dual lattices,

= fl* (5.24)

Therefore, the two representations(5.7) (the “sum over paths”) and (5.19) (the eigenfunctionexpansion)of the heatkernel aresumsover dual lattices in iT, andthe problemarisesof showingtheirequivalence.A relatedproblemis that the spherical functions are known explicitly only on a fewsymmetricspaces,e.g.,in the rank-onecase,andin the Lie-groupcase.We would like to determinethesphericalfunctionsin the generalcaseas well, regardingthem as polynomialsin I variables,orthogonalwith respectto the radial measureJ2 dth.


5.4. The intertwining operatormethod

These problemsare solved by a new approachto the solution of the radial heat equation(5.17),basedon the methodof the intertwining operators. The methodhas beenrecently appliedto constructthe heatkernelandthe sphericalfunctionson all of therank-oneandmanyof the rank-twoSS, andcanbe applied, in principle,to any SS [35, 4].

One looks for an operatorD, not necessarilydifferential, intertwining the radial Laplacianon thesymmetricspaceto the ordinaryLaplacianon the maximal torus, i.e..

LRD= D(a2 + p2). (5.25)

The constantp2 will be seento coincide with the norm squaredof the vectorp. definedin (5.23). If theoperatorD is normalizedso that it relatesthe invariant deltafunction 6R to the ordinaryone on thetorus, i.e.,

= D5’57. (5.26)

thenD relatesthe heat kernelsof (ia, + LR) and (i8, + a2~.

K = e’502DKT. (5.27)

KT is a “flat” kernel on the maximal torus (see section 6), and satisfies appropriate boundaryconditions,

KT(h, s) = (4iris)~2~ exp[i(h + 2irn)2/4s+ iq~(n)], (5.28)

where the phaseexp[i~(n)] of the “indirect geodesics”dependson the boundaryconditions. Forexample,if this phaseis 1 for anyvector in the lattice, thenKT is the periodic propagatoron the torus.The boundaryconditionsaredeterminedby the structureof D and, in fact, by the multiplicities of theroots of G/H (see section 8).

The important point is that KT hasalso a representationin termsof eigenfunctionson the torus(basicallycomplexexponentials).The relation betweenthe two forms is given by applyingthe Poissonsummationformula to the maximal torus of the symmetric space,and will be discussedin the nextsection.Thus, the sum over F in (5.28) is transformedinto a sum over the dual lattice of sphericalweights and the first problemis solved.

Furthermore,from (5.25) and (5.22)we seethat if we know D explicitly we also know the sphericalfunctionsin terms of elementaryfunctionson the torus,

q~5(h)=a5D ~ exp[iw(A+p).h], (5.29)

wE W

whereaA is a normalization constantand we useda Weyl invariant notation. In sections7 and 8, weshall see that the relation (5.29) can be explicitly demonstratedon Lie groupsand on rank-onesymmetricspaces.Comparing(5.19) to (5.27) and using (5.29) we get the value of the constanta5,

a5 = VG/H/VTdA , (5.30)

R. Camporesi,Harmonicanalysisandpropagaforson homogeneousspaces 27

whered5 is the dimensionof the representationA, and V,. is the volume of the maximal torus,givenby(seesection6)

VT = (21r)’[det(â,. ~)]1/2 . (5.31)

It is clear, from the abovediscussion,that the heatkernelon any SS can be explicitly calculatedif wecan find the form of the operatorD. Some generalpropertiesof D areknown, basedon the casesthathavebeensolvedso far. Since an intertwining relationcan be iterated,the operatorD can be built upas a product of partial operators,intertwining the radial Laplacianon the given SS to the radialLaplacianson other SS of the samerank, but with lower multiplicities.

That is, the transformationfrom the radial Laplacianon the symmetricspaceto the flat Laplacianonthe maximal torus takesthe form of multiplicity reduction,andit will be seenin sections7, 8 and9 (seealso ref. [4]), that a differential intertwining operator“reduces” by two units the multiplicity of therootsin a Weylset,a set of roots with the samelengthandmultiplicity, connectedto oneanotherby theactionof the Weyl group. Once the multiplicities of all roots havebeenreducedto zero, the problembecomesan ordinaryproblemon a flat torus andcan be solved exactlyby standardmethods.

If the multiplicities areall eventhe SS is calledsplit-rank,becausethis is equivalentto the conditionthatthe rank of the groupG is equalto the rank of the groupH plus that of the symmetricspaceG/H[6, 7, 76]. This is the only casein which the operatorD is purely differential. From eqs. (5.27) and(5.28)weseethat the exactkerneltakesthe form of anMPSDexpansion(2.8),which terminatesuponfactoring the phaseexp(isp

2).It is now clear why the WKB approximationis exact on Lie groupsand odd spheres:they are

split-rankSS,with rn, = 2 for Lie groups,andma = N — 1 for S”. In section9, the remainingsplit-rankSS will be considered,and the intertwining operatorcalculatedin the rank-two case[4].

In the generalcase,the multiplicities can be odd, and a pseudodifferential(fractional) operatorisneededto reducema by one. Fractionaloperatorsin one dimensionare knownto havea Riemann—Liouville integralrepresentation,andare applied in section8 to solve the rank-onecase.

Anothercasethatcan be solvedexactlyis whenthereare“doubleroots” 2a in the root system,orwhen there is only one simple root in the Dinkin diagram with odd multiplicity. Looking at theclassificationin ref. [76], p. 532, we see that theseare the symmetricspacesof type Alli, CII, DIII,Eli! and EVIl. On thesespacesthe problemcan be reduced,by meansof a differentialoperator,tothat of a product of rank-onespaces.However, the only casein which the intertwining operatorhasbeenexplicitly calculated,for arbitraryrank, is for the Grassmanianmanifolds SU(p + q)/S(U~x Uq)(typeAll!) [15,5, 35, 36, 4]. In the rank-twocasetheintertwiningoperatorandthe sphericalfunctionshavebeen calculatedin ref. [132], see also refs. [90, 35, 4].

The spaceswherethe solutionwith this approachis mostdifficult, are the SS of normal real form,i.e., thoseof typeI, with ma = 1 for any root [76]. In this caseD is the fractionalpower, of order 1/2,of a partial differentialoperatorin manyvariablesandno explicit knowledgeof its convolutionkernelisavailable,exceptwhen the rank of the spaceis one.

5.5. Thenoncompactcase

As we havealready remarked,the compactand the noncompactsymmetricspacesare relatedbyanalyticcontinuation.Giventhe noncompactSS G’/H, dual to GIH, we havethe Cartandecomposi-tion of the Lie algebra¶~‘of G’

(5.32)


~‘ is a noncompactLie algebra, since the Killing form y(X, Y) = Tr ad(X)ad(Y) (assumingG’semisimple)is negative-definiteon ~ andpositive-definite on At’.

Considerthe new direct sum,

~Q=iAt’E&~C. (5.33)

Clearly, ~ is a compactLie algebra,and can be identifiedwith the Lie algebraof G. In this way, (5.33)

is nothing but the decomposition(5.2) consideredearlier, with At iAt’.The spectrumof the radial Laplacianis now continuousand the sphericalfunctionssatisfy [78, 64]

LRcb~= -(~A~2+ p~2)~, (5.34)

where A is an arbitrary vector in iT’ (a maximal abeliansubspaceof At’), and p is given in (5.23).

The spherical(or Harish-Chandra)transformof a zonal function f is

j(A)~ff(h)~5(h)J2(h)d1h, (5.35)

wherethe integral is over the fundamentalWeyl chamberin iT’, and the densityfunction is

J2(h) = LI (sinh(a . h))’~’ . (5.36)

We can invert (5.35), by using the Planchereltheorem[78, 64]

f(h)=c f J(A)~(h)~H(A)~2d’A, (5.37)

where c is a normalization constant and H( A) is the Harish-Chandrafunction [78]. This is givenexplicitly by the Gindikin—Karpelevicformula [69], in terms of a productover the positive roots,

B(~m ~rn , +iA.a/a)H(A)=J1 .. a 1 s2 2 (5.38)

+ B(~m~,grna2+p~a/a)

whereB is the Eulerbetafunction,B(a, b) = F(a)F(b)/F(a+ b). We shall see,in section10, that theanalyticstructureof H(A)L

2 (e.g., the locationof its polesandthe residuesat thesepoles)is relatedtothe dimensionsof the spherical representations,and to the compactpartition function.

The sphericaltransformK of the heatkernel satisfies

[ia5— (A

2 + p2)]k(A, s) = 0 (5.39)

(where A2 Al2, etc.) and the eigenfunctionexpansionof the heat kernel in the noncompactcasebecomes[63]

K(h, s)= cf &(h)IH( A)l -2 ~ d’A. (5.40)


The relation betweenthe compactand the noncompactsphericalfunctionsis

45(h) = ~~)(5+~)(ih), (5.41)

andvice versa

4~(h) = ~~5_~(—ih). (5.42)

For examplethe sphericalfunctionson thetwo-spherearethe Legendrepolynomials4~(0) = P1(cos0),and, by (5.42), we obtain the sphericalfunctionson the two-dimensionalhyperbolicspaceH

2

q5~(x)= P~5~112(coshx), (5.43)

i.e., the Legendrefunctions, solutionsof

[a~+ coth(x)a~]~= —(A2 + 1/4)~ . (5.44)

The intertwining operatorD hasa nice group-theoreticinterpretation,in the noncompactcase,asthe inverseof the Abel transform ~. (For the definition and a discussionof the propertiesof ~ see,e.g., ref. [64].) The basicpropertyof the Abel transformis that it factorizesthe sphericaltransform,accordingto

(5.45)

where.S7~is the ordinary Fourier transform

(~f)(A)=f f(h) e~d1h. (5.46)

Taking the inversesphericaltransformin (5.45), we have

f=s~iT~J. (5.47)

Applying this to k(A, s)= exp[—is(A2 + p2)], andperforminga Gaussianintegrationgives

K(h, s) = e_~2~_1[(41ris)_hl2e~1z2/4s]. (5.48)

This formula is the noncompactanalogof eqs. (5.27)and (5.28),with the right changeof sign in the p2term (proportionalto thecurvature),andwith ~ 1 in placeof the intertwiningoperatorD. The preciserelation betweenD and zsi1 will be illustrated in sections7 and 8 in the caseof Lie groupsandrank-onesymmetricspaces,respectively.

The problemof calculatingthe heat kernel and the spherical functionsis thereforeequivalenttofinding the inverseAbel transformon the SS. The remarkson the generalpropertiesof D apply to ~as well, and will not be repeatedhere.We only give the relationscorrespondingto (5.29) and (5.30),for the sphericalfunctionsin the noncompactcase,


cf~(h) = b5.~J ~ehfc5~h1 . (5.49)

The constantb5, determinedby comparing(5.40) and (5.48), is given by

b~= clWl(2iT)11H(A)1. (5.50)

where WI is the order of the Weyl group, and c is the constantin (5.40).

5.6. Completeintegrability of the one-dimensionalquantumsystem

The Hamiltonian of the one-dimensional quantum system associated with a given SS is [115, 116]

H= _~a2+~ g~v(a.h). (5.51)

wherea2 = ~ a1a1. a1 a/ah, and h = (h’, h h’) is the 1-dimensionalvectorof the coordinates

of the particles.The sum is over the positive roots and the potentialfunction v is given in eq. (5.1). Inorder to preserveWeyl invariance,the couplingconstantsg~mustbe the samefor roots thatarein thesame Weyl set, i.e., roots that are connectedby transformationsin the Weyl group. The simplestpossiblecase is a systemof I + I particles, interactingpairwise with the potential

U(h) = g ~ v(a(h’ — h’)). (5.52)

wherea is a parameter with dimension length’. This is just the potential term in (5.51), for the rootsystem of type A1 [the same as for the group SU(1 + 1)]. Indeed, in the center-of-masssystem, theconfigurationspaceis the 1-dimensionalhyperplaneh’ + h

2 + . . + h’~= 0, and, given an orthonormalbasis {e

5} in R’~1,the positive roots of the systemA

5 are given by {a(e1 — e1), i <j} [76]. The Weylgroupis simply the permutationgroupof the coordinatesof theparticles,andits order is (1 + 1)!. Sincethe roots are all in the sameWeyl set, therecan only be one couplingconstantg.

The proofof the integrability, in the generalcase,is basedon the following simple relationbetweenthe Hamiltonian (5.51) and the radial LaplacianLR on the symmetricspace[116]:

H= —J[~(LR±p2)]ol/J, (5.53)

wherep is definedin (5.23) andthe density function J2 is given by (5.14) [or (5.36)] in the compact(ornoncompact)case, and by T = [1÷ (a . h)m~ in the zero-curvaturecase. The minus sign in (5.53)correspondsto the compactcase,the plus to the noncompactone, and in the flat casethe p~term issimply omitted. The relation betweenthe couplingconstantsg~and the multiplicities of the roots is

g~= ~rn,, (m21, + ~ m1, — I )a

2, (5.54)

wherea2 is the norm squaredof the root vector a.Next, one considers the radial parts of I algebraically independent Laplace operators,

D1, D2, . . . , D1, with D1 = LR. Since they all commute,it follows that the operators

Ik_J’~k0h/J (5.55)


commuteas well and provide 1 integralsof motion for the one-dimensionalsystemwith Hamiltonian(5.51).Therefore,the systemis completelyintegrableprovidedthecouplingconstantsg~satisfy(5.54).

However,we can imagine that the couplingconstantsassumearbitrary valuescompatiblewith Weylinvariance (i.e., roots with the same length must have the same gj, and conjecturethat theone-dimensionalquantumsystemremainsintegrablefor all such valuesof g~[116].

From the mathematicalpoint of view, we expect that the Laplace operatorson the SS will stillcommuteif the multiplicities are “analytically continued” to arbitrary Weyl-invariant values.We cansay that the completeintegrabilityof thesesystemsis inherent to the root structureof the symmetricspace (i.e. to the Dinkin diagram, or equivalently to the Weyl group), but not to the multiplicityfunction.

The proofof (5.53), e.g., in the compactcase,is the following. We startfrom the radial Laplacianwritten in the form (5.13),andprove that, on anyroot system,the “potential” term V— (a2J/J)canbewritten as

a2rn (m +2rn —2)V(h)=_p2+~ a ~2 2a (5.56)

+ 4sin(a.h)

The following important identitiesareused [116]:

a~Iimarnp[1 + cot(a . h) cot(!3 .h)] = 0, (5.57)

valid for any reducedroot system(i.e., onewith no doubleroots, m2a =0), and

a~J3rnm[1 + cot(a . h) cot(~. h)]a �f3>0

= 4 ~ a2m~m

2,,[1+ cot(a. h) cot(2a. h)], (5.58)aE~

valid for a nonreducedone, where.~ is the setof positive roots a suchthat 2a is a root.Using (5.56) we have

LR—p2= ~ a2~J—2~~ (5.59)

from which (5.53) follows at once.A similar proofworks in the noncompactand in the flat case.It isclear, from (5.53), that the eigenfunctions‘p

5 of the Hamiltonian can be expressedin termsof thesphericalfunctionson the SS as

‘p5 = J~5. (5.60)

The correspondingeigenvaluesare (A + p)2/2, in the compactcase,and A2/2 in the noncompactone.

The propagatorof the one-dimensionalsystem, i.e., the Green’sfunction of (ia~— H), is

K(h, h’, t) = J(h)J(h’) ~ d5çb5(h)çb5(h’) et(5+~S/2, (5.61)


in the compactcase,and

K’(h, h’, t) = cJ(h)J(h’) f ~(h)&(h~H(A)L2 ehfA2/2 d’A, (5.62)

in the noncompactone.

6. The heat equation on a torus and the Poissonsummation formula

We haveseenthat thepropagatorof a free particleon a symmetricspacecanbe written in termsof aflat kernelon the maximal torus of the spacesatisfyingappropriateboundaryconditions.In this section,we considerthe heat (Schrödinger)equationon a torus.

A remarkableidentity, the Poissonsummationformula [17, 28, 134], relatesthe spectrumof theLaplacianto the classicalpaths(geodesics)on the torus. It is usedhereto showthe equivalenceof theeigenfunctionand the “sum over paths” representationsof the heat kernel.

We also considernonperiodic(twisted) boundaryconditionson the torus, The Poissonformulacanbe applied in this caseas well andgives a nontrivial phasefor the “indirect geodesics”in the sumoverpaths.

In 1-dimensionalEuclideanspace,considera lattice F of rank I. This is a discretesubgroupof theabelian group R’ (undervectortranslations)with a set of 1 linearly independentgenerators.To retainthe notationsof the previoussection,we shall denote the generatorsby {2ir&

5}~ and write F as

F={2~~nj&,. njinteger}. (6.1)

A torus T is associatedwith F by taking the quotient

(6.2)

that is, the points h andh + 2irn, with 2irn E F, are identified. T is compactand isomorphicto (S’)’.the topological productof 1 circles. The region

~5={hE~’:h=~h’&,. 0~h’~2~} (6.3)

is mappedby the exponentialmappingExp: ~‘ —* T (which coincideswith the projectionof E~’onto~‘IF) diffeomorphically into T, exceptfor a zero-measureset (the points in the boundaryof V0). The{ h’} arethe canonicalcoordinatesof the point Exp(h)C T, andthe invariant volume elementon T is

dw = ~ dh’ . - - dh’ = [det(&5 ~)J1/2 LI dh’. (6.4)

The volume of the torus is

I / 1/2VT = j dw = (2ir) [det(a . a1)] . (6.5)


A function on T can be regardedas a function on 111 invariant undertranslationsin the lattice F, i.e.,f(h + 2irn) = f(h). It is easyto constructa basis for suchfunctions.

Definethe latticeA, dual to FI2ir, as theset of vectorsA suchthat A n is integer,for any n C F/2rr.Then clearly

A={~nj~rj, n5integer}, (6.6)

where{ITJ is the basis dual to {&~},

(6.7)

Theseare, in general,oblique basesin l~.The covariantandcontravariantcomponentsof the metrictensorin canonicalcoordinatesareg~1= &j, andg” = ir,~~r1respectively,andthe Laplacianon T is

LT = (1/v~)a~(v~g”a1)= 7~ ir1a1a1 . (6.8)

The orthonormalized eigenfunctions and the spectrumof LT are given by {—A2, ‘p~},where A =

~ n1ir, C A, and ‘p~are the complexexponentials

(6.9)

The multiplicity m5 of the eigenvalue— A2 is the number of vectors in the lattice A thathavethe same

norm squared.Except for the zero eigenvalue(for which m0 = 1), m5 is always even, because

—Al2 = Al2.

Therefore,the heat kernel KT(h, s) KT(O, Exp(h), s) hasthe following eigenfunctionexpansionover the latticeA:

1~ . -2

KT(h, s) = ~ exp(iA~h — isA ). (6.10)‘~TAEJI

Alternatively, wecan useageometricapproachto solve the heatequationon the torus.Since T is flat,as a manifold, the function

KdP(h, s) = (4lTis)°2exp(ih2/4s) (6.11)

is a local solutionto the propagatorequation,as in Euclideanspace.The periodic propagatoron T isthenobtainedby summingthis “direct-path” term over the lattice F

KT(h,s) = (4iris)°2 ~ exp[i(h + 2irn)2/4s]. (6.12)2irnEF

The sum in (6.12) is over all classicalpaths (geodesics)y connectingthe given point to the identity, withy(t) = Exp[t(h + 2irn)], tE [0, 1]. Each term in the sum is separatelya solutionof the heatequation,.and (h + 2irn)2 is just the geodesicdistancesquaredalong the multiple geodesiclabelledby n.

The equivalenceof (6.10)and(6.12) can be provedby usingthe Poissonsummationformula.Givendual latticesF andF” in !~‘, let f be a function with Fourier transform


f(y) = J f(x) eXY d’x, (6.13)

and such that the sum ~kEF f(k) converges.Then, accordingto Poissonformula

E f(k)=-~- ~ f(2i~m), (6.14)kEF T mET’

where VT is the volume of the torus l~’!F. More generallyone has

E f(x + k) = ~ f(2~m)e2~tm~1. (6.15)kEF T mEl’

The proofof (6.14) and (6.15) is the following [17].Considerthe function

~ f(x+k). (6.16)kEI’

Being periodic in F, this definesa function on the torus T. Thereforeit can be expandedas a Fourierseriesover the dual lattice

2irimxg(x)= cme . (6.17)mEF’

As usual in Fourier analysis,the coefficientsc~are given by

cm = i f g(x) e2~1mx d’x = ~ f f(x + k) e2~m1d1x. (6.18)T TkEF

T ‘1

If {~,} are the generatorsof F, we can write x = ~xe,, and the integral in (6.18) is over thefundamentalregion 0 < x’ <1. Doing a simple changeof variables,and expandingk= ~ k’e

1, we get

cm = ~ f f(z) e2~1mzd’z = ~ f f(z) e2~mzd1z = f(2~m). (6.19)

k’<z’Kk’+l

Inserting this in (6.17) we get (6.15), and settingx = 0 gives (6.14).We can now apply(6.14)and(6.15) to the dual latticesFin (6.1) andF* = (A/2ir) in (6.6) to obtain

~ f(2~n)-~-~ f(A), (6.20)2irnEF T SEA

~ f(h + 2lTn) = ~ f(A) e~. (6.21)2irnET T SEA

Considernow the function

f(h) = elh2/45. (6.22)

R. Camporesi.Harmonicanalysis andpropagatorson homogeneousspaces 35

(More properly, weshouldconsidere_hS/41,which satisfiesthe conditionsof applicability of the Poissonformula, and t~enWick-rotate t—* is.) Its Fourier transformcan be found by completingthe squaresand using $ e~d1x = (i~i)1’2. This result is

f(y)= (4iris)”~ e”~’. (6.23)

Applying (6.21) to this function we get preciselythe equality (6.10)= (6.12). By takingthe s —~0 limitin (6.10)—(6.12), we obtain the representationsof the (periodic) delta function on T in terms ofeigenfunctionsand of a “sum over paths”:

~T(h)= -~-- ~ e’5’5 = ~ ~(h — 27rn), (6.24)VT SEA 2irnEF

ô1(h — 2i~n)= fl 6(h’ — 2~n’)/[det(&1â~)]”

2. (6.25)

Equation(6.24) is equivalentto the Poissonformula. Indeed, if we integrateover Il” both sides of(6.24) with respectto a function f, we get exactly (6.20). We can also define a multidimensional(Jacobi)thetafunction by [11]

03(z, T) = m~-~exp(2im. z + i~m. Tm), (6.26)

wherethe sum is over all possible l-tuples of integers,and T is a symmetricnonsingularI x 1 matrix.Using the Poissonformula, we see that 03 satisfiesthe fundamentalidentity

03(z, T) (—i)1’2(det T)”2 exp(—iz. T~z/~)t9

3(—T~z,— T1) . (6.27)

Thus, the equivalenceof (6.10) and (6.12) simply reads

03(h/2,—sA/IT)= (4iris)

112 eIh2/4503(Ahlth/2s,~rA~/s), (6.28)

A~1=~5.i~s(A~)11=&5.&1. (6.29)

The zetafunction on a torus, given by a Mellin transformof the heat kernel (section4), can becalculatedin termsof Epsteinzetafunctions[54,55, 47]. Thesearemultidimensionalgeneralizationsofthe Riemann zeta function. They can be analytically continued to meromorphicfunctions in thecomplexplane,with at most a single pole.

Geometrically, this can be understoodby rememberingthat the heat kernelexpansionon a torusterminatesto the first coefficient (just as on Euclideanspace,sincea torusis flat), seeeq. (6.12).Sincethe zetafunction haspolesat z,, = 1/2 — n, with residuesproportionalto theheatkernelcoefficientsa~,the fact that an = 0 for n � 1 implies that the zetafunction on a torus canonly haveonepoleat z = 1/2.

The basic functionalrelation satisfiedby Epsteinzetafunctions(andgiving their analytic continua-tion) can be obtainedby taking the Mellin transformin eqs. (6.10) and (6.12), and equatingthe tworesults.For the physical applicationsconsideredin this paper,we shall only needthe zetafunction inthe rank-onecase,i.e., on a circle, whereit is easilycalculatedin termsof the Riemannzetafunction(seesection11). Furtherdetails, in the higher rank case,can be foundin the original papersby Epstein[54].


Let us now discussthecaseof differentboundaryconditionson thetorus.For theapplicationsto themultiply connectedLie groupsandto the rank-oneSS, consideredin the next two sections,we needaheatkernel on T which satisfies the “twisted” boundaryconditions

KT(h + 2i~n,s) = e2’~”~KT(h,s) , (6.30)

for some vector p. If p C A the phase is one, and we obtain the periodic propagator.Since theeigenfunctionsmust satisfy the same boundaryconditions as the heat kernel, we now need theexponentials

= (1 I~V~)e1(S~. (6.31)

Using (6.28), with h —* h — 2sp, we obtain the following equivalentexpansionsfor the heat kernel

KT(h, s)= ~ exp[i(A + p) . h — is(A + p)2]T SEA

= (41ris)~2E exp[i(h + 27rn)2/4s— 2irip~n]. (6.32)2~nEF

As in the periodic case, the sum over F can be interpretedas a sum of ordinary kernels over themultiple geodesicsof T, weightedby the phaseexp(—2lTip . n). In the next section,we shall see ageometricinterpretationof this factoron the multiply connectedLie groups,whenp is half the sum ofthe positive root vectorsof the group. Let us specializeour formulasto the one-dimensionalcase.Inthiscasep is just a number,andwe shall beinterestedin either integeror half-integervaluesof p, withcorrespondingkernelsK~andK~given by

K~(0,s) = (4~is)1~2 ~ (±1)n exp[i(0 + 2~n)2I4s]. (6.33)

K~ is the periodic propagatoron the circle and K~ the antiperiodicone, i.e., satisfies

KT(0 + 2irn, s)= (—1)~K~(0,s) . (6.34)

The eigenfunctionexpansionsof thesekernelsare

K~(0,s)= ~_ (i + 2 ~ cos(nO)exp(_isn2))

= ~ exp(in0— isn2) = 03(0/2, —sI7r), (6.35)

K~(0,s) = -~ ~ cos[(n + 1/2)0] exp[—is(n + 1/2)2]

= ~ exp[i(n + 1/2)0— is(n + 1/2)2] = 02(0/2, —s/~), (6.36)

where03 and 02 are ordinaryone-dimensionalJacobi theta functions.


7. The groupmanifold case:theequivalenceof theeigenfunctionexpansionandthe sum overclassicalpaths

The quantumdynamicsof a free particleon a Lie grouphasbeenconsideredby Schulman[127],inthe caseof the rotationgroup,and by Eskin and Dowker [56,44, 45], in the generalcase.They haveshown the remarkablepropertythat the WKB approximationto thepropagatoris exact.

In sections2 and5, we arguedthat this propertyof the Lie groupsholds, more generally,on thesymmetricspaceswith evenmultiplicities for the rootvectors (split-rankSS), of which the Lie groupsarea particularcase,for ma = 2, andwhich include,e.g.,theodd spheresand thecosetsSU(2n)/Sp(n)(seesection9).

Nevertheless,the groupmanifolds arethe only spaceswherethe full WKB approximationcoincideswith the Gaussianone (or leading WKB), i.e., the MPSD heat kernelexpansionterminates(up to aphase)to its first coefficient. This fact characterizesthe compactLie groups U, as well as their dualnoncompactpartners,namelythesymmetricspacesG/ U, with G a complexsemisimpleLie groupandU a maximal compactsubgroup.

In this sectionwe review the results obtained in refs. [56,57, 45], using both the intertwiningoperatormethodand thespectrum.The equivalencebetweenthetwo methodswill bedemonstratedforanarbitrarycompactsemisimple(simply connected)Lie groupby using thePoissonformuladevelopedin theprevioussection.This will generalizeto any compactgrouptheproof consideredby Dowker inthe caseof G = SU(n).

We will also considerthemultiply connectedcompactLie groups,anddeterminethephasefactorofthe multiple geodesicsin termsof the vectorsin the unit lattice of the group. We shall seethat thisfactorcoincideswith thephaseassociatedwith theindirect pathsby the index MorsetheoremappliedtotheJacobioperatorin the Gaussianapproximation[128,93]. The sectionwill be endedby somebriefcommenton the noncompactcase.

7.1. The radial Laplacian on Lie groupsand the intertwining operator

What makesit possibleto solve the Schrödingerequationon a compactLie group G is the simplestructureof the radial Laplacian.This is given by (seesection5)

Ln=a2+~cot(a.hI2)a.O (7.1)

=(1/J)a2oJ—(a2J)/J, (7.2)

where a2 = E~ ~ with a. aiah’, is the flat Laplacianon themaximal torus T, h is thepositionvector in the Lie algebra iT of T, andthe sum is over thepositive roots. The functionJ is given in(5.16). Apart from a different normalization,specific of Lie groups, LR coincides with the radialLaplacian on a symmetricspacewith multiplicities ma = 2 for eachroot.

We anticipated,in section5, that on Lie groupsJ is an eigenfunctionof a2 with eigenvalue— RI6,and,in section2, that thesquarerootof theVanVleck—Morettedeterminant ~/2 is an eigenfunctionof LR with eigenvalueR/ 6, where R is the curvaturescalarof the Lie group.The proof of the firststatementmakesuseof the following remarkableidentity (see ref. [78],p. 270):

J(h) 112 sin(a h/2) = wEW (—1)”~~ (7.3)


where the sum is over the Weyl group, andp is half the sum of the positive root vectors

p=~>a. (7.4)

Furthermore,~ givesthe numberof positiveroots andequals(N — 1)12, whereN is the dimensionandI the rank of G. The factor (—1)”~is the determinantof the linear transformationw C W, wheren(w)is the numberof positive roots that aremappedby w into negativeones.

By (7.3) we immediately get

a2J = —p2J, (7.5)

wherewe used the Weyl invarianceof the Killing metric to write (wp)~ (wp) = p p = p2. Therefore,the radial Laplacianon a Lie group becomes[13, 78]

LR= (1IJ)a2oJ+P2~ (7.6)

which coincideswith (5.15), provided that

p2 = R16. (7.7)

This equalitycan beprovedin general,as well as by explicit computationusingthe Cartanclassificationof the classical groups. The generalproof is the following (ref. [78], p. 544). We know that thecurvaturescalar of the Killing—Cartan metric is given by R= N14 (see appendixA). ConsidertheCasimir numberC(A) of an irreducible representationA of G. This is the valuetakenby the operator— ~ T~T~,where { T~}are the antihermitiangeneratorsof A. For the adjoint representationthegeneratorsf are the structureconstantsof group

(f)k = f~, (7.8)

and a simplecalculation,usingthe definition of Killing metric = —Tr ff1, gives the valueonefor the

Casimir number,

C(A=Ad)=1. (7.9)

(Here,weassumedG simple,so that the adjoint representationis irreducible;asemisimplegroupis theproductof simple groups,and a similar argumentcan be usedfor each factor.)

The Casimir number of an arbitrary irreducible representationcan be expressedin termsof thehighestweight vectorA as [62]

C(A)=(A+p)2—p2, (7.10)

and alsoin termsof the weightsp of A [78]

(7.11)

wheremE is the multiplicity of ~.L.


Specializing(7.11) to the adjoint representation,whose weights are just the roots of G, and using(7.9) gives

~N(a.p)22~(a.p)2. (7.12)

Finally, we recall that the Killing metric on iT can be written in termsof the roots as

XY=—TradXadY=2~(aX)(a.Y). (7.13)

Thereforethe right handside of (7.12) is justp2, and(7.7) is proved.Next, considerthe quantity~1/2

on a Lie group (seeappendixA)

~1J2fh\_lJ a 714“ 1’+’ 2sin(a.h/2)

wherethe product is over all positive roots.Using (7.6) and (7.7) we get

= ~ a~(JJa h) + ~R~”2, (7.15)

andwe now showthat the first term in the right handside of this equationvanishes(ref. [78], p. 271).Indeedthe polynomial A(h) [i~ a . h is skewunder the Weyl group, i.e.,

A(wh)= (—1)’~’~’~A(h), (7.16)

ascan beeasilyproved. Sincea2 is W-invariant, it follows that ~2Ais a skewpolynomial in iT, of orderlessthanthe order of A. But according to a well known result (see,e.g.,ref. [135],p. 367), any skewelementin the spaceof polynomialsover iT is divisible by A, i.e., hasthe form

Pskew = ~ (7.17)

wherep~is a polynomial invariant underthe Weyl group. It follows that~2Amustvanish identically,and that

= (R/6)1.1U2, (7.18)

as claimedearlier.We are now ready to illustrate how the intertwining operatormethodworks. Given the radial

Laplacian(7.6), it can immediately be shownthat the operator

D=(k/J)fla8, (7.19)

wherek is a normalizationconstantanda 0 the derivativein the direction of a, satisfies

LRD = D(a2 + p2), (7.20)


i.e., intertwinesthe radial Laplacianto theflat one.Actually any operatorof the form (1IJ)a(w),wherea(~)is a constant-coefficientspartial differentialoperatoron iT, satisfies(7.20). HoweverD is the onlyoperatorthat relatesthe invariant deltafunction on the torus, 6Th to the ordinaryone, ~T’ This can beseenby eitherusing the spectrum(seelater), or directly from the definition of ~R

f ôR(h)f(’~WG/Tj(’~) d’h =f(0), (7.21)

wherethe integral is over a fundamentaldomainof W in iT (a Weyl alcove), andVG/T is the volume ofthe spaceGIT, orthogonalto the maximal torus (ref. [78], p. 192).

If we substitute the ansatz

= D8T (7.22)

in the left-handside of (7.21), and integrateby parts p. times, we get

(—1)~’kVG/TIIWlJ~.(iia a)(fJ)d’h, (7.23)

wherethe integrationhasbeenextendedto the whole torus, and WI is the order of the Weyl group.Doing the integral over 6T’ we see that we only get a nonzeroresult wheneachderivative actson adifferentsinefactor in J, andnoneactson f. Thus,the result is in fact proportionalto f(0), and (7.21)gives the value of the normalizationconstantk,

k = [(_lyvG/T 11 (a . p)] , (7.24)

wherewe used the following result (ref. [78], p. 329):

(fJa.OJ)(fJa.a)(flh)=IWIfla.p. (7.25)

The sameproof showsthat any other intertwiningoperatorwhich is not proportionalto D doesnotsatisfy (7.22). Indeed,given a Weyl invariant operatorof the form D’ = (1 IJ)a(w), sinceJ(h) is skew[just like A(h)], a(w) must be skewas well, andit follows from (7.17) that a(w) = ‘~flVfl~a 0, where~ is a Weyl invariant, constant-coefficientsdifferential operator.

However, doingthe integralover ‘~Tin (7.23),with the replacementfI~a ~ H~a 0, we getterms proportional to the derivativesof f at 0 (for example when ~ acts on f), as well as termsproportionalto f(0). Therefore,C,,~must be a constant,i.e., D’ must be proportional to D.

We concludethat the operatorD that satisfieseqs. (7.20) and (7.22) is unique. Using (7.20) and(7.22) we obtainthe heatkernel on G,

1

KG= —1 EVe H . Jfla.0KT~ (7.26)( ) G/T ~(a p) +

whereKT is a flat propagatoron the maximaltorus,consideredin section6. If G is simply connectedKT


is the ordinary(periodic)propagatoron T, given by eqs.(6.10) and(6.12).We will showlater that, if Gis multiply connected,KT is given by eq. (6.32).

7.2. Thespectrumand the “sum over classicalpaths”

Let usnowprove the result (7.26)usingthe spectrum.Considerfirst the caseof G simplyconnected.The eigenfunctionexpansionof KG hasbeengiven in (4.33) in termsof the charactersof the irreduciblerepresentations.Since the heat kernel is a class function (i.e., is invariant under conjugation),oneendpointin the propagatorcan be rotatedinto the maximal torus T, while the otheris held fixed at theidentity, and we get

KG(h,s)=KG(e,eh,s)=~ ~ dsXs(eh)e_15c(S), (7.27)

whereVG is the volume of the group. We nowrecall Weyl dimensionandcharacterformulas(see,e.g.,ref. [78])

(7.28)+ ap

h ~w (~iy’~”~exp[iw(A + p) . h]~5(e ) = ~ (—1y’~ exp(iwp.h) (7.29)

(7.30)

where we used (7.3) and (7.16). We see that (7.30) is nothing but (5.29), namely the intertwiningoperator(7.19) relatesthe sphericalfunctionson the group to theelementary functions on the torus. Itis clear, from (7.28), that the constantin (7.30) is proportional to lid5, in agreementwith (5.30).

We can now substitute(7.10), (7.28) and (7.30) in (7.27).The dimensionfactor simplifies and weget

Isp2

KG(h,s) = —1 EV H ~ LI a~0 ~ exp[iw(A + p) h — is(A + p)2], (7.31)( ) G ~a P + AEA÷wEW

where A~is the setof the dominant (highest)weights. It is well known (see,e.g.,ref. [78]) that if G issimply connecteda dominant weight A is defined by the integrality condition that A . 2a/a2 is anon-negativeinteger, for any positive root a. Thus

A÷= n~tI5, n~non-negativeinteger}, (7.32)

and similarly the lattice A of all weights,obtainedby applying the transformations in Wto A~,is givenby

A = WA~= n~co~,n~integer}. (7.33)


Here {w,} are the fundamentalweights, definedby

= 8~ (7.34)

where = 2a1/a~and {a1.} are the simple roots [78]. The sum in (7.31) can now be shifted byremoving the elementp. Indeed, from the following alternativeexpressionof p in terms of thefundamentalweights [76]

we see thatp is a dominantweight, andthat the dominantweights that are not of the form A + p arejust the weights in the walls of the fundamentalWeyl chamberiTt, definedby the conditiona . h > 0for anypositiveroot a. The fundamentalweightsform the edgesof this chamber,andthe walls arethehyperplanesa, . h = 0. Then the weights on the walls satisfy

JJa~A0, (7.36)

and do not contributewhen we shift the sum in (7.31). Finally, becauseof (7.33), we can changethesum over the dominantweights and the Weyl group into the sumover all weights, andwe get

KG(h,s)= — e”° I] a~O~ exp(iA.h—isA2)

( ) ~ + a p SEA

VTe°~ 1 ~ exp[i(h + 2lTn)214s]= _____________— aO L~ . I~2 (7.37)

(—1) VGH+ a~p‘ 2irnEI’ (4iris)

wherewe used the Poissonsummationformula [seeeqs. (6.10) and (6.12)]. In eq. (7.37) FI2rr is thedual lattice to A and gives the periodicity of the closedgeodesicson G. For G simply connected,F,which is the kernel of the exponentialmapping, is given by the integer linear combinationsof thevectors &~,

F={2~~n1a~j,n1integer}, (7.38)

and is called the unit lattice of G [76]. Comparing (7.37) to (7.26), we see that they are the sameprovided the volumessatisfy

Vc~VTVc;T . (7.39)

Therefore, we have obtained the same result by either using the spectrum or the intertwiningoperator.It is evidentthat the latter leadsto the result in a somewhatsimpler way. It is alsoclear thatno otherintertwiningoperator,besidesD, relatesthe heatkernelsandthe deltafunctionson GandonT. For examplethe operator(1IJ)H~(a .0)~satisfies (7.20), and relatesthe characterson G to theWeyl invariant combinationsof exponentialson the torus, but with a constantproportional to l/d~.Therefore,the dimensionfactor in (7.27) will not simplify (leaving a 1Id~),andthe given operatordoesnot relate the Green’sfunctions.

The volumes in (7.39) can be calculatedin closed form. For the torus we alreadyknow, from the


discussionin section6, that

VT = (2ir)1[det(&~ _)]1/2 (7.40)

The volume of the compactgroup G is given by [98, 94, 58]

V = VT(2~-)’~’~2= ~ ~)]1/2 41G H~ap H~a”p

from which we get the volume of the spaceG/ T

VG/T = (2w) 1~2/fl a p. (7.42)

Using thesevalues for the volumes,alongwith (7.28) for the dimensions,andeqs. (7.19)—(7.24)for theintertwiningoperator,we seethat formula (7.30) for the characterscoincideswith the generalformulas(5.29) and (5.30) for the sphericalfunctionson symmetricspaces,in the caseGIH = G.

In order to explicitly calculatethe heatkernel in (7.37), we have to apply the differentialoperatorll~a ~Oto the exponentialunderthe sum. The following result is needed[63]:

[I a ~Oe152/4s= LI (~—a h) e~2/4s. (7.43)

Indeed the left-hand side of this equation equals exp(ih2/4s) times a skew polynomial of orderp. = (N—1)12, which thereforemust be proportionalto A(h) = H~ a h, see eq. (7.17).

We finally get the heat kernel on a simply connected(compact) Lie group as a sum over classicalpaths (geodesics)

KG(h,s) = e ~ fl (a .(~ ~)exp[i(h + 2~n)2I4s]. (7.44)(4iris) 2,rnEF + 2 slnka

Comparing(7.44) to (5.8),we see that the Gaussian(or leadingWKB) approximationis exact on Liegroups.The direct-pathn = 0 term in (7.44) coincideswith the Schwinger—DeWittresult (2.19).Thesum over F producesthe requiredperiodicity of K undertranslationsin the lattice F. Indeed,using(7.3), (7.38) andthe fact that p is a weight, it is easyto show that the function J in the denominatorof(7.44) is invariant under the replacementh —~ h + 2lTn. This is preciselywherethe differencebetweenthe simply andmultiply connectedcasescomesin.

7.3. Multiply connectedLie groupsandthe phaseof the indirect geodesics

Let G be a compactmultiply connectedLie group. Then G GI~1(G),whereG is the universal

covering group, and i~(G) is the fundamentalgroup of G. It is knownthat ii~(G) is a finite groupcontainedin the centerZ of G. The pathsbetweenanytwo pointsfall into distinct homotopyclasses,which can contributeto the “sum overpaths”with different relativephasefactors.It was shownin refs.[91, 46] that these phase factors belong to a one-dimensionalunitary representationof 1T1(G).

Dependingon which representationis chosen,we obtain propagatorssatisfying different boundaryconditions.Unlike the simply connectedcase,the periodicpropagatoron a multiply connectedgroupisnot necessarilydefinedby the trivial representationof 1T1(G).


For example,on SO(3) SU(2)/{1, —1} we havetwo homotopyclasses,which correspondton evenor odd in eq. (4.35). Theyenterthe sum in KSQ(3) with oppositesigns, due to the (—1)~factor,whichcorrespondsto the nontrivial representationof {1, —1). Notice that this phasefactor (—1)~,associatedwith the multiple geodesiclabelled by n, can be determinedby looking at the transformationof thefunction J= 2 sin(0/2) under a lattice translation: if 0—~0 + 2ii-n then J—* (—1)~J.For SU(2) thetranslationsin the unit lattice are 0—~0 + 4i~-n,andJ is invariant.

Let us nowgeneralizethis to an arbitrarygroup. Let FandFbethe unit latticesof GandG, i.e., thekernelsof the correspondingexponentialmappingsrestrictedto iT. F is given by eq. (7.38) in termsofthe roots and F is normally largerthan F. It can be shown (see,e.g., ref. [137], p. 95) that

EcFcF1, (7.45)

whereF1 is the lattice of vectorsh suchthat exp(h) is in the centerZ of G, and is given explicitly by

I~= {h C iT: a . h = 2irn, n integer, for eachroot a} . (7.46)

We also have the identifications

2~F1lf, (7.47)

ir1(G)~-~FIF. (7.48)

Given a lattice F that satisfies(7.45), thereis a connectedLie groupGwith the sameLie algebra~ asG, andunit lattice F. Vice versa,a!! Lie groupswith Lie algebra~ areobtainedby selectingall possiblesublatticesof Tj. The smallest is F, with trivial homotopygroup, and centergiven by eq. (7.47). Thelargest is [ itself, with ir1(G) = Z, and no center. (The correspondinggroupis known as the adjointgroup, see ref. [43].) The numberof nonisomorphicconnectedLie groupswith Lie algebra¶~can befound, e.g., in ref. [76], p. 525, for both the classicaland the exceptionalLie algebras.

Now from the definition of J [seeeq. (7.3)] we have

J(h) = i~ ePhfl (1 — e1ah). (7.49)

If we let h —~h + 2lTn, with 2irn C F, the terms in the product are invariant, because F c T.

Therefore,under a unit lattice translationthe f-function transformsas

J-~e2~”J, (7.50)

with a phasefactor which is generallydifferent from one, except in the simply connectedcase.Using

(7.50), it is easyto show that the (periodic) heatkernel on a multiply connectedLie group isIsp -

KG(h, s) = (4iris)~2 2,rnEF± ~ exp[i(h + 2irn)214sj (7.51)

= e~°2 ~ ~ (a ~(h+ 2~n))exp[i(h + 2~n)2/4s— 2~ip. n], (7.52)

(4lTls) 2,rnEF + 2 sin(a . h12)

wherethe sum is over the unit lattice of the group.


The phasefactorexp(—2irip n) hasa simple geometricalinterpretation.For 2~nC F, considerthemultiple closedgeodesicy(t) = exp(2irtn),with t C [0, 1]. A point alongy is saidto be conjugateto theidentity if the VVM determinantLI is infinite at that point (or equivalently if the measurei.I1 inRiemann normal coordinatesvanishes). Remembering(7.14), we see that the total number ofconjugatepoints along y equals the number of intersectionsof the vector 2irn with the singularhyperplanesdefinedby the roots a

{h C iT: a h = 2lTn, n nonzerointeger}. (7.53)

It is not difficult to see that this numberis given by

M~=2~la~nl, (7.54)

the sum being over the positive roots. Indeed,by (7.45) the number ‘~a a • n is an integerandthevectors

2lTflIlflaI, 2~2irn/lnaI,. . . , (~n~— 1)2lTflilflaI , 2i~n, (7.55)

give na I conjugatepoints along y when exponentiated,eachwith multiplicity equalto 2, dueto thesymmetricspacestructureof G. Summingoverall positiveroots gives (7.54). Onan arbitrary compactsymmetricspaceG1Hwe would get

My=2~mala•nl. (7.56)

It is now a standardresult (see, e.g., ref. [128]) that the phaseof the Gaussianapproximationassociatedwith the multiple geodesicy is justexp(—i~M

7/2),i.e., the Gaussianpropagatorpicks up aphaseexp(—iir/2) at eachconjugatepoint. This result can be provedby applying the index Morsetheoremto the Jacobi operatoron y. Using (7.54), we get for the Gaussianphase

e~M~’2= LI (—1)~=11 (—1)~= exp(—ilT ~ a’ ~), (7.57)

which coincidespreciselywith the phaseexp(—2irip . n) in (7.52) [seeeq. (7.4)]. This resultwas to beexpectedbecausethe Gaussianapproximationon Lie groupsis exact. We alsonotice that sinceM~in(7.54) is clearlyWeyl invariant,it follows from (7.57)that exp(—2irip.n) is Weyl invariant as well, aresult that is not obviousa priori, but is requiredin order for (7.52) to yield a Weyl invariant kernel.

7.4. Thenoncompactsymmetricspace GIU

The noncompactsymmetricspacesdual to the compactsimple Lie groupsU (with Lie algebraOil) arethe spacesGIU, whereG is a complexLie groupwith Lie algebra~ = iOu ~ 0.1. U is then a maximalcompactsubgroupof G, and 0/is calleda compactrealform of ~.The classificationof thesespacescanbe found, e.g.,in ref. [76],p. 516.The radial Laplacianon GlUis obtainedfrom (7.6) by letting h—i.ihandchangingsign. The result is thereplacementof the trigonometricfunctionswith the hyperboliconesin the f-function, and a changeof sign in the p2 term. Since the curvaturechangessign too, we haveLR = (1IJ)a2of + R16, as in the compactcase.


The analytically continuedintertwiningoperatorcan be identifiedwith the inverseAbel transformonthe symmetricspaceGIU [5].The spherical functionsare obtainedby using eq. (5.42) in (7.29)

,~(w) iwAh

~(h) = 1~(w) :WPh’ (7.58)

and the heatkernel is obtainedfrom (7.44) by keepingonly the direct-pathterm,replacingh —* ih ands—~—s, and multiplying by (l)N/2

_~lSp / •e a ih2/4s

K~1~(h,s) = (4~is)~

2~“ 2 sinh(a h12))e . (7.59)

8. The finite propagator on spheresand rank-one symmetric spaces: fractional derivatives

In this section we construct the heat kernel on the rank-one symmetric spaces(SS) using theintertwiningoperatormethod (seesection5). This methodhasbeenrecentlyapplied to manyrank-twoandsomehigher rank SS as well [35,4, 5]. The result for the noncompactrank-oneSS was originallyderivedin ref. [92], by usingthe inverseAbel transform.The heatkernel in the compactcasehasbeencalculatedin ref. [4]. We beginwith the most familiar rank-onesymmetricspace,the N-sphere.

8.1. Thespherecase

Let G/H = SO(N + 1)/SO(N)~S1”. As usual,we fix oneendpointin the propagatorat the origin ofthe sphere,the north pole. Since the rank is one, a maximal torus T is just a greatcircle throughthenorth pole,and the propagatorcan only be a function of the distancefrom the northpole alongT. Forsimplicity, we normalize the radius of the sphere to one, and call 0 the geodesicdistanceon a greatcircle. The radial Laplacian is given by (seesection5)

LN=a~+(N—1)cot0a6, (8.1)

wherea6 alao, andthe factorN — 1 is just themultiplicity m0 of the single positiveroot a on S1”. The

orbits of SO(N) are (N — 1)-spheresorthogonalto T. The densityfunction is f2(0) = (sin 0)”’”, andtheradial invariant deltafunction can be definedas

J~N0f9)QN~lsin 9)v~ldO =f(0). (8.2)

where

~N-l =2ir1”2/F(N12) (8.3)

is the (N — 1)-volume of SN_I. It is easyto verify that the operator

D = kala cos 0 = (—k/sin 0)a~, (8.4)


wherek is a constant,satisfies

LND= D(LN_2 + N— 2), (8.5)

i.e., intertwinesthe radial Laplacianon the N-sphereto theradial Laplacianon the (N — 2)-sphereplus

a constant.In order to iterate(8.5),we define the elementp (one-halfthe sum of the positiveroots),

pN_(N1)12ma12 (8.6)

(a is just a numberhere,normalizedto one) andnotice that

2 2

N—2=pN—pN_2. (8.7)For N odd, after applying (8.5) (N — 1)/2 times, we get

L (N—1)/2 — (N—1)I2j~2 + 2N — ~0 PNJ’

wherea~= L1 is the ordinaryLaplacianon the circle andp~= 0 since S’ is flat.Similarly for N evenwe can stepdown to S

2 andwe get

LND~’~2~12= D~2~2(L2+ — p~), (8.9)

where L2 = a~+ cot 0 a0 is the radial Laplacian on the two-sphere, and p~= 1/4. Using now thedefinition (8.2) it is easyto verify, doing an integrationby parts,that D relatesthe deltafunctionsonSN and SN_

2 provided that k= l/27r, i.e.,

~N(0) = 2ir sin 0 aO~N2(e). (8.10)

Therefore,D also relatesthe heatkernelsKN of (ia~+ LN) and KN_2 of (ia5 + LN_2), andwe gettheresult

—e1~~“ ~ P~/—2)

KN(O, s) = 2ir sin 0 aOKN_2(6,s). (8.11)

Iterating this relationwe obtain [3]

1 a (N1)12KN(0, s)= ei5~_1 2]2(_ a cos K1(0, s), N odd, (8.12)

1 a (N2)/2K~(O,s) =e~(~— a cos K2(0, s), N even. (8.13)

This basicallysolvesthe problemof finding K on S1”, sincethe heatkernelsK

1 on the circle andK2 onthe two-spherehavebeencalculatedalreadyin (6.33) (wherewe needK~)and (4.38), respectively.

However, the ideathatan intertwiningoperatorcan fully reducethe problemto the maximal torusof the spacehasbeenso far implementedfor odd spheresonly. Forevenspheresweneedafurtherstep


to relateK2 to K1. It is at this point that the conceptof fractionalderivativecomesin. The aboveresultscan be interpretedby sayingthat the operatorD reducesthe dimension,andhencethe multiplicity m~,by two units andsince D is independentof the multiplicity, the nth powerof D changesm~by 2n. It isreasonableto expect that the fractional power of D of order 1/2 changesm~by 1, and that D’~

2intertwinesL-, to L

1 and relatesK2 to K1.But how do we define the fractionalpowerof a (one-dimensional)differentialoperator?Looking at

(4.38), we see that K. and K1 are relatedby an integraloperator,and we expectthat D~2will be a

pseudo-differentialoperator.Fractionalderivatives were first investigatedby Riemann,Liouville andWeyl, andwere furtherdevelopedby Riesz[121], whoconsideredthefractionalpowerof the Laplacianandof thewave operatorin pseudo-Euclideanspace.We shall reviewbelow the definition andthe mainpropertiesof fractional derivativesin one dimension.

8.2. Fractional derivativesin one dimension:the Riemann—Liouvilleintegral

Following Oldham andSpanier[114], we define the fractionalderivativea~ (a/ax)” by an integralrepresentationknown as the Riemann—Liouvilleintegral. Given a real numbern <0, define

a~~f(x)us F(1) ~ (8.14)

wherea < x is a fixed number,referredto as the boundarypoint. Since a<0, the integral in (8.14) isconvergent,provided that f is well behaved.For a � 0 we define

ax”_a—=a~a~i,,” , (8.15)

where n is a positive integer chosen large enough so that a — n <0. It is not difficult to showthat thisdefinition doesnot dependon n.

For example, for a =0 and ct = 1/2 we can choose n = 1, and we get

a~f(x) = ar f f(x’) dx’ = f(x), (8.16)

a112 —a ~ ff(x’)dx’ 817x-af(~ x\/~j Vx-x’ (. )

If at x = a all the derivativesf~(a)exist, up to ordern, we can integrateby partsn times in (8.15), andthen apply a. The result is

n—I — \k—a~(k)f~ ~x a) j ~aa~~f(x)=a~~a

1f(x)+ ~ ru — L1\ (8.18)k~’O ‘ ~ a

which is the definition given by Riesz for a>0. For example,for a = 1/2 eq. (8.17) is equivalentto

a~2~f(x) =a~~2a~f(x)+ (x — a)~2f(a)/’../~ (8.19)


— 1 ff’(x’)dx’ f(a) 820— c~J Vx — ~‘ + vY~~ ( . )

provided that f(a) exists. We can also define a fractional derivativewith respectto a function g. Fora = 1/2 this is

1/2 -1/2 f f’(x’)dx’ f(a)ag(x)g(a) f(x) = ~ J ~g(x) — g(x’) + ~~g(x) — g(a)~ (8.21)

1ff doesnot haveenoughderivativesat the boundarypoint a, eq. (8.18) hasno meaningandwe willretain the definition (8.15), which is properly referred to as a differo-integral representationforfractional derivatives. It is easyto see that, for a = n, a positive integer, a1” reducesto the ordinaryn-fold derivative. For a = —n, a negative integer,we get from (8.14) Cauchy’sformula for n-foldintegration,

a;~~f(x)= ~ J (x - x’)~f(x’) dx’ (8.22)

=JdXni J~jf(xo)dxo. (8.23)

Therefore,fractional derivatives of negativeintegerorder coincide with repeatedintegrals,and fora<0 a~”is also called afractional integral.

A very importantpropertyof a~” is the semigroupproperty,

a~aa~_af=~ (8.24)

which holds (with suitableassumptionson f) for any a and f3, exceptwhen /3 is a positive integer.Indeed,from (8.15)and (8.18)we seethat the operatorsa”a

5 anda5a” do not commuteand differ by afinite numberof boundaryterms,

n—i k—n—A (k)

a~aa~af=a~f=a~a~f+ ~o ~- n - A + (8.25)

This formula holds for any A. Another important result is the fractional derivative of the function(x—a)~

— a)~= [F(/3+ 1)1F(13 — a + 1)](x — a)’~” , (8.26)

which is valid for any a and for /3>—l (seeref. [114],p. 67): For a = n, a positive(negative)integer,this formula reducesto the well knownrelationsfor n-fold differentiation(integration)of powers.

Finally, we needthe generalizationof the Leibniz rule to fractional derivatives,

a~~(fg)= ~ (~)(ai~f)(a~ag), (8.27)j=O I


which holds for both f andg analytic, or for f equalto a polynomial andg arbitrary (in which casethesum on the right-handside is finite). Although not apparentfrom (8.27), it can be shown that thisformula is symmetricin ~ g.

We can now apply this formalismto the problemof relatingthe heatkernelon the two-sphereto theheat kernel on the circle.

N. .8.3. Theheat kernelon S in terms offractional derivatives

Considerthe operatora~= L1 andthe radial LaplacianF, on the two-sphereexpressedin termsofthe algebraiccoordinatex = cos 0,

L1(x)= (1 — x2)a~— xa

5 , (8.28)

L2(x) = (1 — x2)a~— 2xa~. (8.29)

The idea is that the operatora~2intertwinesL2 and L1. More precisely, taking the boundarypoint at

O = IT, i.e., at x = —1, and using (8.25), (8.26) and (8.27), it is straightforwardto show that theoperatora~~1satisfiesthe following relation:

L2(x)a~2

1f(x) = a~2

1[L1(x)+ flf(x) - (x + 1)~~F(-1/2)’ (8.30)

or, returning to the 0 variable,

(a~+cotoa0)a 6~1f(0)=a~0+1(a~+ ~)f(0)- (1+cosO)~2F(-1/2) (8.31)

Fromtheseformulaswe seethat our guessis almostright, namelytheoperatorD1/2 intertwinesL2 and

L1, but only whenappliedto functionsthat vanishat 0 = IT, becauseof the boundarytermin eq. (8.31).If we now use the ansatz

K2 = k e’4a~.~

0~1k1

for the Green’s function to (ia, + L2) and use (8.31) we see that k1(IT, s) = 0, i.e., k1 must satisfyantiperiodicboundaryconditionson the circle. We concludethat K1 = KT, given in eq. (6.33), andweget

K2(0, s)= e5~(2ir)h/2a~o+iK~(O,s). (8.32)

Thus, the heat kernel on the two-sphereis given by a fractional operatorapplied to the antiperiodicpropagatoron the circle. Using the integral representation(8.20), we see that the boundarytermvanishes,andwe obtain our earlier result (4.38).

We can checkour resultusingthespectrum.The eigenfunctionexpansionof K2 in termsof Legendrepolynomialshasbeengiven in eq. (4.39). We can write down the Mehler—Dirichletintegralrepresenta-tion of P1(cos0) as a fractional derivativeof order 1/2


P1(cos0) = ~ [ciq~ (cosO~~cos~)h/2 (8.33)

=4(2IT)_V2a~

6+1[cos(l+112)0]/(2l+ 1). (8.34)

This is nothing but the general relation (5.29), i.e., the intertwining operatorrelatesthe sphericalfunctionson the symmetricspaceto elementaryfunctionson the torus.If we now substituteeq. (8.34)in eq. (4.39), and useeq. (6.36) to identify the sum as a Jacobi theta function 02, we get exactlyeq.(8.32). Theinvariant delta-functionon the two-sphereis relatedto the antiperiodiconeon thecircle byour fractionaloperator

552(0) = (2ITy~’2a~

0+,~ (—lyS(o — 2ITn) (8.35)

= (2ITyU2a~

0+,~ ~ cos(n+ 1/2)0. (8.36)IT n’O

Notice that if we did not knowthe spectrumon S2, we could haveobtainedboththe eigenfunctionsand

the eigenvaluesof the Laplacianby (8.31). In the general rank-onecase,we shall derive, usingthisapproach,fractional representationsof the sphericalfunctions,the Jacobi polynomials.

The stepfrom S2 to S” is now easy.Doing acalculationsimilar to the onethat led usto eq. (8.31),we get the following intertwining relationbetweenthe radial Laplacianon S” andon S1:

LNa~O~2f(0)= a~01~2(a~+ (N_ l)

2)f(e) - (1 + cos0)~2F((1- N)I2)~ (8.37)

For N odd the boundaryterm vanishes,becausethe F function in the denominatorbecomesinfinite,and we obtain (8.8). For N eventhe boundaryterm is eliminatedby chosingantiperiodicboundaryconditionson the maximal torus. We concludethat the heat kernel on S1” is given by

1 a (N1)12KsN(0,s)=eI5I2]2(~ a(coso+1)) K~(0,s), (8.38)

with K~for N odd, and K~for N even. The intertwining operatoris differential for N odd andpseudodifferential(fractional) for N even.

8.4. Thegeneral rank-onecase

The rank-onesymmetricspaceshavetwo positiveroots, /3 with multiplicity m~,andthehalf root /3/2with multiplicity m

4~2. In table 8.1 we give a complete list of the compact spaces,including themultiplicities of the roots.

The dimensionof the spaceis relatedto the multiplicities by

N~dimGIH= mp + m412 + 1.


Table 8.1Compact rank-onesymmetricspaces(n � 2)

G/H m5

S’ =SO(n+ 1)ISO(n) n — I 0P~(R)= 50(n+ 1)10(n) n — 1 0P

2~(C)= 5U(n + l)/U(n) 1 2(n — 1)

P4~(H)=Sp(n+l)/Sp(n)XSp(l) 3 4(n—l)P”‘(Cay) = F

4/SO(9) 7 8

The spacesare all simply connected,except for the real projective spacesP~(R),which will beconsideredat the end. The measuredensityand the radial Laplacianare given by

J2(O) = (sin O)”’~(sino/2)”’~~, (8.39)

LR = (IT/L)2~ + (m~cot 0 + ~m~/2cot O/2)a6], (8.40)

where0 = irriL is the geodesicdistancer in the maximal torus (a circle throughthe origin) in units ofLIir, and L is the diameterof the space,i.e., the maximum distancebetweenany two points. Themagnitudeof the root ~3is related to L by /3 = iriL, and one-halfthe sum of the positive roots is

p = ~(m~+ ~m~12)/3= ~(N+ m~— 1)j3. (8.41)

Hereafterwe shall normalizeL = ir, i.e., /3 = 1. Fromtable8.1 we seethat p is integerfor oddspheresand for the complex projectivespacesP

4(C), P8(C),. . . , and is half-integerotherwise.This will beimportant in determining the boundaryconditions on the torus.The invariant delta-functionon thetorus is definedby

f SN(0)f(0)VJ(0)dO =f(0), (8.42)

where

= 2”’/3/2Q (8.43)

is the volume factor from the orbits of H, orthogonalto the torus.

Proceedingas for5N we find that the operator

D = ka”’~~2amp,22 (8 44

cosO+1 cos0/2 ‘.

satisfies

LRDf= D(a~+ (1 ~~~)~mO12 (a~f)I5~. (8.45)

For evenm~the boundaryterm vanishesand we obtain the odd-spheresresult. For the remaining


spacesm~is odd, m~~2is even, and the boundaryt~rmis eliminated by an appropriatechoiceofboundaryconditionson the torus.Looking for K = e~PDKTas the Green’sfunction to (ia~+ LR), weseethat the propagatorKT on the circle must satisfy the boundarycondition

a~’~KT~= 0. (8.46)

It is not difficult to show that this is satisfied by KT = K~for ~mp12odd (‘~p is integer), and byKT = K~for ~m4~2even(‘~‘p is half-integer),whereK~aregiven in eq. (6.33).We concludethat theheat kernelon the simply connected,compact,rank-onesymmetricspacesis [4]

~ 1 a mp/2 1 a m612/2

K(0, s) = e’~~(~a(cos0+ 1)) (~a cos0/2) KT(O,s), (8.47)

wherethe kernel

KT(O, s) = (4ITis)”2 ~ exp[i(0 + 2irn)2/4s— 2iripn] (8.48)

coincideswith K~for p integer[odd spheresandP4(C),P8(C), . . . ,], andwith K~for p half-integer.The normalizationconstantin (8.47) has been determined,as usual,by requiring that D relatesthedelta function on GIH to the delta function on the circle.

It is remarkablethat the phase factor exp(—2iripn) in KT coincideswith the Gaussianphaseassociatedwith the multiple closedgeodesicy(t) = Exp(2ITnt) (asin the caseof Lie groups,seesection7). Indeed,the total numberof conjugatepoints along y, including the multiplicity, is

= n!(2m,3 + m412)= 4~n~p, (8.49)

and the Gaussianphaseof y,

e_I~~MY2= e_25TiPh1~, (8.50)

coincideswith exp(—2ITipn), for both p integerand half-integer.For the Lie groupsthis was easytounderstand,becausethe Gaussianapproximation is exact there. On the rank-one SS, where theGaussianapproximation is not exact,exceptfor S3 and P3(R), we do not havea simple reasontoexplain this fact. It is temptingto conjecturethatthis generalizesto the higher rank symmetricspaces,i.e., that the phaseof the indirect geodesicsin the exact representationof the propagatoras a “sumover paths” always coincideswith the Gaussianphase.In the next section,this conjecturewill beproved for the symmetricspacesof split-rank type, whereasthe proof in the generalcase is not yetavailable.

The structureof the intertwining operatorD is determinedby the multiplicities of the roots of thesymmetric space.The only casein which D is purely differential is when m

4 is even, i.e., for oddspheres.OtherwiseD containsa fractionalderivative

a”’~2 = a(m0_I~2ah/2 ~851

cosO+1 cosO cos0+1 ‘ ‘.

wherethe power 1/2 is definedin eq. (8.20). As odd spheresaresplit-ranksymmetricspaces,like Lie

54 R. Camporesi,Harmonicanalysis andpropagatorson homogeneousspaces

groups,we can arguethat the WKB approximationto the free particlepropagatoris exacton all thesymmetricspaceswith evenmultiplicities for the root vectors.Indeed,when the differential intertwin-ing operatoris applied to the flat kernelon the maximal torus,it generatesa kernelwith the structureof an MPSDexpansion(seesection2). For examplefrom (8.38)we can derive the following expressionfor the heat kernelon the odd spheres

52m+1:

iS/fl +~ I \ rne ~ (O+2ITnK2m+i(O,5)= . (2m+1)/2 L

(4ITIs) ,,=—~ sin0

rn-I

x exp[i(0 + 2ITn)2/4s] (i + ~ ak(O + 2ITn)(is)k), (8.52,

whereak(0) are the heat kernel coefficientsof the operatorLR — p2 and can be explicitly calculatedfrom (8.38). Eachterm in the sumover n is a Minakshisundaramexpansionterminatingatami, e.g.,at a

0 on S3, a

1 on ~5, and so on. In the next sectionwe shall derive the intertwiningoperatorandtheheat kernelon the split-rankSS otherthanLie groupsand odd spheres.

8.5. Fractional representationof the facobipolynomialsand the spectrum

It is importantto note,at this point, that the intertwiningoperatormethodalso works for valuesofthe multiplicities m~that are not compatiblewith anysymmetricspace.For example,the intertwiningrelation(8.45) holdsfor arbitrary oddvaluesof mp andevenvaluesof mp12. The intertwiningoperatorrelatesthe eigenfunctionsof LR to elementaryfunctionson the circle in this moregeneralcaseas well.Identifying LR with the differentialoperatorof the JacobipolynomialsP~,”

t’~(cos0), with a = (N — 2)/2and b = (m~— 1)/2, we obtainthe following fractional representationof p(a.b).

p(a~b) ~ — 2~ a+3/2F(n + b + 1) a~2aa_b cos[(n + 853(cos ~— \~F(n+ 2~) cosO+1 cos0/2+1 2(n+ ) ( . )

which is valid for a and b positive integersor half-integers,with a � b, andp = (a + b + 1)12. As onecan see from table8.1, the symmetricspacescorrespondto integervaluesof both a andb, andto a = bhalf-integer(odd spheres).

As on the two-sphere,the fractional representation(8.53) can be usedto derive the result (8.47) forthe heatkernelusingthe spectrumof the radial LaplacianLR. The eigenfunctionexpansionof the heatkernel on the compactrank-onesymmetricspacesis

K(0, s)= —~-— ~ d~q~(O)e~, (8.54)

VG/H n0

where

—C(n) = —n(n+ 2p)= —[(n + p)2 — p2] (8.55)

arethe eigenvaluesof LR, and

~ o~= n!F(N12) p((N_2)/2.(m$_l)/2)(cos 0~ ~856‘t’n~ / Fn+N12 n \ /


are the spherical functions. The degeneraciesd~coincide with the dimensions of the sphericalrepresentationsof the group G with respectto H, and VG/H is the volume of the space. Explicitcalculationgives the following valuesof dn and VG /H in terms of the multiplicities (seesection 10):

F(m~+ 1)/2)F(n+ N/2)F(n+ (N+ m4 — 1)/2)dn = [2n+(N+ m4 — 1)/2] n!F(N12)F((N+ m4+ 1)/2)F(n+(m~+ 1)/2) (8.57)

VG/H = 2”IT’~2F((m

4 + 1)12)IF((N+ m~+ 1)/2). (8.58)

[If the diameterof the spaceis not normalizedto L = ir, a factor (L/IT)1” must be insertedin the

right-handside of eq. (8.58).]The volumes coincide with the values given in ref. [17].The dimensionshavebeencalculatedfor

eachrank-onespaceby Cahn and Wolf [23], and we havecheckedthat our generalformula (8.57)reducesto their resultsin eachcase.For example,on the N-spherewe get

d — (2n+N—1)(n+N—2)! ~859‘I n!(N—1)! ‘

in agreementwith the formula for the scalardegeneraciesof the Laplacianon S’s’ [123].Using eqs. (8.53), (8.57) and (8.58), we find that the fractional representationof the spherical

functionscan be written as

çb~(0)= (VG/H/dflVT)D(e1’~’~°+ e~~’~°), (8.60)

whereD is the operatoracting on KT in (8.47), andVT = 2IT is the volume of the maximaltorus.Thisformulacoincideswith the generalsymmetricspaceformulas(5.29)and(5.30). Usingthisin eq. (8.54),we see that the dimensionfactor simplifies, and we get

K(0, s)= ~— D (e~”~8+ e’~’~’~°)e_I5~~2. (8.61)

The sumover n can now be extendedto

~ exp[i(n + p)O — is(n + p)2]. (8.62)

Indeed,sincep is either integeror half-integer,the two sumsonly differ by a finite numberof terms,and it is possibleto show that all theseterms give zero when acted upon by the operatorD. Theeigenfunctionexpansionbecomes

K(0, s)= e~2D~- ~ exp[i(n + p)0 — is(n+ p)2], (8.63)

andthenour result (8.47) follows after applying (6.32).Let us nowconsiderthe doubly connectedcaseof P~’(R).This is obtainedfrom S” by identifying the

antipodalpoints. The diameteris IT/2, andthe periodof the closedgeodesicsis ir. As the points0 and— 0 on the maximal torus of S’s’ are identified, the periodic propagator,on pN(R) is


KPN(R)(O,s) = KSN(O, s) + K5N(IT — 0, s) . (8.64)

For N odd we get immediately, from (8.38),

KPN(R)(O, s) = exp{is[(N — 1)/2]2} (I a~056)~~2~ (±1)”exp[i(O + ITn)2/4s] (8.65)

fl== (4ITIs)

wherethe plus sign is for N = 1, 5, 9,. . . , and the minus for N = 3, 7, 11 For examplefor thethree-dimensionalprojectivespace,which is just S0(3),we obtain, aftera changein the normalization,Schulman’sresult (4.35). For N evenwe get

(N”-1)/2 += . ~(~—~) . 2

KPN(R)(0,s) = exp{is[(N— 1)/2]2} (I a~0~6±1) (—i) exp[i(0+ ITn) 14s]

IT (4ITls)

(8.66)

wherethe plus sign is for the termsin the sumover n with n evenandthe minusfor thosewith n odd.In other words, the contributions from the two homotopy classeson P~’(R),for N even, have adifferent boundaryterm in the fractional operator,one at 0 = ir (south pole) and the other at 0 = 0

(north pole).Once again the phasefactors of the indirect pathsin eqs. (8.65) and (8.66) equal the Gaussian

phases.Indeed,on pN(R) thereare no “intermediate” conjugatepoints betweenthe origin and the“origin back” and if y is the multiple closed geodesiclabelledby n, M

5 = nlm~= nI(N — 1), withGaussianphase

exp(—iirM~I2)= exp[—iirlnl(N— 1)/2].

Beforeconsideringthe noncompactcase, let us comparethe phasefactorexp(isp2)in eq. (8.47) to

the phaseexp(isR/6).This is importantbecauseit was shownin ref. [86]thatthe heatkernelexpansionon an arbitrarymanifold formally containsall powersof R16and that factoringthe phaseexp(isR/6)eliminatesany dependenceon the curvaturescalarR in that expansion.The curvaturescalarof therank-onesymmetricspacescan be easily computed,with the result [27]

R= ~N(N + 3m,3— 1)(rrIL)

2 (8.67)

[e.g.,R= N(N— 1)/a2on SN]. Using eq. (8.41), we find that p2 ~ R/6 unlessN = 3, i.e., eitherfor S3or P3(R) SO(3). In fact, thesearethe only rank-oneSS that are groupmanifoldsat the sametime.Thus, the phasefactor in the exact propagatordoesnot coincide with exp(isR/6),exceptin the Liegroupcase,andwe see that the “partially summed”form of the heat kernelin ref. [86] is not the mostnatural oneon homogeneousspaces.

For exampleon5N by writing p

2 = ~R we get ~= (N— 1)/4N, the conformal coupling in N + Idimensions,andexplicit calculationshowsthat this is only true for spheresandfor the real projectivespaces.The reasonis probablyrelatedto the fact that spheresarethe only conformallyflat (compact,simply connected)rank-onesymmetricspaces.


8.6. Thenoncompactcaseand the Weylfractional integral

A completelist of the noncompactrank-onesymmetricspacesis given in table8.2. The multiplicitiesof dualsymmetricspacesare,of course,the same,andwe seethat the real, complex,quaternionicandCayley hyperbolic spacescorrespondto S”, P2”(C), p4n(H) and P’6(Cay), respectively.

We have already seen in a number of examples, as well as from general theory, that theeigenfunctionsand the heat kernel in the noncompactcasearerelatedto the correspondingcompactquantitiesby analyticcontinuation.Letting n—kiA — p and 0—~ix in eq. (8.56) (seesection5), we seethat the sphericalfunctionson the noncompactrank-oneSS arethe Jacobifunctionsof the first kind(see, e.g., ref. [78],p. 484)

= F(iA- p+1)F(N!2) p((N2)/2~(m~l)/2) (coshx) (8.68)

= F(iA + p, —iA + p, N/2, —sinh2x!2), (8.69)

whereF(a, b, c,z) is the hypergeometricfunction. They satisfy

LR4’A= —(A2+ ~2)& , (8.70)

wheretheradial LaplacianLR is given by eq. (8.40),with hyperbolicfunctionsin placeof trigonometricones.

The intertwining operatorcan be identified with the inverseAbel transform[5, 92] andrelatestheheatkernelon GIH to the ordinarykernelon the line. The differencefrom the compactcaseis in thedefinition of fractional derivative.We now needthe Weylintegral

F(-a) J (x’-x)~’f(x’)dx’ (8.71)

(a<0), i.e., the boundarypoint in the Riemann—Liouville integral becomesinfinity. This eliminatesthe boundaryterms in the fractional commutations(8.25), and for a>0 a

5” is defined by analyticcontinuation,by usingthe semigroupproperty

a5”a~f=a5”’”~f, (8.72)

which now holds for arbitrary valuesof a and/3.

Table8.2Noncompact rank-one symmetricspaces(n � 2)

GIH m5 m512

H”(R) = SO(n,1)/SO(n) n — 1 0H

2~(C)= SU(n,1)/U(n) 1 2(n —1)H”(H) = Sp(n,1)/Sp(n)x Sp(1) 3 4(n —1)H’6(Cay)= F~/SO(9) 7 8

58 R. Camporesi, Harmonic analysis andpropagators on homogeneous spaces

The heat kernel on the noncompactrank-one symmetricspacesis [5, 35, 92]

~ 1 a n~~/2 1 a n~I3/2/2 e~~2’~”

K(x,s)=e’5~(~ acoshx) (8IT acoshx/2) (4ITis)U2~ (8.73)

andcan be obtainedfrom (8.47)by keepingonly the n = 0 term, replacingO—~ix, and interpretingthefractional term as a Weyl operator.The sign of the curvaturemust also be changed,or, equivalently,one can replaces—~—s and multiply by an additionaloverall phase.

For example,on the two-dimensionalhyperbolicspaceH2(R), using a~2= a,~H2a5, we get

KH2(R)(x, s)= e”4(1 /2ITa~

0~~)I12 ex2145/(4ITis)~2 (8.74)

r —isi4 ~ iv2/4sv2e i ye’ dy— . 3/2 I 1/2’

(4ITls) (coshy—coshx)

in agreementwith our early result (4.42).

9. Exactnessof the WKB approximationon the split-ranksymmetricspaces

Dowker’s result [44, 45] that the WKB approximationto the free particlepropagatoris exacton Liegroupscan be generalizedto a wider class of symmetric spaces(SS), known as split-rank. On thesespacesthe rank of the symmetry group G equals the rank of G/H plus the rank of the isotropysubgroupH.

It is a standardresult ([6, 76]) that a SS is split-rank if and only if the multiplicities m~of the rootvectorsare all even. Fromthe classificationin ref. [76], p. 532, we see that the compactirreducibleSSof split-ranktypearethe compactsimpleLie groups,with m~= 2, the odd spheres52n+t with m~= 2n,the spacesSU(2n)/Sp(n),with m

0 = 4 and rank n—i, and the rank-two spaceE6/F4, with m~= 8.

Sincea differentialoperatorchangesthe multiplicity by two units, the intertwiningoperatorreducingm~to zero, i.e., relatingthepropagatoron G/H to the one on the maximaltorus, is purely differentialon thesespaces.Acting with this operatoron the flat kernelwe obtain, by unicity, the Minakshisun-daram—PleijelSchwinger—DeWitt(MPSD) expansion,which preciselyarisesas a perturbationof theEuclideankernel,seesection2. Therefore,the WKB proper-timeexpansionof the heatkernelis exact.(This is strictly true in the noncompactcase.If the spaceis compacta sum over the lattice, i.e. over theindirect geodesics,is also neededin order to “periodicize” the propagator,see section 5.)

Having alreadydiscussedLie groupsand odd spheres,we shall review herethe results in ref. [4],wherethe heat kerneland the sphericalfunctionson the othersplit-rankspaceswere calculated.As inref. [4] we shall consider,for simplicity, the rank-twocase,althoughthe resultcan be easilygeneralizedto the higher-rankcase.

The rank-twosplit-rankspaceshaveDinkin diagramo~o, i.e., the two simpleroots a1 anda2 forma 120°angle. There are three positive roots, that we normalize according to a~= (1/2, \/~/2),

a2 = (1/2, —\/~/2),and a3 = a1 + = (1,0). They are all connectedby the transformationsin theWeyl groupand thushavethe samemultiplicity m~ m. For m = 2 we havethe Lie groupSU(3) (witha different normalizationfrom that in section7); m = 4 correspondsto SU(6)/Sp(3), and m = 8 toE6/F4.


In order to preserveWeyl invariance,the multiplicities of the roots mustbe reducedtogether.Sinceafirst orderdifferentialoperatorchangesma by two for a single root (section8), we expectthat a thirdorderoperatorDmreducesthethreepositive-rootmultiplicities by two at eachstep.This will resultin areductionpatternof the form

D8D6 D2

E6/F4—*SU(6)/Sp(3)——--~SU(3)-——~T2,

where T2 S1 x S’ is the maximal torus.The density function is given by

J~(h)= H [sin(a~,. h)]”’, (9.1)

and half the sum of the positive roots i~

Pm2m~a~,,m(1,0). (9.2)

Using the resultsin section5, we can write down the radial Laplacian

Lm =a2+ m~cot(a~h)a~•~ (9.3)

where a2 = 1~L~a1a1 is the ordinary Laplacian on T

2. We already know, from section 7, that theoperatorD

2, intertwining the radial Laplacianon SU(3) to (a2+ p~),is the third order operator

D2= III a~,ô. (9.4)

H~=1sin(a~h)~

If thespaceis not a Lie group,the operatorDm reducingm—~ m — 2 hasleadingterm D2, but containslower order derivativesas well.

The result in ref. [4] is thatthe operatorD~intertwiningLm to Lrn_2 with constantp~,,— p~2,i.e.,

LmDrn = Dm(Lm_2+ p~,,— p~,,_2), (9.5)

can be explicitly calculatedin termsof the roots,and is given by

~

+ ~(a2.O)3+ ~(m —2)cot(a

2~h)(a2.ô)2—~(m —2)cot2(a

2.h)a2~a

— ~(a3.O)~— ~(m —2) cot(a3 . h) (a3.0)2 + ~(m—2) cot2(a

3 h) a3 0] , (9.6)

where

J2(h) = sin(a~,h), (9.7)


andkm is a normalizationconstant.The leadingterm in D~is proportionalto H~a ~O,becauseof theidentity

1 3 3 33[(a1 ~8) + (a2.8) — (a5 ~ô) ] = —(a1 0)(a2 O)(a3.0) (9.8)

We also see that D~dependsexplicitly on the multiplicity through the lower order terms.For m = 2thesetermsdisappear,and we obtaineq. (9.4). As usual,the constantk~,is determinedby requiringthat Drn relatesthe delta functionson the two spaceslabelledby m andm — 2.

Iteration of (9.5) gives the heat kernel of (Ia, + Lm) on the compactsplit-rank symmetric spaces,

Krn(h, s)= e’5 D,nD~

2~. D,K~(h,s), (9.9)

whereKT is the periodic propagator(seelater) on the maximal torus, given by eqs. (6.10)—(6.12).ThelatticesA of the sphericalweightsand F, the kernelof the exponentialmapping,aregiven explicitly by

A= {n1 ir1 + n2rr, , n. integer} , (9.10)

F= {2ir(n1a1 + n2a2), n. integer} , (9.11)

where 171 = (1, 1 /V~)and ir2 = (1, —1 /\/~)are the fundamentalweights, and satisfy ~ a1 =

i, j= 1,2.

As in the rank-one case,it is important to note that Drn intertwines Lm to Lm_2 and relatestheireigenfunctionsfor arbitraryevenvaluesof m. This meansthatthereis an entireclass of polynomialsintwo variables,orthogonalwith respectto the measureJ~,,which for m = 2, 4 and 8 are the sphericalfunctionson the split-rank SS.

Some propertiesof thesepolynomials (for m arbitrary) were studied by Koornwinder [90] whohoweverwas unableto obtain an explicit form. Using the intertwiningoperatorwe get the followingexpressionof the Weyl invariant eigenfunctionsof Lrn in termsof thoseon T

2, valid for any evenvalueof m:

~j~(rn)(h)= N,,iDrnDrn_2~. . D2 ~ e~Pn>h (9.12)

WE W

whereNm is a normalizationconstant,andthe sum is over theWeyl group. The vectorA is an arbitrarydominantweight, given by eq. (9.10) with n1 andn2 non-negativeintegers.Sincethe vectorp~,can bewritten as~m(171+ 177), it is adominantweightaswell. Therefore,thefunctionsin (9.12) areinvariantunder translationsin the lattice F, and this implies that KT must be the periodic propagatoron T

2.The phaseof the multiple geodesicsis equalto one and coincides, onceagain,with the Gaussian

phase.Indeed, the total number of conjugatepoints M~,along the path y is proportional to thesymmetricspacemultiplicity m, seeeq. (7.56). Since the Gaussianphaseexp(—iITM

5/2) is onefor the(simply connected)Lie group SU(3), wherem = 2 (seesection7), it will be one, a fortiori, for theremainingsplit-rank SS, wherem = 4, 8. We also remarkthat the intertwining operatorDmD,~i_2~~D2, with the appropriateanalytic continuation obtained by replacingtrigonometricwith hyperbolicfunctions, gives an explicit formula for the inverse Abel transform on the noncompactsplit-ranksymmetricspaces(section5). As far as we know this is a new mathematicalresult.


From eq. (9.9), we get the following expressionof Km [compareto the odd spheresresult in eq.(8.52)]:

e’°~’ 3 fa .(h+2ITn)\m12

Km(h, s) = . (3m+2)12 ~ [‘I (~~. • L\ ) exp[i(h + 217n)2/4s](41715) 2irnEF~=1 sin~a~,~

/ 3(m—2)/2

X 1 + ak(h + 2ITfl)(1S) , .1k=i

wherethe quantitiesak(h) arethe heat kernelcoefficientsof the operatorLm — p2. Eachterm in thesum over F is an exact MPSD expansion terminating to the coefficient a

3(m_2)/2, i.e. to a3 onSU(6)/Sp(3)and to a9 on E6/F4. Theterm in a0 = 1, containingthelowestpowerof s, is obtainedwhenall of the third orderderivativesin the operatorsD1 in eq. (9.9) acton the exponentialin KT [seeeq.(6.12)]. Using eqs. (9.8) and (7.43), we see that they generatethe VVM determinant,as well as thecorrectpowerof s.

The coefficient a1 hasa fairly simple form on the spaceSU(6)/Sp(3),

~ 1—a .hcot(a •h)a1(h) = —2 ~ 2 (9.14)

(a~,.h)

It is interestingto comparethis to the coefficient a1 in eq. (8.52) for the five-sphere(wherema = 4 aswell)

a~5(0)= —2(1 — 0 cot 0)/02. (9.15)

We seethat thecoefficient a1 on SU(6)/Sp(3)is justthe sum of threea1-coefficientsfor S

5,onefor eachpositive root.

Similarly, the highestcoefficient a3(~_2)/2is obtainedwhen noneof the operatorsD. in eq. (9.9),

exceptD2, actson the exponentialin KT, and is given by

~ sinahm/2 ~ ah

a3(~2)/2(h)= n ( ) D~D~2•. D~fl . • . (9.16)

~=i a~, ~ sin(a~ )

Explicit evaluationof thea~andof the sphericalfunctionsis a straightforward,but long computation,andwill not be given here.

We shall concludethis sectionby making somecommentson the exactnessof the WKB approxima-tion. The exact free particlepropagatorin eqs. (9.9)—(9.13) hasthe form of a “sum over classicalpaths”, i.e., a sum over the set ‘~~1~ geodesicsgoing from x to x’ in “time” s,

K(x, x’, s) = (4ITis)’~2~ e~”~F[y], (9.17)yEP~i

whereSEy] is the actionevaluatedon the classicalpathy andF[y] is somefunctionalof the path.TheMPSD proper-timeexpansionconsideredin section2 is also of this form, since the action for a freeparticle on a Riemannianmanifold is proportional to the geodesicdistancesquareddivided by s,


S[y] = u2(y)14s [38]. The functional F is then

F[y] = ~2(x, x’) ~ a~(x,x’)(is)’, (9.18)a = 1)

where is the VVM determinantalong y and the heat kernel coefficientsa,, are given by (2.14) interms of an integral over ‘y.

Thus, if the heat kernelexpansionis exact,the “sum over classicalpaths” is exactas well. Our caseof a free particle on a split-rank SS is a particular example of this, with a proper-timeexpansionterminating(up to aphase).If the symmetricspaceis not split-rank the appearanceof integraloperatorsrelatedto the fractional derivatives(section8) spoils the exactnessof the proper time expansion.

It is interestingto ask whethertheremaybe otherRiemannian(nonsymmetric)manifolds with theproperty that free motion is exactly describedby the WKB approximation.It is possible that thisproperty be relatedto the topology of the manifold.

For example,it is known [94] that a compactLie group G hasthe samecohomology,apartfromtorsion, as the productof odd spheres

52rn1+l x 52rn,’~’I x

where 1 is the rank of G and the numbersm1 arethe so-calledexponentsof the group. Theyarerelatedto theordersof theinvariantsof the Weyl group i-’, by m1 = i.’, — 1. The productof the i~equalstheorderof W and the sum of the m, equalsthe numberof positive roots [95, 98, 116].

Furthermore,Araki has shown [71that the remaining split-rank symmetric spacesbehaveverysimilarly to the compactLie groupsfrom the homological point of view.

It would be important to better understandthis connectionbetweenthe exactnessof the WKBapproximationon thesemanifolds andtheir similar topologicalproperties.This point deservesfurtherinvestigation.

10. The partition function and the dimensions of the spherical representations

In this sectionwe considerthe partition function on symmetricspaces(SS). We obtain an explicitformula for the scalar degeneraciesof the Laplacian on compactsymmetric spacesin termsof themultiplicities and of the root vectors.Specializingto the caseof rank one,gives an interestingrelationbetweenthe partition functionson the compactandnoncompactspaces,which is bestseenby writingthesequantitiesas contour integralsin the complexplane.

This representsa generalizationof a well known symmetry for the partition functions on thetwo-sphereandon the two-dimensionalhyperbolicspace(see,e.g.,ref. [117]).In the nextsectionthissymmetrywill be used to derive recursionrelationsfor the partition functionsand explicit formulasforthe heatkernel coefficientsand for the zetafunction.

10.1. The Plancherelmeasureand the spectrumof a compactsymmetricspace

The partition function of a compactRiemannianmanifold M hasbeendefined, in section4, as theintegral over M of the coincidencelimit of the heat kernel K. In this and the remainingsections,by


“heat kernel” we shall meanthe Green’s function of the heat operator(—a/at+ L), L being theLaplace—Beltramioperatoracting on scalarfields.

The partition function hasthe following eigenvaluerepresentation

Z(t) = m. e~Ai, (10.1)

wherethe sum is over differenteigenvaluesof L, with multiplicities m. If M is a compactsymmetricspaceGIH, wecan usethe eigenfunctionexpansion(5.19),with (5.22) for theeigenvalues,andobtainthe coincidencelimit of the heat kernel,

KG/H(O, t) = -s-—— ~ d~e_t+~2, (10.2)0/H AEIt~

wherep is half the sumof the positiveroot vectors,andthe sum is over thedominantsphericalweightsof G with respectto H. ThedegeneracydA coincideswith the dimensionof the sphericalrepresentationlabelled by A (section 5). Integrating over G/H merely eliminates the volume factor VG/H’ andhereafterwe shall call K(0, t) “partition function” as well.

The coincidencelimit of the heatkernel on the noncompactdualspaceG’/H is obtainedfrom eq.(5.40),

KG/H(O, t) = cet~2J H(A)L2 etA2 d~ (10.3)

(t > 0), whereH( A) is the Harish-Chandrafunction andtheintegral is over the1-dimensionalEuclideanspace~ (a maximal abeliansubspacein the tangentspaceat the origin, I being the rank of the SS).The normalization constantc can be determined,e.g., by comparing eq. (10.3) to the MPSDasymptoticexpansionof the heatkernel in the t—~0 limit (seelater).

By eqs. (10.2)and (10.3)weseethat the discretesumin the compactcasebecomesan 1-dimensionalintegral in the noncompactone, accordingto the replacement

~ dA—4j~H(A)L2dlA. (10.4)

The dimensionsof the irreduciblerepresentations(irreps) of a compactgroup can be calculatedbyusingthe Weyl dimensionformula(7.28).For exampleCahnandWolf [23]havedeterminedin thiswaythe degeneraciesdA on the rank-onesymmetricspace.However the Weyl formula involves the rootsystemof the group G, andit is simpler to havea formula that gives dA directly in termsof the rootsand of the multiplicities of the symmetric spaceG/H. This can be done by using the Gindikin—Karpelevichproductformula (5.38) for H( A) andwill allow us to derivea generalexpressionfor dA ~anycompactSS.

By eq. (10.4) andrememberingthe resultsin section5 aboutthe symmetrybetweenthecompactandthe noncompactcase[seein particular eq. (5.41)], we mayexpectthat dA is relatedto the Plancherel(or Harish-Chandra)measureI H( A) -2 by analyticcontinuationthrough A—~i(A + p). Although this isvery close to being true, it turns out that the vectors ±i(A+ p), in the complexified space ~

~ iY’, are singular points of H( A)12 and thereforea more careful limiting proceduremust beconsidered.

64 R. Camporesi,Harmonic analysis and propagalors on homogeneous spaces

For A E /~ the propergeneralizationof the Plancherelmeasureis

p(A) [H(A)H(—A)]’ , (10.5)

which coincideswith H(A)L2 only for A real (i.e., A E f’). It is convenientto rewrite formula (5.38)for H(A) in termsof a productover the set .~÷of the positive unmultipliable roots [78], i.e., the roots/3 >0 such that 2/3 is not a root. A simple calculation using the duplication formula for the gammafunction gives

H(A)H(A)= 11 ~ , A

0=A./3//3~, p0p.fi//32, (10.6)

~ I3~ ~PI

3)

H0(z) = F(2iz)/[22~F(iz + 15

0)F(iz+ 4m,3/2+ ~)], (10.7)

and ~ + 2m13/2) is the p-factorfor the rank-onespacecorrespondingto the root /3, m0 beingthe root multiplicity. It is not difficult to show that the numbersp0 and can take either positiveintegeror half-integervalues,andcoincidewhenever/3 is a simple root(or twice a simple root). Noticethat H(A) in eq. (10.6) is normalizedto H(—ip) = 1.

Using now eqs. (10.6) and(10.7) in (10.5),we find that ~i(A) is singularfor thosevaluesof A suchthat A0 = ±i(n+ p5,,) for someroot /3 with odd m0, n beinga non-negativeinteger.More precisely,ifm0 is odd, the numbers±i(n+ ~ aresimplepoles of [H0(±z)1’ in eq. (10.5),whereasfor evenm0the functions[H0(±z)] 1 are analytic in the z-plane. In particularif GIH is split-rank (~‘m0 is evenfor each root), p(A) is analytic on the whole .~,and in fact it reducesto a polynomial. (For therank-onecasethis will be discussedin more detail later.)

The formula for the dimensionsdA can now be written as

d~1= lim p~(A)/p(A—iA), (10.8)A’—’I(/l +p)

wherethe denominatorgives the right normalizationdo = 1 for the singlet A = 0. Fromeq. (10.8) andthe previousdiscussionwe get

d4 = res~t(A)~I(A+P)/res~ (10.9)

in the non-split rank case,and

d4 = 1L(i(A + p))/~(ip), (10.10)

in the split-rankcase.We haveobtainedthe following result: if G/H is a compactSS not of split-ranktype, the dimensionsdA are given by the residuesof the Plancherelmeasureat A = i(A + p), up to anormalizationconstantindependentof A anddeterminedby the conditiond0 = 1. In the split-rankcase1L(A) is analytic and dA is given by eq. (10.10).

It is now straightforwardto calculatethe residuesin eq. (10.9) [usingeqs. (10.6) and (10.7)] andweget the following expressionof dA on a compactSS in termsof the roots and the multiplicities:

dA”° [I F0(A0+p0)/F0(p0), (10.11)


2zF(z+15)F(z+~m/2~2)F0(z)= F(z — 15~+1)F(z — + ~ (10.12)

andthe numbersA0 A. p1f32 arenon-negativeintegers,in view of the integrality condition for the

dominantweight vectors.In the split-rankcasewe obtainfrom eq. (10.10) the sameformulas(10.11)and(10.12). As far as we knowtheseformulas arenew, althoughthe relationbetweenthe residuesofthe Plancherelmeasureandthe dimensionsof the sphericalrepresentationsis, probably,well known.

10.2. Examples:rank-oneSS,normal real form SS,and Lie groups

In the rank-onecasethereis only one root /3 in ~ p0 = p0(usp), andwe get, settingA0 = n EZ’~,

— 2(n + p)F(n + 2p)F((m0+ 1) /2)F(n + N/2)

d~— n!F(2p + 1)F(N/2)F(n+(m0+ 1)/2) (10.13)

whereN = m0 + m012 + 1 is the dimensionof G/H. This is the resultalreadygiven in section8 [seeeq.(8.57)].Usingeq. (10.13)andthevaluesof the multiplicities in table1 of section8, we easilyderive thefollowing explicit formulasfor the degeneracieson the rank-onespaces:

= (2n + q —1)F(n + q — 1) = 2n+q— 1 n + k , for S9, (10.14)

d~= 2n+ q (F(n+q))2 = 2n+ q ~ (n + k)2 for P2~(C), (10.15)

d = (n+2q)(2n+2q+1) (F(n+2q)\1

2 (n+2q)(2n+2q+1) 2h1 (n+k~2 for P4~(H)2q(2q+ 1)(n + 1) \ n!F(2q) I 2q(2q+ 1)(n + 1) k=1 \ k ,~ ‘ (10 16)

n!F(8)F(12)F(n+4) — 11 k=1 k r”4~ r I s=8

for P16(Cay). (10.17)

Thesevaluesof d~coincidewith the onescalculatedby CahnandWolf, by usingthe Weyl dimensionformula [23]. It is clearthatour derivationusing (10.13) is much simpler.

In the higherrank caseeqs. (10.11) and(10.12) takea simple form whenthereareno doubleroots,i.e., whenm

012 = 0 for any /3 E2~.Then .~ coincideswith the set .~‘÷ of all positiverootsandwe get

d ~ (A0 + p0)F(A0 + p0 + m012)F(p0— m012+ 1) 1018A + p0F(p0+m012)F(A0+p0—m0/2+1)

A particularly simplecaseis givenby the symmetricspacesof normal real form [76], wherem0 = 1 forany root /3. From eq. (10.18)we get in this case

dA=H~’3, (10.19)

+ p13


which is formally the sameas the Weyl formula applied to the root systemof the symmetric space.Fromthe classificationin ref. [761,p. 532, we seethat eq. (10.19)holds for the SS of typeAl, CI, DI(the third type), El, EV, EVIII, Fl and G. Simpleexamplesare the two-sphereandthe rank-twospaceSU(3)/SO(3), with dual spacesH2 — SL(2,R)/SO(2) and SL(3,R)/SO(3) respectively.The spaceSL(3, R)/SO(3) plays an important role in the so called “strong coupling limit” of quantumgravity[120], as well as in Misner’s minisuperspacefor Bianchi type IX cosmologicalmodels[107, 125].

On a compactLie group U we havem0 = 2 for any root andeq. (10.18) gives

d1=fI((~H~P), (10.20)

in agreementwith the Weyl formula for theirreducible(spherical)representationsof U x U actingon Uby left andright multiplications(seeappendixB). The partition function on a compactLie groupcan bewritten down in terms of both the eigenvaluerepresentation,eqs. (10.2) and (10.20),and the “sumover classicalpaths”,consideredin section7. Taking the h—*0 limit in eq. (7.44) we get for a simplyconnectedgroup (N = dim U),

K0(0, t) = (4ITt)~2 etP2[1 + ~

n�0

x (b0 + ~ (a• )2 + (a . n)

2(fl. )2 +... + 1~I(a ~)2)] (10.21)

whereb0, b1,. . . , b~,areconstantsindependentof n (~is the numberof positiveroots). Thesumover

n representsthe “exponentiallysmall” termsassociatedwith the indirect geodesics,which disappearonthe noncompactdualspaceGI U,

KG/U(O, t) = (417t)N/

2 e’~2. (10.22)

In particular, for U = SU(2), the result (10.21) has beenused in quantum(one-loop)calculationsonthe static Einstein universe[50].

10.3. Contour representationof the partition function

We shall now illustrate formula (10.9), by deriving an important symmetry relation betweenthecompactand the noncompactpartition functions.For simplicity we shall considerthe rank-onecase.Let GIH and G’IH be (simply connected)dual rank-one symmetric spacesof the compact andnoncompacttypesrespectively.Thesearelisted in tablesI and2 of section8. By eqs. (10.2) and(10.3)we havethe eigenvaluerepresentationof the partition functions,

KG/H(O, t) = V~Her~ d~e_t~~2,t>0, (10.23)

KG/H(O, t) = c~e~2 J e~2~(A)dA, t>0. (10.24)


The degeneraciesare given in eq. (10.13), the volumes in eq. (8.58), and the factor p equals(N + m0 — 1)/4, N beingthe dimensionof the space.We can determinethe normalizationconstantcN

by requiring that K agreeswith the MPSD asymptoticexpansionas t—~0. The result is

cN = 2m~_3IT_(~2)/2F(N/2). (10.25)

Furthermore,by using eqs. (10.6) and (10.7) for H(A) we get the following expressionsfor thePlancherelmeasure~s(A):

(N—3) / 2

= 22(N2)[~(N/2)]2 fT (A2+j2), N odd, (10.26)

/2ITAtanh(ITA) (N-3) (A2 +j2), N even (10.27)

= 22~21[F(N/2)]21=1/2,3/2

(for N = 2 the product is omitted);

3 N/4—1ITA coth(ITA) n (A2+j2)2, N=4,8,12,... (10.28)~HN(C)(A) = 2N2[F(N/2)]2 j=1

(for N = 4 the product is omitted);

N/4—1irAtanh(irA) i~j (A2+J2)2, N=2,6,l0,... (10.29)~

11N(C)(A) = 2N_2[F(N12)]2 1=1/2,3/2

[for N= 2 the productis omittedandwe get the previousresult for H2(R), in view of the isomorphismH2(C) — H2(R)];

(N—6)/42IT{lt + [(N—2)/4]2) A tanh(irA) fT (A2~J2)2 N= 4,8,12,... (10.30)~HN(H)(A)= 2N[F(N/2)]2 j=1/2,3/2

[for N=4 the product is omitted and we get the previous result for H4(R)~H4(H)]; finally onH16(Cay)we get

ITA tanh(irA)~(A) = 220[1(8)]2 (A2 + ~)(A2 + ~)(A2 + ~)(A2 + ~)2(A2+ ~)2• (10.31)

Theseformulashold for A complexas well, when~L(A)is definedby eq. (10.5). Using eq. (10.10) in(10.26) and (10.9) in (10.27)—(10.31),we obtain other expressionsfor the scalardegeneraciesof theLaplacianin termsof polynomials in the variable n + p:

(1) S”, N odd, p(N—1)/2,

‘1

dn= (N”l)’ fl[(n+p)2_j2], (10.32)

68 R. Camporesi, Harmonic analysisandpropagatorson homogeneousspaces

(2) 5”, N even,p = (N — 1)/2 (product omitted for N = 2),

p—i2(n+p) 1~E 2 .2

= ~‘M ~ ii [(n + p) —] ] , (10.33)~ 1). j=1/2.3/2

(3) pN(C) N = 4, 8,. . . , p = N14 (product omitted for N = 4),

3 p—l2(n+p) 1~E 2 .22d~=(N/2—1)!(N/2)! ~ [(n+p) —J ] , (10.34)

(4) PN(C) N6,10 pNI4,

p—i2(n+p) 2 22

d~= (N/2 — 1)!(N/2)! j=i/2,3/2 [(n + ~) ~i] (10.35)

(5) pN(H) N=8, 12,..., p =(N+2)/4,

= 2(n (N/2-1)!(N/2+1)! j=~2,3/2 [(n + ~)2 J2]2, (10.36)

(6) P’6(Cay), p = 11/2,

3/2 9/2

d~= ~ l~I [(n + ~)2 - J2]2 fT [(n + p)2 - J2] (10.37)1=1/2 j=5/2

Theseformulasfor d~areequivalentto eqs. (10.14)—(10.17)andwill be used,in the next section,tocomputethe zetafunction on the rank-onesymmetricspaces.

The analytic structure of1s(A) is easily investigated from eqs. (10.26)—(10.31).For the odd-

dimensional(real) hyperbolic spaces(i.e., in the split-rank case),p.(A) is just a polynomial in A, andthereforeit is analytic. In the remainingcases~(A) is a meromorphicfunction,with simplepoleson theimaginary axis at

A~=±i(n+p),n=0,1,2 (10.38)

It is not difficult to showthat the residuesat thesepoles are given by

resp~(A)Ia(fl+P) = [(—1)”22ITicNVG/H]’1dfl, (10.39)

i.e., coincide, up to a factor independentof n, with the dimensions (10.13) of the sphericalrepresentations.This is, of course, nothing but our general result (10.9), and indeed the constantmultiplying d~in (10.39) is simply the residueof ~(A) at ip, as can be checkeddirectly.

This polestructureof j.~(A) suggeststhat the compactpartition function (10.23)can be obtainedfromthe noncompactone (10.24) by properlydeformingthe contourof integrationin the complexA-plane.Indeed,let us considerthe function

K(t) e(t)cN e’~2I ~(A) et52 dA, (10.40)


where6(t) = 1 for t>0, e(t) = (_1)N12 for t<0, and C is the contourof fig. 1.

For t >0, C can bedeformedto the real axisby takingthe R—* limit of theclosedcontourin fig. 2.Therefore,for t> 0 eq. (10.40) reproducesthe noncompactpartition function (10.24),

K(t) = KG,/H(O, t), t>0. (10.41)

For t <0, considerthe contourof fig. 3. The integraloverthe arcvanishesas R—* co, andthe integrandin (10.40)hassimplepoleson the imaginaryaxis at i(n + p), n = 0, 1,.. ., with residuesgiven by [seeeq. (10.39)]

res e(t)cNp~(A)e_t~24~2)Ij(fl+P)= d~exp{t[(n + p)2 — p2]}(2ITiVG/H)1 . (10.42)

Using the theoremof residueswe obtain, for t <0, the compactpartition function (10.23),

K(t) = KGIH(O, —t), t<0. (10.43)

Summarizing:the function K(t) in eq. (10.40) gives the compact(noncompact)partition function fort <0 (t> 0). We can say thatthe compactpartition function can beobtainedby analyticallycontinuingthe integral representationof the noncompactone to negativevalues of t. In the particular caseofGIH= S2 and G’IH= H2(R), this resultwas given in ref. [117].

Extension to the higher-rankcaseis also possible. Given the integral in (10.3) (with G/H notsplit-rank)wecan changethe integrationregionin the complexifiedspace~ so as to include,for t <0,

Fig. 1. Fig. 2.

Fig. 3.

70 R. Camporesi,Harmonicanalysis and propagators on homogeneous spaces

the poles of /.L(A) at A = i(A + p), wherethe A arethe dominantsphericalweight vectors.Using eq.(10.9) we get a contourrepresentationof the compactpartition function.

If the SS is split-rank,the function ~i( A) is analyticandthe previousdiscussionneedsto be modified;in particular eq. (10.40) does not give the compactpartition function for t <0. From our results insection9 we know that the Minakshisundaram—Pleijelexpansionof the heat kernel is strictly exact onthe noncompactsplit-ranksymmetricspaces.On the compactspacesit is exactonly up to the additionof the “indirect geodesics”contributions,which give the “exponentiallysmall” termsin the propagator.From this we find that replacingt—~— t in the noncompactpartition function [and multiplying by thephase(_i)N/2] gives the “direct path” (n = 0) term in the compactpartition function.

This remark applies to the MPSD expansionon the other symmetric spacesas well, when thisexpansionis only asymptotic(valid for t—~0).

11. The keat kernel coefficientsand the zeta function in the rank-one case

In this sectionwe calculatein closedform the coincidencelimit of the heatkernelcoefficientsfor therank-onesymmetricspaces.The compactcasehasbeenconsideredpreviouslyby Cahn and Wolf [23].Our methodis moregeneralbecausewetreatthe compactandthe noncompactspacestogetherandwerederive, using a simpler method, Mulholland’s result [109] on the asymptotic expansionof thetwo-spherepartition function [and a similar one for the spaceP4(C)]. As the integral representation(10.3) is more amenableto computation than the discrete sum (10.2), it is easierto obtain theasymptotic expansionof K on H2(R) and H4(C) first, and then analytically continueto the compactcaseusing the resultsin the previoussection.

Recursionrelationsfor the partition function areobtainedandallow us to computethe zetafunctionof thc compactspacesin termsof a sum of Riemannzetafunctions.As far asweknow theseresultsarenew, and representa generalizationof previouswork by Candelasand Weinbergand by Dowker onspheres[26,47].

11.1. Recursionrelationsfor the partition function

Let K(0, t) be the coincidencelimit of the heat kernel (for short, the partition function) on therank-onesymmetricspaces,given by eqs. (10.23) and(10.24).Sincethe factore~t~is naturally singledout of the sum andof theintegral, it is convenientto definethe modified heat kernelcoefficientsa’i,, by

K~(0,t) (4ITt)N/

2 e±tP2(1 + ~ a~t~), t~0, (11.1)

wherethe plus (minus) sign is for the positive curvaturecompact(negativecurvaturenoncompact)spaces.The relation betweenã~and the usualheatkernelcoefficientsa~is obtained by multiplying theexponentialseriesfor the term exp(±tp2)with the asymptoticexpansionin (11.1). The result is

= ~ ~f(±p2)~1/(n — 1)!. (11.2)

Accordingto ourdiscussionin section10, the compactpartition function is obtainedby analyticallycontinuingthe noncompactoneto negativevaluesof t andmultiplying by (

1)N/2 Sincethe asymptotic


expansion(11.1) is analytic in the changet —~ — t, the relationbetweenthe heat kernel coefficientsinthe compactandthe noncompactcasesis simply

= (~1)?~c~~, (11.3)

which, becauseof (11.2), holds for the coefficientsan as well. (As alreadymentioned,the “indirectpaths” contributionson the compactspacesareneglectedin the MPSD asymptoticexpansion,as theyare “exponentiallysmall” whencomparedto the “direct path” term.)

Therefore,it is sufficient to calculatethe t~ on the hyperbolicspacesHN. Defining for simplicity themodified partition functionK as

K~(t)= e+IPSK±(0,t) (11.4)

[this is the coincidencelimit of the Green’s function of the operator(— a1 + L ~ p2), whereL is the

Laplacian], andusing the values(10.26)—(10.31)of the measurep.(A) in eq. (10.24),we easilyderivethe following recursionrelationsfor K

— —a +((N—3)/2)2 —KHN(R)(t) = 2ir(N —2) KHN-

2(R)(t), N= 3, 4, 5,. .. , (11.5)

— [—a+ (N/4 — 1)2]2 —KHN(c)(t) = 4

2(N — 2)(N — 4) KHN-4(c)(t), N = 6, 8, 10,. . ., (11.6)

— [—a1+ (N/4 — 1/2)

2][—a + (N/4 — 3/2)2] —KHN(H)(t) = 4172(N — 2)(N —4) KHN_

4(H)(t), N= 8, 12, 16,. . . , (11.7)

- (—a +81/4)(—a+49/4)KH1S(Cay)(t)= 672172 KHI2(H)(t) , (11.8)

where a, = a / at. The result for the real hyperbolic spacesHN(R) was discussedin ref. [117].The recursionsfor the compactdual spacesis obtainedby replacingt —~ — t (in the caseof S” an

overall changeof sign is also required). A simple iteration of eqs. (11 .5)—(11.8) gives the partitionfunctions on the noncompactrank-oneSS in terms of the partition functionson the spacesH

3(R),H2(R) and H4(C),

(N —3)/2

KHr~’(R)(t) = [2217””’22F(N/2)]” [I (—a, + j2)KH3(R)(t), N odd>3, (11.9)

(N—3)/ 2

1(H~~’(R)(t)= [2N_2IT~~/2~F(N/2)]_lLI (—a,+j2)kH2(R)(t), N even>2, (11.10)

1=1/2,3/2

N/4—1

KH~’(C)(t)= [2~4IT””22F(N/2)]1 LI ~—a

1+j2)2~4(c)(t), N= 8,12,. . . , (11.11)

N/4—1

KHS1(C)(t) = [2N_

217~2~F(NI2)]~ 1] (—a, + i2)2KH2(R)(t), N= 6, 10,. . . , (11.12)

/=1/2,3/2


— N—2 N/2—i —1KHN(H)(t) = [2 IT F(N/2)]

2 (N—6)/4

x [—a1+ (~2)] ~j ~ + j2)2R

112(R)(t), N = 8, 12,..., (11.13)/=1/2,3/2

KHI6(Cay)(t) = [2’4177F(8)]1(—a,+ ~ )(—a, + ~ )(—a, + ~)(—a,+ ~)~(—a, + ~)2K(t)

(11.14)

For the compact spaceswe obtain similar relations by letting t—~—t and multiplying each KN by

(_1)~~2.For exampleon5N N odd, we get

(N—3)/2

KSN(t) = (_1)~3~2[2~2ITN/22F(N/2)]~ U (a, + 2)R(t) (11.15)

/=1

The correspondingformulascan also be obtained,moredirectly, from eq. (10.23),by using the values(10.32)—(10.37)for the degeneracies.

11.2. Theheatkernelcoefficients

Fromthe recursionsderivedabovewe seethat the heatkernelcoefficientson therank-oneSS can berecursively calculatedfrom the correspondingquantitieson the spacesH3(R), H2(R) and H4(C).We shall nowderive the asymptoticexpansionof K(t) for t —+0 on thesespaces,andprove thefollowingresults:

KH3(R)(t) = (417t)312 , (11.16)

KH2(R)(t) (417t)1(1+ ~ B~(21~2~— 1)tnln!), (11.17)

KH4(C)(t) (417t12(1 + ~ B2~(1— n)tf/n!), (11.18)

where B~are the Bernoulli numbers[55, 71]. The result for H3 is actually exact, as can be seenby

taking the coincidencelimit of the heat kernel derived in section 3. The result for H2(R) can beobtainedby eitherusingthe “sum over paths” form (4.42)or moresimply the eigenfunctionexpansion(10.24),

KH2(R)(t) = J A tanh(ITA)e’~2dA, (11.19)

whereeqs. (10.25) and (10.27) havebeen used.The integral in eq. (11.19) is well suited to deriveasymptoticexpansionsfor both small and larget. For small t we use

tanh(irA) = 1 — 2/(e2’~+ 1) . (11.20)

When this is substitutedin eq. (11.19),the constantterm gives the leadingterm (4ITt)~ in eq. (11.17).


The other term in eq. (11.20) is integratedby expandingthe exponentiale’~2in powerseries.Using

[71]2n —1

I e2’~~ = (~1)~’(1 — 2’~2~)B2n/4fl, (11.21)

we get eq. (11.17),andremembering(11.3) we find the heat kernelcoefficients a), on the two-sphere,

= (~1Y’B2n(2~2”— 1)/n! . (11.22)

This is the resultderivedby Mulholland in 1929 [109],in hisinvestigationof the statisticalmechanicalpartition function of a diatomic molecule (rigid rotator) which formally is the same as the partitionfunction on the two-sphere.His methodconsistedin a careful asymptotic evaluationof the contourrepresentation(10.40) for S2.

The spaceH4(C) can be handledin a similar way. The integral representation(10.24) gives afterusing eq. (10.28),

KHS(c)(t)=872 ~-JAcoth(ITA)e_tASdA. (11.23)

We thenuse

coth(ITA) = 1 + 2/(e2’~— 1), (11.24)

i x2~dx — (~1)~~’B2n 11

J e2~~x— 1 — 4n ( .25)

0

andproceedas for H2 to obtaineq. (11.18).Notice that the coincidencelimit of a) vanisheson H4(C)and on P4(C). This can be easily checkedby using the well known result a

1 = R/6, where R is thecurvaturescalarof the space.From eq. (11.2) we havefor the compactspaces~1= p

2 — R/6 whichvanisheson P4(C) because,by eqs. (8.41) and (8.67), p = 1 and R= 6 in that case.

Let us now calculate a~),in the general case. In order to use the recursions(11.9)—(11.14)thedifferential operatorsin the productsneedto be expressedin polynomial form. Therefore,we definenumbersakN, 13k,N’ ‘Yk,N’ ôk.N~,.Lk,N, and ~qk, by

(N—3)/2 (N—3)/2

LI (—a, + j2) = a,,~,a~, N odd>3, (11.26)j=1 k=0

(N—3)/2 N/2—1

LI (—a,.+j2)= ~ Pk,Na~, Neven>2, (11.27)/=1/2,3/2 k=0

N/4—1 N/2—2

LI (—a, + J2)2 = Yk,N a~, N = 8, 12, 16, . . . , (11.28)j=i


N/4—1 N/2—i

If (—a, + = ~ ~k.N a~, N = 6, 10, 14 (11.29)j=i/2,3/2 k=0

2i (N—6)/4

[—a,+ (N~2)j if (—a, + J2)2 = ~k,N a~, N = 8, 12, 16 (11.30)j=1/2,3/2 k=0

(—a, + ~ )(—a, + ~ )(—a,+ ~ )(—a,+ ~)2(—a,+ 1)2 = ~ ~ (11.31)

Similar definitions of numbersakN, f3~,Netc. can be given to put in polynomial form the operatorin eq.(11.15), and in the analogousrecursionsfor the compact spaces.The two classesof numbersareobviously relatedby 4N = (—1)”akN etc. The valuesof the first few coefficients in eachcasearegivenin tables11.1—11.6.

We can now substituteeqs. (11.26)—(11.31)and eqs. (11.16)—(11.18)in the recursions (11.9)—(11.14).Using the generalrelation for the iteratedderivativesof a power,

a~t9= [F(q + 1)/F(q — p + 1)]t9” (11.32)

Table 11.1 Table 11.2Values of 0k N Values of ~AN

k N=3 5 7 9 k N=2 4 6 8

0 1 1 4 36 0 1/4 9/16 225/641 —l —5 —49 1 —1 —5/2 —259/162 1 14 2 I 35/43 —1 3 —I

Table 11.3 Table 11.4Valuesof ‘/k,N Valuesof ~

k N=4 8 12 k N=2 6 10

0 1 1 16 0 1 1/16 81/2561 —2 —40 1 —1/2 —45/162 1 33 2 1 59/83 —10 3 —54 1 4

Table 11.6Table 11.5 Values of ~

,

Valuesof p~ 8037225 18455239

k N=4 8 12 ‘1,, 16384 ‘~‘= — 4096

0 1/4 9/64 2025/1024 — 13020525 — 28643231 —1 —19/16 —4581 /256 1024 ‘ — 2562 11/4 1665/32 262075 104373 —l —313/8 ~14 644 45/45 —1 776_175/4. 777—I•


and comparingwith eq. (11.1), we get the following expressionsfor the coefficientsin:(1) H”(R), N odd,

— ITa(N_3)/2_n,N N — 3

~F(N/2)F(n+1—NI2)’ 0~n� 2 (11.33)

ã),=0, n>(N—3)/2; (11.34)

(2) H”(R), N even,

n-I F(N/2—n) 0 N= (_1)N/2 /3N/2-n-1,N F(N/2) , ~ n ~ ~- —1, (11.35)

1—2k 1’

n B2k(2 — ‘)I3N/2+k—n—1,N N

n�— (11.36)kn+1-N/2 kF(N/2)F(n+1—N/2) ‘ 2

(3) HN(C), N=4,8,12,..., -

F(N/2—n) Na,=(—1) YN/2-n-2,N F(N/2) , 0sn~~- —2, (11.37)

n —B2kYN/2+k_fl_2,N N —aN/

2_I = 0; (11.38)kn+2-N/2 kF(N/2)F(n+ 1— N/2)’ n �

(4) HN(C), N=6, 10, 14,.

F(N12—n) Na~=(—1)ON/2_n_l,N F(N/2) ‘ 0~n~~-—1, (11.39)

a B2k(21~2”—

~/ N/2+k—n—1,N N(11.40)

k-n+1-N/2 kF(N/2)F(n+1—N/2). ‘ 2 ‘

(5) H”’(H), N= 8, 12, 16,. .

n+lan—(1) ~N/2_n_l,NT(2)1T(~2), n�N/2—1, (11.41)

1—2kB2k(2 — 1)Mw/2+k—n—l,N N

an n�— (11.42)kn+1-N/2 kF(N/2)F(n+1—N/2) ‘ 2

(6) H16(Cay),

~ n+1

â’n=(~1) Th_nT(8fl)hT(8), 0�ns7, (11.43)

an ~ B (

21_2k — l)n7+k~— — 2k’ n�8. (11.44)kn-7 kF(8)F(n—7)

These relationscan be checkedin several ways. In eachcasewe have, as expected,a,0 = a0 = 1.


SettingN = 3, 2, 4 in eqs. (11.33),(11.35) and(11.37),we reobtaineqs. (11.16),(11.17) and (11.18),respectively.We can also checkthe relationi1 = p

2 + R’/6, by taking for the curvaturethe oppositeofthe valuesgiven in eq. (8.67) for the compactspaces.

The heatkernel coefficientsfor the compactrank-oneSS areobtainedby applying eq. (11.3), i.e.,multiplying by (—1)~.From eq. (11.34) we see that the heat kernel expansionof K on odd spheresterminatesat ~(N

3)/2’ in agreementwith our earlier result (8.52). The coefficientsa~,calculatedfromeq. (11.2), agreewith the valuesobtainedin ref. [23].

11.3. Scalarzetafunctionsfor arbitrary coupling

We nowpassto the calculationof the zetafunction on the compactrank-onesymmetricspaces.Thecaseof sphereshas been consideredin refs. [26, 47], in one-loop calculations for minimally andconformallycoupledscalarfields.

The zetafunction of an operatorA with eigenvaluesan and degeneraciesd~is the Dirichiet series

= ~‘ d~Ia~, (11.45)

wherethe primemeansthat the zero eigenvalue(if thereis any) must be omitted. It is relatedto thepartition function

ZA(t)=~dfle’~, t>0, (11.46)

by a Mellin transform

~ ft~[ZA(t)—do]dt, (11.47)

wherewe assumea0 = 0 with multiplicity d0.The zetafunctionof a compactRiemannianmanifold M, as definedin section4, is the zetafunction

of the non-negative,self-adjoint operator— L, whereL is the Laplace—Beltramioperatoracting onscalarfields. As will be seenin the nextsection,this zetafunction allows oneto calculatethe one-loopvacuum energydensity of a minimally coupledscalarfield on a static spacetimeR x M, and in aKaluza—Klein backgroundM

4 x M. In the case of arbitrary coupling ~, the relevant operator isA = — L + ~R, whereR is the curvaturescalarof the space.

Given a compact (simply connected)rank-one symmetricspaceM = G/H, we shall consider thefollowing operatorsand zetafunctions:

2 2 d~A—L, a~”(n+p) —p , ~(z)=L, 2 2z’ (11.48)[(n+p) —p1

in the minimal case;

A = —L+ p2, a~= (n + p)2, ~(z) = ~ 2z (11.49),,=~(n + p)


for the coupling 4 = p2/R;

A—L+~R, a~(n+p)2—p2+~R, ~ (11.50)

in the caseof arbitrarycoupling. Thedegeneraciesdn havebeengiven in eq. (10.13), and theelementp = (N + m

0 — 1)/4can be explicitly calculatedby usingthevaluesof m0 in table8.1 of section8. Thecaseof the coupling 4 = p

2/R hasbeensingled out becausethe correspondingzetafunction ~ takesaparticularly simple form.

The series(11.48)—(11.50)convergein the semiplaneRe z> N/2, whereN = dim G/H, and thebasicproblemis to find the appropriateanalyticcontinuationfor the othervaluesof z, in particularforRez <0. We shall first calculate~ in termsof a (finite) sum of ordinary Riemannzetafunctions, andthenexpress~ and ~ in terms of ~. Notice that on N-spheres,and only in that case,the couplingp2/R coincideswith the conformal couplingin N + 1 dimensions.Indeed,usingthe valuesR = N(N — 1)andp=(N—1)/2 we get on SN

(p2/R)ISN= (N — 1)/4N = ~conformai(N+ 1) . (11.51)

Therefore,~‘ is basicallythe zetafunction for aconformallycoupledscalarfield on the staticspacetimeR x 5”.

Since ~ is just the Mellin transformof the partition function

Z(t) = VG/HK(t) (11.52)

(VG/H being the volume of the space),we can substitutein eq. (11.47) the recursions(11.15) (andsimilar ones for the other compactspaces),to derive recursionrelationsfor ~.Eventually, twill berelatedto the samequantityon the spaces~3 ~2 and P4(C),whereit can be easilycalculatedin termsof the Riemannzetafunction ~R’ Let usseethis first. On ~3 we havep = 1, dn = (n + 1)2, and(11.49)gives

2z-2 =~(2z—2). (11.53)n0 (n + 1)

This is relatedto the zetafunction on the circle by ~s(z)= ~ ~‘~i(Z — 1), sinced~= 2, an = n2, andp = 0

on ~1, so that

= ~5i(z)= 2~R(2z). (11.54)

On ~2 we havep = ~, dn = 2n + 1, andwe get

1 2z-1 =2~(2z—1,~), (11.55)n0 (n + ~)

where~R(z, q) is the incompleteRiemannzetafunction,

1 ~, Rez>1. (11.56)n0 (n + q)


Using [71]

~R(z, ~)= (2z — 1)~R(z), (11.57)

we get on the two-sphere

~2(Z) = 2(22~— 1)~(2z—1). (11.58)

On P4(C) we have p = 1, d~= (n + i)~,and we get

~P4(c)(z) = 2z-3 = ~R(2z —3). (11.59),,=o (n + 1)

Considernowthe caseof S”’, N odd. We substituteeqs. (11.52) and (11.15) in (11.47) (with d0 = 0,

since there is no zero eigenvaluefor the operatorL — p2), and integrateby parts each differential

operatorin the product

1 (N_3)/2V = (N—3)/2~N(z) = 22~22F(N!2)V~s f t~ TI (a, + 12)2

5s(t)dt

1 (N_3)/2V (N—3)/2

= 2N-2~22F(NI2)Vc, I ( ~, —~ + 2)tzl)2 (t) dt. (11.60)

The boundarytermsall vanishat infinity, becauseof the factorse’~”in Z. At zerotheyvanishas well,becauseof the conditionRe z> N12, requiredfor the convergenceof thesum in (11.49).For example,after the first integration by parts has been performed, the boundary term is proportional tot~1ZSN_2(t),which, for t--+0, behaveslike tzN/2, in view of the asymptoticform ZSN_2(t) t_(N_2)/2,

valid nearzero. —We now use eqs. (11.26) and (11.32) to write explicitly the polynomial multiplying Z~sin eq.

(11.60).Puttingin the valuesof thevolumes,andusingeq. (11.53),we finally obtain~on S”, Nodd, intermsof a finite sum of Riemannzetafunctions,

— 2(— l)(N3)/2 (N—3)/2~N(z) = 1(N) akN~R(2z— 2k — 2). (11.61)

The other rank-one spacesare treatedin a similar way. The recursionsfor the compact partitionfunctions [analogousto (11.15)] are substitutedin eq. (11.47), multiple integrationsby parts areperformed, and then the definitions (11.27)—(11.31)are used,along with (11.58) and (11.59). Theresultsare given below.

2’— 1 \(N_2)12 N/2—~N(Z) = ~ ~ ~N(22~~ — 1)~R(2z— 2k—i), N= 2,4,..., (11.62)

‘ ~ k=0

2 N/2-2~pN(c)(Z) F(N12)F(N12+1) ~ Yk,N~R(2z 2k 3), N4,8,12 (11.63)


N/2—1

~pN(c)(Z) = F(N/2)F(N/2+ 1) ~o 6kN(2 — 1)~(2z —2k—1), N =6, 10, 14,.(11.64)

N/2—I

~pN(H)(Z) = F(N/2)F(N/2+2) ~o jhk,N(2 — 1)~(2z— 2k —1), N=8,12,16,...,

(11.65)

~pi6(cay)(z) = 420F(12) ~~k(22~2~1 — 1)~(2z—2k—i). (11.66)

Thesameresultsfor ~areobtained,moredirectly, by expressingthe degeneraciesdn aspolynomialsinthe variable (n + p), by comparing eqs. (10.32)—(10.37) with the definitions (11.26)—(11.31).Forexample,on

5N N odd, we get

if_i \P~ ~

1~ -‘.7 2k+2dn= (N—i)! k-O’ , (11.67)

with similar relationsfor the otherspaces.Insertingeq. (11.67) in (11.49),andremembering(11.56),we obtain~asa sumof incompleteRiemannzetafunctionsof orderp, which is equivalentto (11.61),aswe shall seelater. —

It is interestingto investigatethe analyticstructureof ~‘. Rememberingthat ~R is only singular atz = 1, where it hasa simple pole with residue1, we find that ~is a meromorphicfunction with simplepoles at

z= ~N, ~N—1,. ~,~, sN, N odd, z= ~N, iN—i,... ,2, pN(C) N=4,8,12,...,(11.68)

z= ~N, ~N—1,. . . ,1, otherwise.

The residuesat zn = ~N — n are

res~NIz~= ü’~/[(4IT)”2F(N/2 — n)], (11.69)

in” I a~dx=VG/Ha. (11.70)G/H

As we can see,the locationof the polesis differentfrom that of ~(z)[givenin eq. (4.17) for a compactRiemannianmanifold, see also eq. (11.74) below]. In particular for N odd, i.e., in the split-rankodd-spherecase,thereis only a finite numberof poles,correspondingto the fact that the Minakshisun-daramexpansionof K terminates[seeeq. (11.34)].

Let us now consider the minimally coupled case. The structure of the series (11.48) is morecomplicatedthanthat for ~,but, with simplemanipulations,we canfind the analyticcontinuationof ~interms of a seriesof Riemannzetafunctions.Directly from eq. (11.48) we have,for Rez> ~N,

~(z)= ~ (n +)2Z [~- (n ~ p)2]~ (11.71)


Since pI(n + p) < 1 we can usethe binomial expansion,

(1- x)2 = T(z+n)f (11.72)

which convergesfor any z provided x~<1. Using this in eq. (11.71),exchangingthe two series,andrememberingthe definition (11.49) of ~, we obtain

~ F(z + n) 2n,~ \ [p ~(z+n)—p ~-]. (11.73)

,,0 n.~

The term ~ 2z is subtractedoff in (11.73) becausein the minimally coupledcasethere is a zeroeigenvalue,which is omitted from the definition of zetafunction. As we shall see later, this term iscrucial in proving the convergenceof theseries(11.73).Notice that the analyticstructureof ~‘ is exactlyas we wouldexpectfrom generaltheory. Indeedusingeqs. (11.68)and (11.73)we easilyfind that ~hassimple poles at

~ Nodd, z=~N,~N—1 1, Neven, (11.74)

in agreementwith the result (4.17) for a compact manifold without boundary. The residues atz,, = ~N—n are given by the sameformulas (11.69) and (11.70) with a~in place of a),, in agreementwith standardzeta function theory [105, 130]. Furthermoreusing eq. (11.61) we find that on oddspheres~(z) vanishesat z = —1, —2,. . . , a result which was first demonstratedby Minakshisundaramin 1949 [1011and which holds for arbitrary odd-dimensionalmanifolds [105]. -

In order to prove the convergenceof (11.73),we first observethat the relations(1l.61)—(11.66)for ~remainvalid if we makethe following replacements:

— 2k — 2)—p ~‘R(2Z— 2k —2, ~) ~N N odd,

— 2k — 3)—* ~R(2z — 2k —3, p), PN(C), N = 4,8,..., (11.75)

~R(2Z2kl,

2)~~R(2Z2kl,P), otherwise,

where ~R(Z, q) is the incompleteRiemannzeta function (11.56). Take, e.g., SN for N odd. Thenp = (N—1)/2 andby (11.26) we see that the polynomial

p—i p—i

P(t) = ~ akNt = if (2_ t) (11.76)k=0 j=1

satisfies

P(1)=P(4)=P(9)=.~.P((p—1)2)—0. (11.77)

From this it is clear that the terms by which the ordinaryand the incompleteRiemannzetafunctionsdiffer, do not contribute to the sum over k in (11.61),proving eq. (11.75) in this case. A similar proofworks for the otherspaces.


Thusif (11.61)—(11.66)are substitutedin (11.73)we get “the” zetafunction of the rank-oneSS intermsof a seriesof incompleteRiemannzetafunctionsof order p.

(1) SN,Nodd,p(N_1)/2,

1 Pl__________ __________ 2k+2—2z2(—1)’~ ~ ak ~ F(z+n) 2n‘(z) 1(N) k=0 N n0 n!F(z) ~ + 2n — 2k — 2, p) — p ]; (11.78)

(2) SN,Neven,p=(N_i)/2,

—1/2 ~112 ‘~ F(z + n) 2n_____________ __________ 2k+1—2z2(—i~ ~ ~ n!F(z) ~ ~R(2z+2n2k1,p)p ]; (11.79)1(N)

(3) pN(C) N=4,8,. . . , pN/4,

—2 ________________________________ __________ 2k+3—2z

___________ F(z+n) [p2n~(2z+2n_2k3, p)—pF(N/2)1(N/2+i) k0 n0 n!F(z)

(11.80)

(4) pN(C) N=6,i0,..., pN/4,

2p—i2 _______________________________ __________ 2k+ 1 —2z~(z)= ~ 8k,N ~ F(z+ n) [2n~(2Z + 2n —2k — 1, p) — p ];

F(N/2)1(N/2+1) k0 n0 n!F(z)(11.81)

(5) pN(H) N8,12,..., p(N+2)/4,

_______________________ __________ 2k+1—2z—2 ~k N ~ F(z + n) [2n~(2Z + 2n — 2k —1, p) — p ];~(z)= _____________ ______

F(N/2)1(N/2+2) k=0 n=0 n!1(z)(11.82)

(6) P16(Cay), N= 16, p = 2~

_______ F(z+n) ~i 2n__________ _________ 11’ (1l\2k+i2Z— 71k = n!F(z) [(T) ~j2z+2n—2k—l,T)—~T) ]. (11.83)~(z) — ________ _______4201(12)k=O n

In eq. (11.78) the identity

_________ 2k+akNp 21 (11.84)1(N) k=0

[obtainedby settingn = 0 in eq. (11.67)] hasbeenused,with similar onesfor the otherformulas.The convergenceof theseseriescan now beprovedby rememberingthat, for n—~ co, the incomplete.

zetafunction ~R(2n + a, p) behaveslike p~2”~”[1+ B(n)], whereB(n) is a sumof termsof the form~ +a with IxI <1. Therefore, the constantterms (independentof n) in the squarebracketsin eqs.

82 R. Camporesi,Harmonicanalysisandpropagatorson homogeneousspaces

(il.78)—(11.83) drop out, and the series convergelike the binomial expansion(11.72), for bothRez<0 and in the strip 0~Rez<~N [except,of course,at the poles (11.74)].

On the other hand for Rez> ~N the argumentsof the incompletezetafunctions havereal partslargerthan 1 for anyvalueof k, andwe can use(11.56)for ~R(z, q) to prove that eqs. (li.78)—(ii.83)areequivalentto (11.71). In conclusion,we haveobtainedthe analyticcontinuationof ~ for anyvalueof z. In the case of spheres, an equivalent analytic continuation in terms of integralswith Besselfunctionswas obtainedin refs. [101, 103], see also ref. [26].

Let us briefly considerthe caseof general coupling. Starting from (11.50), we proceedas in theminimal case, i.e., we factor (n + p)2 out of the squarebracket,and try to use the binomial expansionfor the remainingterm. The following discussionis needed:

(1) 0< ~R <2p2 in this case we have p2 — ~ <p2, implying p2 — ~ <(n + p)2 for any n, andifwe use (11.72) we get

~(z) = (p2 - ~R)~(z + n), (11.85)

which convergesfor any z exceptat the poles (11.74) (in the same way as for ~‘, exceptfor the value= p2IR).The convergenceis provedby rememberingthat, for n—* ~,~‘(z+ n) ~2nHZ Substituting

the values (11.61)—(11.66)of ~‘, we get a series of incomplete Riemannzetafunctions of order p,analogousto (li.78)—(ii.83).

(2) ~R� 2p2 in this case we define ii~as the smallestinteger such that p2 — ~ <(n~+ p)2, and weuse(11.72)in (11.50) for the termswith n � ñ. Forexample,for ~R= 2p2 we haveh = 1. The result is

d~ ‘~ 1(Z+n)(p2—~R)” 7- ~‘ d,~(z) = ~o [(n + p)2 — p2 + ~R]Z + n!1(z) ~~(z + n) — i=o (I + p)2Z+2fl

(11.86)

Convergenceof the seriesis againclear from the fact that the term in largeparenthesesbehaves,forn—~co, as (ñ~+ p)_2fl2Z, Substitutingthe values of ~,we get a seriesof incomplete Riemannzetafunctionsof order ,~+ p.

11.4. Spinorzetafunctionson

The zetafunction for (massless)spinorfields on G/H can be obtainedin a similarway. The resultissomewhateasierto derive than that for minimally (or arbitrarily) coupled scalars,as the calculationparallels the one carried out for ~. We shall restrictourselvesto the case of

5N for both odd andevenN.

Theeigenvaluesandthe degeneraciesof the squaredDiracoperatoron odd sphereshavebeengivenby Candelasand Weinberg[26],

= (n + ~N)2, (11.87)

2(~~÷1)/2 N 1

= n!(N— , n 0,1,2 (11.88)

For N even the only change is the replacement2(N+1)/2~2N/2+1 in d~

1’2~(see ref. [87]). Thedegeneraciescoincide with the dimensionsof the spinor representationsof SO(N+ 1), up to an extra


factor of two. This is becausethe numbers±(n + ~N) areboth eigenvaluesof the Diracoperator[8]. Asimple calculationrevealsthat

d~112~= ~ + ~N) , (11.89)

(N+1)where[N/2] = (N — 1)/2 for N odd, [N/2] = N12 for N even,anddn arethe scalardegeneraciesofthe Laplacian on

5N-f 1 Using eq. (11.67) and the analogousrelation for the evenspheres,we canexpress~ as a polynomial in the variable (n + N/2). The result is

2~+1~2—1 (N—1)/2 (N—1)/2d~”

2~= (N—1)! i~o 13k,N+l(n + ~N)21’, N odd, (11.90)

N/2+1 N/2—1 N/2—1

d~”2~= (N—i)! ~ akN+l(n + ~N)2’~ , N even. (11.91)

Using theserelationsin the definition of zetafunction,

= ~ d2~/A~, (11.92)

we obtain the spinor zeta function on5N in the form of a finite sum of incompleteRiemannzeta

functionsof order N/2,

2~+1~~2—1 (N—l)/2 (N—1)/2= (N— 1)! ~ 13k,N+l~R(2z —2k, ~N), N odd, (11.93)

2~~~2+1N/2—1 N/2—1~~

2~(z) (N—i)! akN+l~R(2z 2k—i, ~N), N even. (11.94)

As in the scalarcasewe can replace

~R(z, ~N)—*~R(z,fl, N odd, ~R(z, ~N)—~~R(z), N even, (11.95)

without alteringthe results. Notice that ~/2) hassimple poles at

~ Nodd, z=~N,1N—1,...,2,i, Neven. (11.96)

Sincefor odd N ~ /2) hasonly a finite numberof poles, thespinor heatkernelexpansionon theoddspheres

52m+1 terminates(to the coefficient am) and is exact, in analogyto the result for conformallycoupledscalars,in N + 1 dimensions.Furthermorefor N odd ~~ij”

2~(z)vanishesatz = 0, —1, —2

Specializingto N = 1, 2, 3 gives

= 2~R(2z,~), (11.97)

= 4~R(2z— 1), (11.98)

~~1/2)(z) = 2[~R(2z—2, ~)— ~~R(2z, i)]. (11.99)

84 R.Camporesi, Harmonic analysis and propagators on homogeneous spaces

12. Finite-temperature quantum field theory in higher dimensions

In this sectionwe calculatethe finite-temperatureone-loop effective potentialfor scalarand spinorfields in ultrastaticspacetimesMD = R x M” usingzetafunction regularisation.The caseof scalarfieldshasbeenconsidered,e.g., in refs. [51, 47]. Here we shall generalizethe formalism to spinor fields aswell (seealso ref. [66]). The Kaluza—Klein spacetimeMD = M4 x V”~,where M4 is Minkowski spaceand V”’ an N-dimensionalcompact manifold, will then be considered,and the (zero-temperature)Casimirenergycomputedin closedform whenV” is a rank-onesymmetricspace,generalizingpreviouswork with spheresas internal manifolds [26, 47, 8, 87]. Consistencywith the quantum-correctedEinstein equationswill be discussedat the end.

12.1. Thescalar case

Considera (d + 1)-dimensionalultrastaticspacetimeR x M”, with signature(—1, 1, . . . , 1), and a

scalarfield ~ which propagatesaccordingto(LII—~R—m2)~=0, (12.1)

where R is the curvaturescalar and D = —a~+ Ld, Ld being the Laplace—Beltramioperatoron Ma.The pointsin spacetimeare labelledby x = (x

0, y), wherey E Md. For simplicity, we shallassumethatMd is compactwithout boundary.

The finite-temperature(T = /3 ~1) one-loop effective potential is given by [51, 47]

V(f3) = ~ urn ~2z~a+i(z,f3)Iz , (12.2)

where ~ is the tracedfinite-temperaturezetafunction on spacetime,and p. is anarbitrary massscale.[As will be seenlater~‘(z,/3) hasdimensionsof (mass)l_2z.]V(/3) can be identified with the free energyF of the quantumfield; the internal energyEand entropyS arethen relatedto V( /3) by the standardthermodynamicidentities

E(13)=af3V(f3)laj3, (12.3)

S(f3)= /32 aV(/3)/af3. (12.4)

At zero temperaturewe have

E(co) = F(ca) = V(~), (12.5)

for the one-loopCasimirenergy.The relationbetweenthis approachandthe conventionalzeta-functiontechnique[73,65, 66] can be seenby rememberingthat in the latter the field theoreticdivergencesareavoidedfrom the beginning,defining V as the finite quantity

V= ~i[~’(0) + ~(0) ln(p.2)] , (12.6)

where~is thezetafunction constructedfrom the eigenvaluesof thedifferentialoperatorthatappearsin


the pathintegral. For finite-temperaturefield theory, the pathintegral is over field configurationsthatare periodic in imaginarytime [18].

On the other hand, in the approachto the zeta function usedin refs. [51,47] (which is the samefollowed here),the infinities aredisplayedin a single pole, with residueproportionalto ~(0).Indeedtaking the limit in (12.2) gives

V(f3)’ ~i[C’(0,/3)+~(0,/3)(2lnp.+i/z)]. (12.7)

It is well known(andwill be provedlater) thatthe T ~ 0 partof thezetafunctionvanishesat z = 0,and that the same is true for the zero-temperaturepart, provided the total dimensionality D ofspacetimeis odd.

Howeverwhen D is eventhe zero-temperaturezeta function doesnot generallyvanish at z = 0(exceptionsto this will be consideredlater). In this casetheconventionalzetafunctionrenormalization[73]simply amountsto droppingthepole termin (12.7),ascanbe seenby comparingwith (12.6).Thisprocedurewill beusedwhenwe calculatethe vacuumenergydensityon M4 X V~’for evenN. For thetime being we adopt the definition (12.2),and keepthepole termsin the effective potential.

The zeta function can be convenientlycalculatedin termsof a Mellin transformof the partitionfunctionZ~,the integraloverM’ ofthecoincidencelimit of the finite-temperatureheatkernelK~.Thiskernel satisfiesthe samedifferential equationas thezero-temperatureone,

(—a, + LI — — m2)K~(x, x’, t) = —6(t)3(x, x’) (12.8)

(t is here theproper-timeparameterin the heatequation,not to be confusedwith the real timex0) with

periodic boundaryconditionsin imaginary time with period /3

K~(x0—x~+in/3,y,y’,t)=K~(x0—x~,y,y’,t). (12.9)

It follows thatupon a Wick rotationof x0 to imaginary time, the finite-temperatureheat kernel is thesame as the zero-temperatureoneon the background~1 X Md, i.e., omitting an overall stepfunction0(t)

/ .~ exp[(i/4t)(x0—x~+in/3)2]\

K~(x0—x~,y,y’,t)~~ 1/2 )K~(y~y’~t). (12.10)

(—4irt)

The quantity K~is the heat kernelon the operatorLd — — m2 on M”, with partition function

Zd(t) = ~ d. e’°’~ (i2.ii)

where,given the eigenvalues— A. of Ld with degeneraciesd1, we havedefined

w~=A1+m2+~R. (12.12)

(We assume � 0 hereafter.)The finite-temperaturepartition function is then


Z~(t)= (—4ITt)~2 ~ dj:xp(_~_— tw~) (12.13)

= ~ ~d1exp[_t(

4~ + ~ (12.14)

wherea theta-functioninversion hasbeenused(see section 6). We shall assumefor generality thatthereis a zero-mode,i.e., that w

0 = 0 with multiplicity d0. UnlessspecialmanifoldsM’~arechosen,thiswill only happenwhenm

2 = 0 = ~, i.e., in the “massless”minimally coupledcase.It is knownthat thezero-modesmustbeomittedfrom the definition of zetafunction andconsideredseparately.This impliesthat the Mellin transformdefining ~(z, /3) is

~d+l@, /3) = ~ Jdt tz-1(Z (t) - ~o) (12.15)

Indeed,if we substituteeq. (12.14) in (12.15) we get the familiar eigenvaluerepresentation[47, 73]

d-2 2 2 2 (12.16)

/3fl.1 (4irn//3 +w~)

wherethe primemeansthat the term with n = 0 andj = 0 is omitted. Notice that since/3_i and havedimensionof mass,~(z, /3) hasdimension(mass)l_2z.

As usualthe series(12.16)will convergein the semiplaneRe z> (1 + d)/2, andweneedto find theappropriateanalyticcontinuationfor the othervaluesof z. In orderto do this we return to the Mellintransform(12.15) and use the following identity:

Z~(t)— id0/f3 = (—4ITt)”

2 ~ ~ d1 exp(_~ — ~ + ~ exp(—t 4~i2), (12.17)

j~0 n�0 /3

which is easily proved from eq. (12.14). In the zero-temperaturelimit /3 = ~, and we only get acontributionfrom the n = 0 term in the first summationin (12.17). Using eq. (12.17) in (12.15) andseparatingout the zero-temperaturepart we get

~d+l(z, /3) = ~d+l(z, ~)+ (2idOI/3)(~)~R(2z)

d 22+ p fdttz_3/2exp(_~_~__tw~) , (12.18)

(

41T) n1 ~ (z) 4t0

where~R is the Riemannzetafunction. The zero-temperaturezetafunction is given explicitly by

~d+l(z,x) = [i/(4IT)112] 1(z — ~)~d(z — ~)IF(z) , (12.19)


where

= ~ d1/w~, (12.20)j~~0

is the zetafunction of the operator—Ld + ~R + m2 on M”.

Thelast term in eq. (12.18) representsthe T ~ 0 partof the zetafunction which doesnot dependonthe zero modes,andwill be denotedby ~~+

1(z,/3). Using [71]

J t~ e’~2~’dt = 2(x/2)~K~(x), (12.21)

whereKR(x) is a Besselfunction with imaginary argument,we obtainthe following expressionfor ~:

5/2—s d —

= V41(z) ~ ~-‘ ~ (n$’)’2KI/2~Z(n/3~). (12.22)

This is alreadyan analyticcontinuationof the finite-temperaturezetafunction. Anotherone, thatwillbe usefullater, can be obtainedfrom the following integralrepresentationof K~[71]:

K~(z)= ~ f e~(x2— 1)~2 dx, (12.23)

which holdsfor Re i.> —1/2. Using this in (12.22) we can do the sum over n and we get

- — 2i sin(lrz) ,~, d, ~ (x2 — 1)~dx~d+l(Z, /3) — ~ 2z—1 j ~xco. ‘ ( 4)‘iT froW

1 1 e

valid for Re z< 1. Fromeq. (12.24) or (12.22) it follows immediately that

~d+1(P,/3)”°, p=O,l,2 (12.25)

In agreementwith our earlier discussion,thisimplies that the divergencesin the effectivepotentialarecontainedeither in the zero-temperaturepart or in the zero-modepart. Indeed,if eqs. (12.18) and(12.24) [or eq. (12.22)] aresubstitutedin (12.2),we find the statistical-mechanicsform of V(f3) [47],

V(/3)=V(cc)+ ~ (~+2ln(~I3))+~~d1ln(1—e~’1), (12.26)

from which it is clear that the last term is the ordinary free energyof the bosonfield. For examplethefree energyof a scalarfield on the Einstein universeR x S

3 is [1]

F(f3) = ~ (n + 1)2 ln(1 — e~), (12.27)

88 R. Camporesi, Harmonic analysis and propagasors on homogeneous spaces

with eigenvalues

= (n(n +2) + 6~+ m2), (12.28)

in termsof the radiusa of S3 and of the massm of the scalarfield.The zero-temperatureeffectivepotential on R x M” is

1 . p.2Z1(Z_~a(Z_~)V(~)=_

2(4)l/2 ~ F(z+1) (12.29)

Rememberingthe analyticproperties(11.74) of the zetafunction on a compactmanifold,we seethat ifd is even(odd-dimensionalspacetimes),the one-loop (T = 0) effective potential is finite. If d is odd(even-dimensionalspacetimes),thereis an ultraviolet divergencein V(cc), since ~ hasa simplepoleatz = — 1/2, with residueproportionalto the heat kernelcoefficient a(d+ 1)/2~As alreadyremarked,in theconventionalzeta-functiontechniqueonesimply dropsthe pole term.

Concerningthe zero-modepart in eq. (12.26),we seethat thepresenceof zero-modeshasimportanteffects, namely the appearanceof a temperaturedependentpole (to be removedby renormalisation)andof a term dependingon the arbitraryscalep.. Notice howeverthat the internalenergy[givenby eq.(12.3)] is not affected by this pole. Moreover (as we shall see later) the pole disappearson thespacetimeM’~= M

4 x VN, i.e., when thereis a noncompact(flat) componentin Ma.

12.2. The spinor case

Formulasfor spinorfinite-temperaturefield theory on R x M” can be obtainedin a similar way. Theone-loopeffectivepotentialis definedas in eq. (12.2),with achangeof sign coming from the functionalintegral over anticommutingfields, and with the spinorzetafunction ~(1/2) in place of the scalarone.~(1/2) is the tracedfinite-temperaturezetafunction constructedfrom the eigenvaluesof the squaredDirac operator,~‘2 It is known [39] that

50’2=L1—R/4, (12.30)

whereR is the curvaturescalar,and in the ultrastaticcase

(12.31)

where Ld is the Laplace—Beltrami operator on Ma acting on spinors. The eigenvaluesand thedegeneraciesof the operator~La + R/4will be denotedby w~andd.. For exampleif Ma is a compacthomogeneousspaceGIH, we can expressw~in termsof the second-orderCasimirinvariantson G andH (ref. [133], see also appendixB)

= C2(A1) — C2(’r) + R14, (12.32)

whereT is the spinorrepresentationof H (defining spinorfields on G/H) andA1 aretherepresentationsof G which, restrictedto H, contain ‘r. (We areassumingherethat Ma is orientableandadmitsa spinorstructure,i.e., that the secondStiefel—Whitneyclassof Md vanishes,see,e.g., refs. [53, 112].)


The finite-temperature(massless)spinor Green’sfunction K~/2) satisfiesthe heat equation

(—a, + ~ x’, t) = —5(t)ô(x, x’) , (12.33)

with antiperiodic boundaryconditionsin imaginary time,

K~2~(x0— x~+ in$, y, y’, t) = (—i)”K~

112~(x0— x~,y, y’, t) . (12.34)

The spinorpartition function is then

Z~(t)= (—4~t)112n~ ~ (_i)nd

1~ (12.35)

= ~ d1 exp{—t[(4ir/f3)2(n + ~)2 + w~]}, (12.36)

wherea theta-functioninversionhasbeenused [seeeq. (6.36)].We shallassumethat thereis a zero-mode~ = 0, with degeneracyd

0. Thiswill in generalnot bethecaseif R>0 and the Levi-Civita connectionis usedon M”, sinceon a compactRiemannianmanifold—Ld is positivesemidefinite,so that —Ld + R/4 is strictly positiveif R >0. Howeverallowing a torsionon M’~we can havezero-modes,for exampleif the manifold is parallelizable(e.g., Lie groups,S

7) andthe Cartan—Schoutenconnectionis used [37, 24].

Inserting eq. (12.36) in the Mellin transform

~z, /3) = ~ f dt t~Z~(t), (12.37)

we find the eigenvaluerepresentationof the spinorzetafunction,

d.~ /3) = — 2 2 2 2 z (12.38.13 n,j [(4ir /13 )(n + ~) + w.]

No termsare now omitted from the sum becausethere are no zerosof the denominator,even for

= 0. However the zero modescan be dealt with separatelyand we easily get a formula which issimilar to the one derivedin the scalarcase,

~ 13) = ~ oo) + (2ido//3)(/3/21T)2~~R(2z,~)d

+ ~±:~~_ ~ 2z—1 ~ (—i)~(n/3w1)~

112K112_5(n/3w1), (12.39)

v4irI(z) j~é0 w• n=1

where~R(z, q) is the incompleteRiemannzetafunction and the zero-temperaturepart is given by thesameformulas(12.19) and (12.20),with the spinoreigenvaluesanddegeneraciesin placeof the scalarones.Using eq. (12.23) in (12.39) we can do the sum over n andrewrite the last term as

90 R. Camporesi, Harmonic analysis and propagatorson homogeneous spaces

- —2isin(lTz) d. f(x2—i)~dxd-1-i ‘ — ‘~.‘ 2z—l I $xw

‘i7~1�0W1 ~ e ‘+1

which holdsfor Re z< 1. Usingthis and(12.39) in the definition (12.2) of the effectivepotential(with achangeof sign), we get the statistical-mechanicsform of V”’

2~(/3),

= v~’~2~(~)— d0 ln 2 — ~ d1 ln(1 + e~’), (12.41)

13 13~

wherethe last term is clearly the free energyof the fermion field. For examplethe free energyof amasslessspinorfield on the Einstein universeis [1]

F~112~(/3)= — ~ 2(n+ 2)(n + 1) ln(1 + ~ (12.42)

All of the divergencesare now containedin the zero-temperaturepart and the commentsmadeabout eq. (12.29) arerelevanthereaswell. There is no polecomingfrom the spinorzero modes,whichdo not contributeto the internal energybut only to the entropywith the constantamount d

0 ln 2.

4 N12.3. Theeffectivepotential on M X V

We now specializeto the casewhereM” is the productR’’ X VN, or equivalently,spacetimeis theglobal productM’ x V”’, whereM’ is an 1-dimensionalMinkowski spaceand V”’ is an N-dimensionalcompactmanifold without boundary.We areinterestedin the free energydensityon M’, andthereforethe partition andzetafunctionsareintegratedover the “internal” spaceV”’. The correspondingresultscan be obtainedby properlymodifying the previousformulasto allow a noncompactcomponentin Ma[i.e., integratingover the (1— 1)-dimensionalwavevectorin R’~].

We can proceedmore directly in the following way. Considerfirst the caseof scalar fields. Thepartition function on Ma now takesthe form

Zd(t) = ZIi(t)ZN(t) , (12.43)

Z1_1(t) = (4irt)Ul~2 , (12.44)

ZN(t) = ~ d. e”°~. (12.45)

Herew and d1 arethe eigenvaluesandthe degeneraciesof — LN + ~R + m

2, whereLN is the Laplacianon V”’. Allowing a zero mode in V”’, with degeneracyd

0, we now have, in place of (12.15),

~~1(z, /3) = ~ Jdtt21Zi1(t)(~ ~ — ~). (12.46)

Using the identity ZN = ZN — + d0, doing a theta-functioninversion, and separatingout the n = 0


term,we get

~d+l(z, 13) = (4~y(ll)/2 1(z —(1—1)12) ~N+IZ — (1—1)12,13), (12.47)

where ~N+l(z,/3) is definedby (12.18) replacingthe dimensiond by N. Using eq. (12.24) for thefinite-temperaturecorrection ‘N+1’ we obtain the following expressionfor ~a+i:

~d+l(z, ~ /3)+~d+1(z, /3), (12.48)

where

i F(z — 1/2)1/2 rI ‘~ ~N(z1/2) (12.49)(4~) .i

is the zero-temperaturepart [~N is definedasin eq. (12.20)with d—~N, i.e., it is the zetafunctionof2 Nthe operator—LN+ ~R+ m on V ],

(d0) = 2id0 132Z_1F(l/2 — z)~R(1 — 2z)~d+I(z, /3) z 1/2 ri ( )i~Z

is the zero-modepart, and

- 2i d1 I (x2 — 1)(11)/2z dx

~~1(z, ~ = (4~~

2F(z)1((l+ 1)/2— z) j~ow~ ~ e~’— i (12.51)

(Rez <(1 + 1) / 2) is the finite-temperaturecorrection. ‘d+ can also be written in termsof a seriesofBesselfunctionsK~,with v= 1/2 — z, and vanishesat zero and at negativeintegers.

Similar formulascan beobtainedfor spinorfields. Assuming(asin ref. [26])that thereareb masslessscalarfields andf Dirac masslessspinorfields in 1 + N dimensions,we havefor the effectivepotential

V(f.3)= ~i lLrn (1/z)[bp.~~~d+I(z, 13) — 2’21fp.~~(z, /3)] , (12.52)

wherethe massscalep.,, for scalars(coming from the path-integralmeasure)is different, in principle,from the correspondingquantity p.

1 for fermions.The factor 2[1/21 comesfrom the trace of the Diracmatricesin M’ ([1/2] is the integerpart of 1/2). Substitutingthe expressionsof the zetafunctions,weobtain the following form of the finite-temperatureone-loop effective potentialon M’ x

V(f3) = V(oo) + V64°~(/3)+ V(13), (12.53)

wherethe zero-temperature,zero-mode,andfinite-temperatureparts are as follows:

V(ce~)= 2(4’iT)’72 ~ .l~~(~+l)~[L~p.~~~(z— 1/2) fp.~2~’21~~21(z— 1/2)], (12.54)

V~0)(/3)= ~ —fd~(~2~’21(21’— 1)] , (12.55)


2 (1-1)12

V(f3) = —1 (b ~ d~w~ (x —1) dx(4~.)(1_1)/2F((l+ 1)/2) ~ ‘ ‘ J exp(/3xw~1’~)— 1

= / 2 1\(/_1)!2 d ~+ 2~21f~ — X) (12.56)

/+0 ~ exp(f3xw1 )+1

Although it is possible to work in general, we now specialize to the physical Minkowski space, I = 4.

Thus, the zero-temperatureCasimir energydensityis finite for N odd andis given by

E(o~) = (—a4I64ir2)[b~(—2)— 4f~~2~’(—2)] , N odd, ~‘(z) ar/az . (12.57)

ForN eventhereis a (ultraviolet) divergence,displayedin simplepoleswith residuesproportionaltothe valuesof the zeta functionsat z = —2,

E(cc) = (—a4I64’iT2){b[~~(—2)+ (liz + ~+ 2 ln(p.~a))~~(—2)]

— 4f[~~2~’(—2) + (liz + ~ + 2 ln(p.fa))~’2~(—2)j}, N even. (12.58)

In eqs. (12.57) and (12.58) the scaling length a gives the size of the internal space. Indeed,sincethezetafunction ~N(z) hasdimensionof (mass)_2z[seeeq. (12.20)], we can scale out a factor a2~4whentaking the limit in (12.54). This factor combines with the mass scales p. to leave a dimensionlessargumentfor the logarithms.

The zero-modescontribution to the free energydensity is

V~°~(/3)= —(‘ii~2I90)/34[bd~+ ~fdV~] , (12.59)

and for the internalenergy,using (12.3), we just needto multiply by a factor —3. Thezero modesdonot producenow any temperaturedependentpole or any term dependingon the scalingparameters.

Finally, the finite-temperaturecorrectionto the free energydensity is

= —~ (b ~ d~b)w~b)4j (x2 — 1)3/2 dx + 4f ~ ~ f (x2 — 1)32dx) (12.60)6’ir j�0 e~°”— 1 j�0 e~’~’+ 1

where,accordingto our previousdiscussion,w~andd. arethe eigenvaluesandthe degeneraciesof theoperator —LN + ~R for bosons,and —L~2~+ Ri4 for fermions, where L is the Laplace—Beltramioperatoracting on scalar or spinor fields on VAT. For completenesswe give the expressionfor theinternal energyE(/3), which follows from (12.3),

E(/3) = E(~)+ (~2I30)/34(bd~ + ~ + ~ (b ~ d~b)w~b)4f x2(x2 dx2’ir /+0 I e13~’ — 1

+ 4f ~d5~~w~4j x2(x2 1)1/2dx) (12.61)j�0 ~ + I


Specific examplesof “internal” spacesfor which the vacuumenergydensitycan be calculatedin closedform will be consideredlater. We now continuewith the general formalism in the case of hightemperature.

12.4. The high-temperatureexpansion

The finite-temperaturecorrectionsin eqs. (12.60) and (12.61)are exact.Howevertheywork betterat low temperatures,where the seriesconvergequickly and can be evaluatednumerically, given aspecific structurefor the internal spaceV”’.

The high-temperaturelimit needsa separatediscussion.We can get an asymptoticexpansionof theeffective potential in inverse powers of the temperatureif we substitute the Schwinger—DeWittexpansionof the partition functionZ~in the Mellin transformdefining ~ + For t—~ 0 we write

ZN(t) (4irt)”~2 ~ Uktk, (12.62)

where uk = ~Nak,~N being the volume of V”’, and ak are the coincidencelimits of the heat kernelcoefficientsfor theoperatorLN — ~R. Forspinorswe havea similar formulawith uk—~~ the tracedheat kernelcoefficientsof L ~ /2) — RI 4. (Thesecan be expressedas polynomialsin the curvature,see,e.g.,ref. [19],p. 172.) If eq. (12.62) is substitutedin (12.46)we get thefollowing expansionfor thezetafunction on M’ x

/ \ — ~ ~—(1—l)/2 1(z —(1—1)/2) /

~d+I~,13)—~1T) p ~‘N~Z 1—i /2)

— 2z+2k—d

+ ~! (4~.)_~2 ~ u~(~_) 1(z +k— d12) ~(2z + 2k — d), (12.63)

whered = 1 + N — 1 andthe first term comesfrom the n = 0 term in eq. (12.46). For spinorswe get asimilar expression:

C~1~(z,/3) = ~ (4~)~a/2~ u~i/2)(~) F(z +k— d/2) ~2z + 2k — d, 1/2). (12.64)

Inserting theserelations in eq. (12.52), taking the limit and specializing to 1 = 4 we get the high-temperatureexpansionof the effective potential.For N odd the result is finite and is given by

(N + 1) / 2

a4V(/3) (4~yD/2[ ~ buk — 4f(22kd— i)u~2~] ~(D — 2k)F(D12 — k)(/3I2a)2~

+ ~ [buk — 4f(22kd— 1)u 2)j 112~(2k— d)F(k — d/2)(/3I2ira)2~]k = (N +5)/ 2

— (/3/a) (4~y~2[bud/2(~— ln(2$/a))+ 4fu~ ln 2] — (b/12~)(/3/a)~pN(3),

(12.65)


whereD = N + 4, d = D — 1, andPN(3) is the finite part of the scalarzetafunction on VN near—3/2,i.e., we write for z—+ 0

- ~)= ~ (4 )N~F( 3/2) + p~(3). (12.66)

Sincethe heatkernelcoefficient Uk hasdimension(mass)2~’~(being proportionalto the residueof ~N atz = N12 — k), a factor~ hasbeenscaledout from eachUk, giving the correctoverall powera4 andleaving the dimensionlessparameter/3/a = (aT) ~. Similarly a factora3 hasbeenscaledfrom PN(3).

For N evenwe get, as expected,a divergencein the form of a simple polewith residueproportionalto UD,2,

(N + 2)/ 2

a4V(/3) (4yDl2( ~ [bu~ — 4f(22~— 1)u~’2~]~(D- 2k)F(D/2— k)(/3/2a)2~

k =0

+ ~ [buk—4f(22~— l)u~2~]~2~R(2k— d)F(k — d/2)(/3/2~a)2kDk = (N -~-6)/2

+ buD2[li2z + y + ln(/3p.6i4~)]— 4fu~[i/2z + y + ln(/3p.1/~)])

- (b/12ir)(/3/a)~,(-4), (12.67)

where y is the Euler constant.As in the zero-temperaturecasethe infinities must be removed byrenormalisation.The first two terms in the high-temperatureexpansion[k = 0, 1 in eqs. (12.65) and(12.67)] are then the same for both odd andevenN andwe find

a4V(/3) —7rm2{[bu() — 4f(2~3— 1)u2~]~(D)F(D/2)(aT)’3

+ [bu1— 4f(2”~ — 1)u

2~]~(D— 2)F(Di2 — 1)~(aT)’~2+ .. }, (12.68)

where the heat kernel coefficientsare [19]

UOQN, Ui—QN(6flR, (12.69)

u~21= QN2~2l, u~2)= —QN2~21Rii2. (12.70)

(Here,R1a2is the curvaturescalarandaNQN the volume of V”.) The scalarpart of the leadingterm ineq. (12.68)coincideswith the high-temperatureterm obtainedin refs. [26, 47]. Thesetermsprovideagood approximation to the effective potentialwhen the temperatureT (or ratherthe dimensionlessparameteraT, wherea is the typical size of the “internal” space)becomevery large. Roughlyspeakingwe havethreebasicregions.For high temperaturetheenergydensitybehaveslike T’3, D being the totaldimensionof spacetime;for smaller T the zero-modecontributionin (12.61) is predominant[sincethefinite-temperaturecorrectionV(f3) vanishesexponentiallyas aT—+0], and E(T) T4 finally for verysmall T we are left with the Casimir energy.

12.5. Explicit zero-temperaturecalculations

From now on spacetimewill be the global productM4 X VN, with VN a compactmanifold withoutboundary. By eq. (12.58) we see that for N even the divergent parts in the Casimir energy are


proportionalto the valuesof the scalarandspinorzetafunctionson V”’ at z = —2. It is astandardresultin zeta-functiontheory that ~N(—2)is proportionalto the heat kernel coefficient UN/2+2. The preciserelation in the scalarcaseis [105]

CN(2) = 2(4~)~2u(N+4)/2, (12.71)

with an analogousstatementfor spinors.We shall first addressthe following question: are there even-dimensionalmanifolds such that

~N(

2) = 0? Besidesthe trivial caseof a torus which, beingflat, hasu,, 0 Vn > 0, anotherpossibility isgiven by a scalarfield on a split-ranksymmetricspacewith a coupling~ = p2/R, wherep is half the sumof the positive roots of the space. Indeed from our discussionin sections7—9, we know that theSchwinger—DeWittexpansionof the heatkernelof the operatorL — p2 terminates,on thesespaces,tothe coefficient a

1, wherej = 0 for Lie groups,j = (N — 3)/2 for odd spheres5”’, j = n(n — 1)/2 forGIH = SU(2n)/Sp(n)and j = 9 for GIH= E6/F4. In eachcasewe have1< (N + 4)/2 andthe scalarzetafunction vanishesat —2.

The sameis trueon a productof suchspaces.For exampleon S”’ x SM, with N andM both odd, thetotal dimensionis even,but the MPSDexpansionfor L — p

2 terminatesat a., with j = (N + M — 6)/2,

and therefore~s~’~sM(2) = 0.

A similar resulton S”’ x SMhasbeenfoundin ref. [70] in the caseof spinors,namely~N~~M(—2) = 0

whenN andM areboth odd. It is possiblethat thisresult for spinorsholdson other(even-dimensional)split-ranksymmetricspacesas well, but this point will not be investigatedfurther.

Rememberingthat the coupling p2/R equals 1/6 for group manifolds (Gaussiancoupling) and(N — i)/4N for Si”’ (conformalcoupling in N + 1 dimensions),we concludethat the one-loopvacuumenergydensityfor scalarsis finite not only for odd-dimensionalinternal manifolds (with anycoupling),but alsofor evendimensionalLie groupswith theGaussiancoupling, andfor the products5” x

5M (NandModd) with the coupling

— (N—i)2+(M—i)2

~4[N(N-i)+M(M-1)]~

In all thesecasesthe scalarpart of E(cc) is given by eq. (12.57).Similar statementsmight be true forspinorsas well.

We nowperformsomeexplicit calculationof the Casimirenergy.By eqs.(12.57) and(12.58) weseethat for N odd E(oo) dependsonly on the internal “radius” a, whereasin the even-dimensionalcaseitalso depends,in general, on the arbitrary mass scale p., coming from the path-integralmeasure.Thereforefor N even,afterthe expressionfor E(c’3) hasbeenrenormalised[i.e., the pole termsin eq.(12.58) dropped],we are left with the expression

a4E(c’s) = A,, + B,, ln(p.,,a) + A1 + Bf ln(p.1a), (12.72)

wherethe coefficientsA,,,Af, etc.can bereadoff from eq. (12.58) in termsof the bosonandfermion

zetafunctionson V”’. In the odd-dimensionalcaseB,, = 0 = Bf. From the definition of ‘N’

= ~ d1Iw~, (12.73)j+0

we see that the calculationof E(m) requiresexplicit knowledgeof the spectrumof the Laplacianand,


moreimportantly, a techniqueto find the analyticcontinuationof the zetafunction to negativevaluesof z. This is not easyto achievein generaland a simplecaseis whenthe sum in eq. (12.73) reducesto aone-dimensionalsummation. In particular the zetafunction for a compactrank-onesymmetricspacehasbeen calculatedin closed form, in the previoussection,in termsof a seriesof Riemannnzetafunctions. The caseof sphereshasbeen treatedby Candelasand Weinberg [26] for minimally andconformallycoupledscalarfields. We shall now generalizethe techniquedevelopedin appendixD ofref. [26] to arbitrarily coupledscalarfields on M4 x G/H, whereG/H is a (simply connected)compactrank-one symmetric space. The caseof spinorswill be consideredfor odd- and even-dimensionalspheres.

(1)5N N odd. Considerthe scalarcasefirst. For simplicity attentionwill be restrictedto valuesof

the coupling ~ that satisfy 0< ~R<2p2, wherep = (N — 1)/2 and R= N(N— 1). The zetafunction is

thengiven by eqs. (11.85)and (11.61), i.e., it is obtainedfrom eq. (11.78) (the minimal case)by justdroppingthesecondterm in squarebracketsandreplacingp2—+ p2 — ~R in the first term. However,theseriescorrespondingto the secondterm vanishesfor z <0, as can be provedby comparingit with thebinomial expansion(11.72) for x = 1. Thus for z <0 the secondterm in squarebracketsin (11.78) canbe droppedfrom ~ (or addedto ~ without modifying the zetafunctions. Sincewe areinterestedin the derivativeof ‘N at z = —2, we can usethe same‘N for both0< ~R<2p2 and in the minimallycoupledcase~ = 0.

As ~N(2) = 0, the effective potential for a single scalarfield is given by

—a4 ~ 2(1)~ak N = (p2 — ~R)~ / F(z + n)

E(~c)=64’ir2 ~ (N—i)!’ ~ n! z-~-2~(z+2)F(z)~2z+2n2k2)

(12.74)wherethe coefficientsakN havebeendefinedin theprevioussection.The termswith n = 0, 1, 2 in eq.(12.74)are only apparentlydivergent,sincethe simple zerosof the Riemannzetafunction at negativeevenintegerscancel the poles coming from the first term in curly large parentheses.For the samereasonthe sum over n for the remainingtermscan be takento run from k+ 3 to ~ insteadof 3 to ~

Separatingout the first threeterms and taking the limit gives

a4E(~)= ~ 2(—iy~a~((_1)k-1~-2k-13/2[~(k + 3)!F(k + 7/2)~R(2k+ 7)321T k=0 (N—i).+ (k + 2)!’ir2(p2 — ~R)F(k + 5/2)~R(2k+ 5)

+ ~(k+ 1)!(p2 — ~R)2’ir41(K+ 3/2)~R(2k+ 3)]

= (p2 — ~R)~~R(2n—2k—6)

n=k+3 nn— n—For ~ = 0, i.e., in the minimally coupledcase, we obtain the results of ref. [26]. Similarly, we canconsider a conformally coupled scalar field in 4 + N dimensions,since the correspondingcoupling

= (N + 2)/[4(N + 3)] satisfiesthe condition0< ~ <2p2. Explicit numericalcalculationgives thesamevaluesobtainedin ref. [26] using anotheranalytic continuationmethod.For examplefor N = 3

and ~R = 5/4 (the conformal case),we get (p = 1)

4 — —1 / ~ ~~B(~) — ~R(~) = (—1 /4)n~(2 —6) \aE(co)— 2~ + +~ ). (1276)

321T 8ir6 8’ir4 64’iT2 ,,~ n(n — 1)(n —2)


The seriesconvergesvery quickly andwe get,after afew terms,the valuea4E(x)= 7.15 x 106, in goodagreementwith the quotedresults.A particularlysimplecaseis for ~ = p2/R, the conformalcouplinginN + 1 dimensions.Only the first termin squarebracketsin (12.75)contributesin thiscase.Forexamplefor N = 3 and ~ = 1/6 we get

a4E(co)= 45~R(7)/2561T8 1.867x i0~. (12.77)

The spinor caseis very similar to the scalarone with coupling ~ = p2/R. Using the spinor zetafunction given in eq. (11.93)we get for a single spinorfield

2~+1~2/_i\15(N—1)/2a4E”’2~(co)= 2 ~ [/3k.N+i(221’4 —

(N— l).i6’iT k=0

x ~._2k_9/2(k+ 2)!F(k + 5/2)~R(2k + 5)], (12.78)

in agreementwith formula (D.i9) of ref. [26].(2) S”, N even,scalarcase.Comparingeq. (12.72) to (12.58) we have(for a single scalar)

A,, = (—1/Mn~2)[~,~,(—2)+ 2~N(2)], (12.79)

B,, = (~i/32’iT2)~N(~2). (12.80)

For 0 ~ ~R<2p2[whereagainp = (N — 1)/2] the valueof thezetafunction at z = —2 can becalculatedusing eqs. (11.85) and (11.62). The only terms that contributeto ~N(—2) in the sum over n in eq.(11.85) arethosefor n=0,1,2, andn=k+3, giving the following value for B,,:

N/2—i /

B,, = 2 ~ Pk.N(~R(2k5,~)—2(p2— ~R)~R(—2k—3, ~)16ir (N—i). k=0

+(p2-~R)2~R(-2k-1,~)+ (k+~)(k~(;+3))’ . (12.81)

wherethe coefficientsf3kN aredefinedin eq. (11.27).For examplefor N = 2 wehavep = ~, R= 2, andfor 0~ ~< ~we get

Bb( ~)— —8.04 X i0~+ 8.44X i0~ — 4.22 x 1O_3~2+ 8.44x io3e, (12.82)

in agreementwith the resultsobtainedin ref. [87].The coefficientA b is calculatedin a similar way andwe get

11\N/2 N/2—i ,‘

Ab= 2 “ ~ ~3

2’iT (N—i). k=0

—2(p2 — ~R)[~R(—2k —3, ~)+ 2~(—2k—3, ~)]+ 2(p2— ~R)2~~(—2k—1, ~)

+ (/( ~f )(k+2(k+3) [ifr(k+ i)+3y +4ln2]

n(n—1)(n—2)~R(2~21~5,2)), (12.83)

98 R. Camporesi, Harmonicanalysis andpropagatorson homogeneousspaces

wherethe prime in the sumover n meansthat the term with n = k + 3 is omitted and ~i(z) = 1’(z) /1(z). The seriesconvergesquickly and numericalcomputation is straightforward.

In the spinorcasethe coefficientsAf and Bf in eq. (12.72) are given by

A1 = (1 /16’ir2)[~ ~(U2)(2) + ~(U2)~(2)] (12.84)

Bf = (i/8~2)~~2)(_2). (12.85)

Using the spinorzetafunctioncalculatedin eq. (11.94) we get immediately

N/2—iBf 2 ~ akN+l~R( 2k 5). (12.86)

8’iT (N—i). k=0

Similarly for A1 we get

‘—2 N/2+1 N/2—IA1 = 2 ~ a~fl[2 ~R(2/( —5) + 2~~(—2k—5)]. (12.87)

16ir(N—1)! k=0

For examplefor N = 2 we have

B1 = ~R( 5)/2ir —2 x i0~ (12.88)

(in agreementwith the resultsin ref. [87]), and

A1 = (1 /42)[3 ~R(—~) + 2~~(—5)]. (12.89)

(3) pN(C) N = 4,8, 12,. .. , p = N/4, R= N(N + 2)/4. The dimensions of these and of the

remainingrank-one symmetricspacesare evenand thereforeE(x) is given by eq. (12.72), and thecoefficientsA,,, B,, by eqs. (12.79) and(12.80). The zetafunction, for 0 ~ ~R<2p

2andz <0, is givenby eqs. (11.85)and (11.63).The calculationof ~N(—2) is similar to the onethat we did for

5N Theonlytermsthat contributein the sum over n in eq. (11.85) are thosefor n = 0, 1,2, and n = k+ 4. Takingthe limit for z—~—2 we get

N/2—2

Bb = 16~2(N/2—1)!(N/2)! ~o (~2k —7) — 2(p2 — ~R)~R(—2k—5)

+ (p2 - ~R)2~R(-2k-3) + (k 4))’ (12.90)

wherethe coefficientsYk,N are definedin eq. (11.28). For A,, we get

1 N/2—2A,, = 32~2(N/2— 1)!(N/2)! ~o ~,N(2 ~R(2k —7) + 2~~(—2k—7)

—2(p2— ~R)[~R(—2k— 5) + 2~~(—2k—5)] + 2(p2 — ~R)2~~(—2k— 3)

+ (k+2;(k+3)(k+4) [~(k+2)+3y]+~’ n(n-1)(n-2) ~R(2n 2k 7)), (12.91)


wherethe prime in the sumover n meansthat the term n = k+ 4 is omitted. For exampleon P4(C) wehavep = 1, R= 6, and we get

B,, = (—1 /321r2)[~R(—7)— 2(1 — 64~)~R(—5)+ (1 — 6~)2~R(_3)+ (1 — 6~)~/24], (12.92)

Ab = _12 (~~R(7) + 2~~(-7)- 2(1 - 6~)[~R(-5)+ 2~~(-5)]64 ‘i~

+ 2(1 - 6~)2~~(_3)+ ~(1 - 6~)~[2y+1] + ~‘ n(n- 1)(n-2) ~R(2~~-7)). (12.93)

The spinorcasecannotbe consideredbecausethereis no spinorstructureon thesemanifolds.Indeed(seeref. [112],p. 217, or ref. [53]) the secondStiefel—Whitneyclasson the complexprojectivespacesP4(C),P8(C),. . . , doesnot vanish.

(4) pN(C) N=6, 10,..., p = N14, R= N(N+2)14. The coefficients B,, andA,, can be obtainedby making the replacement

(—1)”~’2f3k,N/(N— i)!—3 ~ — 1)!(N/2)! (12.94)

in eqs. (12.81) and (12.83), respectively,wherethe numbers~k,N havebeendefined in eq. (11.29).(5) pN(H) N = 8, 12,. .., p = (N + 2)14, R= N(N + 8)/4. B,, and A,, are obtainedfrom eqs.

(12.81) and (12.83) by replacing

(—1)””2/3kN/(N — 1)! —* p.k,NI(N/2 — 1)!(N/2 + 1)! , (12.95)

wherethe numbersp.k,N aredefinedin eq. (11.30).(6) P16(Cay),p = 11/2, R = 144. B,, andA,, areobtainedfrom eqs. (12.81) and(12.83)by replacing

(—1)”’”2f3kN/(N— 1)!—~’7)k/i1!.420 (12.96)

(the numbersTlk are given in table 11.6 of section 11), and letting the sum over k run from 0 to 7.A spinor structureexists on the symmetric spacesin (4), (5) and (6) but we do not know the

eigenvaluesand the degeneraciesof the spinorLaplacian,althoughthesecan be calculatedfrom eq.(12.32) and from the dimensionsof the spinor representationsof G. We defer this problem to ourfuture work.

12.6. Self-consistentEinstein field equations

Let us briefly discussthe consistencyof the one400pcalculationswith the quantum-correctedEinstein equationsfor zerotemperature.It was shownin ref. [26] that if the “internal” spaceV”’ is aone-parametermanifold, the Einstein equationson M4 X V”’ can be written as

VCff(aO)= 0, (aVe,tIaa)~a= 0, (12.97)

wherea is the “radius” of V”’ (in our casea = L/ir, whereL is the diameterof the symmetricspace,


i.e., the maximum distancebetweenany two points) and Veff is the quantumeffective potential

N(a — —4VVeti(a) = ~.. y—~-+ A) + a ,~.,[A, + B. ln(p.1a)]. (12.98)1u7r~.JD a

The first term in eq. (12.98) is the classical potential, where R/a2 is the curvature scalar and

aN(a) = a”’QN(l) the volume of VN, and GD, A are the gravitationaland cosmologicalconstantsin4 + N dimensions.The secondterm in eq. (12.98) is theone-loopcorrectiongiven by (12.72)(the sumis over the different types of matterfields, e.g., scalar,spinor, etc). The gravitationalcontributiontothe Casimirenergycan also be included[110],with a correspondingterm Ag + Bg ln(p.ga).

Thefirst equationin (12.97) simply statesthat thereis no cosmologicalconstantin four dimensions;the secondequationimplies that the one-loopeffective potential is stationaryat a

0. The solution isstableonly if a0 is a minimum of Veff.

In the even-dimensionalcase(B, ~ 0) we can setA = 0 and,as first suggestedin ref. [110],try to usethe dimensionfulparametersp., to obtain the four-dimensionalMinkowski spaceas a solution of theEinstein equations.However, the arbitrary mass scalesp.,, coming from the measurein the path-integral, need not be the same, in principle, for different matter fields, and we would needsomeadditionalrenormalizationconditionto uniquelydetermineboth a0 andthe p.~.Still we can solve for a0in termsof the coefficients B, only, with the result

a~2= 16ITGD ~ B,[uIN(i)(N + 2)R]’ , (12.99)

showingthat a positive a0 can be obtainedonly if ~,B, > 0. However,evenif a solution for a0 can be

found, it is not a minimumof the effectivepotential.For example,supposethat thereis only onekindof fields (scalarsor spinors).SettingA = 0 we then havetwo equationsin the two unknownsa0 and p..The massscalep. is fixed by

A + B ln(p.a0)= B/(N + 2), (12.100)

which may be viewed as a renormalizationcondition. The value of a0 is given by (12.99),providedB>0. By looking at the asymptoticform of Veff we see that whena becomesvery largethe classicalterm dominatesand Veti~~~—ci~ whena —~ 0 the quantumterm takesover and again ~ —~, sinceB >0 and ln( p.a) <0 for small a. Therefore, thereis a unique stationarypoint which howeveris amaximum of the effective potential. As a check we can calculate the secondderivative of Veff at a0,

(a2VffIaa2)~= —a~6(N+ 2)B , (12.101)

and we see that the condition B >0 always implies that the given solution is a maximum. This quitegeneralconclusionholdsin the even-dimensionalcase,for A = 0. It was first establishedin ref. [110]byconsidering the purely gravitational contributionto the Casimirenergy.

A similar result in even dimensionshasbeenobtainedin ref. [70] for a two-parameterinternalmanifold, namely for V”’ equalto a productof two spheresof different radii. A solution was foundwhich, for a sufficiently large number (=~i0~)of matter fields, is consistent with the one-loopapproximation,i.e., the size of the internal dimensionsis largerthanthe Plancklength. Howeverthissolution turns out to be a saddlepoint of V~,andthereforeit is unstable.

By addinganonzerocosmologicalconstantA a minimumof theeffectivepotentialcan be obtained,but thereis no uniquesolution for the parametersa

0, A and p..


ForN odd wehave~N(2) = 0 andthecoefficientsB,,, B

1in eq. (12.72)vanish.In this casetherearetwo equationsfor the two unknownsa0 andA, andwe get

A= (N+2)~ (12.102)a~(N+4)

a”’~2= ~ (12.103)

0 QN(i)R

in agreementwith ref. [26].A stablesolutioncan be found for E, A, >0 and A >0. Indeedfor largeaand A>0, ~ is dominatedby the cosmologicalterm and becomeslarge and positive; for a —~ 0,Veff a ~A which is again largeand positive for ~ A >0. Thus Veff hasa minimumat somefinitevalue a

0.For further discussionof the stability of thesesolutions, including also the finite-temperature

correctionsto Vetg~see,e.g., refs. [26, 122, 47].

Appendix A. Geometry of cosetspaces

In appendicesA and B we shall discussin some detail the geometryof compacthomogeneousspaces.Standardreferencesin the mathematicalliteratureare refs. [16, 20, 27, 89, 113] (differentialgeometry),refs. [9, 43, 79, 100, 135, 136, 141] (group theory), refs. [64, 72, 75, 76—78] (symmetricspacesandsphericalfunctions), refs. [134,137, 138] (harmonicanalysis),and [17, 28, 67, 84, 85, 88,104, 105, 130] (heatkernel andspectralgeometryof Riemannianmanifolds).

In the physical literature the geometryof homogeneousspaceshas been considered,e.g., inapplicationsto generalrelativity [39,53, 83, 106, 125, 139], quantumfield theory andmultidimensionalunified theories[26, 52, 61, 119, 126], andintegrablesystems[115,116].

We shalluseherea coordinate-freenotation,by treatingG as aprincipal fibre bundleoverG/Hwithstructuregroup H [89]. The Killing vectors and the vielbein coframe on GIH will be identified byprojectingand pulling backinvariant vectorsand one-formsfrom the symmetrygroup G.

Two invariant connectionsare naturally definedon any reductivehomogeneousspace.One is, ofcourse,the Levi-Civita connectionof the G-invariantmetric on G/H inducedby a bi-invariantmetricon G. The other is the so called H-connection,which is metric but carries, in general, a nonzerotorsion. The spectrumof the Laplacianof this connectionis given by a simple formula [133]in termsofthe secondorderCasimir invariantson G andH, which will be derivedin appendixB.

We shall computeherethe curvatureof theseconnectionsand show that their geodesicsare theprojections of one-parametersubgroupsof G. We shall also calculate the Van Vleck—Morettedeterminantin terms of the structureconstantson G and of the canonical (or Riemannnormal)coordinateson G/H. This result is obtainedby computing the differentialof the exponentialmapping,andit is equivalentto solvingthe Jacobiequationon G/H. Using theH-connectionthe equationfor theJacobi fields can be solvedin closed form on any reductivehomogeneousspace[16, 142, 93].

A.1. Killing vectors and the vielbeinframe

Let G be a compactLie group, H a closed subgroup,and G/H the correspondingcosetspace,GIH= {gH, g E G}. Theprojection ir and the left-action Pg of G on G/H aredefinedby ir(g) = gH


and PgX= gg’H, wherex = g’H. We shall also use the notation gx for pgx. It is easy to verify the

following relations, for anyg E G andh E H:1TOLg=PgO1T~ i.e. ir(g1g2)g1ir(g2), (Al)

iroR6 = ‘iT, i.e. IT(gh) = ‘ir(g), (A.2)

whereLg and Rg denotethe left and right translationson G.Let o-(x) E G be an elementof the cosetx, definedfor anyx in someopenset U c GIH. Then by

definition ‘iT(o’(x)) = x and p,~(X)xO= x, wherex0 = H= ‘iT(e) is called the origin of the homogeneousspace(e is the identity elementin G). From eq. (A. 1) we have IT( gu(x))= glT(o-(x)) = gx, i.e., gu(x)belongsto the cosetdefinedby gx. If gx E U, we can write

go-(x) = ff(gx)h(g, x), (A,3)

whereh(g, x) E H is a function of both g andx, h: G x GIH—~H. If we regardG as the principal fibrebundleG(G/H, H), with basespaceGIH and typical fibre H (seeref. [89]), then o-(x) is just a localsectionandh(g, x) is equivalentto a gaugefunction.

The explicit form of the function h andof the local sectiona- will bederivedlater. We only noteherethat h satisfies:h(e,x) = e, and the cocyclecondition,

h(g1g2,x) = h(g1, g2x)h(g2,x), for anyg1, g2 in G. (A.4)

The Killing vector fields {K,}, i = 1, . . - , N = dim G, generatetheleft actionof Gandaredefinedby

K1(x) = (dldt)g,(t)x(0 , (A.5)

whereg,(t) = exp(tT~)arethe one-parametersubgroupsof G. (T, arethe abstractgeneratorsin the Liealgebraof G.) Onecan showby direct calculationthat

= ir~R,, (A.6)

i.e., the Killing vectors on G/H are the projection of the right-invariant vector fields on G,R.(g) = Rg*Ti.

If G is semisimple,it is well known [89, 113] that the Killing—Cartan metric inducesa metric g onG/H invariant under G, i.e.,

Pg9=9 ~ ~~1g—0, (A.7)

where~ is the Lie derivative.SinceG andHarecompact,GIH is areductivehomogeneousspace,i.e.,we can decomposethe Lie algebra~ Te(G) of G accordingto

(A.8)

where~ is the Lie algebraof H, and.A~a subspacesuch that


~ [~r,At]ç.41, (A.9)

whereAd(H) is the adjoint representationof G restrictedto H. This decompositionis not unique,ingeneral.However, for G semisimplewe can define ~ to be the orthogonalsubspaceof ~‘ in ~ withrespectto the Killing metric. Indeed,using the invarianceof this metric underthe adjoint representa-tion, it is easyto showthat the condition(A.9) is satisfiedby .A~ ~‘ ~.

If G is not semisimple,we can fix a bi-invariantmetric ‘y on it, by meansof a scalarproducton theLie algebrainvariant underthe adjoint representation.This is alwayspossiblefor acompactgroup(see,e.g.,ref. [20], p. 244).Defining .A~as , eq. (A.9) is againsatisfied,andtherestrictionof ‘yto 4~givesan Ad(H)-invariant scalarproduct, which generatesa G-invariantmetric on GIH (seeref. [113], p.52). We shall assume,onceand for all, that such an invariant metric hasbeenfixed on G/H. Thedescriptionof all metrics on G/H that are invariant under G will be given later.

The reductivedecompositionof the Killing vectorsis

K,=(K~,K1), (A.10)

wherethe conventionabout indices running over the generatorsof G, G/H and H, is the following:

i=1,... ,N~dimG, a=i,...,n~dimG/H, 1=i,...,dimH. (A.li)

The Killing vectorssatisfythe samecommutationrules as the vectorsR., i.e.,

[K,, K1] = _..kKk, (A.i2)

wherethe structureconstants1k are totally antisymmetricwhen the last index is lowered with thebi-invariantmetric on G,

f1jJ,,/~ filk - (A.13)

Moreover from eq. (A.9) the only nonvanishingf..k are

(fap” fit~faP). (A.14)

The tangentspaceat the origin x0 of G/H can be identified with the subspace4~of ~, and theorthonormalbasis { Ta) = {Ra (e)) can be chosen,wherewe assumethat the right-invariantframe{R,}is avielbein.

Fromthe transformationproperty of the vectorfields R, underleft translationon G,

Lg.,Rj=Ad(g)’jRj, (A.15)

we get the transformationof the Killing vectorson G/H underthe action of G

Pg*Ki = Ad(g)’, K1. (A.16)

A simple calculationgives the componentsof the metric on GIH in the frame {Ka }

104 R. Camporesi,Harmonic analysisand propagatorson homogeneousspaces

g~p(x)= Ad(a-’(X))~aAd(a-~(x))~6~, a-’(x) (a-(x)Y’. (A.17)

It is easilyprovedthat (A.17) doesnot dependon the local sectiona-. To seethis we just takeanothersection o-’(x) = a-(x)h(x), with h: G/H—~H a gauge function, and use the invariance of the scalarproducton .11t underAd(H).

To anysectiona- we can associatea vielbein {ea } of the G-invariantmetric on G/H by

ea(x) = P,~(5)~T0- (A.18)

Indeed,using eq. (A.7), we haveimmediatelyg(e0,~ = From eq. (A.18) we also have

e0(X)= ‘iT*La(a-(x)) , (A.19)

where{L,} is the left-invariant frame on G conjugateto {R,}, i.e.,

L,(g) =Ad(g)1

1R1(g). (A.20)

From eq. (A.20) we get the relation betweenthe vielbein frameand the Killing vectorson GIH

e0(x) = Ad(a-(x))’0 K,(x), (A.21)

and conversely

K,(x) = Ad(a-_l(x))0,e0(x). (A.22)

Notice that there are no G-invariant vector fields on GIH. Under the action of G the vielbein

transformsaccordingto

Pg*ea(x) = Ad(h(g,X))~aep(gx), (A.23)

whereh(g, x) is the function definedin eq. (A.3).An important relation is the expressionof the differentialof the function a-°‘ii-. Since ‘ir° a- is the

identity mapping,the two vectorsL0(a-(x)) and (a-s *La(0@~)) projectontothe samevector, namelythe vielbein ea(x) [seeeq. (A.i9)].

Therefore,theycan only differ by “vertical vectors”, i.e., vectorsthat areannihilatedby IT ~. Theseare just the fundamentalvectorfields, generatingthe right-action of the structural group H on theprincipal bundle G(G/H, H). Since this action coincideswith Rh, the right translation on G byelementsof H, it follows that the verticalvectorsarejust the left-invariantvectors{L1’}. (Werememberthat the left-invariant vectorfields generateright-translationson G, andvice versa,see,e.g.,ref. [29].)In, fact, it is easyto showfrom the definitionof IT that ir~L1= 0. We concludethat thereexistfunctionsK’~(x) on G/H, such that

(a-o IT)~L (a-(x)) = L(a-(X)) + K’(x)L~(a-(x)). (A.24)

Let now {L’) be the left-invariantMaurer—Cartanone-formson G that are dual to {L,}. Thesecan


bepulledbackwith the linearmapa-’, to giveone-formson G/H. Usingeqs. (A.i9) and(A.24) we seethat the one-forms

e(x) a-*L~x(u(x)) (A.25)

give the vielbein coframe on G/H, i.e., e’3(ea) = 6~.Similarly, if we denoteby e’(x) the one-forms

a-*Li(o.(x)), and calculatetheir value on thevielbein ea, we gete’(x) = K’a(x)e”(x). (A.26)

We shall see later that the one-forms e’ transform as connection forms and define the so calledH-connectionon GIH.

Let us now usethe Cartanequationsfor the forms {L’} on G,

dL”~= — lf k~, A L’, (A.27)

apply o~’to both sides andset k=y. After somealgebrawe get

de~=_~CapYeaA e~ (A.28)

Cap7 =~fap7~2K~afpi~7. (A.29)

The functionsCap” arejust the anolonomycoefficientsof the vielbein frame, i.e.,

F 1—C ~

Lea, e~1— a~e7 -

Using eqs. (A.30) and (A.24) we can computethe Lie derivativeof the vielbein with respectto theKilling vectors,with the results

~‘K,ea = [K,, ea] = W~fatep, (A.31)

= ~w’f~fe~ , (A.32)

wherethe functionsW~are given by

W~(x)= Ad(a-~(x))’,— K~(x) Ad(a-~(x))”~. (A.33)

This relationcan be “inverted” and gives the two conditions

K~(x) = —Ad(u(x))’a W~(x), (A.34)

Ad(a-(x))’k W~(x)= ~. (A.35)

On the otherhand, the commutator[K,, ea] can also be calculateddirectly by the definition of Liederivative,


(~‘K.ea)(x) = 1in~(1 It)[pg_l(t~*ea( g,(t)x) — e,, (x)] - (A.36)

Using here eq. (A.23) and comparingthe result with (A.31) gives the explicit form of the function

h(g,(t), x)h(g,(t), x) = exp[tW~(x)T1]. (A.37)

In general, if g = exp(gT,),where{g’} are the canonicalcoordinatesof g in G, we get

h(g,x) = exp[g’W(x)T,]. (A.38)

Since in canonicalcoordinatesg~= exp(—g’T,), it follows from eq. (A.38) that

h(g’,x)=(h(g,x))* (A.39)

From this, using (A.4), we obtain the following G-invariancepropertyof the functionsh andW:

h(g, gtx)=h(g,x)=h(g, gx), (A.40)

g’W~(gx)= g’W~(x), (A.41)

or, for the one-parametersubgroups,

W~(g1(t)x)= W~(x) (no sum over i). (A.42)

We seethat the functionsW~(with i fixed) areconstanton the integralcurveg,(t)x of K,, and therefore

K~W~= 0 (no sum over i). (A.43)

More generally, from the expansion

W~(gx)= W~(x)+ g~K~W~(x)+ o(g2), (A.44)

if we multiply by g’ and sum over i we get

K(flW~)=0. (A.45)

The antisymmetricpart K1flW~],obtainedwith similar manipulations[34, 61], is given by

K~W1—K,W’~=f~~’W’W~fnimW’,n. (A.46)

This relationis equivalentto giving the transformationof the one-formse’ under the action of G (seelater), or their Lie derivativewith respectto the Killing vectors

= —dW~+ f,~1W?1ek. (A.47)

R. Camporesi,Harmonicanalysisand prepagatorson homogeneousspaces 107

The two equations(A.47) and(A.32) can be written in acompactway, definingthe canonical¶~-valuedone-formon G/H

e=ea®Ta+ei®Ti. (A.48)

The one-form e(x) is often denoteda-’(x) do-(x). This form is just the pull-back, through a-, of thecanonicalMaurer—Cartanone-formon G, 0 = L’ 0 T~(see,e.g.,ref. [29]).ThentheLie derivativeof eis given by

~K~e= —dIV, + [W,, e], (A.49)

whereW~is the ~-valuedfunction W~T1,andthecommutatoris in W As alreadyseen,the terme” 0 Tain eq. (A.48) definesthe vielbein coframe on G/H. The other term e’ 0 T1 is the pull-back of theMaurer—Cartanone-formof the groupH

OHLI®T (A.50)

This form satisfies,for h E H,

R~O’1=Ad(h1)OH (A.5i)

as follows immediately from its left-invariance. Therefore0H defines a bundle connection on

G(G/H, H) [89]. According to the generalYang—Mills theory, the pull-back of a bundleconnectionwith respectto alocal sectiondefinesgaugefields on the basespaceof the bundle[29,53]. We shall seethat the “gaugefields” {e’) of the “gauge group” H define a linear connectionon G/H.

A.2. Invariant connectionson homogeneousspacesand their curvature

A connectionw on a manifold M is said to be invariant under a diffeomorphismf if

Vf~Y=f~V~Y, (A.52)

for any vector fields X and V on M. An affine transformation (i.e., one leaving w invariant) mapsgeodesicsinto geodesics.By invariant connectionon GIHwe shall meana connectionwhich is invariantunder the transformationsp5, for any g E G. The Levi-Civita connectionof the G-invariantmetric isclearly an invariant connection.It will be denotedby ~i. The connectioncomponentsin the vielbeinframe aregiven by

o 7—]~ ~ ~ + ~~0p 2 af3 ~ a~ na)’

wherethe indices in the anolonomycoefficientsare raisedand lowered with the vielbein metricUsing eq. (A.29) we obtain the vielbein componentsof the Levi-Civita connection

= ~fap7 + ~ (A.54)

108 R. Camporesi,Harmonic analysis andpropagatorson homogeneousspaces

The connectionone-formsare

us ~~ap7~° = ~f~7e0+ e’f

1~7, (A.55)

and satisfythe usual relation for a metric connection

= (z)~, (A.56)

as can be easilyverified usingeq. (A.i3).Another invariant connection(referredto as the H-connection)which is still metric, but carriesa

nonvanishingtorsion is definedby the one-forms

= e’f1~

7, (A.57)

with vielbein components

= K~f~7. (A.58)

By writing

~ ~ K,.~7, (A.59)

whereK~7are the contorsionforms, we get the vielbein componentsof the contorsiontensorK

Kap7= ~~fap7’ (A.60)

This contorsionis clearly totally antisymmetric,and thereforethe correspondingtorsiontensorTis just2K, and,in the vielbein, T

0~7= fap”’ The torsionand curvaturetensorR of w (to be computedlater

on) areparallel, i.e., VT = 0 = yR. More generally,the covariantdifferentialof any G-invarianttensorfield with respectto the H-connectionvanishes(seeref. [89],vol. 2, p. 193). On theotherhandwith theLevi-Civita connectionwe have

(A.61)

unlessGIN is a symmetricspace.In this case

[~,At]C~~fap7=O, (A.62)

andthe two connections~ and~ coincide as T = 0. In fact, the two conditionsof vanishingtorsion andparallelcurvaturetensorare necessaryand sufficient for a manifold to be locally symmetric[113,76].The usual definition of a symmetricmanifold is that the geodesicsymmetry 5: Exp(X)—~Exp(—X),whereExp is the exponentialmappingbasedat anypoint, is an affine transformationof the connection(an isometry in the Riemanniancase).

The geometricalmeaningof the connectionw can be understoodin the following way. A gaugetransformationon the bundleG(G/H, H) can be seen,from a “passive”point of view, as a changeof

R. Camporesi,Harmonic analysisandpropagaforson homogeneousspaces 109

sectiona-—~a-’, with a-’(x) = a-(x)h(x),andh: G/H—~H (a gaugefunction). The one-formse’ definedin(A.26) are mapped,by such a change,into new forms e” = a-F *1]. A simplecalculationshowsthat

e”(x) = Ad(h~(x))’~e’(x) + h*L~(h(x)) , (A.63)

wherethe last term is just the Ith componentof the ~C-valuedone-formh’(x) dh(x), i.e.,

[h*Li(h(x))] 0 T~= h1(x)dh(x). (A.64)

(Since G andH arecompact,theycan alwaysbe identified with groupsof matricesandg1 dg is awelldefined~-valuedone-form.)

Fromthe “active” point of view we have the following transformationof the one-formse’ undertheaction of G:

(pe’)(x) = Ad(h(g,x))’1 e

3(x)+ (hg(X) dh~(x))’, (A.65)

wherehg: x—* hg(X)us h(g, x) [definedin eq. (A.3)], andthe secondterm,usingeq. (A.38), can alsobewritten as —g’ dW1

1(x).It is clear from eqs. (A.63) or (A.65) that the one-formse’ transformas“gaugefields” on the “basespace” G/N of the bundle G(G/H, H). The structureconstantsfiap allow theisotropy groupH to be embeddedinto the tangentspacerotationgroup SO(n) andrelate the “gaugefields” e’ to the H-connectionone-formsw_~[seeeq. (A.57)].

We can seehow this embeddingworks if we recall that the adjoint representationof G in ~, whenrestrictedto H, reducesto the direct sum

Ad(H)~ =Ad(H)~Ef3Ad(H)I~ . (A.66)

Obviously, Ad(H)~,,is the adjoint representationof H on its Lie algebra,with matrix generators

(]~)‘~1=f~. (A.67)

On the other hand, Ad(H)[,,, coincides with the isotropy representationr of H on .ilt, whereT(h) = Ph* L0~see appendixB. The generatorsare the n x n antisymmetricmatrices

(h)ap = (J})[apj f’pa , (A.68)

which belongto the Lie algebraof S0(n) andcan be expressedin termsof the generators~ (of thevectorrepresentation)of SO(n)

(,~aP) = ~ — ~ (A.69)

Indeedwe have

1z ap~”

1~’ (A.70)

and thusthe embeddingof H into SO(n) is fixed by


h—+Ad(h)~,,=exp(h~f~)ESO(n). (A.71)

We can saythat the “gaugefields” of the“gaugegroup” H on G(G/H, H) definea linear connectionon the “basespace”G/H and that the gaugederivative

D=d+Q1e’ (A.72)

is the covariantderivativewith respectto this connection(seeappendixB).We now calculate the curvature of w and w. The simplest way is to use the Cartan structure

equationsfor the curvaturetwo-forms, i.e.,

= ~ A e~= dw~7— A . (A.73)

Insertinghereeqs. (A.55)and(A.57) andusingJacobi’sidentity, we get thevielbein componentsof theRiemanntensorof the Levi-Civita and of the H-connection,

7 =_(], 7 F~f F 7 7p pff ‘.2 r~ p,y 2Jp[p (T]F ip pu )

~ = —J~~7f~0

1. (A.75)

Theseareonly a function of the structureconstantsf~1/(of the symmetrygroupG. We remarkthat eqs.

(A.74) and (A.75) hold, in the vielbein, at any point of GIH, and not just at the origin. This is aconsequenceof the reductivity of GIH. On a nonreductivehomogeneousspacethe curvaturewoulddependexplicitly on the “gauge fields” K~(x)and it would not be constantin the vielbein frame.

If G is semisimple,we can use the normalization

= —Tr ff = ffl3rrn (A.76)

andwe get for the Ricci tensorsR6,1 R137

70~

~ = — ~fpF’f~i = + ~fpF

7fU

7’’ (A.77)

R~= + ~fpFYfUyF= ~~7f’ = —Uf~,. (A.78)

Fromeq. (A.78) we see that the Ricci tensorof the H-connectioncoincides,for G semisimple,withthe Casimiroperatorof the isotropy representationr of H. If r is irreducibleon the tangentspace~1l,GIH is called an isotropy irreducible homogeneousspace [89, 140]. In this casewe get with ournormalisation [C(T) being the Casimir number]

Rp(, = C(r)~~, , (A.79)

= + ~C(r)]ô~ = ~ , (A.80)

i.e., an isotropy ireducible homogeneousspaceis an Einstein space. For isotropy irreducible homo-geneousspacesthe metricg = 6

0~e00 e’~’is the only metricon G/Ninvariant underG (up to a constant

factor, see ref. [140]).

R. Camporesi, Harmonic analysis and propagators on homogeneous spaces 111

If ‘r is reduciblewe havethe direct sum decomposition~ = ~ ~ ~‘ - ~ ~r’ whereAd(H) isirreducibleon 41 ~,.. 41 r andtrivial on 41,~.A scalarproducton 41 invariant by Ad(H) hasthe form

K’ 9o+~ Aa’y~, (A.8i)

whereg0 is an arbitrarymetric on ~ Aa arepositivenumbers,andy~~ is the restrictionof the givenbi-invariantmetric on ~ to the subspace~a Then ( ,) generatesap-parameterfamily of G-invariantmetrics on GIN, wherep = ~m0(m0+ 1) + r, with m0= dim 41g.

The metric that has been usedso far (known as a normal homogeneousmetric) is obtainedbyrestricting ‘y to 41 with the samescalefactor for each irreduciblesubspace,i.e., with )ta = 1 Va, and

= ~L~- A motivation for choosingthis metric is that the correspondingLevi-Civita connectioncoincideswith the so calledcanonicalconnectionof thefirst kind (seeref. [113],theorem13.2) whichsharesnice geometricalpropertieswith the H-connection(or canonical connection ofthe secondkind),as will be seenlater.

Having discussedhow to generateall G-invariantmetrics on GIH, for G fixed, thereremainsthepossibility that different groupsG, G’, etc., act transitively on the given manifold M, with isotropysubgroupsH, H’, etc. Startingwith the mostsymmetricconfiguration,i.e., the one with the maximumnumberof Killing vectors, sayGIN, it is possiblefor Ad(H) to be irreducibleon 41 [e.g., the roundn-spherewith H= SO(n)]. When we lower the symmetriesand consider,e.g., G’/H’, the isotropyrepresentationof H’ will (in general)no longer be irreducible.The lesssymmetricconfiguration(theonewith the lowestdimensionaltransitivegroup)will give rise to thefamily of homogeneousmetricsonM with the largestnumberof parameters.This family mayincludethe more symmetricconfigurationsas particularcasesandby varying the parameterswe obtain a “squashed”homogeneousspace.

For example the groupsacting transitively on spheresand rank-one symmetricspacesare known(see,e.g.,ref. [143]).On evenspheres~ the only homogeneousmetric is, up to afactor, the standardmetric inducedby SO(2n+ 1), i.e., no “squashing”is possiblewith evenspheres.The sameconclusionholdsfor the projectivespacespN(H) F”

6(Cay), and P4(C),P8(C),... (seeref. [143]).On odd spheresand on the complex projective spacesP6(C),P10(C),..., we can have other

homogeneousmetrics besidesthe standardone. For examplethe groupSU(n + 1) acts transitivelyon~2n +1 and inducesa two-parameterfamily of invariant metrics. On ~ + ~we also havethe transitivegroupsSp(n+ 1) X Sp(l), Sp(n+ 1) X U(i), andSp(n+ 1), andall homogeneousmetricson ~4~z+3 canbeobtainedas specialcasesof the seven-parameterfamily inducedby Sp(n+ 1) (exceptfor the Spin(9)invariant metrics on S15, seeref. [143]).

Of course the “squashing” (or symmetry breaking) mechanismis well known in physics.For examplein homogeneoustype IX cosmologicalmodelswe start with a Friedmann—Robertson—Walkeruniverse(one parameter)and by squashinga three-spherewe get the Taub universe (two parameters),thediagonalMixmasteruniverse(threeparameters),andthe generalMixmasteruniverse(six parameters)[106].Anotherwell known exampleis the seven-spherein Kaluza—Klein supergravity[52].

Let us concludethis sectionwith a brief remark on the curvatureof symmetricspaces.From eqs.(A.77) and (A.78) we see that a symmetric GIN (fap7 = 0) with G semisimpleis an Einstein space

= Rap = ~. (A.82)

[Noticethat in the normalization(A.76) we haveC(r) = ~ in this case.]

112 R. Camporesi,Harmonicanalysis andpropagatorson homogeneous spaces

An importantexampleof symmetricspaceis that of a semisimpleLie group G equippedwith theKilling—Cartan metric (see,e.g., ref. [20], p. 349). Then G/H~—[GLx GR]/Gdiag, whereGLand GRindicateG actingon itself by left andright translations,and Gdlag us {(g, g), g E G} is just the isotropygroup of this action at the identity (see appendixB). In this casewe get from the previousformulas

Ri’mn = — ~fk/frnn~’ , (A.83)

= R = N/4, (A.84)

where N = dim G. Thereforea semisimpleLie group manifold with the Killing—Cartan metric is anEinstein space,with curvaturescalar equalto N/4 [39].

A.3. Geodesicson GIN and the Van Vieck—Morettedeterminant

What singles out w and ~ amongall possible G-invariantconnectionson GIH is the importantproperty that their geodesicsare just the projectionof suitableone-parametersubgroupsof G. Morepreciselywe havethe following resultdue to Nomizu (for the proof seeref. [113]):

TheoremA.1. The geodesicof both w and ~ leaving the origin x0 with tangentvectorX is

4x(t) = ‘ir(exp(tX)) = Pexp1txy~o’ (A.85)

i.e., it is the projection of the one-parametersubgroupg~(t)= exp(tX), where exp denotestheexponentialmappingon G. Thereforethe exponentialmappingon G/H, basedat the origin, is given by

ExpX= IT(expX), XE41~T~0(GIH), (A.86)

i.e., Exp= ‘ii-o exp[,,. Moreoverthe paralleldisplacementof anyvectorYE 41 along4~(t)with respectto the H-connection(not the Levi-Civita!) is just the translationof V by g~(t),i.e., Pexp(tX)*~-

The fact that o and ~i havethe samegeodesicsis an easyconsequenceof the fact that the torsiontensorof w is totally antisymmetric.Clearly to and ~i also havethesameJacobifields (becausetheyshare the sameexponential mapping). Using the H-connectionthe Jacobi equation becomesanordinaryconstant-coefficientsdifferential equationandcan be solved exactly [142].

A local section a-: G/N—~G can naturally be definedfrom this theoremas

u(ExpX)=expX, (A.87)

or equivalentlya-(expXx0) = expX, wherenow XE 41~,the largestopen domainof 41 containingtheorigin and suchthat the exponentialmapping,restrictedto 41~,is a diffeomorphism(seeref. [17], p.59).

The sectiona- mapsG/N— A0 into exp(410)C G, whereA0 = Exp(8410) is called the cut locusofthe origin x0 (a410is the boundaryof 41~).It is known(seeref. [89],vol. II, p. 97, theorem7.1) thatA,)consistseitherof pointsconjugateto x0 [i.e., suchthatL1~given in eq. (A.89) vanishes],or of pointsthat can be joined to x0 by morethanone minimizinggeodesic.SinceA0 hasalwayszero measure,we


can fix on G/H asystemof canonical(or Riemannnormal)coordinatesthat coversthe wholemanifoldminusa zero-measureset.Thusif x = Exp X with X = X”Ta~the Riemannnormalcoordinates(RNC) ofx are just {f).

We shall nowcomputethe Van Vleck—Morettedeterminant4 usingthesecoordinates.We recallthat4(x, x’) is definedby

—1/2 (ad(x,x) \x ) = [g(x)g(x )] det~,, a ~, ) (A.88)ax ax

whereg is the determinantof the metric tensorand d(x, x’) is the geodesic(Riemannian)distancebetweenx andx’. If weknow ~ 1/2 we can solve recursivelythe differential relationsfor the heatkernelcoefficients (see section2). In particular, the propagatorof a free particle on a manifold M in theGaussianapproximationis justthe productof 41/2 times (4ITis)”2 exp(id2/4s+ isRI6),whereR is thecurvaturescalarands the “proper time”.

In a RNC frame with one of the two points atthe origin, 41 reducesto the squareroot of g andgives the measurefunction in that frame. Let us first derivean intrinsic definition of 41, namelylet usshowthat4’(x

0, ExpX) is the determinantof the differentialof the exponentialmap atX, i.e.,

4’~(x0,ExpX) = detExp~1~. (A.89)

Notice that Exp~1~is a linear map from the tangentspaceat the origin into the tangentspaceat thepoint x [identifyingT~(41)with 41 itself], andthereforeits determinantis well defined.To prove (A.89)in a RNC patch,we notethat\/~is just the determinantof thevielbein matrix. But from the definitionof a RNC frame we have

(alax”)15 = Exp*jxTa , (A.90)

andwe seethat the vielbein matrix coincideswith the matrix of the linear transformationExp~11,in theorthonormalbasis Ta andea (x) -

Next,we stateanimportantresult (for the proofseeref. [76],p. 36) thatgivesExp~in termsof Liederivatives.

TheoremA2. Let Mbe amanifold with an analyticconnectionto, andlet U bea fixed convexnormalneighborhoodof a point x0. Givena tangentvectorX at x0 define the vectorfield X* at anypoint of Uto be the parallel transportof X alongthe uniquegeodesicjoining that point to x0. Then for V E Ti,,

I1—exp(—~ ) 1Exp~11~V = “

5~’tX* tX* ~ EXp ix’ (A.91)

where~ is the Lie derivative and

i—e’~ (n+1)! - (A.92)

This result is essentiallya formal solutionof the Jacobiequationon the manifold M. Indeed(see,

114 R. Camporesi,Harmonic analysis andpropagatorson homogeneousspaces

e.g., ref. [17] or ref. [28]) Exp~is also given by

Exp~11~V = (1 /t)Y(t), (A.93)

whereY(t) is the Jacobifield alongthe curve Exp(tX) satisfying the initial conditions

Y(0)=0, (dY(t)Idt)10=Y. (A.94)

We shall now specify eq. (A.91) first for the groupmanifold G itself, andthen for the homogeneousspaceGIN. On G we get [76]

- ad(X)exp*ix=Lexpx*(l—e )/ad(X), (A.95)

wheread(X)(Y) = [X, Y], andLg is the usualleft translationon G. Thus,the quantity4’ on a groupisgiven by

4’(e, g) = det[(1 — e’~)/g~], (A.96)

whereg = exp(g’T1) and (f)~~, usfk are the matrix generatorsof the adjoint representation(see,e.g.,ref. [39]).Sincethe determinantof the adjoint representationmatricesis equalto one, 4_i can alsobewritten as

4’(e, g) = det(sinh(g~/2)/g’f1/2). (A.97)

This resultcan be further specifiedin termsof a root systemof ‘~with respectto a Cartansubalgebra,namely

exph) = ~ (sin(a.h12))2 (A.98)

whereh is in the Cartansubalgebraand the product is over the positive roots a.On G/N we use theoremA.1 to write Exp~= ir~,expl~.andwe get

(A.99)

wherex = Exp(X), X E 41, and adX is now the restrictionof the map ad(X) to the subspace41 of ~.

Since [41, 41] C cc and ‘ir~:‘~—~ 41, the term in squarebracketsin (A.99) is a well defined linearoperatoron 41. Using det A = expTr(log A) andexpandinglog(1 + B), or directly from

det(1+K)=l+TrK—~TrK2+~(TrK)2+.-.,

we can deducefrom eq. (A.99) the expansionof 41 in Riemannnormal coordinates.For simplicity let us considerthe caseof a symmetricspace.Fromthe condition [41,41] C ~‘we have

that the odd powersof adX in eq. (A.99) give zero when actedupon by ‘iT,, because

(adX)2~~’:41—+~C,IT~(~’)=O. (A.100)


Thus we get

4_i(xo,x)=det(1+~(2fl+1)!)’ (A.101)

where Ax (adX)2 is a well defined linear operatoron 41. By writing the vector X in “polar”coordinates[78], X = Ad(h) h, we have

Ax = Ad(h) Ah Ad(h~), (A.102)

where h EH and h E~, a maximal abelian subspaceof 41. By ref. [76], p. 288, lemma2.9, theeigenvaluesof the operatorsAh’ for h E ~, arethe numbers— (a - h)2, wherea runsover the positiveroot vectors of the symmetric space, with multiplicity ma (the root multiplicity), and zero, withmultiplicity 1, the rank of GIH. Thereforefrom eqs. (A.101) and (A. 102)we get a formula similar to(A.98)

4~(x0,x) = ~ (sin(a.h))rn~ (A.103)

wherethe componentsof the vectorh E ~ definethe so called “flat coordinates”of x on the maximaltorus T= Exp .~T,i.e., we write

x=ExpX=ExpAd(h)h=phExph.

The geometryof symmetric spacesis consideredfurther in section5 in connectionwith integrablesystems.For more information aboutsymmetric spacesand root systemssee refs. [35, 76, 78, 116].

Appendix B. Harmonic analysis on homogeneousspaces

In this appendix we consider the harmonic expansion of arbitrary fields defined on compacthomogeneousspaces.The harmoniccomponentsare identified with the matrix elementsof suitableirreduciblerepresentations(irreps) of the symmetry group. When the cosetspaceGIN reducesto thegroup manifold G itself (i.e., H = e), all the irreps of G enterin the harmonicexpansion,a resultknown as the Peter—Weyl theorem.

For nontrivial isotropy groupsH only certain irreps appear for a given field, according to a result inrepresentation theory known as Frobeniusreciprocity (see, e.g., refs. [79, 96, 137, 138]). Thus oneobtainsa straightforwardgeneralizationof the ordinary expansionin sphericalharmonicsfor fieldsdefinedon the two-sphere.

We shall use here in a simple way the concepts of homogeneousvector bundle and of inducedrepresentation,as being the most natural framework for studyingthe transformationpropertiesofphysicalfields under the actionof a symmetrygroup [96]. The resultson harmonicanalysisgiven bySalamand Strathdee[126]in applicationsto Kaluza—Klein theorieswill be derivedusingthis approach.

The resultsobtainedin this appendixcan be generalizedto the noncompacthomogeneousspaces.Inparticular the theory of spherical functions on semisimple Lie groups, developedby Harish-Chandra[72], can be usedto defineaFourier transformon thenoncompactsymmetricspaces,in analogyto the

116 R. Camporesi. Harmonic analysis and propagators on homogeneousspaces

harmonicexpansionin the compactcase [75, 78, 64, 138]. In section 5 the conceptof “duality” forsymmetricspacesand the inversionformula for the sphericaltransform are briefly reviewedand areappliedto the calculationof the heat kernel on theseRiemannianmanifolds.

B. 1. Homogeneous vector bundles and inducedrepresentation

Consideran arbitraryfield cli on a compactcosetspaceGIN. This is asectionof a vectorbundleE onG/N defined as follows. For x E GIN, cli(x) belongsto a vector spaceE~.These vectorspacesare allisomorphic to a typical fibre E0, which we identify with E1, x0 beingthe origin of GIN. The vector

bundleE is the union of the E~,

E= U Er., (B.1)xEG/H

to which we give the structureof differentiablemanifold. Thus a sectionof E assignsto a point x anelementof thefibre E~.If {~a)is a set of linearly independentsections,suchthat {Oa(x)} is a basisof E~for any point x E U C GIN, an arbitrary section cli of E can be written as

cli(x) = ~//‘(x)Oa(x), (B.2)

wherethe sum over a is understoodand runsfrom 1 to the dimensionof E0. For examplethe scalarfields on GIN arethe sectionsof the trivial bundleGIN x E~,the vector fields arethe sectionsof thetangentbundleT(GIH) = U T~, andsoon. The space1(E) of sectionscanbe given thestructureof an(infinite-dimensional)vectorspace,defining, for real numbersa and f3,

(alIIi + f3~/J2)(x)= açli1(x) + f3~lJ2(x). (B.3)

This constructioncan be repeated,of course, for an arbitraryRiemannianmanifold. However thehomogeneityof GIN entersnowin a preciseway. Takefor exampleE = T(G/H), the tangentbundle.Then G acts on E in a naturalway; given avectorV(x) E T~we can define,for anyg E G, anew vector

V’(x’) = Pg*1”(X), (B.4)

where x’ = gx. Since Pg* is a well definedlinear mappingof T,~ into T1, this action is, by definition,fibre-preserving.

Abstractingfrom this example,we consideran arbitraryvector bundleE on G/N and we assumethat G acts on E with transformations(still denotedby g) that satisfy: (i) gE~= Egx; and (ii)g: E1—~ Egx is an isomorphismof vectorspaces,for anygE G andx E GIN. Then,the vectorbundleEis called a homogeneous vector bundle (hvb).

As hx0 = x0 for any h in N, we have a linear invertible mapping r(h) on E0, i.e. a linearrepresentationT of H on the typical fibre. Vice versa, it is possible to prove that any linearrepresentationof H on avectorspaceE0 determinesa uniquehvb on G/N, with fibers E~isomorphictoE0, and with E5 = E0. Thereforewe havethe following result [79, 137].

Theorem B. 1. There is aone-to-onecorrespondencebetweenthe homogeneousvectorbundlesE on acosetspaceGIN and the linear (finite dimensional)representationsr of the subgroup H.


We shall write E = ET for the hvb’s on GIN. For examplethe singletof H definesthe trivial bundle11 x GIN. The mapping Ph*JX

0 is the isotropy representationofN on TX(GIH), anddefinesthe tangentbundleof GIN. The spin representationof N is obtainedby embeddingN into S0(n), n = dim GIN(seeappendixA), andhenceinto Spin(n),the two-fold coveringof SO(n), anddefinesspinorfieldsonGIN. We recall that spinors can be globally definedon an orientedmanifold M only if the secondStiefel—Whitneyclassof M vanishes[53, 112].

We now consideran alternativerealizationof the spaceof sections,in termsof functionsdefinedonG, ratherthanon G/N. Let us showthat thereis a one-to-onecorrespondencebetweenthe sectionscl’of the hvb E

T andthe mappingscl’ of G into the typical fibre E0 such that, for anyh in H,

~i(gh)=r(h~)~i(g). (B.5)

Indeed, given cl’ E 1(ET), define cli: G—~’E

0 by

~/i(g)= g~/i(gx0), (B.6)

whereg’ denotesthe actionof G betweenthe fibres Egx0 and E0 of ET. Then cl’ verifies(B.5) as one

can check immediately.Vice versa,given i/i satisfying(B .5), define

çfr(x)=gçb(g), (B.7)

wherex = gx0. If g’ = gh, with h E N, thenx = g’x0 as well, but g’i/i(g’) = gçli(g) becauseof (B.5) and

cl’ is well defined.The spaceof the functionscl’ will be denotedby C(G, r). In termsof componentsif{Ea) is a fixed basis of E0 we shall write

T(h)Ea = T(h)~’aEb, (B.8)

clI(g) = ~fra(~ (B.9)

The fields on GIN definedby the linear representationr of H are thenin one-to-onecorrespondencewith the functions ~ on G suchthat for anyh in N

~a(gh) = ~(~1)a~~(g) (B.10)

For example the scalar functions on GIN correspondto the scalarfunctionsf on G that are constantonthe äosetsgH, i.e.,

f(gh)=f(g)~f(gN), (B.11)

or f—~ fo IT, whereIT is the projectionof G onto GIN. As a basisfor the sectionscl’ the following choiceis convenient.Let a-(x) EG be an elementof the cosetx = gH for any x E U C GIN, i.e., a localsectionof the principal fibre bundleG(G/N, H) (seeappendixA). SincePU(x)Xo = x, we can chooseasbasisfor the fibre E~the fixed basis {Ea) of E0 transformedthrough a-(x),

00(x) = a-(x)Ea. (B.12)

118 R.Camporesi,Harmonicanalysis andpropagatorson homogeneousspaces

In the caseof the tangentbundlethis coincideswith the definition of vielbein frame associatedto agiven in appendixA [seeeq. (A.18)]. Settingg = u(x) in eq. (B.7) and using eq. (B.9) gives

= ~/i°(a-(x)). (B.13)

If a-’(x) = a-(x)h(x) is anothersection,the componentsof cli in the new basis {O~}arerelatedto theold onesby

,a —1 a bcl’ (x)=r(h (x))bcli (x). (B.14)

The inducedrepresentationir of G on the spaceof sections1(ET) is defined by

[ITT(g)çfi](x) = g~i(g~x), (B.15)

whereg actsbetweenthe fibres at the pointsg’x andx. It is easyto showthat eq. (B.15) definesarepresentation,i.e., that

‘irt(g1g2) = ir

T(g1)ir

T(g2) . (B.16)

For examplefor scalarfields we have

[ITT(g)çb](x) = 4(g’x), (B.17)

which is called the regular representationof G. For vectorfields ir( g) associatesto a field V the newfield V’, with

V’(x) = pg*V(g’x). (B.18)

Clearly IT is the productof two representationscommutingwith one another,one acting on theargumentof the fields by g1 and the other “rotating” their componentsby g. This is the usualtransformationlaw of physicalfields underthe actionof a symmetrygroup G [96]. The correspondingrepresentation~ inducedon the spaceof the functions i/i is simply a left translationof the argument,

(*T(g)çfi)(g’) = çli(g’g’). (B.19)

Denotingthe field ‘irT(g)çli by ~i’ and defining the componentsof cl’ and cl” as in (B.2), we get thefollowing transformation(x’ = gx):

cl,Pa(x~)= °r(h(g,x))~2b~/Jb(x), (B.20)

where h(g, x) EN has beendefined in appendixA. The transformation(B.20) is just the “active”counterpartof (B.14) and tells us, oncemore, that in order to define fields on GIN it is sufficient togive a linear representationof N.

Supposenowthat a scalarproductis givenon eachfibre E~so that the isomorphismsg: E~—~ Egx areisometric, i.e.,


(gV, gW)I~ = (V, W)~E, (B.21)

for anyV andW in E~.For example if we have a scalar product on E0, we can definethescalarproducton Egxo by (B.21). Then (T, E0) will be an orthogonal(unitary) representationof N.

As in appendix A, we assumethat a G-invariant metric is induced on GIN by restricting abi-invanantmetric on G to the tangentspace41 at the origin of GIN. The volume elementof thismetric is clearly invariant under G andthe integralover GIN of a scalarfield ~ is given by

J çb(x)dx= J ~(ExpX)Iz1~(x0,ExpX)ldX, (B.22)GIH

wherethe notationis the sameas in appendixA, i.e., we write x = Exp X with X E 41~,the maximalopen set in 41 such that the exponentialmapping is one-to-one,and dX is the ordinary Euclideanmeasureon 41.

A pre-Hilbertstructureis definedon the spaceof sections1(ET) by

(~1’~2)= J (~l(x),~2(x))IE~dx. (B.23)

Gil-I

Let L2(ET) denote the corresponding Hilbert space.Thenwe have

(ITT(g)~, ITT(g)~ = J (g~(g~x), g~(g~x))~~dxG/H

= J (~(g’x),~(g~x))l~ dx = J (~(y), ~(Y))lE~dy = (~,~),G/H GIH

(B.24)

wherethe G-invarianceof the measuredx has beenusedin the shorthandnotation:x = gy ~ dx = dy.Thus the operators‘ir( g) of the inducedrepresentationare unitary for anyg E G, and we havethefollowing result [137]:

Theorem B.2. The inducedrepresentation(ITT, 1(ET)) extendsto a unitary representationof G onL2(ET).

A similar result can be proved for the fields ~/‘on G. Given a bi-invariant measure0 = dg on G,define

(cl~~2) =J Kc~1(g),~2(g))lE0dg. (B.25)

Then

(.~r(g)~j ~r(g)~j) = (~,~), (B.26)


i.e., ~T(g) is unitary. Denotingby L2(G, T) the correspondingHilbert space,the inducedrepresenta-tions (ITT, L2(ET)) and (ITT, L2(G, r)) are unitarily equivalent, i.e., there exists a unitary invertibleoperatorA: L2(ET)~~+L2(G, r), such that ~T(g)= A’irT(g)A_l for any g E G. IndeedA is given by(B.6) and A’ by (B.7).

B.2. Frobeniusreciprocity

Let O be the set of the equivalenceclassesof the unitary irreduciblefinite dimensionalrepresenta-tions (irreps) of the compactgroupG. For anyA in G we choosea representative(UA, VA), whereforg E G, U”( g) is a unitary operatoron the vectorspaceV

5, andwe set dA = dim VA.

Fromnow on we shall assumethat the representation(r, E0) of H, defining the vectorbundleET, is

irreducible.The restriction of U5 to N is, in general,a reduciblerepresentation,and T will appearacertain number of times, m(r, U5IH), in its reduction in irreps of H. Eventually we can havem(T, U5IH) = 0.

On the otherhand,the inducedrepresentation‘iT can bewritten as a direct sumof irreps of G, as forany representationof a compactgroup. The multiplicity of the irrep U5 in this reduction will bedenotedm(U5, irT) Thenwe havethe following result,knownas Frobeniusreciprocity (see,e.g., ref.[96]):

TheoremB.3. For any irrep T of H and any A in G

m(T, U5~H)= m(U5, ‘iTt) (B.27)

i.e., any A appearsin the decompositionof IT’ in irreps of Gexactlythe samenumberof times that T

appearsin the decompositionof U5IH in irreps of H. In particularif U5~Hdoesnot containr, the irrepU5 doesnot appearin the harmonicexpansionof the given fields (the sectionsof ET).

This theoremtells us that inducing a representationand restricting it to a subgrouparein a sense“dual” operations.It is clear that in order to determinethe decompositionof the spacesL2(ET) orL2(G, T) we needto know the numbers~ us m(T, U5IH), for anyA in G. Clearly this must be studiedcaseby case,dependingon the groupG andon the fields in question.Whatwe aregoing to shownowis that the “harmonics” of the fields in L2(ET) (i.e., the componentsirreducible under G) can beexpressedin termsof the matrix elementsof the irreps A for which ~ ~ 0.

Indeed,let A E G with ~ ~ 0, and let

u5lH=Ul~U2~---~U. (B.28)

be the decompositionof U5~Hin irreduciblerepresentationsof H, (U,, V,). By definition ~ of these

irreps areequivalentto (r, E0). For examplesupposetheyarethe first ~ andlabel them by (Ut, Ye)’

for ~ = - ~ Then,on eachspaceV~wecan definean intertwiningoperatorLg: V~—~ E0 suchthat

U5(h)l~ = L

4~’ r(h)L~ , (B.29)

for anyh in N. Given anorthonormalbasis {Ea} in E0, the matrix elementsof r(h) aredefinedin eq.


(B.8), and we choosein each V~the basis {la~ L~1Ea}, so that

U5(h)la~)= T(h)balb5~). (B.30)

An orthonormal basis is chosenin V5 accordingto

{I’)} = {la~),Irest)} , (B.31)

where“rest” refers to the subspacesof V5 in which U5IH is not equivalentto r, and the indices areas

follows:

1=1,... ,d5, a=1,. . - ,drusdimEo=dimVg, ~1,. - . ,~ - (B.32)

The matrix elementsof the operatorsU5( g) are definedby

U5(g)lI) = UA(g)Jj~J), (B.33)

and eq. (B.30) can also be written in the form

UA(h)btag = T(h)”a , for any ~. (B.34)

Considernow the following functionson

a~ 5 —1 agf51:g—~U(g ) ~. B. 5

These are the rows of the matrix U5(g~),correspondingto theindex a in thespaceVg, for any fixed A,

a and ~. It is easy to verify, using (B.30), that for h in H

f/(gh) = r(h~Y’~f,~~’f(g). (B.36)

Hence, the functions fJ belong to C(G, r), where the induced representationacts just as a lefttranslationof the argument.By eq. (B.19) we get

[~T(g)f~f](gF) = U5(g)~1fJ(g’), (B.37)

i.e., the matrix representingthe operator*T( g) in this subspaceis preciselythe matrix of the irrep A ofG. We concludethat the d5 — dimensionalspaceV~of functions cl’~:G—~E0, with

= ~ U5(g’)~jE~ (B.38)

(cl’~E R or C), is a subspaceof C(G, T), invariant and irreduciblewith respectto *~.In agreementwith the Frobenius theorem the irrep A is containedin C(G, T) exactly ~ times, becausethe inducedrepresentation~ is equivalentto A in eachsubspace1’~,for ~ = 1,. . . , ~. Therefore we have the

122 R. Camporesi, Harmonic analysis andpropagators on homogeneous spaces

orthogonaldirect sumdecomposition

C(G, r) = ~ , (B.39)A ~=i

and any function cl’ can be expandedaccordingto

~a(g) = E ~U5(g~)~1, (B.40)

5 JF

wherethe sumconvergesboth in L2(G, T) and in the uniform topologyof C(G, r). The expansionfor

the correspondingfields cli(x) on GIN is obtainedby applying (B.13), i.e.,

= ~ U5(a-~(x))~1, (B.41)

A l.~

whereo~’(x) us (a-(x)) [126].In particularfor scalarfields the spacesyE areall one-dimensionaland(B.41) becomes

q~(x)= ~ UA(a~(x))Ei, (B.42)A I.E

whereA runs over the socalledsphericalrepresentationsof G, i.e., thosecontainingthe singletof H.Let us summarizeour results.We have found that the completeset of “harmonics” for a field on

G/H (definedby the irreduciblerepresentationr of H) is given by the matrix elements(a~jU5 o a-~l’~

of thoseirreps U5 of GcontainingT at leastonetime. The supplementaryindex ~ labelsthe multiplicityof r in U5.

The extensionof this resultto the casein which T is not irreducibleis obvious. We just determinethecombinationsof thefields i/i irreducibleunderH, andwrite down anexpansionlike (B.41)for eachoneof them.

The harmonicexpansion(B.40) and(B.41) can beinverted by using theorthonormalityformula forthe irreps of the compact group G [137, 138, 141]

J UA(g)K,UA*(g)~dg = (B.43)

where a star meanscomplex conjugationand VG is the volume of G. Multiplying both sides of eq.(B.40) by U5 *(g_l)bE~ integratingwith respecttog, and using (B.43), we get

~ ~1~ub =JUA*(gl)~,~(g)dg. (B.44)

We now multiply by 6ab and sumover a and b to get

= f U5(g)~,~(g)dg. (B.45)G r a

R. Camporesi, Harmonic analysis and propagators on homogeneous spaces 123

In order to obtain the inversion formula on GIN, we note that given a function f on GIN withassociatedfunction f = fo IT on G [seeeq. (B.il)] we have the integral formula

~— J f(x)dx=i~-ff(g)dg, (B.46)GIl-f GIN GG

whereVG/H is the volume of G/H. It is immediateto checkthat thefunction ~a US*(g_~)~~,c(/1(g)in eq.(B.45) is constanton eachcosetx= gH. The correspondingfunction on G/H is obtainedby replacingg—~a-(x),seeeq. (B.13). Using eq. (B.46) in (B.45) we finally get

= vdsd ~ I US*(a-(x))a~(x)dx. (B.47)GIH T GIN

If we substituteeq. (B.41) back into (B.47), we also obtain the orthonormality condition on G/H,

~f UA*(ui(x))~jUS(a-i(x))~j,dx = VGIHdT 56ô~~’ (B.48)GIN

B.3. Peter—Weyltheorem

As a simple but importantexamplelet us specializeto the group manifold case,i.e., let us takeH= {e} so that G/N = G. For scalarfields L

2(ET) coincideswith L2(G), the spaceof squareintegrablefunctionson the group,and r is just the singlet of {e}. It is clear that anyirrep A containsT exactlyd

5times,since U

5(e) is just the d5 x d5 unit matrix. Therefore~ = d5 and~is afull representationindex,

running from 1 to d5. Application of eq. (B.42) gives

I A —1 Kf(g)= fAKU (g ) ~, (B.49)

A I,K

wherethe sum is over all the irreduciblerepresentationsof G. This is the Peter—Weyltheorem.Theinvariant and irreduciblesubspacesof L

2(G) are the d5-dimensionalspacesgeneratedby the elements

(U5)’<

1, (U5)’~

2,- . - , (U5)’<~ (B.50)

of the Kth row of the matrix U5. SinceK can go from 1 to d5, we haved5 subspaces,in agreementwith

the Frobeniustheorem.The group G can also be consideredas a nontrivial cosetspaceof G x G. Considerthe following

left-actionof GX G on G:

= g1gg~, (B.51)

i.e., P(g,,g2) = Lg,ORg~1~The groupG x G is denotedGL x GR, to indicatethat it acts on Gby left andright multiplications. The stability group at the origin is just the subgroupof Gx G consistingof thecouples(g, g) with equalarguments.This is called the diagonal of G x G and is denotedGdiag. Sincethe action (B.51) is clearly transitive, we haveG=[GL x GR]/Gdjag.

124 R. Camporesi. Harmonic analysis and propagalors on homogeneous spaces

The irreduciblerepresentationsof GL X GR arethe tensorproductsA ® ~s,with A and~.t irreduciblerepresentationsof G. The irreps appearingin the harmonicexpansionof scalarfields on G are of theform A* 0 A, whereA* is the contragradientrepresentationof A. One can show that the singlet ofGdiag G is containedin A* ® A exactly one time. The Frobeniusreciprocity gives the direct sumdecomposition

(IT, L2(G)) = ~ (U5’ 0 U5, V5’ 0 V5), (B.52)

from which it is easyto reobtain the expansion(B.49). The invariant and irreducible subspacesaregeneratedby all the matrix elements(U5)’K.

Considernow the vector fields on G. As already seen, the tangentbundle is generatedby theisotropy representationof H, (T, E

0) = (Ph*H’ Tk(G/H)). In our casethis is (x0 = e)

P(g.g)* =(LgoRg~i)*1:T~(G)—~T~(G), (B.53)

i.e., just the adjoint representationof G (identifying the Lie algebraccwith the tangentspaceat theidentity). Assuming that G is simple, so that the adjoint representationis irreducible, the irreps of

x GRcontainingthe adjoint of G are of the form A®~.whereA is containedin Ad®js and ~i iscontainedin A 0 Ad. For a vector field V we obtain the two equivalentexpansions

V(g)= ~ ~ ~ (B,54)AEG M.N

= ~ ~ (B.55)~ M.N

in termsof conjugateright (left) invariant frames {R,} ({L,}) on G. In particular for a right-invariant(or left-invariant) vector field only the irrep (Ad, singlet) [or (singlet,Ad)] of GL x GR contributestothe expansion,i.e., we haveV= V’R, (or V= V’L~) with V = constant.

For exampleon G = SU(2) the “scalar harmonics”of SU(2) x SU(2) are given by

(0,0)(~,~)(1,1)(~,4)---, (B.56)

and the “vector harmonics”by

(0, 1)(l, 0)(1, 1)(~,k)( ~, 4 )(~,~)( 4,4)(1, 2)(2, 1)---. (B.57)

Similarly, the “spinor harmonics” are the representationsof SU(2) X SU(2) which contain the spinrepresentationj = 1/2 when restrictedto the diagonalof SU(2), i.e., seenas tensorproductsof onlyone factorSU(2). They aregiven by

(0, ~)(~,0)(~, 1)(1, ~)(1, 4)(4,1)---. (B.58)

In the well known (local) isomorphismsSO(4) SU(2)x SU(2) and SU(2) SO(3), the relationSU(2) [SU(2)x SU(2)]/SU(2) becomes

SU(2) SO(4)/SO(3) S3. (B.59)


Therefore,the irreps given in eqs. (B.56), (B.57) and (B.58), are the SO(4) scalar,vectorandspinorharmonicson the three-sphere.

The generalizationof the Peter—Weyltheoremto spinor and tensorfields on a group manifold isobvious. Given an arbitrary field, written as ~fr= ~//

2Ø in, e.g., a left-invariant basis °a’ we havetheharmonic expansion

a a 5 --1cl1

5,MNU(g )NM’ (B.60)A M,N

wherethe sumover A runs over all the inequivalentunitary irreduciblerepresentationsof G.

B.4. Laplacianson GIN and their spectrum

We haveseen in appendixA that the Levi-Civita connectionto of the G-invariantmetric and theH-connectionto coincide if and only if G/H is a symmetricspace. Therefore in generalthe metricLaplacianL (or Laplace—Beltramioperator)andthe LaplacianL of to aredifferent. Only whenactingon scalarfields, do they give the sameresult, and a simple calculationshows that

Lf= Lf= ~ K,K1f, (B.61)

where{K,} arethe Killing vectors.It is possibleto see that the operator~, K,K, equalsthe oppositeofthe CasimirnumberC2(A) whenactingon the matrix elements(U

5 °o~’)~of the sphericalrepresenta-tions of G [seeeq. (B.42)]. Howeverthe metric Laplacianis in generalnot diagonalon the irreps of Grelativeto tensoror spinorfieldson GIN. Forexamplefor avectorfield V= V~e~we get in the vielbein

8a~~7~ = (~~~4V) + I~VY+f~V~, (B.62)

whereRap is the Ricci tensor(A.77), ..9? is the Lie derivative, andthe operator

~ ‘~‘K,4,= —~ C2(A)ld5Xd5 (direct sum) (B.63)

is the Casimir Laplacian of G, acting in a naturalway on the fields of GIN and equalto minus theCasimir numberwhen acting on any irreps A of G. It will also be denotedby — C2(G).

On the otherhand usingthe N-connectionwe get

VaVaVP= (~4,4V) — (ñf1Y

7V~, (B.64)

wherethe secondterm in theright-handsideis just the Casimiroperatorof theisotropy representation‘r, with generators(J)a~ f,~ai.e.,

C2(T) = —J~f’. (B.65)

The formula (B.64) is a particular caseof a more generalresult (see,e.g., ref. [133]), accordingtowhich the Laplacianof the N-connectioncan be expressedin termsof the quadraticCasimir operators


on G andon H as

L = ~ + C2(r) = —C2(G)+ C2(r), (B.66)

wherer is the representationof H defining the given fields. We shall now prove this result.Considerthecovariant derivative in the H-connection of any field ~j/’on Gil-I. This is defined by the exteriorcovariantderivativeD

DclJa = d~’ + Qi’h~e’= (e,,~i”+ K~,Q~hi~)e”(~i~”)e’, (B.67)

where Q, are the generatorsof i-. In the embeddingof H into SO(n) describedin appendixA, the Q;

aregiven in termsof SO(n)-generatorsby a formula similar to (A.70). We can write eq. (B.67) in thesymbolic form

D = d + Q~e1= eaVa , (B.68)

showingthat the “gauge” derivativewith respectto the “gauge” group H coincideswith the covariantderivative in the H-connection.

Expanding the field ~//‘accordingto eq. (B.41) and comparing to (B.67) we get the covariantderivativeof the matrix elements(U5 °a

Va(U5°a-~’)~= e~(U5°a’ )~+ K~,Qr6(U

5°a~)~. (B.69)

Then, we have the following result:

TheoremB.4. The covariantderivativeandthe Laplacianof the matrix elements(U5 °a - )“~ with theH-connectionare given by (dropping a- ‘for simplicity)

= _(T~,)~~EK(UA)~J- (B.70)

= [C2(r) — C2(A)](U

5Y~, (B.71)

whereC2( A) is the Casimirnumberof the irrep A of G, T~= (T~,T~)is the reductivedecompositionof

the generatorsof A, and

C2(r)8~= —(~Q~Q~), (B.72)

is the Casimir operatorof the (irreducible)representationr of H defining the fields cl”

Proof. Define on G the function ~: g—~ g ~‘. One can verify that the differential of this functioninterchangesthe left with the right invariant vectorfields, accordingto

= —R,. (B.73)

Moreover, as the right-invariant vectorsR~generateleft translationson G, we have

R~U5= T~U5. (B.74)

R. Camporesi.Harmonic analysisandpropagatorson homogeneousspaces 127

Rememberingthe definition (A.18) of vielbein frameandeq. (A.24), andusingeqs. (B.73)and (B.74),we get, after a straightforwardcalculation,

ea(U5 oa-’)’~ = —(T~)’K(U5 oa1)~ — K’a(T~)’K(U5oa-1)”~. (B.75)

Now if U5~His decomposedin irreps of H accordingto (B.28) the generatorsT~of H satisfy

(T~Y’~K= 0 if K ~ b~in V~, (T~y2EK= Qib if K = b~in V~- (B.76)

Using this in eq. (B.75) with I = a~and comparingwith (B.69) we get (B.70). Next, computing theLaplacianandusing eqs. (B.70), (B.75) and (B.76), we get

VOVa(UAoa-’)”~ = (T~T~K(U5oa-~)K~+ K’a(T~ T~— T~T~+ ~?T~ )~K(U5~

(B.77)

wherethe sum overa is understood.The secondterm in the right-handside vanishes,becauseof thecommutation[T~, T~]= fatT~,. Concerningthe first term, we noticethat the Casimir operatorof A isdefinedby

—C2(A) = T~T~+ ~ (B.78)

whereT~are the antihermitiangeneratorsof U5. Moreover from eq. (B.76) we have

1 0 ifR~b~inV,

(T~T~Y’~R= t(QIQIYb = —C,(r)ô~ if R= b~in V (B.79)

wherethe irreducibility of T hasbeenused.Using this and eq. (B.78) in (B.77) we finally get eq.(B.71). QED

Accordingto this resultanydifferential relationon GIN reducesto algebraicmanipulationswith thematricesU5 and T~. The extensionto the casein which T is not irreducibleis obvious.

This theoremgives a useful formulafor calculatingthe eigenvaluesof the Laplacianof to, actingonanyfield on GIN, in terms of group-theoreticquantities.By the Frobeniusreciprocitytheoremit is alsoclear that the degeneraciesof theseeigenvaluesare just d

5 ~, whered5 is thedimensionof U5 and ~ is

the multiplicity of r in U5. In this way the spectralpropertiesof the compacthomogeneousspacescanbe deducedfrom the classificationof the finite-dimensionalirreduciblerepresentationsof the compactLie groups.

B.5. Examples:scalar and vector harmonicson G/H and the n-sphere

Let us considersomesimple applicationof the aboveresults. For scalarfields r is the singlet of Hand C

2(T) = 0. Thus, the eigenvaluesof the Laplacianreduceto — C2(A0), where A0--is a spherical irrepof G [seeeq. (B.42)]. (As we havealreadynoticed this is true for both to andc~.The reason is thatsince the torsion tensoris totally antisymmetricthe two LaplaciansL and L coincide whenacting onscalarfields.)

Concerning~, it is known[78, 84, 138] that if GIH is asymmetricspace~A0 = 1, i.e., eachspherical


representationcontains the singlet of H exactly once. Thereforeon symmetricspacesthe degeneracyfactor reducesto the dimensiond

5. This result generalizesto the tensorand spinor harmonicsonsymmetricspaces.

For vectorfields r is the isotropy (or vector) representationof H. For simplicity let usassumethat T

is irreducible. It is knownthat if GIN is simply connected(andwithout boundary),any vectorfield canbe uniquely written as the sum of a transverseand a longitudinal part,

1”,, = T~+ L , (B.80)

where V”Ta = 0 and La = ~ f for some function f. Using for f eq. (B.42) we see that the “vectorharmonics”split into (A0, A,), whereA1 refersto the transversepart, andA0 containsboth the singletandthe vectorialof H. (Obviouslythe singlet of G, A0 = 0, mustbe excluded.)For the three-spherethiscan be seenby comparingeqs. (B.56) and (B.57).

The harmonics

L~OJE= V~(U5°oa-1)E

1 = _(T~0)EK(USoo~ (B.81)

T~’~1= (U5’ oa-’)”~, (B.82)

givecompletesetsfor longitudinal andtransversevectorson GIH. For T5’ this is clearalreadyfrom eq.(B.48). The completenessof the set {L5°} and its orthogonalityto {T5’}, i.e.,

G/H d~x~ = C2( AO)VGIH ~ (B.83)

I dnx~g~(x)L~0*(x)T~I(x)=0,VA,, A,, (B.84)G/H

follow immediately upon an integrationby parts. The eigenvaluesof the Laplacianof to actingon T5’

are, by theorem(B .4), [C2(‘r) — C2(A1 )]. Using the commutationrules for the covariantderivativesof

scalarand vector fields,

VaVpf~VpVaf=~Tap~Vyf~ (B.85)

VQV~V0 VpVaV~R”papVyTapWyVp, (B.86)

whereR and Tarethe Riemannandtorsion tensorsof to (see appendixA), we can calculatethe valueof the Laplacianon the longitudinal harmonics(B.81). As one expects,the result is

gaPv V,,L5° = [C

2(r) — C2(A0)]L5. (B.87)

Togetherwith (B.83) and (B.84) this implies that the longitudinalharmonicscoincidewith the matrix

elements(U5~°a-_’ )“~,

~a-~)~= (C2(A0)/d~)~

2(U5°o a-1 )aE , (B.88)


or equivalently that

(T~0)E~= _[C2(Ao)Id~]U2o~- (B.89)

Let usspecializetheseresultsto the familiar caseof then-spherewith thestandardmetric inducedby~n+1 GIN = SO(n + 1)ISO(n). This is a symmetricspaceandthe two connectionsto andt~coincide.

The irreducible representationsof a compact Lie group G of rank r are labelled by a set of rnon-negativeintegersgiving the componentsof the highest (or dominant) weight vectorsin “the”Cartansubalgebra.For the orthogonalgroupswe have

rank SO(2n)= rank SO(2n+ 1) = n, (B.90)

andthe dominantvectorsof the irreps of SO(n+ 1) havea numberof componentsequalto the integerpartof (n + 1)/2. The irrepsof G that appearin theharmonicexpansionof a field on asymmetricspaceGIN can be labelled by a set of 1 non-negativeintegers,where1 is the rank of GIN. In otherwordsthedominantweight vectors of theseirreps live in “the” Cartan subspaceof GIN (a maximal abeliansubspaceof 41).

In our caseS” hasrank one(the only Casimiroperatorscommutingwith the Laplacianarepowersofthe Laplacian itself) and each irreducible tensorharmonic of a given type (e.g., longitudinal ortransversevectors)is labelledby only one indexp - In table B.i we havecollectedthe irreps r of SO(n)defining scalar,vectorandrank-two, symmetrictracelesstensorfields on S”, n � 4, andthe correspond-ing irreps A of SO(n + 1) which contain r when restrictedto SO(n).

The secondrank symmetrictracelesstensorscan be decomposedin a “longitudinal—longitudinal”(LL), “longitudinal—transverse”(LT), and a “transverse” part [123, 124], in analogy with thedecomposition(B .80) for vectors. Having in mind quantumfield theory (one-loop) calculationswithzeta-function regularisationwe shall only give here the eigenvaluesand the degeneraciesof theLaplacian.A descriptionof the irreducibletensorharmonicson S~,i.e., of the matrix elements(U5)~,

can be found,e.g., in refs. [80, 123, 136].The Casimir number of the irreps (p, q) (p, q,0,. . . , 0) of SO(n+ 1) is [9]

C2(p, q)=p(p + n — 1)+ q(q + n —3). (B.91)

From this formula we can calculate C2(A,) and C2(r,), i = 0, 1,2, and then the corresponding

eigenvaluesof the Laplacian[C2(r,)— C2(A1)], i �j. The resultsare

Table B.1Scalar,vectorand2nd-ranktensorharmonicson S’, n � 4

SO(n) T SO(n+ 1) A

f r,(0,0 0) A,(p,0 0) p�O

V, I,,, r,(1,0 0) A,=(p,0 0) p�lT r,=(1,0 0) A,=(p,1,0 O)p�l

A,, LL i,(2,0 0) A,=(p,0,... 0) p�2LT r,(2,0 0) A,(p,1,0 O)p�2T r,”(2,0 0) A2=(p,2,0 O)p�2

130 R. Camporesi, Harmonic analysis and propagator.c on homogeneous spaces

C2(r,,)—C2(A,,)—C2(A,)—p(p+n—l). (B.92)

C2(r1) — C7(A,,)= —p(p + n—i) + n —1 , (B.93)

C5(r1) — C2(A,)= —p(p + n — 1)+ 1, (B.94)

C,(r2) — C.,(A,,) = —p(p + n — 1) + 2n (B.95)

C5(r2)—C,(A1)=---p(p+n—1)+n+2, (B.96)

C2(r2)—C,(A,)”~—p(p+n—1)+2. (B.97)

As S” is a symmetricspace,the degeneraciesof theseeigenvaluesaregiven by thedimensionsd5 of the

correspondingrepresentationsof SO(n+ 1). The degeneraciesof the longitudinal vectorharmonicsandof the “longitudinal—longitudinal” rank-twotensorharmonicscoincide with thescalardegeneraciesd5,.

Similarly the degeneraciesof the “longitudinal—transverse”rank-two harmonicscoincide with thetransversevector degeneraciesd5. Therefore we just needthe values of d5, d5 and d5~.From theknown formulas for the dimensionsof SO(n+ 1) [100], we get [123,124] -

d — (2p+n—1)(p+n---2)! B98)p!(n—1)!

d p(p+n—1)(2p+n—1)(p+n—3)! (B99)— (p + 1)!(n —2)!

d (p+n)(2p+n—1)(p+n—3)!(p—1)(n+1)(n—2) B100— 2(p+ 1)!(n —1)! ( - )

The scalar degeneraciesof the Laplacian of an arbitrary compactsymmetric spaceare calculatedinsection 10.

Note added in proof

After this work was completedwe becameaware of ref. [144],where the formula (10.8) for thedimensionsof the spherical representationson a compact symmetric space is derived. We thankSigurdur Helgasonfor providing us with a copy of his paper.

Acknowledgements

We thank Arlen Anderson,Bei-Lok Hu, and Claudio Orzalesi for many helpful discussions.Thiswork was supportedin part by NSF grant no. PHY-87-17155.

R. Camporesi,Harmonicanalysis andpropagazorson homogeneousspaces 131

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