Alberta-Thy 0-97 Euclidean and Canonical Formulations of Statistical Mechanics in the Presence of Killing Horizons Dmitri V. Fursaev Theoretical Physics Institute, Department of Physics, University of Alberta Edmonton, Canada T6G J and Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 4 980 Dubna, Russia e-mail: dfursaev@phys.ualberta.ca Abstract The relation between the covariant Euclidean free-energy F E and the canonical statistical-mechanical free energy F C in the presence of the Killing horizons is studied. F E is determined by the covariant Euclidean eective action. The denition of F C is related to the Hamiltonian which isthegenerator of the evolution along the Killing time. At arbitrary temperatures F E acquires additional ultraviolet divergences because of conical singularities. The divergences of F C are dierent and occur since the density dn of the energy levels of the system blows up near the horizon in an infrared way. We show that there are regularizations that make it possible to remove the infrared cuto in dn C. After that the divergences of F become identical to the divergences of F E. The latter property turns out to be crucial to reconcile the covariant Euclidean and the canonical formulations of the theory. The method we use is new and is based on a relation between dn and heat kernels on hyperbolic-like spaces. Our analysis includes spin 0 and spin / elds on arbitrary backgrounds. For these elds the divergences of dn, F C and F E are presented in the most complete form.
Introduction There are two approaches how to describe quantum thermal eects in the gravitational eld. The approach by Gibbons and Hawking [],[] denes the partition function of the system as an Euclidean path integral. It enables one to express the free energy F E [g; ]in terms of the eective action W [g; ], as F E [g; ] = W[g; ]. Functionals W [g; ] are given on Euclidean manifolds M with the period in the Euclidean time. is considered as the inverse temperature. The elds are assumed to be periodic or antiperiodic in, depending on their statistics. The Gibbons-Hawking approach is a straightforward generalization of the nite-temperature theory in the Minkowsky space-time. Its advantage is that it is manifestly covariant in the Euclidean sector and enables one to consider the gravitational eld on the equal footing with matter elds. This approach is especially convenient for the thermodynamics of black holes []-[5], it reproduces the entropy, temperature and other characteristics of a black hole in the semiclassical approximation. When the space-time is static statistical-mechanical quantities can be also described in a canonical way. The canonical partition function has the form C =Tre ^H ; (.) where the Hamiltonian ^H is the generator of the time evolution of the system. C is well dened when ^H is a normally ordered operator [6]. In this case the free-energy takes the form F C [g; ] = ln = 0 dn(!) ln ( e! ) ; (.) dn(!) where = + for Bose elds and = for Fermi elds. is the density of eigenvalues! of quantum-mechanical Hamiltonians of the elds [6]-[4]. The advantage of denition (.) is that it is given in accordance with the unitarity evolution of the system. However, as distinct from the Gibbons-Hawking approach, it is not manifestly covariant. There is no special terminology to distinguish F E and F C. In this paper we call F E and F C the covariant Euclidean and the canonical free energies, respectively. The corresponding formulations of the nite-temperature theory will be called the covariant Euclidean and the canonical formulations. Sometime we will also say "Euclidean" instead of "covariant Euclidean", for simplicity. For static space-times without horizons comparing F E and F C shows [6] that these functionals dier only by the vacuum energy which is not included in F C. Thus, in this case the Euclidean and canonical formulations are, in fact, equivalent. In space-times with Killing horizons the quantum theory has a number of specic properties. On one hand, in the Euclidean formulation there is a distinguished value of the period, corresponding to the Hawking temperature, for which M is a regular Allen [6] called exp( F E ) and exp( F C ) the "quantum" and "thermodynamic" partition functions. These names are not very suitable at least because the both partition functions are essentially quantum.
space. At other values of the space M has conical singularities which result in additional ultraviolet divergences [5]-[8]. On another hand, the canonical formulation runs into diculties because the time evolution is not dened at the bifurcation surface of the Killing horizons. The density dn of the energy levels in Eq. (.) blows up near the horizon [9]-[] atany temperature in an infrared way. As a result, in the presence of horizons the Euclidean and the canonical free energies look dierent and nding the relation between them becomes a problem. This problem has not been analysed before and our aim is to investigate it for the case of scalar and spinor elds in some details. Comparison of the covariant Euclidean and the canonical formulations is important for dierent reasons. The main of them is statistical-mechanical interpretation of black hole thermodynamics which can be naturally dened in the framework of the Gibbons-Hawking approach []-[5]. The paper is organized as follows. In Section we describe the Euclidean and the canonical formulations of statistical mechanics of scalar and Dirac elds on static backgrounds. The ultraviolet divergences appearing in F E because of conical singularities are given in Section 3 in the most complete form. We use the dimensional and Pauli-Villars regularization procedures. In Section 4 we develop a method how to nd the divergences of the density of levels dn dn. We rst apply this method to study in the presence of the spatial cuto near the horizon. Then we show, in Section 5, that in dimensional and Pauli-Villars regularizations the spatial cuto can be removed and one can dene dn on the complete space. In these regularizations the divergences of dn have the ultraviolet character. It means that the corresponding divergences of the covariant free energy F C coincide exactly with the divergences of the Euclidean free energy F E. In Section 6 we discuss a hypothesis that (for spinors and scalars) the entire bare functionals F C and F E must coincide. We illustrate it with some examples. Technical details are given in Appendixes. In Appendix A we remind the reader how to relate the canonical free energy on ultrastatic spaces to the eective action. Appendix B is devoted to the calculation of the spinor heat coecients on conical singularities, some of the coecients represent the new result. Denitions and basic relations Let us consider scalar elds described by the Klein-Gordon equation and spinor elds described by the Dirac equation, ( r r + R + m ) =0 ; ( r +m) =0 ; (.) This is the property of nonextremal black hole backgrounds. The extremal black holes will not be considered here. 3
where R is the scalar curvature and r are the covariant derivatives, dened according with the spin of the elds 3. The Dirac -matrices = ( 0 ; a ) obey the standard commutation relations f ; g = g, 0 is the anti-hermitean matrix. It is supposed that the space-time is static ds = g 00 dt + g ab dx a dx b ; a; b =;;3 : (.) The component g 00 is a nonpositive function and g 00 = at spatial innity. The temperature measured at innity is.on space (.) equations (.) can be rewritten in the form g 00 (@ t + H s ) =0 ; i 0 (i@ t + H d ) =0 ; (.3) H s = jg 00 j( r a r a w a r a + m + R) ; (.4) H d = i 0 a (r a + w a)+m : (.5) Here r a is the covariant derivative computed with the help of the metric g ab of the three-dimensional surface of constant time t = const. We denote this surface B. Index a is up and down with the help of g ab. The vector w a = r a ln jg 00 j is the vector of acceleration. The operators H s and H d are called the one-particle Hamiltonians because their eigen-values coincide with the frequencies of one-particle excitations 4. To calculate the canonical free energies Fi C (i = s; d) with the help of Eq. (.) one has to know the densities dn i of the energy levels of H s and H d. Let us dene now the Euclidean free energies for the elds described by Eq. (.). To this aim we consider the Euclidean manifold M which is the Euclidean section of the Lorentzian geometry (.) ds = g d + g ab dx a dx b ; 0 ; (.6) where g = jg 00 j. The Euclidean eective actions for the elds and are W s [g; ] = log det L s ; W d [g; ] = log det L d ; (.7) L s = r r + R + m ; L d = 5 ( r + m) : (.8) It is assumed that W i are regularized functionals. The operators L s act on scalar elds on M which are periodic in, L d act on spinors which change the sign when is increased by. The Euclidean matrix is i 0, the matrix 5 anticommutes with the other 's and it is "normalized" as 5 =. Both operators (.8) are Hermitean with respect to the standard inner product ('; ' 0 ) = R ' + ' 0 p gd4 x. According to Eq. (.), the canonical 3 We dene the spinor derivative asr =@ +, where = 8 [ ; ] V i r V i is the connection and V i are the tetrades. 4 It is easy to check R that H s and H d are Hermitean operators with respect to the following inner products ( 0 ;)= Bp R (3) p gjg 00 j d 3 x ( 0 ),( 0 ; )= (3) g d 3 x ( 0 ) + where (3) g = det g B ab and ( 0 ) + denotes Hermitean conjugation. 4
free energy vanishes at zero temperature. It is convenient to dene the Euclidean free energy so that it have the same property, i.e., as F E i [g; ] = W i [g; ] E 0 i [g] ; (.9) E 0 i [g] = lim W i [g; ] : (.0)! The quantities Ei 0 [g] have the meaning of the vacuum energy. Note that Fi E [g; ] and Ei 0 [g; ] are covariant functionals of the metric, because the Euclidean actions W i [g; ] are covariant atanyvalues of. Our aim is to nd the relation between F E and F C. The important property off C is that it can be represented in the form similar to Eq. (.9) F C i [g; ] = Wi [g; ] E 0 i [g] ; (.) The functionals W i [g; ] are the following eective actions E 0 i [g] = lim Wi [g; ] : (.)! W s [g; ] = log det Ls ; Wd [g; ] = log det Ld : (.3) The operators Li are related to L i, Eq. (.8), by the conformal transformations L s = e 3 L s e ; Ld = e 5 L d e 3 ; (.4) where e = g. Representation (.) is well known [6]-[] but for the sake of completeness we give its derivation in Appendix A. The key point is that Li are expressed in terms of the one-particle Hamiltonians L s = H s @ ; (.5) L d = 5 ( Hd + @ ) ; L d = H d @ ; (.6) H s = e H s e ; Hd = e 3 H d e 3 ; (.7) where = e. Note that according to Eqs. (.7) the spectra of the operators Hi and H i, Eqs. (.4), (.5), coincide, which means that Hi are simply an another representation of H i. The operators Li act on the elds on the ultrastatic space M with the metric ds = d +g ab dx a dx b ; 0 (.8) which is conformally related to metric (.6), g = g =jg 00 j. One can show that where f ; g =g, and H d = i 0 ( a r a + e m) ; (.9) H i = r a r a + e m + V i : (.0) 5
The derivatives r are dened with the help of the metric g ab of the surface = const in ultrastatic space (.8). This surface is conformally related to B and we denote it B. The "potential terms" V i are V s = R + e ( 6)(r w w w ) ; (.) V d = 4 R + e m w : (.) Here w =(0;w a ) is the four vector of acceleration and R is the curvature of the ultrastatic background R = e (R +6(r w w w )) : (.3) The formulation of statistical mechanics in terms of the theory on ultrastatic spaces was developed by Dowker and Kennedy [9] and by Dowker and Schoeld [3],[4]. Relations (.9)-(.3) coincide with those used in Refs.[3],[4]. The actions W i and Wi are determined by the conformally related wave operators, see Eqs. (.4). In static space-times without horizons the renormalized functionals W i and Wi dier by the anomalous terms computed in [3],[4]. These terms, however, are proportional to, so they result in the dierence between vacuum energies (.0) and (.). The Euclidean and the canonical free energies in this case coincide. When there is a horizon the conformal transformation to the ultrastatic metric becomes singular and this case requires a special analysis. 3 Covariant Euclidean formulation: conical singularities and divergences To begin with we describe the class of space-times with Killing horizons which will be discussed here. We assume that the metric (.) near the bifurcation surface has the following form ds = g 00 (; )dt + d + pq (; )d p d q ; p; q =; : (3.) In this representation the location of is determined by the equation = 0, p are the coordinates on this surface. We also assume that near = 0 the components of the metric are decomposed as g 00 (; ) = 6 R ijij ()+O( 4 ) ; (3.) pq (; ) =~ pq ()+ h pq ()+O( 4 ) ; (3.3) where ~ pq is the metric tensor on. The constant is called the surface gravity. Metrics which obey the properties (3.)-(3.3) correspond to static nonextremal black holes. It 6
can be shown that the quantities R ijij, R ii are the projections of the Riemann and Ricci tensors calculated on on the directions normal to this surface. Namely, R ii = R n i n i ; R ijij = R n i n j n i n j : (3.4) Here n i are two unit orthonormal vectors orthogonal to and the summation over the indexes i; j =; is assumed. It can be also shown that h pq ~ pq = (R ijij R ii ) : (3.5) Near the Euclidean section M of space-time (3.) looks as This space is regular when = H ds ' dt + d +~ pq d p d q ; 0 : (3.6) =, where the constant H is called the Hawking temperature. For arbitrary there are conical singularities and M looks as C, where C is a cone. As a result of conical singularities, the Euclidean free energy F E is divergent even after subtracting from it the vacuum energy. The divergent part F div E of F E can be calculated with the help of dierent regularizations. We begin with the dimensional regularization and consider D-dimensional space-time. It is assumed that when going to arbitrary dimensions the background space holds its Killing structure and equations (3.)- (3.3) do not change. The dierence now is that the tensors R, R are calculated in D dimensions and is a (D ) dimensional surface. The divergent part of the Euclidean free energy is D F div E [g; ; D] = m D 4 (4) D= 3 "f m p 4 P + p R + p 3 R ii! ; (3.7) where D is considered as a complex parameter and the integral is taken over the bifurcation surface, R R p ~d D. We put P =R ijij R ii and introduce the constants f and p k which are listed in Table : spin f p p p 3 0 60 0 6 r d 7 480 r d 4 r d 6 r d Table. Here r d is the dimensionality of the spinor representation, r d = 4 for Dirac spinors and r d = for massless Weyl spinors. As follows from Eq. (3.7), in the dimensional regularization F E has a simple pole at D =4.The dimensional regularization reproduces the divergences 7
of the logarithmical type only. For this reason it is also worth studying F div E in Pauli- Villars regularization which usually gives all divergent terms. The Pauli-Villars method is based on introduction of several, say, 5 additional elds. elds with masses M k have the same statistics as the original eld, while other 3 ones with masses M 0 r have the wrong statistics, i.e., they are fermions for scalars and bosons for spinors. The latter elds give contribution to F E with the sign opposite to that of physical elds. To eliminate the divergences two restrictions are imposed m p + X k M p k X r (M 0 r) p =0 ; p=;4 : (3.8) These equations can be resolved by choosing [] M ; = p 3 + m, M 0 ; = p + m, M 0 3 = p 4 + m. The divergences in Pauli-Villars regularization can be obtained from Eq. (3.7). By adding contributions of the regulator elds with the sign corresponding to their statistics and taking the limit D! 4, which is nite due to restriction (3.8) with p =,one nds F div E [g; ; ] = 48 "bf + a p 4 X X a = a(m; ) = ln m ln M k + r b = b(m; ) =m ln m + X k k M k ln M k P + p R + p 3 R ii! X r ; (3.9) ln(m 0 r) ; (3.0) (M 0 r) ln(m 0 r) : (3.) The parameter plays the role of the ultraviolet cuto. The regularization is removed when!. In this limit a ln( =m ) and b (a; b > 0). Thus, in general, F E div includes both logarithmic and quadratic divergences. The derivation of Eq. (3.7) is standard. In the dimensional regularization the Schwinger- DeWitt proper-time representation [3] gives the divergent part of the eective action in the form W E div [g; ] = 0 ds s e m s (4s) D= B 0 + sb + s B : (3.) Here B k are the Hadamard-Minackshisundaram-DeWitt-Seeley (or heat) coecients of the heat kernel asymptotic expansion 5 Tr e s4 The Laplacians 4 look as 4 = B (4s) D= 0 + sb + s B + ::: : (3.3) r r + X, where r is the covariant derivative dened according with the spin; X =(=6 )Rfor scalars, and X = R for spinors. The relation 4 between 4's and the operators L s and L d, Eqs. (.8), is L s = 4 s + m ; L d = 4 d + m : (3.4) 5 We do not take into account the boundaries of space-time. 8
On singular spaces M the coecients B k can be represented as the sum of two terms B k = A k + A ;k : (3.5) A k have the form of the standard coecients dened on the regular domain of M, A ;k, k, are functionals on which appear because of conical singularities. For scalars and spinors the rst two coecients have the form A ; = 3 A ; = f 3 ( )A ; (3.6) h ( 4 )p P +( )(p R + p 3 R ii ) i ; (3.7) where = H and the numbers f and p k are given in Table. Expression (3.9) for F div E [g; ] follows from Eqs. (.9), (.0) and (3.),(3.5)-(3.7). For scalars the coecient A ; was found by Cheeger [4], see also Refs. [7],[5]. The spinor coecient A ; follows from the results of Refs. [5],[6]. The scalar coecient A ; and the general structure of the higher coecients A ;k were analysed in Refs. [6],[7],[8]. The calculation of the spinor coecient A ; is our new result. Its derivation is similar to that of Ref. [6] but has new features related to the spin. The reader can nd the details in Appendix B. 4 Canonical formulation: infrared divergences Our aim now is to investigate the divergences of the canonical free energy and compare them with the results (3.7),(3.9) found in the Euclidean formulation. We begin with remarks concerning specic features of quantum systems in the presence of horizons. In this case the one-particle oscillators of elds have a continuous spectrum of frequencies 0! <. The eigenvalues of the operators Hi, Eqs. (.9),(.0), run down to! = 0 even for massive elds, i.e., the usual mass gap is absent. This property has the simple explanation. Hi are given on the space B, which is the spatial part of ultrastatic space (.8) related to the original space (.6) by the conformal transformation. On the ultrastatic space the location of the horizon is mapped at innity and B turns out to be non-compact. At the same time, the masses m i of the elds can be neglected near the horizon because they are multiplied by the vanishing factor e = jg 00 j. Regarding the "potential" terms in Hi, see Eqs. (.),(.), they are constant and negative on the horizon, V s =, V d = 3, and look as tachionic masses [0],[]. As a result, the densities dn i of the eigen-values of Hi blow up near the horizon in an infrared way. To investigate this divergence the following method can be used. By the denition, the trace of the heat kernel of the operator H i is H Tr e i t = 0 dn i(!) e! t : (4.) 9
By making use of the inverse Laplace transform one obtains dn i (!) =! + d ei!tr e i H i : (4.) The diagonal matrix elements hxj exp( H i t)jxi h exp( H i t) i diag are well dened but the traces involve the integration over the non-compact space B and diverge. The key observation is that the divergent parts of dn i(!) can be found by making use of the asymptotic properties of the traces at small t, which are very well known. Let us begin with massless elds in the Rindler space ds = dt + d + dx + dy ; <x;y< ; >0 : (4.3) The Rindler horizon is the plane IR, is the proper distance to the horizon. The Rindler space can be considered as an approximation to the black hole geometry near the horizon. The metric on the corresponding space B is dl = (d + dx + dy ) : (4.4) As can be shown, B coincides with the hyperbolic manifold IH 3 having the constant curvature R = 6 [9]. A review of the heat kernels of Laplace operators on such spaces can be found in Refs. [9],[30]. It is remarkable that for the massless elds the heat kernels are known explicitly and their diagonal elements read 6 h e t H s idiag = ; (4t) 3= h e t H d idiag = r d + (4t) 3= t : (4.5) The number r d appears in the spinor case after tracing over the spinor indexes. Obviously, the traces of these operators diverge at =0. Let us restrict the integration in the traces by values where is a proper distance to the horizon. Such a method is called the volume cuto. With the help of Laplace transform (4.) (see Ref. [3]) one easily obtains from (4.5) the divergences of the densities of levels at! 0 " dns (!; ) = A "! dnd (!; ) ; 4 3 div div " A! = r d 4 3 + 4 (4.6) where A formally stands for the area of the horizon 7. For scalar elds our result agrees with previous computations by the WKB method, see, for instance Ref. [9]. A generalization of (4.6) to massive scalars was explicitly found in Refs. [0],[]. Let us consider how do Eqs. (4.6) modify when the geometry deviates from the Rindler form. The spaces B can be approximated by IH 3 only in the limit! 0. For this reason, the diagonal elements of the heat kernels of H i on B are represented by the Taylor series 6 The expression for the scalar kernel is given in Ref. [9], the spinor kernel follows from the function which is also given there. 7 Since the Rindler horizon is a plane, only the densities of levels per unit area have a strict meaning. 0
in converging at! 0 to expressions (4.5). Presumably, the coecients in these series should be the local functions of the curvature and so to nd them it is sucient to use the asymptotic form of the traces. One has H he i ti diag ' r i +a (4t) 3= i; t +a i; t + ::: ; (4.7) where r s = and a i;n are the diagonal elements of the corresponding heat coecients. As before the summation over the spinor indexes in (4.7) is assumed. With the help of Eqs. (.)-(.3) one nds a s; = e R m 6 a d; = e r d R + (r w w w )+m ; (4.8) : (4.9) These expressions can be decomposed in powers of by making use of Eqs. (3.)-(3.5) r w = R t t = R ii + O( ) ; (4.0) w w = 3 R ijij + O( 4 ) R m ; (4.) a s; = + O( 4 ) ; (4.) 6 a d; = r d + R ii R ijij 6 R m + O( 4 ) ; (4.3) where R is evaluated at =0. For the massless elds in the Rindler space a s; = 0, while a d; = r d, in agreement with Eqs. (4.5). Decompositions (4.), (4.3) are written explicitly up to the terms of the order, other terms do not contribute to the divergence of the traces. Analogous decompositions can be found for the heat coecients a i;n with n, however, a i;n vanish at small faster than. To see this let us consider the operators L s, L d which are given on the ultrastatic space M, Eq. (.8). Ls, L d are related to the three dimensional Hamiltonians H i by Eqs. (.5),(.6). Because M = S B the diagonal parts of the heat coecients of Ls, L d coincide with a i;n. On the other hand, Ls, L d are related by conformal transformation (.4) to the covariant Euclidean operators L s, L d. In this case, according to Dowker and Schoeld [3],[4], a i; = e 4 (a i; + r J i ) ; (4.4) where a i; are the heat coecients corresponding to L s, L d. The currents J i have the form 8 J s = 5 45 r (w )+w (w ) w 3 rw r rw R w 8 Note that our signature of the metric is dierent from that in Refs. [3],[4].
J d = r d 45 + 3 w R +5 R+m w 6 4 r (w )+w (w ) w rw ; (4.5) +8r rw+r w (5R +30m )w o ; (4.6) where w = w w and rw = r w. The coecients a i; are regular at! 0. With the help of Eqs. (4.0),(4.) one can check that r J i are regular as well. Therefore, as follows from (4.4), a i; e 4 4. The other coecients a i;n with n>are determined from a i; by recursion relations [3] and so a i;n vanish near at least as fast as a i;. Our conclusion is that only a i; contribute to the divergences of the traces. From Eqs. (4.),(4.7),(4.),(4.3) we obtain the divergent part of the densities of levels " dns (!; ) = (!! div 4 P ln ln 3 4 l l 6 " (! dnd (!; ) = r d! div 4 P ln + 3 4 l 4 ln l 8 R ii ) R m R m ) ; (4.7) : (4.8) Here P is the quantity dened after Eq. (3.7) and l is an additional infrared cuto parameter imposed at a large distance from the horizon. For massive elds l ' m. The following remarks concerning expressions (4.7) and (4.8) are in order. As compared to the computations on the Rindler space, Eqs. (4.6), there are logarithmic corrections from the mass of the elds and non-zero curvature near the horizon. Because of the logarithmic terms the densities dn i have a dierent behavior at small frequencies for scalar and spinor elds. Spinor density (4.8) is positive at small in the whole range of frequencies. Contrary to this, scalar density (4.7) at!! 0 is proportional to m ln =l and is negative (note that curvature corrections are negligible with respect to the mass of the eld). Such a feature indicates that for scalars the description of the modes with low frequencies, the so called soft modes, may need a modication Ref. [3]. From Eqs. (.),(4.7),(4.8) we get the divergent parts of the canonical free energies F s;div C [g; ; ] = ( ; 3 80 4 " 70 4 P + d F d;div C [g; ; ] = r " 3 7 5760 P + 4 96 8 R ii ( 7 48 6! 440 + 4 9 ) R m ln l R m ln l ) (4.9) : (4.0) These equations enable one to calculate the divergences of the other characteristics of canonical ensembles. In particular, one can see that the statistical-mechanical entropy
diverges near the Killing horizon and in the leading asymptotic it is proportional to the area A of the horizon. For scalar elds this leading asymptotic coincides with the WKB results by t'hooft [9] and many other authors. Equations (4.9) and (4.0) also follow from the high-temperature expansions obtained by Dowker and Schoeld [3],[4]. Application of these results to our case is justied because when approaching the horizon the local temperature unlimitedly grows. 5 Canonical formulation: ultraviolet divergences Comparison of Eq. (3.9) with Eqs. (4.9),(4.0) shows that the divergences of the Euclidean and canonical free energies are expressed in terms of the similar geometrical quantities but have dierent dependence on the temperature. Also the nature of the divergences is dierent: F E diverges in an ultraviolet way while the -divergence in F C has an infrared origin. Finally, there is one more important dierence between the regularizations of F E and F C. The ultraviolet regularizations are usually applied to operators and functionals but not to the background eld itself. Contrary to this, the volume cuto regularization makes the space incomplete and modies the background eld essentially. In the presence of the horizon the densities of levels have a remarkable property. Namely, there are regularizations of dn which make it possible to remove the infrared cuto near the horizon and to dene the densities on the complete background. As a result, dn acquire new divergences which correspond exactly to the ultraviolet divergences of the covariant Euclidean theory. As the rst example, let us consider the dimensional regularization. The power of the leading divergency in Eqs. (4.7),(4.8) is determined by the dimensionality of the space B. In D-dimensional space-time the leading divergence is D, if D 6=,and at D > one can take the limit! 0. After the analytical continuation to the complex values of D the quantities dn dn, have a pole at D =4. The method how to investigate near the pole is the following. As before we use relation (4.). By taking into account the form of the operators H i, see Eq. (.0), we can write H he i ti diag ' (4t) e m t r (D i + b )= i; t + b i; t + ::: bd; = r d bs; = 6 + R ii ; (5.) R + O( 4 )+O(D 4) ; (5.) R ijij 6 R + O( 4 )+O(D 4) ; (5.3) bs; = O( 4 )+O(D 4; ) ; bd; = r d 4 m + O( 4 )+O(D 4; ) : (5.4) The coecients b i;n are found with the help of Eqs. (4.7),(4.)-(4.4) The terms O(D 4), O(D 4; ) denote additions which appear in formulas (4.)-(4.4) when D 6= 4. Wedo 3
not write these terms explicitly because they do not result in singularities when D! 4. The integration measure in the traces is obtained from Eqs. (3.)-(3.5) B p gd D x ' D 0 D d + 4 P +O(D 4; ) : (5.5) After substitution of Eqs. (5.)-(5.4) in Eq. (5.) and making use of (5.5) we can derive the singular part of the traces h Tre H s ti div = D h Tre H d ti div = r d m D 4 (4) (D )= D m D 4 (4) (D )= t 3= t 3= m 6 m + R R t R ii t 8 P 4 P 4 ; (5.6) : (5.7) The -functions appear as a result of the integration over. It should be noted that for the spinors the contribution of the coecient b d; cancels the pole caused by the term r d = in b d;. The divergent part of density of energy levels is obtained from (5.6),(5.7) with the help of Eq. (4.) " dns (!; D) D m D 4 "! = m R (4) D= 6 P ; (5.8) " dnd (!; D) div D m D 4 = r d (4) D= " m + R R ii 8! P : (5.9) div The divergence represents a simple pole at D = 4. Finally, from Eqs. (.),(5.8),(5.9) one can nd the divergent part of the canonical free energy. We do not write this divergence explicitly because it is exactly the same as that of the Euclidean free energy, Eq. (3.7), computed in the dimensional regularization F div C [g; ; D] =FE div [g; ; D] : (5.0) This key equality can be also established in the Pauli-Villars regularization. The regularized density of states in this method is the following quantity dn i (!; ) dn i(!) + X k dn i (!; M k ) X r dn i (!; M 0 r) : (5.) Denition (5.) takes into account that in the Pauli-Villars method each eld is replaced by the "multiplet" of elds. The quantities dn i(!;m k ), dn i(!;mr 0 ) are the densities of levels of the Pauli-Villars partners and the elds with the wrong statistics give negative contributions. The number of such elds equals the number of the elds with the correct statistics and the leading -divergences in Eqs. (4.7),(4.8) are cancelled. Regarding logarithmical divergences ln, they disappear because of constraint (3.8) with p =. As a result, regularized densities (5.) are left nite when! 0 and can be dened on the 4
complete space. When the Pauli-Villars cuto is removed (! ), dn i(!;) diverge in an ultraviolet way. The divergences can be inferred from Eqs. (5.8),(5.9) by taking into account constraints (3.8) " dns (!; ) " dnd (!; ) div = "b + a! (4) P + 6 = r d "b + a! (4) P! R R 6 + R ii 4! ; (5.) : (5.3) div Constants a and b are dened by Eqs. (3.0),(3.) and diverge in the limit of innite. With the help of Eqs. (5.),(5.3) one can show that F div C [g; ; ] =FE div [g; ; ] ; (5.4) where F div E [g; ; ] is given by Eq. (3.9). It should be noted in conclusion that Pauli- Villars regularization of the canonical free energy was rst suggested by Demers, Lafrance and Myers [] who considered a scalar eld on the Reissner-Nordstrom black hole background. The authors used the WKB method. Although our method is dierent the results for F div C [g; ; ] in this particular case coincide. 6 Discussion We are interested in nding the relation between the covariant Euclidean and the canonical formulations of statistical mechanics on curved backgrounds with horizons. Let us discuss rst why these formulations are equivalent for spaces without horizons. As we showed in Section, the canonical formulation is equivalent to the Euclidean theory on ultrastatic background, M, Eq. (.8), conformally related to the original space-time M, Eq. (.6). The Euclidean actions W i are determined by the operators L i on M, see Eq. (.8). Analogously, in canonical theory the functionals Wi are determined by operators L i on M. The classical actions corresponding to these two types of operators are I i [g; ' i ]= M ' + i L i' i p gd 4 x ; Ii [g; ' i ]= M ' + i L i ' i p gd 4 x ; (6.) where the notation ' i is used for scalars or spinors. As a result of Eq. (.4), I i [g; ' i ]= I i [g; ' i ] (6.) for = e and = e 3. The transformation from one action to another is not singular and the classical theories on M and M are equivalent. In case of massless spinors and massless scalars with = the operators L 6 i and Li have the same form, which means that the classical theories are conformally invariant. In general case this invariance does not exist. However, it is still possible to introduce an auxiliary conformal charge in the classical actions and interpret Eq. (6.) in terms of a pseudo conformal 5
invariance [3],[4]. According to a common point of view [34],[35] the bare quantum actions respect the classical symmetries. Thus, for the bare regularized functionals there is the same equality asfor the classical actions 9, W i [g; ] bare = Wi [g; ] bare : (6.3) This relation is not valid for the renormalized quantities because the conformal symmetry is broken by the quantum anomalies [34],[35]. The dierence between the renormalized actions W i and Wi for scalar and spinor elds was found explicitly bydowker and Schoeld [3],[4]. The anomaly, which is an integral over the Euclidean space, is proportional to and so it contributes to the vacuum energy only. As a result, the free energies Fi E Fi C, Eqs. (.9),(.), coincide before and after renormalization. In case of the horizons M and M have the dierent topologies, IR and S B, respectively. The transformation which relates M and M is singular on the bifurcation surface. So the classical theories are not quite equivalent. On the quantum level the horizons result in the divergences of Fi E and Fi C which cannot be eliminated by subtracting the vacuum energy. Moreover, the divergences of Fi E and Fi C have the dierent origins. Our results suggest a way how can the covariant Euclidean and the canonical formulations be reconciled. We showed that there are regularizations which are applicable to both Fi E and Fi C. In such regularizations the canonical free energy Fi C can be dened on the complete background M and its divergences are identical to the divergences of Fi E, see Eqs. (5.0),(5.4). When the functionals Fi E and Fi C are well dened, it becomes possible to carry out the transformation from M to M and to interpret it as a conformal symmetry. In analogy with the case without horizons, we can make ahypothesis that (at least for scalars and spinors) the regularized bare free energies are identical and F E i [g; ] bare = F C i [g; ] bare : (6.4) It is assumed that the both functionals in (6.4) are considered in the same regularization. To give a strict proof of this equality may be a rather dicult problem. There are examples which enable one its direct check. In Ref. [0] Cognola, Vanzo and erbini obtained the free energy of massive scalar elds in the Rindler space in an explicit form, see also Ref. []. It can be shown that these results, rewritten in the Pauli-Villars regularization, conrm relation (6.4). Another direct conrmation is possible in two dimensions. Two dimensional massless scalar elds were analysed in Ref. [33] and these results support our hypothesis as well. Equality (6.4) enables one to apply the methods of quantum eld theory to statistical mechanics with the horizons. As a consequence, it justies the ultraviolet renormalization of statistical-mechanical quantities [36]. Our results are also important for studying the statistical-mechanical foundation of the thermodynamics of black holes. It was realized 9 It is true if the regularization itself does not break classical symmetries 6
in the last years that statistical-mechanical computations in this case require an oshell procedure [37], i.e., considering a black hole at temperatures dierent from the Hawking value. From this point of view, Eq. (6.4) demonstrates the equivalence of two o-shell methods, the canonical and the conical singularity methods. However, the comparison of the o-shell and on-shell results goes beyond the subject of this paper. As non-minimally coupled scalars show [3], the o-shell and on-shell computations are not always equivalent. Finally, several remarks about the range of validity of our results are in order. We dealt with static nonextremal black hole backgrounds. The method described in Sections 4, 5 is applicable to the extremal black holes as well, but rotating black holes require an additional analysis. For the extremal black holes the density of levels dn has the same property as in nonextremal case. Namely, the Pauli-Villars and dimensional regularizations eliminate the -divergences. For the scalar elds the corresponding calculations were done in Ref. []. Our consideration was also restricted to the scalar and spinor elds. The method to calculate the quantity dn can be generalized to the elds of other spins. Finding the correspondence between the canonical and the Euclidean formulations for these elds would be an interesting extension of this work. Acknowledgements: I am grateful to V. Frolov, Yu. Gusev, and A. elnikov for helpful discussions. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada. 7
A Canonical free-energy and eective action Here we give the details how to obtain relation (.). We suppose that the system has a discrete spectrum of frequencies. Then the canonical free energy is F C i [g; ] = i X! d i (!)ln i e! ; (A.) where s =, d = and d i (!) is the degeneracy of the level!. Equation (.) for Fi C [g; ] is obtained in the limit when intervals between the frequencies! go to zero. The basic identities we use are [38] ln e!! =! X +ln! + ln +! ; (A.) 4k Note that ln +e! = d X = lim 4 z!0 dz k= q! +ln+ X X k= q k=0 k= ln +! (k + q)! (k + q)! z +! X ln +! (k +)! k= q (k + q) : (A.3)! z3 5 (A.4) where q = 0 or. By using the properties of the Riemann function [38] we nd lim z!0 d dz X k= (k)! z d = ln ; lim z!0 dz X k=0 Thus, Eqs. (A.) and (A.3) are represented in the form ln i e! =! i (z j!; ) = X k= (k +)! z = ln : (A.5) lim d z!0 dz i(z j!; ) ; (A.6) 4 (k + l i)! 3 +! 5 z ; (A.7) where l s = 0 and l d =. The series (A.7) converge at Rez >, so the functions i(z j!; ) can be dened at z! 0 with the help of the analytic continuation. By taking into account Eqs. (A.) and (A.6) we get the canonical free energy in form (.) W i [g; ] = i X! F C i [g; ] = Wi [g; ] E 0 i [g] ; (A.8) d X d i (!) lim z!0 dz i(z j!; ) ; E 0 i [g] = i d i (!)!! ; (A.9) where E 0 i [g] is the vacuum energy. The quantities W i [g; ] and E 0 i [g] diverge at large! although the free energy itself is nite. To see that Wi [g; ] is eective action (.3) one 8
can regularize this functional with the help of the function method. It is enough to make the replacement X! d d i (!) lim z!0 dz d i(zj!; )! lim z!0 dz i(zj) (A.0) where, according to Eq. (A.7), i (zj) = X! X k= d i (!) 4 (k + l i)! 3 +! 5 z : (A.) i (zj) are the generalized functions of the operators Li, see Eqs. (.5),(.6). For operators on manifolds without boundaries i (z j) can be dened as a meromorphic function with simple poles at z =; [30]. B Spinor heat coecients on conical singularities Let us consider the heat kernel of the spinor Laplace operator 4 d = r r + R on 4 spaces M with conical singularities. For simplicity we put =, so M are regular when =. The heat kernel obeys the equation (4 d ) x K (x; x 0 ;s)+@ s K (x; x 0 ;s)=0 ; K (x; x 0 ; 0) = (x; x 0 ) ; (B.) where (x; x 0 ) is the delta function on M. We rst describe the heat kernel on a simple cone C, which will be required for us later. It is known since Sommerfeld that heat kernels on C can be expressed in terms of the corresponding heat kernels on the plane IR. A suitable generalization of the Sommerfeld representation for integer and half-odd-integer spins was given by Dowker [39],[40]. By making use of the results of Ref. [40] we can represent the spinor heat kernel on C in the following form K (x();x 0 (0);s)= i A sin (z + )U(z)K (x(z);x0 (0);s) : (B.) Here is the polar-angle coordinate on C. A is the contour in the complex plane which has two parts. In the upper half-plane it runs from ( )+i to ( + )+i and in the lower half-plane from ( + ) i to ( ) i. K (x; x 0 ;s) is the spinor heat kernel which obeys the problem (B.) on IR. The operators K (x; x 0 ;s) and K (x; x 0 ;s) correspond to the dierent spin structures and have the dierent periodicity. The kernel on IR is unchanged when going around the origin of the polar coordinate system. (There is no dierence between the origin of the polar coordinates and other points on the plane). Contrary to this, the kernel on C changes the sign when is increased by. The covariant derivatives on IR are trivial, r = @, but the the covariant derivatives on C are dened by the polar tetrades and nontrivial, r = @ +. In the basis = ( ; ), where k are the Pauli matrices, the spinor connection -form is = i 3d. The role of the 9
matrix U in relation (B.) is to make a gauge-like transformation from the derivative on IR to that on C, i U()@ U () =r ; U() = exp 3 : (B.3) In fact, when is real U() is the unitary matrix which is the spinor representation of the rotation on the angle. To nd corrections to the heat coecients from the conical singularities we follow the method suggested in Ref. [6]. According to this method, it is sucient to work in a narrow domain ~ of the singular surface. The rest region of M does not have conical singularities and the heat kernel expansion on it has a standard form. ~ can be approximated as C. So here one can relate K (x; x 0 ;s) with the kernel K(x; x 0 ;s) on the regular space M = by the formula analogous to Eq. (B.). The contribution Tr ~ K (s) from ~ to the trace can be written as Tr ~ K (s) =Tr ~ K(s)+ i A 0 sin z ~ p gd D xtr i [U(z)K (x(z);x(0);s)] ; (B.4) where Tr i stands for the trace over the spinor indexes. The points x(z) and x(0) are connected by the integral line of the Killing eld @. The two terms in r.h.s. of (B.4) appear when contour A is deformed to a small loop around the origin and contour A 0 which consists of two vertical curves. The eect of conical singularities is related to the second term in (B.4). The asymptotic form of K(x; x 0 ;s)is K(x; x 0 ;s)' e (x;x 0 )=4s (4s) D= 4= (x; x 0 X ) n a n (x; x 0 )s n ; (B.5) where (x; x 0 ) is the geodesic distance between points x; x 0 and 4(x; x 0 ) is the Van Vleck determinant. The coecients a n are determined in terms of the Riemann tensor and its derivatives. Let be the proper distance from the points x(z) and x(0) to. One can nd the following relations [6] (x(z);x(0)) ' 4 sin z 6 4 R ijij sin z ; (B.6) 4 = (x(z);x(0)) ' + 6 R ii sin z ; (B.7) where R ii and R ijij are dened in (3.4). The integration measure on ~ can be derived from Eqs. (3.)-(3.3) ~ p gd D x ' p d D dd For the rst coecients in Eq. (B.5) one nds + 6 R ijij 4 R ii : (B.8) a 0 (x 0 ;x)'i + 8 R (x 0 ) x ; a (x 0 ;x)' RI ; (B.9) 0
where I is the unit matrix in the spinor representation and = 4 [ ; ]. The matrix U(z) corresponds to the rotation of a vector normal to the surface on the angle z. So from Eq. (B.9) we get Tr i [U(z)a 0 (x(z);x(0))] ' r d cos z + 4 sin z R ijij : (B.0) Here r d is the dimensionality of the spinor representation. Now, it follows from Eqs. (B.5)-(B.7),(B.9) that Tr i [U(z)K (x(z);x(0))] " exp s ' r sin z d cos z + (4s) D= 4 R ijij + 6 R ii sin z + 4 4s R ijij sin z Rs : (B.) It can be shown that approximation (B.) is sucient to nd all corrections for the rst three heat coecients due to conical singularities. By integrating (B.) with measure (B.8) we obtain an integral over the surface. Its integrand is an expression linear in the quantities R, R ii and R ijij with coecients proportional to cos z sin q z, q = ;. Finally, we have to integrate this expression in the complex plane, see. Eq. (B.4). This can be done with the help of formulas [8] where = (B.4) we nd i A 0 i A 0 dz sin z cos z sin z dz sin z cos z sin 4 z = = 3 ( ) ; (B.) 80 ( )(7 + 7) ; (B.3) (or, in general case, = H ). By making use of these relations and Eq. Tr ~ K (s) Tr ~ K(s) ' (4s) D= A ; + sa ; + O(s ) ; (B.4) where A ;k are given by Eqs. (3.6),(3.7) and Table for spinors. Formula (B.4) determines the dierence between asymptotic expansions on regular and singular spaces. It is easy to see that A ;k are the corrections to the spinor heat coecients from conical singularities. References [] G.W. Gibbons and S.W. Hawking, Phys. Rev. D5 (976) 75. [] S.W. Hawking, In: General Relativity: An Einstein Centenary Survey. (eds. S.W. Hawking and W. Israel), Cambridge Univ.Press, Cambridge, 979. [3] J.W. York, Phys. Rev. D33 (986) 09.
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