The Quantization of Proca Fields on Globally Hyperbolic Spacetimes: Hadamard States and Møller Operators

This paper deals with several issues concerning the algebraic quantization of the real Proca field in a globally hyperbolic spacetime and the definition and existence of Hadamard states for that field. In particular, extending previous work, we construct the so-called Møller ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-isomorphism between the algebras of Proca observables on paracausally related spacetimes, proving that the pullback of these isomorphisms preserves the Hadamard property of corresponding quasifree states defined on the two spacetimes. Then, we pull back a natural Hadamard state constructed on ultrastatic spacetimes of bounded geometry, along this ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-isomorphism, to obtain an Hadamard state on a general globally hyperbolic spacetime. We conclude the paper, by comparing the definition of an Hadamard state, here given in terms of wavefront set, with the one proposed by Fewster and Pfenning, which makes use of a supplementary Klein–Gordon Hadamard form. We establish an (almost) complete equivalence of the two definitions.


Introduction
The (algebraic) quantization of a quantum field propagating in a globally hyperbolic curved spacetime pM, gq [7,56] and the definition of meaningful quantum states has been and continues to be at the forefront of scientific research. Linearized theories are the first step of all perturbative procedures, so the definition of physically meaningful states for linearized field equations is an important task.
Gaussian, also known as quasifree, states ω : A Ñ C on the relevant CCR or CAR unital˚algebra A of observables of a given quantum field are an important family of (algebraic) states [40]. They are completely defined by assigning the two-point function, a bi-distribution ω 2 px, yq on the sections used to smear the field operator.
A crucial physical requirement on ω is the so-called Hadamard condition, which is needed, in particular, for defining locally-covariant renormalization procedures of Wick polynomials [18,40] and for the mathematical formulation of locally covariant perturbative renormalization in quantum field theory [52].

Generalized Klein-Gordon vector fields
All the notations and conventions used in this section to briefly summarize our results will be defined precisely later. For a charged (i.e. complex) Klein-Gordon field A, possibly vector-valued, the construction of Hadamard states amounts to finding distributional bi-solutions Λ2 px, yq of the Klein-Gordon equation NA " 0 describing the two-point functions 1 ωpâpfqâ˚pf 1 qq " ż MˆM Λ2 px, yq cd γ ca pxqγ db pyqf a pxqf 1 b pyqvol g b vol g ": Λ2 pf, f 1 q , and ωpâ˚pf 1 qâpfqq " ż MˆM Λ2 px, yq cd γ ca pxqγ db pyqf 1 a pxqf b pyqvol g b vol g ": Λ2 pf,f 1 q .
Above, the generators of the CCR˚-algebra of the Proca fieldâpfq andâ˚pf 1 q "âpfq˚are the (algebraic) field operators smeared with smooth compactly supported complex sections f, f 1 of the relevant complex vector bundle E Ñ M. That bundle is equipped with a non-degenerate Hermitian 2 fiberwise scalar product (not necessarily positive) γ. In case of the standard complex vector Klein-Gordon field over pM, gq constructed out the 1-form Hodge D'Alembertian or the Levi-Civita vector D'Alembertian, the vector bundle E is the one of smooth 1-forms T˚M C :" T˚M`iT˚M and the Hermitian scalar product γ is the indefinite one induced by the metric g in T˚M C , i.e., γ " g 7 . In the general case, a Klein-Gordon operator N is by definition a second-order operator on the smooth sections of E which is normally hyperbolic [3,4]: its principal symbol σ N satisfies σ N pξq "´g 7 pξ, ξq Id E for all ξ P T˚M, where Id E is the identity automorphism of E.
N is also required to be formally selfadjoint with respect to the Hermitian scalar product (generally non-positive!) induced on the space of complex sections f by γ and the volume form vol g , pf|gq : " The scalar complex Klein-Gordon field is encompassed by simply taking C as canonical fiber of E and using the trivial positive scalar product. The requirements on the bi-distributions Λ2 are, where G N is the causal propagator of N, p1q N x Λ2 px, yq " Λ2 px, yqN y " 0 and Λ2´Λ2 "´iG N ; p2q Λ2 pf, fq ě 0 , where Λ˘pf, fq " 0 implies f " Ng for a compactly supported section g; p3q W F pΛ2 q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u .
The second part of (1) corresponds to the canonical commutations relations, the first part is the "on-shell" condition, while condition (2) is the positivity requirement on two-functions. Then, the Gelfand-Naimark-Segal construction gives rise to a˚-representation of A g in terms of densely defined operators in a Hilbert space which, as a consequence of the above requirements (1) and (2) and the Wick rule, is a Fock space. Here ω is the expectation value referred to vacuum state and the action of the image of the representation on the vacuum state produces the dense invariant domain of the representation itself. Requirement (3) is the celebrated Hadamard condition (also known as the microlocal spectrum condition) which ensures the correct short-distance behavior of the n-point functions of the field. This condition has a long history which can be traced back to [22], passing to [41] and [50,51] (see [40] for a review). It plays a crucial role in various contexts of quantum field theory in curved spacetime. In particular, but not only, in perturbative renormalization and semiclassical quantum gravity. More recently, Gérard and Wrochna in [25,27], proved that condition (1)- (3) can be controlled at the same time by using methods of pseudodifferential calculus in spacetimes of bounded geometry (see also the subsequent papers [28][29][30][31][32]). When dealing with real quantum fields, as in this work, for instance the Klein-Gordon real vector field A, a single bidistribution ω 2 px, yq is sufficient to define a quasifree state ω: whereâpfq "âpfq˚is the (algebraic) field operators smeared with smooth real compactly supported sections f of a relevant real vector bundle E Ñ M, equipped with a fiberwise real symmetric nondegenerate (but not necessarily positive) scalar product γ. As before, a Klein-Gordon operator N is by definition a second-order differential operator on the smooth sections of E which is normally hyperbolic (same definition as for the complex case) and formally selfadjoint with respect to the real symmetric scalar product (not necessarily positive) pf|gq :" ż M f a pxqγ ab pxqg b pxqvol g pxq .
In the case of the standard real vector Klein-Gordon field (constructed out of the Hodge D'Alembertian or the Levi-Civita D'Alembertian) the bundle is exactly T˚M, equipped with a real symmetric non-degenerate but indefinite fiberwise scalar product induced by the metric g on T˚M, namely γ " g 7 . The theory of the scalar real Klein-Gordon field is encompassed simply by taking R as canonical fiber of E with trivial positive scalar product.
The apparently new continuity condition (3)' for the real case is actually embodied in the positivity condition (2) for the complex case [23]. As a matter of fact (2)' and (3)' together give rise to positivity of the whole state ω on A g induced by ω 2 in the real case. Once again, the GNS construction gives rise to a representation of the (complex) unital˚-algebra A g generated by the field operatorsâpfq exactly as in the complex case.
Since the Klein-Gordon equations are normally hyperbolic, not only they are Green hyperbolic so that the Green operators GP and the causal propagator G P " GP´GṔ can be therefore defined, but the Cauchy problem is also automatically well posed [3,4]. An important implication of this fact is that the two-point function of a quasifree state can be defined as a Hermitian or real bilinear form -in the complex and real case respectively -on the Cauchy data of solutions of the Klein-Gordon equation (e.g., see [48]). We follow this route in the present paper and, to this end, we will translate (1)-(3) and (1)'-(4)' in the language of Cauchy data.

Issues with the quantization of the Proca field
Most of the quantum theories are described by Green hyperbolic operators [3,4], as Klein-Gordon operators N discussed above or the Proca operator [16,55], studied in this work, P " δd`m 2 acting on smooth 1-forms A P Ω 1 pMq and where m 2 ą 0 is a constant. These operators are usually formally self-adjoint w.r.t. a (Hermitian or real symmetric) scalar product induced by the analog γ on the fibers of the relevant vector bundle. In general γ is not positive definite. Very common and physical examples are: the standard vector Klein-Gordon field, the Proca field, the Maxwell field, more generally, the Yang-Mills field and also the linearized gravity. Referring to the Proca, and in general all 1-form fields, we have that γ " g 7 is the inverse (indefinite!) Lorentzian metric of the spacetime pM, gq.
Unfortunately, in those situations, the Hadamard condition (4) and (5)' are in conflict with the positivity of states, respectively, (3) and (2)'-(3)'. It is known that for a vectorial Klein-Gordon operator that is formally self-adjoint w.r.t. an indefinite Hermitian/real symmetric scalar product, the existence of quasifree Hadamard states is forbidden (see the comment after [53,Proposition 5.6] and [29,Section 6.3]).
The case of a (real) Proca field seems to be even more complicated at first glance. In fact, on the one hand differently from the Klein-Gordon operator, the Proca operator is not even normally hyperbolic and this makes more difficult (but not impossible) the proof of the well-posedness of the Cauchy problem, in particular. On the other hand, similarly to the case of the vectorial Klein-Gordon theory, the Proca theory deals with an indefinite fiberwise scalar product. Actually, as we shall see in the rest of the work, these two apparent drawbacks cooperate to permit the existence of quasifree Hadamard states. Positivity of the two-point function ω 2 is restored when dealing with a constrained space of Cauchy conditions that make well-posed the Cauchy problem.
In the present paper, we study the existence of quasifree Hadamard states for the real Proca field on a general globally hyperbolic spacetime. A definition of Hadamard states for the Proca field was introduced by Fewster and Pfenning in [16], to study quantum energy inequalities, with a definition more involved than the one based on conditions (3) and (4)' above. They also managed to prove that such states exist in globally hyperbolic spacetimes whose Cauchy surfaces are compact.
Differently from Fewster-Pfenning's definition, here we adopt a definition of Hadamard state which directly relies on conditions (3) and (4)' above and we consider a generic globally hyperbolic spacetime. At the end of the work, we actually prove that the two definitions of Hadamard states are substantially equivalent.
Before establishing that equivalence, using the technology of the Møller operators we introduced in [48] for normally hyperbolic operators, and here extended to the Proca field, we prove the existence of quasifree Hadamard states in every globally hyperbolic spacetime, also in the case in which their Cauchy hypersurfaces are not compact.
As a matter of fact, it is enough to focus our attention on ultrastatic spacetimes of bounded geometry. In this class of spacetimes, we directly work at the level of initial data for the Proca equation and we establish the following, also by taking advantage of some technical results of spectral theory applied to elliptic Hilbert complexes [5].
1. The initial data of the Proca equations are a subspace C Σ of the initial data of a couple of Klein-Gordon equations, one scalar and the other vectorial, however both defined on bundles with fiberwise positive real symmetric scalar product; 2. The difference of a pair of certain Hadamard two-point functions for two above-mentioned Klein-Gordon fields becomes positive once that its arguments are restricted to C Σ . There, it defines a two-point function ω 2 for a quasifree state ω of the Proca field; 3. ω is also Hadamard since it is the difference of two two-point functions of Klein-Gordon fields which are Hadamard. They are Hadamard in view of known results of microlocal analysis of pseudodifferential operators on Cauchy surfaces of bounded geometry, for more details the interested reader can refer to [23].
Every field theory defined on a globally hyperbolic spacetime pM, gq is connected to one defined on an ultrastatic spacetime of bounded geometry pRˆΣ,´dt 2`h q through a Møller operator: the associated Møller˚-isomorphism between the algebras of Proca observables preserves the Hadamard condition. We therefore conclude that every globally hyperbolic spacetime pM, gq admits a Hadamard state for the Proca field. This state is nothing but the Hadamard state on pRˆΣ,´dt 2`h q pulled back to pM, gq by the Møller˚-isomorphism. One novelty of this paper is in particular a direct control of the positivity of the two-point functions, obtained by spectral calculus of elliptic Hilbert complexes. Some microlocal property of the Møller operators then guarantees the validity of the Hadamard condition without exploiting the classical so called deformation argument, or better, by re-formulating it into a new form in terms of Møller operators.

Main results
We explicitly state here the principal results established in this paper referring, for the former, to the notions introduced in the previous section. Below, GP denote the retarded and advanced Green operators of the Proca equation (2.3), we shall discuss in Section 3. The symbol κ g 1 g denotes a linear fiber-preserving isometry from the spaces of smooth sections ΓpV g q to ΓpV g 1 q constructed in Section 3. Here, V g indicates the vector bundle of real 1-forms over the spacetime pM, gq whose sections are the argument of the Proca operator P.
Theorem 1 (Theorems 3.2 and 3.7). Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, with associated real Proca bundles V g and V g 1 and Proca operators P, P 1 . If the metric are paracausally related g » g 1 , then there exists a R-vector space isomorphism R : ΓpV g q Ñ ΓpV g 1 q, called Møller operator of g, g 1 (with this order), such that the following facts are true.
The next result (Theorem 2) permits us to promote R to a˚-isomorphism R of the algebras of field operators A, A 1 respectively associated to the paracausally related metrics g and g 1 , with the associated P, P 1 , and generated by respective Hermitian field operators apfq and a 1 pf 1 q with f, f 1 compactly supported smooth real sections of V. We will introduce these notions in Section 4. These field operators satisfy respective CCRs rapfq, aphqs " iG P pf, hqI , ra 1 pf 1 q, a 1 ph 1 qs " iG P 1 pf 1 , h 1 qI 1 and the said unital˚-algebra isomorphism R : A 1 Ñ A is uniquely determined by the requirement The final important result regards the properties of R for the algebras of a pair of paracausally related metrics g, g 1 when it acts on the states ω : A Ñ C, ω 1 : A 1 Ñ C of the algebras in terms of pull-back. ω 1 " ω˝R .
As is known, the most relevant (quasifree) states in algebraic QFT are Hadamard states characterized by the microlocal spectrum condition valid for the wavefront set of their two-point functions or, equivalently, an universal short distance structure of the distribution defining the two-point function. A definition of Hadamard state for the Proca field was first stated by Fewster and Pfenning in [16] and corresponds to Definition 6.1 in this paper. That definition requires the existence of a bisolution of the Klein Gordon field satisfying the microlocal spectrum condition. This bisolution is next used to construct the two-point function of the Proca field. Differently, in this work we adopt a direct definition (Definition 4.5) which only requires the validity of the microlocal spectrum condition directly for the two-point function of the Proca two-point function.
We also prove that our definition, exactly as it happens for Fewster and Pfenning's definition, satisfies some physically relevant properties. In addition to these general results, we also prove that the Hadamard property is preserved by the Møller operators as one of main results of this work.
Theorem 2 (Theorem 4.9). Let g, g 1 be paracausally related metric and consider the corresponding Proca operators P, P 1 . Finally refer to the associated on-shell CCR-algebras A and A 1 .
Attention is next focused on the existence problem of quasifree Hadamard states for the real Proca field in a generic globally hyperbolic spacetime. The technology of Møller operators allows us to reduce the construction of Hadamard states for the real Proca field to the special case of an ultrastatic spacetime pRˆΣ,´dt 2`h q. In this class of spacetimes, if assuming the further geometric hypothesis of bounded geometry, we provide a direct construction of a Hadamard state just working on the space of initial data C Σ for the Proca equation PA " 0 where A P ΓpV g q has compact Cauchy data. Here, A decomposes as A " A p0q dt`A p1q , where A p0q and A p1q and are, respectively, a 0-form and a 1-form on ttuˆΣ. As we shall prove, this space of initial data is actually constrained in order to satisfy the existence and uniqueness theorem for the Cauchy problem: where pa p0q , π p0q q :" pA p0q , B t A p0q q| t"0 and pa p1q , π p1q q :" pA p1q , B t A p1q q| t"0 .
Theorem 3 (Propositions 5.8 and 5.10). Consider the˚-algebra A g of the real Proca field on the ultrastatic spacetime pM, gq " pRˆΣ,´dt b dt`hq, with pΣ, hq a smooth complete Riemannian manifold. Let η 0 :"´1, η 1 :" 1 and h 7 pjq denote the standard inner product of j-forms on Σ induced by h. Then the two-point function ω µ papfqapf 1 qq " ω µ2 pf, f 1 q :" µpA, A 1 q`i 2 σ pP q pA, A 1 q defines a quasifree state ω µ on A g where f, f 1 P Γ c pV g q. Above A " G P f , where ∆ pjq is the Hodge Laplacian for compactly supported real smooth j-forms on pΣ, hq. Finally, ω µ is Hadamard if pΣ, hq is of bounded geometry.
Above the bar denotes the closure of the considered operators defined in suitable L 2 -spaces of forms according to the theory of elliptic Hilbert complexes.
Using the fact that every globally hyperbolic spacetime is paracausally related to an ultrastatic spacetime with bounded geometry and combining the two previous Theorems, we can conclude that Proca fields can be quantized in any globally hyperbolic spacetime and admit Hadamard states.
Theorem 4. Let pM, gq be a globally hyperbolic spacetime and refer to the associated CCRalgebras A g of the real Proca field. Then there exists a quasifree Hadamard state on A g .
Eventually, coming back to the alternative definition of Hadamard states proposed by Fewster and Pfenning in [16], we prove an almost complete equivalence theorem, which is the last main achievement of this work.
Theorem 5 (Theorem 6.6). Consider the globally hyperbolic spacetime pM, gq and a quasifree state ω : A g Ñ C for the Proca algebra of observables on pM, gq with two-point function ω P Γ 1 c pV g b V g q. The following facts are true.
(a) If ω is Hadamard according to Fewster and Pfenning, then it is also Hadamard according to Definition 4.5.
(b) If pM, gq admits a Hadamard state according to Fewster and Pfenning and ω is Hadamard according to Definition 4.5, then ω is Hadamard in the sense of Fewster-Pfenning's definition.
The existence of Hadamard states according to Fewster-Pfenning's definition was proved in [16] for spacetimes with compact Cauchy surfaces. For these spacetimes the equivalence of the two definitions is complete.

Structure of the paper
The paper is structured as follows. In Section 3 we will provide a detailed analysis of the Møller maps and the Møller operator for classical Proca fields. In particular, we will analyze the relation between the Møller operators and the causal propagators of Proca operators on paracausally related globally hyperbolic spacetimes. In Section 4 we will extend the action of the Møller operators to a˚-isomorphism of the CCR-algebras of free Proca fields. This will allow us to pullback quasifree Hadamard states preserving the microlocal spectrum condition. In this section we also analyze the general properties of Hadamard states including their existence. The explicit construction of Hadamard states in an ultrastatic spacetime is performed in Section 5. In Section 6 we show that the microlocal spectrum condition is essentially equivalent to the definition of Hadamard states proposed by Fewster and Pfenning. Finally, we conclude our paper with Section 7, where open issues and future prospects are presented.

Acknowledgments
We are grateful to Nicolò Drago, Chris Fewster, Christian Gérard, and Igor Khavkine for helpful discussions related to the topic of this paper. This work was written within the activities of the INdAM-GNFM

Conventions and notation of geometric tools in spacetimes
Throughout all the paper the symbols Ă and Ą allow the case ". We explicitly adopt the signature p´,`,¨¨¨,`q for Lorentzian metrics.
Throughout pM, gq denotes a spacetime, i.e., a paracompact, connected, oriented, timeoriented, smooth, Lorentzian manifold M, whose Lorentzian metric is g. As in [48], the Lorentzian metrics g of spacetimes are hereafter supposed to be equipped with their own temporal orientation.
All considered spacetimes pM, gq are also globally hyperbolic. In other words, a (smooth) Cauchy temporal function t : M Ñ R exists. By definition dt is timelike, past directed and pM, gq is isometric to pRˆΣ, g 1 q with metric where β : RˆΣ Ñ R is a smooth positive function, h t is a Riemannian metric on each slice Σ t :" ttuˆΣ varying smoothly with t, and these slices are smooth spacelike Cauchy hypersurfaces. By definition they are achronal sets intersected exactly once by every inextensible timelike curve (see [47] for a recent up-to-date survey on the subject).
According to [48], given two globally hyperbolic metrics g and g 1 on M , g ĺ g 1 means that V gp Ă V g 1p for all p P M, where V gp Ă T p M is the open cone of future directed timelike vectors at p in pM, gq.
Two globally hyperbolic metrics g and g 1 on M are paracausally related , written g » g 1 , if there exists a finite sequence of globally hyperbolic metrics g 1 " g, g 2 . . . , g n " g 1 on M such that for each pair of consecutive metrics either For a discussion on this notion, its properties, and examples we refer to [48,Section 2]. We henceforth denote by ΓpEq the real vector space of smooth sections of any real vector bundle E Ñ M. More precisely, as in [48], we denote with Γ c pEq, Γ f c pEq, Γ pc pEq, Γ sc pEq the space of sections respectively with compact support, future-compact (i.e. whose support stays before a smooth spacelike Cauchy surface), past-compact (i.e. whose support stays after a smooth spacelike Cauchy surface), and spatially-compact support (i.e. whose support on every smooth spacelike Cauchy surface is compact).
If E Ñ M and E 1 Ñ M 1 are two vector bundles, E b E 1 denotes the external tensor product of these vector bundles. This vector bundle has base MˆM 1 and fiber at pp, p 1 q given by the tensor products of the respective fibers at p P M and p 1 P M 1 respectively. If f P ΓpEq and The tensor product of linear operators acting on sections of an external product bundle are denoted by b.

Smooth forms, Hodge operators, and the Proca equation
In this work we frequently deal with real smooth k-forms f, g P Ω k pMq, where k " 0, . . . , n " dim M (and one usually adds Ω n`1 pMq " Ω´1pMq " t0u). The Hodge real inner product can be computed by integrating the fiberwise product with respect to the volume form induced by g: where at least one of the two forms has compact support and g 7 pkq is the natural inner product of k-forms induced by g. This symmetric real scalar product p¨|¨q g,k is always non-degenerate but it is not positive when g is Lorentzian as in the considered case. It is positive when g is Riemannian.
If k " 1, we simply write In this context, d pkq : Ω k pMq Ñ Ω k`1 pMq is the exterior derivative and δ pkq g : Ω k pMq Ñ Ω k´1 pMq is the codifferential operator acting on the relevant spaces of smooth k-forms Ω k pMq on M depending on the metric g on M. d pkq and δ pk`1q g are the formal adjoint of one another with respect to the Hodge product (2.1) in the sense that pd pkq f|gq g,k`1 " pf|δ pk`1q g gq g,k , @f P Ω k pMq , @g P Ω k`1 pMq if f or g is compactly supported.
In the rest of the paper we will often omit the indices g,k and pkq referring to the metric and the order of the used forms, when the choice of the used metric and order will be obvious from the context.
If pM, gq is globally hyperbolic, we call Proca bundle the real vector bundle V g :" pT˚M, g 7 q obtained by endowing the cotangent bundle with the fiber metric given by the dual metric g 7 (also appearing in (2.1)) defined by g 7 pω p , ω 1 p q :" gp7ω p , 7ω 1 p q for every ω, ω 1 P ΓpT˚Mq and p P M, where 7 : ΓpT˚Mq Ñ ΓpTMq is the standard musical isomorphism. By construction ΓpV g q " Ω 1 pMq and Γ c pV g q " Ω 1 c pMq. Here and henceforth Ω k c pMq Ă Ω k pMq is the subspace of compactly supported real smooth k-forms on M.
The formally selfadjoint Proca operator P on pM, gq is defined by choosing a (mass) constant m ą 0, the same for all globally hyperbolic metrics we will consider on M in this work, P " δd`m 2 : ΓpV g q Ñ ΓpV g q, (2.2) where d :" d p1q , δ :" δ p2q g . Actually P depends also on g, but we shall not indicate those dependencies in the notation for the sake of shortness.
The Proca equation we shall consider in this paper reads where, as said above, Γ sc pV g q is the space of real smooth 1-forms which have compact support on the Cauchy surfaces of the globally hyperbolic spacetime pM, gq.

Møller Maps and Møller Operators
The construction of the so-called Møller operator for hyperbolic PDEs (coming from the realm of quantum field theories on curved spacetimes) has been studied extensively in various contexts in Quantum Field Theory, see e.g. [9,10,12,14,48,49]. The key idea was to inspired by the scattering theory: Starting with two "free theories" described by the space of solutions of normally hyperbolic operators (see (3.3) below) N 0 and N 1 in corresponding spacetimes pM, g 0 q and pM, g 1 q, respectively, we connected them through an "interaction spacetime" pM, g χ q with a "temporally localized" interaction defined by interpolating the two metrics by means of a smoothing function χ. Here we need two Møller maps: Ω`connecting pM, g 0 q and pM, g χ q -which reduces to the identity in the past when χ is switched off -and a second Møller map connecting pM, g χ q to pM, g 1 q -which reduces to the identity in the future when χ constantly takes the value 1. The "S-matrix" given by the composition S :" Ω´Ω`will be the Møller map connecting N 0 and N 1 . As remarked in [48,Section 6], all the results concern vector-valued normally hyperbolic operators acting on real vector bundles whose fiber metric does not depend on the globally hyperbolic metrics g chosen on M. These operators are also assumed to be formally selfadjoint with respect to the associated real symmetric scalar product on the sections of the bundle. As already pointed out in the introduction, to quantize the theory defining quantum states on an associated˚-algebra of observables, the fiberwise metric on E should be assumed to be positive.
This section aims to extend the construction of the Møller operators to Proca fields. The main difficulties we have to face with respect to the case of the Klein-Gordon equation are the following: • the fiber metric of the Proca bundle depends on the underlying globally hyperbolic metrics g chosen on M (and it is not positive definite); • Proca operators are not normally hyperbolic.
The next two sections are devoted to tackle these technical issues before starting with the construction of the Møller maps.

Linear fiber-preserving isometry
As said above, to construct Møller maps for the Proca field we should be able to compare different fiberwise metrics on T˚M when we change the metric g on M. This will be done by defining suitable fiber preserving isometries. If g and g 1 are globally hyperbolic on M and g » g 1 , it is possible to define a linear fiberpreserving isometry from ΓpV g q to ΓpV g 1 q we denote with κ g 1 g and we shall take advantage of it very frequently in the rest of this work. In other words, if f P ΓpV g q, then κ g 1 g f P ΓpV g 1 q, the map κ g 1 g : ΓpV g q Ñ ΓpV g 1 q is R linear, and g 17 ppκ g 1 g fqppq, pκ g 1 g gqppqq " g 7 pfppq, gppqq @p P M .
Let us describe the (highly non-unique) construction of κ gg 1 . If χ P C 8 pM; r0, 1sq and g 0 ĺ g 1 , then is a Lorentzian metric globally hyperbolic on M (see [48,Section 2] for details) and satisfies Now consider the product manifold N :" RˆM, equipped with the indefinite non-degenerate metric where g t " p1´f ptqqg 0`f ptqg χ and f : R Ñ r0, 1s is smooth and f ptq " 0 for t ď 0, f ptq " 1 for t ě 1. Notice that g t is Lorentzian according to [48] because g 0 ĺ g χ and h is indefinite nondegenerate by construction. At this point r κ χ0 : TM Ñ TM is the fiber preserving diffeomorphism such that r κ χ0 px, vq is the parallel transport form p0, xq to p1, xq of v P T x M Ă T p0,xq N along the complete h-geodesic R Q t Þ Ñ pt, xq P N. Standard theorems on joint smoothness of the flow of ODEs depending on parameters assure that r κ χ0 : TM Ñ TM is smooth. Notice that r κ χ0 | TxM : T x M Ñ T x M is also a h-isometry from known properties of the parallel transport and thus it is a g 0 , g χ -isometry by construction because h pt,xq pv, vq " g t pv, vq if v P T x M Ă T pt,xq N. Taking advantage of the musical isomorphisms, r κ χ0 induces a fiber-bundle map κ χ0 : T˚M Ñ T˚M which can be seen as a map on the sections of ΓpV g 0 q and producing sections of ΓpV gχ q, preserving the metrics g 7 0 , g 7 χ . Then the required Proca bundle isomorphism κ g 1 g " κ g 1 g 0 is defined by composition: where κ 1χ from ΓpV gχ q to ΓpV g 1 q is defined analogously to κ χ0 . The general case g » g 1 can be defined by composing the fiber preserving linear isometries κ g k`1 g k or κ g k ,g k`1 .

Klein-Gordon operator associated to a Proca operator and Green operators
We pass to tackle the issue of normal hyperbolicity of P. As we shall see here, it is not really necessary to construct the Møller maps, and the weaker requirement of Green hyperbolicity is sufficient.
Let N be the Klein-Gordon operator associated to the Proca operator P (2.2) acting on 1-forms N :" δd`dδ`m 2 : ΓpV g q Ñ ΓpV g q . Notice that this operator is normally hyperbolic: its principal symbol σ N satisfies σ N pξq "´g 7 pξ, ξq Id Vg for all ξ P T˚M, where Id Vg is the identity automorphism of V g . (3.3) Therefore the Cauchy problem for N is well-posed [3,4]. Both N and P are formally selfadjoint with respect to the Hodge scalar product (2.1) on Ω 1 c pMq " Γ c pV g q.
Since m 2 ą 0 and δ p1q g δ p2q g " 0, it is easy to prove that the Proca equation (2.3) is equivalent to the pair of equations As already noticed, differently from N, the Proca operator is not normally hyperbolic. However, it is Green hyperbolic [3,4,6] as N, in particular there exist linear maps, dubbed advanced Green operator GP : Γ pc pV g q Ñ ΓpV g q and retarded Green operator GṔ : Γ f c pV g q Ñ ΓpV g q uniquely defined by the requirements The causal propagator of P is defined as All these maps are also continuous with respect to the natural topologies of the definition spaces [6]. As a matter of fact (see [6,Proposition 3.19] and also [4]), the advanced and retarded Green operator GP : Γ pc{f c pV g q Ñ Γ pc{f c pV g q can be written as here GN are the analogous Green operators for the Klein-Gordon operator N. Therefore The fact that P is normally hyperbolic can be proved just by checking that the operators above satisfy the requirements which define the Green operators as stated above, using the analogous properties for GN . Eq. (3.7) and the analogous properties for G N entail On account of [48, Proposition 3.6], for any smooth function ρ : M Ñ p0,`8q also ρP is Green hyperbolic and Gρ P " GP ρ´1.

Proca Møller maps
A smooth Cauchy time function in a globally hyperbolic spacetime pM, gq relaxes the notion of temporal Cauchy function, it is a smooth map t : M Ñ R such that dt is everywhere timelike and past directed, the level surfaces of t are smooth spacelike Cauchy surfaces and pM, gq is isometric to pRˆΣ, hq. Here, t identifies with the natural coordinate on R and the Cauchy surfaces of pM, gq identify with the sets ttuˆΣ.
From now on we indicate by N 0 , N 1 , N χ the Klein-Gordon operators (3.2) on M constructed out of g 0 , g 1 and g χ respectively, where the globally hyperbolic metric g χ is defined as in (3.1) (and thus g 0 ĺ g χ ĺ g 1 [48, Theorem 2.18]) and depends on the choice of a function χ P C 8 0 pM, r0, 1sq. Similarly, P 0 , P 1 , P χ denote the Proca operators (2.2) on M constructed out of g 0 , g 1 and g χ respectively.
We can state the first technical result.
(3) If f P ΓpV g 0 q or ΓpV gχ q respectively, then pR`fqppq " fppq for tppq ă t 0 , (3.10) pR´fqppq " fppq for tppq ą t 1 . (3.11) Proof. First of all, we notice that the operator R`is well defined on the whole space ΓpV g 0 q since for all sections f P ΓpV g 0 q we have that pP χ κ χ0 ρ´κ χ0 ρ P 0 qf P Γ pc pV g 1 q: indeed by definition, there exists a t 0 P R such that on t´1p´8, t 0 q and we have that P χ " P 0 , κ χ0 " Id and t is a smooth g 1 -Cauchy time function. Moreover, since g χ ĺ g 1 it follows that Γ pc pV g 1 q Ă Γ pc pV gχ q " DompG Pχ q.
To prove (1), we can first notice that R´1˝R`"´κ 0χ`GP 0 pρκ 0χ P χ´P0 κ 0χ q¯˝´κ χ0´Gρ Pχ pρP χ κ χ0´κχ0 P 0 q" Id´κ 0χ Gρ Pχ pρP χ κ χ0´κχ0 P 0 q`GP 0 pρκ 0χ P χ´P0 κ 0χ qκ χ0 GP 0 pρκ 0χ P χ´P0 ρκ 0χ qGρ Pχ pρP χ κ χ0´κχ0 P 0 q . To conclude it is enough to show that everything cancels out except the identity operator, but that just follows by using basic properties of Green operators and straightforward algebraic steps. We easily see that the last addend can be recast as: Gρ Pχ pρP χ κ χ0´κχ0 P 0 q, which fulfills its purpose. A specular computation proves that R´1 is also a right inverse. Almost identical reasonings prove that R´1 is a two sided inverse of R´which is also well defined, then bijectivity of R is obvious.
(2) follows by the following direct computation (3) Let us prove (3.10). In the following P˚denotes the formal dual operator of P acting on the sections of the dual bundle Γ c pVg q.
Using Proposition 3.1, we can pass to the generic case g » g 1 .
Theorem 3.2. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, with associated Proca bundles V g and V g 1 and Proca operators P, P 1 . If g » g 1 , then there exist (infinitely many) vector space isomorphisms, (1) referring to the said domains, for some smooth µ : M Ñ p0,`8q (which can always be chosen µ " 1 constantly in particular), and a smooth fiberwise isometry κ gg 1 : ΓpV g 1 q Ñ ΓpV g q.
Proof. Since g » g 1 , there exists a finite sequence of globally hyperbolic metrics g 0 " g, g 1 , .., g N " g such that at each step g k ĺ g k`1 or g k`1 ĺ g k . For all k P t0, .., N u we can associate to the metric a Proca operator P k . At each step the hypotheses of Proposition 3.1 are verified, in fact we can choose functions ρ k and ρ 1 k and the Møller map is given by R k " R k´˝Rk`. The general map is then built straightforwardly by composing the N maps constructed step by step: Regarding (1), by direct calculation we get that µ "

Causal propagator and Møller operator
The rest of this section is devoted to study the relation between Møller maps and the causal propagator of Proca operators. To this end, we use a natural extension of the notion of adjoint operator introduced in [48, Section 4.5].
Let g and g 1 (possibly g ‰ g 1 ) globally hyperbolic metric and let V g and V g 1 be a Proca bundle on the manifold M. Consider a R-linear operator where DompTq Ă ΓpV g q is a R-linear subspace and DompTq Ą Γ c pV g q.
is said to be the adjoint of T with respect to g, g 1 (with the said order) if it satisfies ż M g 17 ph, Tfq pxq vol g 1 pxq " ż M g 7´T: gg 1 h, f¯pxq vol g pxq @f P DompTq , @h P Γ c pEq.
When g " g 1 , we use the simplified notation T : :" T :gg .
As in [48], the adjoint operator satisfies a lot of useful properties which we summarize in the following proposition. Since the proof is analogous to the one of [48,Proposition 4.11], we leave it to the reader. Though the rest of this paper deal with the real case only, we state the theorem encompassing the case where the sections are complex and the fiber scalar product is made Hermitian by adding a complex conjugation of the left entry in the usual fiberwise real g 7 inner product, which becomes g 7 pf, gq, where the bar denotes the complex conjugation. Definition 3.3 extends accordingly. For this reason K will denote either R or C, and the complex conjugate c reduces to c itself when K " R. We keep the notation V g for indicating either the real or complex vector bundle T˚M or T˚M`iT˚M corresponding to two possible choices of K. (1) If the adjoint T : gg 1 of T exists, then it is unique.
(2) If T : ΓpV g q Ñ ΓpV g 1 q is a differential operator and g " g 1 , then T :gg exists and is the restriction of the formal adjoint to Γ c pEq. (In turn, the formal adjoint of T is the unique extension to ΓpEq of the differential operator T : as a differential operator.) (3) Consider a pair of K-linear operators T : DompTq Ñ ΓpV g 1 q, T 1 : DompT 1 q Ñ ΓpV g 1 q with DompTq, DompT 1 q Ă ΓpV g q and a, b P K. Then provided T : gg 1 and T 1: gg 1 exist.
Now we are ready to prove that the operators R admit adjoints and we explicitly compute them.
Proposition 3.5. Let g 0 , g 1 be globally hyperbolic metrics satisfying g 0 ĺ g 1 . Let R`, R´and R be the operators defined in Proposition 3.1 and fix, once and for all, ρ " c χ 0 and ρ 1 " c 1 0 where c χ 0 , c 1 0 are the unique smooth functions on M such that: Then we have: and can be recast in the form and can be recast in the form (3) The map R :g 0 g 1 : Γ c pV g 1 q Ñ Γ c pV g 0 q defined by R :g 0 g 1 :" R :g 0 gχ˝R:gχg 1 is invertible and We call it adjoint Møller operator. Moreover R :g 0 g 1 is a homeomorphism with respect to the natural (test section) topologies of the domain and of the co-domain.
Proof. We start by proving points (1) and (2). For any f P DompR`q " ΓpV g 0 q and h P Γ c pV gχ q we have ż We now split the problem and compute the adjoint of the two summands separately. The adjoint of the first one follows immediately by exploiting the properties of the existing isometry and Equations (3 For the second summand the situation is trickier and we cannot split the calculation in two more summands since it is crucial that the whole difference pc χ 0 P χ κ χ0´κχ0 P 0 q acts on a general f P ΓpV gχ q before we apply the Green operator whose domain is Γ pc pV gχ q.
use the properties of standard adjoints of Green operators for formally self-adjoint Green hyperbolic differential operators to get ż Now we are tempted to exploit the linearity of the integral and of the fiber product, but first, to ensure that the two integrals individually converge, we need to introduce a cutoff function: • We notice again that there is a Cauchy surface of the foliation Σ t 0 such that for all leaves with t ă t 0 the operator´P χ κ χ0´κ χ0 c χ 0 P 0¯" 0; • So take a t 1 ă t 0 and define a cutoff smooth function s : M Ñ r0, 1s such that s " 0 on all leaves with t ă t 1 .
In this way we are allowed to rewrite our last integral and split it in two convergent summands without modifying its numerical value.
where in the last identities we have used properties of the standard adjoints of the formally selfadjoint operators, of the isometries and of the cutoff function.
Since the domain of the operator is just made up of compactly supported sections, we may exploit the inverse property of the Green operators to immediately obtain that To see that the image of the operators is indeed compactly supported we can focus on R :g 0 gχ , the rest follows straightforwardly. The first summand c χ 0 κ 0χ does not modify the support of the sections, whereas the second does. Let us fix f P Γ c pV gχ q, then supp pGṔ χ fq Ă Jǵ χ psupp fq which means that GṔ χ f P Γ sf c , i.e it is space-like and future compact. The thesis follows by again observing that there is a Cauchy surface such that in its past´P χ κ χ0´κ The computation of the adjoint of R´is almost identical to the one just performed.
The first part of (3) is an immediate consequence of property (4) in Proposition 3.4, while the invertibility of the adjoint can be proved by explicitly showing that the operator serves as a left and right inverse of R :g 0 gχ . An analogous argument can be used for R The continuity of both the adjoint and its inverse comes by the same arguments used in the proof of [48,Theorem 4.12] (with the only immaterial difference that this time the smooth isometry κ χ0 is included in the definition of the Møller operator.) Remark 3.6. An interesting fact to remark is that having defined the adjoints over compactly supported sections makes the dependence on the auxiliary volume fixing functions disappear.
We conclude the section, by proving the second part of Theorem 1.
Theorem 3.7. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, with associated Proca bundles V g and V g 1 and Proca operators P, P 1 . If g » g 1 , it is possible to specialize the R-vector space isomorphism R : ΓpV g q Ñ ΓpV g 1 q of Proposition 3.2 such that the following further facts are true.
(1) The causal propagators G P and G P 1 (3.6), respectively of P and P 1 , satisfy RG P R : gg 1 " G P 1 .
(2) It holds R as above is called Møller operator of g, g 1 (with this order).
Proof. Since g » g 1 and the Møller map is defined as the composition R " R N˝. ..˝R 1 , we can use properties (4) in Proposition 3.4 and reduce to the case where g " g 0 ĺ g 1 " g 1 . With this assumption, (2) can be obtained following the proof of Proposition 3.1 and (3) is identical to [48,Theorem 4.12 (5)]. So we leave it to the reader. It remains to prove (1). Decomposing R as above, we define the maps R g 0 gχ , R gχg 1 by choosing the various arbitrary functions as in Proposition 3.5. We first notice where the first equality follows by definition, in the second one we have used the properties of Green operators, while in the third one we have just equated the two expressions for the adjoint operator according to (1) in Proposition 3.5 and performed some trivial algebraic manipulations. Another analogous computation can be performed for the retarded Green operator yielding Therefore, subtracting the two terms we get Applying now R´and its adjoint we get the claimed result.

The CCR algebra of observables of the Proca field
We now pass to introduce the algebraic formalism to quantize the Proca field [16,55]. Let pM, gq be a globally hyperbolic spacetime, V g be a Proca bundle and denote by P : ΓpV g q Ñ ΓpV g q the Proca operator. Following [40], we call on-shell Proca CCR˚-algebra, the˚-algebra defined as -A g is the free complex unital algebra finitely generated by the set of abstract elements I (the unit element), apfq and apfq˚for all f P Γ c pV g q, and endowed with the unique (antilinear) -involution which associates apfq to apfq˚and satisfies I˚" I and pabq˚" b˚a˚.
-I g is the two-sided˚-ideal generated by the following elements of A f : 1. apaf`bhq´aapfq´baphq , @a, b P R @f, h P Γ c pV g q; 2. apfq˚´apfq , @f P Γ c pV g q; 3. apfqaphq´aphqapfq´iG P pf, hqI , @f, h P Γ c pV g q; 4. apPfq , @f P Γ c pV g q.
The four entries of the list respectively implement linearity, hermiticity of the generators, canonical commutation relations and the equations of motion for the quantum field.
Remark 4.1. As in [16], we adopt in this paper the interpretation of apfq is pa|fq, where the pairing is the Hodge inner product of 1-forms (2.1). An equivalence class in A g is denoted by rapfqs "âpfq, the equivalence class corresponding to the identity is denoted by rIs " Id. The hermitian elements of the algebra A g are called observables.
Remark 4.2. Requirement 4, when we pass to the quotient algebra corresponds to the distributional relation Pâ " 0, when we take Remark 4.1 into account and the fact that P is formally selfadjoint. Since every solution of the Proca equation is a co-closed solution of the Klein-Gordon equation and vice versa, we conclude that δâ " 0, i.e.âpdfq " 0 for every f P Γ c pV g q, must be valid.
If, moreover, we deprive the ideal I g of the generators in 4, the quotient algebra is said to be off-shell, however it would still be convenient to assume the-closedness constraint when defining the off-shell algebra. That is when defining the relevant ideal of the free off-shell algebra, we should keep 1-3, we should drop 4, and we should replace it with the weaker condition 4'.âpdfq , @f P Γ c pV g q.
This work however deals with the on-shell algebra only, we shall indicate by A g throughout. A study of the off-shell algebra, which may result in some relevance in perturbative renormalization procedure will be done elsewhere.

Møller˚-isomorphism and Hadamard states
From now on let X be a topological vector space, we indicate by X 1 its topological dual. For example Γ 1 c pV g q represents the space of distributions acting on compactly supported test functions, and shall not be confused with the space of compactly supported distributions.
Having built the CCR-algebra, the subsequent step in quantization consists in finding a way to associate numbers to the abstract operators in A g by identifying a distinguished state. For sake of completeness, let us recall that a state over the Proca algebra A g a C-linear functional ω : A g Ñ C which is (i) Positive: ωpa˚aq ě 0 @a P A g , (ii) Normalized: ωpIq " 1 A generic element of the CCR-algebras A g of a quantum field can be written as a finite polynomial of the generatorsâpf q, where the zero grade term is proportional to I. To specify the action of a state it is sufficient to know its action on the monomials, i.e its n-point functions: If we impose continuity with respect to the usual topology on the space of compactly supported test sections we can uniquely extend the n-point functions to distributions in Γ 1 c pV nb g q we shall hereafter indicate by the symbol r ω n . Among all possible states the physical ones are the so-called quasifree (or Gaussian) Hadamard states. Quasifree means that the n-point distributions of the state have a structure resembling the one of a free theory, i.e they all can be recovered just by knowing the two-point distribution.
Definition 4.3. Consider the globally hyperbolic spacetime pM, gq and a state ω : A g Ñ C for the Proca algebra of observables on pM, gq. ω is called quasifree, if for all choices of f i P Γ c pV g q (i) ω n pf 1 , . . . , f n q " 0, if n P N is odd, where Π refers to the class of all possible decompositions of the set t1, 2, . . . , 2nu into n pairwise disjoint subsets of 2 elements ti 1 , i 2 u, ti 3 , i 4 u, . . ., ti n´1 , i n u with i 2k´1 ă i 2k for k " 1, 2, . . . , n.
Regarding the notion of Hadamard state for the Proca field, which is a vector field, we adopt the notions of microlocal analysis for vector-valued distributions introduced in [53].
Remark 4.4. The interpretation of the action of a distribution on test sections is formalized in the sense of the Hodge product (2.1). This interpretation is necessary in order to agree with the interpretation of the symbolâpfq stated in Remark 4.1, since some of the distributions we shall consider in the rest of the paper arise from field operators, as the two-point functions ω 2 pf, gq :" ωpâpfqâpgqq. If Γ c pV g q Q g Þ Ñ ω 2 p¨, gq P Γ 1 c pV g q is well-defined and continuous, ω 2 actually defines a distribution of Γ 1 c pV g b V g q and vice versa, as a consequence of the Schwartz kernel theorem as clarified below. From now on, if F P Γ 1 c pV g q and f P Γ c pV g q, the action of the former on the latter is therefore interpreted as the Hodge product (2.1) F pfq " pF |fq " pf|F q " ż M g 7 pF, fqvol g .
With a straightforward extension of the Definition 3.3, operators working on a generic space of k test-forms T : Ω k c pMq Ñ Ω k c pMq can be extended to the topological duals, i.e the associated distributions, in terms of the action T : on the argument of the distribution: pTF qpfq :" F pT : fq .
If S : Γ c pV g q Ñ Γ 1 c pV g q is continuous (in particular if S : Γ c pV g q Ñ Γ c pV g q is continuous), the standard Schwartz kernel theorem permits to introduce the distribution indicated with the same symbol S P Γ 1 c pV g b V g q, which is the unique distribution such that Spf b gq :" Spf, gq :" pSgqpfq " " pf|Sgq 2 .
Conversely, a distribution of Γ 1 c pV g b V g q defines a unique map Γ c pV g q Ñ Γ 1 c pV g q that fulfills the identity above. In the rest of the work we shall take advantage of these facts and notations above. Furthermore, we adopt the notion of wavefront set of a distribution on test sections of a vector bundle on M as defined in [53].
Definition 4.5. Consider the globally hyperbolic spacetime pM, gq and a state ω : A g Ñ C for the Proca algebra of observables on pM, gq. ω is called Hadamard if it is quasifree and its two-point function ω 2 P Γ 1 c pV g b V g q satisfies the microlocal spectrum condition 3 , i.e.
W F pω 2 q " H :" tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u . Above, px, k x q " py, k y q means that x and y are connected by a lightlike geodesic and k y is the co-parallel transport of k x from x to y along said geodesic, whereas k x Ź0 means that the covector k x is future pointing.
As for Klein-Gordon scalar field theory, Hadamard states for Proca fields have two important properties which were also established in [16] for the notion of Hadamard state adopted there. We present here independent proofs only based on Definition 4.5. Indeed, [16] uses a definition of Hadamard states which is apparently different from our definition. A comparison of the two definitions and an equivalence result appear in Section 6. The first property of Hadamard states is the fact that the difference between the two-point functions of a pair of Hadamard states is a smooth function. This fact plays a crucial role in the point-splitting renormalization procedure (for instance of Wick polynomials and time-ordered polynomials [36][37][38][39] and of the stress-energy tensor [35,45,56]) and is, in fact, one of the reasons for assuming that Hadamard states are the physically relevant ones. Proposition 4.6. Let ω, ω 1 P Γ 1 c pV g b V g q be a pair of Hadamard states on the algebra A g of the Proca field according to Definition 4.5. Then, ω´ω 1 P ΓpV g b V g q, i.e., ω´ω 1 is smooth. More generally, ω´ω 1 is smooth if ω, ω 1 are distributions satisfying (4.1) such that their antisymmetric parts coincide mod. C 8 .
Proof. Let us first prove the second statement. Let us define ω2 pf, gq :" ω 2 pf, gq and ω2 pf, gq :" ω 2 pg, fq, N`:" tpx, kq P T˚Mzt0u | k a k a " 0 , k Ź 0u , N´:" tpx, kq P T˚Mzt0u | k a k a " 0 , k Ÿ 0u , Γ 1 :" tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x ; y, k y q P Γu . for every Γ Ă T˚M 2 zt0u. If both distributions satisfy (4.1), then With the hypotheses of the proposition define R˘:" ω2´ω 12 . Since ω2´ω2 " ω 12´ω12`F where F is a smooth function, we have that R`"´R´mod. C 8 . At this juncture, (4.3) yields W F pR`q 1 X W F pR´q 1 " H because N`X N´" H. Since W F pR`q " W F p´R´`F q " W F p´R´q " W F pR´q, we conclude that the wavefront set of the distributions R˘is empty and thus they are smooth functions. This is the thesis of the second statement. The latter statement implies the former because, since both ω and ω 1 are states on the Proca˚-algebra, their antisymmetric part must be identical and it amounts to iG P , furthermore ω and ω 1 satisfy (4.1) in view of Definition 4.5.
The second property regards the so called propagation property of the Hadamard singularity or also the local-global feature of Hadamard states. It has a long history which can be traced back to [22] passing through [41], [50,51] and [53] (and the recent [44]) at least. Proposition 4.7. Consider a globally hyperbolic spacetime pM, gq and a globally hyperbolic neighborhood N of a smooth spacelike Cauchy surface Σ of pM, gq. Finally, let ω N be a quasifree state for the on-shell algebra of the Proca field in pN , g| N q. The following facts are valid.
(a) There exists a unique a quasifree state ω : A g Ñ C for the Proca field on the whole pM, gq which restricts to ω N on the Proca algebra on N . (b) If ω N is Hadamard according to Definition 4.5, then ω is.
Proof. (a) According to (3.9), G P f " 0 for f P Γ c pV g q if and only if f " Pg for some g P Γ c pV g q. We will use this fact to construct ω out of ω N . Consider two other smooth spacelike surfaces (for both M and N ) Σ`in the future of Σ and Σ´in the past of Σ. Let χ`, χ´: M Ñ r0, 1s be smooth maps such that χ`ppq " 0 if p stays in the past of Σ´and χ`ppq " 1 if p stays in the future of Σ`and χ``χ´" 1. Then, defining Tf :" Pχ`G P f , f P Γ c pV g q (4.4) we have that Tf P Γ c pV g | N q (more precisely supppTfq stays between Σ´and Σ`), and Tf´f " Pg for some g P Γ c pV g q , (4.5) because by standard properties of Green operators: Therefore we can apply (3.9) obtaining (4.5).
With these results, let us define ω 2 pf, gq :" ω N 2 pTf, Tgq , f, g P Γ c pV g q . Taking the continuity properties of G P into account, we leave to the reader the elementary proof of the fact that there is a unique distribution Γ 1 c pV g b V g q such that its value on f b g coincides with 4 ω 2 pf, gq. (We will indicate that distribution by ω 2 with the usual misuse of language.) Furthermore, in view of the definition of T, it is obvious that ω 2 is also a bisolution of the Proca equation, since G P P " PG P " 0. Using Definition 4.3 to construct a candidate quasifree state ω on A g out of its two-point function ω 2 , it is clear that the positivity requirement is guaranteed because ω N satisfies it. We conclude that there is a quasifree state ω on A g , whose two point function is (4.6), and this two point function is a distribution which is also bisolution of the Proca equation. Finally, observe that ω extends to the whole A g the state ω N since the states are quasifree and the two-point function of the former extends the two point function of the latter. Indeed, This is because, specializing (3.9) and (4.4)-(4.5) to the globally hyperbolic spacetime pN , g| N q since f P Γ c pV g | N q, we have that Tf´f " Pg with g P Γ c pV g | N q and ω N 2 vanishes when one argument has the form Pg, because it is a bisolution of the Proca equation in N .
There is only one such quasifree state which is an extension of ω N to the whole algebra A g , and such that its two-point function is a bisolution of the Proca equation. In fact, another such extension ω 1 would satisfy ω 1 2 pf, gq " ω 1 2 pTf, Tgq " ω N pTf, Tgq " ω 2 pTf, Tgq " ω 2 pf, gq , for all f, g P Γ c pV g q.
(b) We pass to the proof that ω is Hadamard if ω N is. We have to prove that (4.1) is valid if it is valid for ω N in pN , g| N q. Interpreting the two-point functions as distributions of Γ 1 c pV g b V g q, The wavefront sets of G P and Pχ`G P can be computed as follows. First of all, from (3.7), where Q " I`m´2dδ g . It is known that W F pG N q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y qu Notice that, in particular k x ‰ 0 and k y ‰ 0 nor simultaneously by definition, nor separately since they are connected by a coparallel transport. So, since Q is a differential operator we immediatly deduce by 4.8 that W F pG P q Ă W F pG N q.
Then we associate to the two operator their distributional kernels G P px, yq and G N px, yq and recast equation 4.8 in the form: which, by standard microlocal analysis results, implies that However explicit computations give that CharpId x b Q y q " tpx, k x ; y, 0q|px, k x q P T˚M, y P Mu which does not intersect W F pG N q at any point, implying So G P and G Q have the same wavefront set. Therefore, since Pχ`is a smooth differential operator, W F pPχG N q Ă tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y qu A this point, a standard estimate of composition of wavefront sets in (4.7) yields (see, e.g., [40]) where the Hadamard wavefront set H is the one in (4.1). To conclude the proof, it is sufficient to establish the converse inclusion. To this end, observe that, since the antisymmetric part of ω 2 is ω2´ω2 " iG P , where we adopted the same notation as at the beginning of the proof of Proposition 4.6: ω2 " ω 2 , ω2 pf, gq " ω 2 pg, fq. If, according to that notation, the prime applied to wavefront sets is defined as in (4.2), the above inclusion can be re-phrased to tpx, k x ; y, k y q P T˚M 2 zt0u | px, k x q " py, k y qu " W F pG P q 1 Ă W F pω2 q 1 Y W F pω2 q 1 (4.9) Above W F pω2 q 1 Ă H 1 " tpx, k x ; y, k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u and, with a trivial computation, W F pω2 q 1 Ă tpx,´k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k y Ź 0u , Now suppose that px, k x ; y, k y q P H 1 does not belong to W F pω2 q 1 . According to (4.9), px, k x ; y, k y q R W F pG P q 1 (notice that H 1 Q px, k x ; y, k y q R W F pω2 q 1 since the two sets are disjoint). This is impossible because every px, k x ; y, k y q P H 1 belongs to W F pG P q 1 as it immediately arises by comparing the explicit expressions of these two sets written above. In summary H 1 Ă W F pω 2 q 1 , that is H Ă W F pω 2 q, concluding the proof.
Hadamard states turned also out to be relevant in the study of quantum energy conditions [16,17,19] and in black hole physics [13,24,42,46,54] (see references in [44] for a summary) We are finally ready to extend the Møller operator to the quantum algebras, proving that they are indeed isomorphic. To this end, for any paracausally related metric g » g 1 , we define an isomorphism of the free algebras R gg 1 : A g 1 Ñ A g as the unique unital˚-algebra isomorphism between the said free unital˚-algebras such that where R is a Møller operator of g, g 1 and the adjoint R : gg 1 is defined as in Proposition 3.5.

Møller˚-isomorphism and the pullback of Hadamard states
When we pass to the quotient algebras, the preservation of the causal propagators discussed in the previous sections, immediately implies that the induced map on the quotient algebras is an isomorphism, that we call Møller˚-isomorphism. Proposition 4.8. Let now R gg 1 : A g 1 " A g 1 {I g 1 Ñ A g " A g {I g be the quotient morphism constructed out of R gg 1 . Then R is well defined and is indeed a˚-algebra isomorphism.
Proof. The proof of this statement is identical to the one performed in [48,Proposition 5.6]. Indeed it just relies on the preservation of the causal propagators proved in Theorem 3.7, which implies that the associated CCR-ideals are˚-isomorphic.
The final step in our construction is to define a pullback of the Møller˚-isomorphism to the states and then to prove that the Hadamard condition is preserved, as done in [48,Theorem 5.14] for normally hyperbolic field theories. Theorem 4.9. Let R gg 1 be the Møller *-isomorphism and let ω : A g Ñ C be a quasifree Hadamard state, we define the pull-back state ω 1 : A g 1 Ñ C by ω 1 " ω˝R gg 1 . The following facts are true: 1 ω 1 is a well-defined state; 2 ω 1 is quasifree; 3 ω 1 is a Hadamard state.
Proof. The proof of 1-2 is trivial and discussed in [48,Proposition 5.11]. The proof of 3 follows from the Hadamard propagation property stated in Proposition 4.7. To prove the statement we can just focus on the case in which the Møller operator is constructed out of two spacetimes such that g ĺ g 1 , the reasoning can then be iterated at each step of the paracausal chain. The two-point function of the pullback state can be written as ω 1 2 pf, hq " ω 1 pâ 1 pfqâ 1 phqq " ωpR gg 1 pâ 1 pfqâ 1 phqqq " ωpâpR : gg 1 fqâpR : gg 1 hqq " ω 2 pR : gg 1 f, R : gg 1 hq.
We recall that the operator is the composition of two pieces R : gg 1 " R :gg χ˝R : gχg 1 and split the proof in two steps. First we focus on the bidistribution ω χ 2 pf, hq :" ω 2 pR :gg χ f, R :gg χ hq on pM, g χ q defining a quasifree state therein. By the property 3.10, in the region in which g χ " g, there is a t 0 a Cauchy surface Σ t 0 in common for g and g χ , a common globally hyperbolic neighborhood N of that Cauchy surface such that ω χ 2 pf, hq " ω 2 pf, hq when the supports of f and g are chosen in N and thus the corresponding state is Hadamard in pN , g χ q. Now Proposition 6.3 implies that ω χ 2 is Hadamard in the whole pM, g χ q. Secondly, the same argument can be used once again for the operator R : gχg 1 on the Hadamard state ω χ on pM, g χ q, proving that the state induced by ω 2 pR : gg 1¨, R : gg 1¨q " ω χ 2 pR : gχg 1¨, R : gχg 1¨q is Hadamard as well on pM, g 1 q. In other words the full Møller operator preserves the Hadamard form.

Existence of Proca Hadamard states in globally hyperbolic spacetimes
This section is devoted to the construction of Hadamard states for the real Proca field in a generic globally hyperbolic spacetime. Actually, the technology of Møller operators, in particular Theorem 4.9, allows us to reduce the construction of Hadamard states for the Proca equation to the special case of an ultrastatic spacetime with Cauchy hypersufaces of bounded geometry. Indeed, as shown in [48,Corollary 2.23], for any globally hyperbolic spacetime pM, gq, there exists a paracausally related globally hyperbolic spacetime pM, g 0 q which is ultrastatic. In other words, first of all pM, g 0 q is isometric to RˆΣ where pΣ, h 0 q is a t-independent complete Riemannian manifold and g 0 "´dt b dt`h 0 , where t is the natural coordinate on R and dt is past directed. We also denote by B t the tangent vector to the submanifold R of RˆΣ. In view of the completeness of h, these spacetimes are globally hyperbolic (see e.g. [20]) and Σ is a Cauchy surface of this spacetime. In turn, it is possible to change the metric on Σ in order that the final metric, indicated by h is both complete and of bounded geometry [34]. By construction, the final ultrastatic spacetime pM,´dt b dt`hq is paracausally related to pM, g 0 q because the intersection of the corresponding open cones is non-empty as it always contains B t . By transitivity pM, gq is paracausally related with pRˆΣ,´dt b dt`hq. Hence, we assume without loss of generalities, that pM, gq " pRˆΣ,´dt b dt`hq is a globally hyperbolic ultrastatic spacetime, with dt past directed, whose spatial metric h is complete. When dealing with the construction of Hadamard states we also assume that the spatial manifold pΣ, hq is also of bounded geometry. In the final part of the section, we will come back to consider a generic globally hyperbolic spacetime pM, gq

The Cauchy problem in ultrastatic spacetimes
We study here the Cauchy problem for the Proca (real and complex) field in ultrastatic spacetimes pM, gq " pRˆΣ,´dt b dt`hq, where pΣ, hq is complete. A more general treatise appears in [55] where the Cauchy problem is studied, also in the presence of a source of the Proca field, in a generic globally hyperbolic spacetime and the continuity of the solutions with respect to the initial data is focused.
Let us consider the Proca equation (2.3) (where m 2 ą 0) on the above ultrastatic spacetime. As observed in [16], every smooth 1-form A P Ω 1 pMq naturally uniquely decomposes as Apt, pq " A p0q pt, pqdt`A p1q pt, pq (5.1) where A piq pt,¨q P Ω i pΣq for i " 0, 1 and t P R. By direct inspection and taking the equivalence of (2.3) and (3.4)-(3.5) into account, one sees that Eq. (2.3) is equivalent to the constrained double Klein-Gordon system h is the Hodge Laplacian on pΣ, hq for k-forms and the last condition (5.4) is nothing but the constraint δ p1q g A " 0 arising from (2.3). The theory for the fields A p1q and A p0q is a special case of the theory of normally hyperbolic equations on corresponding vector bundles with positive inner product over a globally hyperbolic spacetime [3,4]. In our case, (1) there is a real vector bundle V p1q g with basis M, canonical fiber isomorphic to Tq Σ, and equipped with a fiberwise real symmetric scalar product induced by h 7 q . A p1q P ΓpV p1q g q; (2) there is another real vector bundle V p0q g with basis M, canonical fiber isomorphic to R, and equipped with a positive fiberwise real symmetric scalar product given by the natural product in R. A p0q P ΓpV p0q g q.
Equations (5.2) and (5.3) admit existence and uniqueness theorems for smooth compactly supported Cauchy data and corresponding smooth spacelike compact solutions in Γ sc pV p0q g q and Γ sc pV p1q g q respectively, as a consequence of very well-known results in the theory of normally hyperbolic equations [3,4,33]. However, when viewing A p0q and A p1q as parts of the Proca field A, we have also to deal with the additional constraint (5.4). Notice that (5.4) imposes two constraints on the Cauchy data of A p0q and A p1q on Σ: The second constraint is only apparently of the second order. Indeed, taking (5.2) into account, it can be re-written as a condition of the Cauchy data At this juncture we observe that, with some elementary computation (use ∆ which, in turn, implies Equation (5.4) , if the initial condition of that scalar Klein-Gordon equation for pB t A p0q`δ p1q h A p1q q are the zero initial conditions. This exactly amounts to In summary, we are naturally led to focus on this Cauchy problem with initial data A p0q p0,¨q " a p0q p¨q, B t A p0q p0,¨q " π p0q p¨q, A p1q p0,¨q " a p1q p¨q, B t A p1q p0,¨q " π p1q p¨q (5.9) where a p0q , π p0q , a p1q , π p1q are pairs of smooth compactly supported, respectively 0 and 1, forms on Σ, and the constraints are valid  4), for the fields A p0q P Γ sc pV p0q g q and A p1q P Γ sc pV p1q g q, with the same initial data (5.9) and constraints (5.10). As a consequence, (1) every smooth spacelike compact solution of the Proca equation A P Γ sc pV g q (2.3) defines compactly supported smooth Cauchy data on Σ which satisfy the constraints (5.10); (2) if the Cauchy data are smooth, compactly supported and satisfy (5.10), then there is a unique smooth spacelike compact solution of the Proca equation A P Γ sc pV g q (2.3) associated to them; (3) the support of a solution A P Γ sc pV g q with smooth compactly supported initial data satisfies supppAq Ă J`pSqYJ´pSq, where S Ă Σ is the union of the supports of the Cauchy data.
Remark 5.2. (1) All the discussion above, and Proposition 5.1 in particular, extends to the case of a complex Proca field and corresponding associated complex Klein Gordon fields. The stated results can be extended easily to the case of the non-homogeneous Proca equation and also considering continuity properties of the solutions with respect to the source and the initial data referring to natural topologies. (See [55] for a general discussion.) (2) A naive idea may be that we can freely fix smooth compactly supported Cauchy data for A p1q and then define associated Cauchy conditions for A p0q by solving the constraints (5.10). In this case the true degrees of freedom of the Proca field would be the vector part A p1q , whereas A p0q would be a constrained degree of freedom. This viewpoint is incorrect, if we decide to deal with spacelike compact solutions, because the second constraint in Equation (5.10) in general does not produce a compactly supported function a p0q when the source δ p1q h π p1q is smooth compactly supported (the smoothness of a p0q is however guaranteed by elliptic regularity from the smoothness of δ p1q h π p1q ). a p0q is compactly supported only for some smooth compactly supported initial conditions π p1q . Therefore the linear subspace of initial data (5.9) compatible with the constraints (5.10) does not include all possible compactly supported initial conditions π p1q which, therefore, cannot be freely chosen.
(3) However this space of constrained Cauchy data is non-trivial, i.e., it does not contain only zero initial conditions and in particular there are couples pa p0q , π p1q q such that both elements do not vanish. This is because, for every smooth compactly supported 1-form f p1q (with δ p1q f p1q ‰ 0 in particular) and for every smooth compactly supported 2-form f p2q , are smooth, and compactly supported, they solve the nontrivial constraint in (5.10) δ p1q h π p1q " p∆ p0q`m 2 qa p0q and f p1q , f p2q can be chosen in order that neither of a p0q and π p1q vanishes.
The easier constraint π p0q "´δ p1q h a p1q is solved by every smooth compactly supported 1-form a p1q by defining the smooth compactly supported 0-form π p0q correspondingly.

The Proca symplectic form in ultrastatic spacetimes
Consider two solutions A, A 1 P Γ sc pV g q X KerP of the Proca equation in our ultrastatic spacetime, choose t P R and consider the bilinear form where we are referring to the Cauchy data on Σ of the smooth spacelike compact solutions of the Proca equation. Σ is viewed as the time slice at time t. As is well known, it is possible to define a natural symplectic form for the Proca field in general globally hyperbolic specetimes [6] with properties analogous to the ones we are going to discuss here. In this section we however stick to the ultrastatic spacetime case which is enough for our ends. According to [6] (with an argument very similar to the proof of Propositions 3.12 and 3.13 in [48]) we have immediately that σ pPq t pA, A 1 q " σ pPq t 1 pA, A 1 q @t, t 1 P R , and, omitting the index t as the symplectic form is independent of it, where A, f (resp. A, f 1 ) are related by A :" G P f (resp. A 1 :" G P f 1 ).
Remark 5.3. The important identity (5.12) is also valid in a generic globally hyperbolic spacetime when σ pP q is interpreted as the general symplectic form of the Proca field according to [6].
Let us suppose to deal with the Cauchy data of the real vector space C Σ Ă Ω 0 c pΣq 2Ω 1 c pΣq 2 of smooth compactly supported Cauchy data pa 0 , π 0 , a 1 , π 1 q subjected to the linear constraints (5.10), . (5.13) Not only the Cauchy problem is well behaved in that space as a consequence of Proposition 5.1, but we also have the following result which, in particular, implies that the Weyl algebra of the real Proca field has trivial center.
Proposition 5.4. The bilinear antisymmetric map σ pP q : C ΣˆCΣ Ñ R defined in (5.11) is non-degenerate and therefore it is a symplectic form on C Σ .
where the smooth compactly supported sections are complex. We have used the same symbols as for the real case for the causal propagators since the associated operators commute with the complex conjugation. As a consequence, a standard argument about the uniqueness of Green operators implies that the causal propagators for the real case are nothing but the restriction of the causal propagator of the complex case which, in turn, are the trivial complexification of the real ones.

The Proca energy density in ultrastatic spacetimes
Starting from the Proca Lagrangian in every curved spacetime (see, e.g, [15]) and referring to local coordinates px 0 , . . . , x n´1 q adapted to the split M " RˆΣ of our ultrastatic spacetime, where x 0 " t runs along the whole R and x 1 , . . . , x n´1 are local coordinates on Σ, the energy density reads in terms of initial conditions on Σ of the considered Proca field 7 pa p1q , a p1q q`a p0q a p0q¯ě 0 .

(5.15)
Above h 7 p2q is the natural scalar product for the 2-forms on Σ induced by the metric tensor. It is evident that the energy density is non-negative since the metric h and its inverse h 7 are positive by hypothesis. The total energy at time t is the integral of T 00 on Σ, using the natural volume form, when replacing A p0q and A p1q for the respective Cauchy data. As B t is a Killing vector and the solution is spacelike compact, the total energy is finite and constant in time.

(5.16)
Using Hodge duality of d and δ and the definition of the Hodge Laplacian, the expression of the total energy can be re-arranged to h a p1q q`m 2 pa p0q a p0q`h7 pa p1q , a p1q q˘¯vol h . Using again the Hodge duality of d and δ the third term in the integral can be rearranged tó ż The term δ p1q π p1q above and the term δ p1q h a p1q δ p1q h a p1q appearing in the expression for the total energy can be worked out exploiting the constraints (5.10). Inserting the results in the found formula for the total energy, we finally find, with the notation already used for the symplectic form, when the used Cauchy data belong to the constrained space C Σ . It is now clear that the total energy of the Proca field is the difference between the total energies of the two Klein-Gordon fields composing it exactly as it happened for the symplectic form. This difference is however positive when working on smooth compactly supported initial conditions satisfying the constraints (5.10), because the found expression of the energy is the same as the one computed with the density (5.15).
Remark 5.6. We notice that the negative energy component of the field can be interpreted as a ghost, in this case however no issues arise since dynamical constraints covariantly remove such a state. A different approach to the problem by generalizing to curved spacetime the Stuckelberg lagrangian, can be found in [2], where it is appearently argued the no Hadamard states exist for the Proca field, contrarily to the results of [16] and of this work.
Remark 5.7. With the same argument, the found result immediately generalizes to the case of complex k-forms and one finds where the bar over the forms denotes the complex conjugation and pa p0q , π p0q , a p1q , π p1q q are complex forms of C Σ`i C Σ .

Elliptic Hilbert complexes and Proca quantum states in ultrastatic spacetimes
We can proceed to the construction of quasifree states. As we shall see shortly, this construction for the Proca field uses some consequences of the spectral theory applied to the theory of elliptic Hilbert complexes [5] defined in terms of the closure of Hodge operators in natural L 2 spaces of forms. Some of the following ideas were inspired by [16]. However we now work in the space of Cauchy data instead of in the space of smooth supportly compacted forms and/or modes. Furthermore we do not assume restrictions on the topology of the Cauchy surfaces used in [16] to impose a pure point spectrum to the Hodge Laplacians.
To define quasifree states for the Proca field we observe that, as P is Green hyperbolic, the CCR algebra A g is isomorphic to the analogous unital˚-algebra A psympq g generated by the solution-smeared field operators σ pPq pâ, Aq, for A P Ker sc pP q, which are R-linear in A, Hermitian, and satisfy the commutation relations 5 " σ pPq pâ, Aq, σ pPq pâ, The said unital˚-algebra isomorphism F : is completely defined as the unique homomorphism of unital˚-algebras that satisfies The definition is well-posed in view of (5.12), (3.8), (3.9), and the definition of A g . Within this framework, the two point function ω 2 is interpreted as the integral kernel of ω´σ pPq pâ, Aqσ pPq pâ, A 1 q¯.
In particular, its antisymmetric part is universally given by i 2 σ pP q pA, A 1 q due to (5.19). The specific part of the two point function is therefore completely embodied in its symmetric part µpA, A 1 q.
According to this observation, a general recipe for real (bosonic) CCR in generic globally hyperbolic spacetimes to define a quasifree state on the˚-algebra A g (e.g., see [40,41,56] for the scalar case and [23, Chapter 4, Proposition 4.9] for the generic case of real bosonic CCRs) is to assign a real scalar product on the space of spacelike compact solutions µ : Ker sc pPqˆKer sc pPq Ñ R satisfying (a) the strict positivity requirement µpA, Aq ě 0 where µpA, Aq " 0 implies A " 0; (b) the continuity requirement with respect to the relevant symplectic form σ pP q (see, e.g., [23,Proposition 4.9]), σ pPq pA, The continuity requirement directly arises form the fact that the quasifree state induced by µ on the whole˚-algebra A g " A symp g according to Definition 4.3 is a positive functional. The converse implication, though true, is less trivial [23,41]. The two mentioned requirements are nothing but the direct translation of (2)' and (3)' stated in the introduction. (Regarding the latter, observe that σ pP q corresponds to the causal propagator at the level of solutions -Eq. (5.12) in our case -as discussed in Section 5.2.) At this point, it should be clear that the quasifree state defined by µ has two-point function, viewed as bilinear map on Γ c pV g qˆΓ c pV g q, However, since the Cauchy problem is well posed on the time slices Σ of an ultrastatic spacetime pRˆΣ,´dt b dt`hq, as proved in Proposition 5.1, we can directly define µ (and σ pPq ) in the space of Cauchy data C Σ on Σ, for smooth spacelike compact solutions, viewed as the time slice at t " 0, µ : C ΣˆCΣ Ñ R .
In view of the peculiarity of the Cauchy problem for the Proca field as discussed in Section 5.1, the real vector space of the Cauchy data C Σ is constrained. We underline that working at the level of constrained initial data does not affect the construction of quasifree states. Indeed, it is sufficient that the space of constrained initial conditions is a real (or complex) vector space and that the constrained Cauchy problem is well posed. With this in mind, referring to the canonical decomposition A " A p0q dt`A p1q of a real smooth spacelike compact solution A of the Proca equation, we remember that ) .
Above pa p0q , π p0q q :" pA p0q , B t A p0q q| t"0 and pa p1q , π p1q q :" pA p1q , B t A p1q q| t"0 . With the said definitions and where A denotes both a solution of Proca equation and its Cauchy data on Σ, we have the first result.
Proposition 5.8. Consider the˚-algebra A g of the real Proca field on the ultrastatic spacetime pM, gq " pRˆΣ,´dt b dt`hq, with dt past directed and pΣ, hq a smooth complete Riemannian manifold. Let η 0 :"´1, η 1 :" 1 and h 7 pjq denote the standard inner product of j-forms on Σ induced by h. The bilinear map on the space C Σ of real smooth compactly supported Cauchy data (5.13) µpA, A 1 q :" pjq pπ pjq , p∆ pjq`m2 q´1 {2 π pjq 1 q`h 7 pjq pa pjq , p∆ pjq`m2 q 1{2 a pjq 1 q vol h (5.21) is a well defined symmetric positive inner product which satisfies (5.20) and thus it defines a quasifree state ω µ on A g completely defined by its two-point function where f, f 1 P Γ c pV g q satisfy The bar over the operators in (5.21) denotes the closure in suitable Hilbert spaces of the operators originally defined on domains of compactly supported smooth functions. To explain this formalism, before starting with the proof we have to permit some technical facts about the properties of the Hodge operators at the level of L 2 spaces. Given the complete Riemannian manifold pΣ, hq, with n :" dimpΣq consider the Hilbert space H h :" À n k"0 L 2 k pΣ, vol h q, where the sum is orthogonal and L 2 k pΣ, vol h q is the complex Hilbert space of the square-integrable k-forms with respect to the relevant Hermitian Hodge inner product: where a denotes the pointwise complex conjugation of the complex form a. The overall inner product on H h will be indicated by p¨|¨q and the Hilbert space adjoint of a densely-defined operator A : DpAq Ñ H h , with DpAq Ă H h , will be denoted by A˚: DpA˚q Ñ H h . The closure of A will be denoted by the bar: If Ω c pΣq C :" À n k"0 Ω k c pΣq C denotes the dense subspace of complex complactly supported smooth forms Ω k c pΣq C :" Ω k c pΣq`iΩ k c pΣq, define the two operators (we omit the index h for shortness) d :" ' n k"0 d pkq : Ω c pΣq C Ñ Ω c pΣq C , δ :" ' n k"0 δ pkq : Ω c pΣq C Ñ Ω c pΣq C with d pnq :" 0 and δ p0q :" 0. Finally, introduce the Hodge Laplacian as ∆ :" n ÿ k"0 ∆ pkq : Ω c pΣq C Ñ Ω c pΣq C with ∆ pkq :" δ pk`1q d pkq`dpk´1q δ pkq .
Since pΣ, hq is complete, ∆ can be proved to be essentially selfadjoint, for instance exploiting the well-known argument by Chernoff [8] (or directly referring to [1]). Since ∆ is essentially selfadjoint, if c P R, also ∆`cI is essentially selfadjoint. In particular, its unique selfadjoint extension is its closure ∆`cI.

(5.24)
A trivial generalization of the decomposition as in (5.24) holds for ∆`cI " ∆`cI with c P R.
We are now prompt to prove a preparatory technical lemma -necessary to establish Proposition 5.8 -that will be fundamental for showing that the bilinear map µ is positive on the space C Σ . Lemma 5.9. For every given k " 0, 1, . . . , n, c ą 0, and α P R, the identites hold p∆ pk`1q`c Iq α d pkq x " d pkq p∆ pkq`c Iq α x , @x P Dpp∆ pkq`c Iq α q X Dpp∆ pk`1q`c Iq α d pkq q p∆ pk´1q`c Iq α δ pkq y " δ pk´1q p∆ pkq`c Iq α y , @y P Dpp∆ pkq`c Iq α q X Dpp∆ pk´1q`c Iq α δ pkq q .
Proof. Since dd " 0 and δδ " 0, from (5.23), we also have d dx " 0 if x P Dpdq and δ δy " 0 if y P Dpδq, and thus (5.24) yields 6 d However, if Dpd ∆q Ľ Dpd δ dq, we would have x P Dp∆q " Dpδ dq X Dpd δq such that ∆x " δdx`dδx P Dpdq, but x R Dpdδdq, namely δdx R Dpdq. This is impossible since δdx`dδx P Dpdq, Dpdq is a subspace and d δx P Dpdq (and more precisely d d δx " 0 as stated above). Therefore and the same result is valid with δ in place of d. Evidently, in both cases ∆ can be replaced by the selfadjoint operator ∆`cI = ∆`cI for every c P R: We henceforth assume c ą 0. In that case, as ∆ is already positive on its domain, the spectrum of the selfadjoint operator ∆`cI is strictly positive and thus ∆`cI´1 : H h Ñ Dp∆`cIq is well defined, selfadjoint and bounded. The former identity in (5.25) also implies that Dpd ∆`cIq " Dp∆`cI dq, so that ∆`cI´1d ∆`cI| Dpd ∆`cIq x " d| Dpd ∆`cIq x .
By construction, we can choose x " ∆`cI´1y with y P Dpdq in view of the definition of the natural domain of the composition d ∆`cIq. In summary ∆`cI´1dy " d ∆`cI´1y , @y P Dpdq .
Since the argument is also valid for δ, we have established that Iterating the argument, for every n " 0, 1, . . ., This result extends to complex polynomials of ∆`cI´1 in place of powers by linearity. Using the spectral calculus (see e.g. [43]) where µ xy pEq " px|P E yq and P is the projector-valued spectral measure of ∆`cI´1, the found result for d can be written ż r0,bs ppλqdµ x,dy pλq " ż r0,bs ppλqdµ δx,y pλq (5.26) for every complex polynomial p, where r0, bs is a sufficiently large interval to include the (bounded positive) spectrum of ∆`cI´1, x P Dpδq, y P Dpdq, and where we have used δ " d˚. Since the considered regular Borel complex measures are finite and supported on the compact r0, bs, we can pass in (5.26) from polynomials p to generic continuous functions f in view of the Stone-Weierstrass theorem. At this juncture, PE " P E and the uniqueness part of Riesz' representation theorem for regular complex Borel measures, implies that pP E δy|xq " pP E y|dxq for all x P Dpδq, y P Dpdq, and every Borel set E Ă R.
which means P E δ Ă d˚P E , namely P E δ Ă δP E . Analogously, we also have P E d Ă dP E . If f : R Ñ C is measurable and bounded, the standard spectral calculus and (5.23), with a procedure similar to the one used to prove P E δ Ă δP E and taking into account the fact that Dpf p∆`cI´1qq " H h , yields If f is unbounded, we can choose a sequence of bounded measurable functions f n such that f n Ñ f pointwise. It is easy to prove that (see, e.g. [43]) x P Dp ş R f dP q entails This is the case for instance for f pλq " λ β with β ă 0 restricted to r0, bs. Referring to this function and the pointed out result for some sequence of bounded functions with f n Ñ f pointwise, the latter of (5.27) implies that 7 , p∆`cIq α dx " dp∆`cIq α x if x P Dpp∆`cIq α q X Dpdq and dx P Dpp∆`cIq α q, where we used also the fact that d is closed. The case of δ can be worked out similarly. Summing up, we have proved that, if α P R, p∆`cIq α dx " dp∆`cIq α x , @x P Dpp∆`cIq α q X Dpp∆`cIq α dq p∆`cIq α δy " δp∆`cIq α y , @y P Dpp∆`cIq α q X Dpp∆`cIq α δq .
Let us remark that for α ď 0 it is sufficient to choose x P Dpdq and y P Dpδq. For every given k " 0, 1, . . . , n, c ą 0, and α P R, taking the decomposition of H h into account the above formulae imply p∆ pk`1q`c Iq α d pkq x " d pkq p∆ pkq`c Iq α x , @x P Dpp∆ pkq`c Iq α q X Dpp∆ pk`1q`c Iq α d pkq q p∆ pk´1q`c Iq α δ pkq y " δ pk´1q p∆ pkq`c Iq α y , @y P Dpp∆ pkq`c Iq α q X Dpp∆ pk´1q`c Iq α δ pkq q .
That is the thesis.
We are now prompted to prove that the bilinear map defined by Equation (5.21) defines a quasifree state defined by the two-point function given by (5.22) establishing the thesis of Proposition 5.8.
Proof of Proposition 5.8. To continue with the proof of the proposition, we now demonstrate that µ is well-defined and positive. That bilinear form is well-defined because Ω pjq c pΣq Ă Dp∆ pjq`m2 I α q for α ď 1 as one immediately proves from spectral calculus. Furthermore, the integrand in the right-hand side of Equation (5.21) is the linear combination of products of L 2 functions (of which one of the two has also compact support). Let us pass to the positivity issue. Our strategy is to re-write µpA, Aq, where A " pa p0q , π p0q , a p1q , π p1q q P C Σ , as the quadratic form of the energy µpA, Aq " E pP q pA o q, where the right-hand side is defined in Equation (5.16), for a new set of initial data A o which are not necessarily smooth and compactly supported but such that E pP q pA o q is well defined. If A P C Σ , define for j " 0, 1 Notice that the definition is well posed and the forms a pjq o and π pjq o belong to the respective Hilbert spaces of j-forms, because Ω pjq c pΣq Ă Dp∆ pjq`m2 I α q for α ď 1 as said above. Furthermore the new forms are real since the initial ones are real and ∆ pjq`m2 I α commutes with the complex conjugation 8 . At this juncture, we have from (5.21) µpA, Aq " Furthermore, the new Cauchy data, though they stay outside C Σ in general, they however satisfy the natural generalization of the constraints defining C Σ in view of Lemma 5.9: These identities arise immediately from Definitions (5.28), the constraints (5.10), and by applying Lemma 5.9 and paying attention to the fact that Ω pjq c pΣq Ă Dpp∆ pj´1q`c Iq α δ pjq q for every α ď 1 and also using p∆ pjq`m2 Iqp∆ pjq`m2 Iq´1 {4 " p∆ pjq`m2 Iq´1 {4 ∆ pjq`m2 I (for, e.g., [43, (f) in Proposition 3.60 ]). Using (5.23) and (5.30) in the right-hand side of (5.29), we can proceed backwardly as in the proof that (5.16) is equivalent to (5.17). Indeed, the only ingredients we used in that proof were the constraint equations which are valid also for A o and the duality of δ and d with respect to the Hodge inner product, which extends to δ and d. In summary, From that identity, it is clear that µpA, Aq ě 0 and µpA, Aq " 0 implies A o " 0, which in turn yields A " 0 because the operators ∆ pjq`m2 I 1{4 are injective. We have established that µ : C ΣˆCΣ Ñ R is a positive real symmetric inner product.

Hadamard states in ultrastatic and generic globally hyperbolic spacetimes
With the next proposition, we show that the quasifree states defined in Proposition 5.8 is a Hadamard state when pΣ, hq is of bounded geometry. To prove the assertion we will take advantage of the general formalism developed in [23] and [26]. An alternative proof, which does not assume that the manifold is of bounded geometry (however we here take advantage of [34]), could be constructed along the procedure developed in [21] and extending it to the vectorial Klein-Gordon field. where H pjq :" ∆ pjq`m2 1{2 , σ pjq are the symplectic forms of the corresponding Klein-Gordon fields taking place in the right-hand side of (5.14), now evaluated on complex fields. Above, a pjq , π pjq P Ω j c pΣq C are the Cauchy data on Σ of A pjq respectively and a pjq 1 , π pjq 1 P Ω j c pΣq C are the Cauchy data on Σ of A pjq 1 respectively. Notice that we are not imposing constraints on these initial data since we are dealing with independent Klein-Gordon fields. λp jq are evidently positive because, if all involved forms in the right-hand side are smooth and compactly supported, then the right-hand side of the identity above is well-defined and λp jq pA pjq , A pjq 1 q :" 1 2 The case of λṕ jq is strictly analogous. Furthermore λp jq pA pjq , A pjq 1 q´λṕ jq pA pjq , A pjq 1 q " iσ pjq pA pjq , A pjq 1 q .
Therefore λp jq satisfy the hypotheses of [23, Proposition 4.14] 9 so that they define a pair, for j " 0, 1, of gauge-invariant quasifree states for the complex Klein-Gordon fields respectively associated to Equations (5.2) and (5.3). We pass to prove that both states are Hadamard exploiting the fact that pΣ, hq is of bounded geometry. By rewriting the covariances λp jq as λp jq "˘qcp jq (q " iσ pjq ) a quick computation shows that We can immediately realize that the operator cp jq is the same Hadamard projector obtained in [26, Section 5.2] 10 -see also [23,Section 11] for a more introductory explanation for the scalar case. This operator belongs to the necessary class of pseudodifferential operators C 8 b pR; Ψ 1 b pΣqq because pΣ, hq is of bounded geometry. Therefore, on account of [26,Proposition 5.4], the two quasifree states associated to λp jq , for A pjq and j " 0, 1, are Hadamard. In other words, the Schwartz kernels provided by the two-point functions λp jq pG pjq¨, G pjq¨q , viewed as distributions of where H is defined in (4.1) and G piq , i " 0, 1 are the causal propagators for the normally hyperbolic operators N piq :" B 2 t`∆ piq h`m 2 I : Γ sc pV piq g q Ñ Γ sc pV piq g q i " 0, 1 . Above and from now on we use the same notation to indicate a bidistribution and the associated Schwartz kernel. Notice that we have used the same symbol G pjq of the causal propagator we used for the real vector field case. This is because the causal propagators for the complex fields are the direct complexification of the scalar case (see Remark 5.5). We pass now to focus on the expression of ω µ2 provided in (5.22) taking the usual decomposition Ω 1 c pMq C Q f " f p0q dt`f p1q into account. It can be written where, comparing (5.21) and (5.22) with (5.31) for real arguments f, f 1 P ΓpV g q, we find ω pjq µ2 pf pjq , f pjq 1 q " λp jq pG pjq f pjq , G pjq f pjq 1 q .
Taking (5.5) into account, we now observe that ω µ2 P ΓpV g bV g q 1 " ΓppV p0q g 'V p1q g qbpV p0q g 'V p1q g qq 1 . As a matter of fact, however, ω µ2 does not have mixed components acting on sections of V g . These are respectively represented by´ω p0q µ2 and ω p1q µ2 whose wavefront set is H in both cases. The remaining two components have empty wavefront set since they are the zero distributions. Applying the definition of wavefront set of a vector-valued distribution [53], we conclude that concluding the proof.
Combining the results obtained so far, we get the main result of this paper.
Theorem 5.11. Let pM, gq be a globally hyperbolic spacetime and refer to the CCR-algebra A g of the real Proca field. Then there exists a quasifree Hadamard state on A g . Proof of Theorem 4. As already explained in the beginning of Section 5, for any globally hyperbolic spacetime pM, gq, there exists a paracausally related globally hyperbolic spacetime pM, g 0 q which is ultrastatic and whose spatial metric is of bounded geometry. In particular, in this class of spacetimes, the quasifree states defined in Proposition 5.8 satisfy the microlocal spectrum condition, as proved in Proposition 5.10. Therefore, since the pull-back along a Møller˚-isomorphism preserves the Hadamard condition on account of Theorem 4.9, we can conclude.

Comparison with Fewster-Pfenning's definition of Hadamard states
Though the paper [16] by Fewster and Pfenning concerns quantum energy inequalities, it also offers a general theoretical discussion about the algebraic quantization of the Proca and the Maxwell fields in curved spacetime. In particular, the authors propose a definition of a Hadamard state which appears to be technically different from ours at first glance, even if it shares a number of important features with ours. This section is devoted to a comparison of the two definitions for the Proca field.

Proca Hadamard states according to Fewster and Pfenning
The definition of Hadamard state stated in [16,Equation (35)] is formulated in terms of causal normal neighborhoods of smooth spacelike Cauchy surfaces (see also below) and the global Hadamard parametrix for distributions which are bisolutions of the vectorial Klein-Gordon equation. Our final goal is to prove an equivalence theorem of our definition of Hadamard state Definition 4.5 and the one adopted in [16].
As a first step, we translate the original Fewster-Pfenning's definition of a Hadamard state into an equivalent form which will turn out to be more useful for our comparison. The equivalence of the version stated below of Fewster-Pfenning's definition and the original one was established in [16, Section III C] (see also the comments under Definition 6.1).
Definition 6.1. [Fewster-Pfenning's definition of Proca Hadamard state] Consider the globally hyperbolic spacetime pM, gq and a state ω : A g Ñ C for the Proca algebra of observables on pM, gq. ω is called Hadamard if it is quasifree and its two-point function has the form ωpâpfqâphqq " W g pf, Qhq (6.1) @f, h P Γ c pV g q, where Q : ΓpV g q Ñ ΓpV g q in the differential operator Q " Id`m´2pdδ g q. Above W g P Γ 1 c pV g b V g q is a Klein-Gordon distributional bisolution such that W g pf, gq´W g pg, fq " iG N pf, gq mod C 8 , Finally, [16] also contains a proof of the existence of Hadamard states for the Proca (and the Maxwell) field in globally hyperbolic spacetimes with compact Cauchy surfaces (whose first homology group is trivial when treating the Maxwell field). This proof establishes first the existence in ultrastatic spacetimes and next it exploits a standard deformation argument [56].

An (almost) equivalence theorem
We are in a position to state and prove our equivalence result. Theorem 6.6. Consider the globally hyperbolic spacetime pM, gq and a quasifree state ω : A g Ñ C for the˚-algebra of observables on pM, gq of the real Proca field. Let ω 2 P Γ 1 c pV g b V g q be the twopoint function of ω. The following facts are true.
(a) If ω is Hadamard according to Definition 6.1, then it is also Hadamard according to Definition 4.5.
(b) If pM, gq admits a Proca quasifree Hadamard state according to Definition 6.1 and ω is Hadamard according to Definition 4.5, then ω is Hadamard in the sense of Definition 6.1.
Proof. Tha following argument is identical to the one used in 4.7 to prove W F pG P q " W F pG N q, but we repeat it here to keep this section self-contained. First of all notice that, since ω 2 pf, gq " W g pf, Qgq, then viewing ω 2 and W g as bidistributions, we have ωpx, yq " pId x b Q y q W px, yq (where we have used the fact that Q is formally selfadjoint) taking Remark 4.4 into account). Now suppose that ω is Hadamard according to Definition 6.1. Since W g satisfies the microlocal spectrum condition and the differential operator I b Q is smooth, we have W F pω 2 q Ă W F pW g q " tpx, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, k x Ź 0u .
Notice that, in particular, k x and k y cannot vanish (simultaneously or separately) if they take part of W F pW g q. Let us prove the converse inclusion to complete the proof of (a). Again from known results, from ω 2 px, yq " pId x b Q y qW g px, yq, we have However, by direct inspection, one sees that CharpI b Qq " tpx, k x ; y, 0q | px, k x q P T˚M , y P Mu , so that W F pω 2 q Ă W F pW g q Ă W F pω 2 q Y tpx, k x ; y, 0q | px, k x q P T˚M , y P Mu . (6.4) However W F pW g q X tpx, k x ; y, 0q | px, k x q P T˚M , y P Mu " H and thus we can re-write the chain of inclusions (6.4) as W F pω 2 q Ă W F pW g q Ă W F pω 2 q so that W F pω 2 q " W F pW g q .
This is the thesis of (a) because we have established that Definition 4.5 is satisfied by ω.
To prove (b), let us assume that ω satisfies Definition 4.5. By hypotheses the antisymmetric part of ω 2 is´iG P . Let Ω be another quasifree state of the Proca field which satisfies Definition 6.1. Also the antisymmetric part of Ω 2 is´iG P . Due to Proposition 4.6, F px, yq :" ω 2 px, yq´Ω 2 px, yq .
is a smooth function. Furthermore it is a symmetric bisolution of the Proca equation. In particular it therefore satisfies 13 F pf, dh p0q q " 0, where h p0q P Ω 0 c pMq, so that F pf, Qgq " F pf, gq`1 m 2 F pf, dpδ g gqq " F pf, gq .
Collecting everything together, we can assert that, for some distributional bisolution of the Klein-Gordon equation W g which satisfies (6.2), (6.3), and is associated to the Hadamard state Ω, it holds ω 2 pf, gq " W g pf, Qgq`F pf, gq " W g pf, Qgq`F pf, Qgq .
If we re-absorb F in the definition of W g , W 1 g pf, Qgq " W g pf, Qgq`F pf, Qgq .
the new W 1 g is again a distributional bisolution of the Klein-Gordon equation which satisfies (6.2), (6.3) and ω 2 pf, gq " W 1 g pf, Qgq . In other words, the Hadamard state ω according to Definition 4.5 is also Hadamard in the sense of Definition 6.1 concluding the proof of (b).
Remark 6.7. Regarding (b), the existence of Hadamard states in the sense of Definition 6.1 has been established in [16] for globally hyperbolic spacetimes whose Cauchy surfaces are compact: in those types of spacetimes at least, the two definitions are completely equivalent. We expect that actually the equivalence is complete, even dropping the compactness hypothesis (see the conclusion section). This issue will be investigated elsewhere.

Conclusion and future outlook
We conclude this paper by discussing some open issues which are raised in this paper and we leave for future works.
On an ultrastatic spacetime M " RˆΣ, the one-parameter group of isometries given by timetranslations has an associated action on A g in terms of˚-algebras isomorphisms α u completely induced by α u pâpfqq :"âpf u q for every f P Γ c pMq, where f u pt, pq :" fpt´u, pq for every t, u P R and p P Σ. It is shall not be difficult to prove that the Hadamard state constructed in Theorem 3 is invariant under the action of α u ω µ pα u paqq " ω µ paq @u P R @a P A g It should be also true that the map R Q u Þ Ñ ω µ pbα u paqq P C is continuous for every a, b P A g which would assure (see, e.g. [43]) that α :" tα h u hPR is unitarily implementable by a strongly continuous unitary representation of R in the GNS representation of ω µ and that the vacuum vector of the Fock-GNS representation is left invariant under the said unitary representation. We expect that the selfadjoint generator of that unitary group has a positive spectrum where, necessarily, the vacuum state is an eigenvector with eigenvalue 0. In other words ω µ should be a ground state of α. We finally expect that ω µ is pure (on the Weyl algebra associated to the symplectic space ppKerPq X Γ sc pMq, σ pPq q and it is the unique quasifree algebraic state which is invariant under α. We can summarize the previous discussion in the following question.
Question 7.1. Is the Hadamard state defined in Theorem 3 a ground state for the timetranslation? More precisely, is it the unique, pure, quasifree algebraic state which is invariant under action of α?
Last, but not least, we have seen in Section 6 that if a globally hyperbolic manifold admits a Proca quasifree Hadamard state according to the definition of Fewster-Pfenning, then Definition 4.5 and 6.1 are equivalent. This is the case for example for globally hyperbolic spacetimes whose Cauchy surfaces are compact. We do expect to extend this result for the whole class of globally hyperbolic spacetime.
As is evident from our quasi equivalence theorem, a complete equivalence would take place if a Hadamard state according to [16] is proved to exist for every globally hyperbolic spacetime. As a matter of fact, we expect that every globally hyperbolic spacetime pM, gq admits a quasifree Proca Hadamard state ω according to Fewster and Pfenning. This state should exist in every paracausally related ultrastatic spacetime pRˆΣ,´dt 2`h q with complete Cauchy surfaces of bounded geometry. With the same argument used for our existence proof of Hadamard states or the deformation argument exploited in [16], it should be possible to export this state to the original space pM, gq. We expect that the Hadamard Klein-Gordon bisolution for the real Proca field on pRˆΣ,´dt 2`h q used to define ω according to (6.1) in Definition 6.1 should have this form.
W g pf, f 1 q :" µpG N f, where N is the Klein-Gordon operator (3.2) associated to P and G N its causal propagator. The bilinear symmetric form µ :`pΩ 0 c pΣqq 2ˆp Ω 1 c pΣqq 2˘ˆ`p Ω 0 c pΣqq 2ˆp Ω 1 c pΣqq 2˘Ñ R is defined as in (5.21), but with the crucial difference that here its arguments are not restricted to C ΣˆCΣ .