A NEW ALGEBRAIC APPROACH FOR CALCULATING THE HEAT KERNEL IN QUANTUM GRAVITY. I. G. AVRAMIDI * x. Department of Mathematics, University of Greifswald

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1 University of Greifswald, EMA-MAT hep-th/ A NEW ALGEBRAIC APPROACH FOR CALCULATING THE HEAT KERNEL IN QUANTUM GRAVITY I. G. AVRAMIDI * x Department of Mathematics, University of Greifswald Jahnstr. 5a, 7489 Greifswald, Germany avramidi@math-inf.uni-greifswald.d400.de Abstract It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e. in symmetric spaces, may be presented in form of an averaging over the Lie group of isometries with some nontrivial measure. Using this representation the heat kernel diagonal, i.e. the heat kernel in coinciding points is obtained. Related topics concerning the structure of symmetric spaces and the calculation of the eective action are discussed. * Alexander von Humboldt Fellow x On leave of absence from Research Institute for Physics, Rostov State University, Stachki 94, Rostov-on-Don 34404, Russia

2 . Introduction The heat kernel, a very powerful tool for investigating the eective action in quantum eld theory and quantum gravity, has been the subject of muchinvestigation in recent years in physical as well as in mathematical literature [-22]. The subject of present investigation is the low-energy limit of the one-loop contribution of a set of quantized elds on a d- dimensional Riemannian manifold M of metric g with Euclidean signature to the eective action, which can best be presented using the -function regularization in the form [] () = 2 0 (0); (:) where (p) = 2p TrF p = 2p dt t p (p) Tr U(t); (:2) 0 F = +Q + m 2 ; (:3) U(t) = exp( tf ); (:4) with = g r r, Tr meaning the functional trace, being a renormparameter introduced to preserve dimensions, Q(x) an arbitrary matrix-valued function (potential term), m a mass parameter and r acovariant derivative. The covariant derivative includes, in general, not only the Levi-Civita connection but also the appropriate spin one as well as the vector gauge connection and is determined by the commutator [r ; r ] = R. The Riemann curvature tensor, the curvature of background connection and the potential term completely describe the background metric and connection, at least locally. In the following we will call these quantities the background curvatures or simply curvatures and denote them symbolic by < = fr ; R ;Qg. Exact evaluation of the heat kernel U(t) isobviously impossible. Therefore, one should make use of various approximations. First of all, let us mention the very important so called Schwinger - De Witt asymptotic expansion of the heat kernel at t! 0 [-5] Tr U(t) =(4t) d=2 exp( tm 2 ) B k = M X k=0 ( t) k B k ; (:5) k! dxg =2 trb k : (:6) This expansion is purely local and does not depend, in fact, on the global structure of the manifold *. Its coecients b k (we call them Hadamard - Minakshisundaram - De Witt - * In manifolds with boundary additional terms in B k as well as new terms of order t d=2+k=2 in form of surface integrals over the appear. For details see [2,3], where all coecients for arbitrary boundary conditions up to terms of order t d=2+ are calculated. 2

3 Seely (HMDS) coecients) are local invariants built from the curvature, the potential term and their covariant derivatives [,5,6,4]. They play avery important role both in physics and mathematics and are closely connected with various sections of mathematical physics [4,22]. Therefore, the calculation of HMDS-coecients is in itself of great importance. Various methods were used for calculating these coecients, beginning from the direct De Witt's method [] to modern mathematical methods, which make use of pseudodierential operators, functorial properties of the heat kernel etc. [5-3]. Very good reviews of the calculation of the HMDS-coecients are given in recent papers [4]. Nowadays, in general case only the rst four coecients are explicitly calculated. The rst three coecients were calculated in [9]. An eective covariant technique for calculating HMDS-coecients is elaborated in [0,4], where also the rst four coecients are computed. In the case of scalar operators the fourth coecient is also calculated in []. Analytic approachwas developed in [7], where a general expression in closed form for these coecients was obtained.the leading terms in all the volume coecients B k quadratic in the background curvatures were calculated completely independently in [5,6]. Although the Schwinger - De Witt expansion is good for small t, (viz. t< ), and thereby in the case of massive quantized elds in weak background elds when <m 2, it is absolutely inadequate for large t in strongly curved manifolds and strong background elds (<m 2 ). For investigating these cases one needs some other methods. A possibility to exceed the limits of the Schwinger - De Witt expansion is to employ the direct partial summation [2]. Namely, one can compare all the terms in HMDS-coecients B k (.6), pick up the main (the largest in some approximation) terms and sum up the corresponding partial sum. There is always a lack of uniqueness concerned with the global structure of the manifold, when doing so. But, hopefully, xing the topology, e.g. the trivial one, one can obtain a unique, well dened, expression that would reproduce the Schwinger -De Witt expansion, being expanded in curvature. Actually, the eective action is a covariant functional of the metric and depends on the geometry of the manifold as a whole, i.e. it depends on both local characteristics of the geometry like invariants of the curvature tensor and its global topological structure. However, we will not investigate in this paper the inuence of the topology but concentrate our attention, as a rule, on the local eects. Then the possible approximations for evaluating the eective action can be based on the assumptions about the local behavior of the background elds, dealing with the real physical gauge invariant variations of the local geometry, i.e. with the curvature invariants, but not with the behavior of the metric and the connection which is not invariant. Comparing the value of the curvature with that of its covariant derivatives one comes to two possible approximations: i) the short-wave (or high-energy) approximation characterized by rr<<<and ii) the long-wave (or low-energy) one rr<<<. The idea of partial summation was realized in short-wave approximation for investigating the nonlocal aspects of the eective action (in other words the high-energy limit of that) in [5,4], where all the terms in the HMDS-coecients B k with higher derivatives (quadratic in the curvature and potential term) are calculated and the corresponding asymptotic expansion is summed up. Another approach to study the high-energy limit of the eective action, so called covariant perturbation theory, is developed in [7]. 3

4 2. Low-energy approximation and its consequences The low-energy eective action, in other words, the eective potential, presents a very natural tool for investigating the vacuum of the theory, its stability and the phase structure [23]. Here only partial success is achieved and various approaches to the problem are only outlined (see, e.g. our recent papers [20,2]). The long-wave (or low-energy) approximation is determined, as it was already stressed above, by strong slowly varying background elds. This means that the derivatives of all invariants are much smaller than the products of the invariants themselves. The zeroth order of this approximation corresponds to covariantly constant background curvatures r R =0; r R =0; r Q=0: (2:) In this case the HMDS-coecients are simply polynomials in curvature invariants and potential term of dimension < k up to terms with one or more covariant derivatives of the background curvatures O(r<) b k = kx n=0 k n Q k n a n + O(r<); (2:2) a k = b k Q=rR=0 = X R k : (2:3) Mention that the commutators [Q; R ] are of order O(rr<) and, therefore are neglected here. Then after summing the Schwinger-De Witt expansion (.5) we obtain for the heat kernel, the -function and the eective action Tr U(t) = (p) = () = M M M dx g =2 (4t) d=2 tr exp t(m 2 + Q) ((t)+o(r<)) ; (2:4) dx g =2 (4) d=2 2p dt t p d=2 tr exp t(m 2 + Q) ((t)+o(r<)) ; (p) 0 (2:5) dx g =2 fv (<)+O(r<)g; (2:6) with V (<) = 2 (4) d=2 ( d 2 +) 0 dt log( d 2 + d 2 + tr exp( t(m 2 + Q))(t) (2:7) 4

5 for even d and V (<) = 2 d=2 ( d +) dtt 2 0 for odd d, where (t) = X k=0 d+ 2 tr exp( t(m 2 + Q))(t) (2:8) ( t) k a k ; (2:9) k! is a function of local invariants of the curvatures (but not of the potential). It is naturally to call the functions (t) and V (<), that do not contain the covariant derivatives at all and so determine the zeroth order of the heat kernel and that of the effective action, the generating function for covariantly constant terms in HMDS-coecients and the eective potential in quantum gravity respectively. Let us mention that such a denition of the eective potential is not conventional. It diers from the denition that is often found in the literature [24]. What is meant usually under the notion of the eective potential is a function of the potential term only Q, because it does not contain derivatives of the background eld (in contrast to Riemann curvature R that contains second derivatives of the metric and the curvature R with rst derivatives of the connection). So, e.g. in [24] the potential term Q is summed up exactly but an expansion is made not only in covariant derivatives but also in powers of curvatures R and R, i.e. the curvatures are treated perturbatively. Thereby the validity of this approximation for the eective action is limited to small curvatures R ;R Q. Such an expansion is called `expansion of the eective action in covariant derivatives'. Without the potential term (Q = 0) the eective potential in such a scheme is trivial. Hence we stress here once again, that the eective potential in our denition contains, in fact, much more information than the usual eective potential does when using the `expansion in covariant derivatives'. As a matter of fact, what we mean is the low-energy limit of the eective action formulated in a covariant way. Mention that the conditions (2.) are local. They determine the geometry of the locally symmetric spaces. However, the manifold is globally symmetric one only in the case when it satises additionally some global topological restrictions (usually it has to be connected) and the condition (2.) is valid everywhere, i.e. at any point of the manifold [25]. But in our case, i.e. in physical problems, the situation is radically dierent. The correct setting of the problem seems to be as follows. The low-energy eective action depends, in general, also essentially on the global topological properties of the space-time manifold. But, as it was mentioned above, we do not investigate in this paper the inuence of the topology. Therefore, consider a complete noncompact asymptotically at manifold without boundary that is homeomorphic to IR d. Let a nite not small, in general, domain of the manifold exists that is strongly curved and quasi-homogeneous, i.e. the invariants of the curvature in this region vary very slowly. Then the geometry of this region is locally very similar to that of a symmetric space. However one should have in mind that there are always regions in the manifold where this condition is not fullled. This is, rst of all, the asymptotic Euclidean region that has small curvature and, therefore, the opposite short-wave approximation is valid. 5

6 The general situation in correct setting of the problem is the following. From innity with small curvature and possibly radiation, where [7] << rr<, we pass on to quasi-homogeneous region where the local properties of the manifold are close to those of symmetric spaces. The size of this region can tend to zero. Then the curvature is nowhere large and the short-wave approximation is valid anywhere. If one tries to extend the limits of such region to innity, then one has also to analyze the topological properties. The space can be compact or noncompact depending on the sign of the curvature. But rst we will come across a coordinate horizon-like singularity, although no one true physical singularity really exists. This construction can be intuitively imagined as follows. Take the at Euclidean space IR d, cut out from it a region M with some and stick to it along the boundary, instead of the piece cut out, a piece of a curved symmetric space with the same Such a construction will be homeomorphic to the initial space and at the same time will contain a nite highly curved homogeneous region. By the way, the exact eective action for a symmetric space diers from the eective action for built construction by a purely topological contribution. This fact seems to be useful when analyzing the eects of topology. Thus the problem is to calculate the low-energy eective action (2.7), (2.8), i.e. the heat kernel for covariantly constant background. Although this quantity, generally speaking, depends essentially on the topology and other global aspects of the manifold, one can disengage oneself from these eects xing the trivial topology. Since the asymptotic Schwinger - De Witt expansion does not depend on the topology, one can hold that we thereby sum up all the terms without covariant derivatives in it. In other words the problem is the following. One has to obtain a local covariant function of the invariants of the curvature (t) (2.9) that would describe adequately the low-energy limit of the trace of the heat kernel and that would, being expanded in curvatures, reproduce all terms without covariant derivatives in the asymptotic expansion of heat kernel, i.e. the HMDS-coecients a k (2.3). If one nds such an expression, then one can simply determine the -function (2.5) and, therefore, the low-energy limit of the eective action (2.7), (2.8). 3. Symmetric spaces In this paper we will get the most out of the properties of symmetric spaces. Let us list below some known ideas, facts and formulae about symmetric spaces presented in the form that is most convenient for calculating the heat kernel and the eective action. So, what is the direct consequences of the condition of covariant constancy of the curvature (2.)? 3. Geometrical framework First of all, to carry out the calculations in the curved space in a covariant waywe need some auxiliary two-point geometric objects, namely the geodetic interval (or world function) (x; x 0 ), dened as one half the square of the length of the geodesic connecting the 6

7 points x and x 0, the tangent vectors (x; x 0 )=r (x; x 0 ) and 0(x; x 0 )=r 0(x; x 0 )to this geodesic at the points x and x 0 respectively and a frame e a(x; x 0 ) which iscovariantly constant (parallel) along the geodesic between points x and x 0, i.e. r e a =0:We denote the frame components of the tangent vector by a (x; x 0 )=g ab e 0 a (x 0 )r 0(x; x 0 ). Any tensor Tb a, can be presented then in the form of covariant Taylor series T a b = X n0 ( ) n n! 0 0 n r( r n )T (x 0 )e a 0 e0 b : (3:) Therefrom it is clear that the frame components of a covariantly constant tensor are simply constant. In the case of covariantly constant curvature one can express the mixed second derivatives of the geodetic interval, i.e. the matrix a b (x; x0 )=e a 0(x0 )e b (x)r0 r (x; x 0 ); (3:2) explicitly in terms of the curvature at a xed point x 0. Introducing a matrix K = fk a b (x; x0 )g K a b = R a cbd c d ; (3:3) one can sum up the Taylor series obtaining a closed form [4] a b = p K sin p K! a b : (3:4) This expression as well as any other similar expressions below should be always understood as a power series in the curvature. 3.2 Curvature Let us consider the Riemann tensor in more detail. The components of the curvature tensor of any Riemannian manifold can be always presented in the form R abcd = ik E i ab Ek cd (3:5) where Eab i,(i=;:::;p;p d(d )=2), is some set of antisymmetric matrices (2-forms) and ik is some symmetric nondegenerate matrix. Then dene the traceless matrices D i = fd a ibg D a ib = ik E k cb gca = D a bi (3:6) so that R a bcd = D a ib Ei cd ; Ra c bd= ik D a ib Dc kd ; (3:7) R a b = ik D a ic Dc kb ; R = ik D a ic Dc ka = ik tr(d i D k ) (3:8) 7

8 where ik =( ik ). Because of the curvature identities we have identically D a j[b Ej cd] =0: (3:9) The matrices D i are known to be the generators of the isotropy algebra H (or restricted holonomy algebra [25]) of dimension dimh = p [D i ;D k ]=F j ik D j; or D a ic Dc kb D a kc Dc ib = F j ik Da jb : (3:0) The structure constants F j ik of the isotropy algebra are completely determined by these commutation relations and satisfy the Jacobi identities F i j[k F j mn] =0; or [F i;f k ]=F j ik F j; (3:) where F i = ff k ilg are the generators of the isotropy algebra in adjoint representation. Mention that the isotropy group H is always compact as it is a subgroup of the orthogonal group (in Euclidean case). Now let us rewrite the condition of integrability of the relations (2.) given simply by the commutator of covariant derivatives [r ; r ]R = R [ R ] + R [R ] =0 (3:2) in terms of introduced quantities. It is not dicult to show that it looks like E i ac Dc bk E i bc Dc ak = E j ab F i jk : (3:3) This equation takes place only in symmetric spaces and is the most important one. It is this equation that makes a Riemannian manifold the symmetric space. From the eqs. (3.0) and (3.3) we havenow ik F k jm + mk F k ji =0; or F T i = F i ; (3:4) that means that the adjoint and coadjoint representations of the isotropy group are equivalent. The eq. (3.3) leads also to some identities for the curvature tensor D a i[b R c]ade + D a i[d R e]abc =0; (3:5) R a c Dc ib = Da ic Rc b (3:6) that means, in particular, that the Ricci tensor matrix commutes with all matrices D i and is, therefore, an invariant matrix of the isotropy algebra. Actually, eq. (3.3) brings into existence a much wider algebra G of dimension dimg = D = p + d, in other words it closes this algebra. Really, let us introduce new quantities C A BC = CA CB,(A=;:::;D) C i ab = E i ab ; Ca ib = D a ib ; Ci kl = F i kl ; (3:7) 8

9 C a bc = C i ka = C a ik =0; forming the matrices C A = fc B ACg =(C a ;C i ) C a = 0 D b ai E j ac 0 and symmetric nondegenerate matrix AB = ; C i = D b ia 0 0 F j ik ; (3:8) gab 0 : (3:9) 0 ik Then one can show, rst, that as a consequence of the identities (3.9)-(3.3) the quantities C A CB satisfy the Jacobi identities C E D[A CD BC] =0; or [C A;C B ]=C C AB C C (3:20) and are, therefore, the structure constants of some Lie algebra G, the matrices C A being then the generators of this algebra in adjoint representation. More precisely, the commutation relations have the form [C a ;C b ]=E i ab C i; [C a ;C i ]=D b ai C b; [C i ;C k ]=F j ik C j: (3:2) As we will see below this algebra is, actually, isomorphic to the Lie algebra of innitesimal isometries. And, second, using the denition of D-matrices and the eq. (3.4) one can show that the structure constants satisfy also the identity AB C B CD + DB C B CA =0; or C T A = C A ; (3:22) meaning the equivalence of the adjoint and coadjoint representations of the algebra G. In other words, the Jacobi identities (3.22) are equivalent to the identities (3.2) that the curvature must satisfy in the symmetric space. This means that the set of structure constants C A BC, satisfying the Jacobi identities, determines the curvature tensor of symmetric space R a bcd. Vice versa the structure of the algebra G is completely determined by the curvature tensor of symmetric space at a xed point x 0. Now consider the curvature of background connection R ab. One can show analogously to (3.2) that because of the integrability conditions of the eq. (2.) the curvature must have the form [r ; r ]R =[R ; R ]+2R [ R ] =0 (3:23) R ab = R i E i ab ; (3:24) where E i ab are the same 2-forms and R i are some matrices forming a representation of the isotropy algebra [R i ; R k ]=F j ik R j: (3:25) 9

10 Finally, from (2.) it follows that the potential term should commute with the curvature R [r ; r ]Q =[R ;Q]=0 (3:26) and, therefore, with all the matrices R i [R i ;Q]=0: (3:27) 3.3 Isometries On the covariantly constant background (2.), i.e. in symmetric spaces, one can easily solve the Killing equations L g =2r ( ) =0; (3:28) where L means the Lie derivative. Indeed, by dierentiating the equation L = r ( r ) + R (jj) =0; (3:29) having in mind rr = 0, and symmetrizing the derivatives we get r ( r 2n ) =( ) n R ( j j 2 R 3 j 2 j 4 R n 2n j n j 2n ) n ; (3:30) r ( r 2n+ ) =( ) n R ( j j 2 R 3 j 2 j 4 R n 2n j n j 2n r 2n+ ) n :(3:3) Therebywehave found all the coecients of the covarianttaylor series (3.) for the Killing vectors of symmetric spaces. Moreover, one can now sum it up obtaining a closed form (x) =e a ( (cos p K) a b b (x 0 ) sin p K p K! a b c b ;c(x 0 ) ) ; (3:32) where b ;c = ; eb e c. Therefore, all Killing vectors at any point x are determined in terms of initial values of the vectors themselves b (x 0 ) and their rst derivatives b ;c(x 0 ) at a xed point x 0. The set of all Killing vectors G = f A g; dimg = D, can be split in two essentially dierent sets: M = fp a g; dim M = d, with P a dened by p b cos K P a (x) =e b and H = fl i g; dim H = p = D d d(d )=2, where c P c a (x0 ) (3:33) L i (x) = e b sin p K p K! b a c L a i;c(x 0 ); (3:34) 0

11 according to the values of their initial parameters P a x=x 0 6=0; L i =0: (3:35) x=x 0 In fact, all odd symmetrized derivatives of P a and all even symmetrized derivatives of L i as well as L i themselves vanish at the point x 0 r P a L i x=x 0 = r ( r 2n+ )P a x=x 0 = r ( r 2n )L i x=x 0 x=x 0 =0; (3:36) =0: (3:37) All the parameters P b a(x 0 ) are independent and, therefore, there are exactly d such parameters. The maximal number of L b i;c is d(d )=2, since they are antisymmetric. However, they are not independent. This can be seen immediately if one mentions the equation L Li R =2fL i;[ R ] + L i;[ R ]g =0 (3:38) that holds in symmetric spaces and that is, actually, the integrability condition for Killing equations (3.26). This equation imposes strict constraints on the possible initial parameters L b i;c (x0 ). One can show that the number of independent parameters L b i;c (x0 ) is equal to p. Thus taking into account (3.5) it is evident that one can put P a b(x 0 )= a b ; La i;b(x 0 )= D a ib : (3:39) The spaces with maximal number of independent isometries, i.e. with p = d(d )=2 and D = d + p = d(d +)=2, are the spaces of constant curvature and only those. The generators of isometries A = A r in symmetric spaces have the form where P a = P ar = pk cot p K b a D b; (3:40) L i = L ir = D b ia a D b ; (3:4) D a =( a b ) e b (3:42) One can show [25] that the generators of isometries (acting on scalar elds) form a representation of the Lie algebra G (3.20) or, more explicitly, [ A ; B ]=C C AB C; (3:43) [P a ;P b ]=E i ab L i; [P a ;L i ]=D b ai P b; [L i ;L k ]=C j ik L j: (3:44)

12 This algebra is just the one generated by the curvature tensor of symmetric space. Hence we conclude that the curvature tensor of the symmetric space completely determines the structure of the group of isometries General structure The locally symmetric space M with covariantly constant curvature tensor is called globally symmetric space (or, simply, symmetric space) if it is simply connected. The symmetric space M is isomorphic to the quotient space of the group of isometries by the isotropy subgroup M = G=H [25]. The eigenvalues of the matrix ik determine the sectional curvatures K(u; v) =R abcd u a v b u c v d = ik (E i ab ua v b )(E k cd uc v d ). The Riemannian symmetric manifold is of compact, noncompact or Euclidean type if all sectional curvatures are positive, negative or zero, i.e. if the matrix ik is positive denite, negative denite or zero. A simply connected symmetric space is, in general, reducible, and has the following general structure [25] M = M 0 M + M (3:45) where M 0, M + and M are the Euclidean, compact and noncompact components. The corresponding algebra of isometries is a direct sum of ideals G = G 0 G + G (3:46) where G 0 is an Abelian ideal and G + and G are the semi-simple compact and noncompact ones. There is a remarkable duality relation between compact and noncompact objects. For any algebra G = M + H = fp a ;L i g one denes the dual one according to G = im + H = fip a ;L k g, the structure constants of the dual algebra being fc A BCg = fe i ab ;Dc dk ;Fj lm g = f E i ab ;Dc dk ;Fj lmg: (3:47) So, the star only changes the sign of E i ab but does not act on all other structure constants. This means also that the matrix (3.9) for dual algebra should have the form AB = gab 0 0 ik = gab 0 0 ik and, therefore, the curvature of the dual manifold has the opposite sign (3:48) R abcd = R abcd : (3:49) We will consider in this paper mostly the case of compact manifolds when all sectional curvatures are positive and, therefore, the matrix ik and the matrix AB (in Euclidean case) are positive denite. It is not dicult to generalize then the results to the general case using the duality relation and analytical continuation. 2

13 4. Heat kernel Below in this paper we restrict ourselves to the case of scalar operators, i.e. R =0. The general case will be investigated in a future work. 4. Heat kernel operator It is not dicult to show that the metric of the symmetric space can be presented in the form g = AB A B = g ab P a P b + ik L i L k : (4:) Indeed, by making use of the eqs. (3.7) and recalling the denition of the matrix K (3.3) it is easy to obtain (4.) using the explicit expressions (3.33), (3.34). Now having the metric (4.) we can build the Laplacian for the scalar (R = 0) case = g r r = AB A B ; (4:2) where A = Ar and the Killing equation (3.5) has been used. It is not dicult to show that the Laplacian belongs to the center of the enveloping algebra, i.e. it commutes with all the generators of the algebra [ ; A ]=0: (4:3) Let us now try to represent the heat kernel in terms of a group average, i.e. let us nd a formula like exp (t )= dk =2 (tjk) exp(k A A ): (4:4) We formulate rst the answer in form of a theorem and prove it below. Theorem : For any compact D-dimensional Lie group generated by A [ A ; B ]=C C AB C (4:5) it takes place the operator identity sinh(k A =2 exp(t )=(4t) D=2 dk =2 C A =2) det k A C A =2 exp 4t ka AB k B + 6 R Gt exp(k A A ); (4:6) where = AB A B, AB =( AB ), = det AB, AB is a symmetric nondegenerate positive denite matrix connecting the generators in adjoint C A =(C B AC ) and co-adjoint CA T representations CA T = C A ; (4:7) R G is the scalar curvature of the group manifold R G = 4 AB C C AD CD BC ; (4:8) 3

14 and the integration is to be taken over the whole Euclidean space IR D. The proof: Let us consider the integral (t) = dk =2 (k; t) exp(k A A ); (4:9) where (tjk) =(4t) D=2 det sinh(k A =2 C A =2) k A exp C A =2 4t ka AB k B + 6 R Gt : (4:0) To prove the theorem we have to show that (t) = exp(t ), in other words, that it satises the operator t = (4:) with initial condition (t) =: t=0 (4:2) First one can show that B exp(k A A )=X B exp(k A A ); (4:3) where X A = X M (4:4) are the left-invariant vector elds on the group that have in canonical coordinates the explicit form X M A (k) = k A M C A exp(k A : (4:5) C A ) Therefore, from the denition of the Laplacian we have exp(k A A )=X 2 exp(k A A ); (4:6) X 2 = AB X A X B : (4:7) Then, introducing the metric on the group manifold G MN = AB X A M X B N (4:8) and its determinant A G = det G MN = det X 2 = det sinh(k A 2 C A =2) k A ; (4:9) C A =2 one can obtain the transposition relation G =2 X 2 G =2 T = X2 : (4:20) 4

15 Now, making use of (4.9), (4.6) and (4.20) and integrating by parts we obtain (t) = dk =2 exp(k A A ) G =2 X 2 G =2 : (4:2) On the other hand, one has from t (t) = dk t exp(k A A ): (4:22) Thus to prove (4.) we havetoshow t =G =2 X 2 G =2 : (4:23) Substituting the explicit expression for (tjk) = =4 G =4 (k)(4t) D=2 exp 4t ka AB k B + 6 R Gt (4:24) and using the relations X 2 G =4 = 6 R GG =4 (4:25) and AG =4 = 2 (D trx)g =4 ; (4:26) where trx = XA A =tr ka C A coth k A C A ; (4:27) that hold on the group manifold, we convince ourselves that the eq. Thereby itisshown that (t) really satises the eq. (4.). Further, from (4.0) it follows immediately (4.23) is correct. (tjk) = =2 (k) (4:28) t=0 and, therefore, the initial condition (4.2). Thus we found (t) = exp(t theorem. ) that proves the 4.2. Heat kernel diagonal So, we have found a very nontrivial representation (4.6) that holds on any compact Lie group. How can we proceed now with this useful theorem? First, we can express the scalar curvature of the group manifold in terms of the scalar curvature of the symmetric space R and that of the isotropy subgroup R H R G = 4 AB C C AD CD BC = 3 4 R + R H; (4:29) 5

16 where R H = 4 ik F m il F l km : (4:30) The representation (4.6) is valid for any generators A, satisfying the commutation relations (4.5), and so it is also valid for the innitesimal isometries (3.40), (3.4) of the symmetric space. In this case is the usual Laplacian and exp(t ) is the heat kernel operator. For further use it is convenient to rewrite the integral (4.6) splitting the integration variables k A =(q a ;! i ) in the form exp(t ) =(4t) D=2 sinh((q a C a +! i =2 C i )=2) exp dq d! =2 =2 det 4t (qa g ab q b +! i ik! k )+ (q a C a +! i C i )=2 8 R+ 6 R H t exp q a P a +! i L i ; (4:3) where = det ik, = det g ab.toget the heat kernel explicitly in coordinate representation we have to act with the heat kernel operator exp(t ) on the delta-function on M exp(t )(x; x 0 ) = exp(t )(x; x 0 )= dq d! =2 =2 (tjq;!) exp q a P a +! i L i (x; x 0 ): (4:32) To learn how the operator exp(k A A ) acts on a scalar function f(x) let us introduce a new function (s; k; x) = exp(sk A A )f(x): (4:33) This function satises the rst order dierential s = k A A = k A A (x)@ (4:34) with the initial condition of the form = f(x): (4:35) s=0 It is not dicult to prove that where x 0 (s; k; x) satises the equation of characteristics (s; k; x) =f(x 0 (s; k; x)); (4:36) with initial condition dx 0 ds = ka A (x 0) (4:37) x 0 s=0 = x : (4:38) 6

17 Therefore, we have exp k A A (x; x 0 )=(x 0 (;k;x);x 0 ): (4:39) Consider now the operator integrals of the form we need I(x; x 0 )= dq d! =2 =2 (q;!) exp q a P a +! i L i (x; x 0 ); (4:40) where (q;!) is some analytic function. Using the eq. (4.39) we have exp q a P a +! i L i (x; x 0 )=(x 0 (;q;!;x;x 0 );x 0 )= =2 J(!; x; x 0 )(q q); (4:4) where q =q(!; x; x 0 ) is to be determined from the equation and J(!; x; x 0 ) is the Jacobian computed at x 0 = x 0 J(!; x; x 0 )=g 0 x 0 (; q;!; x; x 0 )=x 0 a =2 =2 0 So, we can now simply integrate over q in (4.40) to get I(x; x 0 )= q=q;s= : (4:43) d! =2 (q(!; x; x 0 );!)J(!; x; x 0 ): (4:44) If we are interested in coincidence limit then one has to put nally x = x 0 I(x; x) = d! =2 (q(!; x; x);!)j(!; x; x): (4:45) Consider now the equation of characteristics at greater length. Making a change of variables we arrive to the equation of more explicit form x! a 0 = a (x 0 ;x 0 )=e a 0(x0 ) 0 (x 0 ;x 0 ) (4:46) d a 0 ds = p K(0 ) cot p K( 0 ) a b qb! i D a ib b 0 : (4:47) Let a 0 = a 0 (s; q;!; b ) be the solution of the equation (4.47). Then q is to be determined from an equation like (4.42) a 0(; q;!; b )=0 (4:48) and J(!; x; x 0 ) = det a b q=q;s= ; (4:49)

18 where it has been taken into account det(e 0 a )=g 0 =2 =2. Therefore, we have to nd the solution to the equation (4.47) near the zero, i.e. assuming a 0 to be small. Moreover, we consider mostly the case when the points x and x 0 are close to each other that means that a is small too. The equation (4.47) near the point a 0 = 0 looks like d a 0 ds = qa (4:50) meaning that the momentums q a are of the same small order. More precisely, we assume a 0 b q c " (4:5) and look for a solution of the eq. (4.47) in form of a power series in ", i.e. in form of a Taylor series in a and q a. In this way one simply obtains up to quadratic terms 0(s; a q;!; x; x 0 exp( ) = (exp( s! i D i )) a s! i a D i ) b b +! i q b + O(" 2 ) (4:52) D i b With the same accuracy the solution of the eq. (4.48) is q a! i D i exp( s! i a D i ) = exp( s! i b + O( a 2 ): (4:53) D i ) Further, one nds from (4.52) and so, from (4.49) a b q=q;s= = det J(!; x; x 0 ) = det b sinh(! i D i =2)! i + O( a ) (4:54) D i =2 sinh(! i D i =2)! i + O( a ): (4:55) D i =2 Substituting (4.53) and (4.55) in (4.44) and expanding (q;!) we can calculate the integral (4.40) for near points x and x 0 in form of an expansion in a (x; x 0 ). For coincidence limit (4.45) we have, in particular, the exact answer I(x; x) = = dq d! =2 =2 (q;!) exp q a P a +! i L i (x; x 0 ) sinh(! i d! =2 D i =2) (0;!) det! i : (4:56) D i =2 8 x=x 0

19 Using the obtained results (4.53), (4.55) and (4.56) and substituting the explicit form of our integral (4.3) we get the heat kernel in coordinate representation =2 exp(t )(x; x 0 sinh(! i )=(4t) D=2 D i =2) det d! =2 H det sinh(! i C i =2)! i C i =2 exp 4t (!i ik! k + a g ac B c b(!) b )+ 8 R+ 6 R H! i D i =2 t +O( a ); (4:57) where B(!) =fb a b (!)gis a matrix of the form sinh(! i 2 D i =2) B(!) =! i : (4:58) D i =2 Now, from (3.8) it is not dicult to mention that sinh(! i C i =2) sinh(! i D i =2) det! i = det C i =2! i D i =2 det sinh(! i F i =2)! i : (4:59) F i =2 Therefore, the nal result after taking into account (4.59) looks like exp(t )(x; x 0 sinh(! )=(4t) D=2 d! =2 det i =2 F i =2) sinh(!! i det i =2 D i =2) F i =2! i D i =2 exp 4t (!i ik! k + a g ac Bb(!) c b )+ 8 R+ 6 R H t +O( a ): (4:60) The coincidence limit of this heat kernel is then simply derived by putting x = x 0, i.e. a =0, sinh(! i =2 exp(t )(x; x) =(4t) D=2 d! =2 F i =2) det det exp 4t!i ik! k +! i F i =2 8 R + 6 R H t sinh(! i =2 D i =2)! i D i =2 (4:6) Mention, that this formula is exact (up to possible nonanalytic topological contributions, see the discussion in sect. 2). This gives a nontrivial example how the heat kernel can be constructed using only the algebraic properties of the isometries of the symmetric space. One can derive an alternative nontrivial formal representation of this result. Substituting the equation (4t) p=2 =2 exp 4t!i ik! k =(2) p 9 dp exp ip k! k tp k kn p n (4:62)

20 into the integral (4.6) and integrating over! we obtain exp(t )(x; x) =(4t) d=2 exp t 8 R + 6 R H dp exp tp n nk p k det sinh( i@ k F k =2) i@ k F k =2 =2 det sinh( i@ k =2 D k =2) i@ k (p); D k =2 (4:63) k k. Therefrom integrating by parts and changing the integration variables p k! it =2 p k we get nally an expression without any integration exp(t )(x; x) =(4t) d=2 exp t 8 R + 6 R H =2 det det : sinh( p t@ k F k =2) p t@ k F k =2 sinh( p t@ k D k =2) p t@ k D k =2 =2 exp p n nk p k p=0 (4:64) This formal solution should be understood as a power series in the i that is well dened and determines the heat kernel asymptotic expansion at t! Heat kernel asymptotics Using obtained result one can get easily the explicit form of the generating function for HMDS-coecients (2.9) sinh(! i =2 (tjx; x) =(4t) p=2 F i =2) det d! =2 det exp 4t!i ik! k +! i F i =2 8 R + 6 R H t sinh(! i =2 D i =2)! i D i =2 (5:) This formula can be used now to generate all HMDS-coecients a k for any symmetric space, i.e. for any space with covariantly constant curvature, simply by expanding it in a power series in t. over! we get Changing the integration variables!! p t! and introducing a Gaussian averaging (tjx; x) = exp <f(!)>=(4) p=2 * 8 R + 6 R H t det d! =2 exp p sinh( t! i =2 F i =2) p t! i det F i =2 20 4!i ik! k f(!) (5:2) p sinh( t! i =2 + D i =2) p t! i D i =2 (5:3)

21 Using the standard Gaussian averages < >= ; <! i >=0 ; <! i! k >= 2 ik <! i! i 2n+ >=0; (5:4) <! i!i 2n >= (2n)! 2 2n n! (i i 2 i 2n i 2n ) one can obtain now all HMDS-coecients in terms of various foldings of the quantities D a ib and F j ik with the help of matrix ik. All these quantities are curvature invariants and can be expressed directly in terms of Riemann tensor. Thereby one nds all covariantly constant terms in all HMDS-coecients in manifestly covariant way. We are going to obtain the explicit formulae in a further work. 5. Concluding remarks In present paper we continued the study of the heat kernel that we conducted in our papers [4,0,5,2]. Here we have discussed some ideas connected with the point that was left aside in previous papers, namely, the problem of calculating the low-energy limit of the eective action in quantum gravity. We have analyzed in detail the status of the low-energy limit in quantum gravity and stressed the central role playing by the Lie group of isometries that naturally appears when generalizing consistently the low-energy limit to curved space. We have proposed a promising, to our mind, approach for calculating the low-energy heat kernel and realized, thereby, the idea of partial summation of the terms without covariant derivatives in local asymptotic expansion for computing the eective action that was suggested in [2,4]. Of course, there are left many unsolved problems. First of all, one has to obtain explicitly the covariantly constant terms in HMDS-coecients. This would be the opposite case to the high-derivative approximation [5,6] and can be of certain interest in mathematical physics. Then, we still do not know how to calculate the low-energy heat kernel in general case of covariantly constant curvatures, i.e. when all background curvatures ( < = fr ; R ;Qg) are present. Besides, it is not perfectly clear how to do the analytical continuation of Euclidean low-energy eective action to the space of Lorentzian signature for obtaining physical results. Acknowledgments I would like to thank G. A. Vilkovisky for many helpful discussions and R. Schimming and J. Eichhorn for their hospitality at the University of Greifswald. I am also grateful to P. B. Gilkey, H. Osborn, S. Fulling, D. M. Mc Avity, T. Osborn, S. Odintsov and K. Kirsten for correspondence. This work was supported, in part, by a Soros Humanitarian Foundations Grant awarded by the American Physical Society and by an Award through the International Science Foundation's Emergency Grant competition. 2

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