Stueckelberg massive electromagnetism in curved spacetime: Hadamard renormalization of the stress-energy tensor and the Casimir effect 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 quantization; (ii) the wave equations for the massive spin-1 field Aμ, for the auxiliary Stueckelberg scalar field Φ and for the ghost fields C and C ; (iii) Ward identities; (iv) the Hadamard representation of the various Feynman propagators 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-mass limit, the massive vector field contribution reduces smoothly to the result obtained from Maxwell’s theory. As an application of our results, we consider the Casimir effect outside a perfectly conducting medium with a plane boundary. We discuss the results obtained using Stueckelberg but also de Broglie-Proca electromagnetism, and we consider the zero-mass limit of the vacuum energy in both theories. We finally compare the de Broglie-Proca and Stueckelberg formalisms and highlight the advantages of the Stueckelberg point of view, even if, in our opinion, the de Broglie-Proca and Stueckelberg approaches of massive electromagnetism are two faces of the same field theory.
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 photon. 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 quantum field theory 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 Standard Model of particle physics. 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 gauge theory, 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 string theory 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 massive gravity 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 symmetry 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 propagator 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 regularization 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 ;gμν ¼SCl½Aμ; ;gμν þSGB½Aμ; ;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 ;gμν ¼SA½Aμ;gμν þSΦ½ ;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□ − ξm2 Φ 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; x0Þ¼ihψjTC ðxÞCðx0Þjψið26Þ 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 Þ g 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; x0Þ g 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; x0Þ¼hψj½C ðxÞ;Cðx0Þ jψið63Þ 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 ½μν ag 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 ½μν a;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 ½μν a 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 ½μρ aÞ 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: a1 ap ðμνÞa1 ap 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 a1 ap ½μν a1 ap ;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