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

    HARMONIC ANALYSIS AND PROPAGATORS ON HOMOGENEOUS SPACES

    RobertoCAMPORESI Departmentof 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: the 2. The heat kernel and the Schwinger—DeWittexpansion 7 Riemann—Liouville integral 48 3. 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 51 4.2. The heat kernel on a compacthomogeneousspace 16 8.5. Fractional representationof the Jacobipolynomials 4.3. Example:S 19 and the spectrum 54

    5. Freemotion on symmetricspacesandquantum integrahle 8.6. The noncompactcase and the Weyl fractional in- systems 20 tegral 57 5.1. The maximal torus and the lattice 21 9. Exactnessof the WKB approximationon the split-rank 5.2. The radial Laplacian 23 symmetric spaces 58 5.3. The duality spectrum—geodesics 24 10. The partition function and thedimensionsof thespherical 5.4. The intertwining operatormethod 26 representations 62 5.5. The noncompactcase 27 111.1. The Plancherel measureand the spectrum of a 5.6. Complete integrability of the one-dimensionalquan- compactsymmetricspace 62

    tum system 30 1(1.2. Examples:rank-oneSS, normal real form SS. and 6. The heatequationon a torus and the Poissonsummation Lie groups 65

    formula 32 1(1.3, Contour representationof the partition function 66 7. 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 70 7.1. 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 72 7.2. The spectrumand the “sum over classicalpaths” 41 11.3. Scalarzeta functions for arbitrarycoupling 76 7.3. 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 84 spaces:fractional derivatives 46 12.2. The spinor case 88 8.1. The spherecase 46 12.3. The effective potential on M’ x V 9))

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    PHYSICSREPORTS(Review Sectionof PhysicsLetters) 196, Nos. 1 & 2(1990)1—134.

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  • HARMONIC ANALYSIS AND PROPAGATORS ON

    HOMOGENEOUS SPACES

    Roberto CAMPORESI

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

    ii NORTH-HOLLAND

  • R. Camporesi,Harmonicanalysisand propagatorson homogeneousspaces 3

    12.4. The high-temperatureexpansion 93 Appendix B. Harmonicanalysison homogeneousspaces 115 12.5. 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 120 Al. Killing vectors and thevielbein frame 101 B.3. Peter—Weyltheorem 123 A.2. Invariant connectionson homogeneousspacesand B.4. Laplacianson G/H and theirspectrum 125

    their curvature 107 B.5. Examples:scalarandvectorharmonicson GIH and A.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.Severaltopics relevantto physical applicationsare discussed,including the Schwinger—DeWittexpansion,the exactnessof the WKB approximationin curved spaces,theconnectionbetweenfree motion on symmetricspacesandquantumintegrablesystems,andfinite-temperaturequantumfield theoriesin higher dimensions.The paper containssome new resultsof both mathematicaland physical interest; e.g., explicit formulas for the scalar degeneraciesof theLaplacianon acompactsymmetricspace,exactforms of thezetafunctioli on thesymmetricspacesof rankone,extensionof the finite-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 Moretteand DeWitt on the path integral approach[108, 38], and in the sixties with the developmentof the Schwinger—DeWittpropertime formalism and the heatkernel expansion.

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

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

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

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

    Minakshisundaramand Pleijel gaveas early as in 1949 a generalformula for recursivelycalculating the heat kernel coefficients, {a~},in termsof an integral over geodesics.The actual details for an arbitrary spacetimeare rathercomplicated.Expandingthe Laplacian and the Van Vleck—Morette determinantin Riemannnormal coordinates(RNC), the perturbativeexpressionof the first few a~has beenobtainedin 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 usualflat space Feynmanrules, as exemplified by Bunch and Parker’s treatment [22] of A~”theory on an arbitrarycurvedspacetime (seealsoref. [821).However, all of thesetechniquesare local in natureand are not sensitive to the global structureof spacetime.

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

    Exact forms of the propagatorareobtainablefor fields in spacetimeswith high symmetry.andknown eigenvaluesfor the spacepart (e.g., a homogeneousmanifold) andsolutionsto the wave equationfor the 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]. Nonlocal Riemannnormal coordinatesand group theoreticalmethodshave been developedfor the study of homogeneoustype IX cosmologicalmodels [83].

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

    The emphasisthroughout the paper is on exact results. For this reasonwe have restrictedour attentionto the very specialclassof homogeneousmanifolds,whosespectralpropertiesareknownfrom the representationtheory of Lie groups. The most explicit results will be obtainedfor symmetric homogeneousspaces.

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

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