Hadamard Renormalization of the Stress-Energy Tensor for a Quantized Scalar Field in a General Spacetime of Arbitrary Dimension Yves Décanini, Antoine Folacci

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Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension Yves Décanini, Antoine Folacci

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Yves Décanini, Antoine Folacci. Hadamard renormalization of the stress-energy tensor for a quantized scalar field in a general spacetime of arbitrary dimension. Physical Review D, American Physical Society, 2008, 78, pp.044025. 10.1103/PhysRevD.78.044025. hal-00338657

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:gr-qc/0512118v2 12 May 2008 ly eta oei eilsia rvt.Indeed: gravity. semiclassical in role central a plays teseeg-esroperator stress-energy-tensor xetto au ihrsetto respect with value expectation emt st ics n netgt t validity. its investigate gravity, and semiclassical discuss als of to which gravity, us extension permits stochastic an semiclassical for and so-called 7] gravity the semiclassical [6, and of Refs. status for applicability to [5] the of Ref. about domain of account II.B the critical Sec. interesting to very refer a its rather to finally but and We short gravity a semiclassical applications. is to introduction which date [4] Ford to therein refer up by references also review [1], We to recent gravity. a Davies as semiclassical to well and of as aspects Birrell various [3] for of Wald and monographs [2] the Fulling the to time-machines... and refer and Penrose We wormholes and traversable of Hawking existence the of both theorems with connection singularity in violations conditions quantum energy iv) classical and of in effect Casimir physics the cos- hole universe iii) mology, early black ii) radiation, quantum Hawking concern- with i) connection results particularly interesting more very obtain ing has to spacetime, curved us in permitted theory field last quantum usu- the gravity, called quantum In ally at of quantized. approximation field this be years, graviton to thirty assumed the are this to order) on fields one-loop propagating matter fields (from other background the all while classically, † ∗ lsia on fve,ie t metric its i.e. view, of point classical a lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic o unu edi oenraie state normalized some in field quantum a For nsmcasclgaiy pctm scniee from considered is spacetime gravity, semiclassical In o unie clrfil nagnrlsaeieo arbitra of spacetime general a in field scalar quantized a for ASnmes 46.v 11.10.Gh 04.62.+v, numbers: PACS forms. from irreducible constructed on polynomials given Riemann thus for bases effect standard the (iii in on process renormalization and renormalization renormalization the Hadamard in pr space involved by (ii) renormalizati given counterterms anomaly, trace Hadamard a the the for in ambiguities of corresponding explicitly ambiguities the procedure provide the this we considering eleven, perform spat to Ha to seven extra the reader from describe of dimension explicitly physics dimension we spacetime quantum six, for arbitrary the to up of of dimension spacetime aspects spacetime some general treating a in on defined theory edvlpteHdmr eomlzto ftestress-ener the of renormalization Hadamard the develop We .INTRODUCTION I. 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Ref. [1] for the state of affairs of the literature concern- The vacuum expectation value of the stress-energy tensor ing this subject before 1982). Among all the methods and the associated vacuum energy have been called upon employed, the axiomatic approach introduced by Wald to stabilize the size of the extra dimensions. There is an [9] is certainly the most general and the most power- extensive literature on the subject. We refer more partic- ful. It is an extension of the “point-splitting method” ularly to Ref. [31] where back reaction effects are in addi- [10, 11, 12] and it has been developed in connection with tion considered and to Ref. [32] where cosmological con- the Hadamard representation of the Green functions by siderations in connection with the inflationary scenario Wald [9, 13], Adler, Lieberman and Ng [14, 15], Brown are in addition discussed (see also Refs [33, 34, 35, 36, 37] and Ottewill [16] and Castagnino and Harari [17]. We and references therein). refer to the monographs of Fulling [2] and Wald [3] for – In the context of the vacuum polarization induced rigorous presentations of this approach which is usually by topological defects such as monopoles [38, 39, 40] or called Hadamard renormalization. It permitted us to ob- cosmic strings (see Ref. [37] and references therein). tain, in the most general context, the explicit expressions – In the context of the AdS/CFT correspondence of the renormalized expectation value of the stress-energy [41, 42, 43] which asserts the existence of a duality be- tensor for the scalar field theory [18, 19, 20] but also for tween a theory of gravity in the (D + 1)-dimensional some gauge theories such as i) electromagnetism [18], ii) anti-de Sitter space and a conformal field theory liv- quantum gravity at one-loop order [21] (here the the- ing on its D-dimensional boundary (for a review see ories described by the standard effective action as well Ref. [44]) and which could provide a concrete realiza- as by the reparametrization-invariant effective action of tion of the holographic principle [45, 46]. A new renor- Vilkovsky and DeWitt were both considered) and iii) malization procedure, the so-called holographic renor- two- and three-form field theories [22] (in this context, malization, has been developed. More precisely, it has the Hadamard formalism allowed us to treat carefully been shown that the regularized expectation value of the phenomenon of ghosts for ghosts). the stress-energy tensor corresponding to the confor- Hadamard renormalization has been exclusively con- mal field theory living on the boundary can be ob- sidered for field theories defined on four-dimensional tained from the “regularized” action of the gravitational curved spacetimes. (However, it should be noted that field living in the bulk [47, 48] (see also for a review a recent work has been achieved in a two-dimensional Ref. [49] as well as references therein for complements framework [23] but it is incorrect due to a wrong expres- and Refs [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60] for re- sion for the Hadamard representation of the Green func- lated approaches as well as extensions). The counterterm tions.) According to the “recent” physical theories such substraction technique developed in this context permits as supergravity theories, string theories and M-theory, us to obtain the stress-energy tensor, at large distance, which were developed in order to understand gravity in for higher-dimensional black holes such as Kerr-AdS5, a quantum framework and to provide a unified descrip- Kerr-AdS6 and Kerr-AdS7 [61, 62]. tion of all the fundamental interactions, we should live in – In the context of the validity of semiclassical grav- a spacetime with more dimensions than the four we ob- ity but also of the avoidance of the singularities predicted serve, a scenario which is a resurgence of the old Kaluza- by the singularity theorems of Hawking and Penrose [63]. Klein theory [24, 25]. Because all the previously men- Fluctuations of the stress-energy tensor induce Ricci cur- tioned theories are still at an early stage of development vature fluctuations (see, for example, Ref. [64]) or in and are far from being well understood, it is rather dif- other words fluctuations of the gravitational field itself. ficult to make predictions by using them directly. In The existence of these fluctuations places limits on the fact, people studying the consequences of supergravity validity of semiclassical gravity but also could lead to and string theories in cosmology or in black hole physics important effects on the focusing of a bundle of timelike often develop analysis based on semiclassical approxima- or null geodesics. The study of such fluctuations in the tions or more precisely use the methods of quantum field presence of compact extra spatial dimensions has been theory in curved spacetime taking into account the extra discussed more particularly in Ref. [65]. dimensions. In this context, it seems to us crucial to ex- All these works have however been carried out under very tend the powerful Hadamard renormalization procedure strong hypotheses: flat (or conformally flat) spacetimes to be able to deal, as generally as possible, with quantum with extra-dimensions or maximally (or asymptotically fluctuations and with their back reaction effects. In this maximally) symmetric spacetimes as well as massless or paper, we shall take some steps in this direction. conformally invariant field theories. Of course, it is nec- It is important to note that many recent articles have essary, from a physical point of view, to be able to deal already been devoted to the role as well as to the calcula- with situations presenting a lower degree of symmetry. tion of the expectation value of the stress-energy tensor With this aim in view, the Hadamard renormalization in the presence of extra spatial dimensions. For example: procedure could be very helpful. – In the context of the Randall-Sundrum braneworld Finally, it should be noted that some mathematical models [26, 27] introduced in order to solve the hierarchy aspects of the Hadamard renormalization procedure for problem [28, 29, 30], i.e. to eliminate the large hierar- a scalar field in a general “spacetime” of arbitrary di- chy between the electroweak scale and the gravity scale. mension have been already considered by Moretti in a 3 series of recent articles [66, 67, 68, 69, 70]. He has use the commutation of covariant derivatives in the form provided a rigorous proof of the symmetry of the off- T ρ... T ρ... = diagonal Hadamard coefficients, i.e. of the coefficients σ...;νµ − σ...;µν +Rρ T τ... + Rτ T ρ...... (2) corresponding to the short-distance divergent part of the τµν σ... ···− σµν τ... − Hadamard representation of the Green functions for the It is furthermore important to note that all the results Euclidean and Lorentzian scalar field theories [67, 69]. He we provide in Secs. III and IV are given on irreducible has also established a connection between the zeta- and forms: indeed, by using some of the geometrical identities Hadamard- regularization procedures in the Euclidean displayed in our recent unpublished report [73], our re- framework [66, 68] and he has finally discussed the possi- sults have been systematically expanded on the standard ble elimination of the ambiguities plaguing the Hadamard bases constructed from group theoretical considerations renormalization procedure by using microlocal analysis in which have been proposed by Fulling, King, Wybourne the context of the algebraic approach to quantum field and Cummings (FKWC) in Ref. [74]. A reader who theory [70]. In fact, the results we present in this ar- would like to follow or to check our calculations is invited ticle are very different from those of Moretti. We do to have in hand these two papers and more particularly not focus our attention on the mathematical aspects of Ref. [73] which displays, in addition to a list of useful ge- Hadamard renormalization as he did but on its practical ometrical identities, the slightly modified version of the aspects: from our results, the interested reader should be FKWC-bases we used in the present article. able to obtain explicitly the renormalized expression of the expectation value with respect to a given state ψ | i of the stress-energy-tensor operator associated with the II. HADAMARD RENORMALIZED scalar field theory if he knows (exactly or asymptotically STRESS-ENERGY TENSOR: GENERAL in a sense defined below) the Feynman propagator corre- CONSIDERATIONS sponding to ψ . With this aim in view, we have provided | i in Sec. III a step-by-step guide for the reader who simply In this section, we shall describe from a general point wishes to calculate this regularized expectation value and of view the renormalization of the stress-energy tensor is not specially interested in following the derivation of associated with a massive scalar field theory defined on all our results. a general spacetime of arbitrary dimension D 2. We shall assume that the scalar field is in a normalized≥ quan- Our article is organized as follows. In Sec. II, we de- tum state of Hadamard type and we shall consider that velop as generally as possible the Hadamard renormaliza- the Wald’s axiomatic approach (see Refs. [3, 9, 13]) tion of the stress-energy tensor associated with a massive developed in the four-dimensional framework remains scalar field theory defined on a general spacetime of ar- valid in the D-dimensional one. We shall in fact extend bitrary dimension. In Sec. III, we explicitly describe this various considerations previously developed in the four- procedure for arbitrary spacetimes of dimension from 2 dimensional framework (see Refs. [9, 13, 14, 15, 16, 17, to 6. This is done by using recent results we obtained 18, 19, 20, 21, 22]). in Ref. [71] and which concern the covariant Taylor series expansions of the Hadamard coefficients. For spacetime dimension from 7 to 11, we provide the framework A. Some aspects of the classical theory permitting the interested reader to perform this regularization procedure explicitly in a given spacetime. In We begin by reviewing the classical theory of a “free” Sec. IV, we complete our study (i) by considering the am- massive scalar field Φ propagating on a D-dimensional biguities of the Hadamard renormalization of the stress- curved spacetime ( ,g ) in order to emphasize some energy tensor and the corresponding ambiguities for the µν results which shallM play a crucial role at the quantum trace anomaly, (ii) by providing the expressions of the level. We first recall that the associated action is given gravitational counterterms involved in the renormaliza- by tion process (iii) by discussing the connections between Hadamard renormalization and renormalization in the ef- 1 D µν 2 2 2 S = d x√ g g Φ;µΦ;ν + m Φ + ξRΦ (3) fective action. Finally, in Sec. V, we briefly discuss possi- −2 M − ble extensions of our work as well as possible applications. Z where m is the mass of the scalar field and ξ is a di- In a short appendix, we provide the traces of various con- mensionless factor which accounts for the possible cou- served local tensors of rank 2 and orders 4 and 6. These pling between the scalar field and the gravitational back- results are more particularly helpful in order to discuss ground. We furthermore assume that spacetime has the ambiguity problem for the trace anomaly considered no boundary, i.e., that ∂M = . S is a functional of in Sec. IV. ∅ the scalar field Φ and of the gravitational field gµν , i.e. In this paper, we use units with ~ = c = 1 and the geo- S = S [Φ,gµν ]. The functional derivative of S with re- metrical conventions of Hawking and Ellis [72] concerning spect to Φ is given by the definitions of the scalar curvature R, the Ricci tensor δS = √ g m2 ξR Φ (4) Rµν and the Riemann tensor Rµνρσ . We also extensively δΦ − − − 4 and its extremization provides the wave (Klein-Gordon) with equation δΦ= Φ= ǫµΦ (12c) L−ǫ − ;µ m2 ξR Φ=0. (5) δg = g = ǫ ǫ (12d) − − µν L−ǫ µν − µ;ν − ν;µ The functional derivative of S with respect to g permits where denotes the Lie derivative with respect to the µν L−ǫ us to define the stress-energy tensor Tµν associated with vector ǫ. The invariance of the action (3) leads to the scalar field Φ (see, for example, Ref. [72]). Indeed, − we have D δS δS d x δΦ+ δgµν = 0 (13) δΦ δgµν 2 δ ZM T = S [Φ,g ] (6) µν √ g δgµν µν which implies − µν ;µ 2 and by using that in the variation T =Φ m ξR Φ (14) ;ν − − by using (12). Then, from (5) we obtain immediately gµν gµν + δgµν (7) → (10). of the metric tensor we have (see, for example, Ref.[75]) It is also well-known that for 2 gµν gµν + δgµν (8a) m = 0 and ξ = ξc(D) (15) → √ g √ g + δ√ g (8b) − → − − with R R + δR (8c) → 1 D 2 ξ (D)= − (16) c 4 D 1 with − the stress-energy tensor is traceless, i.e. it satisfies δgµν = gµρgνσδg (8d) − ρσ 1 µ δ√ g = √ ggµν δg (8e) T µ =0. (17) − 2 − µν µν ;µν µν ;ρ δR = R δgµν + (δgµν ) (g δgµν ) (8f) This result could be obtained directly from the field equa- − − ;ρ tion (5) by using the expression (9). In fact, from the we can explicitly find that physical point of view, it is more instructive to derive it by noting that for the values of the parameters m2 and 1 ξ given by (15) the scalar field theory is conformally in- T = (1 2ξ)Φ Φ + 2ξ g gρσΦ Φ µν − ;µ ;ν − 2 µν ;ρ ;σ variant (see, for example, Appendix D of Ref. [76]). As a consequence, the action (3) is invariant under the so- 1 2ξΦΦ +2ξg ΦΦ+ ξ R g R Φ2 called conformal transformation − ;µν µν µν − 2 µν 1 Φ Φ=Ωˆ (2−D)/2Φ (18a) g m2Φ2. (9) → −2 µν g gˆ =Ω2g (18b) µν → µν µν It is well-known that the stress-energy tensor is con- and therefore under the infinitesimal conformal transfor- served, i.e. it satisfies mation

T µν =0. (10) Φ Φ=Φ+ˆ δΦ (19a) ;ν → gµν gˆµν = gµν + δgµν (19b) This result could be obtained directly from the field equa- → tion (5) by using the expression (9). However, it is more with instructive from the physical point of view to derive it 2 D from the invariance of the action (3) under spacetime δΦ= − ǫ Φ (19c) diffeomorphisms and therefore under the infinitesimal co- 2 ordinate transformation δgµν =2ǫgµν (19d)

xµ xµ + ǫµ with ǫµ 1. (11) which corresponds to Ω = 1+ ǫ with ǫ 1. The invari- → | | ≪ ance of the action (3) leads to (13) which| | ≪ now implies Indeed, under this transformation, the scalar ﬁeld and D 2 the background metric transform as T µ = − Φ [ ξ (D)R] Φ (20) µ 2 − c Φ Φ+ δΦ (12a) → by using (19). Then, from (5) with (15), we obtain im- g g + δg (12b) mediately (17). µν → µν µν 5

B. Hadamard quantum states and Feynman For D even with D = 2, the Hadamard expansion of propagator the Feynman propagator6 is given by iα U(x, x′) From now on, we shall assume that the scalar field GF(x, x′)= D 2 [σ(x, x′)+ iǫ]D/2−1 theory previously described has been quantized and that the scalar field Φ is in a normalized quantum state ψ of + V (x, x′) ln[σ(x, x′)+ iǫ]+ W (x, x′) (28) Hadamard type. The associated Feynman propagator| i GF(x, x′)= i ψ T Φ(x)Φ(x′) ψ (21) where U(x, x′), V (x, x′) and W (x, x′) are symmetric bis- h | | i calars, regular for x′ x and which possess expansions (here T denotes time ordering) is, by definition, a solution of the form → of D/2−2 m2 ξR GF(x, x′)= δD(x, x′) (22) U(x, x′)= U (x, x′)σn(x, x′), (29a) x − − − n n=0 X with δD(x, x′) = [ g(x)]−1/2(x)δD(x x′). It is sym- +∞ − ′ − ′ ′ n ′ metric in the exchange of x and x and its short-distance V (x, x )= Vn(x, x )σ (x, x ), (29b) ′ behavior is of Hadamard type. Its precise form for x n=0 X near x depends on whether the dimension D of space- +∞ ′ ′ n ′ time is even or odd (see Refs. [77, 78, 79] or the articles W (x, x )= Wn(x, x )σ (x, x ). (29c) by Moretti [66, 67, 68, 69, 70] as well as our recent arti- n=0 cle [71] for more details). It involves the geodetic interval X σ(x, x′) and the biscalar form ∆(x, x′) of the Van Vleck- For D odd, the Hadamard expansion of the Feynman Morette determinant [80]. Here we recall that 2σ(x, x′) propagator is given by is a biscalar function which is defined as the square of the ′ F ′ iαD U(x, x ) ′ geodesic distance between x and x′ and which satisfies G (x, x )= + W (x, x ) 2 [σ(x, x′)+ iǫ]D/2−1 ;µ 2σ = σ σ;µ. (23) (30)

We have σ(x, x′) < 0 if x and x′ are timelike related, where U(x, x′) and W (x, x′) are again symmetric and σ(x, x′) = 0 if x and x′ are null related and σ(x, x′) > 0 regular biscalar functions which now possess expansions if x and x′ are spacelike related. We furthermore recall of the form ′ that ∆(x, x ) is given by +∞ ′ ′ n ′ U(x, x )= Un(x, x )σ (x, x ), (31a) ′ −1/2 ′ ′ ′ −1/2 ∆(x, x )= [ g(x)] det( σ;µν (x, x ))[ g(x )] n=0 − − − − X (24) +∞ and satisfies the partial differential equation ′ ′ n ′ W (x, x )= Wn(x, x )σ (x, x ). (31b) n=0 σ = D 2∆−1/2∆1/2 σ;µ (25a) X x − ;µ In Eqs. (26), (28) and (30), the coefficient αD is given as well as the boundary condition by ′ lim′ ∆(x, x )=1. (25b) 1/(2π) for D =2, x →x α = (32) D Γ(D/2 1)/(2π)D/2 for D =2, For D = 2, the Hadamard expansion of the Feynman − 6 propagator is given by while the factor iǫ with ǫ 0+ is introduced to give to GF(x, x′) a singularity→ structure that is consistent iα GF(x, x′)= 2 (V (x, x′) ln[σ(x, x′)+ iǫ] with the definition of the Feynman propagator as a time- 2 ordered product (see Eq. (21)). ′ ′ +W (x, x )) (26) For D = 2, the Hadamard coefficients Vn(x, x ) and ′ Wn(x, x ) are symmetric and regular biscalar functions. ′ ′ ′ where V (x, x ) and W (x, x ) are symmetric biscalars, reg- The coefficients Vn(x, x ) satisfy the recursion relations ular for x′ x and which possess expansions of the form → 2 ;µ 2(n + 1) Vn+1 + 2(n + 1)Vn+1;µσ −1/2 1/2 ;µ +∞ 2(n + 1)V ∆ ∆ σ − n+1 ;µ V (x, x′)= V (x, x′)σn(x, x′), (27a) 2 n + x m ξR Vn =0 for n N (33a) n=0 − − ∈ X +∞ with the boundary condition W (x, x′)= W (x, x′)σn(x, x′). (27b) n 1/2 n=0 V0 = ∆ . (33b) X − 6

′ F ′ The coefficients Wn(x, x ) satisfy the recursion relations (23) and (25) it is possible to prove that G (x, x ) given by (28)-(29) solves the wave equation (22). This can be 2 ;µ 2(n + 1) Wn+1 + 2(n + 1)Wn+1;µσ done easily by noting that we have 2(n + 1)W ∆−1/2∆1/2 σ;µ n+1 ;µ 2 − ;µ x m ξR V = 0 (40) +4(n + 1)Vn+1 +2Vn+1;µσ − − 2V ∆−1/2∆1/2 σ;µ as a consequence of (38a) and − n+1 ;µ 2 N 2 2 + x m ξR Wn =0 for n . (34) σ m ξR W = m ξR U − − ∈ x − − − x − − D/2−2 ;µ −1/2 1/2 ;µ From the recursion relations (33a) and (34), the bound- (D 2)V 2V;µσ +2V ∆ ∆ ;µσ (41) − − − ary condition (33b) and the relations (23) and (25) it is possible to prove that GF(x, x′) given by (26)-(27) solves as a consequence of (38b) and (39). ′ the wave equation (22). This can be done easily by noting For D odd, the Hadamard coefficients Un(x, x ) and ′ that we have Wn(x, x ) are symmetric and regular biscalar functions. ′ The coefficients Un(x, x ) satisfy the recursion relations 2 x m ξR V = 0 (35) − − (n + 1)(2n +4 D)U + (2n +4 D)U σ;µ − n+1 − n+1;µ as a consequence of (33) and −1/2 1/2 ;µ (2n +4 D)Un+1∆ ∆ ;µσ 2 − −2 σ x m ξR W = + m ξR U =0 for n N (42a) − − x − − n ∈ 2V σ;µ +2V ∆−1/2∆1/2 σ;µ (36) − ;µ ;µ with the boundary condition as a consequence of (33b) and (34). U = ∆1/2. (42b) For D even with D = 2, the Hadamard coefficients 0 ′ ′ 6 ′ ′ Un(x, x ), Vn(x, x ) and Wn(x, x ) are symmetric and reg- The coefficients Wn(x, x ) satisfy the recursion relations ′ ular biscalar functions. The coefficients Un(x, x ) satisfy ;µ the recursion relations (n + 1)(2n + D)Wn+1 + 2(n + 1)Wn+1;µσ −1/2 1/2 ;µ ;µ 2(n + 1)Wn+1∆ ∆ ;µσ (n + 1)(2n +4 D)Un+1 + (2n +4 D)Un+1;µσ − − − 2 N (2n +4 D)U ∆−1/2∆1/2 σ;µ + x m ξR Wn =0 for n . (43) − − n+1 ;µ − − ∈ + m2 ξR U =0 From the recursion relations (42a) and (43), the bound- x − − n for n =0, 1,...,D/2 3 (37a) ary conditions (42b) and the relations (23) and (25) it is − possible to prove that GF(x, x′) given by (30)-(31) solves with the boundary condition the wave equation (22). This can be done easily from

1/2 2 U0 = ∆ . (37b) m ξR W = 0 (44) x − − ′ The coefficients Vn(x, x ) satisfy the recursion relations which is a consequence of (43). ′ ;µ For D = 2, the Hadamard coefficients Vn(x, x ) can (n + 1)(2n + D)Vn+1 + 2(n + 1)Vn+1;µσ be formally obtained by integrating the recursion rela- −1/2 1/2 ;µ ′ 2(n + 1)Vn+1∆ ∆ ;µσ tions (33a) along the unique geodesic joining x to x (it − is unique for x′ near x or more generally for x′ in a con- + m2 ξR V =0 for n N (38a) x − − n ∈ vex normal neighborhood of x). Similarly, for D even ′ with the boundary condition with D = 2, the Hadamard coefficients Un(x, x ) and ′ 6 Vn(x, x ) can be obtained by integrating the recursion ;µ −1/2 1/2 ;µ (D 2)V0 +2V0;µσ 2V0∆ ∆ ;µσ relations (37a) and (38a) along the unique geodesic join- − − ′ + m2 ξR U =0. (38b) ing x to x while, for D odd, the Hadamard coefficients x D/2−2 ′ − − Un(x, x ) can be obtained by integrating the recursion ′ ′ The coefficients Wn(x, x ) satisfy the recursion relations relations (42a) along the unique geodesic joining x to x . As a consequence, all these Hadamard coefficients are ;µ (n + 1)(2n + D)Wn+1 + 2(n + 1)Wn+1;µσ determined uniquely and are purely geometrical objects, −1/2 1/2 ;µ 2(n + 1)Wn+1∆ ∆ ;µσ i.e. they only depend on the geometry along the geodesic ′ − ;µ joining x to x . By contrast, the Hadamard coefficients +(4n +2+ D)Vn+1 +2Vn+1;µσ ′ Wn(x, x ) with n N are neither uniquely defined nor −1/2 1/2 ;µ ∈ 2Vn+1∆ ∆ ;µσ purely geometrical. Indeed, the first coefficient of this − 2 ′ + x m ξR Wn =0 for n N. (39) sequence, i.e. W0(x, x ), is unrestrained by the recursion − − ∈ relations (34) for D = 2, (39) for D even with D = 2 and From the recursion relations (37a), (38a) and (39), the (43) for D odd. As a consequence, this is also6 true for boundary conditions (37b) and (38b) and the relations all the W (x, x′) with n 1. This arbitrariness is in fact n ≥ 7

′ very interesting and it can be used to encode the quan- and V1(x, x ) (see, for example, Ref. [18] and references tum state dependence in the biscalar W (x, x′) by spec- therein for the scalar field). In Ref. [71], we have recently ′ ifying the Hadamard coefficient W0(x, x ). Once it has discussed the construction of the expansions of the geo- ′ ′ been specified, the recursion relations (34), (39) or (43) metrical Hadamard coefficients Un(x, x ) and Vn(x, x ) ′ uniquely determine the coefficients Wn(x, x ) with n 1 of lowest orders in the D-dimensional framework (with and therefore the biscalar W (x, x′). In other words,≥ the D 3) and we intend to use these results later. The Hadamard expansions (26)-(27), (28)-(29) and (30)-(31) case≥D = 2 has not been explicitly treated in Ref. [71] comprise a purely geometrical part, divergent for x′ x but a comparison of Eq. (23) of Ref. [71] with (33) and given by → permits us to express the geometrical Hadamard coefficients V (x, x′) in terms of the mass-dependent DeWitt- iα n F ′ 2 ′ ′ ˜ 2 ′ ′ Gsing(x, x )= (V (x, x ) ln[σ(x, x )+ iǫ]) (45) coefficients An(m ; x, x ) [71]. We have Vn(x, x ) = 2 n n 2 ′ (( 1) /(2 n!)) Añ(m ; x, x ) and this relation together for D = 2, by −with− the covariant Taylor series expansions of the mass- ′ dependent DeWitt-coefficients obtained in Ref. [71] pro- F ′ iαD U(x, x ) Gsing(x, x )= ′ D/2−1 vide the covariant Taylor series expansions of the geo- 2 [σ(x, x )+ iǫ] ′ metrical Hadamard coefficients Vn(x, x ) of lowest orders + V (x, x′) ln[σ(x, x′)+ iǫ] (46) for D = 2. As far as the biscalar W (x, x′) which encodes the state- dependence of the Feynman propagator is concerned, its for D even with D = 2 and by 6 covariant Taylor series expansion is written as ′ F ′ iαD U(x, x ) +∞ p Gsing(x, x )= ′ D/2−1 ( 1) 2 [σ(x, x )+ iǫ] W (x, x′)= w(x)+ − w (x, x′) (50a) p! (p) (47) p=1 X ′ for D odd as well as a regular state-dependent part given where the w(p)(x, x ) with p = 1, 2,... are all biscalars by in x and x′ which are of the form

F ′ iαD ′ ′ ;a1 ′ ;ap ′ G (x, x )= W (x, x ). (48) w(p)(x, x )= wa1...ap (x)σ (x, x ) ...σ (x, x ). reg 2 (50b) It should be noted that, bearing in mind practical applications, it is very interesting to replace the Hadamard The coefficients w(x) and wa1...ap (x) with p =1, 2,... are coefficients by their covariant Taylor series expansions. constrained by the symmetry of W (x, x′) in the exchange Here, we shall provide some associated results which of x and x′ as well as by the wave equations (36), (41) will be helpful afterwards. As far as the geometrical or (44) according to the values of D. The symmetry of ′ ′ ′ Hadamard coefficients Un(x, x ) and Vn(x, x ) which de- W (x, x ) permits us to express the odd coefficients of the termine the singular part of the Feynman propagator are covariant Taylor series expansion of W (x, x′) in terms of concerned, they are usually obtained by looking for the the even ones. We have for the odd coefficients of lowest solutions of the recursion relations defining them as co- orders (see, for example, Refs. [18, 19] or Ref. [71]) variant Taylor series expansions for x′ near x given by wa1 = (1/2) w;a1 (51a)

+∞ p wa1a2a3 = (3/2) w(a1a2;a3) (1/4) w;(a1a2a3).(51b) ′ ( 1) ′ − Un(x, x )= un(x)+ − u (x, x ) (49a) p! n (p) The wave equation satisﬁed by W (x, x′) for D even per- p=1 X mits us to write +∞ p ′ ( 1) ′ Vn(x, x )= vn(x)+ − vn (p)(x, x ) (49b) 2 p! x m ξR W p=1 − − ;µ X = (D + 2)V1 2V1;µσ + O (σ) . (52) ′ ′ − − where the un (p)(x, x ) and vn (p)(x, x ) with p = 1, 2,... are all biscalars in x and x′ which are of the form This relation is valid for D = 2 as well as for D even with D = 2. It is obtained from (36) or (41) by using (27a) ′ ;a1 ′ ;ap ′ 6 un (p)(x, x )= un a1...ap (x)σ (x, x ) ...σ (x, x ) and (33b) or (29b) and (38b) as well as the following two (49c) expansions (see, for example, Refs. [11, 12] or Ref. [71]) ′ ;a1 ′ ;ap ′ vn (p)(x, x )= vn a1...ap (x)σ (x, x ) ...σ (x, x ). ∆1/2 =1+(1/12) R σ;a1 σ;a2 + O σ3/2 (53) (49d) a1a2 This method, due to DeWitt [80, 81], has been used and in the four-dimensional framework to construct the co- ′ ′ σ = g (1/3) R σ;a1 σ;a2 + O σ3/2 . (54) variant Taylor series expansions of U0(x, x ), V0(x, x ) ;µν µν − µa1νa2 8

′ Then, by inserting the expansion of V1(x, x ) given by where gµν′ denotes the bivector of parallel transport from (49b) and (49d) and by using (54), we have x to x′ (see Refs. [80, 81]) which is defined by the partial differential equation 2 x m ξR W − − ;µ ′ ;ρ = (D + 2)v + (D/2) v σ + O (σ) .(55) gµν ;ρσ =0 (62a) − 1 1;µ By inserting the expansion (50a)-(50b) of W (x, x′) up to and the boundary condition order σ3/2 into the left-hand side of (55) and by using (51) as well as (54) we find that lim gµν′ = gµν . (62b) x′→x wρ = (m2 + ξR)w (D + 2)v (56a) ρ − 1 Of course, because of the short-distance behavior of the ρ ρ ρ w a;ρ = (1/4) ( w);a + (1/2) w ρ;a + (1/2)R aw;ρ Feynman propagator, the expression (60) of the expec- 2 tation value of the stress-energy-tensor operator in the (1/2) (m + ξR)w;a + (D/2) v1;a (56b) − Hadamard state ψ is divergent and therefore meaning- | i and by combining (56a) and (56b) we establish another less. This pathological behavior comes from the purely relation geometrical part of the Hadamard expansion given by (45) for D = 2 or (46) for D even with D = 2 or by (47) ρ ρ for D odd. More precisely, for D = 2 the6 terms in ln σ w a;ρ = (1/4) ( w);a + (1/2)R aw;ρ and σ ln σ which are present in (45) induce divergences in +(1/2) ξR w v (57) ;a − 1;a 1/σ and ln σ in the expression (60) of ψ T ψ . For D h | µν | i even with D = 2, the terms in 1/σD/2−1, ...,1/σ, ln σ which will be helpful in the next subsection. The wave 6 equation (44) satisfied by W (x, x′) for D odd can be and σ ln σ which are present in (46) induce divergences D/2 2 worked in the same manner. It leads to in 1/σ ,...,1/σ ,1/σ and ln σ in the expression (60) D/2−1 of ψ Tµν ψ while, for D odd, the terms in 1/σ , ρ 2 h | 1/| 2 i 1/2 w ρ = (m + ξR)w (58a) ..., 1/σ and σ which are present in (47) induce ρ ρ ρ divergences in 1/σD/2,...,1/σ1/2 in this expression. w a;ρ = (1/4) ( w);a + (1/2) w ρ;a + (1/2)R aw;ρ 2 With Wald [3, 9, 13] is possible to cure the patho- (1/2) (m + ξR)w;a (58b) − logical behavior of ψ Tµν ψ given by (60) and to construct from it a meaningfulh | | i expression which can act as and to a source in the semiclassical Einstein equations (1) and which can be considered as the renormalized expectation wρ = (1/4) (w) + (1/2)Rρ w a;ρ ;a a ;ρ value with respect to the Hadamard quantum state ψ +(1/2) ξR;aw. (59) of the stress-energy tensor operator. The Hadamard reg-| i ularization prescription permits us to accomplish this in the following manner: we first discard in the right-hand C. Hadamard renormalization of the stress-energy side of (60) the purely geometrical part (45) or (46) or tensor (47) of GF , i.e. we make the replacement

The expectation value with respect to the Hadamard ′ F ′ lim′ µν (x, x ) iG (x, x ) quantum state ψ of the stress-energy-tensor operator is x →x T − → | i αD ′ ′ formally given as the limit lim µν (x, x )W (x, x ). (63) 2 x′→x T ′ F ′ ψ Tµν (x) ψ = lim µν (x, x ) iG (x, x ) (60) h | | i x′→x T − We then add to the right-hand side of (63) a state- independent tensor Θ˜ which only depends on the pa- F ′ µν where G (x, x ) is the Feynman propagator (21) which is rameters m2 and ξ of the theory and on the local ge- assumed to possess one of the Hadamard form displayed ometry and which ensures the conservation of the result- in the previous subsection. In Eq. (60), (x, x′) is a dif- Tµν ing expression. The renormalized expectation value of ferential operator which is constructed by point-splitting stress-energy tensor operator in the Hadamard state ψ from the classical expression (9) of the stress-tensor. It is therefore given by | i is a tensor of type (0,2) in x and a scalar in x′. It is given by αD ′ ′ ψ Tµν (x) ψ ren = lim µν (x, x )W (x, x )+ Θ˜ µν (x). h | | i 2 x′→x T ′ ′ ν 1 ρσ (64) = (1 2ξ)g ′ + 2ξ g g ′ Tµν − ν ∇µ∇ν − 2 µν ∇ρ∇σ Bearing in mind practical applications, it is also interest- ′ ′ µ ν ρ ing to reexpress the previous result in terms of the lowest 2ξg g ′ ′ +2ξg − µ ν ∇µ ∇ν µν ∇ρ∇ order coeﬃcients of the covariant Taylor series expansion 1 1 of the biscalar W (x, x′). By inserting (50a)-(50b) into +ξ R g R g m2 (61) µν − 2 µν − 2 µν (64) and by using the expansions (54) and (see, for ex- 9 ample, Refs. [11, 12] or Ref. [71]) This result can be also written in the form

′ α 1 ν ;a1 ;a2 3/2 D g σ ′ = g (1/6) R σ σ + O σ ψ Tµν ψ ren = wµν + (1 2ξ)w;µν ν ;µν − µν − µa1νa2 h | | i 2 − 2 − (65) 1 1 as well as the relations (see, for example, Refs. [11, 12]) + 2ξ g w + ξR w +Θ (73) 2 − 2 µν µν µν ′ ρ ′ gµ gνρ = gµν (66a) which is obtained by inserting (58a) into (67) and by ′ ν ;a g gµν′;ρ = (1/2) Rµνρaσ + O (σ) (66b) taking into account (69). In Eqs. (70)-(73), the tensor ν 2 ′ ′ − ν ρ ;a Θµν only depends on the parameters m and ξ of the g g g ′ ′ = (1/2) R σ + O (σ) (66c) ν ρ µν ;ρ − µνρa theory and on the local geometry and it is now conserved, i.e. it satisfies we obtain µν Θ ;ν =0. (74) αD 1 ρ ψ Tµν ψ ren = wµν gµν w h | | i 2 − − 2 ρ To conclude this subsection, we think it is interesting 1 1 1 to recall to the reader that the two coefficients w(x) + (1 2ξ)w + 2ξ g w 2 − ;µν 2 − 2 µν and wµν (x) which appear in the final expressions (71) and (73) and which encode the state-dependence are ob- 1 1 2 +ξ Rµν gµν R w gµν m w + Θ˜ µν . tained as Taylor coefficients of the expansion of the bis- − 2 − 2 ′ calar W (x, x ) but also more directly by the following two (67) formulas

′ Now, by requiring the conservation of ψ T ψ given w(x) = lim W (x, x ) (75a) h | µν | iren x′→x by (67), we ﬁnd that Θ˜ µν must satisfy ′ wµν (x) = lim W (x, x );µν (75b) x′→x µν µν Θ˜ (D/4)αD g v1 = 0 (68) − ;ν which can be derived easily from (50a)-(50b) by using h i (51a) and (54). They are useful to treat practical appli- when D is even and cations. ˜ µν Θ ;ν = 0 (69) D. Ambiguities in the renormalized expectation when D is odd. Equations (68) and (69) are derived by value of the stress-energy tensor using (56a) and (57) for the former and (58a) and (59) for the latter. As we have previously noted, the renormalized expec- It is now possible to provide a deﬁnitive expression for tation value ψ T ψ is unique up to the addition of the renormalized expectation value of the stress-energy µν ren a local conservedh | tensor| i Θ . This problem plagues the tensor operator in the Hadamard state ψ . From (64) µν Hadamard renormalization procedure since its invention and by taking into account (68), we have| fori D even (see Sec. III of Ref. [13]). It has been recurrently dis-

αD ′ ′ cussed in the four-dimensional context: we refer to the ψ Tµν (x) ψ ren = lim µν (x, x )W (x, x ) h | | i 2 x′→x T monographs of Fulling [2] and Wald [3] and to references h D therein as well as to more recent considerations developed + gµν v1 +Θµν (x). (70) in Refs. [70, 82, 83, 84, 85, 86, 87]. In our opinion, this 2 problem cannot be solved in the lack of a complete quan- This result can be also written in the form tum theory of gravity. As a consequence, it induces a se- rious difficulty with regard to the study of back reaction αD 1 effects, the right-hand side of the semiclassical Einstein ψ T ψ = w + (1 2ξ)w h | µν | iren 2 − µν 2 − ;µν equation (1) being ambiguously defined. In the present subsection, we shall not consider the 1 1 + 2ξ gµν w + ξRµν w gµν v1 +Θµν ambiguity problem from a general point of view. We 2 − 2 − shall only discuss the standard ambiguity associated with (71) the choice of a mass scale M - the so-called renormalization mass - introduced in order to make the argu- which is obtained by inserting (56a) into (67) and by ment of the logarithm in Eq. (28) dimensionless. We taking into account (68). From (64) and by taking into intend to provide a more general (but still incomplete) account (69), we have for D odd discussion in Sec. IV. The ambiguity associated with the αD ′ ′ renormalization mass only exists when the dimension ψ Tµν (x) ψ ren = lim µν (x, x )W (x, x )+Θµν(x). h | | i 2 x′→x T D of spacetime is even. It corresponds to the replace- (72) ment of the term V (x, x′) ln[σ(x, x′)+ iǫ] by the term 10

′ 2 ′ ρ V (x, x ) ln[M (σ(x, x )+ iǫ)] and therefore to an inde- which is obtained from (79) by using v0 ρ = D v1 + ′ 2 − terminacy in the function W (x, x ) previously considered (m + ξR)v0, this last relation being easily derived from which corresponds to the replacement (35) or (40). 2 ′ ′ ′ 2 For m = 0 and ξ = ξc(D), i.e. when the scalar field 2 W (x, x ) W (x, x ) V (x, x ) ln M (76) µν M → − theory is conformally invariant, the trace g Θµν van- for which the theory developed in Sec. II.C remains valid. ishes and Eq. (80) yields This indeterminacy is therefore associated with the term µ ψ T ψ ren = αD v1 (83) 2 µ M αD ′ ′ 2 h | | i Θµν (x)= lim µν (x, x )V (x, x ) ln M . (77) − 2 x′→x T for D even. After renormalization, the expectation value By using Eqs. (29b), (49b) and (49d)), we can see also of the stress-energy tensor has acquired a non-vanishing that the transformation (76) leads to the replacement or “anomalous” trace even though the classical stress- energy tensor is traceless [see Eq. (17)]. We refer to 2 w w v0 ln M (78a) the monographs of Birrell and Davies [1], Fulling [2] and → − 2 Wald [3] as well as to references therein for various dis- wµν wµν (v0 µν + gµν v1) ln M (78b) → − cussions and considerations concerning trace anomalies into Eq. (71) and thus we have in quantum field theory in curved spacetime. For D odd, 2 m = 0 and ξ = ξc(D), Eq. (81) yields 2 M αD 1 Θ = (v0 µν + gµν v1)+ (1 2ξ)v0;µν µν − 2 − 2 − ψ T µ ψ = 0 (84) h | µ| iren 1 1 2 + 2ξ gµν v0 + ξRµν v0 ln M . (79) and it appears that the trace anomaly does not exist 2 − 2 when the dimension of spacetime is odd. As a consequence, the knowledge of the first Taylor coefficients of the purely geometrical Hadamard coefficients ′ ′ III. HADAMARD RENORMALIZED V0(x, x ) and V1(x, x ) permits us to treat partially the ambiguity problem. It should be finally recalled that STRESS-ENERGY TENSOR: EXPLICIT the renormalization mass can be fixed by imposing ad- CONSTRUCTION ditional physical conditions on the renormalized expectation value of the stress-energy tensor, these conditions In this section, we shall mainly discuss the practical being appropriate to the problem treated. aspects of the Hadamard renormalization of the expectation value of the stress-energy tensor. This section is written for the reader who simply wishes to calculate this E. Trace anomaly renormalized expectation value in a particular case and is not specially interested in the derivation of all the pre- Here, we shall assume that the renormalized expec- vious general results. We assume that we know the explicit expression of the tation value of the stress-energy tensor ψ Tµν ψ ren is h | | i Feynman propagator GF(x, x′) associated with a given given by (71) for D even with the geometrical tensor Θµν 2 Hadamard quantum state ψ . We first obtain the state- which reduces to ΘM given by (79) and by (73) for D µν dependent Hadamard biscalar| i W (x, x′) from the relation odd with the geometrical tensor Θµν which vanishes. We neglect all the other possible contributions (see however 2 Sec. IV.A for a more general discussion). W (x, x′)= GF(x, x′) GF (x, x′) (85) iα − sing By using (56a), we can show that the trace of D ψ Tµν ψ ren is then given by F ′ h | | i where Gsing(x, x ) is given by (45) or (46) or (47) accord- µ αD 2 ing to the dimension D of spacetime. Of course, we need ψ T ψ = m w + (D 1) (ξ ξ (D)) w ′ h | µ| iren 2 − − − c only the covariant Taylor series expansion of W (x, x ) up 2 µν M to order σ and therefore we do not need to know the terms +2v1]+ g Θµν (80) F ′ of the expansion of Gsing(x, x ) which vanish faster than for D even and by using (58a) that it reduces to σ(x, x′) for x′ near x. For the same reason, the Feynman propagator GF(x, x′) does not need to be known exactly: µ αD 2 ′ ψ T µ ψ ren = m w + (D 1) (ξ ξc(D)) w we need only its asymptotic expansion for x near x and h | | i 2 − − − we do not need to know the terms of this expansion which (81) vanish faster than σ(x, x′) for x′ near x. From the ex- ′ for D odd. Furthermore, we have pansion up to order σ of the biscalar W (x, x ) we then obtain the Taylor coefficients w(x) and wµν (x) either di- 2 µν M αD 2 rectly or by using the relations (75). This permits us g Θ = [ m v0 µν − 2 − to finally construct the renormalized expectation value +(D 1) (ξ ξ (D)) v ] ln M 2 (82) in the Hadamard quantum state ψ of the stress-energy − − c 0 | i 11 tensor by using (71) and (79) or (73) according to the The trace anomaly (83) is obtained by using m2 = 0 parity of D. Of course, for D even, we must in addition and ξ = ξc(2) = 0 into (91). It reduces to [1, 88] 2 M construct the geometrical tensor Θµν from the Taylor coefficients v , v and v in order to do this last step. R 0 0 µν 1 ψ T µ ψ = . (93) In the subsections below, we shall provide for space- h | µ| iren 24π time dimension from D = 2 to D = 6 the explicit ex- F ′ pansion of Gsing(x, x ) and for D =2, 4 and 6 we shall in addition give the explicit expression of the geometrical B. D=3 2 M tensor Θµν as well as of the trace anomaly. We shall use some of the results we obtained in Ref. [71]. We have sim- For D = 3, the expansion of the singular part plified them from the geometrical identities displayed in our unpublished report [73]. These geometrical identities i U(x, x′) are helpful to expand the Riemann polynomials encoun- GF (x, x′)= (94) sing 4√2π [σ(x, x′)+ iǫ]1/2 tered in our calculations on the FKWC-bases constructed from group theoretical considerations in Ref. [74]. They of the Feynman propagator is obtained, up the required have permitted us to provide irreducible expressions for order, for all our results. For spacetime dimension from 7 to 11, we shall describe the method permitting the interested 2 2 F ′ M U = U0 + U1σ + O σ (95) reader to construct explicitly Gsing(x, x ) (as well as Θµν when it is necessary) in a given spacetime by using the with results obtained in Ref. [71]. 1 1 U = u u σ;a + u σ;aσ;b u σ;aσ;bσ;c 0 0 − 0 a 2! 0 ab − 3! 0 abc A. D=2 +O σ2 (96) U = u u σ;a + O (σ) . (97) For D = 2, the expansion of the singular part 1 1 − 1 a i The Taylor coefficients appearing in Eqs. (96)-(97) are GF (x, x′)= (V (x, x′) ln[σ(x, x′)+ iǫ]) (86) sing 4π given by of the Feynman propagator is obtained, up the required u0 =1 (98a) order, for u0 a = 0 (98b)

3/2 u0 ab = (1/6)Rab (98c) V = V0 + V1σ + O σ (87) u = (1/4)R (98d) 0 abc (ab;c) with and ;a 1 ;a ;b 3/2 V0 = v0 v0 aσ + v0 abσ σ + O σ (88) 2 − 2! u1 = m + (ξ 1/6)R (99) − V = v + O σ1/2 . (89) u = (1/2)(ξ 1/6)R . (100) 1 1 1 a − ;a The Taylor coeﬃcients appearing in Eqs. (88)-(89) are given by C. D=4

v0 = 1 (90a) − For D = 4, the expansion of the singular part v0 a = 0 (90b) ′ v0 ab = (1/12)Rgab (90c) i U(x, x ) − GF (x, x′)= sing 8π2 σ(x, x′)+ iǫ and + V (x, x′) ln[σ(x, x′)+ iǫ] (101) v = (1/2) m2 (1/2)(ξ 1/6)R. (91) 1 − − − 2 M The geometrical tensor Θµν which is associated with of the Feynman propagator is obtained, up the required the renormalization mass is obtained from (79) by using order, for (90a), (90c) and (91) and is given by U = U0 (102) 2 2 ln M ΘM = (1/2) m2 g . (92) V = V + V σ + O σ3/2 (103) µν 4 π − µν 0 1 12 with D. D=5 1 1 U = u u σ;a + u σ;aσ;b u σ;aσ;bσ;c 0 0 − 0 a 2! 0 ab − 3! 0 abc For D = 5, the expansion of the singular part 1 + u σ;aσ;bσ;cσ;d + O σ5/2 (104) 4! 0 abcd i U(x, x′) GF (x, x′)= (112) 1 sing ′ 3/2 V = v v σ;a + v σ;aσ;b + O σ3/2 (105) 16 √2 π2 [σ(x, x )+ iǫ] 0 0 − 0 a 2! 0 ab 1/2 V1 = v1 + O σ . (106) of the Feynman propagator is obtained, up the required order, for The Taylor coeﬃcients appearing in Eqs. (104)-(106) are 2 3 given by U = U0 + U1σ + U2σ + O σ (113)

u0 =1 (107a) with u0 a = 0 (107b)

u0 ab = (1/6) Rab (107c) ;a 1 ;a ;b 1 ;a ;b ;c U0 = u0 u0 aσ + u0 abσ σ u0 abcσ σ σ u0 abc = (1/4)R(ab;c) (107d) − 2! − 3! 1 ;a ;b ;c ;d 1 ;a ;b ;c ;d ;e u0 abcd = (3/10)R(ab;cd) + (1/12)R(abRcd) + u σ σ σ σ u σ σ σ σ σ 4! 0 abcd − 5! 0 abcde +(1/15)R Rp q (107e) p(a|q|b c d) +O σ3 (114) and ;a 1 ;a ;b 1 ;a ;b ;c U1 = u1 u1 aσ + u1 abσ σ u1 abcσ σ σ 2 − 2! − 3! v0 = (1/2)m + (1/2)(ξ 1/6)R (108a) − +O σ2 (115) v0 a = (1/4)(ξ 1/6)R;a (108b) − ;a 2 U2 = u2 u2 aσ + O (σ) . (116) v0 ab = (1/12)m Rab + (1/6)(ξ 3/20) R;ab − − (1/120) Rab + (1/12)(ξ 1/6)RRab The Taylor coeﬃcients appearing in Eqs. (114)-(116) are − p −pq +(1/90)R aRpb (1/180)R Rpaqb given by pqr − (1/180)R aRpqrb (108c) − u =1 (117a) and 0 u0 a = 0 (117b) 4 2 v1 = (1/8)m + (1/4)(ξ 1/6)m R − u0 ab = (1/6)Rab (117c) (1/24)(ξ 1/5)R + (1/8)(ξ 1/6)2 R2 u0 abc = (1/4)R(ab;c) (117d) − − pq − pqrs (1/720)RpqR + (1/720)RpqrsR . (109) − u0 abcd = (3/10)R(ab;cd) + (1/12)R(abRcd) 2 M p q The geometrical tensor Θµν which is associated with +(1/15)Rp(a|q|bR c d) (117e) the renormalization mass is obtained from (79) by using u0 abcde = (1/3)R(ab;cde) + (5/12)R(abRcd;e) (108a), (108c) and (109) and is given by p q +(1/3)Rp(a|q|bR c d;e) (117f) 2 2 M ln M 2 Θµν = (1/2)(ξ 1/6)m Rµν 2(2π)2 − − and 2 h +(1/2)[ξ (1/3)ξ +1/30]R;µν (1/120)Rµν 2 − 2 −p u1 = m (ξ 1/6)R (118a) (1/2)(ξ 1/6) RRµν + (1/90)R µRpν − − − − −pq pqr u1 a = (1/2)(ξ 1/6)R;a (118b) (1/180)R Rpµqν (1/180)R Rpqrν µ − −2 − − u1 ab = (1/6)m Rab (1/3)(ξ 3/20)R;ab 4 2 − − − +gµν (1/8)m + (1/4)(ξ 1/6)m R +(1/60)R (1/6)(ξ 1/6)RR − ab − − ab 2 p pq (1/2)[ξ (1/3)ξ +1/40] R (1/45)R aRpb + (1/90)R Rpaqb − − 2 2 pq − pqr +(1/8)(ξ 1/6) R (1/720)R R +(1/90)R Rpqrb (118c) − − pq a pqrs 2 +(1/720)RpqrsR . (110) u1 abc = (1/4)m R(ab;c) − i (1/4)(ξ 2/15)R;(abc) + (1/40)( R(ab);c) The trace anomaly (83) is obtained by using m2 = 0 − − (1/4)(ξ 1/6)R;(aRbc) (1/4)(ξ 1/6)RR(ab;c) and ξ = ξc(4) = 1/6 into (109). It reduces to [1, 88] − − − − p p q (1/15)Rp(aR b;c) + (1/60)Rpq;(aR b c) µ 1 pq − ψ T µ ψ ren = 2 [(1/720) R (1/720)RpqR p q pqr h | | i (2π) − +(1/60)RpqR (a b;c) + (1/30)Rpqr(aR b;c) pqrs +(1/720)RpqrsR ] . (111) (118d) 13 and The Taylor coeﬃcients appearing in Eqs. (123)-(126) are given by u = (1/2)m4 (ξ 1/6)m2R 2 − − − +(1/6)(ξ 1/5)R (1/2)(ξ 1/6)2R2 − − − +(1/180)R Rpq (1/180)R Rpqrs (119a) pq − pqrs u = (1/2)(ξ 1/6)m2R 2 a − − ;a +(1/12)(ξ 1/5)(R) (1/2)(ξ 1/6)2RR − ;a − − ;a +(1/180)R Rpq (1/180)R Rpqrs . pq ;a − pqrs ;a (119b) u0 =1 (127a)

u0 a = 0 (127b) E. D=6 u0 ab = (1/6)Rab (127c) and For D = 6, the expansion of the singular part

i U(x, x′) GF (x, x′)= sing 16 π3 [σ(x, x′)+ iǫ]2 + V (x, x′) ln[σ(x, x′)+ iǫ] (120) of the Feynman propagator is obtained, up the required u0 abc = (1/4)R(ab;c) (127d) order, for u0 abcd = (3/10)R(ab;cd) + (1/12)R(abRcd) p q +(1/15)Rp(a|q|bR c d) (127e) U = U0 + U1σ (121) 3/2 u0 abcde = (1/3)R(ab;cde) + (5/12)R(abRcd;e) V = V0 + V1σ + O σ (122) p q +(1/3)Rp(a|q|bR (127f) c d;e) with u0 abcdef = (5/14)R(ab;cdef) + (3/4)R(abRcd;ef) p q 1 1 +(4/7)Rp(a|q|bR c d;ef) + (5/8)R(ab;cRde;f) U = u u σ;a + u σ;aσ;b u σ;aσ;bσ;c 0 0 − 0 a 2! 0 ab − 3! 0 abc p q +(15/28)Rp(a|q|b;cR d e;f) + (5/72)R(abRcdRef) 1 ;a ;b ;c ;d 1 ;a ;b ;c ;d ;e p q + u0 abcdσ σ σ σ u0 abcdeσ σ σ σ σ +(1/6)R R R 4! − 5! (ab c d |p|e|q|f) 1 +(8/63)Rp Rq Rr (127g) + u σ;aσ;bσ;cσ;dσ;eσ;f + O σ7/2 (123) (a|q|b c|r|d e|p|f) 6! 0 abcdef 1 1 U = u u σ;a + u σ;aσ;b u σ;aσ;bσ;c 1 1 − 1 a 2! 1 ab − 3! 1 abc 1 + u σ;aσ;bσ;cσ;d + O σ5/2 (124) 4! 1 abcd 1 V = v v σ;a + v σ;aσ;b + O σ3/2 (125) 0 0 − 0 a 2! 0 ab 1/2 V1 = v1 + O σ . (126) and 14

u = (1/2)m2 (1/2)(ξ 1/6)R (128a) 1 − − − u = (1/4)(ξ 1/6)R (128b) 1 a − − ;a u = (1/12)m2R (1/6)(ξ 3/20)R + (1/120)R 1 ab − ab − − ;ab ab (1/12)(ξ 1/6)RR (1/90)Rp R + (1/180)RpqR + (1/180)Rpqr R (128c) − − ab − a pb paqb a pqrb u = (1/8)m2R (1/8)(ξ 2/15)R + (1/80)(R ) 1 abc − (ab;c) − − ;(abc) (ab ;c) (1/8)(ξ 1/6)R R (1/8)(ξ 1/6)RR (1/30)R Rp − − ;(a bc) − − (ab;c) − p(a b;c) p q p q pqr +(1/120)Rpq;(aR b c) + (1/120)RpqR (a b;c) + (1/60)Rpqr(aR b;c) (128d) u = (3/20)m2R (1/24)m2R R (1/30)m2R Rp q 1 abcd − (ab;cd) − (ab cd) − p(a|q|b c d) (1/10)(ξ 5/42)R + (1/70)(R ) (1/6)(ξ 3/20)R R − − ;(abcd) (ab ;cd) − − ;(ab cd) (3/20)(ξ 1/6)RR + (1/120)R R (3/70)Rp R + (1/210)Rp R − − (ab;cd) (ab cd) − (a |p|b;cd) (a bc;d)p +(1/70)R Rp q (2/105)R Rp q + (1/70)R ;pqR + (1/105)R Rp q pq;(ab c d) − p(a;b|q| c d) (ab |p|c|q|d) pq (a b;cd) +(2/105)Rpqr R (1/4)(ξ 1/6)R R (11/420)Rp R (a |pqr|b;cd) − − ;(a bc;d) − (a;b |p|c;d) (3/140)Rp R + (17/1680)R ;pR + (1/60)R Rp q + (1/210)R Rp q − (a;b cd);p (ab cd);p pq;(a b c;d) p(a;|q| b c;d) +(1/56)Rpqr R + (1/280)Rp q ;rR (1/24)(ξ 1/6)RR R (1/90)R Rp R (a;b |pqr|c;d) (a b |p|c|q|d);r − − (ab cd) − (ab c |p|d) +(1/180)R RpqR + (1/90)Rp Rq R (1/30)(ξ 1/6)RRp q R (ab |p|c|q|d) (a b |p|c|q|d) − − (a b |p|c|q|d) +(1/180)R Rpqr R + (1/315)R Rrps R (1/315)R Rr p R q (ab c |pqr|d) p(a b |r|c|s|d) − pq (a b |r|c d) (2/315)R Rp q Rr s (1/315)Rp q Rrs R + (4/315)Rp q R rs R (128e) − prqs (a b c d) − (a b |p|c |rsq|d) (a b |p| c |qrs|d) and v = (1/8)m4 (1/4)(ξ 1/6)m2R + (1/24)(ξ 1/5)R 0 − − − − (1/8)(ξ 1/6)2R2 + (1/720)R Rpq (1/720) R Rpqrs (129a) − − pq − pqrs v = (1/8)(ξ 1/6)m2R + (1/48)(ξ 1/5)(R) 0 a − − ;a − ;a (1/8)(ξ 1/6)2RR + (1/720)R Rpq (1/720)R Rpqrs (129b) − − ;a pq ;a − pqrs ;a and v = (1/48)m4R (1/12)(ξ 3/20)m2R + (1/240)m2R (1/24)(ξ 1/6)m2RR 0 ab − ab − − ;ab ab − − ab (1/180)m2R Rp + (1/360)m2RpqR + (1/360)m2Rpqr R − pa b paqb a pqrb +(1/80)(ξ 4/21)(R) (1/3360)R (1/12)(ξ 1/6)(ξ 3/20)RR − ;ab − ab − − − ;ab +(1/144)(ξ 1/5)(R)R + (1/360)(ξ 1/7)R Rp + (1/240)(ξ 1/6)RR − ab − ;p(a b) − ab +(1/1008)R Rp + (1/1680)RpqR + (1/1260)RpqR (1/1680)RpqR p(a b) pq;(ab) p(a;b)q − ab;pq +(1/180)(ξ 3/14)R;pqR (1/2520)(Rpq)R + (1/630)Rpq;r R − paqb − paqb (a |rqp|b) +(1/420)Rp ;qrR (1/1260)RpqrsR (1/16)(ξ 1/6)2R R (a |pqr|b) − pqrs;(ab) − − ;a ;b (1/120)(ξ 3/14)R Rp + (1/120)(ξ 17/84)R R ;p + (1/1440)Rpq R − − ;p (a;b) − ;p ab ;a pq;b (1/5040)Rp R ;q + (1/1008)Rp Rq (1/2520)Rpq;rR (1/1680)Rpq;rR − a;q pb a;q b;p − rqp(a;b) − paqb;r (1/1344)Rpqrs R (1/1680)Rpqr R ;s − ;a pqrs;b − a;s pqrb (1/48)(ξ 1/6)2R2R (1/180)(ξ 1/6)RR Rp + (1/4320)RpqR R − − ab − − pa b pq ab (1/3780)RpqR R + (1/360)(ξ 1/6)RRpqR + (1/7560)RprRq R − pa qb − paqb r paqb +(1/7560)RpqRr R + (1/360)(ξ 1/6)RRpqr R (1/4320)R RpqrsR (a |rqp|b) − a pqrb − ab pqrs (1/1890)Rp Rqrs R (1/3780)RpqRrs R + (1/1890)R RprqsR − (a |p qrs|b) − pa rsqb pq rasb (1/7560)R Rprs Rq + (1/3780)RpqrsR R t − pq a rsb pqta rs b +(1/378)RprqsRt R (1/3780)Rpqr R Rs t (129c) pqa trsb − s pqrt a b 15 and

v = (1/48)m6 (1/16)(ξ 1/6)m4R + (1/48)(ξ 1/5)m2R 1 − − − − (1/16)(ξ 1/6)2m2R2 + (1/1440)m2R Rpq (1/1440)m2R Rpqrs − − pq − pqrs (1/480)(ξ 3/14)R + (1/48)(ξ 1/6)(ξ 1/5)RR (1/720)(ξ 3/14)R Rpq − − − − − − ;pq (1/5040)R Rpq + (1/840)R Rprqs + (1/96)[ξ2 (2/5)ξ + 17/420]R R;p − pq pq;rs − ;p (1/20160)R Rpq;r (1/10080)R Rpr;q + (1/4480)R Rpqrs;t − pq;r − pq;r pqrs;t (1/48)(ξ 1/6)3R3 + (1/1440)(ξ 1/6)RR Rpq + (1/45360)R Rp Rqr − − − pq pq r (1/15120)R R Rprqs (1/1440)(ξ 1/6)RR Rpqrs + (1/2160)R Rp Rqrst − pq rs − − pqrs pq rst (1/5670)R RpquvRrs (11/11340)R Rp q Rrusv. (130) − pqrs uv − prqs u v

2 M The geometrical tensor Θµν which is associated with the renormalization mass is obtained from (79) by using (129a), (129c) and (130) and is given by

2 2 ln M ΘM = (1/8)(ξ 1/6)m4R (1/4)[ξ2 (1/3)ξ +1/30]m2R + (1/240)m2R µν 2 (2π)3 − µν − − ;µν µν h +(1/4)(ξ 1/6)2m2RR (1/180)m2R Rp + (1/360)m2RpqR + (1/360)m2Rpqr R − µν − pµ ν pµqν µ pqrν +(1/24)[ξ2 (2/5)ξ +3/70](R) (1/3360)R (1/4)(ξ 1/6)[ξ2 (1/3)ξ +1/30]RR − ;µν − µν − − − ;µν (1/24)(ξ 1/6)(ξ 1/5)(R)R + (1/360)(ξ 1/7)R Rp + (1/240)(ξ 1/6)RR − − − µν − ;p(µ ν) − µν +(1/1008)R Rp + (1/360)(ξ 2/7)RpqR + (1/1260)RpqR (1/1680)RpqR p(µ ν) − pq;(µν) p(µ;ν)q − µν;pq +(1/180)(ξ 3/14)R;pqR (1/2520)(Rpq)R + (1/630)Rpq;r R − pµqν − pµqν (µ |rqp|ν) +(1/420)Rp ;qrR (1/360)(ξ 3/14)RpqrsR (1/4)(ξ 1/6)2(ξ 1/4)R R (µ |pqr|ν) − − pqrs;(µν) − − − ;µ ;ν (1/120)(ξ 3/14)R Rp + (1/120)(ξ 17/84)R R ;p + (1/360)(ξ 1/4)Rpq R − − ;p (µ;ν) − ;p µν − ;µ pq;ν (1/5040)Rp R ;q + (1/1008)Rp Rq (1/2520)Rpq;rR (1/1680)Rpq;rR − µ;q pν µ;q ν;p − rqp(µ;ν) − pµqν;r (1/360)(ξ 13/56)Rpqrs R (1/1680)Rpqr R ;s − − ;µ pqrs;ν − µ;s pqrν +(1/8)(ξ 1/6)3R2R (1/180)(ξ 1/6)RR Rp (1/720)(ξ 1/6)RpqR R − µν − − pµ ν − − pq µν (1/3780)RpqR R + (1/360)(ξ 1/6)RRpqR + (1/7560)RprRq R − pµ qν − pµqν r pµqν +(1/7560)RpqRr R + (1/360)(ξ 1/6)RRpqr R + (1/720)(ξ 1/6)R RpqrsR (µ |rqp|ν) − µ pqrν − µν pqrs (1/1890)Rp Rqrs R (1/3780)RpqRrs R + (1/1890)R RprqsR − (µ |p qrs|ν) − pµ rsqν pq rµsν (1/7560)R Rprs Rq + (1/3780)RpqrsR R t − pq µ rsν pqtµ rs ν +(1/378)RprqsRt R (1/3780)Rpqr R Rs t pqµ trsν − s pqrt µ ν +g (1/48)m6 (1/16)(ξ 1/6)m4R + (1/4)[ξ2 (1/3)ξ +1/40]m2R µν − − − − (1/16)( ξ 1/6)2m2R2 + (1/1440)m2R Rpq (1/1440)m2R Rpqrs − − pq − pqrs (1/24)[ξ2 (2/5)ξ + 11/280]R + (1/4)(ξ 1/6)[ξ2 (1/3)ξ +1/40]RR − − − − (1/720)(ξ 3/14)R Rpq (1/360)(ξ 5/28)R Rpq + (1/90)(ξ 1/7)R Rprqs − − ;pq − − pq − pq;rs +(1/4)[ξ3 (13/24)ξ2 + (17/180)ξ 53/10080]R R;p − − ;p (1/360)(ξ 13/56)R Rpq;r (1/10080)R Rpr;q + (1/360)(ξ 19/112)R Rpqrs;t − − pq;r − pq;r − pqrs;t (1/48)(ξ 1/6)3R3 + (1/1440)(ξ 1/6)RR Rpq + (1/45360)R Rp Rqr − − − pq pq r (1/15120)R R Rprqs (1/1440)(ξ 1/6)RR Rpqrs + (1/180)(ξ 1/6)R Rp Rqrst − pq rs − − pqrs − pq rst (1/360)(ξ 47/252)R RpquvRrs (1/90)(ξ 41/252)R Rp q Rrusv . (131) − − pqrs uv − − prqs u v i 16

2 The trace anomaly (83) is obtained by using m = 0 and ξ = ξc(6) = 1/5 into (130). It reduces to 1 ψ T µ ψ = [(1/33600)R + (1/50400)R Rpq (1/5040)R Rpq + (1/840)R Rprqs h | µ| iren (2π)3 ;pq − pq pq;rs +(1/201600)R R;p (1/20160)R Rpq;r (1/10080)R Rpr;q + (1/4480)R Rpqrs;t ;p − pq;r − pq;r pqrs;t (1/1296000)R3 + (1/43200)RR Rpq + (1/45360)R Rp Rqr (1/15120)R R Rprqs − pq pq r − pq rs (1/43200)RR Rpqrs + (1/2160)R Rp Rqrst (1/5670)R RpquvRrs (11/11340)R Rp q Rrusv]. − pqrs pq rst − pqrs uv − prqs u v (132)

F. D=7,8,9,10,11 IV. IMPORTANT REMARKS AND COMPLEMENTS

The complexity of the explicit expressions of In this section, we shall complete our study by dis- 2 cussing some aspects of the Hadamard renormalization GF (x, x′) and of the geometrical tensor ΘM greatly in- sing µν of the stress-energy tensor which are more or less directly creases with the dimension D of spacetime. That clearly related to the explicit calculations described in Secs. II appears in the previous subsections. For this reason, and III. They are helpful in order to simplify some of we cannot write them explicitly for spacetime dimen- the results displayed above. Furthermore, they permit sion from D = 7 to D = 11 even though we have at us to discuss more generally the ambiguity problem and our disposal all the tools permitting us to carry out all the trace anomaly as well as to clarify the links existing the necessary calculations. Indeed, in the appendices between the Hadamard formalism and the more popular of Ref. [71], we have obtained the covariant Taylor se- method based on regularization and renormalization in ries expansions of the Van Vleck -Morette determinant ′ 1/2 ′ 11/2 the effective action. U0(x, x ) = ∆ (x, x ) up to order σ and of the bitensor σµν (x, x′) up to order σ9/2. We have also developed the general theory permitting us to construct the covariant derivative and the d’Alembertian of an arbitrary biscalar F (x, x′) symmetric in the exchange of x ′ and x . From a theoretical point of view, all these results A. Ambiguities and trace anomaly could permit us to solve the recursion relations (37) and (38) for D even and the recursion relations (42) for D odd and therefore to obtain the explicit expressions of As already noted in Sec. II, the renormalized expec- F ′ Gsing(x, x ) up to the required order and of the geometri- tation value ψ Tµν ψ ren is unique up to the addition 2 h | | i M of a local conserved tensor Θµν . For D even, we have cal tensor Θµν when necessary. Of course, this could be realized but at the cost of odious calculations in a general been able to construct the standard ambiguity associ- spacetime. ated with the choice of the renormalization mass M [see Eq. (79)] and, in Sec. III, we have explicitly obtained its expression for D = 2, 4 and 6 [see Eqs. (92),(110) and By contrast, in a given spacetime, i.e. if we know ex- (131)]. In the present subsection, following Wald’s argu- plicitly the Riemann tensor Rµνρσ and therefore the Ricci ments of Ref. [13], we shall push further our discussion tensor Rµν and the scalar curvature R, interesting sim- and provide for D = 2, 3, 4, 5 and 6, the bases (i.e., all F ′ plifications may occur, the construction of Gsing(x, x ) the independent conserved local tensors) permitting us 2 M to constructed the most general expression for the tensor and of Θµν done explicitly and the renormalization of Θµν . Here, we adhere to a conventional point of view the expectation value of the stress-energy tensor “easily” 2 achieved. For example, in D-dimensional Schwarzschild [13] by discarding ambiguities diverging as m 0. It should be however noted that a less conventional→ point black hole spacetimes where we have R = 0, Rµν = 0 and more generally in Ricci-flat spacetimes, consider- of view has been considered by Tichy and Flanagan in able simplifications could permit us to obtain explic- Ref. [82]. 2 F ′ M itly Gsing(x, x ) and Θµν even for D > 6. This cer- In order to extend Wald’s arguments, it is important tainly also happens in D-dimensional spacetimes such as to keep in mind that Θµν is a local conserved tensor of AdS S with p + q = D where the covariant derivative dimension (mass)D and that it can be obtained by func- p × q of the Riemann tensor vanishes (Rµνρσ;τ = 0) as well tional derivation with respect to the metric tensor from a as in D-dimensional de Sitter and Anti-de Sitter space- geometrical Lagrangian of dimension (mass)D. We note times, i.e. in maximally symmetric spacetimes, where also that gµν is dimensionless while R, Rµν and Rµνρσ R = [R/D(D 1)](g g g g ) with R = Cte. have dimension (mass)2. µνρσ − µρ νσ − µσ νρ 17

1. D = 2 1 δ H(4,2)(2) dDx√ g R Rpq (137a) For D = 2, there are only two “independent” geometri- µν ≡ √ g δgµν − pq cal Lagrangians of dimension (mass)2 which remain ﬁnite − ZM = R R 2 RpqR in the massless limit: = m2 and = R. However, by ;µν − µν − pµqν L L +g [ (1/2) R + (1/2) R Rpq], functional derivation, the latter does not provide any con- µν − pq tribution to Θµν because, in two dimensions, the Euler (137b) number

d2x√ gR (133) 1 δ (4,2)(3) D √ pqrs M − Hµν µν d x g RpqrsR (138a) Z ≡ √ g δg M − − Z p pq is a topological invariant. Θ is then necessarily pro- = 2 R;µν 4 Rµν +4R Rpν 4R Rpµqν µν − µ − portional to the functional derivative of = m2 and 2Rpqr R + g [(1/2) R Rpqrs]. L − µ pqrν µν pqrs therefore of the form (138b) 2 Θµν = Am gµν (134) Θµν is therefore necessarily of the form where A is a dimensionless constant. Θ = Am4g + Bm2[R (1/2)Rg ] µν µν µν − µν It is interesting to note that Θµν given by (134) van- (4,2)(1) (4,2)(2) (4,2)(3) +C1 H + C2 H + C3 H ishes for m2 = 0 and therefore does not modify the trace µν µν µν anomaly (93). (139)

where A, B, C1, C2 and C3 are dimensionless constants. Here, it should be also noted that it is possible to simplify 2. D = 3 the previous expression because, in a four-dimensional background, the Euler number For D = 3, there are only two independent geometrical 3 4 Lagrangians of dimension (mass) which remain finite d x√ g (2), (140) in the massless limit: = m3 and = mR. So, it M − L L L Z is natural to consider that Θµν is necessarily a linear where is the quadratic Gauss-Bonnet Lagrangian 3 (2) combination of their functional derivatives m gµν /2 and given byL m[(1/2)Rgµν Rµν ], i.e. that − 2 pq pqrs (2) = R 4RpqR + RpqrsR , (141) 3 2 L − Θµν = Am gµν ++Bm [Rµν (1/2)Rgµν ] (135) − is a topological invariant. By functional derivation of where A and B are dimensionless constants. (140) we obtain It should be noted that Θ given by (135) vanishes for µν H(4,2)(1) 4 H(4,2)(2) + H(4,2)(3) =0 (142) m = 0. Thus, it cannot be used in order to modified (84). µν − µν µν In other words, the trace anomaly does not exist for D = which could be helpful in order to eliminate one of the 3 even if we take into account the possible ambiguities of three conserved tensors of rank 2 and order 4 into (139). the Hadamard renormalization process. In other words, without loss of generality it is possible to use C1 =0 or C2 =0 or C3 = 0 into (139). It should be noted that the “basis” exhibited above 3. D = 4 which has permitted us to provide the general form for the tensor Θµν can be used to simplify considerably 2 M For D = 4, there are five “independent” geometrical the expression (110) obtained for Θµν . Indeed, from Lagrangians of dimension (mass)4 which remain finite Eqs. (136)-(138), we can write in the massless limit: = m4, = m2R, = R2, 2 pq L pqrs L L 2 ln M = RpqR and = RpqrsR . By functional deriva- ΘM = +(1/2)(ξ 1/6)2 H(4,2)(1) tionL with respectL to the metric tensor, they define the µν (4π)2 × − µν 4 2 conserved tensors m gµν /2, m [(1/2)Rgµν Rµν ] as well (1/180) H(4,2)(2) + (1/180) H(4,2)(3) − − µν µν as the three conserved tensors of rank 2 and order 4 2 (ξ 1/6)m [Rµν (1/2)Rgµν] 1 δ − − − (4,2)(1) D √ 2 4 Hµν µν d x g R (136a) +(1/4)m gµν . (143) ≡ √ g δg M − − Z = 2 R;µν 2 RRµν − Finally, it is interesting to note that Θµν given by (139) +g [ 2 R + (1/2) R2], (136b) can be used in order to modify the trace anomaly (111). µν − 18

2 pq 3 Indeed, for m = 0 and by using Eqs. (A1)-(A3) with (see Refs. [73, 74]) = RR, = RpqR , = R , D = 4, we obtain = RR Rpq, =LR Rp Rqr,L = R R RprqsL , = L pq L pq r L pq rs L RR Rpqrs, = R Rp Rqrst, = R RpquvRrs , µν pqrs L pq rst L pqrs uv g Θµν = [ 6C1 2C2 2C3] R. (144) = R Rp q Rrusv. By functional derivation, they de- − − − L prqs u v fine the conserved tensors m6g /2, m4[(1/2)Rg R ] For example, by taking C = 1/4320(2π)2 and C = µν µν µν 1 2 and the three conserved tensors of rank 2 and order− 4 C = 0, we can remove the R term from (111). This (4,2)(1) (4,2)(2) (4,2)(3) 3 m2H , m2H and m2H as well as the elimination can be achieved by adding a finite R2 term µν µν µν ten conserved tensors of rank 2 and order 6 to the gravitational Lagrangian [see Eq. (136)] and is in accordance with the discussion we shall develop in {2,0}(1) 1 δ D pq H d x√ g RR (146) Sec. IV.B. On the contrary, the RpqR term and the µν ≡ √ g δgµν − pqrs − ZM RpqrsR term cannot be modified. We refer to Sec. 6.3 1 δ of Ref. [1] for various physical comments concerning the H{2,0}(3) dDx√ g R Rpq (147) µν ≡ √ g δgµν − pq possible modifications of the trace anomaly in a four di- − ZM 1 δ mensional gravitational background. H(6,3)(1) dDx√ g R3 (148) µν ≡ √ g δgµν − − ZM 1 δ D H(6,3)(2) dDx√ g RR Rpq (149) 4. = 5 µν ≡ √ g δgµν − pq − ZM (6,3)(3) 1 δ D p qr For D = 5, there are five independent geometrical La- H d x√ g RpqR R (150) µν ≡ √ g δgµν − r grangians of dimension (mass)5 which remain finite in − ZM 5 3 2 1 δ the massless limit: = m , = m R, = mR , H(6,3)(4) dDx√ g R R Rprqs(151) pq L L pqrs L µν µν pq rs = mRpqR and = mRpqrsR . By functional ≡ √ g δg M − L L 5 − Z derivation, they define the conserved tensors m gµν /2, (6,3)(5) 1 δ D pqrs 3 H d x√ g RR R (152) m [(1/2)Rg R ] as well as the three conserved ten- µν µν pqrs µν µν ≡ √ g δg M − − (4,2)(1) (4,2)(2) Z sors of rank 2 and order 4 mH , mH and − µν µν (6,3)(6) 1 δ D p qrst (4,2)(3) Hµν d x√ g RpqR rstR mHµν .Θµν is therefore necessarily of the form ≡ √ g δgµν − − ZM 5 3 (153) Θµν = Am gµν + Bm [Rµν (1/2)Rgµν] − (6,3)(7) 1 δ D pquv rs (4,2)(1) (4,2)(2) (4,2)(3) H d x√ g RpqrsR R +C1 mHµν + C2 mHµν + C3 mHµν µν ≡ √ g δgµν − uv − ZM (145) (154)

(6,3)(8) 1 δ D p q rusv where A, B, C1, C2 and C3 are dimensionless con- √ Hµν µν d x g RprqsR u vR . (4,2)(1) (4,2)(2) ≡ √ g δg M − stants. Here, the conserved tensors Hµν , Hµν − Z (4,2)(3) (155) and Hµν are still respectively defined by Eqs. (136a), (137a) and (138a) but now, in these equations, D = 4 Here we do not provide the explicit expressions of these must be replaced by D = 5. Their explicit expressions ten tensors. They are very complicated ones and can be (136b), (137b) and (138b) remain unchanged. For D = 5, found in Ref. [89] [see Eqs. (2.22)-(2.31) of this article]. it is not possible to simplify Eq. (145) by using (142). In- (4,2)(1) (4,2)(2) deed, for D > 4 this topological constraint is not valid As far as the conserved tensors Hµν , Hµν and (4,2)(3) because the Euler number (140) does not remain a topo- Hµν are concerned, they are still respectively defined logical invariant. by Eqs. (136a), (137a) and (138a) but now, in these equa- It should be noted that Θµν given by (145) vanishes for tions, D = 4 must be replaced by D = 6. Their explicit m = 0. Thus, it cannot be used in order to modified (84). expressions (136b), (137b) and (138b) remain unchanged. In other words, the trace anomaly does not exist for D = Θµν is therefore necessarily of the form 5 even if we take into account the possible ambiguities of 6 4 the Hadamard renormalization process. Θµν = Am gµν + Bm [(1/2)Rgµν Rµν ] − 2 (4,2)(1) 2 (4,2)(2) 2 (4,2)(3) +C1 m Hµν + C2 m Hµν + C3 m Hµν {2,0}(1) {2,0}(3) (6,3)(1) 5. D = 6 +D1 Hµν + D2 Hµν + D3 Hµν (6,3)(2) (6,3)(3) (6,3)(4) +D4 Hµν + D5 Hµν + D6 Hµν For D = 6, there are fifteen “independent” geometri- (6,3)(5) (6,3)(6) (6,3)(7) cal Lagrangians of dimension (mass)6 which remain fi- +D7 Hµν + D8 Hµν + D9 Hµν 6 4 (6,3)(8) nite in the massless limit: = m , = m R and +D10 Hµν (156) the three Riemann polynomialsL of rankL 0 and order 4 2 2 2 pq 2 pqrs = m R , = m RpqR , = m RpqrsR as well where A, B, C1, C2 and C3 as well as D1, ... D9 and D10 Las the ten RiemannL monomialsL of rank 0 and order 6 are dimensionless constants. Finally, it should be noted 19 that it is possible to simplify the previous expression for Θµν can be used to simplify considerably the expression 2 Θ because, in a six-dimensional background, the Euler M µν (131) obtained for Θµν . By using Eqs. (2.22)-(2.31) of number Ref. [89] and after a tedious calculation, we obtain the compact expression d6x√ g , (157) − L(3) M 2 2 Z M ln M Θµν = 3 where (3) is the cubic Lovelock Lagrangian explicitly (4π) × given byL [(1/12) ξ2 (1/30) ξ +1/336] H{2,0}(1) − µν 3 pq p qr (3) = R 12 RRpqR + 16 RpqR rR {2,0}(3) 3 (6,3)(1) L − +(1/840) Hµν (1/6)(ξ 1/6) Hµν p − − +24 R R Rprqs +3 RR Rpqrs 24 R R Rqrst pq rs pqrs pq rst +(1/180)(ξ 1/6) H(6,3)(2) (4/2835) H(6,3)(3) pquv rs p q− rusv µν µν +4 RpqrsR R 8 RprqsR R , (158) − − uv − u v +(1/945) H(6,3)(4) (1/180)(ξ 1/6) H(6,3)(5) µν − − µν is a topological invariant. By functional derivation of +(1/7560) H(6,3)(6) + (17/45360) H(6,3)(7) (157) we obtain the relation µν µν (6,3)(8) 2 2 (4,2)(1) (1/1620) Hµν (1/2)(ξ 1/6) m Hµν (6,3)(1) (6,3)(2) (6,3)(3) − − − Hµν 12 Hµν + 16 Hµν +(1/180)m2 H(4,2)(2) (1/180)m2 H(4,2)(3) − µν − µν +24 H(6,3)(4) +3 H(6,3)(5) 24 H(6,3)(6) 4 µν µν − µν +(1/2)(ξ 1/6)m [Rµν (1/2)Rgµν] (6,3)(7) (6,3)(8) − − +4 Hµν 8 Hµν =0. (159) (1/12)m6g . (160) − − µν Equations (159) could be helpful in order to eliminate into (156) one of the conserved tensors of rank 2 and Finally, it is interesting to note that Θµν given by (156) order 6. could permit us to modify the trace anomaly (132). In- Of course, the “basis” exhibited above and which has deed, for m2 = 0 and by using Eqs. (A4)-(A13) with permitted us to provide the general form for the tensor D = 6, we obtain

gµν Θ = [ 10D 3D ]R + [ 2D 30D 4D D /2 2D ]RR µν − 1 − 2 − 1 − 3 − 4 − 6 − 7 +[ 8D 4D 6D +2D 4D D +3D /2]R Rpq − 2 − 4 − 5 6 − 7 − 8 10 ;pq +[2D 10D 3D 6D 2D 3D ]R Rpq 2 − 4 − 5 − 6 − 8 − 10 pq +[8D 4D 40D 14D 24D +3D ]R Rprqs 2 − 6 − 7 − 8 − 9 10 pq;rs +[ 2D 3D /2 30D 6D 3D /2 3D /4 4 D7 D /2]R R;p − 1 − 2 − 3 − 4 − 5 − 6 − − 8 ;p +[10D 10D 3D 10D 8D 12D 3D ]R Rpq;r 2 − 4 − 5 − 6 − 8 − 9 − 10 pq;r +[ 18D 6D +9D +6D + 12D +3D ]R Rpr;q + [ 10D 2D 3D +3D /4] R Rpqrs;t − 2 − 5 6 8 9 10 pq;r − 7 − 8 − 9 10 pqrs;t +[ 10D 6D +5D +3D ] (R Rp Rqr R R Rprqs) − 2 − 5 6 10 pq r − pq rs +[ 10D 2D 3D +3D /4](2R Rp Rqrst R RpquvRrs 4R Rp q Rrusv). (161) − 7 − 8 − 9 10 pq rst − pqrs uv − prqs u v

It should be noted that three of the ten scalar Rie- erator for a massive scalar field in a general spacetime 3 pq mann monomials of order 6, namely R , RRpqR and of arbitrary dimension by assuming that the Wald’s ax- pqrs RRpqrsR , do not appear in (161). As a consequence, iomatic approach (see Refs. [3, 9, 13]) remains valid for all it is impossible to remove such terms from the trace dimensions. In the Wald’s axiomatic approach, the treat- anomaly (132). On the contrary, by choosing correctly ment of the divergences present in the formal expression the coefficients Di, it is possible to remove any other term (60) does not necessitate a particular study i.e. absorb- from (132). tion into renormalized gravitational parameters. These divergences are simply discarded and the cosmological constant Λ and the Newton’s gravitational constant G B. Infinities and gravitational actions (as well as the other coupling constants associated with higher-order gravitational terms if we need to consider such terms) appearing in the semiclassical Einstein equa- In the previous sections, we have constructed the tions (1) are directly the physical gravitational parame- renormalized expectation value of the stress-energy op- 20 ters while the expectation value ψ Tµν ψ ren constructed For D = 3, we obtain for the averaged divergent part from the Hadamard biscalar Wh(x,| x′)| isi automatically of (60) an expression of the form the physically meaningful source. gµν [Rµν (1/2)Rgµν] In the present subsection, we shall depart from the ψ Tµν ψ sing A + B − path marked out by Wald. We shall briefly describe one h | | i ∼ σ3/2 σ1/2 3 way to deal with the divergent part of (60) by extending + finite terms in m gµν and m[Rµν (1/2)Rgµν]. − the approach developed by Christensen in Refs. [11, 12] (165) (see also Adler and coworkers in Refs. [14, 15] for a related but slightly different approach). We intend to dis- Here A and B are dimensionless constants. This singular cuss at more length this very technical aspect of our work tensor can be absorbed into a bare gravitational action in a paper in preparation [90]. However, in order to be given by as completed as possible, we shall here provide partial 1 results related to the present work. S = d3x√ g (R 2Λ ) . (166) grav −16πG − − B For a given spacetime dimension, we can formally eval- B ZM uate the divergent part of (60) and express the result of For D = 4, we obtain for the averaged divergent part our calculation as a power series in σ;a(x, x′). By consis- of (60) an expression of the form tently averaging this power series over all the angular di- rections joining x′ and x and by adding to it, for D even, gµν [Rµν (1/2)Rgµν] ψ Tµν ψ sing A 2 + B − the opposite of (79) as well as (D/4)αDgµν v1, we find a h | | i ∼ σ σ final divergent expression constructed− from “simple” con- (4,2)(1) (4,2)(2) (4,2)(3) 2 + C1Hµν + C2Hµν + C3Hµν ln(M σ) served geometrical tensors which can be absorbed into a 4 2 bare gravitational Lagrangian. It is important to note + finite terms in m gµν ,m [Rµν (1/2)Rgµν], − that the averaging process of the direction-dependant H(4,2)(1), H(4,2)(2) and H(4,2)(3). (167) terms adopted in Refs. [11, 12, 14, 15]) must be mod- µν µν µν ified in order to take into account spacetime dimension. Here A, B, C1, C2 and C3 are dimensionless constants. For D = 2, we obtain for the averaged divergent part This singular tensor can be absorbed into a bare gravi- of (60) an expression of the form tational action given by 1 gµν 2 2 4 ψ Tµν ψ sing A + [B1m gµν + B2Rgµν ] ln(M σ) Sgrav = d x√ g R 2ΛB h | | i ∼ σ −16πGB M − − 2 Z + finite terms in m gµν and Rgµν . (162) (1) 2 (2) pq (3) pqrs +αB R + αB RpqR + αB RpqrsR . Here A, B1 and B2 are dimensionless constants. This (168) singular tensor cannot be absorbed into a bare gravitational Lagrangian of Einstein-Hilbert type because, in In this bare gravitational action, the term in (3) pqrs two dimensions, the Euler number (133) being a topo- αB RpqrsR could be removed because, as we have logical invariant, the tensor Rµν (1/2)Rgµν vanishes already noted, the Euler number (140) is a topological identically. However, this tensor− can be absorbed into invariant in four dimensions. Similarly, it would have (4,2)(3) the Polyakov non-local bare gravitational action been possible to remove the Hµν term from (167). For D = 5, we obtain for the averaged divergent part 2 1 of (60) an expression of the form Sgrav = d x√ g aBR R 2ΛB . (163) M − − Z gµν [Rµν (1/2)Rgµν] ψ Tµν ψ sing A + B − It should be noted that the non-local Lagrangian = h | | i ∼ σ5/2 σ3/2 1 L (4,2)(1) (4,2)(2) (4,2)(3) R R provides, by functional derivation with respect to C1Hµν + C2Hµν + C3Hµν the metric tensor, a contribution in 2Rgµν but also a + 1/2 non-local contribution proportional to σ + finite terms in m5g ,m3[R (1/2)Rg ], µν µν − µν 1 1 1 mH(4,2)(1), mH(4,2)(2) and mH(4,2)(3). (169) 2 R + R R µν µν µν − ;µν ;µ ;ν Here A, B, C , C and C are dimensionless constants. 1 1 1 ;p 1 2 3 g R R . (164) This singular tensor can be absorbed into a bare gravi- −2 µν ;p tational action given by

We think that these two contributions must be added to 1 5 Sgrav = d x√ g R 2ΛB (134). In the particular case of a two-dimensional back- −16πGB M − − Z ground, it is not natural to follow Wald’s prescription (1) 2 (2) pq (3) pqrs +αB R + αB RpqR + αB RpqrsR . and to construct the conserved tensor Θµν from a purely local Lagrangian. (170) 21

Of course, this bare gravitational action cannot be simpli- are formal relations: they display on a very condensed ﬁed because, for D> 4, the Euler number (140) does not form the true behavior of the averaged divergent part of (4,2)(3) remain a topological invariant. Similarly, the Hµν (60) in the limit of small σ (for more details, we refer to term cannot be removed from (169). Ref. [90]). Finally, for D = 6, we obtain for the averaged divergent C. Hadamard renormalization versus part of (60) an expression of the form renormalization in the eﬀective action

gµν [Rµν (1/2)Rgµν] Field quantization in curved spacetime can be ad- ψ T ψ A + B − h | µν | ising ∼ σ3 σ2 dressed very efficiently by using the effective action (4,2)(1) (4,2)(2) (4,2)(3) [80, 88, 91]. This basic object contains, in principle, C1Hµν + C2Hµν + C3Hµν + all the information about a given quantum field the- σ ory but, unfortunately, it is not usually possible to ex- {2,0}(1) {2,0}(3) (6,3)(1) press it explicitly. Even in the very simple case of the + D1Hµν + D2Hµν + D3Hµν scalar field theory considered in the present article, we (6,3)(2) (6,3)(3) (6,3)(4) +D4Hµν + D5Hµν + D6Hµν have only an approximation for the associated effective action, the so-called DeWitt-Schwinger approximation +D H(6,3)(5) + D H(6,3)(6) + D H(6,3)(7) 7 µν 8 µν 9 µν [1, 80, 88, 91, 92, 93], which may be represented by the (6,3)(8) 2 +D10Hµν ln(M σ) asymptotic series [88] + finite terms in m6g ,m4[R (1/2)Rg ], µν µν − µν W DS = dDx g(x) 2 (4,2)(1) 2 (4,2)(2) 2 (4,2)(3) − × m Hµν ,m Hµν ,m Hµν , ZM +p∞ 2 {2,0}(1) {2,0}(3) (6,3)(1) (6,3)(2) (6,3)(3) 1 d(is) −m is Hµν , Hµν , Hµν , Hµν , Hµν , e Λ(x; s) (173) 2(4π)D/2 (is)D/2+1 (6,3)(4) (6,3)(5) (6,3)(6) (6,3)(7) 0 Hµν , Hµν , Hµν , Hµν Z and H(6,3)(8). (171) where Λ(x; s) is a purely geometrical object (see Ref. [88]) µν which satisfies

Here A, B, C1, C2, C3, D1, ..., D9 and D10 are dimen- 2 lim e−m isΛ(x; s)=0 (174) sionless constants. This singular tensor can be absorbed s→+∞ into a bare gravitational action given by and which can be formally written for s 0 on the form 1 → 6 √ Sgrav = d x g R 2ΛB +∞ −16πGB − − ZM k (1) 2 (2) pq (3) pqrs Λ(x; s) ak(x)(is) . (175) +α R + α RpqR + α RpqrsR ≈ B B B k=0 (1) 3 (2) pq (3) p qr X +βB R + βB RRpqR + βB RpqR rR Here, ak(x) are the diagonal DeWitt coefficients. The (4) prqs (5) pqrs +βB RpqRrsR + βB RRpqrsR four first ones can be found in Refs. [88, 94, 95] and, +β(6)R Rp Rqrst + β(7)R RpquvRrs because we have previously assumed that spacetime has B pq rst B pqrs uv no boundary, we have for their global (or integrated) ex- (8) p q rusv +βB RprqsR u vR . (172) pressions This bare gravitational action could be simplified by dDx√ g a = dDx√ g, (176) − 0 − using the fact that the Euler number (157) is a topo- ZM ZM logical invariant. This result could be used to re- D D d x√ g a1 = d x√ g [ (ξ 1/6) R] , (177) move from the bare gravitational action a term such as M − M − − − (8) p q rusv Z Z βB RprqsR u vR . Similarly, it would have been pos- D D 2 2 (6,2)(8) d x√ g a2 = d x√ g (1/2) (ξ 1/6) R sible to remove the Hµν term from (171). M − M − − Z Zpq pqrs To conclude this subsection, it is important to note (1/180) RpqR + (1/180)RpqrsR ] (178) that the previous results must be taken with a grain of − salt. Indeed, Eqs. (162), (165), (167), (169) and (171) and 22

dDx√ g a = dDx√ g [(1/12) ξ2 (1/30) ξ +1/336] RR + (1/840) R Rpq − 3 − − pq ZM ZM (1/6) (ξ 1/6)3 R3 + (1/180)( ξ 1/6) RR Rpq (4/2835) R Rp Rqr + (1/945) R R Rprqs − − − pq − pq r pq rs (1/180)(ξ 1/6) RR Rpqrs + (1/7560) R Rp Rqrst + (17/45360) R RpquvRrs − − pqrs pq rst pqrs uv (1/1620) R Rp q Rrusv . (179) − prqs u v

The DeWitt-Schwinger representation (173) of the effec- use of the Hadamard method we have developed. tive action is a purely local geometrical object which con- First, it is important to note that the effective action tains all the information on the ultraviolet behavior of W DS is divergent at the lower limit of the integral over s the quantum theory of the scalar field obeying the wave for all the positive values of the dimension D. For D =2 equation (5) but which does not take into account its and D = 3, this divergent behavior is associated with the state-dependence. By functional derivation of (173) with integrated DeWitt coefficients (176) and (177); for D =4 respect to the metric tensor, we can construct the formal and D = 5, it is associated with the integrated DeWitt stress-energy tensor coefficients (176), (177) and (178); and for D = 6, it is associated with the integrated DeWitt coefficients (176), 2 δW DS T DS = . (180) (177), (178) and (179). As a consequence, from (180) and h µν i √ g δgµν − by using (136)-(138) and (146)-(155), we can very easily Of course, it is also a purely local geometrical object obtain results analogous to those described in Sec. IV.B which is furthermore state-independent. However, in concerning the formal expression of the divergent part of spite of this last drawback, it has been extensively used, the stress-energy tensor. in the four-dimensional context, in order i) to understand The treatment of the divergent behavior of (180) can the regularization and renormalization of the true (i.e., be achieved by first regularizing the effective action (173), state-dependent) stress-energy tensor (see, for example, then by absorbing its divergent part into a bare grav- Ref. [1]) or ii) to provide approximations valid in the large itational action and finally by functionally deriving the mass limit for this true stress-energy tensor (see, for ex- renormalized effective action so obtained. By considering ample, Refs. [89, 91, 96, 97]). In the present subsection, the dimensionality D of spacetime as a complex number, we shall briefly discuss some aspects of the renormal- the effective action W DS can be regularized by analytic ization of the formal stress-energy tensor (180) directly continuation and its divergent part can be extracted co- linked to our propose in order to shed light, from a dif- herently and naturally absorbed into a bare gravitational ferent point of view, on the results obtained above but action [1, 88]. The resulting renormalized effective action also to advocate, with in mind practical applications, the can be written in the form

+∞ D/2+1 1 ∂ 2 W DS = dDx g(x) d(is) ln (4πM 2is) e−m isΛ(x; s) (181) ren − × −2[(D/2)!](4π)D/2 ∂ (is) M " 0 # Z p Z h i for D even and in the form

+∞ D/2+1/2 1 d(is) ∂ 2 W DS = dDx g(x) e−m isΛ(x; s) (182) ren − × 2[Γ(D/2)/√π](4π)D/2 (is)1/2 ∂ (is) M " 0 # Z p Z h i for D odd. Formulas (181) and (182) generalize results (180), we can obtain a renormalized stress-energy tensor displayed in Ref. [88] for D =2, 3, 4. In Eq. (181), M is for D even or D odd. Of course, the object calculated an arbitrary mass scale parameter (the renormalization in that way is only a state-independent approximation mass) which is necessary in dimensional regularization of the true expectation value of the stress-energy opera- because only dimensionless quantities can be analytically tor. Furthermore, because in order to obtain (181) and continued. M remains in the renormalized effective ac- (182) we have discarded not only infinite terms involv- tion for D even. Now, by inserting (181) or (182) into ing the integrated DeWitt coefficients but also finite ones 23 which have been absorbed by finite renormalization, this us to understand the structure of the ultraviolet diver- object is also ambiguously defined. The corresponding gences contained in the unrenormalized expression of the ambiguities are obtained by functional derivation of the stress-energy tensor and to discuss the ambiguity prob- integrated DeWitt coefficients and are those displayed in lem. Because it uses functional derivation with respect to Sec. IV.A. the metric instead of the point-splitting method, it per- Let us now consider the part of the renormalized ef- mits us to obtain very easily the results mentioned above fective action (181) associated with the renormalization with a formalism which is rather independent of the di- mass M. By using (174) we obtain for its expression mension of spacetime. Hadamard formalism, if we depart from the axiomatic point of view advocated by Wald, 2 DS D ln M does not seem so interesting. Unfortunately, calculations W 2 = d x g(x) M − 2[(D/2)!](4π)D/2 based on renormalization in the effective action cannot ZM D/p2 permit us to take into account the state-dependence of ∂ 2 e−m isΛ(x; s) (183) the considered quantum theory and therefore to obtain, ∂ (is) in a general framework, the full renormalized expecta- #s=0 h i tion value of the stress-energy operator. In fact, bearing and, from (180) and (175), it provides a geometrical am- in mind practical calculations, Hadamard formalism is biguity associated with the stress-energy tensor given by much more efficient than the method based on renormalization in the effective action even if, at first sight and 2 2 ln M 2 δ because of its use of the point-splitting method, it seems ΘM = dDx g(x) µν 2(4 π)D/2 × √ g δgµν − × rather heavier. It is also important to note that, in the M − Z p present article, we have achieved the major part of the D/2 k ( 1) 2 k boring job. The reader who simply wishes to calculate − (m ) a (x) . (184) k! D/2−k  the renormalized expectation value of the stress-energy k=0 X tensor in a particular case must only extract from the  available Feynman propagator the first two coefficients This ambiguity is in fact equivalent to that obtained from ′ the Hadamard formalism in Sec. II [see Eq. (79)]. Indeed, of the biscalar W (x, x ) by using the formulas displayed for D = 2 it reduces to in Sec. III. If he wants furthermore to discuss the ambiguities of the renormalized stress-energy tensor obtained, 2 2 ln M he can used the expressions displayed in Sec. IV.A. He ΘM = µν 2(4 π) × has nothing else to do! 2 δ d2x g(x) a (x) m2a (x) √ g δgµν − 1 − 0 − ZM V. CONCLUSION AND PERSPECTIVES p (185) which permits us to recover (92) from (176) and (177). In this article, we have developed the Hadamard renor- For D = 4, it reduces to malization of the stress-energy tensor for a massive scalar field theory defined on a general spacetime of arbitrary 2 2 ln M 2 δ dimension. For spacetime dimension up to 6, we have ex- ΘM = d4x g(x)[a (x) µν 2 (4π)2 × √ g δgµν − 2 plicitly described the renormalization procedure while for M − Z p spacetime dimension from 7 to 11, we have provided the m2a (x) + (m4/2)a (x)] (186) framework permitting the interested reader to perform − 1 0 this procedure explicitly in a given spacetime. Our formalism is very general: we do not assume any which permits us to recover (143) from (176)-(178) by symmetry for spacetime and we do not limit our study using (136)-(138). For D = 6, it reduces to to the massless or the conformally invariant scalar fields. 2 As a consequence, we have provided a powerful formalism 2 ln M 2 δ ΘM = d6x g(x)[a (x) which could permit us to deal with some particular as- µν 2 (4π)3 × √ g δgµν − 3 − ZM pects of the quantum physics of extra spatial dimensions p in a rather general way or, more precisely, in a more gen- m2a (x) + (m4/2)a (x) (m6/6)a (x)] (187) − 2 1 − 0 eral way than usual (see references in Sec. I). We think that this formalism could be immediately used to discuss, which permits us to recover (160) from (176)-(179) by from a more general point of view, the consequence of the using (136)-(138) and (146)-(155). presence of extra spatial dimensions upon: We shall now conclude the present subsection by com- - The stabilization of Randall-Sundrum braneworld paring the respective merits of the Hadamard formalism models of cosmological interest (in connection with the developed in this article and of the approach based on inflationary scenario and the dark energy problem). renormalization in the effective action. Renormalization - The quantum violations of the classical energy con- in the effective action is a powerful tool which permits ditions (in connection with the singularity theorems of 24

Hawking and Penrose) as well as of the averaged null by going beyond existing results. energy condition (in connection with the existence of traversable wormholes and time-machines). - The fluctuations of the stress-energy tensor (in Acknowledgments connection with the validity of semiclassical gravity and again with the singularity theorems of Hawking and Pen- We would like to thank Valter Moretti for valuable rose). comments and for bringing the Hollands and Wald refer- Furthermore, we think it would be very interesting ences [84, 85, 86, 87] to our attention. We are also grate- to revisit holographic renormalization from the point ful to Bruce Jensen for various discussions some years of view of the Hadamard formalism and, above all, ago and to Rosalind Fiamma for help with the English. to use the Hadamard renormalization procedure developed in this article to perform calculations of stress- energy tensors for higher-dimensional black holes. In- APPENDIX A: TRACES FOR THE ANOMALOUS deed, even though such a subject has been a central TRACE OF THE STRESS TENSOR topic of four-dimensional semiclassical gravity, very lit- tle has been realized in the higher-dimensional framework. This is rather incomprehensible since string the- In this appendix, we provide the expressions for the traces of the three conserved tensors of rank ory (or more precisely the so-called TeV-scale quantum (4,2)(1) (4,2)(2) (4,2)(3) gravity [28, 29, 30]) predicts the possibility of produc- 2 and order 4 Hµν , Hµν and Hµν and of the ten conserved tensors of rank 2 and order tion of such black holes at CERN’s Large Hadron Col- {2,0}(1) {2,0}(3) (6,2)(1) (6,2)(2) (6,2)(3) lider [98, 99, 100] with a production rate around 1 Hz 6 Hµν , Hµν , Hµν , Hµν , Hµν , (6,2)(4) (6,2)(5) (6,2)(6) (6,2)(7) (6,2)(8) [101, 102]. In this context, the semiclassical Einstein Hµν , Hµν , Hµν , Hµν and Hµν . equations (1) could permit us to partially describe the From Eqs. (136a), (137a) and (138a), we easily obtain black hole evaporation and to test TeV-scale quantum gravity. gµν H(4,2)(1) = (2D 2) R + (D/2 2) R2, (A1) µν − − − Of course, with the various applications previously gµν H(4,2)(2) = (D/2) R + (D/2 2) R Rpq, mentioned in mind, it is necessary to extend our present µν − − pq work to more general field theories and more particularly (A2) µν (4,2)(3) pqrs to the graviton field. In order to perform such a general- g Hµν = 2 R + (D/2 2) RpqrsR .(A3) ization, it is first of all necessary to carry out the program − − described at the end of the conclusion of Ref. [71], i.e. to From Eqs. (2.22)-(2.31) of Ref. [89], we obtain after te- construct the covariant Taylor series expansions for the dious calculations using results and geometrical identities off-diagonal Hadamard coefficients for these field theories of Ref. [73]

gµν H{2,0}(1) = (2D 2) R 2 RR (D/2 1) R R;p (A4) µν − − − − − ;p

gµν H{2,0}(3) = (D/2) R + (4 2D) R Rpq + (D 4) R Rpq + (2D 4) R Rprqs µν − − ;pq − pq − pq;rs (D/2 3/2) R R;p + (5D/2 5) R Rpq;r (4D 6) R Rpr;q (2D 2) R Rp Rqr − − ;p − pq;r − − pq;r − − pq r +(2D 2) R R Rprqs (A5) − pq rs

gµν H(6,3)(1) = (6D 6) RR (6D 6) R R;p + (D/2 3) R3 (A6) µν − − − − ;p −

gµν H(6,3)(2) = (D/2+1) RR (D 2) R Rpq (2D 2) R Rpq D R R;p µν − − − ;pq − − pq − ;p (2D 2) R Rpq;r + (D/2 3) RR Rpq (A7) − − pq;r − pq

gµν H(6,3)(3) = (3D/2 3) R Rpq 3 R Rpq (3D/8 3/4) R R;p 3 R Rpq;r µν − − ;pq − pq − − ;p − pq;r (3D/2 3) R Rpr;q D R Rp Rqr + (3D/2 3) R R Rprqs (A8) − − pq;r − pq r − pq rs

gµν H(6,3)(4) = (1/2) RR + (D/2 1) R Rpq D R Rpq (D 2) R Rprqs (3/4) R R;p µν − − ;pq − pq − − pq;rs − ;p (2D 2) R Rpq;r + (2D 3) R Rpr;q + (D 1) R Rp Rqr (D/2+2) R R Rprqs (A9) − − pq;r − pq;r − pq r − pq rs 25

gµν H(6,3)(5) = 2 RR 4 R Rpq (8D 8) R Rprqs 4 R R;p (2D 2) R Rpqrs;t µν − − ;pq − − pq;rs − ;p − − pqrs;t +(D/2 3) RR Rpqrs (4D 4) R Rp Rqrst + (2D 2) R RpquvRrs − pqrs − − pq rst − pqrs uv +(8D 8) R Rp q Rrusv (A10) − prqs u v

gµν H(6,3)(6) = R Rpq 2 R Rpq (2D + 2) R Rprqs (1/2) R R;p (D + 2) R Rpq;r µν − ;pq − pq − pq;rs − ;p − pq;r pr;q pqrs;t p qrst pquv rs +D Rpq;rR (D/4+1/2) Rpqrs;tR 4 RpqR rstR + (D/4+1/2) RpqrsR R uv − p q rusv − +(D + 2) RprqsR u vR (A11)

gµν H(6,3)(7) = 24 R Rprqs 12 R Rpq;r + 12 R Rpr;q 3 R Rpqrs;t µν − pq;rs − pq;r pq;r − pqrs;t 6 R Rp Rqrst + (D/2) R RpquvRrs + 12 R Rp q Rrusv (A12) − pq rst pqrs uv prqs u v

gµν H(6,3)(8) = (3/2) R Rpq 3 R Rpq +3 R Rprqs 3 R Rpq;r +3 R Rpr;q µν ;pq − pq pq;rs − pq;r pq;r +(3/4) R Rpqrs;t +3 R Rp Rqr 3 R R Rprqs + (3/2) R Rp Rqrst pqrs;t pq r − pq rs pq rst (3/4) R RpquvRrs + (D/2 6) R Rp q Rrusv. (A13) − pqrs uv − prqs u v

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