Stueckelberg massive in curved spacetime: Hadamard of the stress-energy tensor and the Andrei Belokogne, Antoine Folacci

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Andrei Belokogne, Antoine Folacci. Stueckelberg massive electromagnetism in curved spacetime: Hadamard renormalization of the stress-energy tensor and the Casimir effect. Physical Review D, American Physical Society, 2016, 93 (4), pp.044063. ￿10.1103/PhysRevD.93.044063￿. ￿hal-01316785￿

HAL Id: hal-01316785 https://hal.archives-ouvertes.fr/hal-01316785 Submitted on 18 May 2016

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW D 93, 044063 (2016) 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’Université de Corse, Université de Corse, BP 52, F-20250 Corte, France (Received 19 December 2015; published 24 February 2016) We discuss Stueckelberg massive electromagnetism on an arbitrary four-dimensional curved spacetime and, in particular, (i) the gauge invariance of the classical theory and its covariant ; (ii) the wave equations for the massive -1 Aμ, for the auxiliary Stueckelberg scalar field Φ and for the fields C and C; (iii) Ward identities; (iv) the Hadamard representation of the various Feynman and the 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- 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.

DOI: 10.1103/PhysRevD.93.044063

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

2470-0010=2016=93(4)=044063(29) 044063-1 © 2016 American Physical Society ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) μ Maxwell’s electromagnetism. It is also important to ∇ Aμ ¼ 0 ð4aÞ recall that, in general, it is the de Broglie-Proca theory that is used to impose experimental con- which is here a dynamical constraint (and not a gauge straints on the photon mass [2–4]. condition) as well as the wave equation (ii) The most aesthetically appealing one which, con- 2 ν trarily to the de Broglie-Proca theory preserves the □Aμ − m Aμ − Rμ Aν ¼ 0: ð4bÞ local Uð1Þ gauge invariance of Maxwell’s electro- magnetism, has been proposed by Stueckelberg (see It should be noted that the action (1) is also directly relevant Refs. [12,13] for the original articles on the subject at the quantum level because the de Broglie-Proca theory is and also Ref. [14] for a nice recent review). The not a gauge theory. construction of such a massive gauge theory can be Stueckelberg massive electromagnetism is described by achieved by coupling appropriately an auxiliary a vector field Aμ and an auxiliary scalar field Φ, and its scalar field to the massive spin-1 field. This theory Φ action SCl ¼ SCl½Aμ; ;gμν, which can be constructed from is unitary and renormalizable and can be included in the de Broglie-Proca action (1) by using the substitution the Standard Model of particle physics [14]. More- over, it is interesting to note that extensions of the 1 Standard Model based on predict the Aμ → Aμ þ ∇μΦ; ð5Þ m existence of a hidden sector of particles which could explain the nature of dark matter. Among these is given by exotic particles, there exists in particular a dark Z    photon, the mass of which arises also via the ffiffiffiffiffiffi 1 1 1 4 p− − μν − 2 μ ∇μΦ Stueckelberg mechanism (see, e.g., Ref. [15]). This SCl ¼ d x g F Fμν m A þ M 4 2 m “heavy” photon may be detectable in low energy   experiments (see, e.g., Refs [16–19]). It is also worth 1 × Aμ þ ∇μΦ ð6aÞ pointing out that the Stueckelberg procedure is not m limited to vector fields. It has been recently extended Z  “ ” to restore the gauge invariance of various massive 4 pffiffiffiffiffiffi 1 μ ν 1 μ 2 1 2 μ ¼ d x −g − ∇ A ∇μAν þ ð∇ AμÞ − m A Aμ field theories (see, e.g., Refs. [20,21] which discuss M 2 2 2 the case of massive antisymmetric tensor fields and,  1 μ ν 1 μ μ e.g., Ref. [22] where is considered). − RμνA A − ∇ Φ∇μΦ − mA ∇μΦ : ð6bÞ In the two following paragraphs, we shall briefly review 2 2 these two theories at the classical level. De Broglie-Proca massive electromagnetism is described It should be noted that, at the classical level, the vector field Φ by a vector field Aμ, and its action S ¼ S½Aμ;gμν, which is Aμ and the scalar field are coupled [see, in Eq. (6a), μ directly obtained from the original Maxwell Lagrangian by the last term −mA ∇μΦ]. Here, it is important to note that adding a mass contribution, is given by Stueckelberg massive electromagnetism is invariant under Z   the gauge transformation ffiffiffiffiffiffi 1 1 4 p− − μν − 2 μ S ¼ d x g F Fμν m A Aμ : ð1Þ → 0 ∇ Λ M 4 2 Aμ Aμ ¼ Aμ þ μ ; ð7aÞ

Here, m is the mass of the vector field Aμ, and the Φ → Φ0 ¼ Φ − mΛ; ð7bÞ associated field strength Fμν is defined as usual by for an arbitrary scalar field Λ, so the local Uð1Þ gauge Fμν ¼ ∇μAν − ∇νAμ ¼ ∂μAν − ∂νAμ: ð2Þ of Maxwell’s electromagnetism remains unbro- ken for the spin-1 field of the Stueckelberg theory. As a Let us note that, while Maxwell’s theory is invariant under consequence, in order to treat this theory at the quantum the gauge transformation level (see below), it is necessary to add to the action (6) a gauge-breaking term and the compensating ghost 0 Aμ → Aμ ¼ Aμ þ ∇μΛ ð3Þ contribution. Here, it seems important to highlight some considera- for an arbitrary scalar field Λ, this gauge invariance is tions which will play a crucial role in this article. Let us broken for the de Broglie-Proca theory due to the mass note that the de Broglie-Proca theory can be obtained from term. The extremization of (1) with respect to Aμ leads to Stueckelberg electromagnetism by taking ν 2 μ the Proca equation ∇ Fμν þ m Aμ ¼ 0. Applying ∇ to this equation, we obtain the Lorenz condition Φ ¼ 0: ð8Þ

044063-2 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) We can therefore consider that the de Broglie-Proca theory the point-splitting method [23,28,29] which has been is nothing other than the Stueckelberg gauge theory in the developed in connection with the Hadamard representation particular gauge (8). However, it is worth noting that this is of the Green functions (see, e.g., Refs. [30–42] and, more a “bad” choice of gauge leading to some complications. In particularly, Refs. [32,33,35,38,39] where gauge theories particular: are considered). (i) Due to the constraint (4a), the Feynman Our article is organized as follows. In Sec. II,we associated with the vector field Aμ does not admit a review the covariant quantization of Stueckelberg massive Hadamard representation (see below), and, as a electromagnetism on an arbitrary four-dimensional curved consequence, the quantum states of the de spacetime (gauge-breaking action and associated ghost Broglie-Proca theory are not of Hadamard type. contribution; wave equations for the massive spin-1 field This complicates the and renormali- Aμ, for the auxiliary Stueckelberg scalar field Φ and for the zation procedures. ghost fields C and C; Feynman propagators and Ward (ii) In the limit m2 → 0, singularities occur, and a lot identities). In Sec. III, we focus on the particular gauge for of physical results obtained in the context of the de which the various Feynman propagators and the associated Broglie-Proca theory do not coincide with the Hadamard Green functions admit Hadamard representa- corresponding results obtained with Maxwell’s tion, or, in other words, we consider quantum states of theory. Hadamard type. We also construct the covariant Taylor In this article, we intend to focus on the Stueckelberg series expansions of the geometrical and state-dependent theory at the quantum level, and we shall analyze its coefficients involved in the Hadamard representation of the energetic content with possible applications to the Casimir Green functions. In Sec. IV, we obtain, for a Hadamard effect (in this paper) and to cosmology of the very early quantum state, the renormalized expectation value of the universe (in a next paper) in mind. More precisely, stress-energy-tensor operator, and we discuss carefully its we shall develop the formalism permitting us to construct, geometrical ambiguities. In fact, we provide two alternative for a normalized Hadamard quantum state jψi of the but equivalent expressions for this renormalized expect- ψ ˆ ψ Stueckelberg theory, the quantity h jTμνj iren which ation value. The first one is obtained by removing the denotes the renormalized expectation value of the stress- contribution of the auxiliary scalar field Φ (here, it plays the energy-tensor operator. It is well known that such an role of a kind of ghost field) and only involves state- expectation value is of fundamental importance in quantum dependent and geometrical quantities associated with the field theory in curved spacetime (see, e.g., Refs. [23–27]). massive vector field Aμ. The other one involves contribu- Indeed, it permits us to analyze the quantum state jψi tions coming from both the massive vector field and the without any reference to its particle content, and, moreover, auxiliary Stueckelberg scalar field, and it has been con- it acts as a source in the semiclassical Einstein equations structed in such a way that, in the zero-mass limit, the ˆ massive vector field contribution reduces smoothly to the Gμν ¼ 8πhψjTμνjψi which govern the backreaction of ren ’ the quantum field theory on the spacetime geometry. result obtained from Maxwell s theory. In Sec. V,asan ˆ application of our results, we consider in the Minkowski Let us recall that the stress-energy tensor Tμν is an spacetime the Casimir effect outside of a perfectly con- operator quadratic in the quantum fields which is, from the ducting medium with a plane boundary wall separating it mathematical point of view, an operator-valued distribu- from free space. We discuss the results obtained using tion. As a consequence, this operator is ill defined, and the ˆ Stueckelberg but also de Broglie-Proca electromagnetism, associated expectation value hψjTμνjψi is formally infinite. and we consider the zero-mass limit of the vacuum energy In order to extract from this expectation value a finite in both theories. Finally, in a conclusion (Sec. VI), we and physically acceptable contribution which could act as provide a step-by-step guide for the reader wishing to use the source in the semiclassical Einstein equations, it is our formalism, we briefly discuss and compare the de necessary to regularize it and then to renormalize all the Broglie-Proca and Stueckelberg approaches in the light of coupling constants. For a description of the various the results obtained in our paper, and we highlight the techniques of regularization and renormalization in the advantages of the latter. In a short Appendix, we have context of quantum field theory in curved spacetime gathered some important results which are helpful to do the (adiabatic regularization method, dimensional regulariza- calculations of Secs. III and IV, and, in particular, (i) we tion method, ζ-function approach, point-splitting methods, define the geodetic interval σðx; x0Þ, the Van Vleck-Morette …), see Refs. [23–27] and references therein. determinant Δðx; x0Þ and the bivector of parallel transport 0 In this paper, we shall deal with Stueckelberg electro- gμν0 ðx; x Þ which play a crucial role along our article, and magnetism by using the so-called Hadamard renormaliza- (ii) we discuss the concept of covariant Taylor series tion procedure (for a rigorous axiomatic presentation of this expansions. approach, we refer to the monographs of Wald [26] and It should be noted that, in this paper, we consider a four- Fulling [25]). Here, we just recall that it is an extension of dimensional curved spacetime ðM;gμνÞ with no boundary

044063-3 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) Z  (∂M ¼ Ø), and we use units with ℏ ¼ c ¼ G ¼ 1 and ffiffiffiffiffiffi 1 1 4 p− − ∇μ ν∇ − μ ν the geometrical conventions of Hawking and Ellis [43] ¼ d x g 2 A μAν 2 RμνA A concerning the definitions of the scalar curvature R, the M    Ricci tensor Rμν and the Riemann tensor Rμνρσ as well as 1 2 μ 1 1 μ 2 − m A Aμ þ 1 − ð∇ AμÞ ð13bÞ the commutation of covariant derivatives. It is moreover 2 2 ξ important to note that we provide the covariant Taylor series expansions of the Hadamard coefficients in irreduc- and ible form by using the algebraic proprieties of the Riemann Z   tensor (and more particularly the cyclicity relation and its 4 pffiffiffiffiffiffi 1 μ 1 2 2 SΦ ¼ d x −g − ∇ Φ∇μΦ − ξm Φ ; ð14Þ consequences) as well as the Bianchi identity. M 2 2

II. QUANTIZATION OF STUECKELBERG SGh remaining unchanged and still given by Eq. (11).Itis μ ELECTROMAGNETISM worth noting that the term −mA ∇μΦ coupling the fields Φ In this section, we review the covariant quantization Aμ and in the classical action (6b) has disappeared; of Stueckelberg electromagnetism on an arbitrary four- because spacetime is assumed with no boundary, it is − Φ∇μ dimensional curved spacetime. The gauge-breaking term neutralized by the term m Aμ in the gauge-breaking considered includes an arbitrary gauge parameter ξ, and all action (10). the results concerning the wave equations for the massive The functional derivatives with respect to the fields Aμ, vector field Aμ, for the auxiliary scalar field Φ, for the ghost Φ, C and C of the quantum action (9) or (12) will allow us fields C and C and for all the associated Feynman to obtain, in Sec. II B, the wave equations for all the fields propagators as well as the Ward identities are expressed and to discuss, in Sec. IVA, the conservation of the stress- in terms of ξ. energy tensor associated with Stueckelberg electromagnet- ism. They are given by A. Quantum action 1 δS μν μ ν μν 2 μν At the quantum level, the action defining Stueckelberg pffiffiffiffiffiffi ¼½g □ − ð1 − 1=ξÞ∇ ∇ −R − m g Aν −g δAμ massive electromagnetism is given by (see, e.g., Ref. [14]) ð15Þ Φ Φ Φ S½Aμ; ;C;C ;gμν¼SCl½Aμ; ;gμνþSGB½Aμ; ;gμν for the vector field Aμ, þ SGh½C; C ;gμν; ð9Þ

1 δS 2 where we have added to the classical action (6) the gauge- pffiffiffiffiffiffi ¼½□ − ξm Φ ð16Þ breaking term −g δΦ Z   ffiffiffiffiffiffi 1 for the auxiliary scalar field Φ, as well as 4 p− − ∇μ ξ Φ 2 SGB ¼ d x g ð Aμ þ m Þ ð10Þ M 2ξ 1 δ S 2 pffiffiffiffiffiffi R ¼ −½□ − ξm C ð17Þ and the compensating ghost action −g δC Z ffiffiffiffiffiffi 4 p− ∇μ ∇ ξ 2 and SGh ¼ d x g½ C μC þ m C C: ð11Þ M 1 δ S 2 pffiffiffiffiffiffi L ¼ −½□ − ξm C ð18Þ By collecting the fields in the explicit expression (9), the −g δC quantum action can be written in the form for the ghost fields C and C. It should be noted that, due Φ Φ S½Aμ; ;C;C ;gμν¼SA½Aμ;gμνþSΦ½ ;gμν to the fermionic behavior of the ghost fields, we have introduced in Eq. (17) the right functional derivative and in þ SGh½C; C ;gμν; ð12Þ Eq. (18) the left functional derivative. where Z  B. Wave equations 4 pffiffiffiffiffiffi 1 μν SA ¼ d x −g − F Fμν The extremization of the quantum action (9) or (12) M 4 permits us to obtain the wave equations for the fields Aμ, 1 1 2 μ μ 2 Φ − m A Aμ − ð∇ AμÞ ð13aÞ , C and C . The vanishing of the functional derivatives 2 2ξ (15)–(18) provides

044063-4 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) μν□ − 1 − 1 ξ ∇μ∇ν − μν − 2 μν 0 1 ξ ∇μ A 0 ∇ Gh 0 ½g ð = Þ R m g Aν ¼ ð19Þ ð = Þ Gμν0 ðx; x Þþ ν0 G ðx; x Þ 2 −1 μ ρ A 0 ¼ð1 − 1=ξÞ½□ − ξm ½∇ fRμ G 0 ðx; x Þg: ð28Þ for the vector field Aμ, x ρν

½□ − ξm2Φ ¼ 0 ð20Þ It should be noted that the nonlocal term in the right-hand side of this equation is associated with the nonminimal 1 − 1 ξ ∇ν∇ for the auxiliary scalar field Φ, as well as term ð = Þ μ appearing in the wave equation (23) and includes appropriate boundary conditions. The second 2 2 Ward identity can be obtained directly from the wave ½□ − ξm C ¼ 0 and ½□ − ξm C ¼ 0 ð21Þ equations (25) and (27) by using arguments of uniqueness. We have for the ghost fields C and C. GΦðx; x0Þ − GGhðx; x0Þ¼0: ð29Þ C. Feynman propagators and Ward identities From now on, we shall assume that the Stueckelberg III. HADAMARD EXPANSIONS OF THE GREEN field theory previously described has been quantized and FUNCTIONS OF STUECKELBERG ψ is in a normalized quantum state j i. The Feynman ELECTROMAGNETISM propagator From now on, we assume that ξ ¼ 1.(Forξ ≠ 1, the A 0 ψ 0 ψ various Feynman propagators cannot be represented in the Gμν0 ðx; x Þ¼ih jTAμðxÞAν0 ðx Þj ið22Þ Hadamard form.) For this choice of gauge parameter, the wave equations (23), (25) and (27) for the Feynman associated with the field Aμ (here, T denotes time ordering) A 0 Φ 0 Gh 0 propagators G 0 ðx; x Þ, G ðx; x Þ and G ðx; x Þ reduce to is, by definition, a solution of μν

ν□ − ν − 2 ν A 0 − δ4 0 ν□ − 1 − 1 ξ ∇ν∇ − ν − 2 ν A 0 ½gμ x Rμ m gμ Gνρ0 ðx; x Þ¼ gμρ0 ðx; x Þ; ð30Þ ½gμ x ð = Þ μ Rμ m gμ Gνρ0 ðx; x Þ 4 0 ¼ −gμρ0 δ ðx; x Þð23Þ 2 Φ 0 4 0 ½□x − m G ðx; x Þ¼−δ ðx; x Þð31Þ 4 −1 2 4 with δ ðx; x0Þ¼½−gðxÞ = δ ðx − x0Þ. Similarly, the and Feynman propagator 2 Gh 0 4 0 ½□x − m G ðx; x Þ¼−δ ðx; x Þ: ð32Þ GΦðx; x0Þ¼ihψjTΦðxÞΦðx0Þjψið24Þ As far as the Ward identity (28) is concerned, it takes now associated with the scalar field Φ satisfies the local form

□ − ξ 2 Φ 0 −δ4 0 ∇μ A 0 ∇ Gh 0 0 ½ x m G ðx; x Þ¼ ðx; x Þ; ð25Þ Gμν0 ðx; x Þþ ν0 G ðx; x Þ¼ ; ð33Þ and the Feynman propagator while the Ward identity (29) remains unchanged. Because this last relation expresses the equality of the Feynman Gh G ðx; x0Þ¼ihψjTCðxÞCðx0Þjψið26Þ propagators associated with the auxiliary scalar field and the ghost fields, we shall often use a generic form for associated with the ghost fields C and C satisfies these propagators (and for their Hadamard representation discussed below) where the labels Φ and Gh are omitted. □ − ξ 2 Gh 0 −δ4 0 ½ x m G ðx; x Þ¼ ðx; x Þ: ð27Þ and we shall write

The three propagators are related by two Ward identities. Gðx; x0Þ¼GΦðx; x0Þ¼GGhðx; x0Þ: ð34Þ The first one is a nonlocal relation linking the propagators A 0 Gh 0 Gμν0 ðx; x Þ and G ðx; x Þ. It can be obtained by extending For ξ ¼ 1 the nonminimal term in the wave equation A 0 the approach of DeWitt and Brehme in Ref. [44] as follows: for Gμν0 ðx; x Þ has disappeared [compare Eq. (30) with μ we take the covariant derivative ∇ of Eq. (23) and the Eq. (23)]. As consequence, we can consider a Hadamard covariant derivative ∇ρ0 of Eq. (27); then, by commuting representation for this propagator as well as for the suitably the various covariant derivatives involved and propagators GΦðx; x0Þ and GGhðx; x0Þ. In other words, we μ 4 0 4 0 by using the relation ∇ ½gμρ0 δ ðx; x Þ ¼ −∇ρ0 δ ðx; x Þ, can assume that all fields of Stueckelberg theory are in a we obtain the formal relation normalized quantum state jψi of Hadamard type.

044063-5 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) A. Hadamard representation of the 2 1 2 A 2 1 A σ;a ðn þ Þðn þ ÞVnþ1 μν0 þ ðn þ ÞVnþ1 μν0;a Feynman propagators A −1=2 1=2 ;a − 2ðn þ 1ÞV 0 Δ ðΔ Þ; σ A 0 nþ1 μν a The Feynman propagator Gμν0 ðx; x Þ associated with the ρ ρ 2 ρ A þ½gμ □ − Rμ − m gμ V 0 ¼ 0 ð39aÞ vector field Aμ can be now represented in the Hadamard x n ρν form ∈ N  for n with the boundary condition Δ1=2 0 A 0 i ðx; x Þ 0 0 ; −1 2 1 2 ; Gμν0 ðx; x Þ¼ 2 0 gμν ðx; x Þ 2 A 2 A σ a − 2 A Δ = Δ = σ a 8π σðx; x Þþiϵ V0 μν0 þ V0 μν0; V0 μν0 ð Þ;a  a ρ□ − ρ − 2 ρ Δ1=2 0 A 0 σ 0 ϵ A 0 þ½gμ x Rμ m gμ ðgρν0 Þ¼ ; ð39bÞ þVμν0 ðx; x Þ ln½ ðx; x Þþi þWμν0 ðx; x Þ ;

A 0 ð35Þ while the coefficients Wn μν0 ðx; x Þ satisfy the recursion relations A 0 A 0 where the bivectors Vμν0 ðx; x Þ and Wμν0 ðx; x Þ are sym- ; A 0 A 0 2 n 1 n 2 WA 2 n 1 WA σ a metric in the sense that Vμν0 ðx; x Þ¼Vν0μðx ;xÞ and ð þ Þð þ Þ nþ1 μν0 þ ð þ Þ nþ1 μν0;a A 0 A 0 0 A −1=2 1=2 ;a W 0 ðx; x Þ¼W 0 ðx ;xÞ and are regular for x → x. − 2 1 Δ Δ σ μν ν μ ðn þ ÞWnþ1 μν0 ð Þ;a Furthermore, these bivectors have the following expansions 2 2 3 A 2 A σ;a þ ð n þ ÞVnþ1 μν0 þ Vnþ1 μν0;a Xþ∞ A −1=2 1=2 ;a − 2V 0 Δ ðΔ Þ; σ A 0 A 0 σn 0 nþ1 μν a Vμν0 ðx; x Þ¼ V μν0 ðx; x Þ ðx; x Þ; ð36aÞ n ρ ρ 2 ρ n 0 □ − − A 0 ¼ þ½gμ x Rμ m gμ Wn ρν0 ¼ ð40Þ

Xþ∞ A 0 A 0 σn 0 for n ∈ N. It should be noted that from the recursion Wμν0 ðx; x Þ¼ Wn μν0 ðx; x Þ ðx; x Þ: ð36bÞ n¼0 relations (39) and (40) we can show that

Similarly, the Hadamard expansion of the Feynman propa- ν□ − ν − 2 ν A 0 ½gμ x Rμ m gμ Vνρ0 ¼ ð41Þ gator Gðx; x0Þ associated with the auxiliary scalar field Φ or the ghost fields is given by and  i Δ1=2 x; x0 0 ð Þ 0 σ 0 ν ν 2 ν A Gðx; x Þ¼ þ Vðx; x Þ ln½ ðx; x Þ σ½gμ □ − Rμ − m gμ W 0 8π2 σðx; x0Þþiϵ x νρ  ν ν 2 ν 1=2 ¼ −½gμ □x − Rμ − m gμ ðgνρ0 Δ Þ iϵ W x; x0 ; þ þ ð Þ ð37Þ − 2 A − 2 A σ;a 2 A Δ−1=2 Δ1=2 σ;a Vμρ0 Vμρ0;a þ Vμρ0 ð Þ;a : ð42Þ

0 0 where the biscalars Vðx; x Þ and Wðx; x Þ are symmetric, These two “wave equations” permit us to prove that the 0 0 0 0 i.e., Vðx; x Þ¼Vðx ;xÞ and Wðx; x Þ¼Wðx ;xÞ, regular Feynman propagator (35) solves the wave equation (30). 0 for x → x and possess expansions of the form 0 Similarly, the Hadamard coefficients Vnðx; x Þ and W ðx; x0Þ are also symmetric and regular biscalar functions. Xþ∞ n 0 0 0 n 0 The coefficients Vnðx; x Þ satisfy the recursion relations Vðx; x Þ¼ Vnðx; x Þσ ðx; x Þ; ð38aÞ n¼0 2 1 2 2 1 σ;a ðn þ Þðn þ ÞVnþ1 þ ðn þ ÞVnþ1;a Xþ∞ −1=2 1=2 ;a 0 0 n 0 − 2 n 1 V 1Δ Δ σ Wðx; x Þ¼ Wnðx; x Þσ ðx; x Þ: ð38bÞ ð þ Þ nþ ð Þ;a n¼0 2 þ½□x − m Vn ¼ 0 ð43aÞ In Eqs. (35) and (37), the factor iϵ with ϵ → 0 ensures the þ ∈ N singular behavior prescribed by the time-ordered product for n with the boundary condition introduced in the definition of the Feynman propagators 2 2 σ;a − 2 Δ−1=2 Δ1=2 σ;a [see Eqs. (22), (24) and (26)]. V0 þ V0;a V0 ð Þ;a A 0 A 0 The Hadamard coefficients V 0 ðx; x Þ and W 0 ðx; x Þ 2 1=2 n μν n μν þ½□x − m Δ ¼ 0; ð43bÞ introduced in Eq. (36) are also symmetric and regular VA x; x0 0 bivector functions. The coefficients n μν0 ð Þ satisfy the while the coefficients Wnðx; x Þ satisfy the recursion recursion relations relations

044063-6 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) 2 1 2 2 1 σ;a ðn þ Þðn þ ÞWnþ1 þ ðn þ ÞWnþ1;a Here, it is important to note that, due to the geometrical 0 0 1=2 0 −1=2 1=2 ;a nature of σðx; x Þ, gμν0 ðx; x Þ, Δ ðx; x Þ (see the Appendix) − 2 n 1 W 1Δ Δ σ 2 2n 3 V 1 ð þ Þ nþ ð Þ;a þ ð þ Þ nþ A 0 0 and of Vμν0 ðx; x Þ and Vðx; x Þ (see Sec. III C), the singular 2 σ;a − 2 Δ−1=2 Δ1=2 σ;a þ Vnþ1;a Vnþ1 ð Þ;a parts (48a) and (50a) are purely geometrical objects. By 2 þ½□x − m Wn ¼ 0 ð44Þ contrast, the regular parts (48b) and (50b) are state dependent (see Sec. III D). for n ∈ N. It should be also noted that from the recursion relations (43) and (44) we can show that B. Hadamard Green functions □ − 2 0 ½ x m V ¼ ð45Þ In the context of the regularization of the stress-energy- and tensor operator, instead of working with the Feynman propagators, it is more convenient to use the associated 2 2 1=2 ;a so-called Hadamard Green functions. Their representations σ½□ − m W ¼ −½□ − m Δ − 2V − 2V; σ x x a can be derived from those of the Feynman propagators by 2 Δ−1=2 Δ1=2 σ;a þ V ð Þ;a : ð46Þ using the formal identities

These two “wave equations” permit us to prove that the 1 1 ¼ P − iπδðσÞð51Þ Feynman propagator (37) solves the wave equation (31) σ þ iϵ σ or (32). The Hadamard representation of the Feynman propa- and gators permits us to straightforwardly identify their singular σ ϵ σ πΘ −σ and regular parts (when the coincidence limit x0 → x is lnð þ i Þ¼ln j jþi ð Þ: ð52Þ considered). We can write Here, P is the symbol of the Cauchy principal value, A 0 A 0 A 0 and Θ denotes the Heaviside step function. Indeed, these G 0 ðx; x Þ¼G 0 ðx; x ÞþG 0 ðx; x Þð47Þ μν singμν regμν identities permit us to rewrite the expression (35) of the Feynman propagator associated with the massive vector with  field Aμ as Δ1=2 0 A 0 i ðx; x Þ 0 G μν0 ðx; x Þ¼ gμν0 ðx; x Þ i 1 sing 8π2 σðx; x0Þþiϵ A 0 ¯ A 0 ð ÞA 0  Gμν0 ðx; x Þ¼Gμν0 ðx; x Þþ2 Gμν0 ðx; x Þ; ð53Þ A 0 0 þV 0 ðx; x Þ ln½σðx; x Þþiϵ ð48aÞ μν where the average of the retarded and advanced Green functions is represented by and 1 ¯ A 0 Δ1=2 0 0 δ σ 0 Gμν0 ðx; x Þ¼8π f ðx; x Þgμν0 ðx; x Þ ½ ðx; x Þ A 0 i A 0 G 0 ðx; x Þ¼ W 0 ðx; x Þð48bÞ regμν 8π2 μν − A 0 Θ −σ 0 Vμν0 ðx; x Þ ½ ðx; x Þg ð54Þ as well as and the Hadamard Green function has the representation

0 0 0  Gðx; x Þ¼Gsingðx; x ÞþGregðx; x Þð49Þ 1 Δ1=2 0 ð1ÞA 0 ðx; x Þ 0 G 0 ðx; x Þ¼ gμν0 ðx; x Þ μν 4π2 σðx; x0Þ with   A 0 0 A 0 þV 0 ðx; x Þ ln jσðx; x Þj þ W 0 ðx; x Þ : i Δ1=2ðx; x0Þ μν μν G ðx; x0Þ¼ sing 8π2 σðx; x0Þþiϵ  ð55Þ 0 σ 0 ϵ þVðx; x Þ ln½ ðx; x Þþi ð50aÞ Similarly, we have for the Feynman propagator (37) associated with the auxiliary scalar field Φ or the ghost and fields

i 0 ¯ 0 i ð1Þ 0 G x; x0 W x; x0 : Gðx; x Þ¼Gðx; x Þþ G ðx; x Þ; ð56Þ regð Þ¼8π2 ð Þ ð50bÞ 2

044063-7 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) where straightforwardly identify their singular and purely geo- metrical parts as well as their regular and state-dependent 1 0 1 2 parts (when the coincidence limit x → x is considered). We G¯ ðx; x0Þ¼ fΔ = ðx; x0Þδ½σðx; x0Þ−Vðx; x0ÞΘ½−σðx; x0Þg 8π can write ð57Þ ð1ÞA 0 ð1ÞA 0 ð1ÞA 0 Gμν0 ðx; x Þ¼Gsing μν0 ðx; x ÞþGreg μν0 ðx; x Þð67Þ and  with 1 Δ1=2 0 1 ðx; x Þ  ð Þ 0 0 σ 0 1 2 G ðx; x Þ¼ 2 0 þ Vðx; x Þ ln j ðx; x Þj 1 Δ = 0 4π σðx; x Þ ð1ÞA 0 ðx; x Þ 0  G 0 ðx; x Þ¼ gμν0 ðx; x Þ sing μν 4π2 σðx; x0Þ  þ Wðx; x0Þ : ð58Þ A 0 σ 0 þVμν0 ðx; x Þ ln j ðx; x Þj ð68aÞ It is important to recall that the Hadamard Green function associated with the massive vector field Aμ is defined as the and anticommutator 1 ð1ÞA 0 A 0 G μν0 ðx; x Þ¼ Wμν0 ðx; x Þð68bÞ 1 reg 4π2 ð ÞA 0 ψ 0 ψ Gμν0 ðx; x Þ¼h jfAμðxÞ;Aν0 ðx Þgj ið59Þ as well as and satisfies the wave equation ð1Þ 0 ð1Þ 0 ð1Þ 0 G ðx; x Þ¼Gsingðx; x ÞþGregðx; x Þð69Þ ν□ − ν − 2 ν ð1ÞA 0 0 ½gμ x Rμ m gμ Gνρ0 ðx; x Þ¼ : ð60Þ with   Similarly, the Hadamard Green function associated with the 1=2 1 1 Δ ðx; x0Þ auxiliary scalar field Φ is defined as the anticommutator Gð Þ ðx; x0Þ¼ þVðx; x0Þ ln jσðx; x0Þj sing 4π2 σðx; x0Þ 1 Φ Gð Þ ðx; x0Þ¼hψjfΦðxÞ; Φðx0Þgjψið61Þ ð70aÞ which is a solution of and

2 ð1ÞΦ 0 ½□ − m G ðx; x Þ¼0; ð62Þ ð1Þ 1 x G x; x0 W x; x0 : regð Þ¼4π2 ð Þ ð70bÞ while the Hadamard Green function associated with the ghost fields is defined as the commutator It should be pointed out that the regular part of the Hadamard Green function given by Eq. (68b) [respectively, Gð1ÞGhðx; x0Þ¼hψj½CðxÞ;Cðx0Þjψið63Þ by Eq. (70b)] is proportional to that of the Feynman propagator given by Eq. (48b) [respectively, by Eq. (50b)]. and satisfies the wave equation C. Geometrical Hadamard coefficients and associated 2 ð1ÞGh 0 ½□x − m G ðx; x Þ¼0: ð64Þ covariant Taylor series expansions A 0 Formally, the Hadamard coefficients Vn μν0 ðx; x Þ or The Ward identities (33) and (29) satisfied by the 0 Feynman propagators are also valid for the Hadamard Vnðx; x Þ can be determined uniquely by solving the Green functions. We have recursion relations (39) or (43), i.e., by integrating these recursion relations along the unique geodesic joining x to x0 1 (it is unique for x0 near x or more generally for x0 in a ∇μ ð ÞA 0 ∇ ð1ÞGh 0 0 Gμν0 ðx; x Þþ ν0 G ðx; x Þ¼ ð65Þ convex normal neighborhood of x). As a consequence, all these coefficients as well as the sums given by Eqs. (36a) and and (38a) are of purely geometric nature; i.e., they only 1 Φ 1 x x0 Gð Þ ðx; x0Þ − Gð ÞGhðx; x0Þ¼0: ð66Þ depend on the geometry along the geodesic joining to . From the point of view of the practical applications Similarly, as it has been previously noted in the case of considered in this work, it is sufficient to know the the Feynman propagators, the Hadamard representaion expressions of the two first geometrical Hadamard coef- of the Hadamard Green functions permits us to ficients. Furthermore, their covariant Taylor series

044063-8 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) 1 0 2 expansions are needed up to order σ for n ¼ 0 and σ for v0 ¼ð1=2Þm − ð1=12ÞR; ð74aÞ n ¼ 1. The covariant Taylor series expansions of the bivector coefficients VA ðx; x0Þ and VA ðx; x0Þ are given 0 μν 1 μν 1 12 2 − 1 40 − 1 120 □ by [see Eqs. (A7) and (A8)] v0ab ¼ð = Þm Rab ð = ÞR;ab ð = Þ Rab p − ð1=72ÞRRab þð1=90ÞRapRb A ν0 A V0 μν ¼ gν V0 μν0 − 1 180 p q − 1 180 pqr ð = ÞRpqRa b ð = ÞRapqrRb ; ð74bÞ A − 1 2 A A σ;a ¼ v0 ðμνÞ fð = Þv0 ðμνÞ;a þ v0 ½μνag 1 1 8 4 − 1 24 2 1 120 □ vA vA σ;aσ;b O σ3=2 v1 ¼ð = Þm ð = Þm R þð = Þ R þ 2! f 0 ðμνÞab þ 0 ½μνa;bg þ ð Þ 2 pq þð1=288ÞR − ð1=720ÞRpqR ð71aÞ pqrs þð1=720ÞRpqrsR : ð74cÞ and In order to obtain the expressions of the Taylor coefficients A ν0 A A σ1=2 V1 μν ¼ gν V1 μν0 ¼ v1 ðμνÞ þ Oð Þ: ð71bÞ given by Eqs. (72) and (74), we have used some of the 0 0 1=2 0 properties of σðx; x Þ, gμν0 ðx; x Þ, Δ ðx; x Þ mentioned in Here, the explicit expressions of the Taylor coefficients can the Appendix as well as the algebraic properties of the be determined from the recursion relations (39).Wehave Riemann tensor.

A 2 v ¼ð1=2ÞRμν þ gμνfð1=2Þm − ð1=12ÞRg; ð72aÞ 0 ðμνÞ D. State-dependent Hadamard coefficients and associated covariant Taylor series expansions A 1 6 v0 ½μνa ¼ð = ÞRa½μ;ν; ð72bÞ 1. General considerations A 1 6 1 12 v0 ðμνÞab ¼ð = ÞRμν;ðabÞ þð = ÞRμνRab Unlike the geometrical Hadamard coefficients, the coef- A 0 0 pq 2 ficients W μν0 ðx; x Þ and Wnðx; x Þ are neither uniquely þð1=12ÞRμ Rν þ gμνfð1=12Þm R n pqðaj jbÞ ab defined nor purely geometrical. Indeed, the coefficient − ð1=40ÞR; − ð1=120Þ□R A 0 0 ab ab W0 μν0 ðx; x Þ [respectively, W0ðx; x Þ] is unrestrained by p −ð1=72ÞRRab þð1=90ÞRapRb the recursion relations (42) [respectively, by the recursion − 1 180 p q − 1 180 pqr relations (46)]. As a consequence, this is also true for all the ð = ÞRpqRa b ð = ÞRapqrRb g; A 0 0 ≥ 1 coefficients Wn μν0 ðx; x Þ and Wnðx; x Þ with n and for ð72cÞ the sums (36b) and (38b). This arbitrariness is in fact very interesting, and it can be used to encode the quantum state A 2 v 1=4 m Rμν − 1=24 □Rμν − 1=24 RRμν A 0 1 ðμνÞ ¼ð Þ ð Þ ð Þ dependence of the theory in the coefficients W0 μν0 ðx; x Þ 0 p pqr and W0 x; x . Once they have been specified, the coef- þð1=8ÞRμpRν − ð1=48ÞRμpqrRν ð Þ A 0 0 ≥ 1 4 2 ficients Wn μν0 ðx; x Þ and Wnðx; x Þ with n as well as the μν 1 8 − 1 24 1 120 □ þ g fð = Þm ð = Þm R þð = Þ R A 0 0 bivector Wμν0 ðx; x Þ and the biscalar Wðx; x Þ are uniquely 1 288 2 − 1 720 pq þð = ÞR ð = ÞRpqR determined. pqrs þð1=720ÞRpqrsR g: ð72dÞ In the following, instead of working with the state- dependent Hadamard coefficients, we shall consider the A 0 0 The covariant Taylor series expansions of the biscalar sums Wμν0 ðx; x Þ and Wðx; x Þ, and, more precisely, we 0 0 coefficients V0ðx; x Þ and V1ðx; x Þ are given by [see shall use their covariant Taylor series expansions up to Eqs. (A5) and (A6)] order σ3=2. We have [see Eqs. (A7) and (A8)] 1 − 1 2 σ;a σ;aσ;b σ3=2 V0 ¼ v0 fð = Þv0;ag þ v0ab þ Oð Þ A ν0 A 2! Wμν ¼ gν Wμν0 ð73aÞ ;a ¼ sμν − fð1=2Þsμν;a þ aμνagσ 1 1 and σ;aσ;b − 3 2 þ 2! fsμνab þ aμνa;bg 3! fð = Þsμνab;c 1=2 V1 ¼ v1 þ Oðσ Þ: ð73bÞ ;a ;b ;c 2 −ð1=4Þsμν;abc þ aμνabcgσ σ σ þ Oðσ Þð75Þ Here, the explicit expressions of the Taylor coefficients can be determined from the recursion relations (43).Wehave and [see Eqs. (A5) and (A6)]

044063-9 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) 1 ρ0 ν□ − ν − 2 ν A − 1 2 σ;a σ;aσ;b gρ ½gμ x Rμ m gμ Wνρ0 W ¼ w fð = Þw;ag þ 2! wab −6 A − 2 ρ0 A σ;a σ 1 ¼ V1 μρ gρ V1 μρ0;a þ Oð Þ ;a ;b ;c 2 − 3=2 w ; − 1=4 w; σ σ σ O σ : 3! fð Þ ab c ð Þ abcg þ ð Þ −6 A 2 A 8 A σ;a σ ¼ v1 ðμρÞ þð v1 ðμρÞ;a þ v1 ½μρaÞ þ Oð Þ: ð80Þ ð76Þ Here, we have used the expansions of the geometrical In the expansion (75) we have introduced the notations Hadamard coefficients given by Eqs. (36a) and (71).By A 0 inserting the expansion (75) of Wμνðx; x Þ into the left-hand A sμν ≡ w ð77aÞ side of Eq. (80), we find the following relations: a1ap ðμνÞa1ap ν p 2 A sμρν ¼ R μsρ þ m sμρ − 6v ; ð81aÞ for the symmetric part of the Taylor coefficients and ð Þp 1 ðμρÞ

μ ν pq 2 p A p A sμ ν ¼ R s þ m s − 6v1 ; ð81bÞ aμν ≡ w ð77bÞ pq p p a1ap ½μνa1ap ;ν p aμρν ¼ −R μsρ ; ð81cÞ for their antisymmetric part. ½ p It is important to note that, with practical applications in ;ν 1 4 □ 1 2 p mind, it is interesting to express some of the Taylor sμρνa ¼ð = Þð sμρÞ;a þð = ÞR a;ðμsρÞp − 1 2 ;p − 1 2 p coefficients appearing in Eqs. (75) and (76) in terms of ð = ÞRðμja sjρÞp ð = ÞR ðμsρÞp;a A 0 0 the bitensors Wμν0 ðx; x Þ and Wðx; x Þ. This can be done by p p q þð1=2ÞR sμρ; − R μ s ρ ; inverting these equations. From Eq. (75), we obtain a p ð j a j Þp q − p − p q 1 2 p R ðμjapjρÞa R ðμj aajρÞpq þð = Þsμρp ;a A 0 2 A sμνðxÞ¼limWμν0 ðx; x Þ; ð78aÞ − ð1=2Þm sμρ; þ 2v ; ð81dÞ x0→x a 1 ðμρÞ;a

μ ;ν p q p 1 sμ ν ¼ð1=4Þð□sp Þ; þð1=2ÞR asp ; A 0 − A 0 a a q aμνaðxÞ¼ lim½Wμν0; 0 ðx; x Þ Wμν0; ðx; x Þ; ð78bÞ 2 0→ a a pq pqr p q x x − ð1=2ÞR s ; þ R a þð1=2Þs pq a a pqr p q ;a 2 p A p 1 − ð1=2Þm s þ 2v1 : ð81eÞ A 0 A 0 p ;a p ;a sμνabðxÞ¼ lim½Wμν0; a0b0 ðx; x ÞþWμν0; ab ðx; x Þ: ð78cÞ 2 x0→x ð Þ ð Þ Furthermore, by combining Eq. (81d) with Eq. (81a) and

(Here, the coefficient aμνabc is not relevant because it does Eq. (81e) with Eq. (81b), we also establish that not appear in the final expressions of the renormalized ;ν 1 4 □ 1 2 p stress-energy-tensor operator given in Sec. IV C). Similarly, sμρνa ¼ð = Þð sμρÞ;a þð = ÞR a;ðμsρÞp from Eq. (76), we straightforwardly establish that − 1 2 ;p 1 2 p ð = ÞRðμja sjρÞp þð = ÞR ðμj;asjρÞp p p q 1=2 R sμρ; − R μ s ρ ; wðxÞ¼limWðx; x0Þ; ð79aÞ þð Þ a p ð j a j Þp q 0→ x x − p − p q − A R ðμjapjρÞa R ðμj aajρÞpq v1 ðμρÞ;a ð81fÞ 0 wabðxÞ¼limW;ða0b0Þðx; x Þ: ð79bÞ x0→x and

μ ;ν 1 4 □ p 1 2 q p We shall now rewrite the wave equations (60), (62) and (64) sμ νa ¼ð = Þð sp Þ;a þð = ÞR asp ;q as well as the Ward identity (65) in terms of the Taylor 1 2 pq pqr − A p A 0 0 þð = ÞR ;aspq þ R aapqr v1 p ; : ð81gÞ coefficients of Wμνðx; x Þ and Wðx; x Þ. To achieve the a calculations, we shall use extensively the properties of 0 0 1=2 0 Mutatis mutandis, by inserting the Hadamard represen- σðx; x Þ, gμν0 ðx; x Þ, Δ ðx; x Þ mentioned in the Appendix. tation (58) of the Green function Gð1Þðx; x0Þ into the wave equation (62) or (64), we obtain a wave equation with 2. Wave equations source for the state-dependent Hadamard coefficient 0 By inserting the Hadamard representation (55) of the Wðx; x Þ.Wehave ð1ÞA 0 Green function Gμν0 ðx; x Þ into the wave equation (60),we 2 ;a ð□x − m ÞW ¼ −6V1 − 2V1;aσ þ OðσÞ obtain a wave equation with source for the state-dependent ;a A 0 ¼ −6v1 þ 2v1;aσ þ OðσÞ: ð82Þ Hadamard coefficient Wμν0 ðx; x Þ. We have

044063-10 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) Here, we have used the expansions of the geometrical the Hadamard coefficients associated with the auxiliary Hadamard coefficients given by Eqs. (38a) and (73).By scalar field and the ghost fields. We have inserting the expansion (76) of Wðx; x0Þ into the left-hand side of Eq. (82), we find the following relations: VΦ ¼ VGh ð87Þ ρ 2 − 6 wρ ¼ m w v1; ð83aÞ and

;ρ p wρ 1=4 □w 1=2 w Φ Gh a ¼ð Þð Þ;a þð Þ p ;a W ¼ W : ð88Þ p 2 þð1=2ÞR aw;p − ð1=2Þm w;a þ 2v1;a: ð83bÞ

Furthermore, by combining Eq. (83b) with Eq. (83a),we IV. RENORMALIZED STRESS-ENERGY also establish that TENSOR OF STUECKELBERG ELECTROMAGNETISM ;ρ 1 4 □ 1 2 p − wρa ¼ð = Þð wÞ;a þð = ÞR aw;p v1;a: ð83cÞ A. Stress-energy tensor The functional derivation of the quantum action of the 3. Ward identities Stueckelberg theory with respect to the metric tensor gμν The first Ward identity given by Eq. (65) expressed in permits us to obtain the associated stress-energy tensor Tμν. terms of the Hadamard representation of the Green func- By definition, we have ð1ÞA 0 ð1Þ 0 tions G 0 ðx; x Þ [see Eq. (55)] and G ðx; x Þ [see μν 2 δ μν Eq. (58)] permits us to write a relation between the T ¼ pffiffiffiffiffiffi S½Aμ; Φ;C;C ;gμν; ð89Þ A 0 −g δgμν geometrical Hadamard coefficients Vμν0 ðx; x Þ and Vðx; x0Þ as well as another one between the state-dependent A 0 0 and its explicit expression can be obtain by using that, in Hadamard coefficients W 0 ðx; x Þ and Wðx; x Þ. We obtain μν the elementary variation ν0 A ;μ 0 gν ðVμν0 þ V;ν0 Þ¼ ð84Þ gμν → gμν þ δgμν ð90Þ which is an identity between the geometrical Taylor of the metric tensor, we have (see, for example, Ref. [45]) coefficients (72a)–(72d) and (74a)–(74c) and μν μν μν 0 → δ ν A ;μ − A σ;μ σ σ g g þ g ; ð91aÞ gν ðWμν0 þ W;ν0 Þ¼ V1 μν þ V1 ;ν þ Oð Þ A ;a ffiffiffiffiffiffi ffiffiffiffiffiffi ffiffiffiffiffiffi − v − v1gν σ O σ : p p p ¼ ð 1 ðνaÞ aÞ þ ð Þ ð85Þ −g → −g þ δ −g; ð91bÞ To establish Eq. (85), we have used the expansions of the Γρ → Γρ δΓρ geometrical Hadamard coefficients given by Eqs. (36a), μν μν þ μν ð91cÞ (38a), (71) and (73). By inserting the expansions (75) of A 0 0 Wμνðx; x Þ and (76) of Wðx; x Þ into the left-hand side of with

Eq. (85), we find the following relations: μν μρ νσ δg ¼ −g g δgρσ; ð91dÞ μ ;p aμν ¼ð1=2Þspν þð1=2Þw;ν; ð86aÞ ffiffiffiffiffiffi 1 ffiffiffiffiffiffi δp− p− μνδ μ 1 2 ;p 1 2 p p g ¼ 2 gg gμν; ð91eÞ sμν a ¼ð = Þspν a þð = ÞR asνp þ a ν½a;p − A þ wνa v1 ðνaÞ þ v1gνa: ð86bÞ 1 δΓρ −δ ;ρ δ ρ δ ρ μν ¼ 2 ð gμν þ g μ;ν þ g ν;μÞ: ð91fÞ Furthermore, by combining Eq. (86b) with Eq. (86a),we also establish that The stress-energy tensor derived from the action (9) is given by μ 1 4 ;p 1 2 p 1 2 ;p sμν a ¼ð = Þspν a þð = ÞR asνp þð = Þapνa − 1 4 − A μν μν μν μν ð = Þw;νa þ wνa v1 ðνaÞ þ v1gνa: ð86cÞ T ¼ Tcl þ TGB þ TGh; ð92Þ

Of course, the second Ward identity given by Eq. (66) where the contributions of the classical and gauge-breaking provides trivially the equality of the Taylor coefficients of parts take the forms

044063-11 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) μν μ νρ 2 μ ν Tcl ¼ F ρF þ m A A while the contribution associated with the ghost fields remains unchanged [see Eq. (93c)]. ∇μΦ∇νΦ 2mAðμ∇νÞΦ þ þ By construction, the stress-energy tensor (89) [see also μν ρτ 2 ρ − ð1=4Þg fFρτF þ 2m AρA its explicit expressions (92) and (94)] is conserved, i.e., ρ ρ þ2∇ρΦ∇ Φ þ 4mAρ∇ Φg μν ∇νT ¼ 0: ð96Þ μ ρ ν ðμ νÞ ρ μ ν ρ ¼ ∇ρA ∇ A − 2∇ρA ∇ A þ ∇ Aρ∇ A þ m2AμAν þ ∇μΦ∇νΦ þ 2mAðμ∇νÞΦ Indeed, this property is due to the invariance of the action (9) or (12) under spacetime diffeomorphisms and therefore μν ρ τ τ ρ − ð1=2Þg f∇ρAτ∇ A − ∇ρAτ∇ A under the infinitesimal coordinate transformation 2 ρ ρ ρ þm AρA þ ∇ρΦ∇ Φ þ 2mAρ∇ Φgð93aÞ xμ → xμ þ ϵμ with jϵμj ≪ 1: ð97Þ and Under this transformation, the vector, scalar and ghost μν −2 ðμ∇νÞ∇ ρ − 2 ðμ∇νÞΦ fields as well as the background metric transform as TGB ¼ A ρA mA μν ρ τ ρ 2 − ð1=2Þg f−2Aρ∇ ∇τA − ð∇ρA Þ Aμ → Aμ þ δAμ; ð98aÞ 2 2 ρ þm Φ − 2mAρ∇ Φg; ð93bÞ Φ → Φ þ δΦ; ð98bÞ while the contribution associated with the ghost fields is given by C → C þ δC; ð98cÞ μν −2∇ðμj ∇jνÞ μν ∇ ∇ρ 2 TGh ¼ C Cþg f ρC Cþm C Cg: ð93cÞ C → C þ δC; ð98dÞ We can note the existence of terms coupling the fields Aμ μν and Φ in the expression of T [see Eq. (93a)] as well as in → δ μν cl gμν gμν þ gμν: ð98eÞ the expression of TGB [see Eq. (93b)]. We also give an alternative expression for the stress- The variations associated with the field transformations energy tensor which can be derived from the action (12) or μν μν (98) are obtained by Lie derivation with respect to the μ by summing Tcl and TGB. This eliminates any coupling vector −ϵ : between the fields Aμ and Φ and permits us to straight- ρ ρ forwardly infer that the stress-energy tensor has three δAμ ¼ L−ϵAμ ¼ −ϵ ∇ρAμ − ð∇μϵ ÞAρ; ð99aÞ independent contributions corresponding to the massive vector field Aμ, the auxiliary scalar field Φ and the ghost ρ δΦ ¼ L−ϵΦ ¼ −ϵ ∇ρΦ; ð99bÞ fields C and C. We can write

μν μν μν μν ρ δC L−ϵC −ϵ ∇ρC; T ¼ TA þ TΦ þ TGh ð94Þ ¼ ¼ ð99cÞ

ρ with δC ¼ L−ϵC ¼ −ϵ ∇ρC ; ð99dÞ

μν μ νρ 2 μ ν − 2 ðμ∇νÞ∇ ρ TA ¼ F ρF þ m A A A ρA δgμν ¼ L−ϵgμν ¼ −∇μϵν − ∇νϵμ: ð99eÞ μν ρτ 2 ρ − ð1=4Þg fFρτF þ 2m AρA ρ τ ρ 2 −4Aρ∇ ∇τA − 2ð∇ρA Þ g The invariance of the action (9) or (12) leads to ∇ μ∇ρ ν − 2∇ ðμ∇νÞ ρ ∇μ ∇ν ρ Z     ¼ ρA A ρA A þ Aρ A ffiffiffiffiffiffi 1 δS 1 δS 4 p− ffiffiffiffiffiffi δ ffiffiffiffiffiffi δΦ 2 μ ν ðμ νÞ ρ d x g p Aμ þ p þ m A A − 2A ∇ ∇ρA M −g δAμ −g δΦ     μν ρ τ τ ρ − ð1=2Þg f∇ρAτ∇ A − ∇ρAτ∇ A 1 δ S 1 δ S þ pffiffiffiffiffiffi R δC þ δC pffiffiffiffiffiffi L 2 ρ − 2 ∇ρ∇ τ − ∇ ρ 2 −g δC −g δC þm AρA Aρ τA ð ρA Þ gð95aÞ    1 2 δS þ pffiffiffiffiffiffi δgμν ¼ 0 ð100Þ and 2 −g δgμν μν ∇μΦ∇νΦ − 1 2 μν ∇ Φ∇ρΦ 2Φ2 TΦ ¼ ð = Þg f ρ þ m g; ð95bÞ which implies

044063-12 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016)

μν μ μ μ ν ν ρ 2 ν ρσ0 ∇ ∇ − ∇ − ∇ □ − − clA α0 ρσ0 ρ σ0 αβ0 νT ¼½ Aα αA A νð A R ρA m A Þ T μν ¼ gν g ∇μ∇α0 þ gμ gν g ∇α∇β0 ∇μΦ □Φ − 2Φ − □ − 2 ∇μ ρ α0 βσ0 2 ρ σ0 þ½ ð m Þ ð C m C Þ½ C − 2gμ gν g ∇β∇α0 þ m gμ gν − ∇μ □ − 2 1 ½ C ð C m CÞð101Þ − ρσ0 αβ0 ∇ ∇ − ρα0 βσ0 ∇ ∇ 2 gμνfg g α β0 g g β α0 2 ρσ0 by using Eq. (99) as well as Eqs. (15)–(18). From the wave þm g g; ð104aÞ equations associated with the massive vector field Aμ [see 1 Φ clΦ ν0 αβ0 Eq. (19)], the auxiliary scalar field [see Eq. (20)] and the T μν ¼ gν ∇μ∇ν0 − gμνfg ∇α∇β0 g; ð104bÞ ghost fields C and C [see Eq. (21)], we then obtain 2

Eq. (96). 0 GBA ρσ ρ α0 σ0 T μν ¼ −2gμ gν ∇α0 ∇ 1 B. Expectation value of the stress-energy tensor − −∇ρ∇σ0 − 2 ρα0 ∇ ∇σ0 2 gμνf g α0 g; ð104cÞ At the quantum level, all the fields involved in the Stueckelberg theory as well as the associated stress-energy 1 T GBΦ − 2 tensor [see Eqs. (92) and (94)] are operators. From now on, μν ¼ 2 m gμν ð104dÞ ˆ we shall denote the stress-energy-tensor operator by Tμν ˆ and we shall focus on the quantity hψjTμνjψi which denotes and its expectation value with respect to the Hadamard quantum T Gh −2 ν0 ∇ ∇ αβ0 ∇ ∇ 2 state jψi discussed in Sec. III. μν ¼ gν μ ν0 þ gμνfg α β0 þ m g: ð104eÞ ˆ The expectation value hψjTμνjψi corresponding to the expression (92) of the stress-energy tensor is decomposed It should be noted that we have not included in Eqs. (103a) as follows: and (103b) the contributions which can be obtained by point splitting from the terms coupling Aμ and Φ in Eqs. (93a) and (93b). Such contributions are not present ˆ ˆ cl ˆ GB ˆ Gh hψjTμνjψi¼hψjTμνjψiþhψjTμν jψiþhψjTμν jψi: because, due to the absence of coupling between Aμ and Φ ð102Þ in the quantum action (12), two-point correlation functions involving both Aμ and Φ vanish identically. It should be noted that the absence of these contributions can be also The three terms in the right-hand side of this equation are justified in another way: in the quantum stress-energy- explicitly given by tensor operator (94), any coupling between Aμ and Φ has disappeared. Here, some remarks are in order: 1 ρσ0 1 ψ ˆ cl ψ T clA 0 ð ÞA 0 h jTμνðxÞj i¼ lim μν ðx; x Þ½Gρσ0 ðx; x Þ (i) When we use the point-splitting method, it is more 2 0→ x x ψ ˆ ψ 1 convenient to define the expectation value h jTμνj i clΦ 1 Φ þ limT μν ðx; x0Þ½Gð Þ ðx; x0Þ; ð103aÞ from Hadamard Green functions rather than from 2 0→ x x Feynman propagators. Indeed, this avoids us having to deal with additional singular terms due to the time-ordered product. 1 ρσ0 1 ψ ˆ GB ψ T GBA 0 ð ÞA 0 (ii) Of course, because of the short-distance behavior of h jTμν ðxÞj i¼ lim μν ðx; x Þ½Gρσ0 ðx; x Þ 2 x0→x the Hadamard Green functions, the expressions 1 ˆ GBΦ 1 Φ (103) as well as the expectation value hψjTμνjψi þ limT μν ðx; x0Þ½Gð Þ ðx; x0Þ ð103bÞ 2 x0→x given in Eq. (102) are divergent and therefore meaningless. In Sec. IV C we will regularize these quantities. and (iii) Even if the formal expression (102) of the expect- ation value of the stress-energy-tensor operator is 1 divergent, it is interesting to note that ˆ Gh Gh 0 ð1ÞGh 0 hψjTμν ðxÞjψi¼ limT μν ðx; x Þ½G ðx; x Þ; ð103cÞ 2 x0→x ˆ GB ˆ Gh hψjTμν jψiþhψjTμν jψi¼0: ð105Þ

ρσ0 ρσ0 clA clΦ GBA GBΦ Gh where T μν , T μν , T μν , T μν and T μν are the Indeed, from the definitions (103b) and (103c),we differential operators constructed by point splitting from can obtain Eq. (105) by using Eqs. (65) and (66) as the expressions (93a), (93b) and (93c).Wehave well as the wave equation (64). It should be noted

044063-13 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) that, as a consequence of Eq. (105), Eq. (102) C. Renormalized stress-energy tensor reduces to 1. Definition and conservation ˆ ˆ cl hψjTμνjψi¼hψjTμνjψi: ð106Þ As we have already noted, the expectation value ˆ hψjTμνjψi given by Eq. (102) is divergent due to the short-distance behavior of the Green functions or, more We can also give the alternative expression of the precisely, to the singular purely geometrical part of the ˆ expectation value hψjTμνjψi obtained from Eq. (94).It Hadamard functions given in Eqs. (68a) and (70a) (see the takes the following form, terms in 1=σ and ln jσj). It is possible to construct the renormalized expectation value of the stress-energy- ˆ ˆ A ˆ Φ ˆ Gh hψjTμνjψi¼hψjTμνjψiþhψjTμνjψiþhψjTμν jψi; tensor operator with respect to the Hadamard quantum state jψi by using the prescription proposed by Wald in ð107Þ Refs. [26,30,31]. In Eqs. (103a)–(103c) we first discard the singular contributions; i.e., we make the replacements where the contributions associated with the massive vector field Aμ and the auxiliary scalar field Φ are separated and 1 ð1ÞA 0 ð1ÞA 0 A 0 G 0 x; x → G 0 x; x W 0 x; x ; 111a given by μν ð Þ reg μν ð Þ¼4π2 μν ð Þ ð Þ

1 ρσ0 1 ψ ˆ A ψ T A 0 ð ÞA 0 1 Φ 1 h jTμνðxÞj i¼ lim μν ðx; x Þ½Gρσ0 ðx; x Þ ð108aÞ ð1ÞΦ 0 ð Þ 0 Φ 0 0 G x; x → G x; x W x; x ; 111b 2 x →x ð Þ reg ð Þ¼4π2 ð Þ ð Þ and 1 ð1ÞGh 0 → ð1ÞGh 0 Gh 0 G ðx; x Þ Greg ðx; x Þ¼4π2 W ðx; x Þ; ð111cÞ 1 ˆ Φ Φ 0 ð1ÞΦ 0 hψjTμνðxÞjψi¼ limT μνðx; x Þ½G ðx; x Þ: ð108bÞ 2 0→ ~ x x and we add to the result a state-independent tensor Θμν

A ρσ0 Φ which only depends on the mass parameter and on the local Here, the differential operators T μν and T μν are con- geometry and which ensures the conservation of the final structed by point splitting from the expressions (95a) and expression. In other words, we consider that the renormal- (95b).Wehave ized expectation value of the stress-energy-tensor operator is given by T A ρσ0 α0 ρσ0 ∇ ∇ ρ σ0 αβ0 ∇ ∇ μν ¼ gν g μ α0 þ gμ gν g α β0 1 ˆ clA A clΦ Φ ρ α0 βσ0 2 ρ σ0 hψjTμνjψi ¼ fT μν ½W þT μν ½W g − 2gμ gν g ∇β∇α0 þ m gμ gν ren 8π2 − 2 ρ α0 ∇ ∇σ0 1 gμ gν α0 T GBA A T GBΦ Φ þ 2 f μν ½W þ μν ½W g 1 8π ρσ0 αβ0 ρα0 βσ0 − gμνfg g ∇α∇β0 − g g ∇β∇α0 1 2 T Gh Gh Θ~ þ 2 μν ½W þ μν ð112Þ 2 ρσ0 ρ σ0 ρα0 σ0 8π þm g − ∇ ∇ − 2g ∇α0 ∇ gð109aÞ with and ρσ0 T clA A T clA 0 A 0 μν ½W ðxÞ¼lim μν ðx; x Þ½Wρσ0 ðx; x Þ; ð113aÞ 1 x0→x T Φ ν0 ∇ ∇ − αβ0 ∇ ∇ 2 μν ¼ gν μ ν0 2 gμνfg α β0 þ m g: ð109bÞ clΦ Φ clΦ Φ T μν ½W ðxÞ¼limT μν ðx; x0Þ½W ðx; x0Þ; ð113bÞ 0→ It should be noted that the expressions (102) and (107) of x x ˆ the expectation value ψ Tμν ψ are identical because the ρσ0 h j j i GBA A GBA 0 A 0 0 0 T μν T μν cl ρσ clΦ GB ρσ ½W ðxÞ¼lim ðx; x Þ½Wρσ0 ðx; x Þ; ð113cÞ various differential operators T μνA , T μν , T μν A and x0→x 0 GBΦ A ρσ Φ T μν appearing in (103) and T μν and T μν appearing in GBΦ Φ GBΦ Φ T μν ½W ðxÞ¼limT μν ðx; x0Þ½W ðx; x0Þ ð113dÞ (108) are related by x0→x

A ρσ0 cl ρσ0 GB ρσ0 T μν ¼ T μνA þ T μν A ; ð110aÞ and

Gh Gh Gh 0 Gh 0 Φ clΦ GBΦ T μν ½W ðxÞ¼limT μν ðx; x Þ½W ðx; x Þ: ð113eÞ T μν ¼ T μν þ T μν : ð110bÞ x0→x

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0 0 cl ρσ clΦ GB ρσ Here, the differential operators T μνA , T μν , T μν A , It is therefore suitable to redefine the purely geometrical GBΦ Gh Θ~ T μν and T μν are given by Eqs. (104a)–(104e).In tensor μν by Eqs. (113a)–(113e), the coincidence limits x0 → x are obtained from the covariant Taylor series expansions 1 (75) and (76) by using extensively some of the results ~ A A ρ Θμν → Θμν − f6v1 μν − 2gμνv1 ρ þ 2gμνv1g; ð117Þ displayed in the Appendix. The final expressions can be 8π2 simplified by using the relations (81a), (81b) and (83a) we have previously obtained from the wave equations. We have where the new local tensor Θμν is assumed to be conserved, i.e., T clA A 1 2 ρ 1 2 □ − ρ μν ½W ¼ð = Þsρ ;μν þð = Þ sμν sρðμ;νÞ − ρ − ρ − ρ 2 ρ ;ν aμ ½ν;ρ aν ½μ;ρ sρ μν þ sρðμνÞ Θμν ¼ 0: ð118Þ ρ ;ρτ − ð1=2Þgμνfð1=2Þ□sρ − ð1=2Þsρτ − 1 2 ρτ − ρ;τ ρτ ð = ÞR sρτ aρτ þ sρτ g As a consequence, the renormalized expectation value of A A ρ þ 6v1 μν − 3gμνv1 ρ ; ð114aÞ the stress-energy-tensor operator takes the following form,

T clΦ Φ 1 2 Φ − Φ μν ½W ¼ð = Þw;μν wμν 1 ˆ clA A GBA A Φ 2 Φ hψjTμνjψi ¼ fT μν ½W þT μν ½W − ð1=2Þgμνfð1=2Þ□w − m w g − 3gμνv1; ren 8π2 A A ρ ð114bÞ −6v1 μν þ 2gμνv1 ρ g 1 GB ρ ρ ρ ρ T clΦ Φ T GBΦ Φ T μν A A − − − 2 þ 2 f μν ½W þ μν ½W ½W ¼R ðμsνÞρ aμ ðν;ρÞ aν ðμ;ρÞ sρðμνÞ 8π ;ρτ ρτ 1 − ð1=2Þgμνf−ð1=2Þsρτ þð1=2ÞR sρτ 2 T Gh Gh − 4 þ gμνv1gþ 2 f μν ½W gμνv1g ρ;τ ρτ 8π þaρτ − sρτ g; ð114cÞ þ Θμν; ð119Þ

GBΦ Φ 2 Φ T μν ½W ¼−ð1=2Þgμνm w ð114dÞ and where the various state-dependent contributions are given by Eqs. (114a)–(114e). Gh Gh Gh Gh Gh T μν ½W ¼−w;μν þ 2wμν þð1=2Þgμν□w þ 6gμνv1: 2. Cancellation of the gauge-breaking ð114eÞ and ghost contributions Let us now consider the divergence of the terms given by In Sec. IV B we have mentioned that the formal con- ˆ GB Eqs. (114a)–(114e). By taking into account Eqs. (81) and tributions of the gauge-breaking term hψjTμν jψi and the ˆ Gh (83), we obtain ghost term hψjTμν jψi cancel each other out [see Eq. (105)].

;ν This still remains valid for the corresponding regularized T clA A T GBA A 6 A ;ν − 2 A ρ ð μν ½W þ μν ½W Þ ¼ v1 μν v1 ρ ;μ; ð115aÞ expectation values up to purely geometrical terms. Indeed, by using the first Ward identity in the form (86) as well as clΦ Φ GBΦ Φ ;ν ðT μν ½W þT μν ½W Þ ¼ −2v1;μ ð115bÞ the second Ward identity in the form (88), we obtain and GBA A GBΦ Φ Gh Gh T μν ½W þT μν ½W þT μν ½W Gh Gh ;ν ðT μν ½W Þ ¼ 4v1;μ; ð115cÞ A A ρ ¼ 2v1 μν þ gμνf−ð1=2Þv1 ρ þ 3v1g: ð120Þ and we then have 1 Now, by using this relation in connection with Eqs. (114a) ψ ˆ ψ ;ν 6 A − 2 A ρ ðh jTμνj irenÞ ¼ 8π2 f v1 μν gμνv1 ρ and (114b), we can rewrite the renormalized expectation value of the stress-energy-tensor operator given by 2 ;ν Θ~ ;ν 0 þ gμνv1g þ μν ¼ : ð116Þ Eq. (119) in the form

044063-15 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) 1 1 ˆ ρ ρ ˆ ρ ρ ψ Tμν ψ 1=2 sρ 1=2 □sμν − sρ μ;ν ψ Tμν ψ 1=2 sρ 1=2 □sμν −sρ μ;ν h j j iren ¼ 8π2 fð Þ ;μν þð Þ ð Þ h j j iren ¼8π2 fð Þ ;μν þð Þ ð Þ − ρ − ρ − ρ 2 ρ 1 2 ρ − 1 2 ρ − 1 2 ρ aμ ½ν;ρ aν ½μ;ρ sρ μν þ sρðμνÞ þð = ÞR ðμsνÞρ ð = Þaμ ðν;ρÞ ð = Þaν ðμ;ρÞ − 1 2 1 2 □ ρ − 1 2 ;ρτ − ρ − ρ − ρ ρ ð = Þgμν½ð = Þ sρ ð = Þsρτ aμ ½ν;ρ aν ½μ;ρ sρ μν þsρðμνÞ − 1 2 ρτ − ρ;τ ρτ ρ ;ρτ ρ;τ ð = ÞR sρτ aρτ þ sρτ −ð1=2Þgμν½ð1=2Þ□sρ −ð1=2Þsρτ −aρτ 1 2 Φ − Φ A A ρ þð = Þw;μν wμν þv1 μν −gμνv1 ρ gþΘμν: ð123Þ Φ 2 Φ A −ð1=2Þgμν½ð1=2Þ□w − m w þ2v1 μν We have now at our disposal an expression for the − 3 2 A ρ − 2 Θ ð = Þgμνv1 ρ gμνv1gþ μν: ð121Þ renormalized expectation value of the stress-energy-tensor operator associated with the full Stueckelberg theory which only involves state-dependent and geometrical quantities This expression only involves state-dependent and geo- associated with the massive vector field Aμ. It is the main metrical quantities associated with the quantum fields Aμ result of our article. and Φ. We could consider it as our final result, but, in fact, it It is interesting to note that Eq. (123) combined with is very important here to note that, due to the first Ward Eq. (81b) leads to identity, the decomposition into a part involving the massive vector field and another part involving the aux- 1 ψ ˆ ρ ψ − 2 ρ − 1 2 ρτ iliary scalar field is not unique. In the next sections, we h jTρ j iren ¼ 8π2 f m sρ ð = ÞR sρτ shall provide two alternative expressions which, in our ρτ 3 A ρ Θ ρ opinion, are much more interesting from the physical point þsρτ þ v1 ρ gþ ρ : ð124Þ of view. From now, in order to simplify the notations and because 4. Another final expression involving both the vector field this does not lead to any ambiguity, we shall omit the label Aμ and the auxiliary scalar field Φ Φ Φ Φ for the Taylor coefficients w and wμν. Even if we are satisfied with our final expression (123),it is worth nothing that it does not reduce, in the limit 3. Substitution of the auxiliary scalar field m2 → 0, to the result obtained from Maxwell’s theory. contribution and final result This is not really surprising because it involves implicitly It is possible to remove in Eq. (121) any reference to the the contribution of the auxiliary scalar field. In fact, by 2 auxiliary scalar field Φ. In some sense, it plays the role of a replacing in Eq. (121) the term m w given by Eq. (122a),it kind of ghost field (the so-called Stueckelberg ghost [46]), is possible to split the renormalized expectation value of the but its contribution must be carefully taken into account. By stress-energy-tensor operator in the form using Eqs. (83a), (86a) and (86b) in the form ψ ˆ ψ T A T Φ Θ h jTμνj iren ¼ μν þ μν þ μν; ð125Þ 2 ρ m w ¼ wρ þ 6v1 where the terms associated with the vector and scalar fields ;ρτ ρτ ¼ −ð1=2Þsρτ − ð1=2ÞR sρτ are given by ρ;τ ρτ A ρ þ aρτ þ sρτ þ v1 ρ þ 2v1; ð122aÞ 1 A ρ ρ T μν 1=2 sρ 1=2 □sμν − sρ μ;ν ¼ 8π2 fð Þ ;μν þð Þ ð Þ − ρ − ρ − ρ 2 ρ aμ ν;ρ aν ½μ;ρ sρ μν þ sρðμνÞ ;ρ ρ ½ ;μν − ρ μ 2 ρ μ w ¼ s ð j jνÞ þ a ð j ;jνÞ ð122bÞ ρ ρ;τ −ð1=2Þgμν½ð1=2Þ□sρ − 2aρτ A A ρ þ2v1 μν − gμνv1 ρ gð126aÞ and and ;ρ ρ μν − 1 2 ρ μ − 1 2 μ ν ρ w ¼ ð = Þs ð j jνÞ ð = ÞR ð s Þ 1 T Φ 1 2 − 1 2 ρ 1 2 ρ μν ¼ 8π2 fð = Þw;μν wμν þð = Þaμ ½ν;ρ þð = Þaν ½μ;ρ − 1 4 □ − ρ A − ð = Þgμν w gμνv1g: ð126bÞ þ sρðμνÞ þ v1 μν gμνv1; ð122cÞ A Φ The stress-energy tensors T μν and T μν are separately we obtain conserved (this can be checked from relations obtained

044063-16 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) Φ 1 in Sec. III D), and, moreover, the expression of T μν μν Maxwell A ρ g T μν ¼ f2v1 ρ − 4v1g corresponds exactly to the renormalized expectation value 8π2 of the stress-energy-tensor operator associated with the 1 − 1=20 □R − 5=72 R2 quantum action (14) for ξ ¼ 1 (see, e.g., Refs. [35,36]). As ¼ 8π2 f ð Þ ð Þ A a consequence, it could be rather natural to consider T μν pq pqrs þð7=30ÞRpqR −ð13=360ÞRpqrsR g: given by Eq. (126a) as the renormalized expectation value of the stress-energy-tensor operator associated with the ð130Þ massive vector field Aμ. This physical interpretation is 2 A ’ strengthened by noting that, in the limit m → 0, T μν We recover the trace for Maxwell s theory. reduces to the result obtained from Maxwell’s theory (see Sec. IV D). However, despite this, we are not really E. Ambiguities in the renormalized stress-energy tensor satisfied by this artificial way to split the contributions 1. General expression of the ambiguities of the vector and scalar fields because, as we have already ψ ˆ ψ noted, the first Ward identity allows us to move terms from The renormalized expectation value h jTμνðxÞj iren is one contribution to the other. So, we consider that the only unique up to the addition of a geometrical conserved tensor nonambiguous result is the one given by Eq. (123). Θμν. In other words, even if it takes perfectly into account It is interesting to note that Eq. (125) combined with the quantum state dependence of the theory, it is ambig- Eqs. (81b) and (83a) leads to uously defined (see, Sec. III of Ref. [31] as well as, e.g., comments in Refs. [25,26,41,47,48]). ˆ ρ T A ρ T Φ ρ Θ ρ As noted by Wald [31], Θμν is a local conserved tensor hTρ iren ¼ ρ þ ρ þ ρ ð127Þ of dimension ðmassÞ4 ¼ðlengthÞ−4 which remains finite in the massless limit. As a consequence, it can be con- with structed by functional derivation with respect to the metric tensor from a geometrical Lagrangian of dimension 1 4 −4 T A ρ − ;ρτ − 2 ρ − ρτ ðmassÞ ¼ðlengthÞ . Such a Lagrangian is necessarily a ρ ¼ 2 f sρτ m sρ R sρτ 8π linear combination of the following four terms: m4, m2R, 2 ρ;τ 2 ρτ 4 A ρ 2 pq þ aρτ þ sρτ þ v1 ρ gð128aÞ R , RpqR . It should be noted that we could also take into pqrs account the term RpqrsR . But, in fact, we can eliminate and this term because, in a four-dimensional background, the Euler number 1 T Φ ρ − 1 2 □ − 2 2 Z ρ ¼ 2 f ð = Þ w m w þ v1g: ð128bÞ ffiffiffiffiffiffi 8π 4 p 2 pq pqrs χ ¼ d x −g½R − 4RpqR þ RpqrsR ð131Þ M ’ D. Maxwell s theory associated with the quadratic Gauss-Bonnet action is a 2 A Let us now consider the limit m → 0 of T μν given by topological invariant. Eq. (126a). By using Eq. (122a), it reduces to The functional derivation of the action terms previously discussed provides the conserved tensors 1 Maxwell ρ ρ Z T μν ¼ fð1=2Þsρ þð1=2Þ□sμν − sρ μ;ν 1 δ 8π2 ;μν ð Þ 4 pffiffiffiffiffiffi 4 4 μν pffiffiffiffiffiffi d x −gm ¼ð1=2Þm g ; ð132aÞ − ρ − ρ − ρ 2 ρ −g δgμν M aμ ½ν;ρ aν ½μ;ρ sρ μν þ sρðμνÞ ρ ;ρτ −ð1=2Þgμν½ð1=2Þ□sρ − ð1=2Þsρτ Z 1 δ pffiffiffiffiffiffi ρτ ρ;τ ρτ ffiffiffiffiffiffi 4 − 2 − 2 μν − 1 2 μν − 1=2 R sρτ − aρτ sρτ p d x gm R ¼ m ½R ð = ÞRg ; ð Þ þ −g δgμν M 2 A − 3 2 A ρ þ v1 μν gμν½ð = Þv1 ρ þ v1g: ð129Þ ð132bÞ

This last expression is nothing other than the renormalized Z 1 δ pffiffiffiffiffiffi ð1Þ μν ≡ ffiffiffiffiffiffi 4 − 2 expectation value of the stress-energy-tensor operator asso- H p− δ d x gR ciated with Maxwell’s electromagnetism (see Eq. (3.41b) g gμν M of Ref. [39]). ¼ 2R;μν − 2RRμν It is interesting to note that Eq. (129) combined with þ gμν½−2□R þð1=2ÞR2; ð132cÞ Eq. (81b) leads to

044063-17 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) Z 1 δ ffiffiffiffiffiffi ð2Þ μν 4 p pq → − λ2 H ≡ pffiffiffiffiffiffi d x −gRpqR wab wab ðv0ab þ v1gabÞ lnð Þð137bÞ −g δgμν M ;μν μν μpνq Φ ¼ R − □R − 2RpqR for the scalar field or the ghost fields. By substituting Eqs. (135a)–(135c) into the general expression (112) of the gμν − 1=2 □R 1=2 R Rpq : 132d þ ½ ð Þ þð Þ pq ð Þ renormalized expectation value of the stress-energy-tensor operator, we obtain the general form of the ambiguity The general expression of the local conserved tensor Θμν associated with the scale length. It is given by can be therefore written in the form λ2 1 lnð Þ clA A clΦ Φ 4 2 ΘμνðλÞ¼− fΘμν ½V þΘμν ½V g Θμν αm gμν βm Rμν − 1=2 Rgμν 2 ¼ 8π2 f þ ½ ð Þ 8π λ2 ð1Þ ð2Þ lnð Þ GB GBΦ Φ þγ1 Hμν þ γ2 Hμνg; ð133Þ − Θμν A VA Θμν V 8π2 f ½ þ ½ g α β γ γ λ2 where , , 1 and 2 are constants which can be fixed by lnð Þ Gh Gh − Θμν ½V ð138Þ imposing additional physical conditions on the renormal- 8π2 ized expectation value of the stress-energy-operator tensor, these conditions being appropriate to the problem treated. with

ρσ0 ΘclA A T clA 0 A 0 μν ½V ðxÞ¼lim μν ðx; x Þ½Vρσ0 ðx; x Þ; ð139aÞ 2. Ambiguities associated with the renormalization mass x0→x So far, in order to simplify the calculations, we have clΦ Φ clΦ Φ dropped the scale length λ (or, equivalently, the mass scale Θμν ½V ðxÞ¼limT μν ðx; x0Þ½V ðx; x0Þ; ð139bÞ x0→x M ¼ 1=λ, i.e., the so-called renormalization mass) that ρσ0 should be introduced in order to make dimensionless the ΘGBA A T GBA 0 A 0 μν ½V ðxÞ¼lim μν ðx; x Þ½Vρσ0 ðx; x Þ; ð139cÞ argument of the logarithm in the Hadamard representation x0→x of the Green functions. In fact, in Eqs. (35) and (37) it is GBΦ Φ GBΦ Φ necessary to make the substitution ln½σðx; x0Þþiϵ → Θμν ½V ðxÞ¼limT μν ðx; x0Þ½V ðx; x0Þ ð139dÞ 0→ ln½σðx; x0Þ=λ2 þ iϵ which leads in Eqs. (55) and (58) to x x the substitution and

0 0 2 ln jσðx; x Þj → ln jσðx; x Þ=λ j: ð134Þ Gh Gh Gh 0 Gh 0 Θμν ½V ðxÞ¼limT μν ðx; x Þ½V ðx; x Þ; ð139eÞ x0→x This scale length induces an indeterminacy in the bitensors 0 0 A 0 0 clA ρσ GBA ρσ clΦ Wμν0 ðx; x Þ and Wðx; x Þ which corresponds to the where the differential operators T μν , T μν , T μν , GBΦ Gh replacements T μν and T μν are given in Eqs. (104a)–(104e). It should be noted that ΘμνðλÞ is a purely geometrical object. This is due A 0 → A 0 − A 0 λ2 Wμν0 ðx; x Þ Wμν0 ðx; x Þ Vμν0 ðx; x Þ lnð Þ; ð135aÞ to the geometrical nature of the Hadamard coefficients A 0 0 Vμν0 ðx; x Þ and Vðx; x Þ. Φ 0 → Φ 0 − Φ 0 λ2 W ðx; x Þ W ðx; x Þ V ðx; x Þ lnð Þ; ð135bÞ In order to obtain the explicit expression of the stress- energy tensor ΘμνðλÞ, we can repeat the calculations of Gh 0 Gh 0 Gh 0 2 W ðx; x Þ → W ðx; x Þ − V ðx; x Þ lnðλ Þ; ð135cÞ A A Φ Φ Gh Sec. IV C by replacing Wμν0 by Vμν0, W by V and W by Gh – i.e., in terms of the associated Taylor coefficients, to V . From Eqs. (114a) (114e) it is straightforward to cl clΦ Φ GB GBΦ Φ replacements obtain explicitly ΘμνA ½VA, Θμν ½V , Θμν A ½VA, Θμν ½V Gh Gh and Θμν ½V by using the replacements (136) and (137). → − A λ2 sμν sμν v0 ðμνÞ lnð Þ; ð136aÞ We can then show that

;ν → − A λ2 ΘclA A ΘGBA A 0 aμνa aμνa v0 ½μνa lnð Þ; ð136bÞ ð μν ½V þ μν ½V Þ ¼ ; ð140aÞ

→ − A A λ2 ΘclΦ Φ ΘGBΦ Φ ;ν 0 sμνab sμνab ðv0 ðμνÞab þ v1 ðμνÞgabÞ lnð Þð136cÞ ð μν ½V þ μν ½V Þ ¼ ð140bÞ for the vector field Aμ and and

2 Gh Gh ;ν w → w − v0 lnðλ Þ; ð137aÞ ðΘμν ½V Þ ¼ 0: ð140cÞ

044063-18 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) Equations (140a)–(140c) are similar to Eqs. (115a)–(115c) (137). If we use the form (123) without taking into account but now with the right-hand sides vanishing. This is due to the geometrical terms, we obtain the fact that, unlike the wave equations (42) and (46) for λ2 A Φ Gh A Φ lnð Þ A ρ A W 0 , W and W , the wave equations for V 0, V and Θμν λ − 1=2 v 1=2 □v μν μν ð Þ¼ 8π2 fð Þ 0 ρ ;μν þð Þ 0 μν Gh V [cf. Eqs. (41) and (45)] have no source terms. As a A ρ ρ A ρτ A −v0 ρ μ;ν þð1=2ÞR μv0 ν ρ − ð1=2Þg v0 μρ ν;τ consequence ΘμνðλÞ is a conserved geometrical tensor. We ð Þ ð Þ ½ ð Þ can also check that − 1 2 ρτ A − ρτ A − ρτ A ð = Þg v0 ½νρðμ;τÞ g v0 ½μρ½ν;τ g v0 ½νρ½μ;τ A ρ A ρ A ρ A GBA A GBΦ Φ Gh Gh −v0 þð1=2Þv þð1=2Þv þ v1 μν Θμν ½V þΘμν ½V þΘμν ½V ¼0: ð141Þ ρ μν 0 ðρμÞν 0 ðρνÞμ A ρ A ;ρτ A ρ;τ − gμν½ð1=4Þ□v0 ρ − ð1=4Þv0 ρτ − ð1=2Þv0 ρτ Equation (141) is similar to Eq. (120) but now with the ½ A ρ right-hand side vanishing. This is due to the fact that, unlike þ v1 ρ g: ð143Þ A Gh the Ward identity (85) linking Wμν0 and W , the Ward A Gh Similarly, if we use the alternative form (125) where the identity (84) linking Vμν0 and V has no right-hand side. contributions corresponding to the vector field Aμ and the As a consequence, from Eqs. (138) and (141), we obtain auxiliary scalar field Φ are highlighted [see Eqs. (126a) and λ2 (126b)], we obtain lnð Þ clA A clΦ Φ Θμν λ − Θμν V Θμν V : ð Þ¼ 8π2 f ½ þ ½ g ð142Þ A Φ ΘμνðλÞ¼ΘμνðλÞþΘμνðλÞð144Þ The ambiguities associated with the scale length can now be obtained explicitly from the replacements (136) and with

λ2 A lnð Þ A ρ A A ρ ρτ A ρτ A A ρ ΘμνðλÞ¼− fð1=2Þv þð1=2Þ□v −v − g v − g v −v 8π2 0 ρ ;μν 0 μν 0 ρðμ;νÞ 0 ½μρ½ν;τ 0 ½νρ½μ;τ 0 ρ μν A ρ A ρ 2 A − 1 4 □ A ρ − A ρ;τ A ρ þv0 ðρμÞν þ v0 ðρνÞμ þ v1 μν gμν½ð = Þ v0 ρ v0 ½ρτ þ v1 ρ g ð145aÞ and

λ2 ΘΦ λ − lnð Þ 1 2 − − 1 4 □ μνð Þ¼ 8π2 fð = Þv0;μν v0μν gμν½ð = Þ v0 þ v1g: ð145bÞ

Now, by using the explicit expressions (72) and (74) of the Taylor coefficients of the purely geometrical Hadamard coefficients, we can show that Eq. (143) reduces to

λ2 lnð Þ 2 p Θμν λ − 1=4 m Rμν − 1=20 R;μν 13=120 □Rμν − 1=8 RRμν 2=15 Rμ Rν ð Þ¼ 8π2 fð Þ ð Þ þð Þ ð Þ þð Þ p p q pqr 4 2 þð7=20ÞRpqRμ ν − ð1=15ÞRμpqrRν þ gμν½−ð3=8Þm − ð1=8Þm R − ð1=240Þ□R 2 pq pqrs þð1=32ÞR − ð29=240ÞRpqR þð1=60ÞRpqrsR g; ð146Þ while Eqs. (145a) and (145b) provide

λ2 A lnð Þ 2 p p q Θμν λ − 1=3 m Rμν − 1=30 R;μν 1=10 □Rμν − 5=36 RRμν 13=90 Rμ Rν 31=90 R Rμ ð Þ¼ 8π2 fð Þ ð Þ þð Þ ð Þ þð Þ p þð Þ pq ν pqr 4 2 2 pq −ð13=180ÞRμpqrRν þ gμν½−ð1=4Þm − ð1=6Þm R − ð1=60Þ□Rþð5=144ÞR − ð11=90ÞRpqR pqrs þð13=720ÞRpqrsR g ð147aÞ and

044063-19 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) λ2 Φ lnð Þ 2 p Θμν λ − − 1=12 m Rμν − 1=60 R;μν 1=120 □Rμν 1=72 RRμν − 1=90 Rμ Rν ð Þ¼ 8π2 f ð Þ ð Þ þð Þ þð Þ ð Þ p p q pqr 4 2 þð1=180ÞRpqRμ ν þð1=180ÞRμpqrRν þ gμν½−ð1=8Þm þð1=24Þm R þð1=80Þ□R 2 pq pqrs −ð1=288ÞR þð1=720ÞRpqR −ð1=720ÞRpqrsR g: ð147bÞ

A Φ Of course, it is easy to check that the sum of ΘμνðλÞ and ΘμνðλÞ is equal to ΘμνðλÞ. It is possible to obtain a more compact form for the stress-energy tensors (146), (147a) and (147b) by using the conserved p pqr pqrs tensors (132a)–(132d). It should be noted that the terms in RμpRν , RμpqrRν and RpqrsR which are not present in ð1Þ ð2Þ Hμν and Hμν can be eliminated by introducing Z 1 δ ffiffiffiffiffiffi ð3Þ μν 4 p pqrs H ≡ pffiffiffiffiffiffi d x −gRpqrsR −g δgμν M ;μν μν μ νp μpνq μ νpqr μν pqrs ¼ 2R − 4□R þ 4R pR − 4RpqR − 2R pqrR þ g ½ð1=2ÞRpqrsR ð148Þ and by noting that, due to Eq. (131),

ð1Þ ð2Þ ð3Þ Hμν − 4 Hμν þ Hμν ¼ 0: ð149Þ

We then have

λ2 lnð Þ 4 2 ð1Þ ð2Þ Θμν λ 3=8 m gμν− 1=4 m Rμν − 1=2 Rgμν − 7=240 Hμν 13=120 Hμν ; ð Þ¼ 8π2 fð Þ ð Þ ½ ð Þ ð Þ þð Þ g ð150Þ

ln λ2 ΘA λ ð Þ 1 4 4 − 1 3 2 − 1 2 − 1 30 ð1Þ 1 10 ð2Þ μνð Þ¼ 8π2 fð = Þm gμν ð = Þm ½Rμν ð = ÞRgμν ð = Þ Hμν þð = Þ Hμνgð151aÞ and

ln λ2 ΘΦ λ ð Þ 1 8 4 1 12 2 − 1 2 1 240 ð1Þ 1 120 ð2Þ μνð Þ¼ 8π2 fð = Þm gμνþð = Þm ½Rμν ð = ÞRgμνþð = Þ Hμν þð = Þ Hμνg: ð151bÞ

As expected, we can note that the ambiguities associated with the scale length (or with the renormalization mass) are of the form (133).

V. CASIMIR EFFECT A. General considerations In this section, we shall consider the Casimir effect for Stueckelberg massive electromagnetism in the Minkowski 4 spacetime ðR ; ημνÞ with ημν ¼ diagð−1; þ1; þ1; þ1Þ. We denote by ðT;X;Y;ZÞ the coordinates of an event in this spacetime. We shall provide the renormalized of the stress-energy-tensor operator outside of a perfectly conducting medium with a plane boundary wall at Z ¼ 0 separating it from free space (see Fig. 1). It is worth pointing out that this problem has been studied a long time ago by Davies and Toms in the framework of de Broglie-Proca electromagnetism [49]. We shall revisit this problem in order to compare, at the quantum level and in the case of a simple example, de Broglie-Proca and Stueckelberg theories and to discuss their limit for m2 → 0. It should be noted that the Casimir effect in connection with a massive photon has been considered for various geometries (see, e.g., Refs. [50–54]). From symmetries and physical considerations, we can observe that, outside of the perfectly conducting medium, the renormalized stress-energy tensor takes the form (see Chap. 4 of Ref. [24]) 1 0 ˆ 0 0 ˆ ρ 0 η − ˆ ˆ h jTμνj iren ¼ 3 h jTρ j irenð μν ZμZνÞ; ð152Þ

044063-20 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) 0 ˆ ρ 0 0 h jTρ j iren ¼ ð156Þ

which plays the role of a constraint for α. In the Minkowski spacetime, the Feynman propagator A 0 Gμνðx; x Þ associated with the vector field Aμ satisfies the wave equation (23), i.e.,

2 A 0 4 0 ½□x − m Gμνðx; x Þ¼−ημνδ ðx; x Þ; ð157Þ

and its explicit expression is given in terms of a Hankel function of the second kind by (see, e.g., Chap. 27 of FIG. 1. Geometry of the Casimir effect. Ref. [55])

2 μ 1 ˆ A 0 m ð2Þ 0 where Z is the spacelike unit vector orthogonal to the wall. Gμνðx; x Þ¼− H1 ½Zðx; x Þημν: ð158Þ As a consequence, it is sufficient to determine the trace of 8π Zðx; x0Þ the renormalized stress-energy tensor. From Eq. (124),we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 have Here, Zðx; x0Þ¼ −2m ½σðx; x0Þþiϵ with 2σðx; x0Þ¼ −ðT − T0Þ2 þðX − X0Þ2 þðY − Y0Þ2 þðZ − Z0Þ2. 1 We have (see Chap. 9 of Ref. [56]) ˆ ρ 2 ρ ρτ 4 0 Tρ 0 −m sρ sρτ 3=2 m h j j iren ¼ 8π2 f þ þð Þ g ð2Þ ρ H1 ðzÞ¼J1ðzÞ − iY1ðzÞ; ð159Þ þ Θρ : ð153Þ

ρ where J1ðzÞ and Y1ðzÞ are the Bessel functions of the first The term Θρ encodes the usual ambiguities discussed in and second kinds. By using the series expansions for z → 0 Sec. IV E. In the Minkowski spacetime it reduces to (see Eqs. (9.1.10) and (9.1.11) of Ref. [56])

X∞ 2 k ρ 1 4 z ð−z =4Þ Θρ ¼ fαm g; ð154Þ J1ðzÞ¼ ð160aÞ 8π2 2 k! k 1 ! k¼0 ð þ Þ where α is a constant. From Eq. (153) it is clear that in order and ˆ ρ to evaluate h0jTρ j0i , it is sufficient to take the coinci- ren   ∞ 0 A 0 A 0 2 2 X dence limit x → x of Wμνðx; x Þ and Wμν; ðx; x Þ [see z z ðabÞ Y1ðzÞ¼− þ ln J1ðzÞ − ½ψðk þ 1Þ πz π 2 2π Eqs. (78a) and (78c) and note that, in the Minkowski k¼0 ν0 0 spacetime, the bivector of parallel transport gμ ðx; x Þ is ð−z2=4Þk ν0 0 A 0 þ ψðk þ 2Þ ð160bÞ equal to the unit matrix gμ ðx; x Þ] where Wμνðx; x Þ is the k!ðk þ 1Þ! A 0 regular part of the Feynman propagator Gμνðx; x Þ corre- sponding to the geometry of the Casimir effect. [we note that Eq. (160b) is valid for j argðzÞj < π, and we recall that the digamma function ψ is defined by the recursion relation ψ z 1 ψ z 1=z with B. Stress-energy tensor in the Minkowski spacetime ð þ Þ¼ ð Þþ ψð1Þ¼−γ], we can provide the Hadamard representation A 0 Let us first consider the vacuum expectation value of the of Gμνðx; x Þ given by Eq. (158). We can write stress-energy-tensor operator in the ordinary Minkowski spacetime (i.e., without the boundary wall). This will 2 1 m ð2Þ 0 permit us to establish some notations and, moreover, to − H1 ½Zðx; x Þ 8π Zðx; x0Þ fix the constant α appearing in Eq. (154). Due to symmetry  1 2 considerations, we have i Δ = ðx; x0Þ ¼ 8π2 σðx; x0Þþiϵ  1 0 ˆ 0 0 ˆ ρ 0 η h jTμνj iren ¼ 4 h jTρ j iren μν; ð155Þ þVðx; x0Þ ln½σðx; x0ÞþiϵþWðx; x0Þ ; ð161Þ

0 ˆ ρ 0 where h jTρ j iren is still given by Eqs. (153) and (154).Of where ˆ course, we must have h0jTμνj0i ¼ 0, and we have ren Δ1=2 0 1 therefore ðx; x Þ¼ ; ð162aÞ

044063-21 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) X∞ components of the but much less 0 k 0 Vðx; x Þ¼ Vkσ ðx; x Þð162bÞ natural for its longitudinal component. Indeed, for this k¼0 component, we could also consider perfect transmission instead of complete reflection (see Refs. [49,50,57]). We with shall now consider that the Feynman propagator is given by

2 kþ1 ðm =2Þ ~ A 0 A 0 − A ~0 V ¼ ð162cÞ Gμνðx; x Þ¼Gμνðx; x Þ qνGμνðx; x Þ: ð168Þ k k!ðk þ 1Þ! Here, x0μ ¼ðT0;X0;Y0;Z0Þ and x~0μ ¼ðT0;X0;Y0; −Z0Þ, and while qν ¼ð1 − 2δ3νÞ. It is important to note that, in ν X∞ Eq. (168), the index is not summed. Furthermore, we A 0 0 k 0 remark that the term Gμνðx; x~ Þ which is obtained by Wðx; x Þ¼ Wkσ ðx; x Þð162dÞ 0 ~0 k¼0 replacing x by x in Eq. (158) as well as its derivatives are regular in the limit x0 → x. with By following the steps of Sec. VBand using the relation    2 1 2 ðm =2Þkþ m − 1 2 π −iπν=2 ð2Þ −iπ=2 W ¼ − ψðk þ 1Þþψðk þ 2Þ − ln : KνðzÞ¼ ð = Þi e Hν ðze Þð169Þ k k!ðk þ 1Þ! 2 −π 2 ≤ ≤ π ð162eÞ which is valid for = argðzÞ as well as the properties of the modified Bessel functions of the second By noting that kind K1, K2 and K3 (see Chap. 9 of Ref. [56]), it is easy to show that the Taylor coefficients sμν and sμνab involved in A 0 0 0 ˆ ρ 0 Wμνðx; x Þ¼Wðx; x Þημν; ð163Þ h jTρ j iren are now given by

0 2 2 where Wðx; x Þ is given by Eqs. (162d) and (162e), we are sμν ¼ m ½−1=2 þ γ þð1=2Þ lnðm =2Þημν ˆ ρ now able to express h0jTρ j0i . From Eqs. (78a) and (78c) ren − ðm=ZÞK1ð2mZÞqνημν ð170Þ we have, respectively,

2 2 and sμν ¼ m ½−1=2 þ γ þð1=2Þ lnðm =2Þημν ð164Þ 4 2 sμν ¼ m ½−5=16 þð1=4Þγ þð1=8Þ lnðm =2Þημνη and ab ab 2 2 − ½ðm =Z ÞK2ð2mZÞqνημνð2η3aη3b − ð1=2ÞηabÞ 4 2 sμνab ¼ m ½−5=16 þð1=4Þγ þð1=8Þ lnðm =2Þημνηab: 3 þðm =ZÞK1ð2mZÞqνημνη3aη3b: ð171Þ ð165Þ By inserting Eqs. (170) and (171) in the expression (153) Then, from Eq. (153), we obtain and using the value of α fixed by Eq. (167), we obtain   4 3 2 3 0 ˆ ρ 0 m α 9 4−3γ− 3 2 2 2 ^ ρ m m h jTρ j i ¼ 2 f þ = ð = Þlnðm = Þg; ð166Þ h0jTρ j0i ¼ K2ð2mZÞþ K1ð2mZÞ ; ð172Þ ren 8π ren 8π2 Z2 Z and, necessarily, by using Eq. (156), we have the constraint and from Eq. (152) we have   α ¼ −9=4 þ 3γ þð3=2Þ lnðm2=2Þ: ð167Þ 2 3 ˆ 1 m m h0jTμνj0i ¼ K2ð2mZÞþ K1ð2mZÞ ren 8π2 Z2 Z C. Stress-energy tensor for the Casimir effect ˆ ˆ × ðημν − ZμZνÞ: ð173Þ Let us now come back to our initial problem. The Feynman propagator previously considered is modified It is very important to note that this result coincides exactly by the presence of the plane boundary wall. The new with the result obtained by Davies and Toms in the ~ A 0 Feynman propagator Gμνðx; x Þ can be constructed by the framework of de Broglie-Proca electromagnetism [49]. 2 method of images if we assume, in order to simplify our In the limit m → 0 and for Z ≠ 0, we obtain problem, a perfectly reflecting wall. It should be noted that this particular boundary condition is questionable from the ˆ 1 1 ˆ ˆ h0jTμνj0i ¼ ðημν − ZμZνÞ: ð174Þ physical point of view. It is logical for the transverse ren 16π2 Z4

044063-22 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) 0 ˆ A ρ 0 T A ρ ΘA ρ In the massless limit, the vacuum expectation value of the h jT ρ j iren ¼ ρ þ ρ ð180Þ renormalized stress-energy tensor associated with the Stueckelberg theory diverges like Z−4 as the boundary and surface is approached. This result contrasts with that ’ 0 ˆ Φ ρ 0 T Φ ρ ΘΦ ρ obtained from Maxwell s theory (see also Ref. [49]). h jT ρ j iren ¼ ρ þ ρ ; ð181Þ Indeed, for this theory, the renormalized stress-energy- tensor operator vanishes identically. In order to extract that where the contributions associated with the vector field Aμ result from the Stuckelberg theory, we will now repeat the and the auxiliary scalar field Φ are separated. At first sight, A ρ previous calculations from the expressions (125) and (126) T ρ seems complicated because it involves Taylor coef- [as well as (127) and (128)] given in Sec. IV C 4, where we 1=2 1 A 0 ficients of orders σ and σ of Wμνðx; x Þ. In fact, its have proposed an artificial separation of the contributions ρ;τ expression can be simplified by replacing the sum aρτ þ associated with the vector field Aμ and the auxiliary scalar ρτ sρτ from the relation (86b), and we obtain field Φ. 1 A ρ 2 ρ 2 4 T ρ −m sρ 2m w 1=2 m D. Separation of the contributions associated with the ¼ 8π2 f þ þð Þ gð182Þ vector field Aμ and the auxiliary scalar field Φ In the Minkowski spacetime, Eqs. (127) and (128) which only involves the first Taylor coefficients sμν and w σ0 0 ˆ ρ 0 reduce to of order . So, in order to evaluate h jTρ j iren given by Eq. (175), it is sufficient to take the coincidence limit 0 ˆ ρ 0 T A ρ T Φ ρ Θ ρ x0 → x of the state-dependent Hadamard coefficients h jTρ j iren ¼ ρ þ ρ þ ρ ð175Þ A 0 0 Wμνðx; x Þ and Wðx; x Þ associated with the Feynman A 0 Φ 0 with propagators Gμνðx; x Þ and G ðx; x Þ corresponding to the geometry of the Casimir effect. 1 A Φ A ρ ;ρτ 2 ρ ρ;τ ρτ 4 At first, we must fix the constants α and α appearing in T ρ ¼ f−sρτ − m sρ þ2aρτ þ 2sρτ þ 2m g 8π2 Eq. (178). This can be achieved by imposing, in the ð176Þ Minkowski spacetime without boundary, the vanishing of 0 ˆ A ρ 0 0 ˆ Φ ρ 0 h jT ρ j iren given by Eq. (180) and h jT ρ j iren given by and Eq. (181). In this spacetime, everything related to the A 0 Feynman propagator Gμνðx; x Þ has been already given in 1 Φ T Φ ρ − 1 2 □ − 2 1 4 4 Sec. VB, while the Feynman propagator G ðx; x0Þ ρ ¼ 2 f ð = Þ w m w þð = Þm g: ð177Þ 8π associated with the scalar field Φ satisfies the wave ρ equation (25) and is explicitly given by The term Θρ encodes the usual ambiguities discussed in 2 Sec. IV E. We can split it in the form m 1 2 GΦ x; x0 − Hð Þ Z x; x0 : 183 ð Þ¼ 8π Z 0 1 ½ ð Þ ð Þ ρ A ρ Φ ρ ðx; x Þ Θρ ¼ Θ ρ þ Θ ρ ð178aÞ By using Eqs. (161) and (162), it is easy to see that this with propagator can be represented in the Hadamard form and to 1 obtain ΘA ρ αA 4 ρ ¼ 2 f m gð178bÞ 8π w ¼ m2½−1=2 þ γ þð1=2Þ lnðm2=2Þ: ð184Þ and 0 ˆ ρ 0 We are now able to express h jTρ j iren. From Eqs. (180), 1 (181), (182), (177) and (178), we obtain ΘΦ ρ αΦ 4 ρ ¼ 2 f m g; ð178cÞ 8π 4 ˆ A ρ m A 2 h0jT ρ j0i ¼ fα þ 3=2 − 2γ − lnðm =2Þg ð185Þ where αA and αΦ are two constants associated, respectively, ren 8π2 with the contributions of the vector field Aμ and the auxiliary scalar field Φ. We can then replace Eq. (175) by and 4 ˆ ρ ˆ A ρ ˆ Φ ρ ˆ Φ ρ m Φ 2 h0jTρ j0i ¼h0jT ρ j0i þh0jT ρ j0i ð179Þ 0 T ρ 0 α 3=4 − γ − 1=2 m =2 ; ren ren ren h j j iren ¼ 8π2 f þ ð Þ lnð Þg with ð186Þ

044063-23 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) and, necessarily, the vanishing of these traces provides the interesting to note that the contribution (192) associated two constraints with the vector field Aμ and which has been artificially separated from the scalar field contribution (see Sec. IV C 4) αA ¼ −3=2 þ 2γ þ lnðm2=2Þð187aÞ vanishes identically for any value of the mass parameter m. This result coincides exactly with that obtained from and Maxwell’s theory (see also Ref. [49]).

Φ 2 α ¼ −3=4 þ γ þð1=2Þ lnðm =2Þ: ð187bÞ VI. CONCLUSION

We now come back to the Casimir effect. The two In the context of quantum field theory in curved spacetime Feynman propagators previously considered are modified and with possible applications to cosmology and to black by the presence of the plane boundary wall. The new hole physics in mind, the massive vector field is frequently Feynman propagators can be constructed by the method of studied. It should be, however, noted that, in this particular domain, it is its description via the de Broglie-Proca theory images. Of course, the propagator of the vector field Aμ is which is mostly considered and that there are very few works still given by Eq. (168), while we have dealing with the Stueckelberg point of view (see, e.g., Φ Refs. [58–63], but remark that these papers are restricted G~ ðx; x0Þ¼GΦðx; x0Þ − GΦðx; x~0Þð188Þ to de Sitter and anti-de Sitter spacetimes or to Roberstson- Walker backgrounds with spatially flat sections). In this for the propagator of the scalar field Φ. In the context of the article, in order to fill a void, we have developed the general Casimir effect, Eq. (184) must be replaced by formalism of the Stueckelberg theory on an arbitrary four- 2 2 dimensional spacetime (quantum action, Feynman propa- w ¼ m ½−1=2 þ γ þð1=2Þ lnðm =2Þ gators, Ward identities, Hadamard representation of the − ðm=ZÞK1ð2mZÞ; ð189Þ Green functions), and we have particularly focussed on the aspects linked with the construction, for a Hadamard and sμν is given by Eq. (170). By inserting Eqs. (170) and quantum state, of the expectation value of the renormalized (189) in Eqs. (182) and (177) and taking into account the stress-energy-tensor operator. It is important to note that we constraints (187a) and (187b), we obtain from Eq. (180) have given two alternative but equivalent expressions for this result. The first one has been obtained by eliminating from a 0 ˆ A ρ 0 0 Φ h jT ρ j iren ¼ ð190Þ Ward identity the contribution of the auxiliary scalar field (the so-called Stueckelberg ghost [46]) and only involves and from Eq. (181) state-dependent and geometrical quantities associated with   the massive vector field Aμ [see Eq. (123)]. The other one 3 2 3 ˆ Φ ρ m m involves contributions coming from both the massive vector h0jT ρ j0i ¼ K2ð2mZÞþ K1ð2mZÞ : ren 8π2 Z2 Z field and the auxiliary Stueckelberg scalar field [see Eqs. (125)–(126)], and it has been constructed artificially ð191Þ in such a way that these two contributions are independently From Eq. (152) we can then see that the vacuum expect- conserved and that, in the zero-mass limit, the massivevector ation value of the stress-energy-tensor operator associated field contribution reduces smoothly to the result obtained from Maxwell’s electromagnetism. It is also important to with the vector field Aμ is such that note that, in Sec. IV E, we have discussed the geometrical ˆ A ambiguities of the expectation value of the renormalized h0jTμνj0i ¼ 0; ð192Þ ren stress-energy-tensor operator. They are of fundamental while the vacuum expectation value of the stress-energy- importance (see, e.g., in Sec. V, their role in the context tensor operator associated with the auxiliary scalar field Φ of the Casimir effect). is given by We intend to use our results in the near future in cosmology of the very early universe, but we hope they   1 2 3 will be useful for other authors. This is why we shall now ˆ Φ m m 0 Tμν 0 K2 2mZ K1 2mZ h j j iren ¼ 8π2 2 ð Þþ ð Þ provide a step-by-step guide for the reader who is not Z Z especially interested by the technical details of our work but ˆ ˆ × ðημν − ZμZνÞ: ð193Þ who wishes to calculate the expectation value of the renormalized stress-energy tensor from the expression Of course, the sum of these two contributions permits us to (123), i.e., from the expression where any reference to recover the result (173) of Sec. VCwhich is also the result the Stueckelberg auxiliary scalar field Φ has disappeared. obtained by Davies and Toms in the framework of de We shall describe the calculation from the Feynman Broglie-Proca electromagnetism [49]. It is moreover propagator as well as from the anticommutator function:

044063-24 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) A 0 (i) We assume that the Feynman propagator Gμν0 ðx; x Þ It is interesting to note the existence of a nice paper by Pitts which is given by Eq. (22) and satisfies the wave [65] where de Broglie-Proca and Stueckelberg approaches equation (30) [or that the anticommutator of massive electromagnetism are discussed from a philo- ð1ÞA 0 sophical point of view based on the machinery of the Gμν0 ðx; x Þ which is given by Eq. (59) and satisfies Hamiltonian formalism (primary and secondary con- the wave equation (60)] has been determined in a … particular gravitational background and for a Hada- straints, Poisson brackets, ). Here, we adopt a more mard quantum state. In other words, we consider that pragmatic point of view. We discuss the two formulations A 0 in light of the results obtained in our article. In our opinion: the Feynman propagator G 0 ðx; x Þ can be repre- μν (i) De Broglie-Proca and Stueckelberg approaches of sented in the Hadamard form (35) [or that the massive electromagnetism are two faces of the same ð1ÞA 0 anticommutator Gμν0 ðx; x Þ can be represented in theory. Indeed, the transition from de Broglie-Proca the Hadamard form (55)]. to Stueckelberg theory is achieved via the Stueck- (ii) We need the regular part of the Feynman propagator elberg trick (5) which permits us, by introducing A 0 an auxiliary scalar field Φ, to artificially restore Gμν0 ðx; x Þ [or that of the anticommutator 1 Maxwell’s gauge symmetry in massive electromag- ð ÞA 0 σ Gμν0 ðx; x Þ] at order . To extract it, we subtract netism, but, reciprocally, the transition from Stueck- A 0 from the Feynman propagator Gμν0 ðx; x Þ its singular elberg to de Broglie-Proca theory is achieved by part (48a) in order to obtain its regular part (48b) [or imposing the gauge choice Φ ¼ 0 [see Eq. (8)]. As a ð1ÞA 0 consequence, it is not really surprising to obtain the we subtract from the anticommutator G 0 ðx; x Þ its μν same result for the renormalized stress-energy- singular part (68a) in order to obtain its regular part tensor operator associated with the Casimir effect (68b)]. We have then at our disposal the state- A 0 (see Sec. V) when we consider this problem in the dependent Hadamard bivector Wμν0 ðx; x Þ. Here, it framework of the de Broglie-Proca and Stueckelberg is important to note that we do not need the full formulations of massive electromagnetism. Indeed, expression of the singular part of the Green function we can expect that this remains true for any other considered, but we can truncate it by neglecting the quantum quantity. terms vanishing faster than σðx; x0Þ for x0 → x.Asa (ii) However, we can note that with regularization and consequence, we can construct the singular part renormalization in mind, it is much more interesting (48a) [or the singular part (68a)] by using the to work in the framework of the Stueckelberg 1 2 covariant Taylor series expansion (A9) of Δ = up formulation of massive electromagnetism. Indeed, 2 to order σ , the covariant Taylor series expansion this permits us to have at our disposal the machinery A 1 (71a) of V0 μν up to order σ [see Eqs. (72a)–(72c)] of the Hadamard formalism which is not the case in and the covariant Taylor series expansion (71b) of the framework of the de Broglie-Proca formulation. A 0 V1 μν up to order σ [see Eq. (72d)]. Indeed, due to the constraint (4a), the Feynman A 0 (iii) Finally, by using Eqs. (78a)–(78c), we can construct propagator Gμν0 ðx; x Þ associated with the vector the expectation value of the renormalized stress- field Aμ cannot be represented in the Hadamard energy tensor given by Eq. (123). form (35). It is interesting to note that, in the literature concerning Stueckelberg electromagnetism, some authors only focus ACKNOWLEDGMENTS on the part of the action associated with the massive vector We wish to thank Yves Décanini, Mohamed Ould El field Aμ and which is given by Eq. (13a) (see, e.g., Hadj and Julien Queva for various discussions and the Refs. [58,61,64]). Of course, this is sufficient because they “Collectivité Territoriale de Corse” for its support through are mainly interested, in the context of canonical quantiza- the COMPA project. tion, by the determination of the Feynman propagator associated with this field. However, in order to calculate physical quantities, it is necessary to take into account the APPENDIX: BISCALARS, BIVECTORS contribution of the auxiliary scalar field Φ. It cannot be AND THEIR COVARIANT TAYLOR SERIES EXPANSIONS discarded. This is very clear in the context of the con- struction of the renormalized stress-energy-tensor operator Regularization and renormalization of quantum field as we have shown in our article and remains true for any theories in the Minkowski spacetime are most times based other physical quantity. on the representation of Green functions in momentum To conclude this article, we shall briefly compare the de space, and, in general, this greatly simplifies reasoning and Broglie-Proca and Stueckelberg formulations of massive calculations. The use of such a representation is not electomagnetism and discuss the advantages of the possible in an arbitrary gravitational background where Stueckelberg formulation over the de Broglie-Proca one. the lack of symmetries as well as spacetime curvature

044063-25 ANDREI BELOKOGNE and ANTOINE FOLACCI PHYSICAL REVIEW D 93, 044063 (2016) prevent us from working within the framework of the A 0 A 0 The Hadamard coefficients Vn μν0 ðx; x Þ and Wn μν0 ðx; x Þ Fourier transform. As a consequence, regularization and introduced in Eq. (36) and which are bivectors involved in renormalization in curved spacetime are necessarily based the Hadamard representation of the Green functions (35) on representations of Green functions in coordinate space, 0 and (55) or the Hadamard coefficients Vnðx; x Þ and and, moreover, they require extensively the concepts of 0 Wnðx; x Þ introduced in Eq. (38) and which are biscalars biscalars, bivectors and, more generally, bitensors. Thanks involved in the Hadamard representation of the Green – to the work of some mathematicians [66 69] and of DeWitt functions (37) and (58) cannot in general be determined [23,44,55,70] and coworkers [28,29], we have at our exactly. They are solutions of the recursion relations (39a), disposal all the tools necessary to deal with this subject. (39b) and (40) or (43a), (43b) and (44), and, following In this short Appendix, in order to make a self-consistent DeWitt [44,70], we can look for the solutions of these paper (i.e., to avoid the reader needing to consult the equations in the form of covariant Taylor series expansions references mentioned above), we have gathered some for x0 in the neighborhood of x. This is the method we use important results which are directly related with the 0 in Sec. III. The series defining the biscalars Vnðx; x Þ and representations of Green functions in coordinate space 0 Wnðx; x Þ can be written in the form and, more particularly, with the Hadamard representations of the Green functions appearing in Stueckelberg electro- 0 − σ;a1 0 Tðx; x Þ¼tðxÞ ta1 ðxÞ ðx; x Þ magnetism [see Eqs. (35), (37), (55) and (58)] which is the 1 main subject of Sec. III and which plays a crucial role in þ t ðxÞσ;a1 ðx; x0Þσ;a2 ðx; x0Þ 2! a1a2 Sec. IV. In particular, we define the geodetic interval σ 0 Δ 0 1 ðx; x Þ, the Van Vleck-Morette determinant ðx; x Þ and − σ;a1 0 σ;a2 0 σ;a3 0 ta1a2a3 ðxÞ ðx; x Þ ðx; x Þ ðx; x Þ the bivector of parallel transport from x to x0 denoted by 3! 0 1 gμν0 ðx; x Þ (see, e.g., Ref. [44]), and we moreover discuss ; ; ; þ t ðxÞσ a1 ðx; x0Þσ a2 ðx; x0Þσ a3 ðx; x0Þ the concept of covariant Taylor series expansions for 4! a1a2a3a4 biscalars and bivectors. ×σ;a4 ðx; x0Þþ: ðA5Þ We first recall that 2σðx; x0Þ is the square of the geodesic distance between x and x0 which satisfies By construction, the coefficients ta1…ap ðxÞ are symmetric in … ;μ the exchange of the indices a1 ap, i.e., ta1…ap ðxÞ¼ 2σ ¼ σ σ;μ: ðA1Þ tða1…apÞðxÞ, and, moreover, by requiring the symmetry of 0 0 We have σðx; x0Þ < 0 if x and x0 are timelike related, Tðx; x Þ in the exchange of x and x , i.e., 0 0 0 0 0 Tðx; x Þ¼Tðx ;xÞ, the coefficients tðxÞ and t … ðxÞ with σðx; x Þ¼0 if x and x are null related and σðx; x Þ > 0 if x a1 ap and x0 are spacelike related. We furthermore recall that p ¼ 1; 2; … are constrained. The symmetry of Tðx; x0Þ Δðx; x0Þ is given by permits us to express the odd coefficients of the covariant Taylor series expansion of Tðx; x0Þ in terms of the even ones. We have for the odd coefficients of lowest orders (see, Δ 0 − − −1=2 −σ 0 − 0 −1=2 ðx; x Þ¼ ½ gðxÞ detð ;μν0 ðx; x ÞÞ½ gðx Þ e.g., Ref. [71]) ðA2Þ 1 2 ta1 ¼ð = Þt;a1 ; ðA6aÞ and satisfies the partial differential equation 3 2 − 1 4 ta1a2a3 ¼ð = Þtða1a2;a3Þ ð = Þt;ða1a2a3Þ: ðA6bÞ −1=2 1=2 ;μ □ σ ¼ 4 − 2Δ Δ ;μσ ðA3aÞ x A 0 Similarly, the series defining the bivectors Vn μν0 ðx; x Þ and A 0 as well as the boundary condition Wn μν0 ðx; x Þ can be written in the form

0 0 ν0 0 0 limΔðx; x Þ¼1: ðA3bÞ Tμνðx; x Þ¼gν ðx; x ÞTμν0 ðx; x Þ x0→x − σ;a1 0 ¼ tμνðxÞ tμνa1 ðxÞ ðx; x Þ The bivector of parallel transport from x to x0 is defined by 1 ;a1 0 ;a2 0 þ tμν ðxÞσ ðx; x Þσ ðx; x Þ the partial differential equation 2! a1a2 ;ρ 1 σ 0 ;a1 0 ;a2 0 ;a3 0 gμν0;ρ ¼ ðA4aÞ − tμν ðxÞσ ðx; x Þσ ðx; x Þσ ðx; x Þ 3! a1a2a3 and the boundary condition þ: ðA7Þ

0 By construction, the coefficients tμν … x are limgμν0 ðx; x Þ¼gμνðxÞ: ðA4bÞ a1 ap ð Þ x0→x symmetric in the exchange of indices a1…ap, i.e.,

044063-26 STUECKELBERG MASSIVE ELECTROMAGNETISM IN … PHYSICAL REVIEW D 93, 044063 (2016) 0 0 0 tμνa1…ap ðxÞ¼tμνða1…apÞðxÞ, and by requiring the symmetry of Tμν ðx; x Þ in the exchange of x and x , i.e., 0 0 0 0 1 2 … Tμν ðx; x Þ¼Tν μðx ;xÞ, the coefficients tμνðxÞ and tμνa1…ap ðxÞ with p ¼ ; ; are constrained. The symmetry of 0 0 Tμν0 ðx; x Þ permits us to express the coefficients of the covariant Taylor series expansion of Tμν0 ðx; x Þ in terms of their symmetric and antisymmetric parts in μ and ν. We have for the coefficients of lowest orders (see, e.g., Refs. [35,39])

tμν ¼ tðμνÞ; ðA8aÞ

1 2 tμνa1 ¼ð = ÞtðμνÞ;a1 þ t½μνa1 ; ðA8bÞ

tμνa1a2 ¼ tðμνÞa1a2 þ t½μνða1;a2Þ; ðA8cÞ

3 2 − 1 4 tμνa1a2a3 ¼ð = ÞtðμνÞða1a2;a3Þ ð = ÞtðμνÞ;ða1a2a3Þ þ t½μνa1a2a3 : ðA8dÞ

In order to solve the recursion relations (39a), (39b), (40), (43a), (43b) and (44) but also to do most of the calculations in Secs. III and IV and, in particular, to obtain the explicit expression of the renormalized stress-energy-tensor operator, it is 1=2 −1=2 1=2 ;μ 1=2 necessary to have at our disposal the covariant Taylor series expansions of the biscalars Δ , Δ Δ ;μσ and □Δ and of the bivectors σ;μν0 and □gμν0 but also of some bitensors such as σ;μν, gμν0;ρ and gμν0;ρ0 . Here, we provide these expansions up to the orders necessary in this article (for higher orders, see Refs. [71,72]). We have   1 1 1 1 1 1=2 ;a1 ;a2 ;a1 ;a2 ;a3 p q ;a1 ;a2 ;a3 ;a4 Δ ¼ 1 þ R σ σ − R ; σ σ σ þ R ; þ R R þ R R σ σ σ σ 12 a1a2 24 a1a2 a3 80 a1a2 a3a4 360 a1qa2 a3pa4 288 a1a2 a3a4   1 1 1 p q ;a1 ;a2 ;a3 ;a4 ;a5 3 − R ; þ R R ; þ R R ; σ σ σ σ σ þ Oðσ Þ; ðA9Þ 360 a1a2 a3a4a5 360 a1qa2 a3pa4 a5 288 a1a2 a3a4 a5   1 1 1 1 1 1 1 1=2 p pq pqr ;a1 ;a2 □Δ ¼ R þ □R − R; þ RR − R R þ R R þ R R σ σ 6 40 a1a2 120 a1a2 72 a1a2 30 a1 pa2 60 pa1qa2 60 a1 pqra2  1 1 1 1 1 p p q − − R; þ ð□R Þ; þ RR ; − R R ; þ R ; R 360 a1a2a3 120 a1a2 a3 144 a1a2 a3 45 a1 pa2 a3 180 q a1 a2pa3  1 1 p q pqr ;a1 ;a2 ;a3 2 þ R R ; þ R R ; σ σ σ þ Oðσ Þ; ðA10Þ 180 q a1pa2 a3 90 a1 pqra2 a3

1 −1=2 1=2 ;μ ;a1 ;a2 3=2 Δ Δ ;μσ ¼ R σ σ þ Oðσ Þ; ðA11Þ 6 a1a2 1 ;a1 ;a2 3=2 σ;μν ¼ gμν − Rμ ν σ σ þ Oðσ Þ; ðA12Þ 3 a1 a2

0 1 ν ;a1 ;a2 3=2 gν σ;μν0 ¼ −gμν − Rμ ν σ σ þ Oðσ Þ; ðA13Þ 6 a1 a2

0 1 1 ν ;a1 ;a1 ;a2 3=2 gν gμν0;ρ ¼ − Rμνρ σ þ Rμνρ ; σ σ þ Oðσ Þ; ðA14Þ 2 a1 6 a1 a2

0 0 1 1 ν ρ ;a1 ;a1 ;a2 3=2 gν gρ gμν0;ρ0 ¼ − Rμνρ σ þ Rμνρ ; σ σ þ Oðσ ÞðA15Þ 2 a1 3 a1 a2 and   0 2 1 1 1 ν a1 p pq ;a1 ;a2 3=2 gν □gμν0 ¼ R μ;ν σ þ − R μ;ν þ Rμν R − Rμ Rν σ σ þ Oðσ Þ: ðA16Þ 3 a1½ 6 a1½ a2 6 pa1 a2 4 pqa1 a2

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