Champs Quantiques Massifs En Espace-Temps Courbe Et Tenseur D’Impulsion-Énergie Renormalisé Andrei Belokogne

Champs quantiques massifs en espace-temps courbe et tenseur d’impulsion-énergie renormalisé Andrei Belokogne

To cite this version:

Andrei Belokogne. Champs quantiques massifs en espace-temps courbe et tenseur d’impulsion-énergie renormalisé. Relativité Générale et Cosmologie Quantique [gr-qc]. Université de Corse Pascal Paoli, 2016. Français. tel-01445979

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Thèse présentée pour l’obtention du grade de

DOCTEUR EN PHYSIQUE Mention : Constituants élémentaires

Soutenue publiquement par

Andrei BELOKOGNE

le : 9 décembre 2016

CHAMPSQUANTIQUESMASSIFSENESPACE-TEMPSCOURBE ETTENSEURD’IMPULSION-ÉNERGIERENORMALISÉ

Directeurs : M Antoine FOLACCI, Professeur, Université de Corse M Yves DECANINI, Professeur, Université de Corse

Rapporteurs : M Eric HUGUET, MCF-HDR, Université de Paris VII M Adrian OTTEWILL, Professeur, University College Dublin

Jury : M Yves DECANINI, Professeur, Université de Corse M Antoine FOLACCI, Professeur, Université de Corse M Eric HUGUET, MCF-HDR, Université de Paris VII M Adrian OTTEWILL, Professeur, University College Dublin M Jacques RENAUD, Professeur, Université de Paris-Est M Thomas SCHUCKER, Professeur, Université d’Aix-Marseille

« On ne peut plus expliquer le monde, faire ressentir sa beauté à ceux qui n’ont aucune connaissance profonde des mathématiques. »

Richard Feynman

Remerciements

J’aimerais tout d’abord remercier grandement mes directeurs de thèse, Antoine Folacci et Yves Dé- canini, de m’avoir proposé ce travail doctoral ainsi que les différents sujets qu’aborde ce manuscrit. Je leur exprime ma profonde gratitude pour tout ce qu’ils m’ont apporté sur le plan scientiﬁque et humain. Je tiens tout particulièrement à remercier Antoine Folacci pour les discussions enrichissantes et fructueuses sur les sujets scientiﬁques pertinents ainsi que sa disponibilité durant toute cette période de thèse.

J’aimerais également remercier M. Eric Huguet et M. Adrian Ottewill d’avoir eu l’amabilité d’accep- ter de rapporter sur ce manuscrit. De même, je souhaite témoigner ma gratitude à M. Jacques Renaud et M. Thomas Schucker d’avoir acceptés de faire partie de mon jury de thèse. Je leur exprime toute ma reconnaissance.

J’aimerais remercier sincèrement mon collègue de bureau, Mohamed Ould El Hadj, de toutes les discussions scientiﬁques et amicales que l’on avait pendant ces années. Mes remerciements vont aussi aux docteurs et futurs docteurs, Damien, David, Eric, Gauthier, Jean-Baptiste, Lara, Romain, Sandrine, Tom et Wani ainsi que Aurélia et Damien, pour tous les moments agréables que l’on a partagés ensemble.

Au cours de la dernière année de thèse, j’ai aussi travaillé avec Julien Queva et je tiens à lui expri- mer ma reconnaissance pour les discussions et les conseils. D’autre part, j’aimerais également remercier l’ensemble des chercheurs et personnels de la Faculté des Sciences de l’Université de Corse, du labora- toire SPE (UMR CNRS 6134) pour leur accueil et sympathie et, en particulier, Pierre Simonnet ainsi que Stéphane Ancey, Rachel Baïle, Frederic Bosseur, Benoit Cagnard, Bernard Di Martino, Catherine Du- courtioux, Vincent Ferreri, Jean Baptiste Filippi, Paul Gabrielli, Nicolas Heraud, David Moungar, Jean- François Muzy, Marie-Laure Nivet, Jean-Martin Paoli, Antoine Pieri, Sonia Ternengo et Yves Thibaudat.

Finalement, j’aimerais dédier ce manuscrit à mon frère, Ivan, ainsi qu’à ma soeur, Nicole, et à mes parents. Sans leur soutien et sans leurs encouragements jamais cette thèse n’aurait vu le jour.

Andrei Belokogne Corte, décembre 2016.

iii

Abstract

Quantum field theory in curved spacetime is a semiclassical approximation of quantum gravity which, by treating classically the spacetime metric gµν and considering from a quantum point of view all the other fields (including the graviton field to at least one-loop order for reasons of consistency), avoids the difficulties due to the nonrenormalizability of quantum gravity and provides a framework which permits us to study the low-energy consequences of a hypothetical “theory of everything”. It should be recalled that this approach allowed theoretical physicists to obtain fascinating results concerning early universe cosmology and quantum black hole physics and, in particular, led to the discovery of particle creation in expanding universes by Parker and of black hole radiance by Hawking.

In quantum field theory in curved spacetime, it is conjectured that the backreaction of a quantum field in the normalized quantum state ψ on the spacetime geometry is governed by the semiclassical Ein- | 〉 1 stein equations Gµν 8π ψ Tbµν ψ . Here, Gµν is the Einstein tensor Rµν R gµν Λgµν or some higher- = 〈 | | 〉 − 2 + order generalization of this geometrical tensor while ψ Tbµν ψ is the expectation value of the stress-energy 〈 | | 〉 tensor associated with the quantum field. The quantity ψ Tbµν ψ is ill-defined and formally infinite due to 〈 | | 〉 the “pathological” short-distance behavior of the Green functions of the quantum field. In order to extract from ψ Tbµν ψ a finite and physically acceptable contribution, it is necessary to regularize it and then to 〈 | | 〉 renormalize all the coupling constants of the theory. The corresponding renormalized expectation value

ψ Tbµν ψ ren is of fundamental importance not only because it acts as the source in the semiclassical Ein- 〈 | | 〉 stein equations, but also because it permits us to analyze the quantum state ψ without any reference to | 〉 its particle content.

In the first part of our manuscript, we have obtained a large-mass approximation for the renormalized expectation value of the stress-energy tensor associated with various massive fields (the massive scalar field, the massive Dirac field and the Proca field) propagating on Kerr-Newman spacetime. Our calculations are based on the DeWitt-Schwinger expansion of the Green functions and of the effective action. Our results could be useful because there exists no exact results on this gravitational background.

In the second part of our manuscript, we have discussed Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime (gauge invariance of the classical theory and covariant quantization ; wave equations for the massive spin-1 ﬁeld Aµ, for the auxiliary Stueckelberg scalar ﬁeld

Φ and for the ghost fields C and C∗ ; Ward identities ; Hadamard representation of the various Feynman propagators and covariant Taylor series expansions of the corresponding coefficients). This has permitted us to construct, for a Hadamard quantum state ψ , the renormalized expectation value of the stress-energy | 〉 tensor associated with the Stueckelberg theory. As applications of our results, (i) we have considered, in the Minkowski spacetime, the Casimir effect outside a perfectly conducting medium with a plane boundary, and (ii) we have obtained, in de Sitter and anti-de Sitter spacetimes, an exact analytical expression for the vacuum expectation value of the renormalized stress-energy tensor of the massive vector field.

Keywords : Quantum ﬁeld theory in curved spacetime ; Massive electromagnetism ; Stueckelberg mechanism ; Effective action ; DeWitt-Schwinger expansion ; Hadamard renormalization ; Renormalized stress-energy tensor ; Vacuum energy ; Kerr-Newman, de Sitter and anti-de Sitter spacetimes ; Casimir effect.

Résumé

La théorie quantique des champs en espace-temps courbe est une approximation semi-classique de la gravitation quantique qui, en traitant classiquement la métrique gµν de l’espace-temps et en considérant du point de vue quantique tous les autres champs (on inclut ici même le champ des gravitons à l’ordre au moins une boucle pour des raisons de cohérence), évite les difﬁcultés dues à la non-renormalisabilité de la gravitation quantique et fournit un cadre qui permet d’étudier les conséquences, à basse énergie, d’une hypothétique “théorie du tout”. Il faut rappeler que cette approche a permis aux physiciens d’obtenir des résultats fascinants concernant la cosmologie de l’univers primordial ainsi que la physique des trous noirs et, en particulier, a conduit Parker à la découverte de la création de particules dans des univers en expansion et Hawking à celle du rayonnement quantique des trous noirs.

En théorie quantique des champs en espace-temps courbe, il est admis que la réaction en retour d’un champ quantique dans un état quantique normalisé ψ sur la géométrie de l’espace-temps est régie | 〉 par les équations d’Einstein semi-classiques Gµν 8π ψ Tbµν ψ . Ici, Gµν est le tenseur d’Einstein Rµν = 〈 | | 〉 − 1 R gµν Λgµν ou une généralisation d’ordre supérieur de ce tenseur géométrique alors que ψ Tbµν ψ est 2 + 〈 | | 〉 la valeur moyenne du tenseur d’impulsion-énergie associé au champ quantique. La quantité ψ Tbµν ψ 〈 | | 〉 est formellement infinie du fait du comportement singulier à courte distance des fonctions de Green du champ quantique. Afin d’extraire une contribution finie et physiquement raisonnable de cette quantité, il est nécessaire de la régulariser puis de renormaliser toutes les constantes de couplage de la théorie.

La valeur moyenne renormalisée ψ Tbµν ψ ren est d’une importance fondamentale non seulement parce 〈 | | 〉 qu’elle agit comme source dans les équations d’Einstein semi-classiques mais aussi parce qu’elle nous permet d’analyser l’état quantique ψ sans faire aucune référence à son contenu en particules. | 〉 Dans la première partie de notre manuscrit, nous avons obtenu, dans la limite des grandes masses, une approximation pour la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé à divers champs massifs (le champ scalaire massif, le champ de Dirac massif et le champ de Proca) se propageant sur l’espace-temps de Kerr-Newmann. Notre calcul est basé sur le développement de DeWitt-Schwinger des fonctions de Green et de l’action effective. Nos résultats sont intéressants parce qu’il n’existe aucun résultat exact sur ce background gravitationnel.

Dans la seconde partie de notre manuscrit, nous avons discuté l’électromagnétisme massif de Stue- ckelberg sur un espace-temps courbe arbitraire de dimension quatre (invariance de jauge de la théorie classique et quantiﬁcation covariante ; équations d’ondes pour le champ massif de spin-1 Aµ, pour le champ scalaire auxiliaire de Stueckelberg Φ et pour les champs de fantômes C et C∗ ; identité de Ward ; représen- tation de Hadamard des divers propagateurs de Feynman et développement en séries de Taylor covariantes des coefﬁcients correspondants). Cela nous a permis de construire, pour un état quantique de type Hada- mard ψ , la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé à la théorie de Stueckel- | 〉 berg. Comme applications de nos résultats, (i) nous avons considéré, en espace-temps de Minkowski, l’effet Casimir en dehors d’un milieu parfaitement conducteur et de bord plan, et (ii) nous avons obtenu, dans les espaces-temps de de Sitter et anti-de Sitter, une expression analytique exacte pour la valeur moyenne dans le vide du tenseur d’impulsion-énergie renormalisé du champ vectoriel massif.

Mots clés : Théories quantiques des champs en espace-temps courbe ; Electromagnétisme massif ; Mécanisme de Stueckelberg ; Action effective ; Développement de DeWitt-Schwinger ; Renormalisation de Hadamard ; Tenseur d’impulsion-énergie renormalisé ; Energie du vide ; Espaces-temps de Kerr-Newman, de de Sitter et anti-de Sitter ; Effet Casimir.

vii

TABLEDESMATIÈRES

Page

Table des matières ix

Introduction 1

I Action effective et tenseur d’impulsion-énergie renormalisé 11

1 Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman ...... 13 1.1 Introduction...... 14 1.2 Approximate renormalized stress tensors in Kerr-Newman spacetime ...... 18 1.3 Special cases : approximate renormalized stress tensors in Schwarzschild, Reissner- Nordström and Kerr spacetimes ...... 19 1.4 Conclusion...... 20 µν 1.5 Appendix A : Coefficients A p,q[θ, M/r]...... 22 1.5.1 Massive scalar field ...... 22 1.5.2 Massive Dirac field...... 27 1.5.3 Proca field ...... 32

II Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé 39

2 Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir ...... 41 2.1 Introduction...... 43 2.2 Quantization of Stueckelberg electromagnetism ...... 46 2.2.1 Quantum action ...... 46 2.2.2 Wave equations...... 46 2.2.3 Feynman propagators and Ward identities...... 47 2.3 Hadamard expansions of the Green functions of Stueckelberg electromagnetism...... 47 2.3.1 Hadamard representation of the Feynman propagators...... 47 2.3.2 Hadamard Green functions...... 49 2.3.3 Geometrical Hadamard coefﬁcients and associated covariant Taylor series expansions 50

ix Table des matières

2.3.4 State-dependent Hadamard coefficients and associated covariant Taylor series expansions ...... 51 2.3.4.1 General considerations...... 51 2.3.4.2 Wave equations...... 51 2.3.4.3 Ward identities...... 52 2.4 Renormalized stress-energy tensor of Stueckelberg electromagnetism ...... 52 2.4.1 Stress-energy tensor ...... 52 2.4.2 Expectation value of the stress-energy tensor ...... 54 2.4.3 Renormalized stress-energy tensor ...... 55 2.4.3.1 Definition and conservation ...... 55 2.4.3.2 Cancellation of the gauge-breaking and ghost contributions...... 56 2.4.3.3 Substitution of the auxiliary scalar field contribution and final result ...... 56

2.4.3.4 Another final expression involving both the vector field Aµ and the auxiliary scalar field Φ ...... 57 2.4.4 Maxwell’s theory ...... 57 2.4.5 Ambiguities in the renormalized stress-energy tensor...... 58 2.4.5.1 General expression of the ambiguities...... 58 2.4.5.2 Ambiguities associated with the renormalization mass...... 58 2.5 Casimir effect...... 60 2.5.1 General considerations ...... 60 2.5.2 Stress-energy tensor in the Minkowski spacetime ...... 61 2.5.3 Stress-energy tensor for the Casimir effect...... 62

2.5.4 Separation of the contributions associated with the vector ﬁeld Aµ and the auxiliary scalar ﬁeld Φ ...... 62 2.6 Conclusion...... 63 2.7 Appendix : Biscalars, bivectors and their covariant Taylor series expansions...... 65

3 Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti- de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé ...... 69 3.1 Introduction...... 70 3.2 Two-point functions of Stueckelberg electromagnetism ...... 71 3.2.1 de Sitter and anti-de Sitter spacetimes ...... 71 3.2.2 Stueckelberg theory, wave equations and Ward identities for the Wightman functions 73 3.2.3 Explicit expression for the Wightman functions in dS4 and AdS4 ...... 74 3.2.3.1 General form of the Wightman functions in maximally symmetric backgrounds...... 74 3.2.3.2 Wightman functions for Hadamard vacua ...... 75 3.2.3.3 Wightman functions in dS4 ...... 76 3.2.3.4 Wightman functions in AdS4 ...... 77 3.2.4 Feynman propagators and Hadamard Green functions in dS4 and AdS4 ...... 78 3.2.4.1 In dS4 ...... 79 3.2.4.2 In AdS4 ...... 80 3.3 Renormalized stress-energy tensor of Stueckelberg electromagnetism ...... 81 3.3.1 General considerations ...... 81 x Table des matières

3.3.2 The renormalized stress-energy tensor in dS4 ...... 84 3.3.3 The renormalized stress-energy tensor in AdS4 ...... 85 3.3.4 Remarks concerning the zero-mass limit of the renormalized stress-energy tensor ..... 86 3.4 Conclusion...... 88

Conclusion et perspectives 91

Bibliographie 95

INTRODUCTION

On peut considérer que c’est Michael Faraday qui, en 1849, a introduit en physique le concept de “champ”. Dans la première moitié du XIXe siècle, il a réalisé l’importance de cette notion après avoir mis expérimentalement en évidence des liens étroits entre les phénomènes électriques et magnétiques. C’est, par la suite, James Clerk Maxwell qui a construit la théorie mathématique de l’électromagnétisme et, plus précisément, la théorie dynamique du champ électromagnétique publiée dans son travail de 1865 [1] ainsi que dans son ouvrage de 1873 [2] où, pour la première fois, toutes les équations qui portent son nom sont apparues. L’une des prédictions majeures de cette théorie est que la lumière est une onde 8 1 électromagnétique se propageant dans le vide avec une vitesse ﬁnie c 2.998 10 m.s− . En fait, c’est ≈ × une théorie qui uniﬁe les interactions associées aux champs électrique et magnétique et qui, de plus, rend compte des phénomènes optiques. C’est aussi une théorie de jauge, du fait de l’invariance de ses équations par les transformations associées à la symétrie de jauge locale correspondant au groupe abélien U(1).

Malgré le succès remarquable de l’électromagnétisme de Maxwell aux niveaux théorique et expé- rimental, la ﬁn de XIXe siècle et le début de XXe siècle ont été secoués par de nombreux problèmes de physique restant encore ouverts. Aﬁn d’élucider ces questions, les physiciens ont dû abandonner l’ancienne conception du monde ce qui les a conduit à formuler les bases de deux théories révolutionnaires : (i) la relativité restreinte et (ii) la mécanique quantique.

(i) La formulation de la théorie de Maxwell a été suivie de nombreuses recherches théoriques et expé- rimentales concernant l’éther, ce support matériel introduit dans le but d’expliquer la propagation mécanique des ondes électromagnétiques. Elles n’ont pas donné de résultats convenables. C’est finalement la relativité restreinte formulée par Albert Einstein en 1905 [3] qui a mis fin à la nécessité de tous ces travaux. Il faut noter que, dans les équations de Maxwell, les transformations de Lorentz [3–7] qui sont à la base de la relativité restreinte sont présentes intrinsèquement. Cette théorie ré- volutionnaire fournit une nouvelle vision de l’espace et le temps qui sont réunis en un concept indis- sociable appelé l’espace-temps dont la formulation géométrique a été faite par Hermann Minkowski [8]. Dans l’espace-temps de Minkowski, l’invariant relativiste ds2 correspondant à l’intervalle infini-

1 Introduction

tésimal ds séparant deux évènements est donné par

ds2 η dxµdxν c2dT2 dX 2 dY 2 dZ2 = µν = − + + +

où η diag( 1, 1, 1, 1) est le tenseur métrique et xµ (T, X,Y , Z) sont les coordonnées d’un µν = − + + + = évènement.

(ii) Les théories physiques du XIXe siècle, malgré leurs innombrables succès, ne pouvaient pas expliquer deux problèmes d’une importance fondamentale : le rayonnement du corps noir et l’effet photoélec- trique. Il fallut attendre l’année 1900 pour la résolution du premier problème par Max Planck dans un travail où il faisait l’hypothèse de la quantiﬁcation de l’énergie dans les échanges entre rayonnement et matière [9, 10]. La valeur élémentaire du quantum d’énergie ε est reliée à la fréquence ν de l’onde électromagnétique par la relation

ε hν =

34 où h 6.626 10− J.s est la constante de Planck. En 1905, Albert Einstein a publié un de ses ≈ × trois fameux articles dans lequel il expliquait l’effet photoélectrique en postulant que le rayonnement électromagnétique est composé de quanta d’énergie localisés dans l’espace appelés aujourd’hui les photons [11]. L’extension du comportement dual “onde-corpuscule” à la matière a été faite par Louis de Broglie dans sa thèse publiée en 1924 [12]. C’est, principalement, à la base de ces considé- rations théoriques de de Broglie que, à partir des années 1925, la mécanique quantique a fait son apparition avec les travaux de Werner Heisenberg, Max Born, Pascual Jordan, Wolfgang Pauli, Paul Adrian Maurice Dirac et Erwin Schrödinger (voir les Réfs. [13–19] ainsi que les références qu’elles contiennent).

Toutefois, cette mécanique quantique développée entre 1925 et 1927 présentait une incohérence fondamentale. Lors de la conception des bases de la mécanique quantique, la considération des principes de la relativité restreinte était négligée. Ainsi, l’équation de Schrödinger décrit la dynamique d’une particule massive non relativiste par le moyen de la fonction d’onde associée. De la même façon, lorsque Dirac dans ses premiers travaux ayant pour but d’étudier les processus d’émission et absorption du rayonnement, a quantiﬁé le système physique électrons-champ électromagnétique (voir la Réf. [20, 21]), il l’a fait d’une manière qui s’est avérée par la suite de ne pas être compatible avec la relativité restreinte. En fait, la relativité restreinte a été introduite dans des modèles quantiques avec les travaux, entre 1926 et 1928, de :

– Oskar Klein et Walter Gordon qui ont établi indépendamment l’équation 1 (elle porte maintenant leurs noms) décrivant des particules massives scalaires (spin 0) [22, 23], – Heisenberg, Born, Jordan et Pauli qui ont appliqué les règles de la mécanique quantique au champ électromagnétique dans l’espace vide [17, 24], – Dirac qui a établi l’équation (elle porte maintenant son nom) décrivant la dynamique de l’électron [25].

Il faut noter que la conséquence du dernier travail mentionné a été l’explication du spin-1/2 de l’électron comme résultat de l’uniﬁcation de la mécanique quantique avec la relativité restreinte ainsi que la prédic- tion de son antiparticule appelée positron.

1. En réalité, Schrödinger est le premier qui a écrit l’équation de Klein-Gordon pour tenter de décrire, mais sans succès, l’électron dans l’atome d’hydrogène.

2 Introduction

La mécanique quantique ainsi que sa version relativiste a permis d’expliquer les divers effets obser- vés (tel que les raies spectrales, l’émission spontanée, la structure hyperfine de l’hydrogène, l’effet Zeeman et autres) en décrivant la matière quantifiée par le moyen de sa fonction d’onde (voir, par exemple, les Réfs. [14, 16–18]), c’est-à-dire en “première quantification”. Cependant, dans les travaux sur les processus d’émission et absorption du rayonnement (voir la Réf. [20]), Dirac a déjà commencé à considérer la description du champ électromagnétique du point de vue des processus de création et de destruction des photons, c’est-à-dire en “seconde quantification”. C’est une vision fondamentalement différente par rapport à la première vision de la mécanique quantique appliquée à la matière. Cette incohérence dans la quantification a disparu avec les travaux de Jordan et Eugene Wigner [26], de Heisenberg et Pauli [27, 28] ainsi que d’Enrico Fermi [29, 30] et de Dirac [31], ces auteurs ayant montré que, comme les photons qui sont les quanta du champ électromagnétique, les particules de matière doivent elles aussi être considérées comme les quanta associés à leurs champs. Wendell Furry et Robert Oppenheimer [32] ainsi que Pauli et Victor Weisskopf [33] ont montré que cette approche pouvait être étendue aux antiparticules. Ce concept fondamental de création et de destruction des particules dont la description mathématique est réalisée par des opérateurs est en fait à la base de ce que l’on appelle aujourd’hui la théorie quantique relativiste des champs dont les propriétés peuvent être résumées en citant Steven Weinberg : “Thus, the inhabitants of the universe were conceived to be a set of fields – an electron field, a proton field, an electromagnetic field – and particles were reduced in status to mere epiphenomena. In its essentials, this point of view [. . .] forms the central dogma of quantum field theory : the essential reality is a set of fields, subject to the rules of special relativity and quantum mechanics ; all else is derived as a consequence of the quantum dynamics of these fields.”. Le lecteur intéressé pourrait consulter les références [34–36] sur la description historique détaillée de la naissance de la théorie quantique des champs.

Dans tous les travaux sur l’électrodynamique quantique, les physiciens théoriciens ont rencontré, à un moment ou un autre de leurs calculs, différentes sortes d’inﬁnis. Ceux-ci apparaissent, par exemple, (i) dans le calcul de la self-énergie de l’électron étudiée par Julius Robert Oppenheimer [37] et Ivar Waller

[38, 39], (ii) dans l’effet appelé Lamb-shift correspondant à la différence des niveaux d’énergie 2s1/2 2p1/2 − (voir les Réfs. [40–43]) ou bien encore (iii) dans la détermination de l’anomalie du moment magnétique de l’électron par rapport à celui prédit par Dirac [25] (voir les Réfs. [44–48]). La méthode astucieuse pour pouvoir éliminer ces infinis consiste à les absorber dans les paramètres du problème physique en redéfinissant ceux-ci pour qu’ils soient finis. Cette méthode est appelée renormalisation 2 et elle a été conceptualisée par Weisskopf [49] et Hans Kramers [50]. La formulation moderne de l’électrodynamique quantique, avec des méthodes plus adaptées pour les calculs, a été développée par Julian Schwinger [51–57], Sin-Itiro Tomonaga [58–63] et Richard Feynman [64–69]. Freeman Dyson [70, 71] a montré l’équivalence de trois approches et a donné le critère de renormalisabilité des théories quantiques des champs permettant d’absorber les infinis dans un nombre fini de paramètres physiques.

Il est important de rajouter que les succès de l’électrodynamique quantique sont aussi dus à son invariance de jauge U(1) héritée de l’électromagnétisme de Maxwell. On a étendu ce modèle de théorie des champs pour formuler les théories de jauge non-abéliennes (voir, par exemple, la Réf. [72]) qui se sont révélées adaptées pour décrire les autres interactions fondamentales. Ainsi, aujourd’hui, les trois interactions fondamentales sont décrites par des champs quantiques vectoriels du type bosonique (spin

2. Dans le cas de l’électrodynamique quantique, la procédure de renormalisation consiste à absorber les infinis dans la masse et la charge de l’électron. En fait, ces grandeurs intervenant dans les équations de la théorie sont appelées nues et ce sont elles que l’on redéfinit afin d’avoir les grandeurs associées mesurables dans les expériences.

3 Introduction entier) et on a :

– l’interaction électromagnétique dont le groupe de jauge est U(1) avec le photon comme médiateur, 0 – l’interaction faible [73–76] dont le groupe de jauge est SU(2) avec trois bosons vectoriels Z et W± comme médiateurs, – l’interaction forte [77, 78] dont le groupe de jauge est SU(3) avec huit gluons comme médiateurs.

Il faut noter que c’est dans le cadre de l’unification des deux premières interactions en une seule interaction électrofaible faite par Sheldon Glashow, Abdus Salam et Steven Weiberg [79–82] que l’interaction faible est mieux comprise. Effectivement, contrairement aux interactions électromagnétique et forte qui sont considérées non massives 3, les champs de jauge associés à trois bosons de l’interaction faible sont massifs 4. Leurs masses sont acquises par l’intermédiaire du mécanisme de Brout-Englert-Higgs-Hagen- Guralnik-Kibble [88–90] où un nouveau boson massif appelé Higgs associé au champ quantique scalaire (spin 0) intervient (voir aussi la Réf. [91]). En résumé, les champs fermioniques (spin demi-entier) décrivant toute la matière sont en interaction entre eux par l’intermédiaire de ces champs bosoniques. C’est l’image actuelle du modèle standard des particules élémentaires [86]. Il est aussi naturel d’envisager une théorie de grande unification qui fusionne les interactions électrofaible et forte en une seule (voir, par exemple, les Réfs. [92, 93] pour les premiers travaux sur le sujet). Cependant, il ne reste que la gravitation, la quatrième interaction fondamentale, qui échappe à une telle unification.

En 1915, Einstein a publié sa version déﬁnitive de la relativité générale [94] qui associe la gravitation à la géométrie courbe et dynamique de l’espace-temps. Dans cette théorie, contrairement à l’espace-temps de Minkowski qui est plat et statique, l’invariant

ds2 g dxµdxν = µν est donné en fonction d’un tenseur métrique gµν de signature Lorentzienne vériﬁant les équations d’Ein- stein

8πG G N T . µν = c4 µν

Ici, G R – 1 R g Λg est le tenseur géométrique d’Einstein, Λ est la constante cosmologique tan- µν = µν 2 µν + µν 5 dis que le tenseur d’impulsion-énergie Tµν décrit le contenu matériel se trouvant dans cet espace-temps. Dans le facteur de proportionnalité interviennent la vitesse de la lumière c et la constante de la gra- 11 3 1 2 vitation de Newton GN 6.674 10− m .kg− .s− . Ces équations nous montrent comment la matière ≈ × déforme la géométrie tandis que la géométrie indique à la matière sa dynamique. La théorie de la relati- vité générale d’Einstein a permis de rendre compte de nombreux phénomènes gravitationnels (l’avancée du périhélie de Mercure [95], la déviation de la lumière par des corps massifs [95, 96], l’effet de la gravitation sur le temps [97, 98]). De plus, son formalisme s’est révélé bien adapté pour la construction des modèles d’espace-temps susceptibles de décrire l’univers [99]. Un de ces modèles est l’espace-temps de

3. La masse du photon est habituellement considérée comme nulle, mais, en fait, les expériences terrestres 18 2 et spatiales ne permettent d’évaluer que sa limite supérieure qui est actuellement de l’ordre de 10− eV.c− 2 54 2 ≈ × 10− kg [83–86]. La situation est similaire pour la masse des gluons qui devrait être inférieure à 1MeV.c− 2 42 ≈ × 10− kg [86, 87]. 0 2 27 4. La masse du boson Z est de l’ordre de 90GeV.c− 160 10− kg alors que celle de deux bosons W± est de 2 27 ≈ × l’ordre de 80GeV.c− 140 10− kg [86]. 5. Ici, le contenu≈ matériel× doit être compris dans le sens suivant : ce sont les champs (classiques) décrivant la matière ainsi que les interactions.

4 Introduction

Friedmann-Lemaître-Robertson-Walker [100–103] qui rentre dans le cadre de la théorie du Big Bang dé- veloppée avec la découverte accidentelle en 1964 du fond diffus cosmologique [104, 105]. Ce modèle permet de rendre compte de l’évolution de l’univers durant son expansion [106–108] et de discuter sa composition (la matière du modèle standard de la physique des particules, la matière noire et l’énergie sombre). En particulier, il faut noter que la géométrie de de Sitter [109] décrit l’univers très primordial et la période de l’inﬂation [110–113]. Un autre succès remarquable de la relativité générale est la prédiction de l’existence des ondes gravitationnelles [114] et des trous noirs [115–119] dont la preuve indirecte date des années 1970 [120–125] alors que la détection directe est récente [126]. Il reste que la relativité générale est une théorie classique du champ gravitationnel décrivant les phénomènes astrophysiques à basse énergie pour les corps macroscopiques.

Aujourd’hui, les physiciens théoriciens tentent de construire une théorie quantique de la gravitation et de réaliser son unification avec les autres interactions (voir, par exemple, la gravité quantique à boucles [127], la supergravité [128], la théorie des supercordes et la théorie M [129]), mais il semble que l’on en est encore loin. Cette difficulté prend son origine au niveau conceptuel. En effet, tous les champs quantiques vus auparavant, quelles que soient leurs natures bosoniques ou fermioniques, se propagent sur l’espace-temps tandis que le champ gravitationnel est en fait l’espace-temps lui-même. Une autre com- plication technique vient du fait que la relativité générale est aussi une théorie de jauge mais que son groupe de symétrie local est celui des difféomorphismes de la variété espace-temps (invariance par rapport aux transformations infinitésimales des coordonnées) dont la structure est plus complexe que celles des groupes intervenant dans les théories de Yang-Mills. Par ailleurs, la théorie quantique de la gravitation traitée de manière perturbative avec les méthodes de la théorie quantique des champs n’est pas renormalisable. D’autre part, en supposant que la théorie quantique de la gravitation ne fait intervenir que les trois constantes fondamentales que sont la vitesse de la lumière c, la constante de Planck ré- duite ~ h/(2π) et la constante de la gravitation GN , il est possible de construire d’une manière unique = ¡ 5¢1/2 43 ¡ 3¢1/2 35 un temps tP GN ~/c 10− s (temps de Planck), une longueur lP GN ~/c 10− m (longueur = ≈ = ≈ ¡ 5 ¢1/2 19 de Planck) et une énergie EP c ~/GN 10 GeV (énergie de Planck). Ces ordres de grandeur nous = ≈ laissent croire que les effets de la gravitation quantique ne pourront pas être observés dans les années à venir (à comparer avec l’énergie actuelle atteinte dans l’accélérateur de particules LHC qui est de l’ordre de 104 GeV). Cependant, il faut rappeler que, du point de vue théorique, la relativité générale prédit l’existence d’espaces-temps contenant des singularités géométriques (par exemple, la singularité du Big Bang ou celles au “centre” des trous noirs [130]) et que, de ce point de vue, elle semble être une théorie incomplète. En partant du principe que l’univers ne contient pas d’infinis, on pourrait espérer que la théorie quantique de la gravitation remédie à ce genre des problèmes.

En absence d’une théorie quantique de la gravitation, on peut étudier le comportement des champs quantiques (y compris le champ des gravitons 6) sur un background gravitationnel considéré classique, i.e., l’espace-temps de la relativité générale. On peut ainsi espérer découvrir et comprendre les éventuels effets quantiques qui pourront exister en champ fort, par exemple, au voisinage des trous noirs ou dans l’univers primordial. Cette approche est dite théorie quantique des champs en espace-temps courbe (voir les Réfs. [131–137] pour des revues et monographies sur le sujet) et on peut la considérer comme une approximation semi-classique de la gravitation quantique. Il est intéressant de rappeler qu’une telle approche est

6. Ici, le champ des gravitons (spin 2) est considéré dans le sens de la quantiﬁcation de l’interaction gravita- tionnelle dans le cadre de la théorie quantique des champs habituelle où la particule non massive appelée graviton est associée à ce champ bosonique.

5 Introduction similaire à celle utilisée pour résoudre certains problèmes de la mécanique quantique lorsqu’une théorie complète de l’électrodynamique quantique n’existait pas encore. En effet, dans certains cas, le champ élec- tromagnétique pouvait être considéré classiquement et en interaction avec la matière quantiﬁée comme, par exemple, les électrons dans un atome. Une telle approche consiste en fait à faire une approximation semi-classique de l’électrodynamique quantique. Comme cela s’est avéré par la suite, la plupart des ré- sultats obtenus se sont révélés être en relatif accord avec la théorie complète. Un exemple important qui met en évidence la polarisation du vide dans l’électrodynamique quantique est l’effet de décalage de Lamb mentionné plus haut. Il faut rappeler que la représentation de la polarisation du vide (mais aussi de l’éner- gie du vide) utilisant les diagrammes de Feynman ne fait intervenir que les boucles fermées associées aux propagateurs des champs de la théorie. Le lecteur pourra consulter la Réf. [138] qui est une revue sur le sujet de l’énergie du vide dans le cadre du problème de la constante cosmologique Λ. Par conséquent, on doit chercher à généraliser en espace-temps courbe les concepts et les méthodes de la théorie quantique des champs habituelles.

Les premières études sur les effets de la gravitation quantique remontent aux travaux de Schrödin- ger de 1932 [139]. Depuis, cette approche semi-classique de la gravitation quantique a permis aux physiciens théoriciens d’obtenir les résultats fascinants concernant la création de particules par des champs gravitationnels dans le domaine de la cosmologie de l’univers primordial ainsi dans la physique quantique des trous noirs (voir les Réfs. [131–137] ainsi que les références qu’elles contiennent). En particulier, il faut mentionner séparément la découverte de la création de particules dans des univers en expansion faite par Leonard Parker [140] et l’émission spontanée des particules par des trous noirs et son caractère ther- mique faite par Stephen Hawking [141]. Depuis le milieu des années soixante-dix, la théorie quantique des champs en espace-temps courbe a connu une avancée considérable sur des sujets comme la déﬁnition du vide quantique, le concept de particules et leur création en espace-temps courbe, le développement de méthodes efﬁcaces permettant de construire et d’évaluer le tenseur d’énergie-impulsion renormalisé (voir ci-dessous), l’étude des phénomènes quantiques dans divers espaces-temps importants en cosmologie (les espaces-temps de de Sitter ou Friedmann-Lemaître-Robertson-Walker) ou en physique des trous noirs (les espaces-temps de Schwarzschild, Reissner-Nordstrom, Kerr ou Kerr-Newman).

8πGN Gµν ψ Tbµν ψ . = c4 〈 | | 〉

Ici, G est le tenseur d’Einstein R 1 R g Λg ou une généralisation d’ordre supérieur de ce tenseur µν µν − 2 µν + µν géométrique alors que ψ Tbµν ψ est la valeur moyenne de l’opérateur tenseur d’impulsion-énergie Tbµν 〈 | | 〉 (il est à valeur sur les distributions) construit à partir des champs quantiques 7 de la théorie (y compris celui des gravitons). La quantité ψ Tbµν ψ est formellement infinie du fait du comportement singulier à 〈 | | 〉 courte distance des fonctions de Green du champ quantique. Afin d’extraire une contribution finie et physiquement raisonnable de cette quantité, il est nécessaire de la régulariser puis de renormaliser toutes les constantes de couplage de la théorie. La valeur moyenne renormalisée ψ Tbµν ψ ren est d’une importance 〈 | | 〉 fondamentale en théorie quantique des champs en espace-temps courbe [131–136] non seulement parce

7. Les champs quantiques sont en fait des opérateurs déﬁnis au sens des distributions.

6 Introduction qu’elle agit comme source dans les équations d’Einstein semi-classiques 8 mais aussi parce qu’elle nous permet d’analyser l’état quantique ψ sans faire aucune référence à son contenu en particules 9. A partir | 〉 du milieu des années soixante-dix, de grands efforts ont permis de développer des techniques rigoureuses de construction formelle et de calcul de la valeur moyenne renormalisée du tenseur d’impulsion-énergie

ψ Tbµν ψ ren (voir les Réfs. [131–135] et leurs références sur le sujet). Les méthodes principales valables 〈 | | 〉 sous certaines conditions sont les suivantes : la méthode de la régularisation adiabatique, la régularisation dimensionnelle, l’approche par les fonctions ζ, la méthode du point-splitting, l’approximation de Brown- Ottewill-Page, l’approximation de DeWitt-Schwinger et la renormalisation dite de Hadamard. Dans ce manuscrit, nous utilisons les deux dernières méthodes :

(i) L’approximation de DeWitt-Schwinger est basée sur le développement de DeWitt-Schwinger des fonctions de Green associées aux champs quantiques massifs et de l’action effective de la théorie consi- dérée. Cette approximation est utilisée dans la limite des grandes masses pour les champs se propageant sur un espace-temps courbe arbitraire. Par exemple, le lecteur peut se reporter aux Réfs. [131– 133, 149–154] pour les bases théoriques ainsi qu’aux Réfs. [155–173] en ce qui concerne ses applications aux trous noirs, aux trous de ver, aux cordes noires, à l’univers très primordial décrit par l’espace-temps de de Sitter et aux univers de Friedmann-Lemaître-Robertson-Walker. Il faut noter que, formellement, elle peut être utilisée lorsque la longueur d’onde de Compton associée à ce champ massif est plus petite qu’une longueur caractéristique construite à partir de la courbure de l’espace- temps. De plus, il est important de rappeler que le développement de DeWitt-Schwinger de l’action effective est purement géométrique et que, par conséquent, le tenseur d’impulsion-énergie renorma- lisé qui lui est associé ne prend pas en compte l’état quantique du champ.

(ii) La renormalisation de Hadamard (pour sa formulation axiomatique, nous nous référons aux monographies de Robert Wald [134] et de Stephen Fulling [133]) est une extension de la méthode du point-splitting [131, 150, 151] qui a été développée en liaison avec la représentation de Hadamard des fonctions de Green (voir, par exemple, les Réfs. [174–188] sur le sujet). Il faut noter que cette représentation des fonctions de Green consiste à considérer le comportement singulier du type Ha- damard en espace-temps courbe ou encore un état quantique du type Hadamard, i.e., un état imitant dans le régime ultraviolet le comportement du vide de Poincaré dans l’espace-temps de Minkowski.

De plus, dans ce manuscrit, nous nous intéressons aux théories quantiques des champs massifs en espace- temps courbe à l’ordre une boucle. En particulier, nous ne considérons que la contribution associée aux champs quantiques massifs pour la valeur moyenne du tenseur d’impulsion-énergie en laissant de côté la partie associée au champ des gravitons. En plus du champ scalaire massif et du champ de Dirac massif, nous discutons ici l’électromagnétisme massif de de Broglie-Proca [189–194] et celui de Stueckelberg [195–198]. Contrairement à la théorie massive de de Broglie-Proca, la théorie proposée par Stueckelberg

8. Donnons quelques exemples d’applications pratiques de ces équations qui ont été utilisées – par Alexei Starobinsky [142] pour montrer que, après l’ère de Planck, les effets quantiques conduisent à l’in- ﬂation de l’univers, i.e., à une phase d’expansion exponentielle décrite par l’espace-temps de de Sitter, – pour analyser la dynamique de l’évaporation des trous noirs à cause du rayonnement de Hawking (voir les Réfs. [143, 144] pour des travaux pionniers sur ce sujet), – pour expliquer l’expansion accélérée de l’univers (voir, par exemple, la Réf. [145]).

9. Il faut préciser que, bien que le contenu d’un état en termes de particules soit bien déﬁni en espace-temps de Minkowski pour un observateur inertiel, la notion de particules n’a plus de sens en espace-temps courbe [141, 146, 147] ou même encore en espace-temps de Minkowski pour un observateur accéléré [148]. Pour comprendre le contenu d’un état, il est nécessaire d’utiliser la notion de détecteur de particules introduite initialement par William George Unruh [148].

7 Introduction préserve la symétrie de jauge locale U(1) de Maxwell. Il faut noter que la théorie non-Maxwellienne (voir les Réfs. [83, 85] pour les revues récentes), où un photon massif très léger est considéré, est nécessaire pour restreindre expérimentalement la masse du photon. Par ailleurs, la théorie de Stuekelberg pourrait être incluse aisément dans le modèle standard des particules élémentaires. En outre, certaines extensions du modèle standard basées sur la théorie des cordes incluent l’existence d’un photon noir très massif pouvant expliquer la matière noire de l’univers (voir, par exemple, la Réf. [199]).

Ce manuscrit constitué des trois articles publiés [200–202] est organisé en deux parties. Dans la première partieI de ce manuscrit, nous nous intéressons à l’action effective à l’ordre une boucle associée aux champs quantiques massifs. Nous présentons le formalisme associé au développement de DeWitt- Schwinger des fonctions de Green et de l’action effective. Puis, nous utilisons ce développement dans la limite des grandes masses en liaison avec un package écrit sous Mathematica permettant d’effectuer les calculs tensoriels efficacement afin d’obtenir une approximation de la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé à divers champs massifs (le champ scalaire massif, le champ de Di- rac massif et le champ de Proca) se propageant sur l’espace-temps de Kerr-Newman. Ce résultat nous permet de retrouver les résultats existant dans la littérature pour les trous noirs de Schwarzschild, de Reissner-Nordström et de Kerr (et d’en corriger certains). En absence de résultats exacts en espace-temps de Kerr-Newman, notre approximation de la valeur moyenne renormalisée du tenseur d’impulsion-énergie pourrait être utile, par exemple, pour étudier l’effet de backreaction des champs quantiques massifs sur cet espace-temps ou sur ses modes quasi-normaux. Nous étudions d’ailleurs un problème simplifié de backreaction consistant à déterminer les modifications de la masse et du moment angulaire du trou noir vues par un observateur à l’infini, et dues à la présence des champs quantiques. Dans la deuxième partieII de ce manuscrit, nous nous intéressons à l’électromagnétisme massif de Stueckelberg. Nous discutons cette théo- rie sur un espace-temps courbe arbitraire de dimension quatre (invariance de jauge de la théorie classique et quantification covariante ; équations d’ondes pour le champ massif de spin-1 Aµ, pour le champ scalaire auxiliaire de Stueckelberg Φ et pour les champs de fantômes C et C∗ ; identité de Ward ; représentation de Hadamard des divers propagateurs de Feynman et développement en séries de Taylor covariantes des coefficients correspondants). Cela nous permet de construire, pour un état quantique de type Hadamard ψ , la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé à la théorie de Stueckelberg. | 〉 Nous avons donné deux expressions différentes mais équivalentes du résultat, l’une d’entre elles nous permettant de discuter la limite de masse nulle dans le but de retrouver certains résultats de la théorie de Maxwell. Comme applications de nos résultats, (i) nous considérons, en espace-temps de Minkowski, l’effet Casimir en dehors d’un milieu parfaitement conducteur et de bord plan, et (ii) nous obtenons, dans les espaces-temps de de Sitter et anti-de Sitter, une expression analytique exacte pour la valeur moyenne dans le vide du tenseur d’impulsion-énergie renormalisé du champ vectoriel massif.

Dans ce manuscrit, nous considérons des espaces-temps courbes de dimension quatre (M , gµν) sans bord (∂M ∅) ayant pour signature de métrique ( , , , ). Nous utilisons les conventions géométriques = − + + + de Hawking et Ellis [130] en ce qui concerne les déﬁnitions de la courbure scalaire R, du tenseur de Ricci

Rpq et du tenseur de Riemann Rpqrs ainsi que la commutation des dérivées covariantes sous la forme

T p ... T p ... R p T t ... Rt T p ... . q...;sr − q...;rs = + trs q... + ··· − qrs t... − ···

De plus, nous travaillons en unités telles que c ~ GN 1. Tout au long de ce manuscrit, nous utilisons = = = les concepts de bi-scalaires, de bi-vecteurs et, plus généralement, de bi-tenseurs développés par certains

8 Introduction mathématiciens [203–206] ainsi que par Bryce DeWitt [131, 149, 154, 207] et ses collaborateurs [150, 151]. En espace-temps courbe arbitraire, ces outils sont fondamentaux pour les méthodes de régularisation et renormalisation basées sur les représentations des fonctions de Green en coordonnées spatiales 10. En particulier, nous utilisons la distance géodétique σ(x, x0), le bi-vecteur de transport parallèle gµν0 (x, x0) et le déterminant de van Vleck-Morette ∆(x, x0) (voir, par exemple, la Réf. [207]) ainsi que le concept de déve- loppement en séries de Taylor covariantes des bi-tenseurs [179, 183, 208, 209]. En ce qui concerne leur déﬁnition, le lecteur pourra consulter l’annexe 2.7 du Chap.2 de ce manuscrit.

10. En général, l’absence des symétries et la courbure de l’espace-temps empêchent de travailler avec la transformation de Fourier.

PARTIE I:

ACTION EFFECTIVE ET TENSEUR D’IMPULSION-ÉNERGIERENORMALISÉ

CHAPITRE 1:

TENSEURD’IMPULSION-ÉNERGIERENORMALISÉASSOCIÉAUXCHAMPS MASSIFS EN ESPACE-TEMPSDE KERR-NEWMAN

En espace-temps de dimension quatre, le développement de DeWitt-Schwinger de l’action effective associée à un champ massif, après renormalisation et dans la limite des grandes masses, se réduit aux termes construits à partir du coefficient de Gilkey-DeWitt a3 qui est de nature purement géométrique. Sa variation par rapport à la métrique nous donne une bonne approximation analytique pour la valeur moyenne renormalisée du tenseur d’impulsion-énergie d’un champ quantique massif. Ici, à partir de l’expression générale de ce tenseur, nous avons obtenu analytiquement la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé au champ scalaire massif, au champ de Dirac massif et au champ de Proca se propageant sur l’espace-temps de Kerr-Newman. Ce résultat nous a permis de retrouver les ré- sultats existant dans la littérature pour les trous noirs de Schwarzschild, de Reissner-Nordström et de Kerr (et d’en corriger certains). En absence de résultats exacts en espace-temps de Kerr-Newman, notre approximation de la valeur moyenne renormalisée du tenseur d’impulsion-énergie pourrait être utile, par exemple, pour étudier l’effet de backreaction des champs quantiques massifs sur cet espace-temps ou sur ses modes quasi-normaux. Nous avons d’ailleurs étudié un problème simplifié de backreaction consistant à déterminer les modifications de la masse et du moment angulaire du trou noir vues par un observateur à l’infini, et dues à la présence des champs quantiques.

13 I. Action effective et tenseur d’impulsion-énergie renormalisé

Renormalized stress tensor for massive ﬁelds in Kerr-Newman spacetime

Andrei Belokogne∗ and Antoine Folacci† Equipe Physique Théorique, Projet COMPA, SPE, UMR 6134 du CNRS et de l’Universitéde Corse, Universitéde Corse, BP 52, F-20250 Corte, France (Dated: September 3, 2014) In a four-dimensional spacetime, the DeWitt-Schwinger expansion of the effective action associated with a massive quantum field reduces, after renormalization and in the large mass limit, to a single term constructed from the purely geometrical Gilkey-DeWitt coefficient a3 and its metric variation provides a good analytical approximation for the renormalized stress-energy tensor of the quantum field. Here, from the general expression of this tensor, we obtain analytically the renormalized stress-energy tensors of the massive scalar field, the massive Dirac field and the Proca field in Kerr- Newman spacetime. It should be noted that, even if, at first sight, the expressions obtained are complicated, their structure is in fact rather simple, involving naturally spacetime coordinates as well as the mass M, the charge Q and the rotation parameter a of the Kerr-Newman black hole and permitting us to recover rapidly the results already existing in the literature for the Schwarzschild, Reissner-Nordströmand Kerr black holes (and to correct them in the latter case). In the absence of exact results in Kerr-Newman spacetime, our approximate renormalized stress-energy tensors could be very helpful, in particular to study the backreaction of massive quantum fields on this spacetime or on its quasinormal modes.

PACS numbers: 04.62.+v, 04.70.Dy

I. INTRODUCTION Misner, Thorne and Wheeler [10] and we adopt units such that ~ = c = GN = 1.) In Eq. (1), Gµν is the Ein- 1 Quantum field theory in curved spacetime (for reviews stein tensor Rµν 2 gµν R + Λgµν or some higher-order generalization of− this geometrical tensor while T and monographs on this subject, see Refs. [1–7]) is a h µν iren semiclassical approximation of quantum gravity which, is the renormalized stress-energy tensor (RSET) of the by treating classically the spacetime metric gµν and con- quantum field or, more precisely, the renormalized ex- sidering from a quantum point of view all the other fields pectation value of the stress-energy tensor operator as- (including the graviton field to at least one-loop order for sociated with the quantum field. reasons of consistency), avoids the difficulties due to the The semiclassical Einstein equations (1) have permit- nonrenormalizability of quantum gravity and provides a ted Starobinsky to show that, after the Planck era, quan- framework which permits us to study the low-energy con- tum effects lead to an inflationary universe, i.e., a uni- sequences of a hypothetical “theory of everything”. It verse with an exponentially expanding de Sitter phase should be recalled that this approach allowed theoretical [11]. They have been also used by several authors to physicists to obtain fascinating results concerning early analyze the dynamics of evaporating black holes due to universe cosmology and quantum black hole physics (see Hawking radiation (see, e.g., Refs. [12, 13] for pioneering Refs. [1–7] and references therein) and, in particular, led work on this topic) and, more recently, they have been to the discovery of particle creation in expanding uni- considered to explain the acceleration of the expansion verses by Parker [8] and of black hole radiance by Hawk- of the universe (see, e.g., Ref. [14]). But unfortunately, ing [9]. Furthermore, this approach also provides the and despite the impressive successes mentioned above, natural theoretical framework to analyze the cosmic mi- the backreaction problem is in general difficult to tackle, crowave background (CMB) observations made in recent not only because the semiclassical Einstein equations (1) years. are a set of coupled nonlinear hyperbolic partial differ- In quantum field theory in curved spacetime, it is con- ential equations but also because it is in general difficult jectured that the backreaction of a quantum field on the to define the right-hand side of these equations, i.e., to spacetime geometry is governed by the semiclassical Ein- construct the RSET. Indeed, the expectation value of stein equations the stress-energy tensor operator is formally infinite and it is necessary to regularize it, i.e., to extract from this Gµν = 8π Tµν ren. (1) formally infinite quantity a meaningful finite part, and h i then to renormalize all the coupling constants appearing (In this article, we use the geometrical conventions of in the problem in order to remove the remaining infinite part. Much work has been done since the mid-1970s in connection with this subject (see Refs. [1–7] and refer- ∗ [email protected] ences therein) and, currently, we have at our disposal † [email protected]

14 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

2 some powerful procedures (such as the adiabatic regular- has never been considered previously and this is certainly ization method, the dimensional regularization method, due to the complexity of the calculations involved. Let us the ζ-function approach, the point-splitting method...) remark that the problems treated in the references previ- permitting us to construct theoretically and without am- ously mentioned mainly concern spherically or cylindri- biguity the RSET. cally symmetric spacetimes and/or Einstein spacetimes It is however important to note that, in four- (i.e., spacetimes satisfying Rµν = Λgµν ) and that the dimensional gravitational backgrounds, except if we work Kerr-Newman spacetime does not belong to one of these under very strong hypotheses (e.g., if we consider field categories. To facilitate our work, we shall write the theories in maximally symmetric spacetimes or if we general expression of the RSET in an irreducible form study massless or conformally invariant field theories on in order to reduce substantially the number of its terms very particular spacetimes), it is in general impossible to and the size of calculations and we shall use Mathemat- obtain an analytical expression for the RSET. In fact, ica packages allowing us to perform tensor algebra very in most cases, it is even impossible to construct, from a efficiently. We shall then obtain, in Sec.II (see also practical point of view, the RSET and, when this is possi- AppendixA for details), analytical expressions for the ble, it is necessary in many cases to perform a numerical RSET of the massive scalar field, the massive Dirac field analysis in order to extract the physical content of the and the Proca field in Kerr-Newman spacetime. It should RSET and, of course, this does not simplify the backre- be noted that, even if, at first sight, the expressions we action problem. So, it is interesting to note that various shall display are complicated, their structure is in fact approaches have been developed which permit us to deal rather simple, involving naturally spacetime coordinates with situations presenting a “lower degree of symmetry” as well as the mass M, the charge Q and the rotation and to construct, in this context, accurate analytical ap- parameter a of the Kerr-Newman black hole and permit- proximations of the RSET (see, e.g., Refs. [15–17] for the ting us to recover rapidly, in Sec.III, the results already theoretical bases of the “Brown-Ottewill-Page approxi- existing in the literature for the Schwarzschild black hole mation” which is limited to static Einstein spacetimes or [23, 40], for the Reissner-Nordströmblack hole [26, 30] Refs. [18–21] for the theoretical bases of the “DeWitt- and for the Kerr black hole [24, 25]. We shall moreover Schwinger approximation” which can be used in an ar- correct the result obtained for the Dirac field in the lat- bitrary spacetime but is limited to massive fields in the ter case. In Sec.IV, we shall conclude by mentioning large mass limit [22]). From a theoretical point of view, some possible applications of our results and by consid- such analytical approximations could be helpful to find ering the shift in mass and angular momentum of the self-consistent solutions to Eq. (1). Kerr-Newman black hole dressed with a massive quan- Here, we focus on the DeWitt-Schwinger approxima- tum field. tion of the RSET. It is based on the DeWitt-Schwinger Before entering into the technical part of our work, it expansion of the effective action associated with a mas- seems to us necessary to recall that, due to its fundamen- sive quantum field and, formally, it can be used only when tal importance in connection with the Hawking effect, the Compton length associated with the massive field is the construction of the RSET in black hole spacetimes much less than a characteristic length constructed from has often been discussed in the past 40 years and, in the curvature of spacetime. In this context, an analyt- Schwarzschild and Reissner-Norströmspacetimes (i.e., in ical expression for the RSET is directly obtained from static spherically symmetric black holes), we have now the metric variation of the renormalized effective action at our disposal exact expressions (which, however, must associated with the massive field. Here, it is important be analyzed numerically) for the RSET associated with to recall that the DeWitt-Schwinger expansion of the ef- various quantum fields and various vacuum states (see, fective action is purely geometrical and to note that, as a e.g., Refs. [41–46] for pioneering work concerning mass- consequence, the approximate RSET does not take into less fields in Schwarzschild spacetime and Ref. [26] for account the quantum state of the field. The literature results concerning both massless and massive fields in concerning the DeWitt-Schwinger approximation and its Schwarzschild and Reissner-Nordströmspacetimes). For applications is rich (see, e.g., Refs. [23–37] for applica- stationary axisymmetric black holes, the situation is less tions to black holes, wormholes and black strings and clear and much more complex. In Kerr spacetime (see, Refs. [38, 39] for a recent work concerning the Friedmann- e.g., Refs. [47–51] and references therein), for massless Lemaˆıtre-Robertson-Walker universes) and it is impor- fields, due both to problems linked with the quantization tant to note that the approximate RSETs obtained from process in this spacetime [47, 52] and to the complexity of the DeWitt-Schwinger expansion of the effective action the mode solutions of the wave equation, the RSET has seem to be in good agreement with exact results (see been calculated only in specific locations and, for massive Ref. [26] where this has been discussed for the Reissner- fields, to our knowledge, there exists no result concerning Nordströmblack hole). the RSET. In Kerr-Newman spacetime, the situation is In this article, we intend to use the DeWitt-Schwinger even worse: we have found no results for this tensor in approximation in order to construct the approximate the literature. RSET for massive fields in Kerr-Newman spacetime. De- We now describe the formalism we shall use in the spite its physical importance, this particular spacetime following. We work in a four-dimensional spacetime

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 15 I. Action effective et tenseur d’impulsion-énergie renormalisé

( , gab) without boundary and we consider more par- of the effective action associated with the massive scalar ticularlyM the massive scalar field φ solution of the Klein- field, the massive Dirac field and the Proca field can be Gordon equation constructed from the Gilkey-DeWitt coefficient a3 and reduces to [18–21] ( m2 ξR)φ = 0, (2) − − the massive spinor field ψ solution of the Dirac equation TABLE I. The coefficients c defining the renormalized effec- µ i (γ µ + m)ψ = 0 (3) tive action (5). ∇ and the massive vector field Aµ solution of the Proca Scalar field Dirac field Proca field (s = 0) (s = 1/2) (s = 1) equation 2 c1 (1/2)ξ (1/5)ξ + 1/56 -3/280 -27/280 − (g m2g R )Aν = 0. (4) c2 1/140 1/28 9/28 µν µν µ ν µν 3 − − ∇ ∇ − c3 (ξ 1/6) 1/864 -5/72 − − c4 (1/30)(ξ 1/6) -1/180 31/60 In the wave equations (2)-(4), m denotes the mass of the − c5 8/945 -25/756 -52/63 fields, in the Klein-Gordon equation (2), ξ is a dimen- − sionless factor which accounts for the possible coupling c6 2/315 47/1260 -19/105 c7 (1/30)(ξ 1/6) -7/1440 -1/10 between the scalar field and the gravitational background c − 1/1260− 19/1260 61/140 and in the Dirac equation (3), γµ denote the usual Dirac 8 c9 17/7560 29/7560 -67/2520 matrices. We recall that, after renormalization and in c10 1/270 -1/108 1/18 the large mass limit, the DeWitt-Schwinger expansion −

1 W = d4x√ g c R R + c R Rpq + c R3 + c RR Rpq + c R Rp Rqr ren 192π2m2 − 1 2 pq 3 4 pq 5 pq r ZM prqs pqrs p qrst pquv rs p q rusv +c6 RpqRrsR + c7 RRpqrsR + c8 RpqR rstR + c9 RpqrsR R uv + c10 RprqsR u vR . (5)

Here, the coefficients ci depend on the field and are given the quantum state of the massive field is not taken into in TableI. In Eq. (5) the integrand is constituted by ten account. So, the renormalized effective action Wren is a Riemann polynomials of order six (in the derivatives of purely geometrical object. the metric tensor) and rank zero (number of free indices). By functional derivation of the effective action (5) with 4 6 It should be noted that terms in 1/m , 1/m ... are respect to the metric tensor gµν , we obtain a purely ge- also present in the full expression of the renormalized ometrical approximation for the RSET associated with effective action Wren (see, e.g., Refs. [19–21]) but here, the massive fields. We have because we assume a large enough mass m, we can neglect 2 δW them. It is also important to recall that, in the large mass T µν = ren (6) h iren √ g δg limit, the nonlocal contribution to Wren associated with − µν which can be written explicitly [53]

(96π2m2) T = d ( R) + d R + d RR + d ( R)R + d R Rp + d R R h µν iren 1 ;µν 2 µν 3 ;µν 4 µν 5 ;p(µ ν) 6 µν p pq pq pq ;pq pq +d7 Rp(µR ν) + d8 R Rpq;(µν) + d9 R Rp(µ;ν)q + d10 R Rµν;pq + d11 R Rpµqν + d12 (R )Rpµqν pq;r p ;qr pqrs p +d13 R R rqp ν) + d14 R R pqr ν) + d15 R Rpqrs;(µν) + d16 R;µR;ν + d17 R;pR (µ | | (µ | | (µ;ν) ;p pq pq p ;q p q pq;r +d18 R;pRµν + d19 R ;µRpq;ν + d20 R ;(µRν)p;q + d21 R µ;qRpν + d22 R µ;qR ν;p + d23 R Rrqp(µ;ν) pq;r pqrs pqr ;s 2 p pq +d24 R Rpµqν;r + d25 R ;µRpqrs;ν + d26 R µ;sRpqrν + d27 R Rµν + d28 RRpµR ν + d29 R RpqRµν pq pq pr q pq r pqr +d30 R RpµRqν + d31 RR Rpµqν + d32 R R rRpµqν + d33 R R (µR rqp ν) + d34 RR µRpqrν | | pqrs p qrs pq rs prqs prs q +d35 Rµν R Rpqrs + d36 R (µR p R qrs ν) + d37 R R pµRrsqν + d38 RpqR Rrµsν + d39 RpqR µR rsν | | | | pqrs t prqs t pqr s t +d40 R RpqtµRrs ν + d41 R R pqµRtrsν + d42 R sRpqrtR µ ν pq pq prqs ;p pq;r +gµν [d43 R + d44 RR + d45 R;pqR + d46 RpqR + d47 Rpq;rsR + d48 R;pR + d49 Rpq;rR pr;q pqrs;t 3 pq p qr prqs +d50 Rpq;rR + d51 Rpqrs;tR + d52 R + d53 RRpqR + d54 RpqR rR + d55 RpqRrsR pqrs p qrst pquv rs p q rusv +d56 RRpqrsR + d57 RpqR rstR + d58 RpqrsR R uv + d59 RprqsR u vR ]. (7)

16 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

TABLE II. The coeﬃcients di deﬁning the expansion of the approximate RSET (7) on the FKWC basis.

FKWC basis Coefficients di Scalar field Dirac field Proca field (s = 0) (s = 1/2) (s = 1) 2 2 (R);µν d1 ξ (2/5)ξ + 3/70 1/70 9/70 6,1 − R Rµν d2 1/140 -1/28 -9/28 −2 RR;µν d3 6 (ξ 1/6)[ξ (1/3)ξ + 1/30] -1/120 -1/10 − − − (R)Rµν d4 (ξ 1/6)(ξ 1/5) 1/120 -7/30 p − − − R R d5 (1/15)(ξ 1/7) 23/840 13/35 ;p(µ ν) − RRµν d6 (1/10)(ξ 1/6) 1/40 -7/60 p − Rp(µR ν) d7 1/42 29/420 337/210 pq R Rpq;(µν) d8 (1/15)(ξ 2/7) -19/420 22/105 2 pq − 2,0 R Rp(µ;ν)q d9 2/105 61/420 34/105 R{ } pq R Rµν;pq d10 1/70 -11/105 -107/210 ;pq − R Rpµqν d11 (2/15)(ξ 3/14) -1/105 1/21 pq − (R )Rpµqν d12 1/105 -17/210 -22/35 pq;r − R R rqp ν) d13 4/105 13/105 46/35 (µ | | p ;qr R (µ R pqr ν) d14 2/35 16/105 116/105 pqrs | | R Rpqrs;(µν) d15 (1/15)(ξ 3/14) 1/210 -1/42 − − 2 R;µR;ν d16 6(ξ 1/4)(ξ 1/6) 0 -1/24 p − − − R;pR (µ;ν) d17 (1/5)(ξ 3/14) 19/840 83/210 ;p − − R;pRµν d18 (1/5)(ξ 17/84) -1/420 -41/84 pq − R ;µRpq;ν d19 (1/15)(ξ 1/4) 0 31/60 pq − R ;(µRν)p;q d20 0 -1/60 -14/15 2 p ;q 1,1 R µ;qRpν d21 1/210 -1/140 221/210 R{ } p q − R µ;qR ν;p d22 1/42 3/35 113/210 pq;r R Rrqp(µ;ν) d23 1/105 -1/21 5/21 pq;r − R Rpµqν;r d24 1/70 -11/105 -107/210 pqrs − R ;µRpqrs;ν d25 (1/15)(ξ 13/56) 1/420 -17/840 pqr ;s − − R µ;sRpqrν d26 1/70 -4/105 -29/105 2 − 3 R Rµν d27 3(ξ 1/6) -1/288 5/24 p − RRpµR ν d28 (2/15)(ξ 1/6) -7/360 -2/5 pq − − R RpqRµν d29 (1/30)(ξ 1/6) 1/180 -31/60 pq − − R RpµRqν d30 2/315 -1/252 1/21 pq − RR Rpµqν d31 (1/15)(ξ 1/6) 11/360 -19/30 pr q − R R rRpµqν d32 1/315 13/1260 33/35 pq r R R (µR rqp ν) d33 1/315 97/1260 -139/105 pqr | | 2 RR µRpqrν d34 (1/15)(ξ 1/6) 7/720 1/5 6,3 pqrs − R Rµν R Rpqrs d35 (1/30)(ξ 1/6) 7/1440 1/10 p qrs − R (µR p R qrs ν) d36 4/315 -73/1260 -74/105 pq rs | | | | − R R pµRrsqν d37 2/315 19/504 -5/42 prqs − RpqR Rrµsν d38 4/315 73/1260 74/105 prs q RpqR µR rsν d39 1/315 -97/1260 -71/105 pqrs t − R RpqtµRrs ν d40 2/315 73/2520 37/105 prqs t R R pqµRtrsν d41 4/63 239/1260 97/105 pqr s t R sRpqrtR µ ν d42 2/315 -73/2520 -37/105 0 2 − 6,1 R d43 ξ + (2/5) ξ 11/280 1/280 9/280 R − 2 − RR d44 6(ξ 1/6)[ξ (1/3) ξ + 1/40] -1/240 19/120 pq − − 0 R;pqR d45 (1/30)(ξ 3/14) 1/420 -1/84 2,0 pq − − R{ } RpqR d46 (1/15)(ξ 5/28) 1/105 -223/420 prqs − − Rpq;rsR d47 (4/15)(ξ 1/7) 3/70 79/105 ;p 3 2 − R;pR d48 6[ξ (13/24)ξ + (17/180)ξ 53/10080] -1/672 163/1680 pq;r − − 0 Rpq;rR d49 (1/15)(ξ 13/56) 3/280 -17/56 1,1 pr;q − − R{ } Rpq;rR d50 1/420 -1/280 11/420 pqrs;t − Rpqrs;tR d51 (1/15)(ξ 19/112) 1/168 51/560 3 − 3 R d52 (1/2)(ξ 1/6) 1/1728 -5/144 pq − − RRpqR d53 (1/60)(ξ 1/6) -1/360 31/120 p qr − RpqR rR d54 1/1890 -1/945 1/630 prqs 0 RpqRrsR d55 1/630 1/315 -53/105 6,3 pqrs − R RRpqrsR d56 (1/60)(ξ 1/6) -7/2880 -1/20 p qrst − − RpqR rstR d57 (2/15)(ξ 1/6) 7/360 2/5 pquv rs − RpqrsR R uv d58 (1/15)(ξ 47/252) -61/15120 -263/2520 p q rusv − − RprqsR R d59 (4/15)(ξ 41/252) -43/1512 -106/315 u v − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 17 I. Action effective et tenseur d’impulsion-énergie renormalisé

This formula is the basic cornerstone of our calculations the space of Riemann polynomials of rank r (see Ref. [54] and it is necessary to comment on it briefly. It should for more precision). It should be finally noted that there be noted that the expression of the RSET has been writ- also exist two geometrical identities [see Eqs. (3.21) and ten in an “irreducible form”. Indeed, if we do not care- (3.22) in Ref. [53]] coming from a topological and a ge- fully take into account the symmetries of the Riemann ometrical constraint due to the four-dimensional nature tensor as well as the Bianchi identities, the metric vari- of spacetime. We could have used them in order to elim- ation of the renormalized effective action (5) could lead inate two other terms in (7) but we have chosen to work to an expression with many of the terms which are lin- with the FKWC basis of Riemann polynomials of order early dependent in a nontrivial way. As a consequence, six which can be used in any dimension. the resulting expression contains too many terms (this is the case in most of the articles dealing with the DeWitt- Schwinger approximation) and, in practice, this increases II. APPROXIMATE RENORMALIZED STRESS significantly the size of calculations. For that reason, in TENSORS IN KERR-NEWMAN SPACETIME Ref. [53], we have expanded the RSET on a standard basis constituted by Riemann polynomials of order six In this section, we consider the massive scalar field, in the derivatives of the metric tensor and constructed the massive Dirac field and the Proca field in the Kerr- from group theoretical considerations by Fulling, King, Newman spacetime and we provide, in the large mass Wybourne and Cummings (FKWC) [54]. This basis is limit, the associated explicit expressions of the approxi- µ described in Secs. 2.1 and 2.2 of Ref. [53] and displayed mate RSET T ν ren. We work with Boyer-Lindquist co- in TableII where we use, furthermore, the FKWC nota- ordinates andh thei spacetime metric then takes the form r r tions s,q and λ ... to denote the various subspaces of [10] R R{ 1 }

∆ a2 sin2 θ 2a sin2 θ(r2 + a2 ∆) (r2 + a2)2 a2∆ sin2 θ Σ ds2 = − dt2 − dtdϕ + − sin2 θ dϕ2 + dr2 + Σ dθ2 (8) − Σ − Σ Σ ∆

where ∆ = r2 2Mr + a2 + Q2 and Σ = r2 + a2 cos2 θ. course, due to the complexity of the Kerr-Newman met- Here M, Q and− J = aM are the mass, the charge ric, the calculations involved cannot be done by hand. and the angular momentum of the black hole while For this reason, we have written the package SETArbi- a is the so-called rotation parameter and we assume traryST (available upon request from the ﬁrst author). M 2 a2 + Q2. The outer horizon is located at r = It is based on the suite of Mathematica packages xAct ≥ + M + M 2 (a2 + Q2), the largest root of ∆. [55] which permits us to perform tensor algebra very ef- By using− the general formula (7) and TableII in con- ﬁciently. nectionp with (8), we can obtain explicitly T µ . Of h ν iren

µ The explicit expressions of the nonzero components of T ν ren can be written in the same form for the diﬀerent massive ﬁelds. We have h i

2 10 5 3 2 q 2p t M r tt Q a T t ren = A p,q [θ, M/r] , (9) h i 2 2 2 2 2 9 M 2 r 40320π m (r + a cos θ) p=0 (q=0 ) X X

2 10 5 3 2 q 2p r M r rr Q a T r ren = A p,q [θ, M/r] , (10) h i 2 2 2 2 2 9 M 2 r 40320π m (r + a cos θ) p=0 (q=0 ) X X

2 10 5 3 2 q 2p θ M r θθ Q a T θ ren = A p,q [θ, M/r] , (11) h i 2 2 2 2 2 9 M 2 r 40320π m (r + a cos θ) p=0 (q=0 ) X X

2 10 5 3 2 q 2p ϕ M r ϕϕ Q a T ϕ ren = A p,q [θ, M/r] , (12) h i 2 2 2 2 2 9 M 2 r 40320π m (r + a cos θ) p=0 (q=0 ) X X

2 11 2 5 3 2 q 2p+1 t M r sin θ tϕ Q a T ϕ ren = A p,q [θ, M/r] , (13) h i 2 2 2 2 2 9 M 2 r 20160π m (r + a cos θ) p=0 (q=0 ) X X

18 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

2 9 4 3 2 q 2p+1 ϕ M r ϕt Q a T t ren = A p,q [θ, M/r] , (14) h i 2 2 2 2 2 9 M 2 r 20160π m (r + a cos θ) p=0 (q=0 ) X X

2 11 4 3 2 q 2p+2 r M r sin θ cos θ rθ Q a T θ ren = A p,q [θ, M/r] , (15) h i 2 2 2 2 2 9 M 2 r 360π m (r + a cos θ) p=0 (q=0 ) X X

2 9 3 2 2 q 2p+2 θ M r sin θ cos θ θr Q a T r ren = A p,q [θ, M/r] . (16) h i 2 2 2 2 2 9 M 2 r 360π m (r + a cos θ) p=0 (q=0 ) X X µν 2 Here, the coefficients A p,q are polynomials of the variables cos θ and M/r with coefficients depending on the field. The interested reader can find these coefficients for the massive scalar field, the massive Dirac field and the Proca field in the subsectionsA1,A2 andA3 of the Appendix.

III. SPECIAL CASES : APPROXIMATE of the effective action for the massive Dirac field from RENORMALIZED STRESS TENSORS IN the (bosonic) Lichnerowicz operator. This error does not SCHWARZSCHILD, REISSNER-NORDSTROM¨ exist in Refs. [20, 21] and in TableI. AND KERR SPACETIMES Similarly, by putting a = 0 into the expressions (9)–(16), we recover the results obtained in Reissner- The structure of the expressions (9)–(16) permits us Nordströmspacetime by Anderson, Hiscock and Samuel to recover very quickly the results corresponding to the [26] (for the scalar field) and by Matyjasek [30] (for the special cases of the Schwarzschild, Reissner-Nordström Dirac and Proca fields). The nonzero components of µ and Kerr spacetimes. T ν ren are By taking Q = 0 and a = 0 we recover the results ob- h i tained in Schwarzschild spacetime by Frolov and Zelnikov M 2 3 Q2 q T t = Att [θ, M/r] , (see Ref. [23] and Sec. 11.3.7 of Ref. [40]). The nonzero h tiren 40320π2m2r8 0,q M 2 µ q=0 components of T ν ren are X h i (21) 2 t M tt T t ren = 2 2 8 A 0,0 [θ, M/r], (17) h i 40320π m r M 2 3 Q2 q T r = Arr [θ, M/r] , h riren 40320π2m2r8 0,q M 2 q=0 M 2 X T r = Arr [θ, M/r], (18) (22) h riren 40320π2m2r8 0,0

2 3 2 q 2 θ M θθ Q θ M θθ T θ ren = A 0,q [θ, M/r] , T θ ren = A 0,0 [θ, M/r], (19) h i 40320π2m2r8 M 2 h i 40320π2m2r8 q=0 X (23) 2 ϕ M ϕϕ T ϕ ren = 2 2 8 A 0,0 [θ, M/r], (20) 2 3 2 q h i 40320π m r ϕ M ϕϕ Q T ϕ ren = A 0,q [θ, M/r] , µν h i 40320π2m2r8 M 2 where the coefficients A do not depend explicitly of q=0 0,0 X θ. Due to the spherical symmetry of the Schwarzschild (24) θθ ϕϕ black hole, A 0,0 = A 0,0 and, as a consequence, θ ϕ µν T θ ren = T ϕ ren. It should be noted that, for the where the coefficients A 0,q do not depend explicitly Dirach i field, Frolovh i and Zelnikov have forgotten a multi- of θ. Due to the spherical symmetry of the Reissner- θθ ϕϕ plicative factor 1/2. The absence of this factor seems to Nordströmblack hole, A 0,q = A 0,q and, as a conse- be due to an error of these authors in their construction quence, T θ = T ϕ . h θiren h ϕiren

Finally, by putting Q = 0 into the expressions (9)–(16), we can ﬁnd the results in Kerr spacetime. The nonzero components of T µ are h ν iren 2 10 5 2p t M r tt a T t ren = A p,0 [θ, M/r] , (25) h i 2 2 2 2 2 9 r 40320π m (r + a cos θ) p=0 X

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 19 I. Action effective et tenseur d’impulsion-énergie renormalisé

2 10 5 2p r M r rr a T r ren = A p,0 [θ, M/r] , (26) h i 2 2 2 2 2 9 r 40320π m (r + a cos θ) p=0 X

2 10 5 2p θ M r θθ a T θ ren = A p,0 [θ, M/r] , (27) h i 2 2 2 2 2 9 r 40320π m (r + a cos θ) p=0 X

2 10 5 2p ϕ M r ϕϕ a T ϕ ren = A p,0 [θ, M/r] , (28) h i 2 2 2 2 2 9 r 40320π m (r + a cos θ) p=0 X

2 11 2 5 2p+1 t M r sin θ tϕ a T ϕ ren = A p,0 [θ, M/r] , (29) h i 2 2 2 2 2 9 r 20160π m (r + a cos θ) p=0 X

2 9 4 2p+1 ϕ M r ϕt a T t ren = A p,0 [θ, M/r] , (30) h i 2 2 2 2 2 9 r 20160π m (r + a cos θ) p=0 X

2 11 4 2p+2 r M r sin θ cos θ rθ a T θ ren = A p,0 [θ, M/r] , (31) h i 2 2 2 2 2 9 r 360π m (r + a cos θ) p=0 X

2 9 3 2p+2 θ M r sin θ cos θ θr a T r ren = A p,0 [θ, M/r] . (32) h i 2 2 2 2 2 9 r 360π m (r + a cos θ) p=0 X

It should be noted that the approximate RSETs for the associated with a massive quantum field. As a conse- massive scalar field, the massive Dirac field and the Proca quence, they do not take into account the quantum state field in Kerr spacetime have been obtained a long time of the field and are only valid in the large mass limit. In ago by Frolov and Zelnikov [24, 25]. At first sight, their particular, it is important to note that they neglect the results and ours are different because theirs are given in existence of supperradiance instabilities for massive fields 1 terms of the complex spin coefficient ρ = (r ia cos θ)− in rotating black holes (see Refs. [56–58] for pioneering and its powers. In our opinion, our formulas− − are clearer. work on this subject and, e.g., Ref. [59] for a more recent Moreover, we have checked that both results are in agree- article concerning more particularly the Kerr-Newman ment for the scalar and vector fields while, for the Dirac black hole) which seems quite reasonable because the in- field, they differ by the multiplicative factor 1/2 previ- stability time scale is very long in the large mass limit. ously discussed. We have also shown that our results permit us to recover those already existing in the literature for the RSET in Schwarzschild, Reissner-Nordströmand Kerr spacetimes and we hope they will be useful to people who will want, IV. CONCLUSION in the future, to make the exact calculations by taking into account, in particular, the quantum state of the mas- In this article, we have obtained an analytical approx- sive field. But, in our opinion, the complexity of our re- imation for the RSET of the massive scalar, spinor and sults leads us to think that an exact expression for the vector fields in Kerr-Newman spacetime. To our knowl- RSET of a massive field in Kerr-Newman spacetime is edge, no other result concerning this tensor in Kerr- completely out of reach. Newman spacetime can be found within the literature. However, this should not prevent us from discussing The Mathematica package SETArbitraryST which per- the following fundamental question : what exact result(s) mits us to derive the expressions of the RSET T h µν iren may be associated with our approximation? This is far in Kerr-Newman spacetime as well as another package from obvious and, here, we shall only provide a partial which contains these explicit expressions are available answer to this important question. Let us first consider upon request from the first author. the case of Schwarzschild spacetime. It is well known The approximate expressions obtained are based on that, in this gravitational background, three Hadamard the DeWitt-Schwinger expansion of the effective action

20 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

momentum of the black hole (measured by a distant ob- TABLE III. The coefficients αi and βi appearing in the shift server) due to the RSET. Such an approach has already formulas (35) and (36). been considered by Frolov and Zelnikov for the Kerr black Scalar field Dirac field Proca field hole [24, 25] and its extension to the Kerr-Newman black (s = 0) (s = 1/2) (s = 1) hole is tractable and permits us to emphasize the role α0 392(9ξ 2) 196 -1176 − − of the black hole charge. For a stationary axisymmetric α1 14(63ξ 16) 7 210 − black hole, we recall that the mass MD and the angular α2 66 -162 978 − momentum JD of the black hole dressed with a quantum β0 60(84ξ 17) 480 -1980 − − field can be expressed in terms of its mass M and its β1 9(252ξ 53) 180 -837 − − angular momentum J by (see, e.g., Ref. [60]) β2 20(714ξ 145) -640 19580 − 1 M M = 2 T µ gµ T ρ ξν d D − h ν iren − 2 ν h ρiren Sµ ZS (33) vacua are physically relevant (see Ref. [41] and refer- and ences therein): the so-called Hartle-Hawking ( H ), Un- | i ruh ( U ) and Boulware ( B ) vacua. In Ref [26], it has µ ν | i | i JD J = T ν renψ d µ (34) been shown that the DeWitt-Schwinger approximation − − h i S µ ZS (17)–(20) is a good approximation of H T ν H ren (i.e., h | | i µ of the exact RSET in the Hartle-Hawking vacuum) for where Tµν ren is the RSET of the quantum field, ξ = h i small and intermediate values of s = r/(2M) 1. As (∂/∂t)µ and ψµ = (∂/∂ϕ)µ denote the two Killing vec- noted by Frolov and Zelnikov [25], the differences− be- tors of the Kerr-Newman black hole, is any space- S tween this mean value and the mean values in the Unruh like hypersurface that extends from the outer horizon vacuum U and the Boulware vacuum B are propor- at r+ to spatial infinity and d µ is the associated sur- S tional to| thei factor exp( m/T ) and this| differencei can face element. In the following, we take for the hy- BH S be neglected everywhere− except close to the horizon. As a persurface defined by t = const and we have therefore 2 2 2 consequence, the DeWitt-Schwinger approximation (17)– d µ = (r + a cos θ) sin θdrdθdϕ (dt)µ. By using the S − (20) is certainly a good approximation of U T µ U expressions (9)-(16) and assuming furthermore a M ν ren and B T µ B for intermediate valuesh of |s. Similar| i and Q M (for arbitrary values of the parameters a ν ren considerationsh | | i apply in the Reissner-Nordströmspace- and Q, the expressions obtained are too complicated to time. By contrast, in Kerr and Kerr-Newman spacetimes be interesting), we obtain the problem is much more complicated due, in particu- M lar, to the superradiant modes (see, e.g., Ref. [48]) and to M M = D − 336 7!πm2M 4 the nonexistence of Hadamard states which respect the × symmetries of the spacetime and are regular everywhere a 2 Q 2 [52]. However, it seems that these difficulties are natu- α0 + α1 + α2 + ... (35) × ( M M ) rally eliminated for the fermionic fields [51] and can be, in some sense, circumvented for bosonic fields [47–50] and, and in fact, one can consider that nonconventional Hartle- 1 a Hawking, Unruh and Boulware vacua exist in Kerr space- J J = D 480 7!πm2M 2 M time (and probably in Kerr-Newman spacetime). In our − × 2 opinion, the DeWitt-Schwinger approximation (9)–(16) a 2 Q β0 + β1 + β2 + ... (36) is certainly a good approximation of the RSETs asso- × ( M M ) ciated with these nonconventional vacua, but, of course, the region of space where this approximation can be used where the dots denote terms of fourth-order [i.e., in must strongly depend on the considered vacuum. (a/M)4 or in (Q/M)4 or in a2Q2/M 4]. The coefficients In the absence of exact results in Kerr-Newman space- αi and βi appearing in Eqs. (35) and (36) are given in time, our approximate RSETs could be very helpful, in TableIII. For Q = 0 (i.e., for the Kerr black hole), our particular to study the backreaction of massive quantum results correct some errors made in Refs. [24, 25] for the fields on this gravitational background. In Refs. [28, 29, Dirac field (all the coefficients αi and βi are incorrect) 31, 33, 35, 36] which deal with massive field theories on but also for the scalar and vector fields (the coefficient the Schwarzschild and Reissner-Nordströmblack holes, α1 is incorrect). some aspects of the backreaction problem have been con- Our results could be also used to study the quasinor- sidered. Unfortunately, here, for the Kerr-Newman black mal modes of the Kerr-Newman black hole dressed by a hole, due to the complexity of the RSET, it seems impos- massive quantum field. Similar problems have been con- sible to find self-consistent solutions to the semiclassical sidered in Refs. [35, 36] for spherically symmetric black Einstein equations (1). However, it is possible to sim- holes. The extension to the Kerr-Newman black hole is plify considerably the backreaction problem by limiting far from obvious, not only because of the RSET complex- us to the determination of the shift in mass and angular ity but also because the subject of massive quasinormal

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 21 I. Action effective et tenseur d’impulsion-énergie renormalisé

modes in this spacetime is rather difficult and has been the “CollectivitéTerritoriale de Corse” for its support little studied (see, however, for recent articles dealing through the COMPA project. with this subject and which could be helpful, Ref. [61] where the uncharged massless scalar field is considered and Ref. [62] where the charged massive scalar field is studied).

ACKNOWLEDGMENTS

We wish to thank Yves D´ecanini, Mohamed Ould El Hadj and Julien Queva for various discussions and

µν Appendix A: Coeﬃcients A p,q [θ, M/r]

1. Massive scalar ﬁeld

For the massive scalar ﬁeld, the coeﬃcients Att appearing in the expression (9) of T t are p,q h tiren Att [θ, M/r] = 180(112ξ 25) 8(5544ξ 1237) (M/r) (A1a) 0,0 − − − Att [θ, M/r] = 1080 192(294ξ 41) (M/r) + 4(36456ξ 6845) (M/r)2 (A1b) 0,1 − − − Att [θ, M/r] = 8(5096ξ 883) (M/r)2 48(3164ξ 613) (M/r)3 (A1c) 0,2 − − − Att [θ, M/r] = 52(882ξ 179) (M/r)4 (A1d) 0,3 − Att [θ, M/r] = 4860(112ξ 25) cos2 θ + 288 2(2303ξ 513) cos2 θ + 14ξ 3 (M/r) (A1e) 1,0 − − − − Att [θ, M/r] = 1080 7 cos2 θ 12 + 144 5(1344ξ 275) cos2 θ 224ξ 219 (M/r) 1,1 − − − − − 2 2 16 7(26556ξ 5903) cos θ + 2(462 ξ 769) (M/r) (A1f) − − − tt 2 2 A 1,2 [θ, M/r ] = 16 49(472ξ 95) cos θ 16(140 ξ + 43) (M/r) − − − 2 3 +16 28(4553ξ 1016) cos θ + 1120ξ 1559 (M/r) (A1g) − − tt 2 4 A 1,3 [θ, M/r ] = 4 (109942ξ 24683) cos θ + 4(455 ξ 421) (M/r) (A1h) − − − Att [θ, M/r] = 7560(112ξ 25) cos4 θ 144 cos2 θ (24836ξ 5525) cos2 θ + 18(14ξ 3) (M/r) (A1i) 2,0 − − − − Att [θ, M/r] = 2160 4 cos4 θ + 6 cos2 θ 15 2,1 − − 144 8(546ξ 149) cos4 θ 3(672ξ 173) cos2 θ + 335 (M/r) − − − − 2 2 2 +24 cos θ 5(43680ξ 9941) cos θ + 4(1470ξ + 61) (M/r ) (A1j) − tt 4 2 2 A 2,2 [θ, M/r] = 96 180 cos θ + 2(784ξ 227) cos θ 253 (M/r) − − − 2 2 3 16 cos θ 7(20612 ξ 4833) cos θ + 2(4424ξ + 377) (M/r ) (A1k) − − tt 2 2 4 A 2,3 [θ, M/r] = 4 cos θ (72310ξ 17861) cos θ + 56(175 ξ + 17) (M/r) (A1l) − Att [θ, M/r] = 7560(112ξ 25) cos6 θ + 96 cos4 θ 2(8631ξ 1916) cos2 θ 15(14ξ 3) (M/r) (A1m) 3,0 − − − − Att [θ, M/r] = 2160 4 cos6 θ 24 cos4 θ + 15 cos2 θ + 10 3,1 − 240 cos2 θ 7(768ξ 169) cos4 θ 3(224ξ + 47) cos2 θ + 318 (M/r) − − − 4 2 2 16 cos θ (90636 ξ 20021) cos θ + 6(630ξ 101) (M/r) (A1n) − − − tt 2 4 2 2 A 3,2 [θ, M/r] = 16 cos θ (23128ξ 5195) cos θ 24(392 ξ 1) cos θ + 2700 (M/r) − − − 4 2 3 +16 cos θ 14(1414ξ 291) cos θ + 3472ξ 939 (M/r) (A1o) − − tt 4 2 4 A 3,3 [θ, M/r] = 4 cos θ (2506ξ 9) cos θ + 4(455ξ 253) (M/r) (A1p) − − − Att [θ, M/r] = 4860(112ξ 25) cos8 θ 72 cos6 θ (1456ξ 321) cos2 θ 20(14ξ 3) (M/r) (A1q) 4,0 − − − − − − Att [θ, M/r] = 1080 cos2 θ 7 cos6 θ 12 cos4 θ 30 cos2 θ + 40 4,1 − −

22 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

+720 cos4 θ 2(252ξ 65) cos4 θ (224ξ 107) cos2 θ 39 (M/r) − − − − 6 2 2 +4 cos θ (9240 ξ 2069) cos θ 8(42ξ 17) (M/r) (A1r) − − − tt 4 4 2 2 A 4,2 [θ, M/r] = 8 cos θ 13(392ξ 151) cos θ 8(560ξ 503) cos θ 2364 (M/r) (A1s) − − − − − tt A 4,3 [θ, M/r] = 0 (A1t) Att [θ, M/r] = 180(112ξ 25) cos10 θ (A1u) 5,0 − Att [θ, M/r] = 1080 cos4 θ cos2 θ 2 cos4 θ 10 cos2 θ + 10 (A1v) 5,1 − − − tt A 5,2 [θ, M/r] = 0 (A1w) tt A 5,3 [θ, M/r] = 0. (A1x)

For the massive scalar ﬁeld, the coeﬃcients Arr appearing in the expression (10) of T r are p,q h riren

Arr [θ, M/r] = 252(32ξ 7) + 56(216ξ 47) (M/r) (A2a) 0,0 − − − Arr [θ, M/r] = 216 + 128(147ξ 40) (M/r) 4(8232ξ 2081) (M/r)2 (A2b) 0,1 − − − Arr [θ, M/r] = 8(1456ξ 383) (M/r)2 + 112(252ξ 65) (M/r)3 (A2c) 0,2 − − − Arr [θ, M/r] = 4(1638ξ 421) (M/r)4 (A2d) 0,3 − − Arr [θ, M/r] = 252(32ξ 7) 31 cos2 θ 4 2016(124ξ 27) cos2 θ (M/r) (A2e) 1,0 − − − − Arr [θ, M/r] = 216 3 cos2 θ 8 288 7(186ξ 41) cos2 θ 2(91ξ 27) (M/r) 1,1 − − − − − − 2 2 +48(9744ξ 2053) cos θ (M/r) (A2f) − Arr [θ, M/r] = 8 7(2288ξ 503) cos2 θ 2800ξ + 867 (M/r)2 16(16268ξ 3303) cos2 θ (M/r)3 (A2g) 1,2 − − − − rr 2 4 A 1,3 [θ, M/r] = 4(11326 ξ 2197) cos θ (M/r) (A2h) − Arr [θ, M/r] = 3528(32ξ 7) cos2 θ 11 cos2 θ 8 + 1008(336ξ 73) cos4 θ (M/r) (A2i) 2,0 − − − − Arr [θ, M/r] = 432 6 cos4 θ 6 cos2 θ 5 2,1 − − − 2 2 4 2 +96 cos θ (14826ξ 3187) cos θ 6(2107 ξ 444) (M/r) 8(44520ξ 9851) cos θ (M/r) (A2j) − − − − − rr 2 2 2 A 2,2 [θ, M/r] = 8 cos θ (50064ξ 10603) cos θ 50064 ξ + 10279 (M/r) − − − 4 3 +16(7196ξ 1709) cos θ (M/r) (A2k) − Arr [θ, M/r] = 4(3346ξ 951) cos4 θ (M/r)4 (A2l) 2,3 − − Arr [θ, M/r] = 3528(32ξ 7) cos4 θ 17 cos2 θ 20 + 672(84ξ 19) cos6 θ (M/r) (A2m) 3,0 − − − Arr [θ, M/r] = 432 cos2 θ 4 cos4 θ + 6 cos2 θ 15 3,1 − − 4 2 6 2 32 cos θ (59010ξ 13061) cos θ 18(4025ξ 893) (M/r) 16(4704ξ 1049) cos θ (M/r) (A2n) − − − − − − rr 4 2 2 6 3 A 3,2 [θ, M/r] = 8 cos θ (55664ξ 12563) cos θ 68880ξ + 15649 (M/r) + 16(1036ξ 235) cos θ (M/r) (A2o) − − − rr 6 4 A 3,3 [θ, M/r] = 4(182ξ 81) cos θ (M/r) (A2p) − − Arr [θ, M/r] = 252(32ξ 7) cos6 θ 85 cos2 θ 112 72(392ξ 87) cos8 θ (M/r) (A2q) 4,0 − − − − − Arr [θ, M/r] = 216 cos4 θ 3 cos4 θ 28 cos2 θ + 30 4,1 − 6 2 8 2 +96 cos θ (4410ξ 953) cos θ 18(315ξ 68) (M/r) + 4(1848ξ 367) cos θ (M/r) (A2r) − − − − rr 6 2 2 A 4,2 [θ, M/r] = 8 cos θ 2(2912ξ 577) cos θ 7280 ξ + 1429 (M/r) (A2s) rr − − − A 4,3 [θ, M/r] = 0 (A2t) Arr [θ, M/r] = 252(32ξ 7) cos8 θ 3 cos2 θ 4 (A2u) 5,0 − − rr 6 4 2 A 5,1 [θ, M/r] = 216 cos θ 3 cos θ 12 cos θ + 10 (A2v) rr − A 5,2 [θ, M/r] = 0 (A2w) rr A 5,3 [θ, M/r] = 0. (A2x)

For the massive scalar ﬁeld, the coeﬃcients Aθθ appearing in the expression (11) of T θ are p,q h θiren

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 23 I. Action effective et tenseur d’impulsion-énergie renormalisé

Aθθ [θ, M/r] = 756(32ξ 7) 8(7056ξ 1543) (M/r) (A3a) 0,0 − − − Aθθ [θ, M/r] = 648 16(4116ξ 1093) (M/r) + 12(15288ξ 3649) (M/r)2 (A3b) 0,1 − − − − Aθθ [θ, M/r] = 16(2912ξ 739) (M/r)2 16(11928ξ 2851) (M/r)3 (A3c) 0,2 − − − Aθθ [θ, M/r] = 28(2106ξ 497) (M/r)4 (A3d) 0,3 − Aθθ [θ, M/r] = 252(32ξ 7) 85 cos2 θ 4 + 288(5516ξ 1207) cos2 θ (M/r) (A3e) 1,0 − − − − Aθθ [θ, M/r] = 648 cos2 θ + 4 + 144 (8204ξ 1763) cos2 θ 588ξ + 169 (M/r) 1,1 − − − 2 2 16(223104ξ 48331) cos θ (M/r) (A3f) − − Aθθ [θ, M/r] = 8 (55664ξ 11807) cos2 θ 7280ξ + 1969 (M/r)2 + 112(22048ξ 4741) cos2 θ (M/r)3 (A3g) 1,2 − − − − θθ 2 4 A 1,3 [θ, M/r] = 4(134414 ξ 28727) cos θ (M/r) (A3h) − − Aθθ [θ, M/r] = 3528(32ξ 7) cos2 θ 17 cos2 θ 8 1008(4088ξ 895) cos4 θ (M/r) (A3i) 2,0 − − − − Aθθ [θ, M/r] = 432 4 cos4 θ 14 cos2 θ 5 2,1 − − 2 2 4 2 48 cos θ 7(5796ξ 1261) cos θ 3(10444 ξ 2181) (M/r) + 8(766920ξ 168349) cos θ (M/r) (A3j) − − − − − θθ 2 2 2 A 2,2 [θ, M/r] = 8 cos θ 7(7152ξ 1561) cos θ 68880ξ + 14029 (M/r) − − 4 3 272(10136ξ 2241) cos θ (M/r) (A3k) − − Aθθ [θ, M/r] = 4(88802ξ 19981) cos4 θ (M/r)4 (A3l) 2,3 − Aθθ [θ, M/r] = 3528(32ξ 7) cos4 θ 11 cos2 θ 20 + 672(2772ξ 607) cos6 θ (M/r) (A3m) 3,0 − − − − Aθθ [θ, M/r] = 1296 cos2 θ 2 cos4 θ 2 cos2 θ 5 3,1 − − 4 2 6 2 +16 cos θ (50820ξ 11489) cos θ 9(14980ξ 3363) (M/r) 16(105168ξ 23071) cos θ (M/r) (A3n) − − − − − θθ 4 2 2 A 3,2 [θ, M/r] = 8 cos θ (16016ξ 3845) cos θ 50064ξ+ 11899 (M/r) − − − 6 3 +16(24304ξ 5305) cos θ (M/r) (A3o) − Aθθ [θ, M/r] = 4(3962ξ 773) cos6 θ (M/r)4 (A3p) 3,3 − − Aθθ [θ, M/r] = 252(32ξ 7) cos6 θ 31 cos2 θ 112 3528(32ξ 7) cos8 θ (M/r) (A3q) 4,0 − − − − Aθθ [θ, M/r] = 648 cos4 θ cos4 θ + 4 cos2 θ 10 4,1 − 6 2 8 2 48 cos θ 2(1260ξ 253) cos θ 3(2660ξ 559) (M/r) + 4(10248ξ 2267) cos θ (M/r) (A3r) − − − − − θθ 6 2 2 A 4,2 [θ, M/r] = 8 cos θ (1456ξ 167) cos θ 2800ξ + 327 (M/r) (A3s) − − θθ A 4,3 [θ, M/r] = 0 (A3t) Aθθ [θ, M/r] = 252(32ξ 7) cos8 θ cos2 θ 4 (A3u) 5,0 − − − Aθθ [θ, M/r] = 216 cos6 θ cos4 θ 8 cos2 θ + 10 (A3v) 5,1 − − θθ A 5,2 [θ, M/r] = 0 (A3w) θθ A 5,3 [θ, M/r] = 0. (A3x) For the massive scalar ﬁeld, the coeﬃcients Aϕϕ appearing in the expression (12) of T ϕ are p,q h ϕiren Aϕϕ [θ, M/r] = 756(32ξ 7) 8(7056ξ 1543) (M/r) (A4a) 0,0 − − − Aϕϕ [θ, M/r] = 648 16(4116ξ 1093) (M/r) + 12(15288ξ 3649) (M/r)2 (A4b) 0,1 − − − − Aϕϕ [θ, M/r] = 16(2912ξ 739) (M/r)2 16(11928ξ 2851) (M/r)3 (A4c) 0,2 − − − Aϕϕ [θ, M/r] = 28(2106ξ 497) (M/r)4 (A4d) 0,3 − Aϕϕ [θ, M/r] = 20412(32ξ 7) cos2 θ + 288 10(553ξ 121) cos2 θ 14ξ + 3 (M/r) (A4e) 1,0 − − − − Aϕϕ [θ, M/r] = 216 41 cos2 θ 56 + 288 (3920ξ 953) cos2 θ 4(28ξ 39) (M/r) 1,1 − − − − 2 2 16 9(24892ξ 5541) cos θ 2(462 ξ 769) (M/r) (A4f) − − − − ϕϕ 2 2 A 1,2 [θ, M/r] = 16 2(13216ξ 3305) cos θ 2240 ξ + 1691 (M/r) − − − 2 3 +16 2(77728ξ 17373) cos θ 1120ξ + 1559 (M/r) (A4g) − − ϕϕ 2 4 A 1,3 [θ, M/r] = 4 (136234ξ 30411) cos θ 4(455 ξ 421) (M/r) (A4h) − − − −

24 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

Aϕϕ [θ, M/r] = 31752(32ξ 7) cos4 θ 144 cos2 θ (28868ξ 6319) cos2 θ 18(14ξ 3) (M/r) (A4i) 2,0 − − − − − Aϕϕ [θ, M/r] = 432 22 cos4 θ + 38 cos2 θ 75 2,1 − 48 13(1176ξ 205) cos4 θ 126(48ξ 11) cos2 θ 1005 (M/r) − − − − − 2 2 2 +8 cos θ (784560ξ 167617) cos θ 12(1470ξ + 61) (M/r ) (A4j) − − ϕϕ 4 2 2 A 2,2 [θ, M/r] = 48 414 cos θ 7(448ξ 87) cos θ 506 (M/r) − − − 2 2 3 16 cos θ (181160 ξ 37343) cos θ 2(4424ξ + 377) (M/r) (A4k) − − − ϕϕ 2 2 4 A 2,3 [θ, M/r] = 4 cos θ (98602ξ 19029) cos θ 56(175 ξ + 17) (M/r) (A4l) − − Aϕϕ [θ, M/r] = 31752(32ξ 7) cos6 θ + 96 cos4 θ 2(9597ξ 2102) cos2 θ + 15(14ξ 3) (M/r) (A4m) 3,0 − − − Aϕϕ [θ, M/r] = 432 22 cos6 θ 132 cos4 θ + 75 cos2 θ + 50 3,1 − − 32 cos2 θ 2(23520ξ 5293) cos4 θ 18(280ξ 199) cos2 θ 2385 (M/r) − − − − − 4 2 2 16 cos θ (108948ξ 23677) cos θ 6(630ξ 101) (M/r) (A4n) − − − − ϕϕ 2 4 2 2 A 3,2 [θ, M/r] = 16 cos θ 2(13216ξ 2927) cos θ 3(3136 ξ 1509) cos θ 2700 (M/r) − − − − 4 2 3 +16 cos θ 28(992ξ 223) cos θ 3472ξ + 939 (M/r) (A4o) − − ϕϕ 4 2 4 A 3,3 [θ, M/r] = 4 cos θ 7(826ξ 255) cos θ 4(455 ξ 253) (M/r) (A4p) − − − − Aϕϕ [θ, M/r] = 20412(32ξ 7) cos8 θ 72 cos6 θ (1288ξ 283) cos2 θ + 20(14ξ 3) (M/r) (A4q) 4,0 − − − − − Aϕϕ [θ, M/r] = 216 cos2 θ 41 cos6 θ 76 cos4 θ 150 cos2 θ + 200 4,1 − − − +48 cos4 θ 10(882ξ 169) cos4 θ 6(560ξ + 11) cos2 θ + 585 (M/r) − − 6 2 2 +4 cos θ (9912 ξ 2131) cos θ + 8(42ξ 17) (M/r) (A4r) − − ϕϕ 4 4 2 2 A 4,2 [θ, M/r] = 16 cos θ (2912ξ 253) cos θ (2240ξ + 1009) cos θ + 1182 (M/r) (A4s) ϕϕ − − − A 4,3 [θ, M/r] = 0 (A4t) Aϕϕ [θ, M/r] = 756(32ξ 7) cos10 θ (A4u) 5,0 − ϕϕ 4 2 4 2 A 5,1 [θ, M/r] = 216 cos θ cos θ 2 3 cos θ 50 cos θ + 50 (A4v) ϕϕ − − A 5,2 [θ, M/r] = 0 (A4w) ϕϕ A 5,3 [θ, M/r] = 0. (A4x) For the massive scalar ﬁeld, the coeﬃcients Atϕ appearing in the expression (13) of T t are p,q h ϕiren Atϕ [θ, M/r] = 72(84ξ 17) (M/r) (A5a) 0,0 − − Atϕ [θ, M/r] = 2160 + 7056 (M/r) + 28(672ξ 293) (M/r)2 (A5b) 0,1 − − Atϕ [θ, M/r] = 3444 (M/r)2 32(609ξ 253) (M/r)3 (A5c) 0,2 − − − Atϕ [θ, M/r] = 72(91ξ 32) (M/r)4 (A5d) 0,3 − Atϕ [θ, M/r] = 144 (910ξ 181) cos2 θ 14ξ + 3 (M/r) (A5e) 1,0 − − Atϕ [θ, M/r] = 2160 cos2 θ 6 72 33 cos2 θ 397 (M/r) 1,1 − − − 2 2 16 (18606ξ 3505) cos θ 462ξ + 769 (M/r) (A5f) − − − tϕ 2 2 2 3 A 1,2 [θ, M/r] = 12 13 cos θ + 1191 (M/r) + 8 7(3836ξ 677) cos θ 1120ξ + 1559 (M/r) (A5g) − − − tϕ 2 4 A 1,3 [θ, M/r] = 8 (6118 ξ 1011) cos θ 455ξ + 421 (M/r) (A5h) − − − Atϕ [θ, M/r] = 144 cos2 θ 10(189ξ 37) cos2 θ 9(14ξ 3) (M/r) (A5i) 2,0 − − − − Atϕ [θ, M/r] = 4320 2 cos4 θ 2 cos2 θ 5 72 240 cos4 θ 353 cos2 θ 335 (M/r) 2,1 − − − − − 2 2 2 +8 cos θ (55860ξ 9617) cos θ 6(1470 ξ + 61) (M/r) (A5j) − − tϕ 4 2 2 A 2,2 [θ, M/r] = 12 671 cos θ 1381 cos θ 1012 (M/r) − − 2 2 3 16 cos θ (14014 ξ 2133) cos θ 4424ξ 377 (M/r) (A5k) − − − − tϕ 2 2 4 A 2,3 [θ, M/r] = 8 cos θ (4123ξ 530) cos θ 14(175 ξ + 17) (M/r) (A5l) − − Atϕ [θ, M/r] = 144 cos4 θ (714ξ 139) cos2 θ + 5(14ξ 3) (M/r) (A5m) 3,0 − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 25 I. Action effective et tenseur d’impulsion-énergie renormalisé

Atϕ [θ, M/r] = 2160 cos6 θ + 9 cos4 θ 15 cos2 θ 5 72 cos2 θ 73 cos4 θ + 261 cos2 θ 530 (M/r) 3,1 − − − − 4 2 2 16 cos θ (7266ξ 1525) cos θ 3(630ξ 101) (M/r) (A5n) − − − − tϕ 2 4 2 2 A 3,2 [θ, M/r] = 12 cos θ 289 cos θ + 531 cos θ 1800 (M/r) − 4 2 3 +8 cos θ (4508ξ 1231) cos θ 3472ξ + 939 (M/r) (A5o) − − tϕ 4 2 4 A 3,3 [θ, M/r] = 8 cos θ (364ξ 191) cos θ 455 ξ + 253 (M/r) (A5p) − − − Atϕ [θ, M/r] = 72 cos6 θ (56ξ 11) cos2 θ + 10(14ξ 3) (M/r) (A5q) 4,0 − − − Atϕ [θ, M/r] = 2160 cos2 θ cos6 θ 6 cos4 θ + 10 + 72 cos4 θ 36 cos4 θ 217 cos2 θ + 195 (M/r) 4,1 − − − 6 2 2 +4 cos θ 9(56ξ 11) cos θ + 4(42ξ 17) (M/r) (A5r) − − tϕ 4 4 2 2 A 4,2 [θ, M/r] = 12 cos θ 108 cos θ 721 cos θ + 788 (M/r) (A5s) − − tϕ A 4,3 [θ, M/r] = 0 (A5t) tϕ A 5,0 [θ, M/r] = 0 (A5u) Atϕ [θ, M/r] = 2160 cos4 θ cos4 θ 5 cos2 θ + 5 (A5v) 5,1 − − tϕ A 5,2 [θ, M/r] = 0 (A5w) tϕ A 5,3 [θ, M/r] = 0. (A5x) For the massive scalar field, the coefficients Aϕt appearing in the expression (14) of T ϕ are p,q h tiren Aϕt [θ, M/r] = 144(14ξ 3) (M/r) (A6a) 0,0 − Aϕt [θ, M/r] = 2160 9648 (M/r) 16(462ξ 769) (M/r)2 (A6b) 0,1 − − − Aϕt [θ, M/r] = 4740 (M/r)2 + 8(1120ξ 1559) (M/r)3 (A6c) 0,2 − Aϕt [θ, M/r] = 8(455ξ 421) (M/r)4 (A6d) 0,3 − − Aϕt [θ, M/r] = 1296(14ξ 3) cos2 θ (M/r) (A6e) 1,0 − − Aϕt [θ, M/r] = 2160 cos2 θ 5 72 39 cos2 θ + 335 (M/r) + 48(1470ξ + 61) cos2 θ (M/r)2 (A6f) 1,1 − − − ϕt 2 2 2 3 A 1,2 [θ, M/r] = 12 229 cos θ + 1012 (M/r ) 16(4424ξ + 377) cos θ (M/r) (A6g) − ϕt 2 4 A 1,3 [θ, M/r] = 112(175 ξ + 17) cos θ(M/r) (A6h) Aϕt [θ, M/r] = 720(14ξ 3) cos4 θ (M/r) (A6i) 2,0 − − Aϕt [θ, M/r] = 2160 4 cos4 θ 5 cos2 θ 5 + 720 cos2 θ 24 cos2 θ 53 (M/r) 2,1 − − − − 4 2 48(630ξ 101) cos θ (M/r) (A6j) − − Aϕt [θ, M/r] = 12 cos2 θ 671 cos2 θ 1800 (M/r)2 + 8(3472ξ 939) cos4 θ (M/r)3 (A6k) 2,2 − − − ϕt 4 4 A 2,3 [θ, M/r] = 8(455ξ 253) cos θ (M/r) (A6l) − − Aϕt [θ, M/r] = 720(14ξ 3) cos6 θ (M/r) (A6m) 3,0 − Aϕt [θ, M/r] = 2160 cos2 θ cos4 θ + 5 cos2 θ 10 + 360 cos4 θ 29 cos2 θ 39 (M/r) 3,1 − − − 6 2 16(42ξ 17) cos θ (M/r) (A6n) − − Aϕt [θ, M/r] = 12 cos4 θ 505 cos2 θ 788 (M/r)2 (A6o) 3,2 − − ϕt A 3,3 [θ, M/r] = 0 (A6p) ϕt A 4,0 [θ, M/r] = 0 (A6q) Aϕt [θ, M/r] = 2160 cos4 θ cos4 θ 5 cos2 θ + 5 (A6r) 4,1 − ϕt A 4,2 [θ, M/r] = 0 (A6s) ϕt A 4,3 [θ, M/r] = 0. (A6t) For the massive scalar field, the coefficients Arθ appearing in the expression (15) of T r are p,q h θiren Arθ [θ, M/r] = 72(32ξ 7) 144(32ξ 7) (M/r) (A7a) 0,0 − − − Arθ [θ, M/r] = 36(109ξ 24) (M/r) + 72(141ξ 31) (M/r)2 (A7b) 0,1 − − − Arθ [θ, M/r] = 10(168ξ 37) (M/r)2 4(1821ξ 401) (M/r)3 (A7c) 0,2 − − −

26 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

Arθ [θ, M/r] = 10(168ξ 37) (M/r)4 (A7d) 0,3 − Arθ [θ, M/r] = 72(32ξ 7) 7 cos2 θ 1 + 1008(32ξ 7) cos2 θ (M/r) (A7e) 1,0 − − − − rθ 2 2 2 A 1,1 [θ, M/r] = 12 5(339ξ 74) cos θ 3(109 ξ 24) (M/r) 24(2367ξ 517) cos θ (M/r) (A7f) − − − − − rθ 2 2 2 3 A 1,2 [θ, M/r] = 2 2(1464ξ 319) cos θ 5(168ξ 37) (M/r) + 4(8013ξ 1748) cos θ (M/r) (A7g) − − − − − rθ 2 4 A 1,3 [θ, M/r] = 4(1464 ξ 319) cos θ (M/r) (A7h) − − Arθ [θ, M/r] = 504(32ξ 7) cos2 θ(cos2 θ 1) 1008(32ξ 7) cos4 θ (M/r) (A7i) 2,0 − − − − Arθ [θ, M/r] = 60 cos2 θ (201ξ 44) cos2 θ 339ξ + 74 (M/r) + 24(1677ξ 367) cos4 θ (M/r)2 (A7j) 2,1 − − − − rθ 2 2 2 4 3 A 2,2 [θ, M/r] = 2 cos θ 5(168 ξ 37) cos θ 2(1464ξ 319) (M/r) 20(771ξ 169) cos θ (M/r) (A7k) − − − − − rθ 4 4 A 2,3 [θ, M/r] = 10(168ξ 37) cos θ (M/r) (A7l) − Arθ [θ, M/r] = 72(32ξ 7) cos4 θ cos2 θ 7 + 144(32ξ 7) cos6 θ (M/r) (A7m) 3,0 − − − − rθ 4 2 6 2 A 3,1 [θ, M/r] = 60 cos θ (9ξ 2) cos θ 201ξ+ 44 (M/r) 24(141ξ 31) cos θ (M/r) (A7n) − − − − rθ 4 2 6 3 A 3,2 [θ, M/r] = 10(168ξ 37) cos θ (M/r) + 60(9ξ 2) cos θ (M/r) (A7o) − − rθ A 3,3 [θ, M/r] = 0 (A7p) Arθ [θ, M/r] = 72(32ξ 7) cos6 θ (A7q) 4,0 − − Arθ [θ, M/r] = 60(9ξ 2) cos6 θ (M/r) (A7r) 4,1 − rθ A 4,2 [θ, M/r] = 0 (A7s) rθ A 4,3 [θ, M/r] = 0. (A7t)

For the massive scalar ﬁeld, the coeﬃcients Aθr appearing in the expression (16) of T θ are p,q h riren Aθr [θ, M/r] = 72(32ξ 7) (A8a) 0,0 − Aθr [θ, M/r] = 36(109ξ 24) (M/r) (A8b) 0,1 − − Aθr [θ, M/r] = 10(168ξ 37) (M/r)2 (A8c) 0,2 − Aθr [θ, M/r] = 504(32ξ 7) cos2 θ (A8d) 1,0 − − Aθr [θ, M/r] = 60(339ξ 74) cos2 θ (M/r) (A8e) 1,1 − Aθr [θ, M/r] = 4(1464ξ 319) cos2 θ (M/r)2 (A8f) 1,2 − − Aθr [θ, M/r] = 504(32ξ 7) cos4 θ (A8g) 2,0 − Aθr [θ, M/r] = 60(201ξ 44) cos4 θ (M/r) (A8h) 2,1 − − Aθr [θ, M/r] = 10(168ξ 37) cos4 θ (M/r)2 (A8i) 2,2 − Aθr [θ, M/r] = 72(32ξ 7) cos6 θ (A8j) 3,0 − − Aθr [θ, M/r] = 60(9ξ 2) cos6 θ (M/r) (A8k) 3,1 − θr A 3,2 [θ, M/r] = 0. (A8l)

2. Massive Dirac ﬁeld

For the massive Dirac ﬁeld, the coeﬃcients Att appearing in the expression (9) of T t are p,q h tiren Att [θ, M/r] = 1080 + 2384 (M/r) (A9a) 0,0 − Att [θ, M/r] = 5400 22464 (M/r) + 21832 (M/r)2 (A9b) 0,1 − Att [θ, M/r] = 10544 (M/r)2 21496 (M/r)3 (A9c) 0,2 − tt 4 A 0,3 [θ, M/r] = 4917 (M/r) (A9d) Att [θ, M/r] = 29160 cos2 θ 288 251 cos2 θ + 1 (M/r) (A9e) 1,0 − tt 2 2 2 2 A 1,1 [θ, M/r] = 5400 7 cos θ 12 + 288 185 cos θ 691 (M/r) + 16 7805 cos θ + 7277 (M/r) (A9f) − − − tt 2 2 2 3 A 1,2 [θ, M/r] = 64 497 cos θ 1616 (M/r ) 8 10010 cos θ + 15373 (M/r) (A9g) − − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 27 I. Action effective et tenseur d’impulsion-énergie renormalisé

tt 2 4 A 1,3 [θ, M/r] = (17765 cos θ + 32546) (M/r) (A9h) Att [θ, M/r] = 45360 cos4 θ + 2592 cos2 θ 76 cos2 θ + 1 (M/r) (A9i) 2,0 − Att [θ, M/r] = 10800 4 cos4 θ + 6 cos2 θ 15 + 288 628 cos4 θ 111 cos2 θ 855 (M/r) 2,1 − − − − 2 2 2 240 cos θ 1658 cos θ 535 (M/r) (A9j) − − tt 4 2 2 2 2 3 A 2,2 [θ, M/r] = 96 900 cos θ 422 cos θ 1349 (M/r) + 8 cos θ 30793 cos θ 20378 (M/r) (A9k) − − − − tt 2 2 4 A 2,3 [θ, M/r] = cos θ 48149 cos θ 49476 (M/r ) (A9l) − − Att [θ, M/r] = 45360 cos6 θ 96 cos4 θ 967 cos2 θ 15 (M/r) (A9m) 3,0 − − − Att [θ, M/r] = 10800 4 cos6 θ 24 cos4 θ + 15 cos2 θ + 10 3,1 − 2 4 2 4 2 2 +1440 cos θ 21 cos θ + 247 cos θ 258 (M/r) + 16 cos θ 4751 cos θ 645 (M/r) (A9n) − − tt 2 4 2 2 4 2 3 A 3,2 [θ, M/r] = 64 cos θ 178 cos θ + 2742 cos θ 3375 (M/r ) + 24 cos θ 140 cos θ 687 (M/r) (A9o) − − − tt 4 2 4 A 3,3 [θ, M/r] = cos θ 8773 cos θ 11042 (M/r) (A9p) − − Att [θ, M/r] = 29160 cos8 θ + 144 cos6 θ 43 cos2 θ 10 (M/r) (A9q) 4,0 − Att [θ, M/r] = 5400 cos2 θ 7 cos6 θ 12 cos4 θ 30 cos2 θ + 40 4,1 − − 4 4 2 6 2 2 1440 cos θ 52 cos θ 131 cos θ + 87 (M/r) 8 cos θ 325 cos θ 158 (M/r) (A9r) − − − − tt 4 4 2 2 A 4,2 [θ, M/r] = 16 cos θ 2041 cos θ 7036 cos θ + 5406 (M/r ) (A9s) − tt A 4,3 [θ, M/r] = 0 (A9t) Att [θ, M/r] = 1080 cos10 θ (A9u) 5,0 − Att [θ, M/r] = 5400 cos4 θ cos2 θ 2 cos4 θ 10 cos2 θ + 10 (A9v) 5,1 − − − tt A 5,2 [θ, M/r] = 0 (A9w) tt A 5,3 [θ, M/r] = 0. (A9x) For the massive Dirac ﬁeld, the coeﬃcients Arr appearing in the expression (10) of T r are p,q h riren Arr [θ, M/r] = 504 784 (M/r) (A10a) 0,0 − Arr [θ, M/r] = 1080 6336 (M/r) + 8440 (M/r)2 (A10b) 0,1 − Arr [θ, M/r] = 3560 (M/r)2 8680 (M/r)3 (A10c) 0,2 − rr 4 A 0,3 [θ, M/r] = 2253 (M/r) (A10d) Arr [θ, M/r] = 504 31 cos2 θ 4 + 16128 cos2 θ (M/r) (A10e) 1,0 − − rr 2 2 2 2 A 1,1 [θ, M/r] = 1080 3 cos θ 8 + 144 189 cos θ 169 (M/r) 16160 cos θ (M/r) (A10f) − − − − rr 2 2 2 3 A 1,2 [θ, M/r] = 8 1141 cos θ 1563 (M/r ) + 88 cos θ (M/r) (A10g) − − rr 2 4 A 1,3 [θ, M/r] = 2015 cos θ (M/r) (A10h) Arr [θ, M/r] = 7056 cos2 θ 11 cos2 θ 8 22176 cos4 θ (M/r) (A10i) 2,0 − − Arr [θ, M/r] = 2160 6 cos4 θ 6 cos2 θ 5 144 cos2 θ 429 cos2 θ 193 (M/r) 2,1 − − − − − 4 2 +31440 cos θ (M/r) (A10j) Arr [θ, M/r] = 8 cos2 θ 1313 cos2 θ + 307 (M/r)2 18840 cos4 θ (M/r)3 (A10k) 2,2 − rr 4 4 A 2,3 [θ, M/r] = 4887 cos θ (M/r) (A10l) Arr [θ, M/r] = 7056 cos4 θ 17 cos2 θ 20 2688 cos6 θ (M/r) (A10m) 3,0 − − − Arr [θ, M/r] = 2160 cos2 θ 4 cos4 θ + 6 cos2 θ 15 + 9360 cos4 θ 15 cos2 θ 19 (M/r) 3,1 − − − 6 2 +4064 cos θ (M/r) (A10n) Arr [θ, M/r] = 8 cos4 θ 5275 cos2 θ 7013 (M/r)2 1496 cos6 θ (M/r)3 (A10o) 3,2 − − − rr 6 4 A 3,3 [θ, M/r] = 773 cos θ(M/r) (A10p) Arr [θ, M/r] = 504 cos6 θ 85 cos2 θ 112 + 1584 cos8 θ (M/r) (A10q) 4,0 − Arr [θ, M/r] = 1080 cos4 θ 3 cos4 θ 28 cos2 θ + 30 720 cos6 θ 31 cos2 θ 39 (M/r) 4,1 − − −

28 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

+248 cos8 θ (M/r)2 (A10r) rr 6 2 2 A 4,2 [θ, M/r] = 40 cos θ 22 cos θ 41 (M/r) (A10s) rr − − A 4,3 [θ, M/r] = 0 (A10t) Arr [θ, M/r] = 504 cos8 θ 3 cos2 θ 4 (A10u) 5,0 − − rr 6 4 2 A 5,1 [θ, M/r] = 1080 cos θ 3 cos θ 12 cos θ + 10 (A10v) rr − A 5,2 [θ, M/r] = 0 (A10w) rr A 5,3 [θ, M/r] = 0. (A10x) For the massive Dirac field, the coefficients Aθθ appearing in the expression (11) of T θ are p,q h θiren Aθθ [θ, M/r] = 1512 + 3536 (M/r) (A11a) 0,0 − Aθθ [θ, M/r] = 3240 + 20016 (M/r) 30808 (M/r)2 (A11b) 0,1 − − Aθθ [θ, M/r] = 12080 (M/r)2 + 33984 (M/r)3 (A11c) 0,2 − Aθθ [θ, M/r] = 9933 (M/r)4 (A11d) 0,3 − Aθθ [θ, M/r] = 504 85 cos2 θ 4 99072 cos2 θ (M/r) (A11e) 1,0 − − θθ 2 2 2 2 A 1,1 [θ, M/r] = 3240 cos θ + 4 144 359 cos θ 243 (M/r) + 186272 cos θ (M/r) (A11f) − − − θθ 2 2 2 3 A 1,2 [θ, M/r] = 40 299 cos θ 499 (M/r) 106960 cos θ (M/r) (A11g) − − θθ 2 4 A 1,3 [θ, M/r] = 18817 cos θ (M/r) (A11h) Aθθ [θ, M/r] = 7056 cos2 θ 17 cos2 θ 8 + 256032 cos4 θ (M/r) (A11i) 2,0 − − Aθθ [θ, M/r] = 2160 4 cos4 θ 14 cos2 θ 5 + 144 cos2 θ 749 cos2 θ 139 (M/r) 2,1 − − − 4 2 393360 cos θ (M/r ) (A11j) − Aθθ [θ, M/r] = 8 cos2 θ 2933 cos2 θ + 1087 (M/r)2 + 196128 cos4 θ (M/r)3 (A11k) 2,2 − θθ 4 4 A 2,3 [θ, M/r] = 31959 cos θ (M/r) (A11l) − Aθθ [θ, M/r] = 7056 cos4 θ 11 cos2 θ 20 115584 cos6 θ (M/r) (A11m) 3,0 − − Aθθ [θ, M/r] = 6480 cos2 θ 2 cos4 θ 2 cos2 θ 5 720 cos4 θ 113 cos2 θ 269 (M/r) 3,1 − − − − 6 2 +108448 cos θ (M/r) (A11n) Aθθ [θ, M/r] = 8 cos4 θ 2761 cos2 θ 7793 (M/r)2 23888 cos6 θ (M/r)3 (A11o) 3,2 − − θθ 6 4 A 3,3 [θ, M/r] = 549 cos θ (M/r) (A11p) − Aθθ [θ, M/r] = 504 cos6 θ 31 cos2 θ 112 + 7056 cos8 θ (M/r) (A11q) 4,0 − − Aθθ [θ, M/r] = 3240 cos4 θ cos4 θ + 4 cos2 θ 10 720 cos6 θ 4 cos2 θ + 13 (M/r) 4,1 − − 8 2 3032 cos θ (M/r) (A11r) − Aθθ [θ, M/r] = 8 cos6 θ 635 cos2 θ 1137 (M/r)2 (A11s) 4,2 − θθ A 4,3 [θ, M/r] = 0 (A11t) Aθθ [θ, M/r] = 504 cos8 θ cos2 θ 4 (A11u) 5,0 − Aθθ [θ, M/r] = 1080 cos6 θ cos4 θ 8 cos2 θ + 10 (A11v) 5,1 − − θθ A 5,2 [θ, M/r] = 0 (A11w) θθ A 5,3 [θ, M/r] = 0. (A11x) For the massive Dirac field, the coefficients Aϕϕ appearing in the expression (12) of T ϕ are p,q h ϕiren Aϕϕ [θ, M/r] = 1512 + 3536 (M/r) (A12a) 0,0 − Aϕϕ [θ, M/r] = 3240 + 20016 (M/r) 30808 (M/r)2 (A12b) 0,1 − − Aϕϕ [θ, M/r] = 12080 (M/r)2 + 33984 (M/r)3 (A12c) 0,2 − Aϕϕ [θ, M/r] = 9933 (M/r)4 (A12d) 0,3 − Aϕϕ [θ, M/r] = 40824 cos2 θ 288 345 cos2 θ 1 (M/r) (A12e) 1,0 − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 29 I. Action effective et tenseur d’impulsion-énergie renormalisé

Aϕϕ [θ, M/r] = 1080 41 cos2 θ 56 576 370 cos2 θ 341 (M/r) + 16 18919 cos2 θ 7277 (M/r)2 (A12f) 1,1 − − − − ϕϕ 2 2 2 3 A 1,2 [θ, M/r] = 160 617 cos θ 667 (M/r) 8 28743 cos θ 15373 (M/r) (A12g) − − − ϕϕ 2 4 A 1,3 [θ, M/r] = (51363 cos θ 32546) (M/r) (A12h) − Aϕϕ [θ, M/r] = 63504 cos4 θ + 288 cos2 θ 898 cos2 θ 9 (M/r) (A12i) 2,0 − − Aϕϕ [θ, M/r] = 2160 22 cos4 θ + 38 cos2 θ 75 288 382 cos4 θ + 168 cos2 θ 855 (M/r) 2,1 − − − 2 2 2 240 cos θ 1104 cos θ + 535 (M/r) (A12j) − ϕϕ 4 2 2 2 2 3 A 2,2 [θ, M/r] = 96 1035 cos θ 21 cos θ 1349 (M/r) + 16 cos θ 2069 cos θ + 10189 (M/r) (A12k) − − ϕϕ 2 2 4 A 2,3 [θ, M/r] = 3 cos θ 5839 cos θ 16492 (M/r ) (A12l) − Aϕϕ [θ, M/r] = 63504 cos6 θ 96 cos4 θ 1189 cos2 θ + 15 (M/r) (A12m) 3,0 − − Aϕϕ [θ, M/r] = 2160 22 cos6 θ 132 cos4 θ + 75 cos2 θ + 50 3,1 − − 2 4 2 4 2 2 +2880 cos θ 53 cos θ 143 cos θ + 129 (M/r) + 16 cos θ 6133 cos θ + 645 (M/r) (A12n) − ϕϕ 2 4 2 2 A 3,2 [θ, M/r] = 32 cos θ 1195 cos θ 6687 cos θ + 6750 (M/r ) − − 4 2 3 8 cos θ 5047 cos θ 2061 (M/r) (A12o) − − ϕϕ 4 2 4 A 3,3 [θ, M/r] = cos θ 10493 cos θ 11042 (M/r) (A12p) − ϕϕ 8 6 2 A 4,0 [θ, M/r] = 40824 cos θ + 144 cos θ 39 cos θ + 10 (M/r) (A12q) Aϕϕ [θ, M/r] = 1080 cos2 θ 41 cos6 θ 76 cos4 θ 150 cos2 θ + 200 4,1 − − − 4 4 2 6 2 2 +720 cos θ 41 cos θ 232 cos θ + 174 (M/r) 8 cos θ 221 cos θ + 158 (M/r) (A12r) − − ϕϕ 4 4 2 2 A 4,2 [θ, M/r] = 16 cos θ 1675 cos θ 6830 cos θ + 5406 (M/r) (A12s) ϕϕ − − A 4,3 [θ, M/r] = 0 (A12t) Aϕϕ [θ, M/r] = 1512 cos10 θ (A12u) 5,0 − ϕϕ 4 2 4 2 A 5,1 [θ, M/r] = 1080 cos θ cos θ 2 3 cos θ 50 cos θ + 50 (A12v) ϕϕ − − A 5,2 [θ, M/r] = 0 (A12w) ϕϕ A 5,3 [θ, M/r] = 0. (A12x) For the massive Dirac ﬁeld, the coeﬃcients Atϕ appearing in the expression (13) of T t are p,q h ϕiren tϕ A 0,0 [θ, M/r] = 576 (M/r) (A13a) Atϕ [θ, M/r] = 10800 + 37296 (M/r) 26320 (M/r)2 (A13b) 0,1 − − Atϕ [θ, M/r] = 20160 (M/r)2 + 27740 (M/r)3 (A13c) 0,2 − Atϕ [θ, M/r] = 7425 (M/r)4 (A13d) 0,3 − Atϕ [θ, M/r] = 144 93 cos2 θ 1 (M/r) (A13e) 1,0 − − tϕ 2 2 2 2 A 1,1 [θ, M/r] = 10800 cos θ 6 576 39 cos θ 256 (M/r) + 8 3837 cos θ 7277 (M/r) (A13f) − − − − tϕ 2 2 2 3 A 1,2 [θ, M/r] = 96 47 cos θ 817 (M/r) 4 3360 cos θ 15373 (M/r) (A13g) − − − tϕ 2 4 A 1,3 [θ, M/r] = (526 cos θ 16273) (M/r) (A13h) − Atϕ [θ, M/r] = 144 cos2 θ 205 cos2 θ 9 (M/r) (A13i) 2,0 − Atϕ [θ, M/r] = 21600 2 cos4 θ 2 cos2 θ 5 144 635 cos4 θ 774 cos2 θ 855 (M/r) 2,1 − − − − − 2 2 2 +120 cos θ 19 cos θ 535 (M/r) (A13j) − tϕ 4 2 2 2 2 3 A 2,2 [θ, M/r] = 48 949 cos θ 1616 cos θ 1349 (M/r) 4 cos θ 6277 cos θ 20378 (M/r) (A13k) − − − − tϕ 2 2 4 A 2,3 [θ, M/r] = cos θ 8095 cos θ 24738 (M/r) (A13l) − Atϕ [θ, M/r] = 144 cos4 θ 79 cos2 θ + 5 (M/r) (A13m) 3,0 − Atϕ [θ, M/r] = 10800 cos6 θ + 9 cos4 θ 15 cos2 θ 5 3,1 − − 2 4 2 4 2 2 1440 cos θ 13 cos θ + 74 cos θ 129 (M/r) + 8 cos θ 2027 cos θ + 645 (M/r) (A13n) − − tϕ 2 4 2 2 4 2 3 A 3,2 [θ, M/r] = 288 cos θ 50 cos θ + 143 cos θ 375 (M/r) 4 cos θ 3406 cos θ 2061 (M/r) (A13o) − − −

30 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

Atϕ [θ, M/r] = cos4 θ 4112 cos2 θ 5521 (M/r)4 (A13p) 3,3 − tϕ 6 2 A 4,0 [θ, M/r] = 144 cos θ 3 cos θ + 5 (M/r ) (A13q) Atϕ [θ, M/r] = 10800 cos2 θ cos6 θ 6 cos4 θ + 10 + 720 cos4 θ 18 cos4 θ 98 cos2 θ + 87 (M/r) 4,1 − − − 6 2 2 8 cos θ 27 cos θ + 79 (M/r ) (A13r) − tϕ 4 4 2 2 A 4,2 [θ, M/r] = 48 cos θ 135 cos θ 840 cos θ + 901 (M/r) (A13s) − − tϕ A 4,3 [θ, M/r] = 0 (A13t) tϕ A 5,0 [θ, M/r] = 0 (A13u) Atϕ [θ, M/r] = 10800 cos4 θ cos4 θ 5 cos2 θ + 5 (A13v) 5,1 − − tϕ A 5,2 [θ, M/r] = 0 (A13w) tϕ A 5,3 [θ, M/r] = 0. (A13x) For the massive Dirac field, the coefficients Aϕt appearing in the expression (14) of T ϕ are p,q h tiren Aϕt [θ, M/r] = 144 (M/r) (A14a) 0,0 − Aϕt [θ, M/r] = 10800 50256 (M/r) + 58216 (M/r)2 (A14b) 0,1 − Aϕt [θ, M/r] = 26640 (M/r)2 61492 (M/r)3 (A14c) 0,2 − ϕt 4 A 0,3 [θ, M/r] = 16273 (M/r) (A14d) ϕt 2 A 1,0 [θ, M/r] = 1296 cos θ (M/r) (A14e) Aϕt [θ, M/r] = 10800 cos2 θ 5 432 8 cos2 θ + 285 (M/r) + 64200 cos2 θ (M/r)2 (A14f) 1,1 − − − ϕt 2 2 2 3 A 1,2 [θ, M/r] = 48 176 cos θ + 1349 (M/r) 81512 cos θ (M/r) (A14g) − ϕt 2 4 A 1,3 [θ, M/r] = 24738 cos θ (M/r) (A14h) ϕt 4 A 2,0 [θ, M/r] = 720 cos θ (M/r) (A14i) Aϕt [θ, M/r] = 10800 4 cos4 θ 5 cos2 θ 5 + 720 cos2 θ 127 cos2 θ 258 (M/r) 2,1 − − − − 4 2 5160 cos θ (M/r) (A14j) − Aϕt [θ, M/r] = 48 cos2 θ 949 cos2 θ 2250 (M/r)2 8244 cos4 θ (M/r)3 (A14k) 2,2 − − − ϕt 4 4 A 2,3 [θ, M/r] = 5521 cos θ(M/r) (A14l) Aϕt [θ, M/r] = 720 cos6 θ (M/r) (A14m) 3,0 − Aϕt [θ, M/r] = 10800 cos2 θ cos4 θ + 5 cos2 θ 10 + 720 cos4 θ 62 cos2 θ 87 (M/r) 3,1 − − − 6 2 +632 cos θ (M/r) (A14n) Aϕt [θ, M/r] = 48 cos4 θ 570 cos2 θ 901 (M/r)2 (A14o) 3,2 − − ϕt A 3,3 [θ, M/r] = 0 (A14p) ϕt A 4,0 [θ, M/r] = 0 (A14q) Aϕt [θ, M/r] = 10800 cos4 θ cos4 θ 5 cos2 θ + 5 (A14r) 4,1 − ϕt A 4,2 [θ, M/r] = 0 (A14s) ϕt A 4,3 [θ, M/r] = 0. (A14t) For the massive Dirac field, the coefficients Arθ appearing in the expression (15) of T r are p,q h θiren Arθ [θ, M/r] = 144 + 288 (M/r) (A15a) 0,0 − Arθ [θ, M/r] = 279 (M/r) 702 (M/r)2 (A15b) 0,1 − Arθ [θ, M/r] = 125 (M/r)2 + 529 (M/r)3 (A15c) 0,2 − Arθ [θ, M/r] = 125 (M/r)4 (A15d) 0,3 − Arθ [θ, M/r] = 144 7 cos2 θ 1 2016 cos2 θ (M/r) (A15e) 1,0 − − rθ 2 2 2 A 1,1 [θ, M/r] = 9 135 cos θ 31 (M/r) + 3438 cos θ (M/r) (A15f) − − rθ 2 2 2 3 A 1,2 [θ, M/r] = (326 cos θ 125) (M/r ) 1867 cos θ (M/r) (A15g) − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 31 I. Action effective et tenseur d’impulsion-énergie renormalisé

rθ 2 4 A 1,3 [θ, M/r] = 326 cos θ (M/r) (A15h) Arθ [θ, M/r] = 1008 cos2 θ cos2 θ 1 + 2016 cos4 θ (M/r) (A15i) 2,0 − − rθ 2 2 4 2 A 2,1 [θ, M/r] = 45 cos θ 17 cos θ 27(M/r) 2538 cos θ (M/r) (A15j) − − rθ 2 2 2 4 3 A 2,2 [θ, M/r] = cos θ 125 cos θ 326 (M/r) + 1015 cos θ (M/r) (A15k) − − rθ 4 4 A 2,3 [θ, M/r] = 125 cos θ (M/r) (A15l) − Arθ [θ, M/r] = 144 cos4 θ cos2 θ 7 288 cos6 θ (M/r) (A15m) 3,0 − − rθ 4 2 6 2 A 3,1 [θ, M/r] = 45 cos θ cos θ 17 (M/r) + 234 cos θ (M/r) (A15n) − − rθ 4 2 6 3 A 3,2 [θ, M/r] = 125 cos θ(M/r) 45 cos θ (M/r) (A15o) − − rθ A 3,3 [θ, M/r] = 0 (A15p) rθ 6 A 4,0 [θ, M/r] = 144 cos θ (A15q) Arθ [θ, M/r] = 45 cos6 θ (M/r) (A15r) 4,1 − rθ A 4,2 [θ, M/r] = 0 (A15s) rθ A 4,3 [θ, M/r] = 0. (A15t)

For the massive Dirac ﬁeld, the coeﬃcients Aθr appearing in the expression (16) of T θ are p,q h riren Aθr [θ, M/r] = 144 (A16a) 0,0 − θr A 0,1 [θ, M/r] = 279 (M/r) (A16b) Aθr [θ, M/r] = 125 (M/r)2 (A16c) 0,2 − θr 2 A 1,0 [θ, M/r] = 1008 cos θ (A16d) Aθr [θ, M/r] = 1215 cos2 θ (M/r) (A16e) 1,1 − θr 2 2 A 1,2 [θ, M/r] = 326 cos θ (M/r) (A16f) Aθr [θ, M/r] = 1008 cos4 θ (A16g) 2,0 − θr 4 A 2,1 [θ, M/r] = 765 cos θ (M/r) (A16h) Aθr [θ, M/r] = 125 cos4 θ (M/r)2 (A16i) 2,2 − θr 6 A 3,0 [θ, M/r] = 144 cos θ (A16j) Aθr [θ, M/r] = 45 cos6 θ (M/r) (A16k) 3,1 − θr A 3,2 [θ, M/r] = 0. (A16l)

3. Proca ﬁeld

For the Proca ﬁeld, the coeﬃcients Att appearing in the expression (9) of T t are p,q h tiren Att [θ, M/r] = 6660 14664 (M/r) (A17a) 0,0 − Att [θ, M/r] = 48600 276096 (M/r) + 374148 (M/r)2 (A17b) 0,1 − Att [θ, M/r] = 167416 (M/r)2 430064 (M/r)3 (A17c) 0,2 − tt 4 A 0,3 [θ, M/r] = 124228 (M/r) (A17d) Att [θ, M/r] = 179820 cos2 θ + 288 1528 cos2 θ + 5 (M/r) (A17e) 1,0 − Att [θ, M/r] = 48600 7 cos2 θ 12 + 144 10435 cos2 θ 13313 (M/r) 1,1 − − − 2 2 48 36771 cos θ 23602 (M/r) (A17f) − − tt 2 2 2 3 A 1,2 [θ, M/r ] = 16 47215 cos θ 69264 (M/r) + 16 91476 cos θ 81229 (M/r) (A17g) − − − tt 2 4 A 1,3 [θ, M/r] = 4 86153 cos θ 92104 (M/r) (A17h) − − Att [θ, M/r] = 279720 cos4 θ 144 cos2 θ 8261 cos2 θ + 90 (M/r) (A17i) 2,0 − Att [θ, M/r] = 97200 4 cos4 θ + 6 cos2 θ 15 + 144 6824 cos4 θ + 3147 cos2 θ 15845 (M/r) 2,1 − − −

32 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

+120 cos2 θ 8033 cos2 θ + 6732 (M/r)2 (A17j) tt 4 2 2 2 2 3 A 2,2 [θ, M/r] = 96 8100 cos θ + 290 cos θ 13289 (M/r) + 16 cos θ 13713 cos θ 73046 (M/r) (A17k) − − − tt 2 2 4 A 2,3 [θ, M/r] = 4 cos θ 51301 cos θ 97664 (M/r) (A17l) − − Att [θ, M/r] = 279720 cos6 θ + 288 cos4 θ 1922 cos2 θ 25 (M/r) (A17m) 3,0 − Att [θ, M/r] = 97200 4 cos6 θ 24 cos4 θ + 15 cos2 θ + 10 3,1 − 2 4 2 4 2 2 240 cos θ 4403 cos θ 14871 cos θ + 13386 (M/r) 144 cos θ 2837 cos θ + 1030 (M/r) (A17n) − − − tt 2 4 2 2 A 3,2 [θ, M/r] = 80 cos θ 4583 cos θ 24648 cos θ + 24300 (M/r) − 4 2 3 +16 cos θ 10654 cos θ 2425 (M/r) (A17o) − tt 4 2 4 A 3,3 [θ, M/r] = 172 cos θ 337 cos θ 376 (M/r) (A17p) − − Att [θ, M/r] = 179820 cos8 θ 72 cos6 θ 493 cos2 θ 100 (M/r) (A17q) 4,0 − − − Att [θ, M/r] = 48600 cos2 θ 7 cos6 θ 12 cos4 θ 30 cos2 θ + 40 4,1 − − 4 4 2 6 2 2 720 cos θ 362 cos θ 1665 cos θ + 1293 (M/r) + 12 cos θ 675 cos θ + 584 (M/r) (A17r) − − tt 4 4 2 2 A 4,2 [θ, M/r] = 8 cos θ 27673 cos θ 104472 cos θ + 83532 (M/r ) (A17s) − tt A 4,3 [θ, M/r] = 0 (A17t) tt 10 A 5,0 [θ, M/r] = 6660 cos θ (A17u) Att [θ, M/r] = 48600 cos4 θ cos2 θ 2 cos4 θ 10 cos2 θ + 10 (A17v) 5,1 − − − tt A 5,2 [θ, M/r] = 0 (A17w) tt A 5,3 [θ, M/r] = 0. (A17x) For the Proca ﬁeld, the coeﬃcients Arr appearing in the expression (10) of T r are p,q h riren Arr [θ, M/r] = 2772 + 4200 (M/r) (A18a) 0,0 − Arr [θ, M/r] = 9720 41792 (M/r) + 51628 (M/r)2 (A18b) 0,1 − Arr [θ, M/r] = 25768 (M/r)2 67984 (M/r)3 (A18c) 0,2 − rr 4 A 0,3 [θ, M/r] = 21460 (M/r) (A18d) Arr [θ, M/r] = 2772 31 cos2 θ 4 86688 cos2 θ (M/r) (A18e) 1,0 − − rr 2 2 2 2 A 1,1 [θ, M/r] = 9720 3 cos θ 8 288 301 cos θ + 624 (M/r) + 271088 cos θ (M/r) (A18f) − − − rr 2 2 2 3 A 1,2 [θ, M/r] = 8 3633 cos θ + 13213 (M/r ) 236144 cos θ (M/r) (A18g) − rr 2 4 A 1,3 [θ, M/r] = 64076 cos θ (M/r) (A18h) Arr [θ, M/r] = 38808 cos2 θ 11 cos2 θ 8 + 117936 cos4 θ (M/r) (A18i) 2,0 − − Arr [θ, M/r] = 19440 6 cos4 θ 6 cos2 θ 5 + 96 cos2 θ 7673 cos2 θ 8766 (M/r) 2,1 − − − − 4 2 76280 cos θ (M/r) (A18j) − Arr [θ, M/r] = 8 cos2 θ 34243 cos2 θ 48823 (M/r)2 34096 cos4 θ (M/r)3 (A18k) 2,2 − − − rr 4 4 A 2,3 [θ, M/r] = 24796 cosθ (M/r) (A18l) Arr [θ, M/r] = 38808 cos4 θ 17 cos2 θ 20 + 18144 cos6 θ (M/r) (A18m) 3,0 − Arr [θ, M/r] = 19440 cos2 θ 4 cos4 θ + 6 cos2 θ 15 160 cos4 θ 2963 cos2 θ 3384 (M/r) 3,1 − − − − 6 2 18192 cos θ (M/r) (A18n) − Arr [θ, M/r] = 8 cos4 θ 5571 cos2 θ 2977 (M/r)2 1872 cos6 θ (M/r)3 (A18o) 3,2 − − rr 6 4 A 3,3 [θ, M/r] = 4836 cos θ (M/r) (A18p) Arr [θ, M/r] = 2772 cos6 θ 85 cos2 θ 112 9432 cos8 θ (M/r) (A18q) 4,0 − − − Arr [θ, M/r] = 9720 cos4 θ 3 cos4 θ 28 cos2 θ + 30 + 480 cos6 θ 341 cos2 θ 450 (M/r) 4,1 − − 8 2 +5676 cos θ (M/r) (A18r) Arr [θ, M/r] = 8 cos6 θ 4126 cos2 θ 5765 (M/r)2 (A18s) 4,2 − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 33 I. Action effective et tenseur d’impulsion-énergie renormalisé

rr A 4,3 [θ, M/r] = 0 (A18t) Arr [θ, M/r] = 2772 cos8 θ 3 cos2 θ 4 (A18u) 5,0 − rr 6 4 2 A 5,1 [θ, M/r] = 9720 cos θ 3 cos θ 12 cos θ + 10 (A18v) rr − A 5,2 [θ, M/r] = 0 (A18w) rr A 5,3 [θ, M/r] = 0. (A18x)

For the Proca ﬁeld, the coeﬃcients Aθθ appearing in the expression (11) of T θ are p,q h θiren Aθθ [θ, M/r] = 8316 19416 (M/r) (A19a) 0,0 − Aθθ [θ, M/r] = 29160 + 126832 (M/r) 123524 (M/r)2 (A19b) 0,1 − − Aθθ [θ, M/r] = 83632 (M/r)2 + 176272 (M/r)3 (A19c) 0,2 − Aθθ [θ, M/r] = 55916 (M/r)4 (A19d) 0,3 − Aθθ [θ, M/r] = 2772 85 cos2 θ 4 + 545760 cos2 θ (M/r) (A19e) 1,0 − − θθ 2 2 2 2 A 1,1 [θ, M/r] = 29160 cos θ + 4 + 4176 143 cos θ + 59 (M/r) 1519024 cos θ (M/r) (A19f) − − θθ 2 2 2 3 A 1,2 [θ, M/r] = 8 39591 cos θ + 18535 (M/r ) + 1292144 cos θ (M/r) (A19g) − θθ 2 4 A 1,3 [θ, M/r] = 340212 cos θ (M/r) (A19h) − Aθθ [θ, M/r] = 38808 cos2 θ 17 cos2 θ 8 1414224 cos4 θ (M/r) (A19i) 2,0 − − Aθθ [θ, M/r] = 19440 4 cos4 θ 14 cos2 θ 5 48 cos2 θ 17171 cos2 θ 24471 (M/r) 2,1 − − − − 4 2 +2057800 cos θ (M/r ) (A19j) Aθθ [θ, M/r] = 8 cos2 θ 19663 cos2 θ 75877 (M/r)2 766160 cos4 θ (M/r)3 (A19k) 2,2 − − θθ 4 4 A 2,3 [θ, M/r] = 34908 cos θ (M/r) (A19l) Aθθ [θ, M/r] = 38808 cos4 θ 11 cos2 θ 20 + 639072 cos6 θ (M/r) (A19m) 3,0 − − Aθθ [θ, M/r] = 58320 cos2 θ 2 cos4 θ 2 cos2 θ 5 80 cos4 θ 67 cos2 θ + 3159 (M/r) 3,1 − − − 6 2 522864 cos θ (M/r) (A19n) − Aθθ [θ, M/r] = 8 cos4 θ 10947 cos2 θ 24077 (M/r)2 + 110928 cos6 θ (M/r)3 (A19o) 3,2 − θθ 6 4 A 3,3 [θ, M/r] = 12956 cos θ (M/r) (A19p) − Aθθ [θ, M/r] = 2772 cos6 θ 31 cos2 θ 112 38808 cos8 θ (M/r) (A19q) 4,0 − − Aθθ [θ, M/r] = 29160 cos4 θ cos4 θ + 4 cos2 θ 10 240 cos6 θ 542 cos2 θ 993 (M/r) 4,1 − − − 8 2 +9756 cos θ (M/r) (A19r) Aθθ [θ, M/r] = 8 cos6 θ 6499 cos2 θ 11087 (M/r)2 (A19s) 4,2 − θθ A 4,3 [θ, M/r] = 0 (A19t) Aθθ [θ, M/r] = 2772 cos8 θ cos2 θ 4 (A19u) 5,0 − − Aθθ [θ, M/r] = 9720 cos6 θ cos4 θ 8 cos2 θ + 10 (A19v) 5,1 − − θθ A 5,2 [θ, M/r] = 0 (A19w) θθ A 5,3 [θ, M/r] = 0. (A19x)

For the Proca ﬁeld, the coeﬃcients Aϕϕ appearing in the expression (12) of T ϕ are p,q h ϕiren Aϕϕ [θ, M/r] = 8316 19416 (M/r) (A20a) 0,0 − Aϕϕ [θ, M/r] = 29160 + 126832 (M/r) 123524 (M/r)2 (A20b) 0,1 − − Aϕϕ [θ, M/r] = 83632 (M/r)2 + 176272 (M/r)3 (A20c) 0,2 − Aϕϕ [θ, M/r] = 55916 (M/r)4 (A20d) 0,3 − Aϕϕ [θ, M/r] = 224532 cos2 θ + 1440 380 cos2 θ 1 (M/r) (A20e) 1,0 − − Aϕϕ [θ, M/r] = 9720 41 cos2 θ 56 8352 115 cos2 θ 216 (M/r) 1,1 − − −

34 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

16 24133 cos2 θ + 70806 (M/r)2 (A20f) − ϕϕ 2 2 2 3 A 1,2 [θ, M/r] = 16 36322 cos θ 65385 (M/r) 16 470 cos θ 81229 (M/r) (A20g) − − − ϕϕ 2 4 A 1,3 [θ, M/r] = 4 7051 cos θ 92104 (M/r) (A20h) − Aϕϕ [θ, M/r] = 349272 cos4 θ 144 cos2 θ 9911 cos2 θ 90 (M/r) (A20i) 2,0 − − Aϕϕ [θ, M/r] = 19440 22 cos4 θ + 38 cos2 θ 75 48 32549 cos4 θ + 7686 cos2 θ 47535 (M/r) 2,1 − − − 2 2 2 +40 cos θ 71641 cos θ 20196 (M/r) (A20j) − ϕϕ 4 2 2 A 2,2 [θ, M/r] = 48 18630 cos θ 1421 cos θ 26578 (M/r) − − 2 2 3 16 cos θ 120931 cos θ 73046 (M/r) (A20k) − − ϕϕ 2 2 4 A 2,3 [θ, M/r] = 4 cos θ 106391 cos θ 97664 (M/r) (A20l) − ϕϕ 6 4 2 A 3,0 [θ, M/r] = 349272 cos θ + 288 cos θ 2194 cos θ + 25 (M/r) (A20m) Aϕϕ [θ, M/r] = 19440 22 cos6 θ 132 cos4 θ + 75 cos2 θ + 50 3,1 − − 2 4 2 4 2 2 +160 cos θ 970 cos θ 22662 cos θ + 20079 (M/r) 144 cos θ 4661 cos θ 1030 (M/r) (A20n) − − − ϕϕ 2 4 2 2 A 3,2 [θ, M/r] = 16 cos θ 2302 cos θ 117237 cos θ + 121500 (M/r ) − − 4 2 3 +16 cos θ 4508 cos θ + 2425 (M/r) (A20o) ϕϕ 4 2 4 A 3,3 [θ, M/r] = 4 cos θ 12929 cos θ 16168 (M/r) (A20p) − Aϕϕ [θ, M/r] = 224532 cos8 θ 72 cos6 θ 439 cos2 θ + 100 (M/r) (A20q) 4,0 − − Aϕϕ [θ, M/r] = 9720 cos2 θ 41 cos6 θ 76 cos4 θ 150 cos2 θ + 200 4,1 − − − 4 4 2 6 2 2 +240 cos θ 2626 cos θ 6054 cos θ + 3879 (M/r) + 12 cos θ 1397 cos θ 584 (M/r) (A20r) − − ϕϕ 4 4 2 2 A 4,2 [θ, M/r] = 16 cos θ 16643 cos θ 56115 cos θ + 41766 (M/r ) (A20s) ϕϕ − − A 4,3 [θ, M/r] = 0 (A20t) ϕϕ 10 A 5,0 [θ, M/r] = 8316 cos θ (A20u) ϕϕ 4 2 4 2 A 5,1 [θ, M/r] = 9720 cos θ cos θ 2 3 cos θ 50 cos θ + 50 (A20v) ϕϕ − − A 5,2 [θ, M/r] = 0 (A20w) ϕϕ A 5,3 [θ, M/r] = 0. (A20x) For the Proca ﬁeld, the coeﬃcients Atϕ appearing in the expression (13) of T t are p,q h ϕiren Atϕ [θ, M/r] = 2376 (M/r) (A21a) 0,0 − Atϕ [θ, M/r] = 97200 + 359856 (M/r) 248836 (M/r)2 (A21b) 0,1 − − Atϕ [θ, M/r] = 219660 (M/r)2 + 303168 (M/r)3 (A21c) 0,2 − Atϕ [θ, M/r] = 90072 (M/r)4 (A21d) 0,3 − Atϕ [θ, M/r] = 144 367 cos2 θ 5 (M/r) (A21e) 1,0 − tϕ 2 2 2 2 A 1,1 [θ, M/r] = 97200 cos θ 6 72 4467 cos θ 19223 (M/r) + 16 7687 cos θ 35403 (M/r) (A21f) − − − − tϕ 2 2 2 3 A 1,2 [θ, M/r] = 12 8221 cos θ 66601 (M/r) 8 10717 cos θ 81229 (M/r) (A21g) − − − tϕ 2 4 A 1,3 [θ, M/r] = 8 275 cos θ 23026 (M/r) (A21h) − Atϕ [θ, M/r] = 2160 cos2 θ 52 cos2 θ 3 (M/r) (A21i) 2,0 − − Atϕ [θ, M/r] = 194400 2 cos4 θ 2 cos2 θ 5 72 12200 cos4 θ 11363 cos2 θ 15845 (M/r) 2,1 − − − − − 2 2 2 +440 cos θ 1243 cos θ 918 (M/r) (A21j) − tϕ 4 2 2 A 2,2 [θ, M/r] = 12 39001 cos θ 53339 cos θ 53156 (M/r) − − 2 2 3 16 cos θ 30799 cos θ 36523 (M/r) (A21k) − − tϕ 2 2 4 A 2,3 [θ, M/r] = 8 cos θ 15007 cos θ 24416 (M/r) (A21l) − tϕ 4 2 A 3,0 [θ, M/r] = 144 cos θ 297 cos θ + 25 (M/r ) (A21m) Atϕ [θ, M/r] = 97200 cos6 θ + 9 cos4 θ 15 cos2 θ 5 3,1 − −

Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 35 I. Action effective et tenseur d’impulsion-énergie renormalisé

360 cos2 θ 223 cos4 θ + 3091 cos2 θ 4462 (M/r) 144 cos4 θ 397 cos2 θ 515 (M/r)2 (A21n) − − − − tϕ 2 4 2 2 A 3,2 [θ, M/r] = 12 cos θ 7615 cos θ + 40317 cos θ 81000 (M/r) − 4 2 3 8 cos θ 3721 cos θ 2425 (M/r) (A21o) − − tϕ 4 2 4 A 3,3 [θ, M/r] = 8 cos θ 2813 cos θ 4042 (M/r) (A21p) − Atϕ [θ, M/r] = 72 cos6 θ 23 cos2 θ + 50 (M/r) (A21q) 4,0 − Atϕ [θ, M/r] = 97200 cos2 θ cos6 θ 6 cos4 θ + 10 4,1 − − 4 4 2 6 2 2 +360 cos θ 324 cos θ 1519 cos θ + 1293 (M/r) + 12 cos θ 69 cos θ 292 (M/r) (A21r) − − tϕ 4 4 2 2 A 4,2 [θ, M/r] = 12 cos θ 4860 cos θ 27055 cos θ + 27844 (M/r ) (A21s) − − tϕ A 4,3 [θ, M/r] = 0 (A21t) tϕ A 5,0 [θ, M/r] = 0 (A21u) Atϕ [θ, M/r] = 97200 cos4 θ cos4 θ 5 cos2 θ + 5 (A21v) 5,1 − − tϕ A 5,2 [θ, M/r] = 0 (A21w) tϕ A 5,3 [θ, M/r] = 0. (A21x) For the Proca field, the coefficients Aϕt appearing in the expression (14) of T ϕ are p,q h tiren ϕt A 0,0 [θ, M/r] = 720 (M/r) (A22a) Aϕt [θ, M/r] = 97200 476496 (M/r) + 566448 (M/r)2 (A22b) 0,1 − Aϕt [θ, M/r] = 277980 (M/r)2 649832 (M/r)3 (A22c) 0,2 − ϕt 4 A 0,3 [θ, M/r] = 184208 (M/r) (A22d) Aϕt [θ, M/r] = 6480 cos2 θ (M/r) (A22e) 1,0 − Aϕt [θ, M/r] = 97200 cos2 θ 5 + 72 1227 cos2 θ 15845 (M/r) + 403920 cos2 θ (M/r)2 (A22f) 1,1 − − − ϕt 2 2 2 3 A 1,2 [θ, M/r] = 12 1499 cos θ + 53156 (M/r) 584368 cos θ (M/r) (A22g) − ϕt 2 4 A 1,3 [θ, M/r] = 195328 cos θ (M/r) (A22h) Aϕt [θ, M/r] = 3600 cos4 θ (M/r) (A22i) 2,0 − Aϕt [θ, M/r] = 97200 4 cos4 θ 5 cos2 θ 5 + 720 cos2 θ 1220 cos2 θ 2231 (M/r) 2,1 − − − − 4 2 74160 cos θ (M/r) (A22j) − Aϕt [θ, M/r] = 12 cos2 θ 39001 cos2 θ 81000 (M/r)2 19400 cos4 θ (M/r)3 (A22k) 2,2 − − − ϕt 4 4 A 2,3 [θ, M/r] = 32336 cos θ (M/r) (A22l) ϕt 6 A 3,0 [θ, M/r] = 3600 cos θ (M/r) (A22m) Aϕt [θ, M/r] = 97200 cos2 θ cos4 θ + 5 cos2 θ 10 + 360 cos4 θ 871 cos2 θ 1293 (M/r) 3,1 − − − 6 2 +3504 cos θ (M/r) (A22n) Aϕt [θ, M/r] = 12 cos4 θ 17335 cos2 θ 27844 (M/r)2 (A22o) 3,2 − − ϕt A 3,3 [θ, M/r] = 0 (A22p) ϕt A 4,0 [θ, M/r] = 0 (A22q) Aϕt [θ, M/r] = 97200 cos4 θ cos4 θ 5 cos2 θ + 5 (A22r) 4,1 − ϕt A 4,2 [θ, M/r] = 0 (A22s) ϕt A 4,3 [θ, M/r] = 0. (A22t) For the Proca field, the coefficients Arθ appearing in the expression (15) of T r are p,q h θiren Arθ [θ, M/r] = 792 1584 (M/r) (A23a) 0,0 − Arθ [θ, M/r] = 972 (M/r) + 2736 (M/r)2 (A23b) 0,1 − Arθ [θ, M/r] = 250 (M/r)2 1472 (M/r)3 (A23c) 0,2 − rθ 4 A 0,3 [θ, M/r] = 250 (M/r) (A23d)

36 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 I.1. Tenseur d’impulsion-énergie renormalisé associé aux champs massifs en espace-temps de Kerr-Newman

Arθ [θ, M/r] = 792 7 cos2 θ 1 + 11088 cos2 θ (M/r) (A23e) 1,0 − − rθ 2 2 2 A 1,1 [θ, M/r] = 12 635 cos θ 81 (M/r) 20784 cos θ (M/r) (A23f) − − rθ 2 2 2 3 A 1,2 [θ, M/r] = 2 1334 cos θ 125 (M/r) + 12956 cos θ (M/r) (A23g) − − rθ 2 4 A 1,3 [θ, M/r] = 2668 cos θ (M/r) (A23h) − Arθ [θ, M/r] = 5544(cos θ 1) cos2 θ(cos θ + 1) 11088 cos4 θ (M/r) (A23i) 2,0 − − Arθ [θ, M/r] = 60 cos2 θ 67 cos2 θ 127 (M/r) + 13584 cos4 θ (M/r)2 (A23j) 2,1 − − rθ 2 2 2 4 3 A 2,2 [θ, M/r] = 2 cos θ 125 cos θ 1334 (M/r) 4520 cos θ (M/r) (A23k) − − rθ 4 4 A 2,3 [θ, M/r] = 250 cos θ (M/r) (A23l) Arθ [θ, M/r] = 792 cos4 θ cos2 θ 7 + 1584 cos6 θ (M/r) (A23m) 3,0 − − rθ 4 2 6 2 A 3,1 [θ, M/r] = 60 cos θ cos θ 67 (M/r) 912 cos θ (M/r) (A23n) − − rθ 4 2 6 3 A 3,2 [θ, M/r] = 250 cos θ(M/r) + 60 cos θ (M/r) (A23o) rθ A 3,3 [θ, M/r] = 0 (A23p) Arθ [θ, M/r] = 792 cos6 θ (A23q) 4,0 − rθ 6 A 4,1 [θ, M/r] = 60 cos θ (M/r) (A23r) rθ A 4,2 [θ, M/r] = 0 (A23s) rθ A 4,3 [θ, M/r] = 0. (A23t)

For the Proca ﬁeld, the coeﬃcients Aθr appearing in the expression (16) of T θ are p,q h riren θr A 0,0 [θ, M/r] = 792 (A24a) Aθr [θ, M/r] = 972 (M/r) (A24b) 0,1 − θr 2 A 0,2 [θ, M/r] = 250 (M/r) (A24c) Aθr [θ, M/r] = 5544 cos2 θ (A24d) 1,0 − θr 2 A 1,1 [θ, M/r] = 7620 cos θ (M/r) (A24e) Aθr [θ, M/r] = 2668 cos2 θ (M/r)2 (A24f) 1,2 − θr 4 A 2,0 [θ, M/r] = 5544 cos θ (A24g) Aθr [θ, M/r] = 4020 cos4 θ (M/r) (A24h) 2,1 − θr 4 2 A 2,2 [θ, M/r] = 250 cos θ (M/r) (A24i) Aθr [θ, M/r] = 792 cos6 θ (A24j) 3,0 − θr 6 A 3,1 [θ, M/r] = 60 cos θ (M/r) (A24k) θr A 3,2 [θ, M/r] = 0. (A24l)

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Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 37 I. Action effective et tenseur d’impulsion-énergie renormalisé

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38 Published in Phys. Rev. D 90, 044045 (2014), eprint : arXiv:1404.7422 PARTIE II:

ELECTROMAGNÉTISMEMASSIFDE STUECKELBERGETRENORMALISATIONDU TENSEURD’IMPULSION-ÉNERGIEASSOCIÉ

CHAPITRE 2:

ELECTROMAGNÉTISMEMASSIFDE STUECKELBERG EN ESPACE-TEMPS COURBE : RENORMALISATIONDE HADAMARDDUTENSEUR D’IMPULSION-ÉNERGIE ET EFFET CASIMIR

Dans ce chapitre, nous nous sommes intéressés à l’électromagnétisme massif de Stueckelberg. Contrai- rement à la théorie massive de de Broglie-Proca qui est une généralisation directe de l’électromagnétisme de Maxwell obtenue en ajoutant un terme de masse qui brise la symétrie de jauge locale U(1) originelle, la théorie massive proposée par Stueckelberg la préserve par l’intermédiaire d’un couplage approprié d’un champ scalaire auxiliaire avec le champ vectoriel massif. Il faut noter aussi que cette théorie est unitaire et renormalisable et peut être incluse dans le modèle standard des particules. De plus, certaines extensions du modèle standard basées sur la théorie des cordes incluent l’existence d’un photon noir très massif pouvant expliquer la matière noire de l’univers. Ici, nous avons développé le formalisme général de la théorie de Stueckelberg sur un espace-temps arbitraire de dimension quatre (invariance de jauge de la théorie classique et quantification covariante ; équations d’ondes pour le champ massif de spin-1 Aµ, pour le champ scalaire auxiliaire de Stueckelberg Φ et pour les champs de fantômes C et C∗ ; identité de Ward ; représentation de Hadamard des divers propagateurs de Feynman et développement en séries de Taylor covariantes des coefficients correspondants). Cela nous a permis de construire, pour un état quantique de type Hadamard ψ , la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé à la théorie de | 〉 Stueckelberg. Nous avons donné deux expressions différentes mais équivalentes du résultat. La première expression ne fait intervenir que les caractéristiques du champ vectoriel Aµ. Dans la seconde expression, nous avons séparé de manière artificielle en deux parties les contributions indépendamment conservées du champ vectoriel Aµ et du champ scalaire auxiliaire Φ dans le but de discuter la limite de masse nulle des résultats obtenus dans le cadre de la théorie de Stueckelberg et les comparer avec ceux issus de celle de Maxwell. Notons que l’on peut passer de la deuxième expression à la première en éliminant la contribution du champ scalaire de Stueckelberg Φ par l’intermédiaire de deux identités de Ward, ce champ pouvant être considéré comme un fantôme.

Comme application de ce résultat, nous avons considéré, en espace-temps de Minkowski, l’effet de Casimir en dehors d’un milieu parfaitement conducteur et de bord plan. Nous avons retrouvé le résultat existant déjà dans la littérature pour la théorie de de Broglie-Proca. Notons toutefois que nous ne sommes pas vraiment satisfaits des conditions aux limites prises sur le bord du milieu conducteur et que des conditions plus réalistes pourraient être envisagées.

Pour ﬁnir, nous avons comparé les formalismes de de Broglie-Proca et de Stueckelberg et discuté les

41 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé avantages de l’approche de Stueckelberg bien qu’à notre avis ces deux descriptions de l’électromagnétisme massif sont les deux faces d’une même théorie.

42 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

Stueckelberg massive electromagnetism in curved spacetime: Hadamard renormalization of the stress-energy tensor and the Casimir eﬀect

Andrei Belokogne∗ and Antoine Folacci† Equipe Physique Théorique- Projet COMPA, SPE, UMR 6134 du CNRS et de l’Universitéde Corse, Universitéde Corse, BP 52, F-20250 Corte, France (Dated: October 13, 2016) We discuss Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime (gauge invariance of the classical theory and covariant quantization; wave equations for the massive spin-1 field Aµ, for the auxiliary Stueckelberg scalar field Φ and for the ghost fields C and C∗; Ward identities; Hadamard representation of the various Feynman propagators and covariant Taylor series expansions of the corresponding coefficients). This permits us to construct, for a Hadamard quantum state, the expectation value of the renormalized stress-energy tensor associated with the Stueckelberg theory. We provide two alternative but equivalent expressions for this result. The first one is obtained by removing the contribution of the “Stueckelberg ghost” Φ and only involves state-dependent and geometrical quantities associated with the massive vector field Aµ. The other one involves contributions coming from both the massive vector field and the auxiliary Stueckelberg scalar field, and it has been constructed in such a way that, in the zero-mass limit, the massive vector field contribution reduces smoothly to the result obtained from Maxwell’s theory. As an application of our results, we consider the Casimir effect outside a perfectly conducting medium with a plane boundary. We discuss the results obtained using Stueckelberg but also de Broglie- Proca electromagnetism and we consider the zero-mass limit of the vacuum energy in both theories. We finally compare the de Broglie-Proca and Stueckelberg formalisms and highlight the advantages of the Stueckelberg point of view, even if, in our opinion, the de Broglie-Proca and Stueckelberg approaches of massive electromagnetism are two faces of the same field theory.

PACS numbers: 04.62.+v, 11.10.Gh, 03.70.+k

I. INTRODUCTION of the various massive non-Maxwellian theories. Among these, two theories are particularly important, and we It is generally assumed that the electromagnetic in- intend to discuss them at more length in our article: teraction is mediated by a massless photon. This seems largely justified (i) by the countless theoretical and prac- (i) The most popular one, which is the simplest gener- tical successes of Maxwell’s theory of electromagnetism alization of Maxwell’s electromagnetism, is mainly and of its extension in the framework of quantum field due to de Broglie (note that the idea of an ultralight theory as well as (ii) by the stringent upper limits on massive photon is already present in de Broglie’s the photon mass (see p. 559 of Ref. [1] and references doctoral thesis [5,6] and has been developed by him therein) which have been obtained by various terrestrial in modern terms in a series of works [7–9] where and extraterrestrial experiments (currently, one of the he has considered the theory from a Lagrangian most reliable results provides for the photon mass m the point of view and has explicitly shown the modifi- 18 54 limit m 10− eV 2 10− kg [2]). cations induced by the photon mass for Maxwell’s ≤ ≈ × Despite this, physicists are seriously considering the equations) but is attributed in the literature to its possibility of a massive but, of course, ultralight photon “PhD student” Proca (for the series of his original and are very interested by the associated non-Maxwellian articles dating from 1930 to 1938 which led him to theories of electromagnetism (for recent reviews on the introduce in Ref. [10] the so-called Proca equation subject, see Refs. [3,4]). Indeed, the incredibly small for a massive vector field, see Ref. [11], but note, value mentioned above does not necessarily imply that however, that the main aim of Proca was the de- the photon mass is exactly zero, and from a theoretical scription of spin-1/2 particles inspired by the neu- point of view, massive electromagnetism can be rather trino theory of light due to de Broglie). Here, it easily included in the Standard Model of particle physics. is worth pointing out that, due to the mass term, Moreover, in order to test the masslessness of the photon the de Broglie-Proca theory is not a gauge theory, or, more precisely, to impose experimental constraints and this has some important consequences when we on its mass, it is necessary to have a good understanding compare, in the limit m2 0, the results obtained via the de Broglie-Proca theory→ with those derived from Maxwell’s electromagnetism. It is also important to recall that, in general, it is the de Broglie- ∗ [email protected] Proca theory that is used to impose experimental † [email protected] constraints on the photon mass [2–4].

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 43 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

(ii) The most aesthetically appealing one which, con- Here, m is the mass of the vector field Aµ, and the asso- trarily to the de Broglie-Proca theory preserves the ciated field strength F µν is defined as usual by local U(1) gauge invariance of Maxwell’s electro-

magnetism, has been proposed by Stueckelberg (see Fµν = µAν ν Aµ = ∂µAν ∂ν Aµ. (2) Refs. [12, 13] for the original articles on the subject ∇ − ∇ − and also Ref. [14] for a nice recent review). The Let us note that, while Maxwell’s theory is invariant un- construction of such a massive gauge theory can der the gauge transformation be achieved by coupling appropriately an auxiliary scalar field to the massive spin-1 field. This the- A A0 = A + Λ (3) ory is unitary and renormalizable and can be in- µ → µ µ ∇µ cluded in the Standard Model of particle physics for an arbitrary scalar field Λ, this gauge invariance is [14]. Moreover, it is interesting to note that exten- broken for the de Broglie-Proca theory due to the mass sions of the Standard Model based on string theory term. The extremization of (1) with respect to Aµ leads predict the existence of a hidden sector of parti- ν 2 µ to the Proca equation Fµν + m Aµ = 0. Applying cles which could explain the nature of dark matter. to this equation, we obtain∇ the Lorenz condition ∇ Among these exotic particles, there exists in particular a dark photon, the mass of which arises also via µA = 0 (4a) the Stueckelberg mechanism (see, e.g., Ref. [15]). ∇ µ This “heavy” photon may be detectable in low energy experiments (see, e.g., Refs [16–19]). It is also which is here a dynamical constraint (and not a gauge worth pointing out that the Stueckelberg procedure condition) as well as the wave equation is not limited to vector fields. It has been recently extended to “restore” the gauge invariance of var- A m2A R ν A = 0. (4b) µ − µ − µ ν ious massive field theories (see, e.g., Refs. [20, 21] which discuss the case of massive antisymmetric It should be noted that the action (1) is also directly tensor fields and, e.g., Ref. [22] where massive grav- relevant at the quantum level because the de Broglie- ity is considered). Proca theory is not a gauge theory. Stueckelberg massive electromagnetism is described by In the two following paragraphs, we shall briefly review a vector field A and an auxiliary scalar field Φ, and its these two theories at the classical level. µ action S = S [A , Φ, g ], which can be constructed De Broglie-Proca massive electromagnetism is de- Cl Cl µ µν from the de Broglie-Proca action (1) by using the substi- scribed by a vector field A , and its action S = µ tution S[Aµ, gµν ], which is directly obtained from the original Maxwell Lagrangian by adding a mass contribution, is 1 given by Aµ Aµ + µΦ, (5) → m∇ 1 1 S = d4x √ g F µν F m2AµA . (1) is given by − −4 µν − 2 µ ZM

1 1 1 1 S = d4x√ g F µν F m2 Aµ + µΦ A + Φ (6a) Cl − −4 µν − 2 m ∇ µ m ∇µ ZM 1 1 1 1 = d4x√ g µAν A + ( µA )2 m2AµA R AµAν − −2 ∇ ∇µ ν 2 ∇ µ − 2 µ − 2 µν ZM 1 µΦ Φ mAµ Φ . (6b) −2 ∇ ∇µ − ∇µ

It should be noted that, at the classical level, the vec- symmetry of Maxwell’s electromagnetism remains unbro- tor field Aµ and the scalar field Φ are coupled [see, in ken for the spin-1 field of the Stueckelberg theory. As a µ Eq. (6b), the last term mA µΦ]. Here, it is impor- consequence, in order to treat this theory at the quan- tant to note that Stueckelberg− massive∇ electromagnetism tum level (see below), it is necessary to add to the action is invariant under the gauge transformation (6) a gauge-breaking term and the compensating ghost contribution. A A0 = A + Λ, (7a) µ → µ µ ∇µ Here, it seems important to highlight some considera- Φ Φ0 = Φ mΛ, (7b) tions which will play a crucial role in this article. Let us → − note that the de Broglie-Proca theory can be obtained for an arbitrary scalar field Λ, so the local U(1) gauge

44 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

3 from Stueckelberg electromagnetism by taking of this approach, we refer to the monographs of Wald [26] and Fulling [25]). Here, we just recall that it is an exten- Φ = 0. (8) sion of the point-splitting method [23, 28, 29] which has We can therefore consider that the de Broglie-Proca the- been developed in connection with the Hadamard repre- ory is nothing other than the Stueckelberg gauge theory sentation of the Green functions (see, e.g., Refs. [30–42] in the particular gauge (8). However, it is worth not- and, more particularly, Refs. [32, 33, 35, 38, 39] where ing that this is a “bad” choice of gauge leading to some gauge theories are considered). complications. In particular: Our article is organized as follows. In Sec.II, we review the covariant quantization of Stueckelberg mas- (i) Due to the constraint (4a), the Feynman propaga- sive electromagnetism on an arbitrary four-dimensional tor associated with the vector field Aµ does not ad- curved spacetime (gauge-breaking action and associated mit a Hadamard representation (see below), and, ghost contribution; wave equations for the massive spin- as a consequence, the quantum states of the de 1 field Aµ, for the auxiliary Stueckelberg scalar field Φ Broglie-Proca theory are not of Hadamard type. and for the ghost fields C and C∗; Feynman propaga- This complicates the regularization and renormal- tors and Ward identities). In Sec.III, we focus on the ization procedures. particular gauge for which the various Feynman propagators and the associated Hadamard Green functions admit (ii) In the limit m2 0, singularities occur, and a Hadamard representation, or, in other words, we consider lot of physical results→ obtained in the context of quantum states of Hadamard type. We also construct the the de Broglie-Proca theory do not coincide with covariant Taylor series expansions of the geometrical and the corresponding results obtained with Maxwell’s state-dependent coefficients involved in the Hadamard theory. representation of the Green functions. In Sec.IV, we In this article, we intend to focus on the Stueckelberg obtain, for a Hadamard quantum state, the renormal- theory at the quantum level, and we shall analyze its en- ized expectation value of the stress-energy-tensor opera- ergetic content with possible applications to the Casimir tor, and we discuss carefully its geometrical ambiguities. effect (in this paper) and to cosmology of the very early In fact, we provide two alternative but equivalent expres- universe (in a next paper) in mind. More precisely, we sions for this renormalized expectation value. The first shall develop the formalism permitting us to construct, one is obtained by removing the contribution of the auxil- for a normalized Hadamard quantum state ψ of the iary scalar field Φ (here, it plays the role of a kind of ghost | i field) and only involves state-dependent and geometri- Stueckelberg theory, the quantity ψ Tµν ψ ren which denotes the renormalized expectationh | value| i of the stress- cal quantities associated with the massive vector field energy-tensor operator. It is well knownb that such an Aµ. The other one involves contributions coming from expectation value is of fundamental importance in quan- both the massive vector field and the auxiliary Stueck- tum field theory in curved spacetime (see, e.g., Refs. [23– elberg scalar field, and it has been constructed in such 27]). Indeed, it permits us to analyze the quantum state a way that, in the zero-mass limit, the massive vector ψ without any reference to its particle content, and, field contribution reduces smoothly to the result obtained moreover,| i it acts as a source in the semiclassical Einstein from Maxwell’s theory. In Sec.V, as an application of our results, we consider in the Minkowski spacetime the equations Gµν = 8π ψ Tµν ψ ren which govern the backreaction of the quantumh | field| i theory on the spacetime Casimir effect outside of a perfectly conducting medium with a plane boundary wall separating it from free space. geometry. b We discuss the results obtained using Stueckelberg but Let us recall that the stress-energy tensor T is an op- µν also de Broglie-Proca electromagnetism, and we consider erator quadratic in the quantum fields which is, from the the zero-mass limit of the vacuum energy in both the- mathematical point of view, an operator-valued distribu- b ories. Finally, in a conclusion (Sec.VI), we provide a tion. As a consequence, this operator is ill defined and step-by-step guide for the reader wishing to use our for- the associated expectation value ψ T ψ is formally in- h | µν | i malism, we briefly discuss and compare the de Broglie- finite. In order to extract from this expectation value a Proca and Stueckelberg approaches in the light of the finite and physically acceptable contributionb which could results obtained in our paper, and we highlight the ad- act as the source in the semiclassical Einstein equations, vantages of the latter. In a short Appendix, we have it is necessary to regularize it and then to renormalize all gathered some important results which are helpful to do the coupling constants. For a description of the various the calculations of Secs.III andIV, and, in particular, (i) techniques of regularization and renormalization in the we define the geodetic interval σ(x, x0), the Van Vleck- context of quantum field theory in curved spacetime (adi- Morette determinant ∆(x, x ) and the bivector of parallel abatic regularization method, dimensional regularization 0 transport gµν0 (x, x0) which play a crucial role along our method, ζ-function approach, point-splitting methods, article, and (ii) we discuss the concept of covariant Taylor . . . ), see Refs. [23–27] and references therein. series expansions. In this paper, we shall deal with Stueckelberg electro- It should be noted that, in this paper, we consider magnetism by using the so-called Hadamard renormal- a four-dimensional curved spacetime ( , g ) with no ization procedure (for a rigorous axiomatic presentation M µν

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 45 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

boundary (∂ = ∅), and we use units with ~ = c = G = and 1 and the geometricalM conventions of Hawking and Ellis 1 1 [43] concerning the definitions of the scalar curvature R, S = d4x √ g µΦ Φ ξm2Φ2 , (14) Φ − −2 ∇ ∇µ − 2 the Ricci tensor Rµν and the Riemann tensor Rµνρσ as ZM well as the commutation of covariant derivatives. It is SGh remaining unchanged and still given by Eq. (11). It moreover important to note that we provide the covari- µ is worth noting that the term mA µΦ coupling the ant Taylor series expansions of the Hadamard coefficients − ∇ fields Aµ and Φ in the classical action (6b) has disap- in irreducible form by using the algebraic proprieties of peared; because spacetime is assumed with no boundary, the Riemann tensor (and more particularly the cyclic- µ it is neutralized by the term mΦ Aµ in the gauge- ity relation and its consequences) as well as the Bianchi breaking action (10). − ∇ identity. The functional derivatives with respect to the fields Aµ, Φ, C and C∗ of the quantum action (9) or (12) will allow us to obtain, in Sec.IIB, the wave equations for all II. QUANTIZATION OF STUECKELBERG ELECTROMAGNETISM the fields and to discuss, in Sec.IVA, the conservation of the stress-energy tensor associated with Stueckelberg electromagnetism. They are given by In this section, we review the covariant quantization of Stueckelberg electromagnetism on an arbitrary four- 1 δS µν µ ν = [g (1 1/ξ) dimensional curved spacetime. The gauge-breaking term √ g δAµ − − ∇ ∇ considered includes an arbitrary gauge parameter ξ, and − Rµν m2gµν A (15) all the results concerning the wave equations for the mas- − − ν sive vector field Aµ, for the auxiliary scalar field Φ, for the for the vector field Aµ, ghost fields C and C∗ and for all the associated Feynman propagators as well as the Ward identities are expressed 1 δS = ξm2 Φ (16) in terms of ξ. √ g δΦ − − for the auxiliary scalar field Φ, as well as A. Quantum action 1 δRS 2 = ξm C∗ (17) At the quantum level, the action defining Stueckelberg √ g δC − − − massive electromagnetism is given by (see, e.g., Ref. [14]) and S[Aµ, Φ,C,C∗, gµν ] = SCl[Aµ, Φ, gµν ] 1 δLS +SGB[Aµ, Φ, gµν ] + SGh[C,C∗, gµν ], (9) = ξm2 C (18) √ g δC − − where we have added to the classical action (6) the gauge- − ∗ breaking term for the ghost fields C and C∗. It should be noted that, 1 due to the fermionic behavior of the ghost fields, we have S = d4x √ g ( µA + ξmΦ)2 (10) GB − −2ξ ∇ µ introduced in Eq. (17) the right functional derivative and ZM in Eq. (18) the left functional derivative. and the compensating ghost action

4 µ 2 S = d x √ g C∗ C + ξm C∗C . (11) Gh − ∇ ∇µ B. Wave equations ZM By collecting the fields in the explicit expression (9), the The extremization of the quantum action (9) or (12) quantum action can be written in the form permits us to obtain the wave equations for the fields Aµ, S [Aµ, Φ,C,C∗, gµν ] = SA [Aµ, gµν ] Φ, C and C∗. The vanishing of the functional derivatives +SΦ [Φ, gµν ] + SGh [C,C∗, gµν ] , (12) (15)–(18) provides where gµν (1 1/ξ) µ ν Rµν m2gµν A = 0 (19) − − ∇ ∇ − − ν 4 1 µν SA = d x √ g F Fµν − −4 for the vector field Aµ, ZM 1 2 µ 1 µ 2 2 m A Aµ ( Aµ) (13a) ξm Φ = 0 (20) −2 − 2ξ ∇ − 4 1 µ ν 1 µ ν for the auxiliary scalar field Φ, as well as = d x √ g A µAν Rµν A A − −2 ∇ ∇ − 2 2 2 ZM ξm C = 0 and ξm C∗ = 0 (21) 1 2 µ 1 1 µ 2 − − m A Aµ + 1 ( Aµ) (13b) −2 2 − ξ ∇ for the ghost fields C and C∗.

46 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

C. Feynman propagators and Ward identities the Hadamard form.) For this choice of gauge parameter, the wave equations (23), (25) and (27) for the Feynman propagators GA (x, x ), GΦ(x, x ) and GGh(x, x ) reduce From now on, we shall assume that the Stueckelberg µν0 0 0 0 field theory previously described has been quantized and to is in a normalized quantum state ψ . The Feynman ν ν 2 ν A gµ x Rµ m gµ Gνρ (x, x0) | i − − 0 propagator 4 = gµρ0 δ (x, x0), (30) A − G (x, x0) = i ψ TA (x)A (x0) ψ (22) µν0 µ ν0 h | | i 2 Φ 4 x m G (x, x0) = δ (x, x0) (31) associated with the field Aµ (here, T denotes time order- − − ing) is, by definition, a solution of and ν ν ν 2 ν A gµ x (1 1/ξ) µ Rµ m gµ Gνρ (x, x0) 2 Gh 4 − − ∇ ∇ − − 0 x m G (x, x0) = δ (x, x0). (32) 4 − − = g δ (x, x0) (23) − µρ0 As far as the Ward identity (28) is concerned, it takes 4 1/2 4 with δ (x, x0) = [ g(x)]− δ (x x0). Similarly, the now the local form − − Feynman propagator µ A Gh G (x, x0) + G (x, x0) = 0, (33) µν0 ν0 Φ ∇ ∇ G (x, x0) = i ψ T Φ(x)Φ(x0) ψ (24) h | | i while the Ward identity (29) remains unchanged. Be- associated with the scalar field Φ satisfies cause this last relation expresses the equality of the Feyn- man propagators associated with the auxiliary scalar field 2 Φ 4 ξm G (x, x0) = δ (x, x0), (25) x − − and the ghost fields, we shall often use a generic form and the Feynman propagator for these propagators (and for their Hadamard representation discussed below) where the labels Φ and Gh are Gh G (x, x0) = i ψ TC∗(x)C(x0) ψ (26) omitted, and we shall write h | | i Φ Gh associated with the ghost fields C and C∗ satisfies G(x, x0) = G (x, x0) = G (x, x0). (34) 2 Gh 4 x ξm G (x, x0) = δ (x, x0). (27) For ξ = 1 the nonminimal term in the wave equation − − for GA (x, x ) has disappeared [compare Eq. (30) with µν0 0 The three propagators are related by two Ward iden- Eq. (23)]. As consequence, we can consider a Hadamard tities. The first one is a nonlocal relation linking the A Gh representation for this propagator as well as for the prop- propagators G (x, x0) and G (x, x0). It can be ob- Φ Gh µν0 agators G (x, x0) and G (x, x0). In other words, we can tained by extending the approach of DeWitt and Brehme assume that all fields of Stueckelberg theory are in a nor- in Ref. [44] as follows: we take the covariant deriva- µ malized quantum state ψ of Hadamard type. tive of Eq. (23) and the covariant derivative ρ0 of | i Eq. (∇27); then, by commuting suitably the various∇ covariant derivatives involved and by using the relation A. Hadamard representation of the Feynman µ 4 4 [gµρ0 δ (x, x0)] = ρ0 δ (x, x0), we obtain the formal propagators relation∇ −∇ µ A Gh A (1/ξ) G (x, x0) + ν G (x, x0) The Feynman propagator Gµν (x, x0) associated with ∇ µν0 ∇ 0 0 1 the vector field Aµ can be now represented in the 2 − µ ρ A = (1 1/ξ) x ξm Rµ Gρν (x, x0) . (28) − − ∇ 0 Hadamard form 1/2 It should be noted that the nonlocal term in the right- A i ∆ (x, x0) G (x, x0) = gµν (x, x0) hand side of this equation is associated with the nonmin- µν0 8π2 σ(x, x ) + i 0 ν 0 imal term (1 1/ξ) µ appearing in the wave equa- − ∇ ∇ A A tion (23) and includes appropriate boundary conditions. + V (x, x0) ln[σ(x, x0) + i] + W (x, x0) ,(35) µν0 µν0 The second Ward identity can be obtained directly from the wave equations (25) and (27) by using arguments of A A where the bivectors V (x, x0) and W (x, x0) are sym- uniqueness. We have µν0 µν0 metric in the sense that V A (x, x ) = V A (x , x) and µν0 0 ν0µ 0 Φ Gh A A G (x, x0) G (x, x0) = 0. (29) W (x, x ) = W (x , x) and are regular for x x. µν0 0 ν0µ 0 0 − Furthermore, these bivectors have the following expan-→ sions III. HADAMARD EXPANSIONS OF THE + GREEN FUNCTIONS OF STUECKELBERG A ∞ A n V (x, x0) = V µν (x, x0) σ (x, x0), (36a) ELECTROMAGNETISM µν0 n 0 n=0 X + From now on, we assume that ξ = 1. (For ξ = 1, the A ∞ A n W (x, x0) = W µν (x, x0) σ (x, x0). (36b) 6 µν0 n 0 various Feynman propagators cannot be represented in n=0 X

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 47 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

Similarly, the Hadamard expansion of the Feynman prop- Similarly, the Hadamard coefficients Vn(x, x0) and agator G(x, x0) associated with the auxiliary scalar field Wn(x, x0) are also symmetric and regular biscalar func- Φ or the ghost fields is given by tions. The coefficients Vn(x, x0) satisfy the recursion re- 1/2 lations i ∆ (x, x0) G(x, x0) = ;a 2 2(n + 1)(n + 2) Vn+1 + 2(n + 1) Vn+1;aσ 8π σ(x, x0) + i 1/2 1/2 ;a 2(n + 1) V ∆− (∆ ) σ − n+1 ;a + V (x, x0) ln[σ(x, x0) + i] + W (x, x0) , (37) 2 + x m Vn = 0 (43a) − for n N with the boundary condition where the biscalars V (x, x0) and W (x, x0) are symmetric, ∈ ;a 1/2 1/2 ;a i.e., V (x, x0) = V (x0, x) and W (x, x0) = W (x0, x), regular 2 V + 2 V σ 2 V ∆− (∆ ) σ 0 0;a − 0 ;a for x0 x and possess expansions of the form 2 1/2 → + x m ∆ = 0, (43b) + − ∞ n while the coefficients W (x, x ) satisfy the recursion re- V (x, x0) = Vn(x, x0) σ (x, x0), (38a) n 0 n=0 lations X + ;a 2(n + 1)(n + 2) Wn+1 + 2(n + 1) Wn+1;aσ ∞ n W (x, x0) = Wn(x, x0) σ (x, x0). (38b) 1/2 1/2 ;a 2(n + 1) W ∆− (∆ ) σ n=0 n+1 ;a X − ;a +2(2n + 3) Vn+1 + 2 Vn+1;aσ In Eqs. (35) and (37), the factor i with 0+ ensures → 1/2 1/2 ;a the singular behavior prescribed by the time-ordered 2 Vn+1∆− (∆ );aσ − product introduced in the definition of the Feynman 2 + x m Wn = 0 (44) propagators [see Eqs. (22), (24) and (26)]. − A for n N. It should be also noted that from the recursion The Hadamard coefficients Vn µν0 (x, x0) and ∈ A relations (43) and (44) we can show that Wn µν0 (x, x0) introduced in Eq. (36) are also sym- 2 metric and regular bivector functions. The coefficients x m V = 0 (45) A − Vn µν0 (x, x0) satisfy the recursion relations and A A ;a 2(n + 1)(n + 2) V µν + 2(n + 1) V µν ;aσ 2 2 1/2 n+1 0 n+1 0 σ x m W = x m ∆ A 1/2 1/2 ;a − − − 2(n + 1) V ∆− (∆ ) σ ;a 1/2 1/2 ;a n+1µν0 ;a 2 V 2 V σ + 2 V ∆− (∆ ) σ . (46) − − − ;a ;a + g ρ R ρ m2g ρ V A = 0 (39a) µ x − µ − µ n ρν0 These two “wave equations” permit us to prove that the for n Nwith the boundary condition Feynman propagator (37) solves the wave equation (31) ∈ or (32). A A ;a A 1/2 1/2 ;a 2 V + 2 V σ 2 V ∆− (∆ ) σ 0 µν0 0 µν0;a − 0 µν0 ;a The Hadamard representation of the Feynman prop- ρ ρ 2 ρ 1/2 agators permits us to straightforwardly identify their + gµ x Rµ m gµ (gρν0 ∆ ) = 0, (39b) − − singular and regular parts (when the coincidence limit A while the coefficients Wn µν0 (x, x0) satisfy the recursion x0 x is considered). We can write → relations A A A G (x, x0) = G µν (x, x0) + G µν (x, x0) (47) A A ;a µν0 sing 0 reg 0 2(n + 1)(n + 2) W µν + 2(n + 1) W µν ;aσ n+1 0 n+1 0 with A 1/2 1/2 ;a 2(n + 1) Wn+1µν0 ∆− (∆ );aσ 1/2 − A i ∆ (x, x0) A A ;a G (x, x0) = g (x, x0) +2(2n + 3) V µν + 2 V µν ;aσ singµν0 2 µν0 n+1 0 n+1 0 8π σ(x, x0) + i A 1/2 1/2 ;a 2 V µν ∆− (∆ );aσ n+1 0 A − + V (x, x0) ln[σ(x, x0) + i] (48a) + g ρ R ρ m2g ρ W A = 0 (40) µν0 µ x − µ − µ n ρν0 for n N. It should be noted that from the recursion and ∈ relations (39) and (40) we can show that A i A G (x, x0) = W (x, x0) (48b) regµν0 2 µν0 ν ν 2 ν A 8π gµ x Rµ m gµ Vνρ = 0 (41) − − 0 as well as and G(x, x0) = Gsing(x, x0) + Greg(x, x0) (49) ν ν 2 ν A σ gµ x Rµ m gµ Wνρ = − − 0 with ν ν 2 ν 1/2 gµ x Rµ m gµ (gνρ0 ∆ ) 1/2 − − − i ∆ (x, x0) A A ;a A 1/2 1/2 ;a Gsing(x, x0) = 2 2Vµρ 2 Vµρ ;aσ + 2 Vµρ ∆− (∆ );aσ . (42) 8π σ(x, x ) + i − 0 − 0 0 0 These two “wave equations” permit us to prove that the + V (x, x ) ln[σ(x, x ) + i] (50a) Feynman propagator (35) solves the wave equation (30). 0 0

48 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

7 and and i 1/2 1 ∆ (x, x0) Greg(x, x0) = W (x, x0). (50b) (1) 8π2 G (x, x0) = 2 4π σ(x, x0) Here, it is important to note that, due to the geometrical 1/2 nature of σ(x, x0), gµν0 (x, x0), ∆ (x, x0) (see the Ap- +V (x, x0) ln σ(x, x0) + W (x, x0) . (58) pendix) and of V A (x, x ) and V (x, x ) (see Sec.IIIC), | | µν0 0 0 the singular parts (48a) and (50a) are purely geometrical It is important to recall that the Hadamard Green objects. By contrast, the regular parts (48b) and (50b) function associated with the massive vector ﬁeld Aµ is are state dependent (see Sec.IIID). deﬁned as the anticommutator

(1)A G (x, x0) = ψ A (x),A (x0) ψ (59) µν0 µ ν0 B. Hadamard Green functions h | { } | i and satisﬁes the wave equation

In the context of the regularization of the stress- ν ν 2 ν (1)A gµ x Rµ m gµ G (x, x0) = 0. (60) energy-tensor operator, instead of working with the Feyn- − − νρ0 man propagators, it is more convenient to use the asso- Similarly, the Hadamard Green function associated with ciated so-called Hadamard Green functions. Their rep- the auxiliary scalar field Φ is defined as the anticommu- resentations can be derived from those of the Feynman tator propagators by using the formal identities G(1)Φ(x, x ) = ψ Φ(x), Φ(x ) ψ (61) 1 1 0 0 = iπδ(σ) (51) h | { } | i σ + i P σ − which is a solution of and 2 (1)Φ m G (x, x0) = 0, (62) x − ln(σ + i) = ln σ + iπΘ( σ). (52) | | − while the Hadamard Green function associated with the Here, is the symbol of the Cauchy principal value, and ghost fields is defined as the commutator Θ denotesP the Heaviside step function. Indeed, these (1)Gh G (x, x0) = ψ [C∗(x),C(x0)] ψ (63) identities permit us to rewrite the expression (35) of the h | | i Feynman propagator associated with the massive vector and satisfies the wave equation field Aµ as 2 (1)Gh x m G (x, x0) = 0. (64) A A i (1)A G (x, x0) = G (x, x0) + G (x, x0), (53) − µν0 µν0 µν0 2 It should be noted that the expressions (59), (61) and where the average of the retarded and advanced Green (63) can be obtained from the definitions (22), (24) and functions is represented by (26) by noting that a Hadamard Green function is twice the imaginary part of the corresponding Feynman prop- A 1 1/2 G (x, x0) = ∆ (x, x0) gµν (x, x0) δ[σ(x, x0)] agator [see also Eqs. (53) and (56)]. The transition from µν0 8π 0 (22) to (59) or from (24) to (61) is rather immediate A n V (x, x0) Θ[ σ(x, x0)] (54) but it is not so obvious to obtain (63) from (26): it is − µν0 − o necessary to use the fact that the time-ordered product and the Hadamard Green function has the representation appearing in Eq. (26) is defined with a minus sign due 1/2 to the anticommuting nature of the ghost fields [i.e., we (1)A 1 ∆ (x, x0) G (x, x0) = gµν (x, x0) 0 00 00 µν0 4π2 σ(x, x ) 0 have TC∗(x)C(x0) = θ(x x ) C∗(x)C(x0) θ(x 0 0 − − − x ) C(x0)C∗(x)] and to consider that the ghost field C is A A +V (x, x0) ln σ(x, x0) + W (x, x0) . (55) hermitian while the anti-ghost field C∗ is anti-hermitian µν0 | | µν0 (see, e.g., Ref. [45]). Similarly, we have for the Feynman propagator (37) as- The Ward identities (33) and (29) satisfied by the sociated with the auxiliary scalar field Φ or the ghost Feynman propagators are also valid for the Hadamard fields Green functions. We have µ (1)A (1)Gh i (1) G (x, x0) + ν G (x, x0) = 0 (65) G(x, x0) = G(x, x0) + G (x, x0), (56) ∇ µν0 ∇ 0 2 and where (1)Φ (1)Gh 1 1/2 G (x, x0) G (x, x0) = 0. (66) G(x, x0) = ∆ (x, x0)δ[σ(x, x0)] − 8π n Similarly, as it has been previously noted in the case of V (x, x0)Θ[ σ(x, x0)] (57) − − the Feynman propagators, the Hadamard representaion o

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 49 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

of the Hadamard Green functions permits us to straight- covariant Taylor series expansions of the bivector coef- A A forwardly identify their singular and purely geometrical ﬁcients V0 µν (x, x0) and V1 µν (x, x0) are given by [see parts as well as their regular and state-dependent parts Eqs. (A.7) and (A.8)] (when the coincidence limit x0 x is considered). We → A ν0 A A can write V0 µν = gν V0 µν0 = v0 (µν) A A ;a (1)A (1)A (1)A (1/2) v0 (µν);a + v0 [µν]a σ G (x, x0) = G µν (x, x0) + G µν (x, x0) (67) − µν0 sing 0 reg 0 1 + vA + vA σ;aσ;b + O(σ3/2) (71a) with 2! 0 (µν)ab 0 [µν]a;b 1/2 (1)A 1 ∆ (x, x0) and Gsing µν0 (x, x0) = 2 gµν0 (x, x0) 4π σ(x, x0) A ν0 A A 1/2 V1 µν = gν V1 µν0 = v1 (µν) + O(σ ). (71b) A + V (x, x0) ln σ(x, x0) (68a) µν0 | | Here, the explicit expressions of the Taylor coeﬃcients can be determined from the recursion relations (39). We and have

(1)A 1 A A 2 G µν (x, x0) = W (x, x0) (68b) v0 (µν) = (1/2) Rµν + gµν (1/2) m (1/12) R , (72a) reg 0 2 µν0 − 4π A v0 [µν]a = (1/6) Ra[µ;ν], (72b) as well as A v0 (µν)ab = (1/6) Rµν;(ab) + (1/12) Rµν Rab (1) (1) (1) pq G (x, x0) = Gsing(x, x0) + Greg(x, x0) (69) +(1/12)Rµpq(a Rν b) | | 2 with +gµν (1/12) m Rab (1/40) R;ab (1/120) Rab − − 1/2 n p (1) 1 ∆ (x, x0) (1/72) RRab + (1/90) Rap R G (x, x0) = − b sing 4π2 σ(x, x ) 0 (1/180) R R p q (1/180) R R pqr , (72c) − pq a b − apqr b + V (x, x0) ln σ(x, x0) (70a) A 2 o | | v1 (µν) = (1/4) m Rµν (1/24) Rµν (1/24) RRµν p − pqr− +(1/8) Rµp Rν (1/48) Rµpqr Rν and − +g (1/8) m4 (1/24) m2 R + (1/120) R µν − (1) 1 2 pq G (x, x0) = W (x, x0). (70b) +(1/288) R (1/720) R R reg 4π2 pq − pqrs +(1/720) Rpqrs R . (72d) It should be pointed out that the regular part of the } Hadamard Green function given by Eq. (68b) [respec- The covariant Taylor series expansions of the biscalar tively, by Eq. (70b)] is proportional to that of the Feyn- coefficients V0(x, x0) and V1(x, x0) are given by [see man propagator given by Eq. (48b) [respectively, by Eqs. (A.5) and (A.6)] Eq. (50b)]. V = v (1/2) v σ;a 0 0 − { 0;a} 1 + v σ;aσ;b + O(σ3/2) (73a) C. Geometrical Hadamard coefficients and 2! 0ab associated covariant Taylor series expansions and A Formally, the Hadamard coefficients V µν (x, x0) or 1/2 n 0 V1 = v1 + O(σ ). (73b) Vn(x, x0) can be determined uniquely by solving the recursion relations (39) or (43), i.e., by integrating these Here, the explicit expressions of the Taylor coefficients recursion relations along the unique geodesic joining x to can be determined from the recursion relations (43). We x0 (it is unique for x0 near x or more generally for x0 in a have convex normal neighborhood of x). As a consequence, all 2 these coefficients as well as the sums given by Eqs. (36a) v0 = (1/2) m (1/12) R, (74a) − and (38a) are of purely geometric nature, i.e., they only v = (1/12) m2 R (1/40) R (1/120) R 0ab ab − ;ab − ab depend on the geometry along the geodesic joining x to p (1/72) RRab + (1/90) Rap R x0. − b p q pqr From the point of view of the practical applications (1/180) Rpq Ra b (1/180) Rapqr Rb , (74b) considered in this work, it is sufficient to know the expres- − − v = (1/8) m4 (1/24) m2 R + (1/120) R sions of the two first geometrical Hadamard coefficients. 1 2− pq Furthermore, their covariant Taylor series expansions are +(1/288) R (1/720) Rpq R 1 0 − pqrs needed up to order σ for n = 0 and σ for n = 1. The +(1/720) Rpqrs R . (74c)

50 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

In order to obtain the expressions of the Taylor coeﬃ- by inverting these equations. From Eq. (75), we obtain cients given by Eqs. (72) and (74), we have used some A 1/2 s (x) = lim W (x, x0), (78a) of the properties of σ(x, x0), gµν (x, x0), ∆ (x, x0) men- µν µν0 0 x0 x tioned in the Appendix as well as the algebraic properties → 1 A A a (x) = lim W (x, x0) W (x, x0) , (78b) of the Riemann tensor. µνa µν0;a0 µν0;a 2 x0 x − → 1 A A s (x) = lim W (x, x0) + W (x, x0) . µνab µν0;(a0b0) µν0;(ab) 2 x0 x → D. State-dependent Hadamard coeﬃcients and h (78c)i associated covariant Taylor series expansions

(Here, the coefficient aµνabc is not relevant because it 1. General considerations does not appear in the final expressions of the renormalized stress-energy-tensor operator given in Sec.IVC). Unlike the geometrical Hadamard coefficients, the coef- Similarly, from Eq. (76), we straightforwardly establish A that ficients Wn µν0 (x, x0) and Wn(x, x0) are neither uniquely defined nor purely geometrical. Indeed, the coefficient A w(x) = lim W (x, x0), (79a) W0 µν0 (x, x0) [respectively, W0(x, x0)] is unrestrained by x0 x the recursion relations (42) [respectively, by the recur- → wab(x) = lim W;(a b )(x, x0). (79b) x x 0 0 sion relations (46)]. As a consequence, this is also true 0→ for all the coefficients W A (x, x ) and W (x, x ) with n µν0 0 n 0 We shall now rewrite the wave equations (60), (62) and n 1 and for the sums (36b) and (38b). This arbitrari- (64) as well as the Ward identity (65) in terms of the Tay- ness≥ is in fact very interesting, and it can be used to lor coefficients of W A (x, x ) and W (x, x ). To achieve encode the quantum state dependence of the theory in µν 0 0 A the calculations, we shall use extensively the properties the coefficients W0 µν0 (x, x0) and W0(x, x0). Once they 1/2 A of σ(x, x0), gµν0 (x, x0), ∆ (x, x0) mentioned in the Ap- have been specified, the coefficients Wn µν0 (x, x0) and A pendix. Wn(x, x0) with n 1 as well as the bivector W (x, x0) ≥ µν0 and the biscalar W (x, x0) are uniquely determined. In the following, instead of working with the state- 2. Wave equations dependent Hadamard coefficients, we shall consider the A sums W (x, x0) and W (x, x0), and, more precisely, we µν0 By inserting the Hadamard representation (55) of the shall use their covariant Taylor series expansions up to (1)A 3/2 Green function G (x, x0) into the wave equation (60), order σ . We have [see Eqs. (A.7) and (A.8)] µν0 we obtain a wave equation with source for the state- A A ν0 A ;a dependent Hadamard coefficient W (x, x0). We have W = g W = sµν (1/2) sµν;a + aµνa σ µν0 µν ν µν0 − { } 1 ;a ;b 1 ρ0 ν ν 2 ν A + sµνab + aµνa;b σ σ (3/2) sµνab;c gρ gµ x Rµ m gµ Wνρ 2! { } − 3! { − − 0 ;a ;b ;c 2 A ρ0 A ;a (1/4) s + a σ σ σ + O(σ ) (75) = 6 V1 µρ 2 gρ V1 µρ0;aσ + O(σ) − µν;abc µνabc} − − = 6 vA + 2 vA + 8 vA σ;a + O(σ). and [see Eqs. (A.5) and (A.6)] − 1 (µρ) 1 (µρ);a 1 [µρ]a (80) ;a 1 ;a ;b W = w (1/2) w;a σ + wab σ σ Here, we have used the expansions of the geometrical − { } 2! Hadamard coefficients given by Eqs. (36a) and (71). By 1 ;a ;b ;c 2 A (3/2) wab;c (1/4) w;abc σ σ σ + O(σ ). inserting the expansion (75) of Wµν (x, x0) into the left- −3! { − } hand side of Eq. (80), we find the following relations: (76) ν p 2 A sµρν = R sρ)p + m sµρ 6 v1 (µρ), (81a) In the expansion (75) we have introduced the notations (µ − µ ν pq 2 p A p sµ ν = R spq + m sp 6 v1 p , (81b) A − sµνa a w (77a) ;ν p 1 p (µν)a1 ap a = R s , (81c) ··· ≡ ··· µρν − [µ ρ]p ;ν p for the symmetric part of the Taylor coefficients and sµρνa = (1/4) (sµρ);a + (1/2) R a;(µsρ)p ;p p A a w (77b) (1/2) R(µ a s ρ)p (1/2) R (µsρ)p;a µνa1 ap [µν]a1 ap | ··· ≡ ··· − | − p p q +(1/2) R asµρ;p R (µ as ρ)p;q for their antisymmetric part. − | | It is important to note that, with practical applications p p q p R (µ ap ρ)a R (µ aa ρ)pq + (1/2) sµρp ;a in mind, it is interesting to express some of the Taylor − | | − | | (1/2) m2s + 2 vA , (81d) coefficients appearing in Eqs. (75) and (76) in terms of − µρ;a 1 (µρ);a the bitensors W A (x, x ) and W (x, x ). This can be done s µ ;ν = (1/4) ( s p) + (1/2) Rq s p µν0 0 0 µ νa p ;a a p ;q

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 51 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

pq pqr p q (1/2) R spq;a + R aapqr + (1/2) sp q ;a To establish Eq. (85), we have used the expansions of the − geometrical Hadamard coefficients given by Eqs. (36a), (1/2) m2s p + 2 vA p . (81e) − p ;a 1 p ;a (38a), (71) and (73). By inserting the expansions (75) of A Furthermore, by combining Eq. (81d) with Eq. (81a) and Wµν (x, x0) and (76) of W (x, x0) into the left-hand side of Eq. (81e) with Eq. (81b), we also establish that Eq. (85), we find the following relations: s ;ν = (1/4) ( s ) + (1/2) Rp s µ ;p µρνa µρ ;a a;(µ ρ)p aµν = (1/2) spν + (1/2) w;ν , (86a) ;p p µ ;p p p s = (1/2) s + (1/2) R sνp + a (1/2) R(µ a s ρ)p + (1/2) R (µ ;as ρ)p µν a pν a a ν[a;p] − | | | | p q A p +wνa v1 (νa) + v1gνa. (86b) +(1/2) R asµρ;p R (µ as ρ)p;q − − | | p p q A Furthermore, by combining Eq. (86b) with Eq. (86a), we R (µ ap ρ)a R (µ aa ρ)pq v1 (µρ);a (81f) − | | − | | − also establish that and µ ;p p ;p sµν a = (1/4) spν a + (1/2) R asνp + (1/2) apνa µ ;ν p q p sµ νa = (1/4) (sp );a + (1/2) R asp ;q A (1/4) w;νa + wνa v1 (νa) + v1gνa. (86c) pq pqr A p − − +(1/2) R ;aspq + R aapqr v1 p ;a. (81g) − Of course, the second Ward identity given by Eq. (66) Mutatis mutandis, by inserting the Hadamard repre- (1) provides trivially the equality of the Taylor coefficients of sentation (58) of the Green function G (x, x0) into the the Hadamard coefficients associated with the auxiliary wave equation (62) or (64), we obtain a wave equation scalar field and the ghost fields. We have with source for the state-dependent Hadamard coefficient Φ Gh W (x, x0). We have V = V (87) 2 ;a x m W = 6 V1 2 V1;aσ + O(σ) and − − − ;a = 6 v + 2 v σ + O(σ). (82) Φ Gh − 1 1;a W = W . (88) Here, we have used the expansions of the geometrical Hadamard coefficients given by Eqs. (38a) and (73). By inserting the expansion (76) of W (x, x0) into the left-hand IV. RENORMALIZED STRESS-ENERGY side of Eq. (82), we find the following relations: TENSOR OF STUECKELBERG ELECTROMAGNETISM ρ 2 wρ = m w 6 v1, (83a) ;ρ − p wρa = (1/4) (w);a + (1/2) wp ;a A. Stress-energy tensor p 2 +(1/2) R aw;p (1/2) m w;a + 2 v1;a. (83b) − The functional derivation of the quantum action of the Furthermore, by combining Eq. (83b) with Eq. (83a), we Stueckelberg theory with respect to the metric tensor gµν also establish that permits us to obtain the associated stress-energy tensor ;ρ p w = (1/4) ( w) + (1/2) R w v . (83c) Tµν . By definition, we have ρa ;a a ;p − 1;a µν 2 δ T = S[Aµ, Φ,C,C∗, gµν ], (89) 3. Ward identities √ g δgµν − and its explicit expression can be obtain by using that, The first Ward identity given by Eq. (65) expressed in the elementary variation in terms of the Hadamard representation of the Green (1)A (1) functions G (x, x0) [see Eq. (55)] and G (x, x0) [see gµν gµν + δgµν (90) µν0 → Eq. (58)] permits us to write a relation between the geometrical Hadamard coefficients V A (x, x ) and V (x, x ) of the metric tensor, we have (see, for example, Ref. [46]) µν0 0 0 as well as another one between the state-dependent Hadamard coefficients W A (x, x ) and W (x, x ). We ob- µν µν µν µν0 0 0 g g + δg , (91a) tain → √ g √ g + δ√ g, (91b) ν0 A ;µ − → − − g V + V = 0 (84) ρ ρ ρ ν µν0 ;ν0 Γ Γ + δΓ (91c) µν → µν µν which is an identity between the geometrical Taylor co- with efficients (72a)–(72d) and (74a)–(74c) and µν µρ νσ δg = g g δgρσ, (91d) g ν0 W A ;µ + W − ν µν0 ;ν0 1 µν = V A σ;µ + V σ + O (σ) δ√ g = √ g g δgµν , (91e) − 1 µν 1 ;ν − 2 − 1 = vA v g σ;a + O (σ) . (85) δΓρ = δg ;ρ + δgρ + δgρ . (91f) − 1 (νa) − 1 νa µν 2 − µν µ;ν ν;µ

52 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

The stress-energy tensor derived from the action (9) is and given by µν µ ν µν ρ 2 2 TΦ = Φ Φ (1/2) g ρΦ Φ + m Φ , (95b) µν µν µν ∇ ∇ − ∇ ∇ T µν = T + T + T , (92) cl GB Gh while the contribution associated with the ghost fields where the contributions of the classical and gauge- remains unchanged [see Eq. (93c)]. breaking parts take the forms By construction, the stress-energy tensor (89) [see also its explicit expressions (92) and (94)] is conserved, i.e., µν µ νρ 2 µ ν Tcl = F ρF + m A A µν µ ν (µ ν) T = 0. (96) + Φ Φ + 2 mA Φ ∇ν ∇ ∇ ∇ (1/4) gµν F F ρτ + 2 m2A Aρ − ρτ ρ Indeed, this property is due to the invariance of the action ρ ρ (9) or (12) under spacetime diffeomorphisms and there- +2 ρΦ Φ + 4 mAρ Φ ∇ ∇ ∇ fore under the infinitesimal coordinate transformation = Aµ ρAν 2 A(µ ν)Aρ + µA ν Aρ ∇ρ ∇ − ∇ρ ∇ ∇ ρ∇ +m2AµAν + µΦ ν Φ + 2 mA(µ ν)Φ xµ xµ + µ with µ 1. (97) ∇ ∇ ∇ → | | (1/2) gµν A ρAτ A τ Aρ − ∇ρ τ ∇ − ∇ρ τ ∇ Under this transformation, the vector, scalar and ghost 2 ρ ρ ρ +m AρA + ρΦ Φ + 2 mAρ Φ (93a) fields as well as the background metric transform as ∇ ∇ ∇ and A A + δA , (98a) µ → µ µ T µν = 2 A(µ ν) Aρ 2 mA(µ ν)Φ Φ Φ + δΦ, (98b) GB − ∇ ∇ρ − ∇ → µν ρ τ ρ 2 C C + δC, (98c) (1/2) g 2 Aρ τ A ( ρA ) → − − ∇ ∇ − ∇ C∗ C∗ + δC∗, (98d) 2 2 n ρ → +m Φ 2 mAρ Φ , (93b) g g + δg . (98e) − ∇ µν → µν µν o while the contribution associated with the ghost fields is The variations associated with the field transformations given by (98) are obtained by Lie derivation with respect to the vector µ: µν (µ ν) − TGh = 2 |C∗ | C − ∇ ∇ ρ ρ µν ρ 2 δAµ = Aµ = ρAµ ( µ )Aρ, (99a) + g ρC∗ C + m C∗C . (93c) − ∇ ∇ L −ρ ∇ − ∇ δΦ = Φ = ρΦ, (99b) − We can note the existence of terms coupling the fields Aµ L − ρ∇ µν δC = C = ρC, (99c) − and Φ in the expression of Tcl [see Eq. (93a)] as well as L − ∇ρ µν δC∗ = C∗ = ρC∗, (99d) in the expression of TGB [see Eq. (93b)]. L− − ∇ We also give an alternative expression for the stress- δgµν = gµν = µν ν µ. (99e) energy tensor which can be derived from the action (12) L− −∇ − ∇ µν µν The invariance of the action (9) or (12) leads to or by summing Tcl and TGB. This eliminates any coupling between the fields Aµ and Φ and permits us to 4 1 δS 1 δS straightforwardly infer that the stress-energy tensor has d x√ g δAµ + δΦ − √ g δAµ √ g δΦ three independent contributions corresponding to the ZM − − massive vector field Aµ, the auxiliary scalar field Φ and 1 δRS 1 δLS + δC + δC∗ the ghost fields C and C∗. We can write √ g δC √ g δC − − ∗ µν µν µν 1 2 δS T µν = T + T + T (94) + δg = 0 (100) A Φ Gh 2 √ g δg µν − µν with which implies µν µ νρ 2 µ ν (µ ν) ρ TA = F ρF + m A A 2 A ρA µν − ∇ ∇ ν T = µν ρτ 2 ρ ∇ (1/4) g Fρτ F + 2 m AρA [ µA Aµ Aµ ] Aν Rν Aρ m2Aν − ∇ α − ∇α − ∇ν − ρ − ρn τ ρ 2 + [ µΦ] Φ m2Φ 4 Aρ τ A 2 ( ρA ) ∇ − − ∇ ∇ − ∇ 2 µ µ 2 C∗ m C∗ [ C] [ C∗] C m C (101) = Aµ ρAν 2 A(µ ν)Aρo+ µA ν Aρ − − ∇ − ∇ − ∇ρ ∇ − ∇ρ ∇ ∇ ρ∇ 2 µ ν (µ ν) ρ by using Eq. (99) as well as Eqs. (15)–(18). From the +m A A 2 A ρA − ∇ ∇ wave equations associated with the massive vector field µν ρ τ τ ρ (1/2) g ρAτ A ρAτ A A [see Eq. (19)], the auxiliary scalar field Φ [see Eq. (20)] − ∇ ∇ − ∇ ∇ µ n and the ghost fields C and C∗ [see Eq. (21)], we then +m2A Aρ 2 A ρ Aτ ( Aρ)2 (95a) ρ − ρ∇ ∇τ − ∇ρ obtain Eq. (96). o

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 53 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

B. Expectation value of the stress-energy tensor It should be noted that we have not included in Eqs. (103a) and (103b) the contributions which can be At the quantum level, all the ﬁelds involved in the obtained by point splitting from the terms coupling Aµ Stueckelberg theory as well as the associated stress- and Φ in Eqs. (93a) and (93b). Such contributions are not energy tensor [see Eqs. (92) and (94)] are operators. present because, due to the absence of coupling between From now on, we shall denote the stress-energy-tensor Aµ and Φ in the quantum action (12), two-point correla- tion functions involving both Aµ and Φ vanish identically. operator by Tµν and we shall focus on the quantity It should be noted that the absence of these contributions ψ T ψ which denotes its expectation value with re- µν can be also justiﬁed in another way: in the quantum specth | to| i the Hadamardb quantum state ψ discussed in stress-energy-tensor operator (94), any coupling between Sec.III. | i b A and Φ has disappeared. The expectation value ψ T ψ corresponding to the µ µν Here, some remarks are in order: expression (92) of the stress-energyh | | i tensor is decomposed as follows: b (i) When we use the point-splitting method, it is

cl GB Gh more convenient to deﬁne the expectation value ψ Tµν ψ = ψ Tµν ψ + ψ Tµν ψ + ψ Tµν ψ . (102) h | | i h | | i h | | i h | | i ψ Tµν ψ from Hadamard Green functions rather thanh | from| i Feynman propagators. Indeed, this The three terms in the right-hand side of this equation b b b b avoids us having to deal with additional singular are explicitly given by b terms due to the time-ordered product.

cl 1 cl ρσ0 (1)A ψ T (x) ψ = lim A (x, x0) G (x, x0) µν µν ρσ0 (ii) Of course, because of the short-distance behavior h | | i 2 x0 x T → h i of the Hadamard Green functions, the expressions 1 clΦ (1)Φ b + lim µν (x, x0) G (x, x0) , (103a) (103) as well as the expectation value ψ Tµν ψ 2 x0 x T h | | i → h i given in Eq. (102) are divergent and therefore meaningless. In Sec.IVC we will regularizeb these GB 1 GB ρσ0 (1)A ψ T (x) ψ = lim A (x, x0) G (x, x0) quantities. µν µν ρσ0 h | | i 2 x0 x T → h i (iii) Even if the formal expression (102) of the expec- 1 GBΦ (1)Φ b + lim µν (x, x0) G (x, x0) (103b) tation value of the stress-energy-tensor operator is 2 x0 x T → h i divergent, it is interesting to note that and GB Gh Gh 1 Gh (1)Gh ψ Tµν ψ + ψ Tµν ψ = 0. (105) ψ Tµν (x) ψ = lim µν (x, x0) G (x, x0) ,(103c) h | | i h | | i h | | i 2 x0 x T → h i Indeed, fromb the definitionsb (103b) and (103c), we clA ρσ0 clΦ GBA ρσ0 GBΦ Gh whereb µν , µν , µν , µν and µν are the can obtain Eq. (105) by using Eqs. (65) and (66) differentialT operatorsT constructedT T by point splittingT from as well as the wave equation (64). It should be the expressions (93a), (93b) and (93c). We have noted that, as a consequence of Eq. (105), Eq. (102) reduces to clA ρσ0 α0 ρσ0 ρ σ0 αβ0 µν = gν g µ α0 + gµ gν g α β0 T ∇ ∇ ∇ ∇ cl ρ α0 βσ0 2 ρ σ0 ψ Tµν ψ = ψ Tµν ψ . (106) 2 g g g β α + m g g h | | i h | | i − µ ν ∇ ∇ 0 µ ν 1 g gρσ0 gαβ0 gρα0 gβσ0 We can also give theb alternativeb expression of the ex- −2 µν ∇α∇β0 − ∇β∇α0 n pectation value ψ Tµν ψ obtained from Eq. (94). It +m2gρσ0 , (104a) takes the followingh | form| i b o A Φ Gh ψ Tµν ψ = ψ Tµν ψ + ψ Tµν ψ + ψ Tµν ψ , (107) clΦ ν0 1 αβ0 h | | i h | | i h | | i h | | i µν = gν µ ν0 gµν g α β0 , (104b) T ∇ ∇ − 2 ∇ ∇ whereb the contributionsb associatedb with theb massive vec- n o tor field Aµ and the auxiliary scalar field Φ are separated GBA ρσ0 = 2 g ρg α0 σ0 and given by Tµν − µ ν ∇α0 ∇ 1 ρ σ0 ρα0 σ0 A 1 A ρσ0 (1)A g 2 g , (104c) ψ T (x) ψ = lim (x, x0) G (x, x0) (108a) µν α0 µν µν ρσ0 −2 −∇ ∇ − ∇ ∇ h | | i 2 x0 x T n o → h i andb 1 GBΦ = m2 g (104d) µν µν Φ 1 Φ (1)Φ T −2 ψ Tµν (x) ψ = lim µν (x, x0) G (x, x0) . (108b) h | | i 2 x0 x T and → h i A ρσ0 Φ Here,b the differential operators µν and µν are con- Gh = 2 g ν0 + g gαβ0 + m2 . (104e) T T Tµν − ν ∇µ∇ν0 µν ∇α∇β0 structed by point splitting from the expressions (95a) and n o

54 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

(95b). We have tensor operator is given by

1 cl A cl Φ A ρσ0 α0 ρσ0 ρ σ0 αβ0 A Φ = g g + g g g ψ Tµν ψ ren = 2 µν W + µν W Tµν ν ∇µ∇α0 µ ν ∇α∇β0 h | | i 8π T T ρ α0 βσ0 2 ρ σ0 1 2 g g g + m g g GBA A GBΦ Φ µ ν β α0 µ ν b + W + W − ∇ ∇ 8π2 µν µν ρ α0 σ0 T T 2 g g α − µ ν ∇ 0 ∇ 1 Gh Gh + 2 µν W 1 ρσ0 αβ0 ρα0 βσ0 8π T gµν g g α β0 g g β α0 −2 ∇ ∇ − ∇ ∇ + Θ (112) n µν +m2gρσ0 ρ σ0 2 gρα0 σ0 (109a) − ∇ ∇ − ∇α0 ∇ with o e cl A cl ρσ0 A A W (x) = lim A (x, x0) W (x, x0) , and µν µν ρσ0 T x0 x T → 1 (113a) Φ = g ν0 g gαβ0 + m2 . (109b) Tµν ν ∇µ∇ν0 − 2 µν ∇α∇β0 clΦ Φ clΦ Φ n o µν W (x) = lim µν (x, x0) W (x, x0) , T x0 x T It should be noted that the expressions (102) and (107) of → (113b) the expectation value ψ T ψ are identical because the h | µν | i various diﬀerential operators clA ρσ0 , clΦ , GBA ρσ0 and µν µν µν GB A GB ρσ0 A T T T A W (x) = lim A (x, x0) W (x, x0) , GBΦ b A ρσ0 Φ µν µν ρσ0 appearing in (103) and and appearing T x0 x T Tµν Tµν Tµν → in (108) are related by (113c)

A ρσ0 clA ρσ0 GBA ρσ0 GBΦ Φ GBΦ Φ µν = µν + µν , (110a) µν W (x) = lim µν (x, x0) W (x, x0) T T T T x0 x T Φ = clΦ + GBΦ . (110b) → Tµν Tµν Tµν (113d) and

Gh Gh Gh Gh C. Renormalized stress-energy tensor µν W (x) = lim µν (x, x0) W (x, x0) . T x0 x T → (113e) 1. Definition and conservation Here, the differential operators clA ρσ0 , clΦ , GBA ρσ0 , Tµν Tµν Tµν As we have already noted, the expectation value GBΦ and Gh are given by Eqs. (104a)–(104e). In Tµν Tµν ψ T ψ given by Eq. (102) is divergent due to the Eqs. (113a)–(113e), the coincidence limits x0 x are ob- µν → short-distanceh | | i behavior of the Green functions or, more tained from the covariant Taylor series expansions (75) precisely,b to the singular purely geometrical part of the and (76) by using extensively some of the results dis- Hadamard functions given in Eqs. (68a) and (70a) (see played in the Appendix. The final expressions can be the terms in 1/σ and ln σ ). It is possible to construct simplified by using the relations (81a), (81b) and (83a) the renormalized expectation| | value of the stress-energy- we have previously obtained from the wave equations. tensor operator with respect to the Hadamard quantum We have state ψ by using the prescription proposed by Wald in ρ | i clA W A = (1/2) s ρ + (1/2) s s Refs. [26, 30, 31]. In Eqs. (103a)–(103c) we first discard Tµν ρ ;µν µν − ρ(µ;ν) the singular contributions, i.e., we make the replacements a ρ a ρ s ρ + 2 s ρ − µ [ν;ρ] − ν [µ;ρ] − ρ µν ρ(µν) (1/2) g (1/2) s ρ (1/2) s ;ρτ − µν ρ − ρτ (1)A (1)A 1 A ρτ ρ;τ ρτ G (x, x0) G µν (x, x0) = W (x, x0), (111a) (1/2) R sρτ aρτ + sρτ µν0 reg 0 2 µν0 − − → 4π A A ρ +6 v µν 3 gµν v , (114a) (1)Φ (1)Φ 1 Φ 1 1 ρ G (x, x0) G (x, x0) = W (x, x0), (111b) − → reg 4π2 cl Φ Φ Φ (1)Gh (1)Gh 1 Gh Φ W = (1/2) w w G (x, x0) G (x, x0) = W (x, x0), (111c) µν ;µν µν → reg 4π2 T − (1/2) g (1/2) wΦ m2wΦ 3 g v , (114b) − µν − − µν 1 and we add to the result a state-independent tensor Θµν ρ which only depends on the mass parameter and on the GBA W A = Rρ s a ρ a ρ 2 s Tµν (µ ν)ρ − µ (ν;ρ) − ν (µ;ρ) − ρ(µν) local geometry and which ensures the conservatione of ;ρτ ρτ (1/2) gµν (1/2) s + (1/2) R sρτ the final expression. In other words, we consider that − − ρτ the renormalized expectation value of the stress-energy- +a ρ;τ s ρτ , (114c) ρτ − ρτ

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 55 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

GBΦ Φ 2 Φ Gh µν W = (1/2) gµν m w (114d) the ghost term ψ Tµν ψ cancel each other out [see T − Eq. (105)]. This stillh | remains| i valid for the corresponding and regularized expectationb values up to purely geometrical Gh W Gh = wGh + 2 wGh terms. Indeed, by using the ﬁrst Ward identity in the Tµν − ;µν µν form (86) as well as the second Ward identity in the form Gh +(1 /2) gµν w + 6 gµν v1. (114e) (88), we obtain

Let us now consider the divergence of the terms given GBA W A + GBΦ W Φ + Gh W Gh = by Eqs. (114a)–(114e). By taking into account Eqs. (81) Tµν Tµν Tµν A A ρ and (83), we obtain 2 v µν + gµν (1/2) v + 3 v1 . (120) 1 − 1ρ ;ν clA W A + GBA W A = 6 vA ;ν 2 vA ρ , Now, by using this relation in connection with Tµν Tµν 1 µν − 1 ρ ;µ (115a) Eqs. (114a) and (114b), we can rewrite the renormalized expectation value of the stress-energy-tensor opera- ;ν tor given by Eq. (119) in the form clΦ W Φ + GBΦ W Φ = 2 v (115b) Tµν Tµν − 1;µ ψ T ψ = and h | µν | iren 1 ρ ρ Gh Gh ;ν (1/2) s + (1/2) s s W = 4 v , (115c) b 2 ρ ;µν µν ρ(µ;ν) Tµν 1;µ 8π − nρ ρ ρ a a s ρ + 2 s and we then have − µ [ν;ρ] − ν [µ;ρ] − ρ µν ρ(µν) ;ν ρ ;ρτ 1 A A ρ (1/2) gµν (1/2) sρ (1/2) sρτ ψ Tµν ψ ren = 6 v µν 2 gµν v − − h | | i 8π2 1 − 1 ρ ρτ ρ;τ ρτ (1/2) R sρτ aρτ + sρτ +2 g v ;ν+ Θ ;ν = 0. (116) − − b µν 1} µν +(1/2) wΦ wΦ ;µν − µν It is therefore suitable to redefine the purely geometrical Φ 2 Φ e (1/2) gµν (1/2) w m w tensor Θµν by − − A A ρ +2 v1 µν (3 /2) gµν v1 ρ 2 gµνv1 1 A A ρ − − Θµν e Θµν 6 v µν 2 gµν v + 2 gµν v1 , o → − 8π2 1 − 1 ρ + Θµν . (121) (117) e This expression only involves state-dependent and geo- where the new local tensor Θµν is assumed to be con- metrical quantities associated with the quantum fields served, i.e., Aµ and Φ. We could consider it as our final result, but, in fact, it is very important here to note that, due to the ;ν Θµν = 0. (118) first Ward identity, the decomposition into a part involving the massive vector field and another part involving As a consequence, the renormalized expectation value the auxiliary scalar field is not unique. In the next sec- of the stress-energy-tensor operator takes the following tions, we shall provide two alternative expressions which, form: in our opinion, are much more interesting from the phys- 1 ical point of view. ψ T ψ = clA W A + GBA W A h | µν | iren 8π2 Tµν Tµν From now, in order to simplify the notations and be- A A ρ cause this does not lead to any ambiguity, we shall omit 6v µν+ 2 gµν v b − 1 1 ρ the label Φ for the Taylor coefficients wΦ and wΦ . 1 µν + clΦ W Φ + GBΦ W Φ 8π2 Tµν Tµν + 2g v µν 1} 3. Substitution of the auxiliary scalar field contribution and 1 final result + Gh W Gh 4 g v 8π2 Tµν − µν 1

+ Θµν , (119) It is possible to remove in Eq. (121) any reference to the auxiliary scalar ﬁeld Φ. In some sense, it plays the where the various state-dependent contributions are role of a kind of ghost ﬁeld (the so-called Stueckelberg given by Eqs. (114a)–(114e). ghost [47]), but its contribution must be carefully taken into account. By using Eqs. (83a), (86a) and (86b) in the form 2. Cancellation of the gauge-breaking and ghost contributions 2 ρ m w = wρ + 6 v1 = (1/2) s ;ρτ (1/2) Rρτ s In Sec.IVB we have mentioned that the formal con- − ρτ − ρτ tributions of the gauge-breaking term ψ T GB ψ and + a ρ;τ + s ρτ + vA ρ + 2 v , (122a) h | µν | i ρτ ρτ 1 ρ 1 b

56 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

;ρ ρ w;µν = sρ(µ ν) + 2 aρ(µ ; ν) (122b) and − | | | | 1 and Φ = (1/2) w w Tµν 8π2 { ;µν − µν ;ρ ρ wµν = (1/2) sρ(µ ν) (1/2) R (µsν)ρ (1/4) gµν w gµν v1 . (126b) − | | − ρ ρ − − } +(1/2) aµ [ν;ρ] + (1/2) aν [µ;ρ] A Φ The stress-energy tensors µν and µν are separately con- ρ A T T +s + v µν gµν v1, (122c) served (this can be checked from relations obtained in ρ(µν) 1 Φ − Sec.IIID), and, moreover, the expression of µν cor- we obtain responds exactly to the renormalized expectationT value of the stress-energy-tensor operator associated with the ψ T ψ = h | µν | iren quantum action (14) for ξ=1 (see, e.g., Refs. [35, 36]). 1 (1/2) s ρ + (1/2) s s ρ As a consequence, it could be rather natural to consider b 2 ρ ;µν µν ρ(µ;ν) A 8π − µν given by Eq. (126a) as the renormalized expectation ρ ρ ρ T +(1n/2) R s (1/2) a (1/2) a value of the stress-energy-tensor operator associated with (µ ν)ρ − µ (ν;ρ) − ν (µ;ρ) ρ ρ ρ the massive vector field Aµ. This physical interpretation ρ 2 aµ [ν;ρ] aν [µ;ρ] sρ µν + sρ(µν) is strengthened by noting that, in the limit m 0, − − − → ρ ;ρτ ρ;τ A reduces to the result obtained from Maxwell’s the- (1/2) gµν (1/2) sρ (1/2) sρτ aρτ µν − − − oryT (see Sec.IVD). However, despite this, we are not A A ρ +v1 µν gµν v1 ρ really satisfied by this artificial way to split the contribu- − tions of the vector and scalar fields because, as we have +Θ . o (123) µν already noted, the first Ward identity allows us to move We have now at our disposal an expression for the renor- terms from one contribution to the other. So, we con- malized expectation value of the stress-energy-tensor op- sider that the only nonambiguous result is the one given erator associated with the full Stueckelberg theory which by Eq. (123). only involves state-dependent and geometrical quantities It is interesting to note that Eq. (125) combined with associated with the massive vector field Aµ. It is the Eqs. (81b) and (83a) leads to main result of our article. ρ A ρ Φ ρ ρ It is interesting to note that Eq. (123) combined with T ren = + + Θ (127) h ρ i T ρ T ρ ρ Eq. (81b) leads to with b ρ 1 2 ρ ρτ ψ T ψ ren = m s (1/2) R sρτ h | ρ | i 8π2 − ρ − A ρ 1 ;ρτ 2 ρ ρτ ρ = 2 sρτ m sρ R sρτ +s ρτ + 3 vA ρ + Θ ρ. (124) T 8π − − − b ρτ 1 ρ ρ ρ;τ ρτ A ρ +2 aρτ + 2 sρτ + 4 v1 ρ (128a)

4. Another final expression involving both the vector field and Aµ and the auxiliary scalar field Φ 1 Φ ρ = (1/2) w m2w + 2 v . (128b) T ρ 8π2 − − 1 Even if we are satisfied with our final expression (123), it is worth nothing that it does not reduce, in the limit m2 0, to the result obtained from Maxwell’s theory. D. Maxwell’s theory This→ is not really surprising because it involves implicitly the contribution of the auxiliary scalar field. In fact, by Let us now consider the limit m2 0 of A given by 2 µν replacing in Eq. (121) the term m w given by Eq. (122a), Eq. (126a). By using Eq. (122a), it reduces→ T to it is possible to split the renormalized expectation value of the stress-energy-tensor operator in the form 1 Maxwell = (1/2) s ρ + (1/2) s s ρ Tµν 8π2 ρ ;µν µν − ρ(µ;ν) A Φ ψ T ψ = + + Θ , (125) ρ n ρ ρ ρ h | µν | iren Tµν Tµν µν a a s + 2 s − µ [ν;ρ] − ν [µ;ρ] − ρ µν ρ(µν) where the terms associated with the vector and scalar ρ ;ρτ b (1/2) gµν (1/2) sρ (1/2) sρτ fields are given by − − (1/2) Rρτ s a ρ;τ + s ρτ − ρτ − ρτ ρτ 1 ρ A ρ A A ρ µν = 2 (1/2) sρ ;µν + (1/2) sµν sρ(µ;ν) +2 v g (3/2) v + v . (129) T 8π − 1 µν − µν 1 ρ 1 ρ n ρ ρ a a s ρ + 2 s o − µ [ν;ρ] − ν [µ;ρ] − ρ µν ρ(µν) This last expression is nothing other than the renor- ρ ρ;τ (1/2) gµν (1/2) sρ 2 aρτ malized expectation value of the stress-energy-tensor op- − − erator associated with Maxwell’s electromagnetism (see +2 vA g vA ρ (126a) 1 µν − µν 1 ρ Eq. (3.41b) of Ref. [39]). o

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 57 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

It is interesting to note that Eq. (129) combined with The general expression of the local conserved tensor Θµν Eq. (81b) leads to can be therefore written in the form

1 4 2 µν Maxwell 1 A ρ Θµν = 2 α m gµν + β m [Rµν (1/2) R gµν ] g µν = 2 v1 ρ 4 v1 8π − T 8π2 − n +γ (1)H + γ (2)H , (133) 1 2 pq 1 µν 2 µν = (1/20) R (5/72) R + (11/45) Rpq R 8π2 − − o where α, β, γ1 and γ2 are constants which can be ﬁxed (13 /360) R Rpqrs . (130) − pqrs by imposing additional physical conditions on the renormalized expectation value of the stress-energy-operator We recover the trace anomaly for Maxwell’s theory. tensor, these conditions being appropriate to the problem treated.

E. Ambiguities in the renormalized stress-energy tensor 2. Ambiguities associated with the renormalization mass

1. General expression of the ambiguities So far, in order to simplify the calculations, we have dropped the scale length λ (or, equivalently, the mass The renormalized expectation value ψ Tµν (x) ψ ren is scale M = 1/λ, i.e., the so-called renormalization mass) h | | i unique up to the addition of a geometrical conserved ten- that should be introduced in order to make dimensionless sor Θµν . In other words, even if it takesb perfectly into the argument of the logarithm in the Hadamard repre- account the quantum state dependence of the theory, it sentation of the Green functions. In fact, in Eqs. (35) and is ambiguously defined (see, Sec. III of Ref. [31] as well (37) it is necessary to make the substitution ln[σ(x, x0) + 2 as, e.g., comments in Refs. [25, 26, 41, 48, 49]). i] ln[σ(x, x0)/λ + i] which leads in Eqs. (55) and → As noted by Wald [31], Θµν is a local conserved ten- (58) to the substitution 4 4 sor of dimension (mass) = (length)− which remains 2 ln σ(x, x0) ln σ(x, x0)/λ . (134) finite in the massless limit. As a consequence, it can be | | → | | constructed by functional derivation with respect to the This scale length induces an indeterminacy in the biten- A metric tensor from a geometrical Lagrangian of dimen- sors W (x, x0) and W (x, x0) which corresponds to the 4 4 µν0 sion (mass) = (length)− . Such a Lagrangian is neces- replacements sarily a linear combination of the following four terms: A A A 2 4 2 2 pq W (x, x0) W (x, x0) V (x, x0) ln(M ), (135a) m , m R, R , RpqR . It should be noted that we could µν0 → µν0 − µν0 pqrs Φ Φ Φ 2 also take into account the term RpqrsR . But, in fact, W (x, x0) W (x, x0) V (x, x0) ln(M ), (135b) we can eliminate this term because, in a four-dimensional Gh → Gh − Gh 2 W (x, x0) W (x, x0) V (x, x0) ln(M ), (135c) background, the Euler number → − i.e., in terms of the associated Taylor coefficients, to re- 4 2 pq pqrs placements χ = d x√ g R 4 RpqR + RpqrsR (131) − − s s vA ln(M 2), (136a) ZM µν → µν − 0 (µν) A 2 associated with the quadratic Gauss-Bonnet action is a aµνa aµνa v ln(M ), (136b) → − 0 [µν]a topological invariant. s s vA + vA g ln(M 2) (136c) The functional derivation of the action terms previ- µνab → µνab − 0 (µν)ab 1 (µν) ab ously discussed provides the conserved tensors for the vector field Aµ and 2 1 δ 4 4 4 µν w w v ln(M ), (137a) d x√ g m = (1/2) m g , (132a) → − 0 √ g δgµν − w w (v + v g ) ln(M 2) (137b) − ZM ab ab 0ab 1 ab 1 δ → − d4x√ g m2R for the scalar field Φ or the ghost fields. By substituting √ g δgµν − − ZM Eqs. (135a)–(135c) into the general expression (112) of = m2 [Rµν (1/2) R gµν ] , (132b) the renormalized expectation value of the stress-energy- − − tensor operator, we obtain the general form of the ambi- (1) µν 1 δ 4 2 H d x√ g R guity associated with the scale length (or with the renor- ≡ √ g δgµν − − ZM maization mass). It is given by = 2 R;µν 2 RRµν 2 µν − 2 ln(M ) +g 2 R + (1/2) R , (132c) Θ (M) = ΘclA V A + ΘclΦ V Φ − µν − 8π2 µν µν 1 δ (2) µν 4 √ pq 2 H d x g RpqR ln(M ) GB A A GBΦ Φ ≡ √ g δgµν − 2 Θµν V + Θµν V ;µν − µν ZM µpνq − 8π = R R 2 RpqR ln(M 2) µν− − pq Gh Gh +g [ (1/2) R + (1/2) R R ] . (132d) 2 Θµν V (138) − pq − 8π

58 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

17 with The ambiguities associated with the scale length can

cl A cl ρσ0 A now be obtained explicitly from the replacements (136) Θ A V (x) = lim A (x, x0) V (x, x0) , µν µν ρσ0 x0 x T and (137). If we use the form (123) without taking into → (139a) account the geometrical terms, we obtain 2 clΦ Φ clΦ Φ ln(M ) A ρ A Θµν V (x) = lim µν (x, x0) V (x, x0) , Θµν (M) = 2 (1/2) v0 ρ ;µν + (1/2) v0 µν x0 x T 8π → − ρ n ρ (139b) vA + (1/2) R vA (1/2) gρτ vA − 0 ρ(µ;ν) (µ 0 ν)ρ − 0 [µρ](ν;τ) ρτ A ρτ A ρτ A GBA A GBA ρσ0 A (1/2) g v [νρ](µ;τ) g v [µρ][ν;τ] g v [νρ][µ;τ] Θ V (x) = lim (x, x0) V (x, x0) , 0 0 0 µν µν ρσ0 − − − x0 x T ρ ρ → vA ρ + (1/2) vA + (1/2) vA + vA (139c) − 0 ρ µν 0 (ρµ)ν 0 (ρν)µ 1 µν A ρ A ;ρτ A ρ;τ gµν (1/4) v0 ρ (1/4) v0 ρτ (1/2) v0 GBΦ Φ GBΦ Φ [ρτ] Θµν V (x) = lim µν (x, x0) V (x, x0) − − − x0 x T h → +vA ρ . (143) (139d) 1 ρ io and Similarly, if we use the alternative form (125) where Gh Gh Gh Gh the contributions corresponding to the vector ﬁeld Aµ Θµν V (x) = lim µν (x, x0) V (x, x0) , x0 x T and the auxiliary scalar ﬁeld Φ are highlighted [see → (139e) Eqs. (126a) and (126b)], we obtain

cl ρσ0 cl GB ρσ0 A Φ where the differential operators A , Φ , A , Θµν (M) = Θ (M) + Θ (M) (144) Tµν Tµν Tµν µν µν GBΦ and Gh are given in Eqs. (104a)–(104e). It should Tµν Tµν with be noted that Θµν (M) is a purely geometrical object. 2 This is due to the geometrical nature of the Hadamard A ln(M ) A ρ A A Θ (M) = (1/2) v + (1/2) v coefficients V (x, x0) and V (x, x0). µν 2 0 ρ ;µν 0 µν µν0 − 8π In order to obtain the explicit expression of the stress- A ρ ρτ An ρτ A v0 g v0 [µρ][ν;τ] g v0 [νρ][µ;τ] energy tensor Θµν (M), we can repeat the calculations of − ρ(µ;ν) − − A A Φ Φ Gh A ρ A ρ A ρ A Sec.IVC by replacing Wµν by Vµν , W by V and W 0 0 v0 ρ µν + v0 (ρµ)ν + v0 (ρν)µ + 2 v1 µν by V Gh. From Eqs. (114a)–(114e) it is straightforward − cl A cl Φ GB A A ρ A ρ;τ A ρ to obtain explicitly Θ A V ,Θ Φ V ,Θ A V , gµν (1/4) v v + v (145a) µν µν µν − 0 ρ − 0 [ρτ] 1 ρ ΘGBΦ V Φ and ΘGh V Gh by using the replacements µν µν h io (136) and (137). We can then show that and ;ν ln(M 2) ΘclA V A + ΘGBA V A = 0, (140a) ΘΦ (M) = (1/2) v v µν µν µν − 8π2 { 0;µν − 0µν ;ν gµν [(1/4) v0 + v1] . (145b) clΦ Φ GBΦ Φ Θµν V + Θµν V = 0 (140b) − } Now, by using the explicit expressions (72) and (74) and of the Taylor coefficients of the purely geometrical Gh Gh ;ν Θµν V = 0. (140c) Hadamard coefficients, we can show that Eq. (143) reduces to Equations (140a)–( 140c) are similar to Eqs. (115a)– (115c) but now with the right-hand sides vanishing. This ln(M 2) Θ (M) = (1/4) m2R (1/20) R is due to the fact that, unlike the wave equations (42) µν − 8π2 µν − ;µν and (46) for W A , W Φ and W Gh, the wave equations +(13/120) R (1/8) RR + (2/15) R R p µν0 µν − µν µp ν for V A , V Φ and V Gh [cf. Eqs. (41) and (45)] have no p q pqr µν0 +(7/20) RpqRµ ν (1/15) RµpqrRν source terms. As a consequence Θ (M) is a conserved − µν +g (3/8) m4 (1/8) m2R (1/240) R geometrical tensor. We can also check that µν − − − 2 pq +(1/32) R (29/240) RpqR GBA A GBΦ Φ Gh Gh − Θµν V + Θµν V + Θµν V = 0. (141) pqrs +(1/60) RpqrsR , (146) Equation (141 ) is similar to Eq. (120) but now with the right-hand side vanishing. This is due to the fact that, while Eqs. (145a) and (145b ) provide unlike the Ward identity (85) linking W A and W Gh, the µν0 2 A Gh A ln(M ) 2 Ward identity (84) linking V and V has no right- Θµν (M) = (1/3) m Rµν (1/30) R;µν µν0 − 8π2 − hand side. As a consequence, from Eqs. (138) and (141), +(1/10) R (5/36) RR + (13/90) R R p we obtain µν − µν µp ν +(31/90) R R p q (13/180) R R pqr ln(M 2) pq µ ν − µpqr ν clA A clΦ Φ 4 2 Θµν (M) = 2 Θµν V + Θµν V . (142) +g (1/4) m (1/6) m R (1/60) R − 8π µν − − −

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 59 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

+(5/144) R2 (11/90) R Rpq − pq perfectly vacuum pqrs  μ +(13/720) RpqrsR (147a) X conducting and medium  μ ln(M 2) Z ΘΦ (M) = (1/12) m2R (1/60) R µν − 8π2 − µν − ;µν p  μ +(1/120) Rµν + (1/72) RRµν (1/90) RµpRν Y p q − pqr +(1/180) RpqRµ ν + (1/180) RµpqrRν +g (1/8) m4 + (1/24) m2R + (1/80) R µν − 2 pq (1/288) R + (1/720) RpqR − FIG. 1. Geometry of the Casimir effect (1/720) R Rpqrs . (147b) − pqrs Of course, it is easy to check that the sum of ΘA (M) µν V. CASIMIR EFFECT Φ and Θµν (M) is equal to Θµν (M). It is possible to obtain a more compact form for the A. General considerations stress-energy tensors (146), (147a) and (147b) by using the conserved tensors (132a)–(132d). It should be noted p pqr pqrs In this section, we shall consider the Casimir effect for that the terms in RµpR , RµpqrR and RpqrsR ν ν Stueckelberg massive electromagnetism in the Minkowski which are not present in (1)H and (2)H can be elim- µν µν spacetime ( 4, η ) with η = diag( 1, +1, +1, +1). inated by introducing R µν µν We denote by (T,X,Y,Z) the coordinates− of an event in this spacetime. We shall provide the renormalized (3) µν 1 δ 4 pqrs H d x√ g RpqrsR vacuum expectation value of the stress-energy-tensor op- ≡ √ g δgµν − ;µν − µνZM µ νp µpνq erator outside of a perfectly conducting medium with a = 2 R 4 R + 4 R pR 4 RpqR plane boundary wall at Z = 0 separating it from free µ − νpqr µν − pqrs 2 R R + g [(1/2) RpqrsR ] (148) space (see Fig.1). It is worth pointing out that this − pqr problem has been studied a long time ago by Davies and and by noting that, due to Eq. (131), Toms in the framework of de Broglie-Proca electromagnetism [50]. We shall revisit this problem in order to (1)H 4 (2)H + (3)H = 0. (149) compare, at the quantum level and in the case of a sim- µν − µν µν ple example, de Broglie-Proca and Stueckelberg theories We then have and to discuss their limit for m2 0. It should be noted that the Casimir effect in connection→ with a massive pho- ln(M 2) 4 ton has been considered for various geometries (see, e.g., Θµν (M) = 2 (3/8) m gµν 8π Refs. [51–55]). (1/4) m2 [R n(1/2) R g ] − µν − µν From symmetries and physical considerations, we can (7/240) (1)H + (13/120) (2)H , (150) observe that, outside of the perfectly conducting medium, − µν µν the renormalized stress-energy tensor takes the form (see o Chap. 4 of Ref. [24])

2 A ln(M ) 4 1 ρ Θ (M) = (1/4) m gµν 0 T 0 = 0 T 0 η Zˆ Zˆ , (152) µν 8π2 h | µν | iren 3 h | ρ | iren µν − µ ν 2 n (1/3) m [Rµν (1/2) R gµν ] − − where bZˆµ is the spacelikeb unit vector orthogonal to the (1/30) (1)H + (1/10) (2)H (151a) − µν µν wall. As a consequence, it is suﬃcient to determine the o trace of the renormalized stress-energy tensor. From and Eq. (124), we have ln(M 2) Φ 4 ρ 1 2 ρ ρτ 4 Θµν (M) = 2 (1/8) m gµν 0 T 0 = m s + s + (3/2) m 8π h | ρ | iren 8π2 − ρ ρτ 2 n +(1/12) m [Rµν (1/2) R gµν ] ρ − b + Θρ . (153) (1) (2) +(1/240) Hµν + (1/120) Hµν . (151b) ρ The term Θρ encodes the usual ambiguities discussed in o Sec.IVE. In the Minkowski spacetime it reduces to As expected, we can note that the ambiguities associated with the scale length (or with the renormalization mass) 1 Θ ρ = α m4 , (154) are of the form (133). ρ 8π2

60 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

19 where α is a constant. From Eq. (153) it is clear that in [we note that Eq. (160b) is valid for arg(z) < π, ρ | | order to evaluate 0 Tρ 0 ren, it is sufficient to take the and we recall that the digamma function ψ is defined h | | i A A by the recursion relation ψ(z + 1) = ψ(z) + 1/z with coincidence limit x0 x of Wµν (x, x0) and Wµν;(ab)(x, x0) →b ψ(1) = γ], we can provide the Hadamard representa- [see Eqs. (78a) and (78c) and note that, in the Minkowski −A ν0 tion of Gµν (x, x0) given by Eq. (158). We can write spacetime, the bivector of parallel transport gµ (x, x0) is ν0 A 2 1/2 equal to the unit matrix δµ (x, x0)], where Wµν (x, x0) is m 1 (2) i ∆ (x, x0) A H1 [ (x, x0)] = 2 the regular part of the Feynman propagator Gµν (x, x0) − 8π (x, x0) Z 8π σ(x, x0) + i corresponding to the geometry of the Casimir effect. Z + V (x, x0) ln[σ(x, x0) + i] + W (x, x0) , (161) B. Stress-energy tensor in the Minkowski spacetime where 1/2 ∆ (x, x0) = 1, (162a) Let us first consider the vacuum expectation value of the stress-energy-tensor operator in the ordinary Minkowski spacetime (i.e., without the boundary wall). ∞ k This will permit us to establish some notations and, V (x, x0) = Vk σ (x, x0) (162b) moreover, to fix the constant α appearing in Eq. (154). kX=0 Due to symmetry considerations, we have with

1 ρ 2 k+1 0 Tµν 0 ren = 0 Tρ 0 ren ηµν , (155) (m /2) 4 Vk = (162c) h | | i h | | i k!(k + 1)! ρ where 0 T 0bren is still givenb by Eqs. (153) and (154). h | ρ | i and Of course, we must have 0 Tµν 0 ren = 0, and we have b h | | i therefore ∞ k W (x, x0) = Wkσ (x, x0) (162d) ρ b 0 T 0 ren = 0 (156) h | ρ | i kX=0 which plays the role of a constraint for α. with In the Minkowski spacetime,b the Feynman propagator A (m2/2)k+1 m2 Gµν (x, x0) associated with the vector ﬁeld Aµ satisﬁes W = ψ(k + 1) + ψ(k + 2) ln . the wave equation (23), i.e., k − k!(k + 1)! − 2 2 A 4 (162e) m G (x, x0) = η δ (x, x0), (157) x − µν − µν and its explicit expression is given in terms of a Hankel By noting that function of the second kind by (see, e.g., Chap. 27 of A Ref. [56]) Wµν (x, x0) = W (x, x0) ηµν , (163) m2 1 A (2) where W (x, x0) is given by Eqs. (162d) and (162e), we Gµν (x, x0) = H1 [ (x, x0)] ηµν . (158) − 8π (x, x0) Z are now able to express 0 T ρ 0 . From Eqs. (78a) Z h | ρ | iren 2 and (78c) we have, respectively, Here, (x, x0) = 2 m [σ(x, x0) + i] with 2 σ(x, x0) = Z 2 − 2 2 2 b (T T 0) + (X X0) + (Y Y 0) + (Z Z0) . s = m2 1/2 + γ + (1/2) ln(m2/2) η (164) − We− have (see Chap.p− 9 of Ref.− [57]) − µν − µν

(2) and H1 (z) = J1(z) iY1(z), (159) − 4 2 sµνab = m 5/16 + (1/4) γ + (1/8) ln(m /2) ηµν ηab. where J1(z) and Y1(z) are the Bessel functions of the − ﬁrst and second kinds. By using the series expansions (165) for z 0 (see Eqs. (9.1.10) and (9.1.11) of Ref. [57]) → Then, from Eq. (153), we obtain z ∞ ( z2/4)k J1(z) = − (160a) m4 2 k!(k + 1)! 0 T ρ 0 = α + 9/4 3 γ (3/2) ln(m2/2) , kX=0 h | ρ | iren 8π2 − − and (166) 2 2 z b Y (z) = + ln J (z) 1 −πz π 2 1 and, necessarily, by using Eq. (156), we have the constraint z ∞ ( z2/4)k [ψ(k + 1) + ψ(k + 2)] − (160b) −2π k!(k + 1)! α = 9/4 + 3 γ + (3/2) ln(m2/2). (167) kX=0 −

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 61 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

C. Stress-energy tensor for the Casimir effect In the limit m2 0 and for Z = 0, we obtain → 6 1 1 Let us now come back to our initial problem. The 0 Tµν 0 ren = ηµν ZˆµZˆν . (174) h | | i 16π2 Z4 − Feynman propagator previously considered is modified by the presence of the plane boundary wall. The new Feyn- In the masslessb limit, the vacuum expectation value of A man propagator G (x, x0) can be constructed by the the renormalized stress-energy tensor associated with the µν 4 method of images if we assume, in order to simplify our Stueckelberg theory diverges like Z− as the boundary problem, a perfectlye reflecting wall. It should be noted surface is approached. This result contrasts with that ob- that this particular boundary condition is questionable tained from Maxwell’s theory (see also Ref. [50]). Indeed, from the physical point of view. It is logical for the trans- for this theory, the renormalized stress-energy-tensor op- verse components of the electromagnetic field but much erator vanishes identically. In order to extract that result less natural for its longitudinal component. Indeed, for from the Stuckelberg theory, we will now repeat the pre- this component, we could also consider perfect transmis- vious calculations from the expressions (125) and (126) sion instead of complete reflection (see Refs. [50, 51, 58]). [as well as (127) and (128)] given in Sec. IV C 4, where We shall now consider that the Feynman propagator is we have proposed an artificial separation of the contribu- given by tions associated with the vector field Aµ and the auxiliary scalar field Φ. A A A G (x, x0) = G (x, x0) q G (x, x˜0). (168) µν µν − ν µν µ µ Here, x0 = (T 0,X0,Y 0,Z0) andx ˜0 = (T 0,X0,Y 0, Z0), e − D. Separation of the contributions associated with while qν = (1 2 δ3ν ). It is important to note that, in Eq. (168), the− index ν is not summed. Furthermore, we the vector field Aµ and the auxiliary scalar field Φ A remark that the term Gµν (x, x˜0) which is obtained by replacing x0 byx ˜0 in Eq. (158) as well as its derivatives In the Minkowski spacetime, Eqs. (127) and (128) re- are regular in the limit x0 x. duce to By following the steps of→ Sec.VB and using the rela- 0 T ρ 0 = A ρ + Φ ρ + Θ ρ (175) tion h | ρ | iren T ρ T ρ ρ iπν/2 (2) iπ/2 Kν (z) = (1/2) iπ e− H (z e− ) (169) with − ν b which is valid for π/2 arg(z) π as well as the 1 − ≤ ≤ A ρ = s ;ρτ m2s ρ properties of the modified Bessel functions of the second T ρ 8π2 − ρτ − ρ kind K1, K2 and K3 (see Chap. 9 of Ref. [57]), it is easy to ρ;τ ρτ 4 +2 aρτ + 2 sρτ + 2 m (176) show that the Taylor coefficients sµν and sµνab involved in 0 T ρ 0 are now given by and h | ρ | iren 2 2 Φ ρ 1 2 4 sµν = m 1/2 + γ + (1/2) ln(m /2) ηµν = (1/2) w m w + (1/4) m . (177) b − T ρ 8π2 − − (m/Z) K1(2mZ) qν ηµν (170) − ρ The term Θρ encodes the usual ambiguities discussed in and Sec.IVE. We can split it in the form 4 2 sµνab = m 5/16 + (1/4) γ + (1/8) ln(m /2) ηµν ηab ρ A ρ Φ ρ − Θρ = Θ ρ + Θ ρ (178a) (m2/Z2) K (2mZ) q η (2 η η (1/2) η ) − 2 ν µν 3a 3b − ab 3 with +(m /Z) K1(2mZ) qν ηµν η3aη3b . (171) A ρ 1 A 4 By inserting Eqs. (170) and (171) in the expression (153) Θ ρ = 2 α m (178b) and using the value of α fixed by Eq. (167), we obtain 8π 2 3 and ρ 3 m m 0 Tρ 0 ren = 2 2 K2(2mZ) + K1(2mZ) , 1 h | | i 8π Z Z ΘΦ ρ = αΦ m4 , (178c) ρ 8π2 b (172) A Φ and from Eq. (152) we have where α and α are two constants associated, respectively, with the contributions of the vector field Aµ 1 m2 m3 0 T 0 = K (2mZ) + K (2mZ) and the auxiliary scalar field Φ. We can then replace h | µν | iren 8π2 Z2 2 Z 1 Eq. (175) by ˆ ˆ b ηµν ZµZν . (173) 0 T ρ 0 = 0 T A ρ 0 + 0 T Φ ρ 0 (179) × − h | ρ | iren h | ρ | iren h | ρ | iren It is very important to note that this result coincides with exactly with the result obtained by Davies and Toms in b b b the framework of de Broglie-Proca electromagnetism [50]. 0 T A ρ 0 = A ρ + ΘA ρ (180) h | ρ | iren T ρ ρ b

62 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

21 and We now come back to the Casimir effect. The two Feynman propagators previously considered are modified 0 T Φ ρ 0 = Φ ρ + ΘΦ ρ, (181) h | ρ | iren T ρ ρ by the presence of the plane boundary wall. The new Feynman propagators can be constructed by the method where the contributions associated with the vector field b of images. Of course, the propagator of the vector field A and the auxiliary scalar field Φ are separated. At µ A is still given by Eq. (168), while we have first sight, A ρ seems complicated because it involves µ T ρ 1/2 1 A Φ Φ Φ Taylor coefficients of orders σ and σ of Wµν (x, x0). G (x, x0) = G (x, x0) G (x, x˜0) (188) In fact, its expression can be simplified by replacing the − ρ;τ ρτ for the propagator of the scalar field Φ. In the context of sum aρτ +sρτ from the relation (86b), and we obtain e the Casimir effect, Eq. (184) must be replaced by 1 A ρ 2 ρ 2 4 2 2 ρ = 2 m sρ + 2 m w + (1/2) m (182) w = m 1/2 + γ + (1/2) ln(m /2) T 8π − − (m/Z) K1(2mZ), (189) which only involves the first Taylor coefficients sµν and − 0 ρ w of order σ . So, in order to evaluate 0 Tρ 0 ren given and sµν is given by Eq. (170). By inserting Eqs. (170) and by Eq. (175), it is sufficient to take theh coincidence| | i limit (189) in Eqs. (182) and (177) and taking into account the x0 x of the state-dependent Hadamardb coefficients constraints (187a) and (187b), we obtain from Eq. (180) A→ W (x, x0) and W (x, x0) associated with the Feynman µν A ρ A Φ 0 T 0 ren = 0 (190) propagators Gµν (x, x0) and G (x, x0) corresponding to h | ρ | i the geometry of the Casimir effect. and from Eq. (181) At first, we must fix the constants αA and αΦ appear- b ing in Eq. (178). This can be achieved by imposing, in the 3 m2 m3 0 T Φ ρ 0 = K (2mZ) + K (2mZ) . Minkowski spacetime without boundary, the vanishing of h | ρ | iren 8π2 Z2 2 Z 1 A ρ Φ ρ 0 T ρ 0 ren given by Eq. (180) and 0 T ρ 0 ren given (191) byh | Eq. (|181i ). In this spacetime, everythingh | related| i to the b A Feynmanb propagator Gµν (x, x0) has beenb already given in From Eq. (152) we can then see that the vacuum expecta- Φ Sec.VB, while the Feynman propagator G (x, x0) asso- tion value of the stress-energy-tensor operator associated ciated with the scalar field Φ satisfies the wave equation with the vector field Aµ is such that (25) and is explicitly given by A 0 Tµν 0 ren = 0, (192) 2 h | | i Φ m 1 (2) G (x, x0) = H [ (x, x0)] . (183) while the vacuum expectation value of the stress-energy- − 8π (x, x ) 1 Z b Z 0 tensor operator associated with the auxiliary scalar field Φ is given by By using Eqs. (161) and (162), it is easy to see that this propagator can be represented in the Hadamard form and 1 m2 m3 0 T Φ 0 = K (2mZ) + K (2mZ) to obtain h | µν | iren 8π2 Z2 2 Z 1 2 2 w = m 1/2 + γ + (1/2) ln(m /2) . (184) b η Zˆ Zˆ . (193) − × µν − µ ν We are now able to express 0 T ρ 0 . From Eqs. (180), h | ρ | iren Of course, the sum of these two contributions permits (181), (182), (177) and (178), we obtain us to recover the result (173) of Sec.VC which is also b the result obtained by Davies and Toms in the frame- m4 A ρ A 2 work of de Broglie-Proca electromagnetism [50]. It is 0 T ρ 0 ren = 2 α + 3/2 2 γ ln(m /2) h | | i 8π − − moreover interesting to note that the contribution (192) (185) b associated with the vector field Aµ and which has been artificially separated from the scalar field contribution and (see Sec. IV C 4) vanishes identically for any value of the m4 mass parameter m. This result coincides exactly with 0 T Φ ρ 0 = αΦ + 3/4 γ (1/2) ln(m2/2) , h | ρ | iren 8π2 − − that obtained from Maxwell’s theory (see also Ref. [50]). (186) b and, necessarily, the vanishing of these traces provides VI. CONCLUSION the two constraints In the context of quantum field theory in curved space- αA = 3/2 + 2 γ + ln(m2/2) (187a) − time and with possible applications to cosmology and to and black hole physics in mind, the massive vector field is frequently studied. It should be, however, noted that, αΦ = 3/4 + γ + (1/2) ln(m2/2). (187b) in this particular domain, it is its description via the −

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 63 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

de Broglie-Proca theory which is mostly considered and (ii) We need the regular part of the Feynman propa- that there are very few works dealing with the Stueck- gator GA (x, x ) [or that of the anticommutator µν0 0 elberg point of view (see, e.g., Refs. [59–64], but remark (1)A G (x, x0)] at order σ. To extract it, we sub- µν0 that these papers are restricted to de Sitter and anti-de A tract from the Feynman propagator G (x, x0) its Sitter spacetimes or to Roberstson-Walker backgrounds µν0 with spatially flat sections). In this article, in order to singular part (48a) in order to obtain its regular fill a void, we have developed the general formalism of part (48b) [or we subtract from the anticommuta- (1)A tor G (x, x0) its singular part (68a) in order to the Stueckelberg theory on an arbitrary four-dimensional µν0 spacetime (quantum action, Feynman propagators, Ward obtain its regular part (68b)]. We have then at identities, Hadamard representation of the Green func- our disposal the state-dependent Hadamard bivec- tions), and we have particularly focussed on the aspects tor W A (x, x ). Here, it is important to note that µν0 0 linked with the construction, for a Hadamard quantum we do not need the full expression of the singular state, of the expectation value of the renormalized stress- part of the Green function considered, but we can energy-tensor operator. It is important to note that we truncate it by neglecting the terms vanishing faster have given two alternative but equivalent expressions for than σ(x, x0) for x0 x. As a consequence, we can this result. The first one has been obtained by eliminat- construct the singular→ part (48a) [or the singular ing from a Ward identity the contribution of the auxil- part (68a)] by using the covariant Taylor series ex- iary scalar field Φ (the so-called Stueckelberg ghost [47]) pansion (A.9) of ∆1/2 up to order σ2, the covariant A and only involves state-dependent and geometrical quan- Taylor series expansion (71a) of V0 µν up to order 1 tities associated with the massive vector field Aµ [see σ [see Eqs. (72a)–(72c)] and the covariant Taylor A 0 Eq. (123)]. The other one involves contributions com- series expansion (71b) of V1 µν up to order σ [see ing from both the massive vector field and the auxiliary Eq. (72d)]. Stueckelberg scalar field [see Eqs. (125)–(126)], and it has been constructed artificially in such a way that these two (iii) Finally, by using Eqs. (78a)–(78c), we can construct contributions are independently conserved and that, in the expectation value of the renormalized stress- the zero-mass limit, the massive vector field contribution energy tensor given by Eq. (123). reduces smoothly to the result obtained from Maxwell’s electromagnetism. It is also important to note that, in It is interesting to note that, in the literature concern- Sec.IVE, we have discussed the geometrical ambigui- ing Stueckelberg electromagnetism, some authors only fo- ties of the expectation value of the renormalized stress- cus on the part of the action associated with the massive energy-tensor operator. They are of fundamental impor- vector field Aµ and which is given by Eq. (13a) (see, e.g., tance (see, e.g., in Sec.V, their role in the context of the Refs. [59, 62, 65]). Of course, this is sufficient because Casimir effet). they are mainly interested, in the context of canonical We intend to use our results in the near future in cos- quantization, by the determination of the Feynman prop- mology of the very early universe, but we hope they agator associated with this field. However, in order to will be useful for other authors. This is why we shall calculate physical quantities, it is necessary to take into now provide a step-by-step guide for the reader who is account the contribution of the auxiliary scalar field Φ. It not especially interested by the technical details of our cannot be discarded. This is very clear in the context of work but who wishes to calculate the expectation value of the construction of the renormalized stress-energy-tensor the renormalized stress-energy tensor from the expression operator as we have shown in our article and remains (123), i.e., from the expression where any reference to the true for any other physical quantity. Stueckelberg auxiliary scalar field Φ has disappeared. We To conclude this article, we shall briefly compare the shall describe the calculation from the Feynman propa- de Broglie-Proca and Stueckelberg formulations of mas- gator as well as from the anticommutator function : sive electomagnetism and discuss the advantages of the Stueckelberg formulation over the de Broglie-Proca one. It is interesting to note the existence of a nice paper by (i) We assume that the Feynman propagator Pitts [66], where de Broglie-Proca and Stueckelberg ap- GA (x, x ) which is given by Eq. (22) and proaches of massive electromagnetism are discussed from µν0 0 satisfies the wave equation (30) [or that the a philosophical point of view based on the machinery of (1)A the Hamiltonian formalism (primary and secondary con- anticommutator G (x, x0) which is given by µν0 straints, Poisson brackets, . . . ). Here, we adopt a more Eq. (59) and satisfies the wave equation (60)] pragmatic point of view. We discuss the two formula- has been determined in a particular gravitational tions in light of the results obtained in our article. In our background and for a Hadamard quantum state. opinion: In other words, we consider that the Feynman A propagator G (x, x0) can be represented in the µν0 (i) De Broglie-Proca and Stueckelberg approaches of Hadamard form (35) [or that the anticommutator massive electromagnetism are two faces of the same (1)A G (x, x0) can be represented in the Hadamard µν0 theory. Indeed, the transition from de Broglie- form (55)]. Proca to Stueckelberg theory is achieved via the

64 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

Stueckelberg trick (5) which permits us, by intro- consult the references mentioned above), we have gath- ducing an auxiliary scalar field Φ, to artificially re- ered some important results which are directly related store Maxwell’s gauge symmetry in massive electro- with the representations of Green functions in coordinate magnetism, but, reciprocally, the transition from space and, more particularly, with the Hadamard repre- Stueckelberg to de Broglie-Proca theory is achieved sentations of the Green functions appearing in Stueckel- by imposing the gauge choice Φ = 0 [see Eq. (8)]. berg electromagnetism [see Eqs. (35), (37), (55) and (58)] As a consequence, it is not really surprising to ob- which is the main subject of Sec.III and which plays a tain the same result for the renormalized stress- crucial role in Sec.IV. In particular, we define the geode- energy-tensor operator associated with the Casimir tic interval σ(x, x0), the Van Vleck-Morette determinant effect (see Sec.V) when we consider this problem in ∆(x, x0) and the bivector of parallel transport from x to

the framework of the de Broglie-Proca and Stueck- x0 denoted by gµν0 (x, x0) (see, e.g., Ref. [44]), and we elberg formulations of massive electromagnetism. moreover discuss the concept of covariant Taylor series Indeed, we can expect that this remains true for expansions for biscalars and bivectors. any other quantum quantity. We first recall that 2σ(x, x0) is the square of the geodesic distance between x and x which satisfies (ii) However, we can note that with regularization and 0 renormalization in mind, it is much more interest- 2σ = σ;µσ . (A.1) ing to work in the framework of the Stueckelberg ;µ formulation of massive electromagnetism. Indeed, We have σ(x, x ) < 0 if x and x are timelike related, this permits us to have at our disposal the machin- 0 0 σ(x, x ) = 0 if x and x are null related and σ(x, x ) > 0 ery of the Hadamard formalism which is not the 0 0 0 if x and x are spacelike related. We furthermore recall case in the framework of the de Broglie-Proca for- 0 that ∆(x, x ) is given by mulation. Indeed, due to the constraint (4a), the 0 Feynman propagator GA (x, x ) associated with µν0 0 1/2 1/2 ∆(x, x0) = [ g(x)]− det( σ;µν (x, x0))[ g(x0)]− the vector field Aµ cannot be represented in the − − − 0 − Hadamard form (35). (A.2) and satisfies the partial differential equation

1/2 1/2 ;µ ACKNOWLEDGMENTS xσ = 4 2∆− ∆ ;µσ (A.3a) −

We wish to thank Yves D´ecanini, Mohamed Ould as well as the boundary condition El Hadj and Julien Queva for various discussions and the “Collectivit´eTerritoriale de Corse” for its support lim ∆(x, x0) = 1. (A.3b) x0 x through the COMPA project. →

The bivector of parallel transport from x to x0 is defined by the partial differential equation Appendix: Biscalars, bivectors and their covariant g σ;ρ = 0 (A.4a) Taylor series expansions µν0;ρ and the boundary condition Regularization and renormalization of quantum field theories in the Minkowski spacetime are most times based lim gµν (x, x0) = gµν (x). (A.4b) on the representation of Green functions in momentum x x 0 0→ space, and, in general, this greatly simplifies reasoning A and calculations. The use of such a representation is The Hadamard coefficients Vn µν0 (x, x0) and A not possible in an arbitrary gravitational background Wn µν0 (x, x0) introduced in Eq. (36) and which are where the lack of symmetries as well as spacetime cur- bivectors involved in the Hadamard representation of vature prevent us from working within the framework of the Green functions (35) and (55) or the Hadamard co- the Fourier transform. As a consequence, regularization efficients Vn(x, x0) and Wn(x, x0) introduced in Eq. (38) and renormalization in curved spacetime are necessarily and which are biscalars involved in the Hadamard repre- based on representations of Green functions in coordinate sentation of the Green functions (37) and (58) cannot in space, and, moreover, they require extensively the con- general be determined exactly. They are solutions of the cepts of biscalars, bivectors and, more generally, biten- recursion relations (39a), (39b) and (40) or (43a), (43b) sors. Thanks to the work of some mathematicians [67–70] and (44), and, following DeWitt [44, 71], we can look for and of DeWitt [23, 44, 56, 71] and coworkers [28, 29], we the solutions of these equations in the form of covariant have at our disposal all the tools necessary to deal with Taylor series expansions for x0 in the neighborhood of x. this subject. This is the method we use in Sec.III. The series defining In this short Appendix, in order to make a self- the biscalars Vn(x, x0) and Wn(x, x0) can be written in consistent paper (i.e., to avoid the reader needing to the form

Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 65 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

;a 1 ;a ;a T (x, x0) = t(x) t (x) σ 1 (x, x0) + t (x) σ 1 (x, x0) σ 2 (x, x0) − a1 2! a1a2 1 ;a ;a ;a t (x) σ 1 (x, x0) σ 2 (x, x0) σ 3 (x, x0) −3! a1a2a3 1 ;a ;a ;a ;a + t (x) σ 1 (x, x0) σ 2 (x, x0) σ 3 (x, x0) σ 4 (x, x0) + . (A.5) 4! a1a2a3a4 ···

By construction, the coeﬃcients ta1...ap (x) are symmetric tµν(a1...ap)(x), and by requiring the symmetry of

in the exchange of the indices a1 . . . ap, i.e., ta1...ap (x) = Tµν0 (x, x0) in the exchange of x and x0, i.e., Tµν0 (x, x0) =

t(a1...ap)(x), and, moreover, by requiring the symmetry Tν0µ(x0, x), the coeﬃcients tµν (x) and tµνa1...ap (x) with

of T (x, x0) in the exchange of x and x0, i.e., T (x, x0) = p = 1, 2,... are constrained. The symmetry of Tµν0 (x, x0)

T (x0, x), the coeﬃcients t(x) and ta1...ap (x) with p = permits us to express the coeﬃcients of the covariant Tay-

1, 2,... are constrained. The symmetry of T (x, x0) per- lor series expansion of Tµν0 (x, x0) in terms of their sym- mits us to express the odd coefficients of the covariant metric and antisymmetric parts in µ and ν. We have for Taylor series expansion of T (x, x0) in terms of the even the coefficients of lowest orders (see, e.g., Refs. [35, 39]) ones. We have for the odd coefficients of lowest orders (see, e.g., Ref. [72]) tµν = t(µν), (A.8a) ta = (1/2) t;a , (A.6a) 1 1 tµνa1 = (1/2) t(µν);a1 + t[µν]a1 , (A.8b)

ta1a2a3 = (3/2) t(a1a2;a3) (1/4) t;(a1a2a3). (A.6b) t = t + t , (A.8c) − µνa1a2 (µν)a1a2 [µν](a1;a2)

A tµνa1a2a3 = (3/2) t(µν)(a1a2;a3) Similarly, the series deﬁning the bivectors Vn µν0 (x, x0) A and W (x, x ) can be written in the form (1/4) t(µν);(a1a2a3) + t[µν]a1a2a3 . (A.8d) n µν0 0 −

ν0 In order to solve the recursion relations (39a), (39b), Tµν (x, x0) = gν (x, x0) Tµν0 (x, x0) ;a (40), (43a), (43b) and (44) but also to do most of the cal- = t (x) t (x) σ 1 (x, x0) µν − µνa1 culations in Secs.III andIV and, in particular, to obtain 1 ;a ;a the explicit expression of the renormalized stress-energy- + t (x) σ 1 (x, x0) σ 2 (x, x0) 2! µνa1a2 tensor operator, it is necessary to have at our disposal

1 ;a1 ;a2 ;a3 the covariant Taylor series expansions of the biscalars tµνa1a2a3 (x) σ (x, x0) σ (x, x0) σ (x, x0) 1/2 1/2 1/2 ;µ 1/2 −3! ∆ , ∆− ∆ ;µσ and ∆ and of the bivectors

+ . (A.7) σ;µν0 and gµν0 but also of some bitensors such as σ;µν , ··· gµν0;ρ and gµν0;ρ0 . Here, we provide these expansions up

By construction, the coeﬃcients tµνa1...ap (x) are symmet- to the orders necessary in this article (for higher orders,

ric in the exchange of indices a1 . . . ap, i.e., tµνa1...ap (x) = see Refs. [72, 73]). We have

1 1 ∆1/2 = 1 + R σ;a1 σ;a2 R σ;a1 σ;a2 σ;a3 12 a1a2 − 24 a1a2;a3 1 1 1 + R + Rp Rq + R R σ;a1 σ;a2 σ;a3 σ;a4 80 a1a2;a3a4 360 a1qa2 a3pa4 288 a1a2 a3a4 1 1 1 R + Rp Rq + R R σ;a1 σ;a2 σ;a3 σ;a4 σ;a5 + O(σ3), − 360 a1a2;a3a4a5 360 a1qa2 a3pa4;a5 288 a1a2 a3a4;a5 (A.9)

1 ∆1/2 = R 6

1 1 1 1 p 1 pq 1 pqr ;a1 ;a2 + Ra1a2 R;a1a2 + RRa1a2 R a Rpa2 + R Rpa1qa2 + R a Rpqra2 σ σ 40 − 120 72 − 30 1 60 60 1 1 1 1 1 p 1 p q R;a1a2a3 + (Ra1a2 );a3 + RRa1a2;a3 R a Rpa2;a3 + R q;a R a pa − −360 120 144 − 45 1 180 1 2 3 1 1 + Rp Rq + Rpqr R σ;a1 σ;a2 σ;a3 + O(σ2), (A.10) 180 q a1pa2;a3 90 a1 pqra2;a3

66 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 II.2. Electromagnétisme massif de Stueckelberg en espace-temps courbe : renormalisation de Hadamard du tenseur d’impulsion-énergie et effet Casimir

1/2 1/2 ;µ 1 ;a ;a 3/2 ν0 ρ0 1 ;a ∆− ∆ σ = R σ 1 σ 2 + O(σ ), g g g = R σ 1 ;µ 6 a1a2 ν ρ µν0;ρ0 −2 µνρa1 (A.11) 1 + R σ;a1 σ;a2 + O(σ3/2) (A.15) 3 µνρa1;a2 1 σ = g R σ;a1 σ;a2 + O(σ3/2), (A.12) ;µν µν − 3 µa1νa2 and

ν0 1 ;a1 ;a2 3/2 ν0 2 a1 1 gν σ;µν0 = gµν Rµa1νa2 σ σ + O(σ ), g g = R σ + R − − 6 ν µν0 3 a1[µ;ν] −6 a1[µ;ν]a2 (A.13) 1 1 + R Rp R R pq σ;a1 σ;a2 6 µνpa1 a2 − 4 µpqa1 ν a2 1 g ν0 g = R σ;a1 3/2 ν µν0;ρ −2 µνρa1 + O(σ ). (A.16) 1 + R σ;a1 σ;a2 + O(σ3/2), (A.14) 6 µνρa1;a2

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68 Published in Phys. Rev. D 93, 044063 (2016), eprint : arXiv:1512.06326 CHAPITRE 3:

ELECTROMAGNÉTISMEMASSIFDE STUECKELBERGDANSLES ESPACES-TEMPSDEDE SITTERETANTI-DE SITTER : FONCTIONSÀDEUX POINTSETTENSEURD’IMPULSION-ÉNERGIERENORMALISÉ

Dans ce chapitre, nous avons considéré de nouvelles applications de l’électromagnétisme massif de Stueckelberg, dont le formalisme a été développé dans le chapitre précédent. Nous avons travaillé dans les espaces-temps maximalement symétriques de de Sitter et anti-de Sitter. Le fait d’avoir la symétrie maximale nous permet de construire analytiquement toutes les fonctions de corrélations à deux points associées à l’électromagnétisme de Stueckelberg. A partir des fonctions de Hadamard, nous avons obtenu les expressions analytiques de la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé au champ vectoriel massif en espaces-temps de de Sitter et anti-de Sitter. Des résultats analogues étaient déjà connus pour le champ scalaire et pour le champ de Dirac mais aucun n’avait été obtenu pour le champ vectoriel massif.

69 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

Stueckelberg massive electromagnetism in de Sitter and anti-de Sitter spacetimes: Two-point functions and renormalized stress-energy tensors

Andrei Belokogne,∗ Antoine Folacci,† and Julien Queva‡ Equipe Physique Théorique- Projet COMPA, SPE, UMR 6134 du CNRS et de l’Universitéde Corse, Universitéde Corse, BP 52, F-20250 Corte, France (Dated: December 8, 2016) By considering Hadamard vacuum states, we first construct the two-point functions associated with Stueckelberg massive electromagnetism in de Sitter and anti-de Sitter spacetimes. Then, from the general formalism developed in [A. Belokogne and A. Folacci, Phys. Rev. D 93, 044063 (2016)], we obtain an exact analytical expression for the vacuum expectation value of the renormalized stress- energy tensor of the massive vector field propagating in these maximally symmetric spacetimes.

I. INTRODUCTION magnetism deﬁned, at the quantum level, by the action [1, 23]

In a recent article, we discussed the covariant quantiza- S [Aµ, Φ,C,C∗, gµν ] = SA [Aµ, gµν ] tion of Stueckelberg massive electromagnetism on an ar- +S [Φ, g ] + S [C,C , g ] , (1) bitrary four-dimensional curved spacetime ( , g ) with Φ µν Gh ∗ µν M µν no boundary and we constructed, for Hadamard quantum where states, the expectation value of the renormalized stress- 1 S = d4x √ g F µν F energy tensor (RSET) [1]. Here, we do not return to the A − −4 µν motivations leading us to consider Stueckelberg massive ZM 1 1 electromagnetism in curved spacetime. The interested m2AµA ( µA )2 (2) −2 µ − 2 ∇ µ reader is invited to consult our previous article, in particular its introduction as well as references therein. The denotes the action associated with the massive vector formalism developed in Ref. [1] permitted us to discuss, field Aµ with mass m (here, Fµν = µAν ν Aµ = as an application, the Casimir effect outside a perfectly ∇ − ∇ ∂µAν ∂ν Aµ is the associated field strength) and conducting medium with a plane boundary. − In the present paper, we shall address a much more 1 1 S = d4x √ g µΦ Φ m2Φ2 (3) difficult problem which could have interesting implica- Φ − −2 ∇ ∇µ − 2 tions in cosmology of the very early Universe or in the ZM context of the AdS/CFT correspondence: we shall ob- is the action governing the auxiliary Stueckelberg scalar tain an exact analytical expression for the vacuum ex- field Φ. The last action term in Eq. (1) is the compen- pectation value of the RSET of the massive vector field sating ghost contribution given by propagating in de Sitter and anti-de Sitter spacetimes. 4 µ 2 It is interesting to note that such results do not exist in SGh = d x √ g C∗ µC + m C∗C , (4) − ∇ ∇ the literature while the RSETs associated with the mas- ZM sive scalar field and the massive spinor field have been where C and C∗ are two fermionic ghost fields. Here, it is obtained quite a long time ago (see, e.g., Refs. [2–8] for important to note that some authors dealing with Stueck- the case of the massive scalar field and Refs. [9–11] for elberg electromagnetism (see, e.g., Refs. [16, 21, 24]) have the case of the massive spinor field). In fact, this void considered Stueckelberg electromagnetism defined from in the literature can easily be explained. Indeed, even if the sole action there exist numerous works concerning the massive vec- 1 tor field in de Sitter and anti-de Sitter spacetimes [12– S = d4x √ g F µν F A − −4 µν 22], the two-point functions are in general constructed in ZM the framework of the de Broglie-Proca theory and, as a 1 1 m2AµA ( µA )2 , (5) consequence, do not display the usual Hadamard singu- −2 µ − 2ξ ∇ µ larity (see the last remark in the conclusion of Ref. [1]) which is a fundamental ingredient of regularization and where ξ is a gauge parameter. In fact, these authors renormalization techniques in curved spacetime. were mainly interested by the determination of the Feyn- In our article, we shall focus on Stueckelberg electro- man propagator associated with the massive vector field Aµ. Of course, in order to calculate physical quantities such as the RSET associated with Stueckelberg electromagnetism, the full action must be considered, i.e., it is

∗ [email protected] necessary to take also into account, in addition to the † [email protected] contribution of the massive vector field, those of the aux- ‡ [email protected] iliary Stueckelberg field and of the ghost fields. In our

70 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

2 article, we shall not consider the case of an arbitrary netism from the Wightman functions associated with the gauge parameter ξ. Indeed, as we have already noted vector field Aµ, the Stueckelberg scalar field Φ and the in Ref. [1], if we want to work with Hadamard quan- ghost fields C and C∗. tum states, it is necessary to take ξ = 1. However, it is interesting to recall that Fröband Higuchi in a recent article [21] have provided, for an arbitrary value of ξ, a A. de Sitter and anti-de Sitter spacetimes mode-sum construction of the two-point functions for the massive vector field by working in the Poincarépatch of Here, we gather some results concerning (i) the geome- de Sitter space. Their results have permitted them to try of the four-dimensional de Sitter spacetime (dS4) and recover, as particular cases, the two-point functions ob- the four-dimensional anti-de Sitter spacetime (AdS4) as tained by Allen and Jacobson in Ref. [15] (they corre- well as (ii) the properties of some geometrical objects de- spond to ξ , i.e., to the de Broglie-Proca theory) fined on these maximally symmetric gravitational back- → ∞ and those obtained by Tsamis and Woodard in Ref. [19] grounds. We have minimized the information on these (they correspond to ξ 0). topics (for more details and proofs, see Refs. [15, 26– → Our article is organized as follows. In Sec.II, we con- 29]). Those results are necessary to construct the Wight- struct the Wightman functions associated with the mas- man functions of Stueckelberg electromagnetism and, in 4 4 sive vector field Aµ, the Stueckelberg auxiliary scalar field Sec.III, will permit us to simplify in dS and AdS the Φ and the ghost fields C and C∗ and, from them, we formalism developed in Ref. [1]. deduce by analytic continuation all the other two-point dS4 and AdS4 are maximally symmetric spacetimes of functions. We do not use a mode-sum construction as constant scalar curvature (positive for the former and in Ref. [21], but we extend the approach of Allen and negative for the latter) which are locally characterized Jacobson in Ref. [15] (see also Refs. [16, 25, 26]). More by the relations precisely, by assuming that the vacuum is a maximally symmetric quantum state, we solve the wave equations Rµνρτ = (R/12) (gµρgντ gµτ gνρ) , (6a) − for the various Wightman functions involved by taking Rµν = (R/4) gµν , (6b) into account, as constraints, two Ward identities; we then fix the remaining integration constants by imposing (i) and Hadamard-type singularities at short distance and (ii) in +12H2 for dS4, de Sitter spacetime, the regularity of the solutions at the R = (6c) 12K2 for AdS4. antipodal point or (iii) in anti-de Sitter spacetime, that (− the solutions fall off as fast as possible at spatial infinity. In Sec.III, from the general formalism developed in Here H and K are two positive constants of dimension 1 Ref. [1], we obtain an exact analytical expression for the (length)− . The relations (6a)–(6c) are useful in order vacuum expectation value of the RSET of the massive to simplify the various covariant Taylor series expansions vector field propagating in de Sitter and anti-de Sitter involved in the Hadamard renormalization process (see spacetimes. The geometrical ambiguities are fixed by Ref. [1]) and will be extensively used in Sec.III. 4 4 considering the flat-space limit and, moreover, we con- dS and AdS can be realized as the four-dimensional sider the two alternative but equivalent expressions for hyperboloids the renormalized expectation value given in Ref. [1] in η XaXb = 12/R (7) order to discuss the zero-mass limit of our results. Fi- ab nally, in a conclusion (Sec.IV), we briefly consider the 5 embedded in the flat five-dimensional space R equipped possible extension of our work to Stueckelberg electro- with the metric magnetism in an arbitrary ξ gauge. It should be noted that, in this article, we use units diag( 1, +1, +1, +1, +1) for dS4, η = − (8) such that ~ = c = G = 1 and the geometrical conventions ab diag( 1, 1, +1, +1, +1) for AdS4. of Hawking and Ellis [27] concerning the definitions of ( − − the scalar curvature R, the Ricci tensor R and the µν Equations (7) and (8) make it obvious that O(1, 4) is the Riemann tensor R as well as the commutation of 4 µνρσ symmetry group of dS and that its topology is that of covariant derivatives. Moreover, we will frequently refer 4 S3, while O(2, 3) is the symmetry group of AdS to our previous article [1] and we assume that the reader R whose× topology is that of S1 3. It is important to has “in hand” a copy of it. R recall that, in order to avoid× closed timelike curves in AdS4, it is necessary to “unwrap” the circle S1 to go 4 onto its universal covering space R1 and then AdS has II. TWO-POINT FUNCTIONS OF the topology of R4. STUECKELBERG ELECTROMAGNETISM In the context of field theories in curved spacetime, the geodetic interval σ(x, x0), defined as one-half the square In this section, we shall construct the various two-point of the geodesic distance between the points x and x0, is functions involved in Stueckelberg massive electromag- of fundamental interest (see, e.g., Refs. [30, 31]) and the

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 71 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

Hadamard renormalization process developed in Ref. [1], ℐ+ (z=∞) which we will exploit in Sec.III, is based on an exten- sive use of this geometrical object. However, in this sec- z<0 z>1 z<0 tion, even though σ(x, x0) is invariant under the symme- z=0 z=1 z=1 z=0 try group of dS4 or AdS4, it is advantageous to consider instead the real quadratic form Px 0

1 a b z(x, x0) = 1 + (R/12) ηabX (x)X (x0) (9) z z z z 2 =0 =1 =1 =0 z<0 z>1 z<0 in order to construct the two-point functions of Stueck- a b elberg electromagnetism. In Eq. (9), X (x) and X (x0) ℐ- (z=∞) are the coordinates of the points x and x0 on the hyperboloid (7) defining dS4 or AdS4 and η is the cor- ab FIG. 1. Carter-Penrose diagram of dS4. Without loss of gen- responding metric given by (8). This quadratic form is erality, the point x can be taken to be any point in space- obviously invariant under the symmetry group of dS4 or 4 time. P x denotes its associated antipodal point. The left AdS and is moreover defined on the whole spacetime, and right edges of this diagram must be identified along the + while σ(x, x0) is not defined everywhere because there is dashed lines. and − denote, respectively, the future and not always a geodesic between two arbitrary points in past spacelikeI infinitiesI for timelike and null geodesics. The these maximally symmetric spacetimes. However, when hatched area is the set of points x0 which cannot be reached σ(x, x0) is defined, we have by geodesics from x and for which σ(x, x0) is not defined. 1 z(x, x0) = 1 + cos (R/6) σ(x, x0) (10a) 2 z(x, x0) = 1 if x and x0 are null related, (16b) h2 p i = cos (R/24) σ(x, x0) (10b) 0 < z(x, x0) < 1 if x and x0 are timelike related, (16c) p z(x, x0) 0 if x and x0 cannot be joined by a geodesic, or, equivalently, ≤ (16d)

2 2 z(x, x0) = cos (H /2) σ(x, x0) (11) in AdS4. All these results can be visualized in the Carter- Penrose diagrams of dS4 (see Fig.1) and AdS 4 (see in dS4 and p Fig.2). 2 2 4 z(x, x0) = cosh (K /2) σ(x, x0) (12) To conclude this subsection, we recall that in dS and AdS4, any bitensor which is invariant under the space- in AdS4. The previous relationsp can be inverted and used time symmetry group (a maximally symmetric tensor) to define σ(x, x0) globally. In fact, it will chiefly help us, can be expressed only in terms of the bitensors z, z;µ, in Sec.III, to reexpress the two-point functions obtained z;µ0 , gµν , gµ0ν0 and gµν0 [15]. Here, gµν0 is the usual bivec- here in terms of σ(x, x0). tor of parallel transport from x to x0 which is defined by We shall now point out some useful properties of ;ρ the differential equation gµν0;ρσ = 0 and the boundary z(x, x ). With respect to the antipodal transformation 0 condition limx0 x gµν0 (x, x0) = gµν (x). For example, a a → which sends the point x with coordinates X (x) on the two-point function G(x, x0) associated with a scalar field hyperboloid (7) to its antipodal point P x with coordi- is necessarily of the form G(z) and a two-point function nates Gµν0 (x, x0) associated with a vector field can be written in the form A(z)g + B(z)z z . This very important Xa(P x) = Xa(x), (13) µν0 ;µ ;ν0 − remark will simplify the construction of the two-point functions of Stueckelberg electromagnetism. Besides, in we have order to handle these functions in connection with the z(x, P x0) = 1 z(x, x0). (14) wave equations and the Ward identities, it will be neces- − sary to use the following geometrical relations We have also ;µ z;µz = (R/12) z(1 z), (17a) z(x, x0) > 1 if x and x0 are timelike related, (15a) ;ρ − gµν0;ρz = 0, (17b) z(x, x0) = 1 if x and x0 are null related, (15b) ;µ gµν0 z = z;ν0 , (17c) 0 < z(x, x0) < 1 if x and x0 are spacelike related, (15c) − ;ν0 gµν z = z;µ, (17d) z(x, x0) 0 if x and x0 cannot be joined by a geodesic, 0 ≤ − (15d) z;µν = (R/24) (1 2z)gµν , (17e) − z;µν = (R/24) gµν + (1/z) z;µz;ν , (17f) 4 0 0 0 in dS and g = (1/z)(g z + g z ) , (17g) µν0;ρ − µρ ;ν0 ρν0 ;µ z(x, x0) > 1 if x and x0 are spacelike related, (16a) g = (1/z)(g z + g z ) , (17h) µν0;ρ0 − µρ0 ;ν0 ν0ρ0 ;µ

72 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

Px in addition, that this quantum state is (i) maximally sym- z z metric and (ii) of Hadamard type. We recall that, in the <0 <0 context of the calculation of the renormalized expecta- z=0 z=0 tion value of the stress-energy-tensor operator with respect to a vacuum 0 , it is convenient to work with the 01 x z>1 it is possible to construct the zoo of the two-point func- (z=∞) (z=∞) tions of the theory from the Wightman functions and, in a ﬁrst step, we shall focus on these particular two-point z=1 z=1 functions. We recall that the Wightman function associated with 0

z z (+)A =0 =0 G (x, x0) = 0 Aµ(x)Aν (x0) 0 (20) µν0 h | 0 | i z<0 z<0 and satisfies the wave equation [see also Eq. (19a)] Px 2 (+)A R/4 m G (x, x0) = 0. (21) x µν0 4 − − FIG. 2. Carter-Penrose diagram of AdS . Without loss of generality, the point x can be taken to be any point in space- Similarly, the Wightman function associated with the time. P x denotes its associated antipodal point. Here, we auxiliary scalar field Φ is given by only display an elementary cell of the universal covering of (+)Φ G (x, x0) = 0 Φ(x)Φ(x0) 0 (22) anti-de Sitter spacetime. The whole manifold is obtained by h | | i gluing along the dashed lines the top edge of one cell with the bottom edge of another one and by replicating, ad nauseam, and is a solution of [see also Eq. (19b)] this process. denotes timelike infinity. The hatched area 2 (+)Φ I x m G (x, x0) = 0, (23) is the set of points x0 which cannot be reached by geodesics − from x and for which σ(x, x0) is not defined. while the Wightman function associated with the ghost fields is defined by

;ρ 2 (+)Gh g = (R/12) [(z 1)/z] gµν + (2/z ) z;µz;ν , (17i) G (x, x0) = 0 C∗(x)C(x0) 0 (24) µν0;ρ − 0 0 h | | i and to note that the d’Alembertian operator acting on and satisﬁes the wave equation [see also Eq. (19d)] biscalar functions is given by 2 (+)Gh x m G (x, x0) = 0. (25) R d2 d − = z(1 z) + (2 4z) . (18) Moreover, we have two Ward identities that relate these 12 − dz2 − dz three Wightman functions. We can write

µ (+)A (+)Gh G (x, x0) + ν G (x, x0) = 0 (26) B. Stueckelberg theory, wave equations and Ward ∇ µν0 ∇ 0 identities for the Wightman functions and (+)Φ (+)Gh G (x, x0) G (x, x0) = 0. (27) From the quantum action (1), we can easily obtain the − wave equations satisfied by the massive vector field Aµ, It should be noted that, in our previous article, we have the auxiliary scalar field Φ and the ghost fields C and derived the Ward identities for the Feynman propagators C∗. They have been derived in our previous article [see [see Eqs. (33) and (29) in Ref. [1]] and for the Hadamard 4 4 Eqs. (19)–(21) in Ref. [1]] and, in dS or in AdS , due to Green functions [see Eqs. (65) and (66) in Ref. [1]]. Here Eq. (6b), they reduce to we have written them for the Wightman functions. They can be derived in the same way, i.e., from the wave equa- R/4 m2 A = 0, (19a) − − µ tions by using arguments of uniqueness [30]. The Ward 2 m Φ = 0, (19b) identity (27) expresses the equality of the Wightman − functions associated with the auxiliary scalar field and m2 C = 0, (19c) − the ghost fields. So, thereafter we shall omit the labels 2 m C∗ = 0. (19d) Φ and Gh and use a generic form for these two Green − functions by writing From now on, we shall assume that the Stueckelberg (+) (+)Φ (+)Gh theory is quantized in a normalized vacuum state 0 and, G (x, x0) = G (x, x0) = G (x, x0). (28) | i

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 73 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

C. Explicit expression for the Wightman functions This equation is invariant under the transformation z in dS4 and AdS4 1 z because the parameters a, b and c satisfy a+b+1→ = 2c−. As a consequence, we can write 1. General form of the Wightman functions in maximally (+) 1 symmetric backgrounds G (x, x0) = C F (3/2 + κ, 3/2 κ; 2; z) G − 2 + CG F (3/2 + κ, 3/2 κ; 2; 1 z) (36) We have previously assumed that the vacuum 0 is − − | i a maximally symmetric state. As a consequence, we where C1 and C2 are two integration constants. (+)A G G can express the Wightman functions G (x, x0) and µν0 The diﬀerential equations (32a) and (32b) which pro- (+) G (x, x0) as a function of the quadratic form z(x, x0) vide the Wightman function (30) are much more compli- [see also the last paragraph of the subsectionIIA] and cated to solve. In order to do so, we introduce the new write function γ(z) deﬁned by G(+)(x, x ) = G(+)(z) (29) 0 γ(z) = α(z) (R/3) z(1 z) β(z), (37) for the scalar Wightman functions (22) and (24) and − −

(+)A (+)A and rewrite the Ward identity (33) in the form G (x, x0) = G (z) = α(z) g + 4 β(z) z z µν0 µν0 µν0 ;µ ;ν0 (30) d 3 R d + γ(z) + β(z) G(+)(z) = 0. (38) dz z 2 − dz for the vector Wightman function (20). By inserting (29) into the wave equation (23) or (25) Then, by using Eqs. (37) and (38), we can combine the and by taking into account the relations (17) and (18), differential equations (32a) and (32b) and we have we obtain the differential equation d2 d 12m2 d2 d 12m2 z(1 z) + (2 4z) G(+)(z) = 0. z(1 z) + (3 6z) 6 γ(z) = − dz2 − dz − R − dz2 − dz − − R (31) d (1 2z) G(+)(z). (39) Similarly, inserting (30) into the wave equation (21) leads − dz to a system of two coupled differential equations for the The general solution γ(z) of this nonhomogeneous hyper- functions α(z) and β(z) given by geometric differential equation is the sum d2 d 2z + 1 12m2 z(1 z) + (2 4z) α(z) − dz2 − dz − z − R γ(z) = γc(z) + γp(z) (40) R + (1 2z) β(z) = 0 (32a) of the complementary solution γc(z) and a particular so- 6 − lution γp(z). γc(z) is the general solution of the hyper- and geometric differential equation (34) with a = 5/2 + λ, d2 d 10z 1 12m2 b = 5/2 λ and c = 3 where z(1 z) + (4 8z) − β(z) − − dz2 − dz − z − R 2 6 1 λ = 1/4 12 m /R. (41) + α(z) = 0, (32b) − R z2 Since the coefficients ap, b and c fulfill again a+b+1 = 2c, while, from the Ward identity (26), we obtain we can therefore write d 3 R d 5R + α(z) z(1 z) + (1 2z) β(z) γ (z) = C1 F (5/2 + λ, 5/2 λ; 3; z) dz z − 3 − dz 6 − c γ − 2 d + Cγ F (5/2 + λ, 5/2 λ; 3; 1 z) (42) G(+)(z) = 0. (33) − − dz − 1 2 where Cγ and Cγ are two new integration constants. Fur- Equation (31) is a hypergeometric differential equation thermore, it is rather easy to check that it is possible to of the form [32–34] take as a particular solution of (39) d2 d z(1 z) + [c (a + b + 1)z] 1 − dz2 − dz γ (z) = p 9/4 κ2 − ab F (a, b; c; z) = 0 (34) d2 d − z(1 z) + (1/2) (1 2z) G(+)(z). × − dz2 − dz with a = 3/2 + κ, b = 3/2 κ and c = 2 where − (43) κ = 9/4 12 m2/R. (35) − p

74 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

We have now at our disposal the general solution of the nonhomogeneous differential equation (39). This permits us to establish the expression of the Wightman function (30) by determining β(z) from Eq. (38) and then α(z) from Eq. (37). After a long but straightforward calculation using systematically, in order to remove higher-order derivatives, the differential relation [32–34] d ab F (a, b; c; z) = F (a + 1, b + 1; c + 1; z) (44) dz c as well as the hypergeometric differential equation (34), we obtain

(+)A 1 2 G (x, x0) = (2/3)C z(1 z) F 0(5/2 + λ, 5/2 λ; 3; z) + (2/3)C z(1 z) F 0(5/2 + λ, 5/2 λ; 3; 1 z) µν0 − γ − − γ − − − h C1 (1 2z) F (5/2 + λ, 5/2 λ; 3; z) C2 (1 2z) F (5/2 + λ, 5/2 λ; 3; 1 z) − γ − − − γ − − − (1/4)C1 F (5/2 + κ, 5/2 κ; 3; z) + (1/4)C2 F (5/2 + κ, 5/2 κ; 3; 1 z) g − G − G − − µν0 8 1 2 i + C F 0(5/2 + λ, 5/2 λ; 3; z) + C F 0(5/2 + λ, 5/2 λ; 3; 1 z) R − γ − γ − − h 3C1 (1/z) F (5/2 + λ, 5/2 λ; 3; z) 3C2 (1/z) F (5/2 + λ, 5/2 λ; 3; 1 z) − γ − − γ − − 1 2 (3/4)C F 0(5/2 + κ, 5/2 κ; 3; z) (3/4)C F 0(5/2 + κ, 5/2 κ; 3; 1 z) − G − − G − − (3/4)C1 (1/z) F (5/2 + κ, 5/2 κ; 3; z) + (3/4)C2 (1/z) F (5/2 + κ, 5/2 κ; 3; 1 z) z z . − G − G − − ;µ ;ν0 i (45) It is important to note that, in Eq. (45) as well in the following of the article, the derivative of the hypergeometric function F (a, b; c; z) with respect to its argument z is denoted by F 0(a, b; c; z). In summary, the general form of the Wightman function associated with the massive vector field Aµ is explicitly given by Eq. (45) while the Wightman function associated with the scalar field Φ and the ghost fields C and C∗ is 1 2 1 2 explicitly given by Eq. (36). In the following subsections, we shall fix the integration constants CG, CG, Cγ and Cγ appearing in the expression of these two-point functions.

2. Wightman functions for Hadamard vacua the Feynman propagators associated with all the fields of the theory can be represented in the Hadamard form Previously, we have assumed that the vacuum 0 [1]. (Without loss of generality, in this discussion, we fo- of Stueckelberg electromagnetism is of Hadamard type.| i cus on Feynman propagators but it would be possible to Here, we shall consider that this assumption can be real- consider, equivalently, Hadamard Green functions.) In- ized by imposing that, at short distance, i.e. for x0 x, deed, this last assumption provides stronger constraints the Wightman function (20) associated with the massive→ on the geometrical coefficients of the singular terms in vector field Aµ and the Wightman function (28) associ- 1/[σ(x, x0)+i] and ln[σ(x, x0)+i] of the Hadamard rep- ated with both the scalar field Φ and the ghost fields C resentations of the Feynman propagators. In fact, due to and C∗ satisfy the choice ξ = 1 of the gauge parameter, we know that the two-point functions of the Stueckelberg theory can

µν0 (+)A 1 1 be represented in the Hadamard form and, as a conse- g G (x, x0) (46) µν0 ∼ 2 x0 x 2π σ(x, x ) quence, if we fix the dominant term of the coefficient of → 0 1/[σ(x, x0) + i], all the other terms are unambiguously and determined [see the differential equation (A3a) and the 1 1 boundary condition (A3b) as well as the recursion rela- G(+)(x, x ) . (47) 0 ∼ 2 tions (39a), (39b), (43a) and (43b) in Ref. [1]]. x0 x 8π σ(x, x ) → 0 It should be noted that, at first sight, the conditions By inserting (10a) or (10b) into (45) and (36), we ob- (46) and (47) are less constraining than assuming that tain the short distance expansions

3072 C1 1 µν0 (+)A γ 2304 CG 1 g G (x, x0) = µν0 −Γ(5/2 + λ)Γ(5/2 λ) − Γ(5/2 + κ)Γ(5/2 κ) Rσ2(x, x ) " − − # 0 2 1 2 1 32 (85/4 λ ) Cγ 72 (19/4 + κ ) CG 1 + − + O [ln σ(x, x0)] (48) −Γ(5/2 + λ)Γ(5/2 λ) − Γ(5/2 + κ)Γ(5/2 κ) Rσ(x, x ) x0 x " − − # 0 →

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 75 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

and the reﬂection formula [32–34]

1 π (+) 24 CG 1 Γ(z)Γ(1 z) = . (52) G (x, x0) = − sin(πz) Γ(3/2 + κ)Γ(3/2 κ) Rσ(x, x ) − 0 + O [ln σ(x, x0)] (49) x x 0→ 3. Wightman functions in dS4

and, by comparing with (46) and (47), we can fix the two 4 1 1 In dS , in order to fix the remaining integration con- integration constants Cγ and CG. We have 2 2 stants Cγ and CG, we require the regularity of the Wight- man functions (45) and (36) at the antipodal point of x, x0 = P x and therefore on its light cone), or, in other 1 R Γ(5/2 + λ)Γ(5/2 λ) Cγ = − (50a) words, for z 0 (see also Fig.1)[29, 35]. We obtain −256π2 1/4 λ2 − immediately → R 9/4 λ2 = − (50b) 2 −256π cos(πλ) Cγ = 0 (53a) and and 2 CG = 0. (53b) R C1 = Γ(3/2 + κ)Γ(3/2 κ) (51a) G 192π2 − By inserting now the integration constants (50), (51), R 1/4 κ2 (53a) and (53b) into the general expressions (45) and = − . (51b) (36), we obtain in dS4 the explicit expressions of the 192π cos(πκ) Wightman function associated with the massive vector field Aµ and the Wightman function associated with both Here, in order simplify the expressions (50a) and (51a) the Stueckelberg scalar field Φ and the ghost fields C and which involve the Gamma function Γ(z), we have used C∗. We have

2 2 2 (+)A H 9/4 λ 3 (9/4 λ ) G (x, x0) = − z(1 z) F 0(5/2 + λ, 5/2 λ; 3; z) + − (1 2z) F (5/2 + λ, 5/2 λ; 3; z) µν0 32π cos(πλ) − − 2 cos(πλ) − − 1 (1/4 κ2) − F (5/2 + κ, 5/2 κ; 3; z) g − 2 cos(πκ) − µν0 1 9/4 λ2 (9/4 λ2) + − F 0(5/2 + λ, 5/2 λ; 3; z) + 3 − (1/z) F (5/2 + λ, 5/2 λ; 3; z) 32π cos(πλ) − cos(πλ) − 1/4 κ2 1/4 κ2 − F 0(5/2 + κ, 5/2 κ; 3; z) − (1/z) F (5/2 + κ, 5/2 κ; 3; z) z z (54) − cos(πκ) − − cos(πκ) − ;µ ;ν0 and 2 2 (+) H 1/4 κ G (x, x0) = − F (3/2 + κ, 3/2 κ; 2; z). (55) 16π cos(πκ) −

It should be noted that (54) is in accordance with the re- tions (45) and (36) fall off as fast as possible at spatial sult obtained recently by Fröband Higuchi from a mode- infinity, i.e., for z (see also Fig.2). We recall that sum construction [see Eq. (25) in Ref. [21]] while (55) is this condition is imposed→ ∞ because the Cauchy problem a closed form which can be found in various works con- is not well posed in AdS4, this gravitational background cerning the massive scalar field in dS4 [2,3, 15, 29, 35]. being not globally hyperbolic [27]. Such a condition permits us to control the flow of information through spatial infinity [28, 36]. 4 4. Wightman functions in AdS In the expressions (45) and (36) of the Wightman functions, the hypergeometric functions are expressed in term In AdS4, in order to fix the remaining integration con- of the variables z and 1 z. In order to impose the 2 2 − stants Cγ and CG, we require that the Wightman func- boundary condition previously mentioned, it is helpful

76 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

8 to reexpress them in term of the variable 1/z. This can be achieved thanks to the connection formulas [32–34]

Γ(c)Γ(b a) a Γ(c)Γ(a b) b F (a, b; c; z) = − ( z)− F (a, a c + 1; a b + 1; 1/z) + − ( z)− F (b, b c + 1; b a + 1; 1/z) Γ(b)Γ(c a) − − − Γ(a)Γ(c b) − − − − − (56a) which is valid for arg( z) < π and | − |

Γ(c)Γ(b a) a Γ(c)Γ(a b) b F (a, b; c; 1 z) = − z− F (a, c b; a b + 1; 1/z) + − z− F (b, c a; b a + 1; 1/z) (56b) − Γ(b)Γ(c a) − − Γ(a)Γ(c b) − − − − which is valid for arg(z) < π. We can then observe (in the lower half plane). This follows from the relation | | a iπa a a that the Wightman functions (45) and (36) approach zero ( z)− = e∓ z− which is a consequence of ( z)− = as fast as possible at spatial infinity if we eliminate the exp[− a ln( z)]. − (5/2 κ) (5/2 λ) − − terms in z− − and z− − in the expression of By inserting now the integration constants (50), (51), (3/2 κ) the former and the term in z− − in the expression (57a) and (57b) into the general expressions (45) and of the latter. (Here, since m > 0, we have that λ = (36), we can obtain in AdS4 the explicit expressions of the 1/4 + m2/K2 > 0 and κ = 9/4 + m2/K2 > 0.) We Wightman function associated with the massive vector then obtain immediately field A and the Wightman function associated with both p p µ 2 iπλ 1 the Stueckelberg scalar field Φ and the ghost fields C and C = i e± C (57a) γ ± γ C∗. Making use of the reflection formula (52) and of the duplication formula [32–34] and 2 iπκ 1 CG = i e± CG. (57b) 22z ∓ Γ(2z) = Γ(z)Γ(z + 1/2) (58) In Eqs. (57a) and (57b), the upper sign (the lower sign) 2√π must be chosen if, in the expressions (45) and (36) of the Wightman functions, z lies in the upper half plane to deal with the Γ-function, we have

2 2 (+)A K 9/4 λ iπλ G (x, x0) = − z(1 z) F 0(5/2 + λ, 5/2 λ; 3; z) i e± F 0(5/2 + λ, 5/2 λ; 3; 1 z) µν0 −32π cos(πλ) − − ∓ − − 2 3 (9/4 λ ) iπλ + − (1 2z) F (5/2 + λ, 5/2 λ; 3; z) i e± F (5/2 + λ, 5/2 λ; 3; 1 z) 2 cos(πλ) − − ± − − 2 1 (1/4 κ ) iπκ − F (5/2 + κ, 5/2 κ; 3; z) i e± F (5/2 + κ, 5/2 κ; 3; 1 z) g − 2 cos(πκ) − ± − − µν0 2 1 9/4 λ iπλ + − F 0(5/2 + λ, 5/2 λ; 3; z) i e± F 0(5/2 + λ, 5/2 λ; 3; 1 z) 32π cos(πλ) − ∓ − − 2 (9/4 λ ) iπλ + 3 − (1/z) F (5/2 + λ, 5/2 λ; 3; z) i e± F (5/2 + λ, 5/2 λ; 3; 1 z) cos(πλ) − ± − − 2 1/4 κ iπκ − F 0(5/2 + κ, 5/2 κ; 3; z) i e± F 0(5/2 + κ, 5/2 κ; 3; 1 z) − cos(πκ) − ∓ − − 2 1/4 κ iπκ − (1/z) F (5/2 + κ, 5/2 κ; 3; z) i e± F (5/2 + κ, 5/2 κ; 3; 1 z) z z − cos(πκ) − ± − − ;µ ;ν0 (59a) 4 K Γ(7/2 + λ)Γ(1/2 + λ) (5/2+λ) = z− (1 z) F (7/2 + λ, 1/2 + λ; 1 + 2λ; 1/z) −32π2m2 Γ(1 + 2λ) − Γ(5/2 + λ)Γ(1/2 + λ) (5/2+λ) 3 z− (1 2z) F (5/2 + λ, 1/2 + λ; 1 + 2λ; 1/z) − Γ(1 + 2λ) −

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 77 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

Γ(5/2 + κ)Γ(1/2 + κ) (5/2+κ) + z− F (5/2 + κ, 1/2 + κ; 1 + 2κ; 1/z) g Γ(1 + 2κ) µν0 2 K Γ(7/2 + λ)Γ(1/2 + λ) (7/2+λ) + z− F (7/2 + λ, 1/2 + λ; 1 + 2λ; 1/z) 32π2m2 Γ(1 + 2λ) Γ(5/2 + λ)Γ(1/2 + λ) (7/2+λ) 6 z− F (5/2 + λ, 1/2 + λ; 1 + 2λ; 1/z) − Γ(1 + 2λ)

Γ(7/2 + κ)Γ(1/2 + κ) (7/2+κ) z− F (7/2 + κ, 1/2 + κ; 1 + 2κ; 1/z) − Γ(1 + 2κ)

Γ(5/2 + κ)Γ(1/2 + κ) (7/2+κ) + 2 z− F (5/2 + κ, 1/2 + κ; 1 + 2κ; 1/z) z z (59b) Γ(1 + 2κ) ;µ ;ν0 and

2 2 (+) K 1/4 κ iπκ G (x, x0) = − F (3/2 + κ, 3/2 κ; 2; z) i e± F (3/2 + κ, 3/2 κ; 2; 1 z) (60a) −16π cos(πκ) − ∓ − − 2 K Γ(3/2 + κ)Γ(1/2 + κ) (3/2+κ) = z− F (3/2 + κ, 1/2 + κ; 1 + 2κ; 1/z). (60b) 16π2 Γ(1 + 2κ)

Here, it is important to recall that, in Eqs. (59a) and Bros, Epstein and Moschella which concern the scalar (60a), the upper sign (the lower sign) must be chosen if field in de Sitter spacetime [39] and general quantum field z lies in the upper half plane (in the lower half plane). theories in anti-de Sitter spacetime [40]. It should be noted that we have provided two equivalent expressions for these two-point functions: the hypergeometric functions appearing in formulas (59a) and (60a) 1. In dS4 are given in term of the variables z and 1 z while those − appearing in formulas (59b) and (60b) are expressed in The expressions (54) and (55) of the Wightman func- term of 1/z. The expression (60) is a result which can tions involve the hypergeometric function F (a, b; c; z) be found in some other works concerning the massive 4 which, in general, has a branch point at z = 1 (i.e., for x0 scalar field in AdS (see, e.g., Refs. [8,9, 15, 37]). To our on the light cone of x) and a branch cut which runs along knowledge, the Wightman function (59) associated with the real axis from z = 1 to + (i.e., for x0 in the light the massive vector field Aµ of the Stueckelberg theory cone of x) [see also Eq. (15)].∞ As a consequence, these is not in the literature. In Ref. [16], Janssen and Dulle- Wightman functions are perfectly defined when x and x0 mond have considered that problem but we are unable to are spacelike related or cannot be joined by a geodesic link their results with ours. but, when they are timelike related, it is important to specify how to approach the branch cut. In fact, it is necessary to replace in Eqs. (54) and (55) the biscalar z by D. Feynman propagators and Hadamard Green 4 4 the biscalar z i (here 0 ) where the minus sign (re- functions in dS and AdS ∓ → + spectively the plus sign) is chosen when x0 lies in the past (respectively the future) of x. Indeed, in dS4, due to the It is well known that, in quantum field theory in flat relation (10) [note that z = 1 (R/24)σ + ... ], the pre- spacetime, we can construct all the interesting two-point scription z z i induces the− change σ σ i which functions, i.e., the retarded and advanced Green func- permits us→ to encode∓ the usual behavior→ of the± Wight- tions, the Feynman propagator and the Hadamard Green man functions in curved spacetime (see also Chap. 4 function, from the Wightman function by taking its real of Ref. [41]) and to recover, in the flat-space limit, the or its imaginary part and, when it is necessary, by us- Wightman functions of Minkowski quantum field theory. ing multiplication by a step function in time. In some A The Feynman propagator G (x, x0) = sense, this remains true in curved spacetime [30, 31, 38] µν0 i 0 TA (x)A (x ) 0 associated with the massive vector and, in this subsection, we shall provide the expressions µ ν0 0 fieldh | A (here, T denotes| i the time-ordering operator) is of the Feynman propagators and the Hadamard Green µ obtained from the Wightman function (54) by writing functions of Stueckelberg massive electromagnetism from A (+)A G (x, x0) = i G (z i) with 0+. Indeed, in the Wightman functions obtained previously. Here, we µν0 µν0 − → shall adopt a pragmatic point of view by following the ap- dS4, the prescription z z i induces the change proach and the arguments of Allen and Jacobson [15, 29]. σ σ + i which permits→ us to− encode the time ordering It is however interesting to note the existence of a more (see→ also Secs. II C and III A of Ref. [1]). Similarly, the Φ rigorous point of view exposed in impressive articles by Feynman propagators G (x, x0) = i ψ T Φ(x)Φ(x0) ψ h | | i

78 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

Gh and G (x, x0) = i ψ TC∗(x)C(x0) ψ associated respec- with tively with the scalarh | ﬁeld Φ and the| i ghost ﬁelds C and (+) (1) (+) (+) C∗ are equals to i G (z i). G (x, x0) = G (x, x0) + G (x0, x). (66) The expression of the Hadamard− Green function

(1)A Formulas (62) and (66) must be taken with a grain of G (x, x0) = 0 Aµ(x)Aν (x0) + Aν (x0)Aµ(x) 0 (61) µν0 h | 0 0 | i salt. Indeed, it is important to recall that Hadamard Green functions are real-valued while Wightman func- associated with the massive vector field A is obviously µ tions are complex-valued and the prescription permitting obtained from (54) by noting that us to define the former and the latter on the branch cut (1)A (+)A (+)A are different. In fact, Hadamard Green functions are av- G (x, x0) = G (x, x0) + G (x0, x). (62) µν0 µν0 ν0µ erage across the cut, i.e., we have In the same way, the Hadamard Green function (1)A (+)A (+)A G (x, x0) = G (z + i) + G (z i) (67) (1)Φ µν0 µν0 µν0 G (x, x0) = 0 Φ(x)Φ(x0) + Φ(x0)Φ(x) 0 (63) − h | | i associated with the auxiliary scalar field Φ and the and

Hadamard Green function (1) (+) (+) G (x, x0) = G (z + i) + G (z i) (68) (1)Gh − G (x, x0) = 0 C∗(x)C(x0) C(x0)C∗(x) 0 (64) h | − | i with 0+. This prescription, which is in total agree- associated with the ghost ﬁelds C and C∗ can be obtained ment with→ that deﬁning the Wightman functions, permits from (55). We have us to have at our disposal Hadamard Green functions

(1)Φ (1)Gh (1) which are real-valued. More explicitly, by inserting (54) G (x, x0) = G (x, x0) = G (x, x0) (65) into (67) and (55) into (68), we obtain

2 2 2 (1)A H 9/4 λ 3 (9/4 λ ) G (x, x0) = − z(1 z) (ReF )0(5/2 + λ, 5/2 λ; 3; z) + − (1 2z) (ReF )(5/2 + λ, 5/2 λ; 3; z) µν0 16π cos(πλ) − − 2 cos(πλ) − − 1 (1/4 κ2) − (ReF )(5/2 + κ, 5/2 κ; 3; z) g − 2 cos(πκ) − µν0 1 9/4 λ2 (9/4 λ2) + − (ReF )0(5/2 + λ, 5/2 λ; 3; z) + 3 − (1/z) (ReF )(5/2 + λ, 5/2 λ; 3; z) 16π cos(πλ) − cos(πλ) − 1/4 κ2 1/4 κ2 − (ReF )0(5/2 + κ, 5/2 κ; 3; z) − (1/z) (ReF )(5/2 + κ, 5/2 κ; 3; z) z z . − cos(πκ) − − cos(πκ) − ;µ ;ν0 (69)

and imaginary part

2 2 F (a, b; c; z + i) F (a, b; c; z i) (1) H 1/4 κ (ImF )(a, b; c; z) = − − G (x, x0) = − (ReF )(3/2 + κ, 3/2 κ; 2; z). 2i 8π cos(πκ) − (71b) (70) and we shall use its derivative (ImF )0(a, b; c; z) with respect to its argument z. Here, we have introduced the average across the cut of the hypergeometric function F (a, b; c; z) deﬁned by 2. In AdS4 F (a, b; c; z + i) + F (a, b; c; z i) (ReF )(a, b; c; z) = − . 2 Mutatis mutandis, the previous discussion can be (71a) adapted to obtain the Feynman propagators and the Hadamard Green functions in AdS4. We ﬁrst note This function is nothing else than the real part of that the branch cut of the Wightman functions (59) F (a, b; c; z) on the branch cut. We have also used its and (60) runs along the real axis from z = to −∞ derivative (ReF )0(a, b; c; z) with respect to its argument z = 1. This appears clearly if we consider the expres- z. It should be noted that below we shall need also its sions (59b) and (60b). Indeed, the functions of the form

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 79 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

a z− = exp( a ln z) and the hypergeometric functions of permits us to encode the time ordering. Similarly, the − Φ the form F (a, b; c; 1/z) involved in these expressions are Feynman propagators G (x, x0) = i ψ T Φ(x)Φ(x0) ψ Gh h | | i respectively cut along the negative axis and the segment and G (x, x0) = i ψ TC∗(x)C(x0) ψ associated respec- [0, 1]. As a consequence, the Wightman functions (59) tively with the scalarh | field Φ and the| i ghost fields C and (+) and (60) are perfectly defined if z > 1, i.e., when x and C∗ are equals to i G (z + i). 4 x0 are spacelike related [see also Eq. (16)] whereas, if In AdS , the expression of the Hadamard Green func- z 1, and in particular when x and x0 are timelike re- tion (61) associated with the massive vector field Aµ is lated≤ (i.e., if z ]0, 1[), it is important to specify how to obtained by inserting (59) into (67) while the Hadamard approach the branch∈ cut. In fact, it is necessary to re- Green function (65) associated with both the massive place in Eqs. (59) and (60) the biscalar z by the biscalar scalar field Φ and the ghost fields C and C∗ is obtained z i (here 0 ) where the plus sign (respectively by inserting (60) into (68). In fact, here we shall con- ± → + the minus sign) is chosen when x0 lies in the past (re- struct the Hadamard Green functions from (59a) and spectively the future) of x. Indeed, in AdS4, due to the (60a) only. Indeed, the resulting expressions are easily relation (10), it is now the prescription z z i which tractable in the context of the renormalization of the induces the change σ σ i permitting→ us± to encode stress-energy tensor, or more precisely, their regular parts the usual behavior of the→ Wightman± functions in curved can be naturally extracted. Of course, in order to obtain spacetime. the Hadamard Green functions, it is then important to

4 A take carefully into account the upper sign (in that case, In AdS , the Feynman propagator G (x, x0) = µν0 we use the variable z + i and we are working in the up- i 0 TA (x)A (x0) 0 associated with the massive vector h | µ ν0 | i per half plane) or the lower sign (in that case, we use ﬁeld Aµ is obtained from the Wightman function (59) the variable z i and we are working in the lower half A (+)A − by writing G (x, x0) = i G (z + i) with 0 . plane) in (59a) and (60a). A straightforward calculation µν0 µν0 + Indeed, in this gravitational background, the prescrip-→ leads to expressions which are explicitly real-valued and tion z z + i induces the change σ σ + i which given by → →

2 2 (1)A K 9/4 λ G (x, x0) = − z(1 z) [(ReF )0(5/2 + λ, 5/2 λ; 3; z) + sin(πλ)(ReF )0(5/2 + λ, 5/2 λ; 3; 1 z) µν0 −16π cos(πλ) − − − − cos(πλ)(ImF )0(5/2 + λ, 5/2 λ; 3; 1 z)] − − − 3 (9/4 λ2) + − (1 2z) [(ReF )(5/2 + λ, 5/2 λ; 3; z) sin(πλ)(ReF )(5/2 + λ, 5/2 λ; 3; 1 z) 2 cos(πλ) − − − − − + cos(πλ)(ImF )(5/2 + λ, 5/2 λ; 3; 1 z)] − − 1 (1/4 κ2) − [(ReF )(5/2 + κ, 5/2 κ; 3; z) sin(πκ)(ReF )(5/2 + κ, 5/2 κ; 3; 1 z) − 2 cos(πκ) − − − −

+ cos(πκ)(ImF )(5/2 + κ, 5/2 κ; 3; 1 z)] g − − µν0 1 9/4 λ2 + − [(ReF )0(5/2 + λ, 5/2 λ; 3; z) + sin(πλ)(ReF )0(5/2 + λ, 5/2 λ; 3; 1 z) 16π cos(πλ) − − − cos(πλ)(ImF )0(5/2 + λ, 5/2 λ; 3; 1 z)] − − − (9/4 λ2) + 3 − (1/z) [(ReF )(5/2 + λ, 5/2 λ; 3; z) sin(πλ)(ReF )(5/2 + λ, 5/2 λ; 3; 1 z) cos(πλ) − − − − + cos(πλ)(ImF )(5/2 + λ, 5/2 λ; 3; 1 z)] − − 1/4 κ2 − [(ReF )0(5/2 + κ, 5/2 κ; 3; z) + sin(πκ)(ReF )0(5/2 + κ, 5/2 κ; 3; 1 z) − cos(πκ) − − −

cos(πκ)(ImF )0(5/2 + κ, 5/2 κ; 3; 1 z)] − − − 1/4 κ2 − (1/z) [(ReF )(5/2 + κ, 5/2 κ; 3; z) sin(πκ)(ReF )(5/2 + κ, 5/2 κ; 3; 1 z) − cos(πκ) − − − −

+ cos(πκ)(ImF )(5/2 + κ, 5/2 κ; 3; 1 z)] z z (72) − − ;µ ;ν0

80 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

and 2 2 (1) K 1/4 κ G (x, x0) = − [(ReF )(3/2 + κ, 3/2 κ; 2; z) + sin(πκ)(ReF )(3/2 + κ, 3/2 κ; 2; 1 z) − 8π cos(πκ) − − − cos(πκ)(ImF )(3/2 + κ, 3/2 κ; 2; 1 z)] . (73) − − −

III. RENORMALIZED STRESS-ENERGY (1)A (1) The expressions of Gsing µν0 (x, x0) and Gsing(x, x0) in- TENSOR OF STUECKELBERG volve the geodetic interval σ(x, x0), the bivector of par- ELECTROMAGNETISM allel transport gµν0 (x, x0), the Van Vleck-Morette determinant ∆(x, x0) (see Ref. [30] or the Appendix of In this section, from the general formalism developed Ref. [1] for its definition and properties) as well as in Ref. [1], we shall obtain exact analytical expressions the geometrical bivector V A (x, x ) and the geometri- µν0 0 for the vacuum expectation value of the RSET of the cal biscalar V (x, x0) which are defined by the expansions 4 4 A + A n massive vector field propagating in dS and AdS . We V (x, x ) = ∞ V (x, x ) σ (x, x ) and V (x, x ) = µν0 0 n=0 n µν0 0 0 0 shall, in particular, fix the geometrical ambiguities in the + n ∞ V (x, x ) σ (x, x ) and by the recursion relations results (see Refs. [42, 43] for interesting remarks on the n=0 n P0 0 satisfied by the coefficients V A (x, x ) and V (x, x ) [see ambiguity problem as well as Sec. IV E of Ref. [1] and ref- n µν0 0 n 0 PEqs. (39a), (39b), (43a) and (43b) in Ref. [1]]. Moreover, erences therein) by considering the flat-space limit and, we have introduced the renormalization mass M permit- moreover, we shall discuss the zero-mass limit of the ex- ting us to make dimensionless the argument of the loga- pressions found. rithm. In fact, in order to construct the RSET, we only need A. General considerations the lower coefficients of the covariant Taylor series expansions for x0 x of the bitensor → In this subsection, we have gathered some results es- A 2 ν0 (1)A tablished in Ref. [1] which will be necessary to construct, Wµν (x, x0) = 4π gν (x, x0)Greg µν0 (x, x0) (78) in the next three subsections, the RSETs associated with Stueckelberg electromagnetism in dS4 and AdS4. By do- and of the biscalar ing so, we hope to alleviate the task of the reader and to prevent him from drowning in the heavy formalism 2 (1) W (x, x0) = 4π Greg(x, x0) (79) developed in our previous article. We have seen in Ref. [1] that, in the context of the or, more precisely, we only need the covariant Taylor renormalization of the stress-energy tensor of Stueck- series expansions of these two quantities up to order elberg electromagnetism, it is necessary to extract the 1 σ (x, x0) [see Sec. IV of Ref. [1]]. As a consequence, we regular and state-dependent parts of the Hadamard are not really interested by the full expressions of (76) (1)A (1) Green functions G (x, x0) and G (x, x0). They µν0 and (77) but by their covariant Taylor series expansions (1)A can be obtained by removing from G (x, x0) and truncated by neglecting the terms vanishing faster than µν0 (1) σ(x, x0) for x0 x. They can be obtained from the co- G (x, x0) their singular and purely geometrical parts → 1/2 (1)A (1) variant Taylor series expansion of ∆ (x, x0) up to order G µν (x, x0) and G (x, x0). We have 2 sing 0 sing σ (x, x0) [see Eq. (A9) in Ref. [1]], the covariant Tay- A 1 (1)A (1)A (1)A lor series expansion of Vµν (x, x0) up to order σ (x, x0) G µν (x, x0) = G (x, x0) G µν (x, x0) (74) reg 0 µν0 − sing 0 [see Eqs. (71a)–(72d) in Ref. [1]] and the covariant Tay- 1 and lor series expansion of V (x, x0) up to order σ (x, x0) [see (1) (1) (1) Eqs. (73a)–(74c) in Ref. [1]]. By inserting these covariant G (x, x0) = G (x, x0) G (x, x0) (75) reg − sing Taylor series expansions into (76) and (77), we can de- A where rive the covariant Taylor series expansions of W (x, x0) 1/2 µν (1)A 1 ∆ (x, x0) defined by (78) [see Eq. (75) in Ref. [1] for its general G (x, x0) = g (x, x0) sing µν0 4π2 σ(x, x ) µν0 expression] and of W (x, x ) defined by (79) [see Eq. (76) 0 0 in Ref. [1] for its general expression]. A 2 + V (x, x0) ln M σ(x, x0) (76) µν0 Fortunately, in maximally symmetric spacetimes, due to the relations (6a)–(6c), the regularization of the and (1)A (1) 1/2 Hadamard Green functions Gµν (x, x0) and G (x, x0) (1) 1 ∆ (x, x0) 0 G (x, x0) = greatly simplifies. Indeed : sing 4π2 σ(x, x ) 0 (i) The covariant Taylor series expansions of the var- + V (x, x ) ln M 2σ(x, x ) . (77) 0 0 ious bitensors involved in the singular parts of the

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 81 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

Hadamard Green functions reduce to expressions simplify considerably. Indeed, the RSET of Stueckelberg electromagnetism can be expressed in term 1/2 ∆ = 1 + (1/24) Rσ of its trace as 2 2 3 + (19/17280) R σ + O(σ ), (80a) 1 0 T 0 = 0 T ρ 0 g (83) h | µν | iren 4 h | ρ | iren µν A 2 4 Vµν = (1/2) m + (1/24) R + (1/8) m and this one reducesb to b 2 2 +(1/24) m R + (1/432) R σ gµν 1 0 T ρ 0 = m2 (1/8) R s ρ 2 3/2 ρ ren 8π2 ρ + (1/1728)R σ;µσ;ν + O(σ ) , (80b) h | | i − − ρτ Aρ ρ b +sρτ + 3 v1 ρ + Θρ (84) and if we focus on the expression which only involves the char- V = (1/2) m2 (1/12) R acteristics of the quantum massive vector field A [see, − µ + (1/8) m4 (1/48) m2R σ in Ref. [1], Eq. (123) as well as Eq. (124)]. If we alter- − natively focus on the expression which involves both the 3/2 + O(σ ). (80c) characteristics of the massive vector field Aµ and of the massive scalar field Φ [see, in Ref. [1], Eqs. (125) and (ii) The covariant Taylor series expansions of the biten- (126) as well as Eqs. (127) and (128)], we have A sors Wµν (x, x0) and W (x, x0) reduce to ρ A ρ Φ ρ ρ 0 Tρ 0 ren = ρ + ρ + Θρ (85) A 1 ;a ;b 3/2 h | | i T T Wµν = sµν + 2 sµνabσ σ + O(σ ) (81) with b and 1 A ρ = m2 (1/4) R s ρ + 2 s ρτ + 4 vA ρ 1 T ρ 8π2 − − ρ ρτ 1 ρ W = w + w σ;aσ;b + O(σ3/2). (82) 2 ab (86a) and In the following, these considerations will facilitate our task. Φ ρ 1 2 ρ = m w + 2 v1 . (86b) In our previous article, we have provided two differ- T 8π2 − ent expressions for the RSET of Stueckelberg electromag- ρ It should be noted that the term Θρ appearing in netism: Eqs. (84) and (85) is a purely geometrical term which encodes the ambiguities in the definition of the RSET [see (i) A first expression which only involves state- Sec. IV E of Ref. [1]] and which involves, in particular, dependent as well as geometrical quantities associ- a contribution associated with the renormalization mass ated with the massive vector field A [see Eq. (123) µ M. We also recall that the term vA ρ in Eqs. (84) and in Ref. [1]]. Here, the contribution of the quantum 1 ρ (86a) and the term v in Eq. (86b) are purely geometri- massive scalar field Φ has been removed thanks to 1 cal quantities which appear in the covariant Taylor series a Ward identity and the result obtained is given in expansions of V A (x, x ) and V (x, x ) [see Eqs. (71b) and terms of the first coefficients of the covariant Tay- µν 0 0 (73b) in Ref. [1]]. From Eqs. (72d) and (74c) of Ref. [1], lor series expansions for x x of the bitensor 0 we can show that, in maximally symmetric backgrounds, W A (x, x ) [see Eq. (75) in Ref.→ [1]]. µν 0 they reduce to

(ii) A second expression where the contributions of A ρ 4 2 2 v1 ρ = (1/2) m + (1/12) m R (1/2160) R (87) the massive vector field Aµ and of the massive − scalar field Φ have been artificially separated [see and Eqs. (125) and (126) in Ref. [1]] and which has been 4 2 2 constructed in such a way that the zero-mass limit v1 = (1/8) m (1/24) m R + (29/8640) R . (88) − of the first contribution reduces to the RSET of ρ Maxwell’s electromagnetism. The result obtained It is crucial to discuss the form of the trace term Θρ is given in terms of the first coefficients of the co- appearing in Eqs. (84) and (85). In an arbitrary gravitational background, such a term is given by Eq. (133) variant Taylor series expansions for x0 x of the A → of Ref. [1] which reduces, in maximally symmetric space- bitensors Wµν (x, x0) and W (x, x0) [see Eqs. (75) and (76) in Ref. [1]]. times, to 1 Of course, these expressions are equivalent but it will Θ ρ = α m4 + β m2R . (89) ρ 8π2 be necessary to consider both in order to clearly discuss the zero-mass limit of the Stueckelberg theory (see Here, α and β are constants which can be fixed by im- Sec.IIID). In maximally symmetric spacetimes, these posing additional physical conditions on the RSET (see

82 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

14 below) or which can be redefined by renormalization of constraint [see Eq. (83a) in Ref. [1]] Newton’s gravitational constant and of the cosmological ρ 2 constant. Indeed, let us recall that, in Eq. (89), the terms w = m w 6 v1. (94) ρ − α m4 and β m2R come from the Einstein-Hilbert action defining the dynamics of the gravitational field (see also By inserting (94) into (93), we then obtain the discussion in Sec. IV E 1 of Ref. [1]). Of course, the s ρτ = (1/8) Rs ρ + m2 w vA ρ 2 v . (95) ambiguities associated with the renormalization mass M ρτ ρ − 1 ρ − 1 are necessarily of this form but their expressions are to- This last equation is very interesting. Indeed, it permits tally determined. In maximally symmetric spacetimes, us to realize that, in maximally symmetric spacetimes, the renormalization mass induces in (84) a contribution the construction of the RSET of Stueckelberg electro- given by magnetism can be accomplished using only the coeffi- ρ 2 cients sρ and w, i.e., the lowest-order coefficients of the ρ ln(M ) 4 2 Θρ (M) = 2 (3/2) m + (1/4) m R . (90) covariant Taylor series expansions (81) and (82). As a 8π consequence, in the regularization process, it would be It can be derived from Eq. (146) of Ref. [1] which is valid sufficient to consider the covariant Taylor series expan- in an arbitrary gravitational background. Similarly, the sions of (76) and (77) truncated by neglecting the terms 0 renormalization mass induces in (85) a contribution given vanishing faster than σ (x, x0) = 1 for x0 x. More- → ρ by over, we can remark that the Taylor coefficient sρ is a trace term. So, in order to obtain its expression, it would ρ A ρ Φ ρ µν0 (1)A Θ (M) = Θ (M) + Θ (M) (91) be sufficient to regularize the trace g (x, x0)G (x, x0) ρ ρ ρ µν0 of the Hadamard Green functions (69) or (72) and this with could be done rather easily by taking into account the ln(M 2) relation (17a). In fact, in the following, we shall not ΘA ρ(M) = m4 + (1/3) m2R (92a) ρ 8π2 use these considerations even if it is obvious they could greatly simplify our job. We intend to determine the full and covariant Taylor series expansions (81) and (82) because this will permit us to control the internal consistency ln(M 2) ΘΦ ρ(M) = (1/2) m4 (1/12) m2R . (92b) of our calculations and, in particular, that the singular ρ 8π2 − terms in ln σ(x, x0) hidden in the expressions (69), (70), | | It can be derived from Eqs. (144), (147a) and (147b) (72) and (73) of the Hadamard Green functions are of of Ref. [1] which are valid in an arbitrary gravitational Hadamard type. We shall use the constraints (94) and background. (95) only to check our results. To conclude this subsection, we would like to remark ρτ that the Taylor coefficient sρτ appearing in the expres- 4 sions (84) and (86a) of the RSET can be related with B. The renormalized stress-energy tensor in dS lower-order Taylor coefficients. Indeed, due to the “Ward A In dS4, the covariant Taylor series expansions (81) and identity” linking the bitensors Wµν (x, x0) and W (x, x0) [see Eq. (85) in Ref. [1]], we can write in a maximally (82) can be obtained by inserting the expressions (69) and symmetric spacetime [see Eq. (86b) in Ref. [1]] (70) of the Hadamard Green functions into (74) and (75) taking into account (i) the relation (11) which links the ρτ ρ ρ A ρ sρτ = (1/8) Rsρ + wρ v1 ρ + 4 v1. (93) quadratic form z(x, x0) with the geodetic interval σ(x, x0) − as well as (ii) the expression of the singular parts (76) Moreover, due to the “wave equation” satisfied by the and (77) constructed by using the covariant Taylor series biscalar W (x, x0) [see Eq. (82) in Ref. [1]], we have the expansions (80a), (80b) and (80c). We then obtain

s = (1/2) m2 (19/144) R + (3/8) m2 + (1/16) R ln(R/24) + Ψ(5/2 + λ) + Ψ(5/2 λ) + 2γ µν − − − n + (1/8) m2 (1/48) R ln(R/ 24) + Ψ(5/2 + κ) + Ψ(5/2 κ) + 2γ g , (96a) − − µν o s = (5/16) m4 (43/288) m2R (691/51840) R2 + µνab − − − +n (5/48) m4 + (5/72) m2R + (5/576) R2 ln(R/24) + Ψ(7/2 + λ) + Ψ(7/2 λ) + 2γ − (1/32) m2R + (1/192) R2 ln(R/24) + Ψ(5/2 + λ) + Ψ(5/2 λ) + 2γ − − + (1/48) m4 + (1/288) m2R (1/864) R2 ln(R/24) + Ψ(7/2 + κ) + Ψ(7/2 κ) + 2γ g g − − µν ab o

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 83 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

+ (1/144) m2R + (19/5184) R2 n (1/24) m4 + (1/36) m2R + (1/288) R2 ln(R/24) + Ψ(7/2 + λ) + Ψ(7/2 λ) + 2γ − − + (1/32) m2R + (1/192) R2 ln(R/24) + Ψ(5/2 + λ) + Ψ(5/2 λ) + 2γ − + (1/24) m4 + (1/144) m2R (1/432) R2 ln(R/24) + Ψ(7/2 + κ) + Ψ(7/2 κ) + 2γ − − 2 2 (1/96) m R (1/576) R ln(R/24) + Ψ(5/2 + κ) + Ψ(5/2 κ) + 2γ gµ(a gν b) (96b) − − − | | o and w = (1/2) m2 + (1/18) R + (1/2) m2 (1/12) R ln(R/24) + Ψ(3/2 + κ) + Ψ(3/2 κ) + 2γ , (96c) − − − w = (5/16) m4 + (13/288) m2R + (1/5760) R2 ab − n + (1/8) m4 (1/48) m2R ln(R/24) + Ψ(5/2 + κ) + Ψ(5/2 κ) + 2γ g . (96d) − − ab In the previous expressions, we have introduced the Digamma function Ψ(z) = (d/dzo) ln Γ(z) and we have used systematically its properties [32–34] and, more particularly, the recurrence formula Ψ(z + 1) = Ψ(z) + 1/z (97) in order to simplify them. We have also introduced the Euler-Mascheroni constant γ = Ψ(1). From these results, we can now write − s ρ = 2 m2 (19/36) R + (3/2) m2 + (1/4) R ln(R/24) + Ψ(5/2 + λ) + Ψ(5/2 λ) + 2γ ρ − − − + (1/2) m2 (1/12) R ln(R/24) + Ψ(5/2 + κ) + Ψ(5/2 κ) + 2γ , (98a) − − s ρτ = (5/4) m4 (19/36) m2R (1/60) R2 ρτ − − − + (3/16) m2R + (1/32) R2 ln(R/24) + Ψ(5/2 + λ) + Ψ(5/2 λ) + 2γ − + (1/2) m4 + (1/12) m2R (1/36) R2 ln(R/24) + Ψ(7/2 + κ) + Ψ(7/2 κ) + 2γ − − (5/48) m2R (5/288) R2 ln(R/24) + Ψ(5/2 + κ) + Ψ(5/2 κ) + 2γ (98b) − − − and w ρ = (5/4) m4 + (13/72) m2R + (1/1440) R2 ρ − + (1/2) m4 (1/12) m2R ln(R/24) + Ψ(5/2 + κ) + Ψ(5/2 κ) + 2γ . (98c) − −

It should be noted that the constraints (94) and (95) are (84) taking into account the geometrical term (87) as well satisﬁed by these coeﬃcients. This can be easily checked as the geometrical ambiguities (89) and (90), we have for using the recurrence formula (97). the trace of the RSET By inserting now the expressions (98a) and (98b) into

2 ρ 4 2 2 8π 0 T 0 4 = (α + 9/4) m + (β + 17/24)m R + (19/1440) R h | ρ | iren dS (3/2) m4 + (1/4) m2R ln(R/(24M 2)) + Ψ(5/2 + λ) + Ψ(5/2 λ) + 2γ (99) b − −

and, of course, the RSET 0 Tµν 0 ren can be obtained all the geometrical ambiguities. However, it is possible to immediately from (83). Thish expression| | i could be consid- go further and to fix the renormalization mass M and the 4 ered as the final result of ourb work in dS . Indeed, by coefficient α by requiring the vanishing of this expression construction, it fully includes the state-dependence of the in the flat-space limit, i.e. for R 0. We first absorb the Stueckelberg theory and, moreover, it takes into account term 2γ into the renormalization→ mass M and we then obtain M = m/√2 and α = 9/4 which leads to −

2 ρ 2 2 8π 0 T 0 4 = (β + 17/24) m R + (19/1440) R h | ρ | iren dS (3/2) m4 + (1/4) m2R ln(R/(12m2)) + Ψ(5/2 + λ) + Ψ(5/2 λ) . (100) b − −

84 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

This last result is not free of ambiguities due to the ar- C. The renormalized stress-energy tensor in AdS4 bitrary coefficient β remaining in the expression of the trace of the RSET. However, it is worth noting that it Mutatis mutandis, the calculations of Sec.IIIB can be would be possible to cancel it or, more precisely, to can- adapted to obtain the RSET of Stueckelberg electromag- 2 cel the term (β + 17/24)m R by a finite renormalization netism in AdS4. We first determine the covariant Taylor of the Einstein-Hilbert action of the gravitational field. series expansions (81) and (82) from the expressions (72) and (73) of the Hadamard Green functions. We have

s = (1/2) m2 + (5/144) R µν − +n (3/8) m2 + (1/16) R ln( R/24) + 2Ψ(1/2 + λ) + 2γ − + (1/8) m2 (1/48) R ln( R/24) + 2Ψ(1/2 + κ) + 2γ g , (101a) − − µν o s = (5/16) m4 (1/18) m2R + (119/51840) R2 µνab − − +n (5/48) m4 + (11/288) m2R + (1/288) R2 ln( R/24) + 2Ψ(1/2 + λ) + 2γ − + (1/48) m4 + (1/288) m2R (1/864) R2 ln( R/24) + 2Ψ(1/2 + κ) + 2γ g g − − µν ab o + (1/144) m2R + (1/5184) R2 n (1/24) m4 (1/288) m2R (1/576) R2 ln( R/24) + 2Ψ(1/2 + λ) + 2γ − − − − 4 2 2 + (1/24) m (1/288) m R (1/1728) R ln( R/24) + 2Ψ(1/2 + κ) + 2γ gµ(a gν b) (101b) − − − | | and o w = (1/2) m2 + (7/72) R + (1/2) m2 (1/12) R ln( R/24) + 2Ψ(1/2 + κ) + 2γ , (101c) − − − w = (5/16) m4 + (25/288) m2R (29/5760)R2 ab − − n + (1/8) m4 (1/48) m2R ln( R/24) + 2Ψ(1/2 + κ) + 2γ g . (101d) − − ab In order to simplify the previous expressions and, in particular, to eliminate the termso in tan(πκ) and in tan(πλ) occurring in the expressions (72) and (73) of the Hadamard Green functions, we have used systematically the relation n 1 − p + 1/2 Ψ(n + 1/2 + z) + Ψ(n + 1/2 z) + π tan(πz) = 2Ψ(1/2 + z) + 2 (1 δ ) (102) − − n0 (p + 1/2)2 z2 p=0 X − (here δnm is the Kronecker delta) which is valid for n N. This relation can be derived from the reﬂection formula [32–34] ∈ Ψ(1 z) = Ψ(z) + π cot(πz) (103) − making use of tan(πz) = cot[π(z + 1/2)] and of the recurrence formula (97). From the results (101a), (101b) and (101d), we can write − s ρ = 2 m2 + (5/36) R ρ − + (3/2) m2 + (1/4) R ln( R/24) + 2Ψ(1/2 + λ) + 2γ − + (1/2) m2 (1/12) R ln( R/24) + 2Ψ(1/2 + κ) + 2γ , (104a) − − s ρτ = (5/4) m4 (11/72) m2R + (1/90) R2 ρτ − − + (3/16) m2R + (1/32) R2 ln( R/24) + 2Ψ(1/2 + λ) + 2γ − + (1/2) m4 (1/48) m2R (1/96) R2 ln( R/24) + 2Ψ(1/2 + κ) + 2γ (104b) − − − and w ρ = (5/4) m4 + (25/72) m2R (29/1440) R2 ρ − − + (1/2) m4 (1/12) m2R ln( R/24) + 2Ψ(1/2 + κ) + 2γ . (104c) − −

It should be noted that the constraints (94) and (95) are satisﬁed by these coeﬃcients.

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 85 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

By inserting now the expressions (104a) and (104b) as well as the geometrical ambiguities (89) and (90), we into (84) taking into account the geometrical term (87) have for the trace of the RSET

2 ρ 4 2 2 8π 0 T 0 4 = (α + 9/4) m + (β + 5/24)m R (11/1440) R h | ρ | iren AdS − (3/2) m4 + (1/4) m2R ln( R/(24M 2)) + 2Ψ(1/2 + λ) + 2γ . (105) b − − Finally, by requiring the vanishing of this expression in the flat-space limit, i.e. for R 0, we can fix the renormalization mass M (after absorption of the term 2γ) and the coefficient α. We then obtain→ M = m/√2 and α = 9/4 which leads to − 2 ρ 2 2 8π 0 T 0 4 = (β + 5/24) m R (11/1440) R h | ρ | iren AdS − (3/2) m4 + (1/4) m2R ln( R/(12m2)) + 2Ψ(1/2 + λ) . (106) b − −

D. Remarks concerning the zero-mass limit of the This “discontinuity” is not really surprising. Indeed, as renormalized stress-energy tensor we have already noted in Ref. [1], in an arbitrary spacetime, due to the contribution of the auxiliary scalar field Because Stueckelberg massive electromagnetism is a Φ, the full RSET of Stueckelberg electromagnetism never U(1) gauge theory which generalizes Maxwell’s the- permits us to recover the RSET of Maxwell’s theory. This ory, we could naively expect to recover, by considering can be alternatively interpreted by noting that the pres- the zero-mass limit of the previous results, the usual ence of a mass term in the Stueckelberg theory breaks trace anomaly for Maxwell’s theory given by [see, e.g., the conformal invariance of Maxwell’s theory. To cir- Eq. (130) in Ref. [1] or Eq. (3.25) in Ref. [44]] cumvent this difficulty, we have proposed in Ref. [1] to split the RSET of Stueckelberg electromagnetism into 2 A ρ A ρ two separately conserved RSETs, a contribution directly 8π 0 T ρ 0 ren = 2v1 ρ 4v1 (107) h | | i − associated with the vector field Aµ and another one corresponding to the scalar field Φ (see also our discussion in which reduces, in ab maximally symmetric gravitational Sec.IIIA), the zero-mass limit of the first contribution background, to reducing to the RSET of Maxwell’s electromagnetism. Even if we consider that this separation is rather artifi- 8π2 0 T A ρ 0 = (31/2160) R2. (108) h | ρ | iren − cial because, in our opinion, only the full RSET is physically relevant, the auxiliary scalar field Φ playing the role In fact, this is notb the case. In dS4, for m2 0, Eq. (100) of a kind of ghost field, it is interesting to test our pro- provides → posal which has nevertheless provided correct results in the context of the Casimir effect (see Sec. V of Ref. [1]). 4 2 ρ 2 In dS , we can obtain the separated RSETs associated 8π 0 Tρ 0 ren dS4 = (19/1440) R , (109) h | | i with the massive vector field Aµ and the massive scalar field Φ which vanish in the flat-space limit by inserting while, in AdS4, forb m2 0, we obtain from Eq. (106) → into Eqs. (86a) and (86b) the Taylor coefficients (98a), (98b) and (96c) and by moreover taking into account the 2 ρ 2 8π 0 T 0 4 = (11/1440) R . (110) geometrical ambiguities (92a), (92b) and (89). We have h | ρ | iren AdS − b 2 A ρ 2 2 8π 0 T 0 4 = (β + 13/18) m R + (59/2160) R h | ρ | iren dS A (3/2) m4 + (1/4) m2R ln(R/(12m2)) + Ψ(5/2 + λ) + Ψ(5/2 λ) b − − + (1/2) m4 (1/12) m2R ln(R/(12m2)) + Ψ(5/2 + κ) + Ψ(5/2 κ) (111) − − for the RSET associated with the massive vector field Aµ and

2 Φ ρ 2 2 8π 0 T 0 4 = (β 5/36) m R + (29/4320) R h | ρ | iren dS Φ − (1/2) m4 (1/12) m2R ln(R/(12m2)) + Ψ(3/2 + κ) + Ψ(3/2 κ) (112) b − − −

for the RSET associated with the massive scalar ﬁeld Φ. It is worth pointing out that the sum of these two RSETs

86 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

18 coincides with the RSET given by Eq. (100). This is a Taylor coefficients involved in the calculations and, in trivial consequence of Eq. (97). We can moreover note particular, of the coefficient w. This assumption is cer- that the RSET (112) associated with the scalar field Φ is tainly valid for a large class of spacetimes but, in the nothing else than the result derived by Bunch and Davies context of field theory in dS4, it is not founded. For ex- in Ref. [3] (see also Refs. [2,4,5]. If we now consider the ample, from Eq. (96c), we can show that w R2/(48 m2) limit m2 0 of the RSET (111) associated with the for m2 0. We here encounter the well-known∼ infrared → → vector field Aµ, we obtain divergence problem which plagues quantum field theories in de Sitter spacetime (see, e.g., Ref [29] and references 2 A ρ 2 8π 0 T 0 4 = (59/2160) R . (113) therein). h | ρ | iren dS Here, we do not recover the usual trace anomaly (108) In AdS4, we can obtain the separated RSETs asso- b of Maxwell’s electromagnetism. In fact, this can be ex- ciated with the massive vector field Aµ and the mas- plained easily. Indeed, in Secs. IV C 4 and IV D. of sive scalar field Φ which vanish in the flat-space limit Ref. [1], we have constructed the RSET associated with by inserting into Eqs. (86a) and (86b) the Taylor coeffi- the massive vector field Aµ which reduces, in the zero- cients (104a), (104b) and (101c) and by moreover taking mass limit, to the RSET of Maxwell’s electromagnetism into account the geometrical ambiguities (92a), (92b) and by assuming tacitely the regularity for m2 0 of the (89). We have →

2 A ρ 2 2 8π 0 T 0 4 = (β + 7/18) m R (31/2160) R h | ρ | iren AdS A − (3/2) m4 + (1/4) m2R ln( R/(12m2)) + 2Ψ(1/2 + λ) b − − + (1/2) m4 (1/12) m2R ln( R/(12m2)) + 2Ψ(1/2 + κ) (114) − − for the RSET associated with the massive vector ﬁeld Aµ and

2 Φ ρ 2 2 8π 0 T 0 4 = (β 13/72) m R + (29/4320) R h | ρ | iren AdS Φ − (1/2) m4 (1/12) m2R ln( R/(12m2)) + 2Ψ(1/2 + κ) (115) b − − − for the RSET associated with the massive scalar field can be used (i.e., ξ = 1). However, in the literature, Φ. We can note that the sum of these two RSETs co- Stueckelberg massive electromagnetism is often consid- incides with the RSET given by Eq. (106) and that the ered for an arbitrary gauge parameter ξ, i.e., by describ- RSET (115) associated with the scalar field Φ is in agree- ing the massive vector field from the action (5) instead ment with the result derived by Camporesi and Higuchi of the action (2). Here, we refer to the articles by Fröb in Ref. [9] (see also Ref. [8]). If we now consider the limit and Higuchi [21] who consider the massive vector field m2 0 of the RSET (114) associated with the vector in de Sitter spacetime and by Janssen and Dullemond → field Aµ, we obtain [16] who work in anti-de Sitter spacetime. The propa-

2 A ρ 2 gators constructed in this context are ξ dependent but 8π 0 T 0 4 = (31/2160) R . (116) h | ρ | iren AdS − have good physical properties and, even if they do not have a Hadamard-type singularity at short distance (for In AdS4, we recover the usual trace anomaly (108) of b ξ = 1), they reproduce in the flat-space limit the stan- Maxwell’s electromagnetism. dard6 Minkowski two-point functions given by Itzykson and Zuber in Ref. [24]. It would be interesting to an- IV. CONCLUSION alyze the physical content of these propagators by con- structing their associated RSETs. Due to the expected In the present article, by focusing on Hadamard vac- gauge independence of the RSET, it is quite likely that uum states, we have first constructed the various two- the results obtained will be identical to those given in point functions associated with Stueckelberg massive the previous section but a proof of this claim certainly electromagnetism in de Sitter and anti-de Sitter space- requires a careful study. We plan to provide it in the near times. Then, from the general formalism developed in future. Ref. [1], we have obtained an exact analytical expression for the vacuum expectation value of the RSET of the massive vector field propagating in de Sitter and anti-de ACKNOWLEDGMENTS Sitter spacetimes. It is worth pointing out that these results have been obtained by working in the unique gauge We wish to thank Yves Décaniniand Mohamed Ould for which the machinery of Hadamard renormalization El Hadj for various discussions, Philippe Spindel for

Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 87 II. Electromagnétisme massif de Stueckelberg et renormalisation du tenseur d’impulsion-énergie associé

19 providing us with a copy of Ref. [13] and the “Collec- tivit´eTerritoriale de Corse” for its support through the COMPA project.

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88 Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 II.3. Electromagnétisme massif de Stueckelberg dans les espaces-temps de de Sitter et anti-de Sitter : fonctions à deux points et tenseur d’impulsion-énergie renormalisé

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Published in Phys. Rev. D 94, 105028 (2016), eprint : arXiv:1610.00244 89

CONCLUSIONETPERSPECTIVES

Ce manuscrit est relatif au domaine de la théorie quantique des champs en espace-temps courbe. Cette théorie est une approximation semi-classique de la gravitation quantique qui consiste à traiter classiquement la métrique gµν de l’espace-temps et à considérer du point de vue quantique tous les autres champs. Il faut préciser que, pour avoir une théorie cohérente, nous devons aussi inclure le champ des gravitons à l’ordre au moins une boucle. Cette approche évite les difﬁcultés dues à la non-renormalisabilité de la théorie quantique de la gravitation et fournit un cadre qui permet d’étudier les conséquences, à basse énergie, d’une hypothétique “théorie du tout”. Il faut rappeler que cette approche a permis aux physiciens d’obtenir des résultats fascinants concernant la cosmologie de l’univers primordial ainsi que la physique des trous noirs et, en particulier, a conduit Parker à la découverte de la création de particules dans des univers en expansion et Hawking à celle du rayonnement quantique des trous noirs.

Dans ce manuscrit, nous avons considéré les théories quantiques des champs massifs en espace- temps courbe à l’ordre une boucle. Par conséquent, la contribution des gravitons négligée doit être rajoutée pour avoir une théorie cohérente. En outre, nous nous sommes intéressés à la contribution “matériel-

91 Conclusion et perspectives le” associée aux champs quantiques massifs des théories considérées. Elle est représentée par la valeur moyenne du tenseur d’impulsion-énergie. Comme cette quantité est mal déﬁnie, aﬁn d’en extraire une valeur moyenne physiquement raisonnable, nous avons utilisé deux méthodes de renormalisation, à savoir, celle fondée sur le développement de DeWitt-Schwinger de l’action effective et la renormalisation dite de Hadamard.

Dans la première partie de ce manuscrit, nous nous sommes intéressés à l’action effective à l’ordre une boucle associée aux champs quantiques massifs. Nous avons plus particulièrement considéré les théo- ries quantiques libres du champ scalaire massif, du champ de Dirac massif et du champ de Proca. La valeur moyenne renormalisée du tenseur d’impulsion-énergie associé aux champs massifs considérés peut être formellement exprimée en utilisant le développement de DeWitt-Schwinger de l’action effective. Il faut rappeler que l’état quantique associé au champ est négligé pour ce développement ce qui est dû à la représentation de DeWitt-Schwinger des fonctions de Green. En espace-temps de dimension quatre, le dé- veloppement de DeWitt-Schwinger de l’action effective associée à un champ massif, après renormalisation et dans la limite des grandes masses, se réduit aux termes construits à partir du coefﬁcient de Gilkey-

DeWitt a3 qui est de nature purement géométrique. Sa variation par rapport à la métrique nous donne une bonne approximation analytique pour la valeur moyenne renormalisée du tenseur d’impulsion-énergie d’un champ quantique massif. A partir de l’expression générale de ce tenseur et en utilisant des packages écrits sous Mathematica qui nous permettent de faire les calculs d’algèbre tensorielle de manière efficace, nous avons obtenu analytiquement la valeur moyenne renormalisée du tenseur d’impulsion-énergie asso- cié au champ scalaire massif, au champ de Dirac massif et au champ de Proca se propageant sur l’espace- temps de Kerr-Newman. Ce résultat nous a permis de retrouver les résultats existant dans la littérature pour les trous noirs de Schwarzschild, de Reissner-Nordström et de Kerr (et d’en corriger certains). Il est important aussi de noter que les expressions associées aux trous noirs en rotation pour les champs massifs négligent l’existence d’instabilités liées à la superradiance. En absence de résultats exacts en espace-temps de Kerr-Newman, notre approximation de la valeur moyenne renormalisée du tenseur d’impulsion-énergie pourrait être utile, par exemple, pour étudier l’effet de backreaction des champs quantiques massifs sur cet espace-temps ou sur ses modes quasi-normaux. Nous avons d’ailleurs étudié un problème simplifié de backreaction consistant à déterminer les modifications de la masse et du moment angulaire du trou noir vues par un observateur à l’infini, et dues à la présence des champs quantiques.

Dans la deuxième partie de ce manuscrit, nous nous sommes intéressés à l’électromagnétisme massif de Stueckelberg. Contrairement à la théorie massive de de Broglie-Proca qui est une généralisation directe de l’électromagnétisme de Maxwell obtenue en ajoutant un terme de masse qui brise la symétrie de jauge locale U(1) originelle, la théorie massive proposée par Stueckelberg la préserve par l’intermédiaire d’un couplage approprié d’un champ scalaire auxiliaire avec le champ vectoriel massif. Nous avons développé le formalisme général de la théorie de Stueckelberg sur un espace-temps arbitraire de dimension quatre (invariance de jauge de la théorie classique et quantiﬁcation covariante ; équations d’ondes pour le champ massif de spin-1 Aµ, pour le champ scalaire auxiliaire de Stueckelberg Φ et pour les champs de fantômes

C et C∗ ; identité de Ward ; représentation de Hadamard des divers propagateurs de Feynman et dévelop- pement en séries de Taylor covariantes des coefﬁcients correspondants). Cela nous a permis de construire, pour un état quantique de type Hadamard ψ et en utilisant la méthode de la renormalisation de Hada- | 〉 mard, la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé à la théorie de Stueckelberg. Nous avons donné deux expressions différentes mais équivalentes du résultat. La première expression ne fait intervenir que les caractéristiques du champ vectoriel Aµ. La seconde expression a été construite dans

92 Conclusion et perspectives le but de discuter la limite de masse nulle des résultats obtenus dans le cadre de la théorie de Stueckel- berg et les comparer avec ceux issus de celle de Maxwell ; ainsi, nous avons séparé de manière artiﬁcielle en deux parties les contributions indépendamment conservées du champ vectoriel Aµ et du champ scalaire auxiliaire Φ. Notons que l’on peut passer de la deuxième expression à la première en éliminant la contribution du champ scalaire de Stueckelberg Φ par l’intermédiaire de deux identités de Ward, ce champ pouvant être considéré comme un fantôme. De plus, nous avons discuté des ambiguïtés géométriques de la valeur moyenne renormalisée du tenseur d’impulsion-énergie qui sont d’une importance fondamentale. Comme première application de ce résultat, nous avons considéré, en espace-temps de Minkowski, l’effet de Casimir en dehors d’un milieu parfaitement conducteur et de bord plan. Nous avons retrouvé le ré- sultat existant déjà dans la littérature pour la théorie de de Broglie-Proca. Notons toutefois que nous ne sommes pas vraiment satisfaits des conditions aux limites prises sur le bord du milieu conducteur et que des conditions plus réalistes pourraient être envisagées. Nous avons aussi comparé les formalismes de de Broglie-Proca et de Stueckelberg et discuté les avantages de l’approche de Stueckelberg bien qu’à notre avis ces deux descriptions de l’électromagnétisme massif ne sont que les deux faces d’une même théorie. Cependant, du point de vue pratique, c’est la théorie de Stueckelberg qui est bien adaptée pour pouvoir utiliser la renormalisation de Hadamard. Par la suite, nous avons considéré de nouvelles applications de l’électromagnétisme massif de Stueckelberg. Nous avons travaillé dans les espaces-temps maximalement symétriques de de Sitter et anti-de Sitter. Le fait d’avoir la symétrie maximale nous a permis de construire analytiquement toutes les fonctions de corrélations à deux points associées à l’électromagnétisme de Stue- ckelberg. A partir des fonctions de Hadamard, nous avons obtenu les expressions analytiques de la valeur moyenne renormalisée du tenseur d’impulsion-énergie associé au champ vectoriel massif en espaces-temps de de Sitter et anti-de Sitter. Des résultats analogues étaient déjà connus pour le champ scalaire et pour le champ de Dirac mais aucun n’avait été obtenu pour le champ vectoriel massif.

Pour ﬁnir, nous concluons par quelques prolongements perspectifs du travail présenté. – Nous pourrions étudier la théorie quantique de l’électromagnétisme massif de Stueckelberg sur les espaces-temps de Friedmann-Lemaître-Robertson-Walker qui sont d’une importance considérable dans le domaine de la cosmologie. – Il serait très intéressant de considérer la théorie de l’électrodynamique quantique en espace-temps courbe où le champ de jauge deviendrait massif par l’intermédiaire du mécanisme de Stueckelberg. Dans le contexte de la renormalisation de Hadamard, on construirait le tenseur d’impulsion-énergie renormalisé total qui pourrait être considéré par la suite sur l’espace-temps de de Sitter et celui de Friedmann-Lemaître-Robertson-Walker.

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