Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a ∗\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 (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of 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.


Introduction
Recently a great deal of progress has been made in comparing the spaces of solutions of hyperbolic partial differential equations on (possibly different) Lorentzian manifolds as well as in the comparison of the associated quantum field theories. More precisely, given a pair N and N 1 of Green hyperbolic differential operators on (possibly different) globally hyperbolic spacetimes pM, gq and pM, g 1 q, a natural issue concerns the existence of a linear isomorphism S : Sol N Ñ Sol N 1 between the linear spaces of the solutions of the equations Nψ " 0 and N 1 ψ 1 " 0. Such an isomorphism, if it exists, is called a Møller map. Since the said space of solutions is the first step in the construction of corresponding (algebraic) free quantum field theories, a natural related issue concerns the possibility to promote the Møller map R to a˚-isomorphism between the associated abstract operator algebras A and A 1 constructed out of N and N 1 respectively on pM, gq and pM, g 1 q, in terms of corresponding generators given by abstract field operators Φpahq and Φ 1 ph 1 q and the associated causal propagators G N , G N 1 . Actually, off-shell linear QFT can be used to build up a perturbative approach to interacting QFT, a final problem would concern the possibility to extend the Møller isomorphism of algebras to an isomorphism of more physically interesting algebras, for instance including Wick powers or time-ordered powers. These problems have been tackled in the past for special cases of metrics g, g 1 and several types of Green hyperbolic field operators which rule the dynamics of bosonic fields [20,27] or fermionic fields [29,65]. In the loc. cit., the pairs of Lorentzian metrics g, g 1 had to satisfy one of the following assumptions: (I) they shared a common foliation of smooth spacelike Cauchy surfaces; (II) they coincided outside a compact set.
In this paper, we do not assume either of the above restrictions and instead, we consider a very wide family of pairs g, g 1 of globally hyperbolic metrics on the same manifold M. As a matter of fact, we consider a new type of relationship between globally hyperbolic metrics, which we call paracausal relationship. To the authors' knowledge this notion represents a complete novelty on the subject. Though the effective definition of paracausal equivalence relation in the set of globally hyperbolic metrics on M (Definition 2. 19) is different and more effective for the issues regarding Møller maps raised above, a complete characterization of it can be stated as follows in terms of elementary Lorentzian geometry: Theorem 1 (Theorem 2.24). The globally hyperbolic metric g on M is paracausally related to the globally hyperbolic metric g 1 on M if and only if there is a finite sequence g 0 :" g, g 1 , . . . , g N :" g 1 of globally hyperbolic metrics on M such that, at each step g k , g k` 1 , the future open light cones of these metrics have non-empty intersection V gk ‰ H at every point x P M.
The class of paracausally related metrics on a given manifold M is very large, though some elementary counterexamples of topological nature can be constructed. A specific study on the properties of this equivalence relation is necessary and it will be done elsewhere.
Equipped with this notion, the paper specializes the analytic setup as follows: N, N 1 : ΓpEq Ñ ΓpEq are (2nd order) normally hyperbolic operators on a real or complex vector bundle E on M, respectively associated to a pair of globally hyperbolic metrics g, g 1 on M.
The first important achievement of this work is the proof of existence of (infinitely many) Møller operators, i.e., isomorphisms R : ΓpEq Ñ ΓpEq, which restrict to Møller maps S between the space of solutions when g and g 1 are paracausally related. The overall idea is inspired by the scattering theory in the special case of a pair of globally hyperbolic metrics g 0 , g 1 over M such that the light cones of g 0 are included in the light cones of g 1 (this is the most elementary case of paracausal relation). We start with two "free theories", described by the space of solutions of normally hyperbolic operators N 0 and N 1 in corresponding spacetimes pM, g 0 q and pM, g 1 q, respectively, and we intend to connect 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 qwhich 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 .
The above construction generalizes to the case of a pair of globally hyperbolic metrics g, g 1 on M which are paracausally related and this fact is denoted by g » g 1 . A summary of the main results obtained is the following where also a special notion of adjoint operator R : gg 1 is used. It will be discussed in details in Section 4.
Theorem 2 (Theorems 4. 5, 4.6, and 4.12). Let E be K-vector bundle over the smooth manifold M with a non-degenerate, real or Hermitian depending on K, fiber metric x¨|¨y. Consider g, g 1 P GH M with respectively associated normally hyperbolic formally-selfadjoint operators N, N 1 . If the metrics are paracausally related g » g 1 , then it is possible to define a (non-unique) K-vector space isomorphism R : ΓpEq Ñ ΓpEq, called Møller operator of g, g 1 (with this order), such that the following facts are true.
(2) The causal propagators G N 1 and G N , respectively of N 1 and N, satisfy RG N R : gg 1 " G N 1 , where R : gg 1 is the adjoint of the Møller operator (see Definition 3.9).
(3) By denoting c 1 the smooth function such that vol g 1 " c 1 vol g , we have c 1 N 1 R " N .
(5) The maps R : gg 1 : Γ c pEq Ñ Γ c pEq and pR : gg 1 q´1 " pR´1q : g 1 g : Γ c pEq Ñ Γ c pEq are continuous with respect to the natural topologies of Γ c pEq in the domain and in the co-domain.
Theorem 2 permits us to promote R to a˚-isomorphism of the algebras of field operators A, A 1 respectively associated to the paracausally related metrics g and g 1 (and the associated N, N 1 ) and generated by respective field operators Φpfq and Φ 1 pf 1 q with f, f 1 compactly supported smooth sections of E. These field operators satisfy respective CCRs rΦpfq, Φphqs " iG N pf, hqI , rΦ 1 pf 1 q, Φ 1 ph 1 qs " iG N 1 pf 1 , h 1 qI 1 and the said unital˚-algebra isomorphism R : A 1 Ñ A is determined by the requirement (Proposition 5.6) RpΦ 1 pf 1 qq " ΦpR : gg 1 fq .
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 a certain wavefront set of their two-point function. To this regard, we prove that the pull-back through R of a Hadamard state ω : A Ñ C is a Hadamard state of the off-shell algebra A 1 , provided the metrics g, g 1 be paracausally related. The result is extended to a generic bidistribution ν (corresponding to the two-function of ω, dropping the remaining requirements included in the definition of state). The proof of the theorem below is both of geometrical and microlocal analytic nature (see also Theorem 5.14).
Theorem 3 (Theorem 5. 19). Let E be an R-vector bundle on a smooth manifold M equipped with a non-degenerate, symmetric, fiberwise metric x¨|¨y. Let g, g 1 P GH M , consider the corresponding formally-selfadjoint normally hyperbolic operators N, N 1 : ΓpEq Ñ ΓpEq and refer to the associated CCR algebras A and A 1 . Let ν P Γ 1 c pE b Eq be of Hadamard type and satisfy νpx, yq´νpy, xq " iG N px, yq mod C 8 , G N px, yq being the distributional Kernel of G N . Assuming g » g 1 , let us define ν 1 :" ν˝R : gg 1 b R : gg 1 , for a Møller operator R : ΓpEq Ñ ΓpEq of g, g 1 . Then the following facts are true.
As this crucial result concerns off-shell algebras, in principle, it could be exploited in perturbative constructions of interacting theories. Indeed the preservation of the Hadamard singularity structure plays a crucial role in the development of the perturbative theory [27]. Another work will be devoted to this investigation.
The work is organized as follows. Section 2 contains a recap on the relevant notions of Lorentzian geometry we exploit throughout. In particular, in Section 2.2 we introduce some (apparently new) results about convex interpolations of globally hyperbolic metrics which are preparatory to Section 2.3 where we present the definition of paracausal relation and we give some basic results about this equivalence relation. Section 3 is completely devoted to recalling some notions and fundamental results about Green hyperbolic operators, normal hyperbolic results and their interplay with convex combinations of Lorentzian metrics. Section 4 is the core of the paper. In the Subsections 4.1 and 4.2 Møller maps are introduced under the additional assumption that the metrics g 0 , g 1 P GH M satisfy g 0 ĺ g 1 . The latter is removed in Subsection 4.3, where the analysis is extended to encompass paracausally deformed metrics g » g 1 . In Subsection 4.4, it is shown that the Møller map preserves the natural symplectic form on the space of initial data and finally, in Subsection 4.5 the Møller operator for paracausally deformed metrics is introduced and analyzed in detail. Section 5 is devoted to the study of free quantum field theories on globally hyperbolic spacetimes. In particular, in Subsection 5.2 we lift the Møller operator to å -isomorphism of algebras of observables and in Subsection 5.3 we show that the pull-back of any quasifree state along the Møller˚-isomorphism preserves the Hadamard condition. Finally, we conclude our paper with Section 6, where open issues and future prospects are presented.

General notation and conventions
-A Ă B permits the case A " B, otherwise we write A Ĺ B.
-The symbol K denotes any element of tR, Cu.
-Tensor fields and sections of K-vector bundles on M are always supposed to be smooth.
-pM, gq denotes a pn`1q-dimensional spacetime (cf. Definition 2.4) and we adopt the convention that g has the signature p´,`. . . ,`q.
-M M , T M Ă M M and GH M Ă T M denote respectively the sets of smooth Lorentzian metrics, time-oriented Lorentzian metric and globally hyperbolic metrics on M; -g ĺ g 1 denotes that g, g 1 P M M and the open light cone V g p of g is a subset of the open lightcone V g 1 p of g 1 at every point p P M; -g » g 1 denotes that g and g 1 are paracausally related (cf. Definition 2.19).

Acknowledgments
We are grateful to Nicolò Drago, Nicolas Ginoux, and Miguel Sánchez for helpful discussions related to the topic of this paper. We are grateful to the referees for useful comments on the manuscript. This work was produced within the activities of the INdAM-GNFM.

Convex interpolation and paracausal deformations of Lorentzian metrics
The aim of this section is twofold. On the one hand we shall investigate the properties of convex interpolation of Lorentzian metrics, on the other hand we introduce the notion of paracausal deformations of globally hyperbolic metrics. As we shall see, these mathematical tools rely on a certain preordering relation in the set of Lorentzian metrics on a given manifold, they are quite interesting on their own right and they will be exploited in the second part of this work to construct Møller operators and Møller˚-isomorphisms of algebras of quantum fields.

Preliminaries on Lorentzian geometry
The aim of this section is to recall some basic results of Lorentzian geometry which we will need later on. For a more detailed introduction to Lorentzian geometry we refer to [3,9,66].

Lorentzian manifolds and cones
Let M be a smooth connected paracompact Hausdorff manifold and assume that M is noncompact or its Euler characteristic vanishes. Under these assumptions, M admits a Lorentzian metric and we denote the space of Lorentzian metrics on M by M M (see e.g. [9]). Once that a Lorentzian metric g is assigned to a smooth manifold M Q p, we can classify the vectors v p P T p M into three different types: • spacelike i.e. gpv p , v p q ą 0 or v p " 0 , • timelike i.e. gpv p , v p q ă 0, • lightlike (also called null) i.e. gpv p , v p q " 0 and v p ‰ 0.
Remark 2.1. Notice that, with our definition, the tangent vector 0 is spacelike.
As usual, we denote as causal vectors any timelike or lightlike vector. Piecewise smooth curves are classified analogously according to the nature of their tangent vectors.
Keeping in mind this classification, the open lightcone of pM, gq at p P M is the set It is not difficult to see that it is an open convex cone made of two disjoint open convex halves defining the two connected components of V g p . The notion of time orientation is defined as in [3]: A smooth Lorentzian manifold pM, gq is said to be time-orientable if there is a continuous timelike vector field X on M.
If pM, gq is time orientable and a preferred continuous timelike vector field X has been chosen as above, the future lightcone V gp Ă V g p at p P M is the connected component of V g p containing X p . The other connected component V gṕ is the past lightcone at p. V gp and V gṕ respectively includes the future-directed and past-directed timelike vectors at p. The terminology extends to the causal (lightlike) vectors which belong to the closures of the said halves. A classification of (piecewise smooth) causal curves into past-directed and future-directed curves (see [3]) arises according to their tangent vectors.
If pM, gq is time orientable, the continuous choice of one of the two halves of V g p for all p P M through a continuous timelike vector field as above defines a time orientation of pM, gq. pM, gq with this choice of preferred halves of cones is said to be time oriented. If pM, gq is connected and time orientable, then it admits exactly two time orientations. Notation 2.2. In the following, we denote with M M , the set of smooth Lorentzian metrics on the smooth manifold M and with T M the class of time-oriented Lorentzian metrics on M.
We have an elementary fact whose proof is immediate if working in a g-orthonormal basis. Proposition 2.3. Assume that g P T M , p P M, and Y p , Z p P V g p . Then (i) Y p P V gp and Z p P V gp if and only if gpY p , Z p q ą 0, (ii) Y p , Z p P V gp if and only if gpY p , Z p q ă 0.
If g P M M , the associated standard (fiberwise) musical isomorphism 7 : ΓpT˚Mq Ñ ΓpTMq is pointwise defined by gp7ω p , v p q " ω p pv p q for every v P ΓpTMq and ω P ΓpT˚Mq and p P M, and we denote the (fiberwise) inverse musical isomorphism by 5 : ΓpTMq Ñ ΓpT˚Mq. The notation g 7 P ΓpTM b TMq indicates the Lorentzian metric induced on 1-forms from 7 as g 7 pω 1p , ω 2p q " gp7ω 1p , 7ω 2p q for every ω 1 , ω 2 P ΓpT˚Mq and p P M.
Once that a Lorentzian metric is introduced on 1-forms, we can distinguish three different type of co-vectors: ω p P TpM is spacelike, timelike, null and causal if, respectively, 7ω p P T p M is spacelike, timelike or null. With the definition, we can define the open lightcone of 1-forms at p P M analogously to the case of vectors V g 7 p :" tω p P TpM | g 7 pω p , ω p q ă 0u .
Analogously, if g P T M , the future and past lightcones of 1-forms at p P M are defined as Let us finally recall that, embedded codimension-1 submanifold Σ Ă M of a Lorentzian manifold pM, gq, also called hypersurfaces, are classified according to their normal covector n: They are spacelike, timelike, null if respectively n is timelike, spacelike, null everywhere in Σ. Notice that an embedded n´1 submanifold Σ Ă M is spacelike if and only if its tangent vectors are spacelike in pM, gq. The restriction of g to the tangent vectors to a spacelike hypersurface Σ defines a Riemannian metric on it.

Spacetimes and causality
Definition 2.4. A spacetime is a pn`1q-dimensional (n ě 1), connected, time-oriented, smooth Lorentzian manifold pM, gq Remark 2.5. Sometimes it is also assumed that M is orientable and oriented, but we do not adopt this hypothesis here. However, when we write that pM, gq is a spacetime we also mean that a time-orientation of pM, gq as Lorentzian manifold has been chosen. In this case, with a little misuse of language, we speak of the time-orientation of the metric g.
Let now A Ă M for a spacetime pM, gq. The causal sets J˘pAq and the chronological sets I˘pAq are defined according to [3]: J˘pAq is made of the points of A itself and all p P M such that there is a smooth future-directed/past-directed causal curve γ : ra, bs Ñ M with γpaq P A and γpbq " p. Notice that J˘pAq Ą A by definition, while I˘pAq is made of the points p P M such that there is a smooth future-directed/past-directed timelike curve γ : ra, bs Ñ M with γpaq P A and γpbq " p. As usual we define JpAq :" J`pAq Y J´pAq.
Let us recall that, on a spacetime pM, gq, a smooth causal curve γ : I Ñ M with I Ă R open interval is said to be future inextendible [66] if there is no continuous curve γ 1 : J Ñ M, defined on an open interval J Ă R, such that sup J ą sup I and γ 1 | I " γ. A past inextendible causal curve is defined analogously. A causal curve is said to be inextendible if it is both past and future inextendible.
We eventually define the future Cauchy development D`pAq of A to be the set of points p P M such that every past inextendible future-directed smooth causal curve passing through p meets A in the past. Similarly, the past Cauchy development D´pAq is the set of points p P M such that every future inextendible future-directed smooth causal curve passing through p meets A in the future.
On a generic Lorentzian manifold, the Cauchy problem for a differential operator is in general ill-posed: This can be a consequence of the presence of closed timelike curves or the presence of naked singularities. Therefore, it is convenient to restrict ourselves to the class of globally hyperbolic spacetimes. Definition 2.6. A globally hyperbolic spacetime is a spacetime pM, gq such that (i) there are no closed causal curves; (ii) for all points p, q P M, J`ppq X J´pqq is compact. Notation 2.7. If M is a smooth connected pn`1q-manifold, GH M Ă T M denotes the class of Lorentzian metrics g such that pM, gq is globally hyperbolic for a time-orientation. Any g P GH M is called globally hyperbolic metric on M.
In his seminal paper [51], Geroch established the equivalence for a Lorentzian manifold being globally hyperbolic and the existence of a Cauchy hypersurface.
Definition 2.8. A subset Σ Ă M of a spacetime pM, gq is called Cauchy hypersurface if it intersects exactly once any inextendible future-directed smooth timelike curve.
In particular, a Cauchy hypersurface is achronal: it intersects at most once every futuredirected smooth timelike curve. It turns out that Cauchy hypersurfaces of pM, gq are closed co-dimension 1 topological submanifolds of M homeomorphic one to each other. As a byproduct of Geroch's theorem, it follows that the globally hyperbolic manifold pM, gq admits a continuous foliation in Cauchy hypersurfaces Σ, namely M is homeomorphic to RˆΣ. The proof of these facts was carried out by finding a Cauchy time function, i.e., a continuous function t : M Ñ R which is strictly increasing on any future-directed timelike curve and such that its level sets t´1pt 0 q, t 0 P R, are Cauchy hypersurfaces homeomorphic to Σ. Geroch's splitting appears at a topological level, and the possibility to smooth them remained an open folk questions for many years. Only recently, in [13] Bernal and Sánchez "smoothened" the result of Geroch by introducing the notion of Cauchy temporal function. and the time-orientation induced from pM, gq through ψ. Above τ is the canonical projection RˆΣ Q pt, pq Þ Ñ t P R and the following facts are valid: is a smooth Riemannian metric on each leaf ttuˆΣ, t P R, (iv) every embedded co-dimension-1 submanifold tt 0 uˆΣ " τ´1pt 0 q is a spacelike (smooth) Cauchy hypersurface.
Finally, if S Ă M is a spacelike Cauchy hypersurface of pM, gq, then we can define an isometry ψ : M Ñ RˆS, and τ , β, h as above in order that S " ψ´1pt0uˆSq.
The characterization given by Bernal and Sánchez permits us to give some relevant definitions.
Definition 2.11. Given a spacetime pM, gq, a smooth surjective function t : M Ñ R with dt past-directed timelike is (a) a Cauchy temporal function if (i) pM, gq is isometric, through some isometry ψ : M Ñ RˆΣ, to a spacetime pRˆΣ, hq with the time-orientation induced from pM, gq, (iii) h has the form (2.1) as in Theorem 2.10 satisfying (i)-(iv); (b) a smooth Cauchy time function if (i) pM, gq is isometric, through some isometry ψ : M Ñ RˆΣ, to a spacetime pRˆΣ, hq with the time-orientation induced from pM, gq, (iii) every Σ t 0 :" t´1pt 0 q " ψ´1ptt 0 uˆΣq is a spacelike Cauchy hypersurface of pM, gq for t 0 P R.
(1) An intrinsic way to write (2.1) for a Cauchy temporal function t without making use to the splitting diffeomorphism ψ is, for p P Σ s " t´1ppq defines the orthogonal projector onto T p Σ s associated to t and g, using 7 g dt as normal (contravariant) vector to Σ s .
(2) If an either smooth time or temporal Cauchy function t exists for pM, gq, the level sets Σ t 0 :" t´1pt 0 q are smooth spacelike Cauchy surfaces diffeomorphic to each other and pM, gq is globally hyperbolic. Theorem 2.10 proves that temporal Cauchy functions -thus also smooth time Cauchy functions -exist for every globally hyperbolic spacetime. Furthermore, every smooth spacelike Cauchy hypersurface can be embedded in the foliation induced by a suitable temporal Cauchy function.
(3) A Cauchy temporal function is always a Cauchy time function, but even a smooth time function may not be a temporal one.
(4) A Cauchy hypersurface may meet a causal curve in more than a point (say, a segment), but this is not the case for the spacelike Cauchy hypersurfaces since they are acausal: they intersect at most once every future-directed smooth causal curve, as easily arises from Theorem 2.10.
We shall now give some notable examples of globally hyperbolic spacetimes to acquaint the reader with some concrete cases.
Examples 2.13. We shall list a few globally hyperbolic spacetimes which appear commonly in general relativity and quantum field theory over curved backgrounds. As one can infer per direct inspection, they all fulfill Theorem 2.10: • the prototype example is Minkowski spacetime which isometric to R n`1 with Cartesian coordinates pt, x 1 , . . . , x n q and equipped with the Minkowski metrić • de Sitter spacetime, that is the maximally symmetric solution of Einstein's equations with a positive cosmological constant Λ. As a manifold it is topologically RˆS 3 and the metric reads: where t P R while pχ, θ, ϕq are the standard coordinates on S 3 ; • the Friedmann-Robertson-Walker spacetime, i.e., an isotropic and homogeneous manifold which is topologically RˆΣ and where k can be either 0 or˘1 and function aptq is smooth and positive valued; • The external Schwarzschild spacetime, i.e., a stationary spherically symmetric solution of vacuum Einstein's equations which is topologically R 2ˆS2 with metric Here M ą 0 is interpreted as the mass of the spherically symmetric source (a blackhole, a star,...) and the domain of definition of the coordinates is t P R, r P p2M,`8q and pθ, ϕq P S 2 ; • Finally, given any n-dimensional complete Riemannian manifold pΣ, hq, an open interval I Ď R and a smooth function f : I Ñ p0,`8q, the Lorentzian warped product defined topologically by IˆΣ with metric g "´dt 2`f ptqh is a globally hyperbolic spacetime.

Convex interpolation of Lorentzian metrics
Due to motivations that will be clear later in the paper and related to the construction of Møller operators, we are now interested in the structure of the set M M of Lorentzian metrics on a given manifold M. In particular, we are interested in the following problem: Are there some natural operations which can be used to produce (globally hyperbolic) Lorentzian metrics starting from (globally hyperbolic) Lorentzian metrics?
Given two globally hyperbolic metrics g, g 1 , a linear combination of them is in general not a Lorentzian metric and, when it is, it fails to be globally hyperbolic in general. However, as shown in [16,Appendix B], if g and g 1 coincide outside a compact set, then there exists a sequence of 5 globally hyperbolic metrics, such that for each neighbouring pair all pointwise convex combinations are globally hyperbolic metrics. Therefore, this section aims to provide sufficient conditions for some kind of linear combination of globally hyperbolic metrics to be a globally hyperbolic Lorentzian metric. We shall see that convex combinations are an interesting case of study under suitable conditions. We point out the recent work [71] where, in addition to several related issues, the convex structure of the space of globally hyperbolic metrics on a given manifold is addressed with a number of results.

A preorder relation of Lorentzian metrics
Definition 2.14. Let g, g 1 P M M and denote We say that g, g 1 P M M are ĺ-comparable if either g ĺ g 1 or g 1 ĺ g (see e.g. Figure 1). (1) Let us remark that the definition above can be generalized by considering the so-called causal diffeomorphisms, namely a time-orientation preserving diffeomorphism ϕ : M Ñ N such that the open light cone V g p of g is included in the open light cone V ϕg 1 p of ϕ˚g 1 for every p P M. For further details and properties we refer to [37,39,40].
(2) The preorder relation introduce in Definition 2.14 has a corresponding for the associated metrics in the cotangent space: If g, g 1 P M M , We observe that if g ĺ g 1 for g, g 1 P T M and the two metrics share the same time-orientationi.e., there is a continuous vector field on M which is timelike for both metrics and defines the same time-orientation for both of them -then V gp Ă V g 1p and V gṕ Ă V g 1ṕ for every p P M. Similar inclusions hold when considering the closures of the considered half cones. As a consequence, we have both inclusions with obvious notations The relation ĺ in M M has several consequences whose most elementary ones are established in the following proposition. We remind the reader that a closed set A Ă M, with pM, gq timeoriented, is past compact if J´ppq X A is compact or empty for every p P M. The definition of future compact is analogous, just replacing J´for J`.
Then next lemma is just a routine computation, so we leave the proof to the reader. (1) For p P M and v P T p M, if g ĺ g 1 then (i) gpv, vq " 0 implies g 1 pv, vq ď 0.
(2) If g ĺ g 1 with g P T M and g 1 P GH M , then pM, gq is globally hyperbolic as well when, e.g., equipped with the same orientation and time-orientation of pM, g 1 q and (i) a spacelike Cauchy hypersurface for pM, g 1 q is also a spacelike Cauchy hypersurface for pM, gq; (ii) a smooth Cauchy time function for pM, g 1 q is also a smooth Cauchy time function for pM, gq; (iii) a closed set A Ă M is past/future compact in pM, gq if it is respectively past/future compact in pM, g 1 q.
(4) If g, g 1 P T M , g ĺ g 1 and p P M, then V gp Ă V g 1p if and only if V g 17p Ă V g 7p (see e.g. Figure 2).  (d) µg and µ´1g are globally hyperbolic if g is and the spacelike Cauchy hypersurfaces of g are also spacelike Cauchy hypersurfaces for µg and µ´1g.

Properties of convex combinations of Lorentzian metrics
A more interesting set of properties arises when focusing on smooth convex combinations of Lorentzian metrics. This is the first main result of this section.
(1) It is sufficient to prove the thesis point by point. Let q, q 1 be quadratic forms in a real n`1 dimensional linear space V of signature p´,`, . . . ,`q such that q 1 pxq ď 0 implies qpxq ď 0. We prove that the strict convex combination q 2 " cq`p1´cqq 1 for c P p0, 1q has signature p´,`, . . . ,`q. Indeed, there is a 1-dimensional subspace L on which q 1 pxq ă 0 if x ‰ 0. So qpxq ď 0 on L and hence q 2 pxq ă 0 on L for x ‰ 0. There is also a n-dimensional subspace H on which qpxq ą 0 if x ‰ 0. Then q 1 pxq ą 0 on H for x ‰ 0 and hence q 2 pxq ą 0 on H if x ‰ 0. By construction, L X H " t0u necessarily, so that V " L ' H. The bilinear form Q 2 : VˆV Ñ R associated to q 2 , in a basis of V made of 0 ‰ e 0 P L and te k u k"1,...,n P H with Q 2 pe k , e h q " δ kh , is represented by the pn`1qˆpn`1q matrix Since the determinant is q 2 pe 0 q´c t c ă 0 and n eigenvalues are`1, its signature is p´,`, . . . ,`q.
(3) g 7 and g 17 are Lorentzian metrics on T˚M and g 17 ĺ g 7 due to Lemma 2.16, we can recast the same argument used to establish (1) with trivial re-arrangements, obtaining that g 7 χ is Lorentzian and g 17 ĺ g 7 χ " p1´χ 1 qg 17`χ1 g 7 ĺ g 7 with χ 1 :" 1´χ. Notice that g χ pv, vq :" g 7 χ p5v, 5vq defines a Lorentzian metric as well, since it has the same signature of h by construction, and g 7 " h trivially (and it is the unique metric with this property since 5 is an isomorphism). (4) It immediately arises from Lemma 2.16 by using g 17 ĺ g 7 χ " p1´χ 1 qg 17`χ1 g 7 ĺ g 7 with χ 1 :" 1´χ. (5) A smooth timelike vector field of pM, gq is also timelike for p1´χqg`χg 1 and g χ for (2) and (4) respectively. Hence these metrics are time-orientable and the thesis follows from Lemma 2.16 point (2).

Paracausal deformation of Lorentzian metrics
The aim of this section is to provide a new definition that shall encode the idea to deform a Lorentzian metric equipped with a time-orientation into another Lorentzian metric with a corresponding time-orientation, taking advantage of a procedure consisting of a finite number of steps. At each step, the light cones of the final metric g k is related to the initial one g k´1 through an inclusion relation, either g k´1 ĺ g k or g k ĺ g k´1 preserving the time-orientation at each step, i.e., the future cone of g k , respectively, includes or is included in the future cone of g k´1 .

Paracausal relation
Definition 2.19. Consider a pair of globally hyperbolic spacetimes on the same manifold M with corresponding metrics g, g 1 P GH M and corresponding time-orientations. We say that g is paracausally related to g -and we denote it by g » g 1 -or equivalently g 1 is a paracausal deformation of g, if there is a finite sequence g 0 " g, g 1 , . . . , g N " g 1 P GH M with corresponding time-orientations, such that for all p P M either where the choice may depend on k " 0, . . . , N´1.
(1) Let us remark that our notion of paracausally deformation implies in particular that g k and g k`1 are always ĺ-comparable.
(2) Evidently, to be paracausally related is an equivalence relation in GH M .
(3) We stress that paracausal deformations explicitly consider the time-orientations of the used sequences of globally hyperbolic spacetimes. So, even if we say that "metrics are paracausally related", the relation actually involves the metrics equipped with corresponding time-orientations.
(3) We shall show below a characterization of the paracausal relationship which seems more natural from a geometric and physical viewpoint. However, the definition above as it stands is much more directly suitable for the applications to Møller operators we shall introduce in the second part of this work.
(1) There two elementary cases of paracausally related (globally hyperbolic) metrics g 0 , g 1 on M which are not directly ĺ-comparable: 1. There is a globally hyperbolic metric g on M such that, simultaneously g ĺ g 0 and g ĺ g 1 and the future lightcones are correspondingly included.
2. There is a globally hyperbolic metric g on M such that, simultaneously g 0 ĺ g and g 1 ĺ g and the future lightcones are correspondingly included.
In both cases, the existence of sequence g 0 , g, g 1 proves that g 0 » g 1 .
(2) Let us give an elementary concrete example of paracausally related metrics. Consider the following smooth manifold R n endowed with the Minkowski metrics where pt, x 1 , . . . , x n q and pτ, y 1 , . . . , y n q are two different systems of Cartesian coordinates on R n`1 . Here t and τ are Cauchy temporal functions associated to the respective Lorentzian metric and defining the time-orientation of the two metrics: dt and dτ are assumed to be past directed for the respective metric. More precisely, we assume that the two coordinate systems are related by means of a physically non-trivial permutation that interchanges space and time, as in Figure 3, τ " x 1 , y 1 " t, and y k " x k for k ą 1. It is not difficult to see that even if η 0 ‰ η 1 evidently, we have η 0 » η 1 : they are paracausally related by the sequence of metrics η 0 , g 1 , g 2 , η 1 whose future cones are given as in Figure 4. It is evident that by further implementing the procedure, it is possible to reverse the time-orientation of pM, η 0 q through a sequence of paracausal deformations leaving the final metric identical to the initial one.
(3) We pass to present a case where a pair of globally hyperbolic metrics are not paracausally related. Consider the 2D Minkowski cylinder M obtained by identifying x and x`L in R 2 with coordinates x, y. The first globally hyperbolic spacetime is pM, η 1 q where η 1 " dy bdy`dxbdx, taking the identification into account, and with time-orientation defined  by assuming that B y is future-directed. The second globally hyperbolic spacetime is pM, η 2 q where again η 2 "´dy b dy`dx b dx, taking the identification into account, but with the opposite time-orientation, i.e., defined by´B y . See also Figure 5. These two metrics are not paracausally related. Any attempt to use the procedure as in the previous example to rotate the former into the latter faces the insurmountable obstruction that one of the auxiliary metrics would have Cauchy hypersurfaces given by the x-constant lines. This Lorentzian manifold is not globally hyperbolic because it admits closed temporal curves as in Figure 6.
Notice that this obstruction does not take place without the identification x " x`L. There is a natural situation where two globally hyperbolic metrics g and g 1 on M are paracausally related. The generalization of the following result leads to a natural characterization of the paracausal relationship.
Proposition 2.22. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes on the same manifold for every x P M, then the metrics g and g 1 are paracausally related.
Proof. To prove the assertion it is sufficient to prove the existence of a Lorentzian metric h P T M such that h ĺ g and h ĺ g 1 . In this case, h would be globally hyperbolic according to (2) in Lemma 2.16 and the same argument as in (1) Examples 2.21 would prove the thesis. Let us start by proving that a smooth vector field X on M exists such that X p P V gp X V g 1p for all p P M. Let us define the smooth functions where Y is a smooth timelike future oriented vector field for g. By construction (with obvious is therefore open, non-empty by hypothesis, and projects onto the whole M by construction. As a consequence, given a local trivialization patch TU around p P U , where pU, ψq is a local chart on M (with dimpMq " n`1), the set´Y pPM V gp X V g 1p¯X TU is diffeomorphic to an open subset A Ă VˆR n`1 with V :" ψpU q Ă R n`1 and π 1 pAq " V (π 1 : R n`1ˆRn`1 Ñ R n`1 being the standard projection onto the former factor). Working in coordinates, it is then trivially possible to pick out a smooth local section X pU q of TU such that X pU q q To conclude, consider a partition of the unity tχ i u iPI of M subordinated to a locally finite covering tU i u iPI of domains of local charts of M and let X be constructed as above when p P U i for every i P I. The smooth vector field constructed as a locally finite convex linear combination X :" are convex sets in a vector space and thus their intersection is also convex. X is the vector field we were searching for.
As the second step we construct a Lorentzian metric h, whose future cones V hp satisfy for every p P M. Notice that it means h P T M since X is future directed for h (and also for the two metrics g and g 1 ) and thus it defines a time-orientation for pM, hq. Since h ĺ g, g 1 , this would conclude our proof.
Let us construct h taking advantage of the vector field X. Consider p P M and define a g-pseudo orthonormal basis e 0 , . . . , e n where e 0 " Xp ?´g pXp,Xpq and the remaining vectors are trivially satisfies (the closure being taken in T p Mzt0u) The strong inclusions are due to the fact that the lightlike boundary of V g ap is made of timelike vectors of g as it arises from the definition of g a . Now note that BV g ap becomes more and more concentrated around the set tλX p | λ ą 0u as a approaches 0 from above. (In particular, the limit and degenerate case g a"0 . This property is locally uniform in a as established in the following technical lemma: We will henceforth deal with U and the coordinates px 0 , . . . , x n q restricted to thereon. We will also take advantage of the compact set K :" ϕpU q Ă R n`1 and identify TU with KˆR n`1 using the coordinates. Finally, we will equip both K and R n`1 (representing T p M at each p P U " K) with the standard Euclidean metric of R n`1 whose norm will be denoted by ||¨||.
Let us start the proof by proving that the family of cones V g 1p of g 1 has a minimal width m ą 0 when p ranges in K. We henceforth view the above future-directed timelike vector field X and g 1 as geometric objects on K using the coordinate system. In particular, if p P K, let us indicate by v p P R n`1 the unique future-directed timelike vector parallel to X p (now viewed as a vector in R n`1 ) such that ||v p || " 1. Consider the set made of future-directed elements of T M C :" tpp, uq P KˆS n | g 1 p pu, v p q ď 0 , g 1 p pu, uq " 0u (above S n :" tz P R n`1 | ||z|| " 1u) and the continuous function which computes the width of BV g 1p (that is of V g 1p itself) around X p along the direction u by using the Euclidean distance induced by ||¨||. Observe that C is compact since it is the intersection of preimages of a pair of closed sets along two corresponding continuous maps and it is included in a compact set. Since this map is continuous and C is compact, there exists m :" min C W ą 0 .
1 As noticed by the referee, a different strategy for proving this lemma would be showing that the function M Q p Þ Ñ appq " supta P p0, 1q : V g a p`Ă V g 1 p`u is continuous. In that case, one canalternatively define aU :" min pPU appq . However the proof of continuity is not technically easy.
In particular, m ą 0, otherwise u " v q for some pq, uq P C and this is not possible since it would imply g 1 pv q , v q q " g 1 pu, uq " 0, but v q is timelike (g 1 q pv q , v q q ă 0) since it does not vanish (||v q || " 1) and it is proportional to the timelike vector X q .
An analogous width-cone function can be defined for the cones of g a (including the degenerated case a " 0) on a set C 1 which also embodies the dependence on a: C 1 :" tpa, p, uq P r0, 1{2sˆKˆS n | g a p pu, v p q ď 0 , g a p pu, uq " 0u . We also define the continuous function Observe that C 1 is again compact since it is the intersection of preimages of two closed sets along a pair of corresponding continuous maps of pa, p, uq and C 1 is included in a compact set.
We want to prove that there exists a m P r0, 1{2s such that W 1 pa m , p, uq ă m for all pp, uq P C. If this were not the case, then for every a n :" 1{n there would be a pair pp n , u n q P C such that W 1 pa n , p n , u n q ě m. Since C 1 is a compact metric space, we could extract a subsequence of triples pa n k , p n k , u n k q Ñ p0, p 8 , u 8 q P r0, 1{2sˆC for k Ñ`8 and some pp 8 , u 8 q P C. By continuity 0 " g an pn pu n , u n q Ñ g 0 p8 pu 8 , u 8 q where ||u 8 || " 1. From (2.2), g 0 p8 pu 8 , u 8 q " 0 would entail that u 8 is parallel to v p8 and thus W 1 p0, p 8 , u p8 q " ||u 8´vp8 || " 0. That is in contradiction with the requirement W 1 pa n , p n , u n q ě m ą 0 for every n " 1, 2, . . . in view of the continuity of W 1 .
We have therefore established that there exists a m P r0, 1{2s such that W 1 pa m , p, uq ă m for all pp, uq P C. From the definition of W and W 1 , we have also obtained that V g a mp Ă V g 1p for all p P K. It is enough to conclude that V g a Up Ă V g 1p for all p P U as wanted simply by taking a U :" a m . This concludes our claim. l Let us go on with the main proof. For every U as in the previous lemma, define the constant function appq " a U for p P U . Since this can be done in a neighborhood of every point p P M, using a partition of the unity tχ i u iPI subordinated to a locally finite covering of charts tU i u iPI , we can construct the metric h, where now every a i :" a U i : U i Ñ p0, 1q is a constant in U and thus it is a smooth function therein.
Since ř i χ i ppqa i ppq P p0, 1q, this metric is still Lorentzian and of the form (2.2) point by point, where now appq " ř i χ i ppqa i ppq. By construction X p P V hp Ă V gp for every p P M, just because it happens point by point with the above choice of appq. In particular, we can endow h with the time-orientation induced by X as it happens for g, g 1 and all local metrics g a i .
because, if h p pv, vq ă 0, at least one of the values g a i 0 ppq pv, vq appearing in for every p P M.
As an immediate byproduct, it is easy to see that for any globally hyperbolic metric g, there exists a paracausal deformation g 1 of g which is ultrastatic.
Corollary 2.23. Let pM, gq be a globally hyperbolic spacetime. Then there exists a paracausal deformation g 1 of g such that pM, g 1 q is an ultrastatic spacetime.
Proof. Let t be a Cauchy temporal function for the globally hyperbolic spacetime pM, gq so that M is isometric to RˆΣ with metric´dt 2`h t . We indicate by B t the tangent vector to the submanifold R. Let h be a complete Riemannian metric on Σ. Then the ultrastatic metric g 1 :"´dt 2`h is globally hyperbolic [70] and the vector B t is contained in the intersection of V gp and V g 1p for any p P M. Proposition 2.22 ends the proof.
The result established in Proposition 2.22 leads to a crucial characterization of paracausally related metrics, which represent the second main result of this section.
Theorem 2.24. Let M be a smooth manifold. Two metrics g, g 1 P GH M are paracausally related if and only if there exists a finite sequence of globally hyperbolic metrics g 1 " g, g 2 . . . , g n " g 1 on M such that all pairs of consecutive metrics g k , g k` ‰ H for every x P M.
Proof. If g, g 1 are paracausally related, then a sequence of metrics as in Definition 2.19 trivially satisfies the condition in the thesis. If that condition is vice versa satisfied, then the metrics of each pair g k , g k`1 of the sequence are paracausally related in view of Proposition 2.22. Since paracausal relation is transitive, g and g 1 are paracausally related.

Paracausal deformation and Cauchy temporal functions
We now study the interplay of the notion of Cauchy temporal function and the one of paracausal deformation.
A first result regards metrics that share a common foliation of Cauchy surfaces. We need a preliminary technical result.
Lemma 2.25. Let pM, gq be a globally hyperbolic spacetime, t : M Ñ R a Cauchy temporal function according to Definition 2.11 and ψ : M Ñ RˆΣ a diffeomorphism mapping isometrically pM, gq to pRˆΣ,´β 2 dτ b dτ ' h τ q. Finally let pM, g α q be a time oriented spacetime with time orientation such that dt is past directed. If ψ maps pM, g α q isometrically to pRˆΣ,´dτ b dτ ' α 2 pτ qβ´2h τ q with α P C 8 pR, p0, 8qq, then pM, g α q is globally hyperbolic.
Proof. We will henceforth omit to write the isometry ψ and consider, without loss of generality, On account of Theorem 2.9, it is enough to prove that Σ, viewed as the t " 0 slice of the temporal function t, is a spacelike Cauchy hypersurface for g α . Evidently Σ is a spacelike hypersurface for g α so that it suffices to prove that it meets exactly once every inextendible future directed g α -timelike curve γ : I Q s Þ Ñ γpsq P M. Since dt ds " g α pB t , 9 γq ă 0 by hypothesis, that γ can be re-parametrized by t itself as γ 1 : J Q t Þ Ñ γ 1 ptq P M for some open interval J Ă R. There must exist a finite a ą 0 such that p´a, aq X J ‰ H. Since γ 1 | p´a,aqXJ is inextendible in the spacetime p´a, aqˆΣ (otherwise it would not be inextendible in the whole spacetime), to conclude it is sufficient to prove that p´a, aqˆΣ equipped with the metric g α and the time-orientation induced by dt admits Σ as a Cauchy surface. Indeed, in that case, γ 1 must meet Σ exactly once in p´a, aqˆΣ and thus Σ would be a Cauchy hypersurface for pRˆΣ, g α q. Moreover, notice that it cannot meet Σ " t´1p0q again outside p´a, aqˆΣ because γ 1 is parametrized by t. Global hyperbolicity of pp´a, aqˆΣ, g α q can be proved as follows. If a ą 0, there exists a positive constant α 0 such that αptq ě α 0 ą 0 for all t P r´a, as. We therefore have g α ĺ g α 0 on pa, bqˆΣ. In particular, with the time-orientation declared in the hypothesis, every future-directed causal tangent vector for g α is a future-directed causal vector for g α 0 . Therefore, according to (2) in Lemma 2.16, it suffices to show that g α 0 is globally hyperbolic on p´a, aqˆΣ and that Σ is a Cauchy hypersurface for g α 0 . To this end, consider an inextendible future-directed timelike curve γ " pγ 0 ,γq in pp´a, aqˆΣ, g α 0 q. The curve r γ :" pα´1 0 γ 0 ,γq is future directed timelike w.r.t. g and still inextendible , therefore it meets Σ " t´1p0q exactly once, but r γ and γ intersect in t " 0. Thus γ intersects Σ once. This shows g α 0 and therefore g α to be globally hyperbolic on p´a, aqˆΣ.
We can now state and prove a first result concerning Cauchy surfaces and the paracausal relation 2 . Proposition 2.26. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes on M which share a Cauchy temporal function t : M Ñ R according to Definition 2.11. Then g » g 1 .
Proof. As before, we will henceforth omit to write the isometries identifying the various spacetimes. However we may have two different isometries from M to RˆΣ for g and g 1 . Proposition 2.17 yields g ĺĝ ĺ g, g 1 ĺĝ 1 ĺ g 1 if where β 2 0 , β 2 1 are the lapse function we choose in accordance with Theorem 2.10. The metricsĝ andĝ 1 are globally hyperbolic for Lemma 2.25 (with α " 1). The proof ends if proving thatĝ andĝ 1 are paracausally related. Referring to the splitting of M as RˆΣ induced by the Cauchy temporal function t, define the globally hyperbolic metric´dt b dt`h, where h is a complete Riemannian metric on Σ (see, e.g., [70]). For every λ P p0, 1q, direct inspection proves that, Since λh is complete, from the former line we conclude that the metric g λ is globally hyperbolic due to (2) Lemma 2.16 and that it is paracausally related to dt b dt`λh. From the latter, sincé dt b dt`p1´λqβ´2 0 h t is globally hyperbolic in view of Lemma 2.25, we have that this metric and g λ are paracausally related. Since p1´λq P p0, 1q, the cones of´dt b dt`p1´λqβ´2 0 h t include the cones of´dt b dt`β´2 0 h t "ĝ so that these metrics are paracausally related as well. Transitivity implies that´dtbdt`λh andĝ are paracausally related. The same argument proves that´dt b dt`λh andĝ 1 are paracausally related so thatĝ »ĝ 1 and the thesis holds. Now we prove another non trivial result about paracausally related metrics for Cauchy compact spacetimes.
Proposition 2.27. Let pM, gq and pM, g 1 q be spacetimes such that g, g 1 P GH M . Suppose that g admits a Cauchy temporal function t : M Ñ R whose spacelike Cauchy hypersurfaces are compact and are also g 1 -spacelike, then g » g 1 up to a change of the temporal orientation of g 1 .
Proof. First of all, by defining the g-normal n g " 7gdt ?´g 7 pdt,dtq , any vector field X can be written as X " X n n g`πg pXq, where X n " gpn g , Xq and π g pXq " Id´gpn g , Xqn g projects on the Cauchy surface. The metric tensor g "´d t 2 g 7 pdt,dtq`h pπp¨q, πp¨qq under the action of the diffeomorphism ψ g gets recast in the orthogonal form g ort "´βdt 2`h t . This metric is obviously, by 2.17, paracausally related to the conformal metric g c "´dt 2`1 βt h t , which is, in turn, paracausally related to the globally hyperbolic metric r g "´dt 2`h with h a complete Riemannian metric on the slice and if we choose coherently the time orientation. The last statement is a consequence of 2.22 which is proved exactly as 2.23. Then we want look at the metric g 1 after the action of the isometric diffeomorphism ψ g and define the r g 1 " ψg g 1 . The proof ends if we are able to find a globally hyperbolic metric g 2 such that r g » g 2 » r g 1 . If we choose a function α P C 8 pR, p0, 8qq, then, by lemma 2.25 the metric tensor g α "´dt 2ὰ ptqh is globally hyperbolic and, by 2.22, paracausally related to r g. We want to tune the fuction α in order to have that the cones of g α intersect the cones of r g 1 . First we define pointwise n 1 the smooth vector field g 1 -normal to the Cauchy hypersurfaces of g and decompose it with respect to the splitting of the tangent space induced by the metric g α through its normal n α . We get n 1 " Zn α`W where Z " g α pn α , n 1 q and W " π gα pn 1 q.
Since the Cauchy hypersurfaces of g and g α are spacelike also for g 1 , we have that Z ‰ 0. The cones of the two metrics intersct if α is such that n 1 is g α -timelike i.e. iff The manifold RˆΣ can be covered by the time-strips T S n " tr´n, nsˆΣu nPN which are obviously compact since Σ is compact by hypotesis. This means that for all n P N the smooth function f : M Ñ R`defined by f :" h |Z| 2 attains a maximum M n and a minimum m n when restricted to the strip T S n . So we construct the required This function is only piecewise constant. The maximum has been increased by one to avoid the possibility that this function gets null: it could happen if the normal n 1 and n α get aligned in the first time-strip and then depart.
The last thing to do is to smoothen the function αptq, something which can of course be done by standard gluing arguments. Now that we know that the cones of g α and r g 1 intersect, if the temporal orientation of g 1 is such that Vg α X Vr g 1 ‰ H we define g 2 :" g α » r g 1 and the proof is concluded. If Vg α X Vr g 1 " H the metric g 2 , and therefore the metric r g, is paracausally related to r g 1 with opposite time orientation.

Normally hyperbolic operators and their properties
One of the main goals of this paper is to realize a geometric map to compare the space of solutions of normally hyperbolic operators defined on possibly different globally hyperbolic manifolds. Before starting to introduce our theory, we remind some general definitions and we fix the notation that will be used from now on. Let E be a vector bundle (always on K and of finite rank in this paper) over a spacetime pM, gq, whose generic fiber (a K vector space isomorphic to a canonical fiber) is denoted by E p for p P M. ΓpEq is the K-space of smooth sections E. ΓpEq has a number of useful subspaces we list below.
(i) Γ c pEq Ă ΓpEq is the subspace of compactly supported smooth sections.
(ii) Γ pc pEq and Γ f c pEq denote the subspaces of ΓpEq whose elements have respectively past compact support and future compact support.
(iii) If pM, gq is globally hyperbolic, Γ sc pEq Ă ΓpEq is the subspace of spatially compact sections: the smooth sections whose support intersects every spacelike Cauchy hypersurface in a compact set.
These spaces are equipped with natural topologies as discussed in [3]. In case there are several metrics on a common spacetime M basis of E, the used metric g will be indicated as well, for instance Γ g pc pEq, if the nature of the space of sections depends on the chosen metric (this is not the case for Γ c pEq).
As usual, E b E 1 denotes the external tensor product of the two K-vector bundles over M. This K-vector bundle has basis MˆM Q pp, qq and fibers given by the pointwise tensor product E p b E 1 q of the fibers of the two bundles. Referring to ΓpE b E 1 q, if f P E and f 1 P E 1 , then f b f 1 P ΓpE b E 1 q denotes the section defined by pf b f 1 qpp, qq :" fppq b f 1 pqq where the tensor product on the right-hand side is the one of the fibers and pp, qq P MˆM.

Normally hyperbolic operators
As for Section 2.1, if g P M M we defined g 7 as the induced metric on the cotangent bundle. If pM, gq is globally hyperbolic, by fixing a Cauchy temporal function t : Definition 3.1. A linear second order differential operator N : ΓpEq Ñ ΓpEq is normally hyperbolic if its principal symbol σ N satisfies Referring to a foliation of pM, gq as in Definition 2.11, in local coordinates pt, xq on M adapted to the foliation so that x " px 1 , . . . , x n q are local coordinates on Σ t , and using a local trivialization of E, any normally hyperbolic operator N in a point p P M reads as where A 0 , A j and B are linear maps E pt,xq Ñ E pt,xq depending smoothly on pt, xq.
Examples 3.2. In the class of normally hyperbolic operators we can find many operators of interest in quantum field theory: • Fix E be the trivial real bundle, i.e. E " MˆR, so that the space of smooth sections of E can be identified with the ring of smooth functions on M. The Klein-Gordon operator N " l`m 2 is normally hyperbolic, where l is the d'Alembert operator and m is a massterm.
• Let now E " Λ k T˚M be the bundle of k-forms and d (resp δ) the exterior derivative (resp. the codifferential). The operator N :" dδ`δd`m 2 is normally hyperbolic and it is used to describe the dynamics of Proca fields, for further details we refer to [4, Example 2.17].
• Let SM be a spinor bundle over a globally hyperbolic spin manifold M g and let ∇ be a spin connection. By denoting with γ : TM Ñ EndpSMq the Clifford multiplication, the classical Dirac operator reads as D " γ˝∇ : ΓpSMq Ñ ΓpSMq, see [29,30,53] for further details. By Lichnerowicz-Weitzenböck formula we get the spinorial wave operator N " D 2 " ∇ : ∇`1 4 Scal g , where Scal g is the scalar curvature.
It is well-known that, once the Cauchy data are suitably assigned, the Cauchy problem for N turns out to be well-posed, see e.g. [3,52]. Theorem 3.3. Let E be a vector bundle (of finite rank) over a globally hyperbolic manifold pM, gq, let N be a normally hyperbolic operator with a N-compatible connection ∇ (see (3.3) below) and Σ 0 a (smooth) spacelike Cauchy hypersurface of pM, gq. Then the Cauchy problem for N is wellposed, i.e. for any f P Γ c pEq, h 1 , h 2 P Γ c pE| Σ 0 q there exists a unique solution Ψ P Γ sc pEq to the initial value problem being n the future directed timelike unit normal field along Σ 0 , and it depends continuously on the data pf, h 1 , h 2 q w.r.to the standard topologies of smooth sections and satisfies As a consequence of the well-posedness of the Cauchy problem of normally hyperbolic operators with "finite propagation of the solutions" stated in (3.1), one may establish the existence of Green operators. In order to recall this result, we need first a preparatory definition.
Definition 3.4. A linear differential operator P : ΓpEq Ñ ΓpEq is called Green hyperbolic if (1) there exist linear maps, dubbed advanced Green operator G`: Γ pc pEq Ñ ΓpEq and retarded Green operator G´: (2) the formally dual operator P˚admits advanced and retarded Green operators as well.
For sake of completeness, let us recall that the formally dual operator P˚: ΓpE˚q Ñ ΓpE˚q is the unique linear differential operator acting on the smooth sections of the dual bundle Es for every f P Γ c pEq and f 1 P Γ c pE˚q (which is equivalent to saying f P ΓpEq and f 1 P ΓpE˚q such that supppfq X supppfq 1 is compact), vol g being the volume form induced by g on M.
(1) The Green operators we define below are the extensions to Γ pc{f c pEq of the analogs defined in [2] and indicated by G˘therein. where GP indicate the Green operators of P and GP˚indicate the Green operators of P˚.
Proposition 3.6. If P is a Green hyperbolic operator on a vector bundle E over the globally hyperbolic spacetime pM, gq and ρ : M Ñ p0,`8q is smooth, then ρP is Green hyperbolic as well and Gρ P " GP ρ´1.
Proof. The thesis immediately follows form the fact that GP ρ´1 and ρ´1GP˚satisfy the properties of the Green operators for ρP and pρPq˚" P˚ρ respectively. Given a Green hyperbolic operator with Green operators G˘, a relevant operator constructed out of G˘is the so-called causal propagator, It satisfies remarkable properties we are going to discuss (see e.g. [3, Theorem 3.6.21]). Theorem 3.8. Let G be the causal propagator of a Green hyperbolic differential operator P : ΓpEq Ñ ΓpEq on the vector bundle E over a globally hyperbolic spacetime pM, gq. The following sequence is exact Proof. Injectivity of Γ c pEq P Ñ Γ c pEq easily arises from the well-posedness of the Cauchy problem stated in Theorem 3.3. Let us pass to the other parts of the sequence. First of all notice that G˘pΓ c pEqq Ă Γ sc pEq since supppG˘pfqq Ă J˘psupppfqq and the first assertion then follows form known facts of globally hyperbolic spacetimes. Let us prove that Γ c pEq G Ñ Γ sc pEq is surjective when the image is restricted to the kernel of Γ sc pEq P Ñ Γ sc pEq. Suppose that PΨ " 0 for Ψ P Γ sc pEq. If t is a smooth Cauchy time function of pM, gq and χ : M Ñ r0, 1s is smooth, vanishes for t ă t 0 and is constantly 1 for t ą t 1 , then f Ψ :" PpχΨq P Γ c pEq is such that Ψ " Gf Ψ . Notice that supppf Ψ q is included between the Cauchy hypersufaces t´1pt 0 q and t´1pt 1 q. Indeed, It is obvious that that f Ψ can be changed by adding a section of the form Ph with h P Γ c pEq preserving the property Gf Ψ " Ψ. This exhaust the kernel of Γ c pEq G Ñ Γ sc pEq as asserted in the thesis. Indeed, if Gf " 0, then G`f " G´f. From the properties of the supports of G˘f, we conclude that G˘f " h˘P Γ c pEq Ă Γ pc pEq X Γ f c pEq. Hence f " PG˘f " Ph˘. To conclude, we prove that Γ sc pEq P Ñ Γ sc pEq is surjective. If f P Γ sc pEq, with χ as above, and G`pχfq`G´pp1´χqfq P Γ sc pEq.

Formally selfadjoint normally hyperbolic operators and their symplectic form
Let E be a K-vector bundle on a globally hyperbolic spacetime pM, gq. As shown in [6, Lemma 1.5.5], for any normally hyperbolic operator N : ΓpEq Ñ ΓpEq there exists a unique covariant derivative ∇ on E such that N "´tr g p∇∇q`c (3.3) for some some zero-order differential operator c : ΓpEq Ñ ΓpEq. In the formula above the left ∇ is actually the connection induced on T˚MbE by the Levi-Civita connection associated to g and the original connection ∇ (the one appearing as the right ∇) given on E Adopting the terminology of [3], we shall refer to ∇ as the N-compatible connection on E.
We stress that, if we suppose that E is equipped with a smooth assignment of a Hermitian fiber metric x¨|¨y p : E pˆEp Ñ K .
then the above ∇ is g-metric but not necessarily metric with respect to x¨|¨y. The physical relevance of the fiber metric is that it permits to equip Ker sc pNq with a symplectic form with important physical properties in the formulation of QFT in curved spacetime. This symplectic form can be derived using the Green identity for a normally hyperbolic operator N and its formal adjoint. For sake of completeness let us remind the definition of formal adjoint.
Definition 3.9. The formal adjoint of a differential operator P : ΓpEq Ñ ΓpEq is the unique differential operator P : : ΓpEq Ñ ΓpEq satisfying ż M xf 1 | Pfy vol g " ż M xP : f 1 | fy vol g for every f, f 1 P Γ c pEq (which is equivalent to saying f, f 1 P ΓpEq such that supppfq X supppfq 1 is compact). If P " P : then N is said to be (formally) selfadjoint. Remark 3.10. If P : ΓpEq Ñ ΓpEq is normally hyperbolic on the bundle E over pM, gq, equipped with a non-degenerate, Hermitian fiber metric x¨|¨y, P is Green hyperbolic as said above. In this case P : has the same principal symbol as P and thus it is Green hyperbolic as well. Taking advantage of the natural (antilinear if K " C) isomorphism ΓpEq Ñ ΓpE˚q induced by x¨|¨y and (3.2), it is not difficult to prove that, if f 1 P Γ c pEq and f P Γ pc pEq or f P Γ f c pEq respectively, where GP indicate the Green operators of P and GP : indicate the Green operators of P : .
Let us pass to introduce a Green-like identity where we explicitly exploit the N-compatible connection ∇.
where n is the outward unit normal vector to BM 0 and vol BM 0 " n vol g is the volume form of BM 0 induced by g.
If the normal vectors to BM 0 are either spacelike or timelike, some computations with the exterior differential of forms yields (3.6). In all cases it is easy to prove that At this juncture, Stokes' theorem for pn`1q-forms,

produces (3.5).
We have the following crucial result when applying the previous lemma to the theory on globally hyperbolic spacetimes. Proposition 3.12. Let Σ Ă M be a smooth spacelike Cauchy hypersurface with its future-oriented unit normal vector field n in the globally hyperbolic spacetime pM, gq and its induced volume element vol Σ . Furthermore, let N be a formally self-adjoint normally hyperbolic operator. Then σ N pM,gq : Ker sc pNqˆKer sc pNq Ñ C such that σ N pM,gq pΨ, Φq " where Ξ N Σ is defined in Equation (3.6), yields a non-degenerate symplectic form (Hermitian if K " C) which does not depend on the choice of Σ.
Proof. First note that, referring to a spacelike Cauchy hypersurface Σ, supppΨq X Σ is compact since supppΨq is spacelike compact, so that the integral is well-defined. The fact that σ N is not degenerate can be proved as follows. If σ N pM,gq pΨ, Φq =0 for all Φ P Γ sc pEq, from the definition of σ N and non-degenerateness of x¨|¨y p (passing to local trivializations referred to local coordinates on Σ re-writing x¨|¨y p in terms of the pairing with Ep), we have that the Cauchy data of Ψ vanishes on every local chart on Σ and thus they vanish on Σ. According to Theorem 3.3, Ψ " 0. The other entry can be worked out similarly. Let Ψ, Φ P Ker sc pNq and Σ 1 t and Σ t 2 be a pair of smooth spacelike Cauchy hypersurfaces associated to a smooth time Cauchy function t with t 2 ą t 1 . Let us focus on the submanifold with boundary M 0 " t´1ppt 1 , t 2 qq. Its boundary is BM 0 " Σ t 1 Y Σ t 2 . The supports of Ψ and Φ between the two Cauchy surfaces are included in the sets of type J`of the compact supports of the Cauchy data on Σ t 1 of Ψ and Φ respectively, and these portions of causal sets are compact as pM, gq is globally hyperbolic (see e.g. [3, Proposition 1.2.56]) we end up with a pair of functions in Γ c pEq and we can apply the Green identity (see Lemma 3.11) to M 0 . Using a smoothly vanishing function as a factor, we can make smoothly vanishing Ψ and Φ before Σ t 1 and after Σ t 2 without touching them between the two Cauchy surfaces. As a matter of fact the resulting sections constructed out Ψ and Φ by this way are smooth, compactly supported and coincide with Ψ and Φ between the two Cauchy surfaces. We can therefore apply Lemma 3.11, obtaining Since N is assumed to be self-adjoint, xΨ | NΦy´xNΨ | Φy " xΨ | NΦy´xΨ | N : Φy " 0. Therefore we can conclude that Finally consider the case of two spacelike Cauchy functions Σ and Σ 1 belonging to different foliations induced by different smooth Cauchy time functions (notice that a spacelike Cauchy hypersurface always belong to a foliation generated by a suitable smooth Cauchy time (actually temporal) function for Theorem 2.10). We sketch a proof of the identity ż Let K Ă Σ a compact set including the Cauchy data of Ψ and Φ. If t is the smooth Cauchy time function such that Σ t 1 " Σ 1 , let T " max K t. If t 1 ă T we can always take t 2 ą T and to consider the symplectic form evaluated on Σ t 2 . In view of the previous part of our proof the symplectic form on Σ t 1 and Σ t 2 coincide, so that our thesis can be re-written As t 2 ą max K t, we conclude that Σ t 2 does not intersect Σ in the set K. Therefore we can define the solid set L K made of the portion of J`pKq between Σ and Σ t 2 . L is compact (see e.g. [3, Proposition 1.2.56]) and is a "truncated cone" whose "lateral surface" is part of the boundary of J`pKq and whose "non-parallel bases" are parts of Σ 2 and Σ. We can include L in the interior of a larger manifold with boundary M 0 whose part of the boundary are portions of Σ and Σ t 2 including the support of the Cauchy data of Ψ and Φ. Notice that M 0 includes the supports of Ψ and Φ between the two Cauchy surfaces according to Theorem 3.3 and these supports do not touch the "lateral surface" of M 0 . We can now apply the Green identity 3.11 to M 0 proving the thesis.
There is a nice interplay of the causal propagator G of N : ΓpEq Ñ ΓpEq as above and the symplectic form σ N pM,gq .
Proposition 3.13. With the same hypotheses as of Proposition 3.12, if f, h P Γ c pEq and Ψ f :" Gf, Proof. If f, h P Γ c pEq, consider a smooth Cauchy time function t and fix t 0 ă t 1 such that the supports of f and h are included in the interior of the submanifold with boundary M 0 contained between the spacelike Cauchy hypersurfaces Σ t 0 :" t´1pt 0 q and Σ t 1 :" t´1pt 1 q. It holds Applying Lemma 3.11, we find where we noticed that G`h vanishes on the remaining part of the boundary Σ t 0 . On the other hand, we can replace G`h for G`h´G´h " Gh in the last integral, since G´h gives no contribution to the integral on Σ t 1 . In summary, and this is the thesis.

Convex combinations of normally hyperbolic operators
Let now N 0 , N 1 be normally hyperbolic operators with respect to different Lorentzian metrics g 0 and g 1 (the former time-orientable and the latter globally hyperbolic) on the same manifold M and assume that they are acting on the smooth sections of the same vector bundle E. It turns out, that a positive (and convex) combination p1´χqN 0`χ N 1 is also (a) normally hyperbolic with respect to the naturally associated metric g χ -the unique Lorentzian metric in TM whose associated metric in T˚M is p1´χqg 7 0`χ g 7 1 according to Theorem 2.18 -and (b) Green hyperbolic with respect to g 1 , everything provided that g 0 ĺ g 1 . This is the main result of this section.
Theorem 3.14. Let E be a K-vector bundle over a smooth manifold M, let be g 0 , g 1 P GM M with g 0 ĺ g 1 , and let N 0 , N 1 : ΓpEq Ñ ΓpEq be normally hyperbolic operator with respect to g 0 and g 1 respectively. If χ P C 8 pM, r0, 1sq, define g χ as the unique Lorentzian metric whose associated metric in T˚M is p1´χqg 7 0`χ g 7 1 according to Theorem 2.18. Then the second order differential operator defined by N χ :" p1´χqN 0`χ N 1 : ΓpEq Ñ ΓpEq (3.8) satisfies the following properties: (1) It is normally and Green hyperbolic over pM, g χ q; (2) It is Green hyperbolic over pM, g 1 q and, with obvious notation, In particular, (2) is true for N 0 by choosing χ " 0.
In particular, we have that N χ is normally hyperbolic with respect to g χ : By Theorem 2.18, the metric g χ is globally hyperbolic and, on account of Proposition 3.7 N χ is Green-hyperbolic over pM, g χ q.
Regarding (2), and referring to the existence of Green operators of N χ in pM, g 1 q we can proceed as follows. Observe that, since g χ ĺ g 1 , we have J gχ pAq Ă J g 1 pAq and, with obvious notation, Γ g 1 pc pEq Ă Γ gχ pc pEq together with Γ g 1 f c pEq Ă Γ gχ f c pEq, in view of (iii) (2) Lemma 2.16. As a consequence, the Green operators of N χ with respect to pM, g χ q are also Green operators with respect to pM, g 1 q.
Finally we pass to the existence of the Green operators of Nχ -where˚is here referred to the volume form of g 1 and not g χ -in pM, g 1 q. Since Nχ has the same principal symbol g 7 χ pξ, ξqId E as N χ it is normally hyperbolic in pM, g χ q and hence Green hyperbolic thereon. With the same argument used above, we see that the Green operators of Nχ (with˚always referred to g 1 ) in pM, g χ q are also Green operators in pM, g 1 q.
Remark 3.15. We stress that, when g 0 ĺ g 1 are globally hyperbolic, N χ and N 0 are therefore Green-hyperbolic second-order differential operators on pM, g 1 q though they are not normally hyperbolic thereon. These are examples of second-order linear differential operators which are Green hyperbolic but not normally hyperbolic in a given globally hyperbolic spacetime.

Møller maps and operators for normally hyperbolic operators
We are in the position to introduce the notion of so-called Møller map, which we shall later specialize to the case of a Møller operator, namely a (geometric) map which compares the space of solutions of different normally hyperbolic operators. The novelty of this approach consists in defining the notion of Møller map in a more general fashion. More in detail, in [20,29,31,65] the Møller operator was constructed once that a foliation of M in Cauchy hypersurfaces was assigned and referring to the family of the metrics which are decomposed as in (2.1) with respect to that foliation. Here we shall see, that the construction of a Møller map still requires the choice of a foliation (associated to some smooth Cauchy time function), but the involved metrics do not have any particular relationship with the choice of the foliation. Instead they should enjoy some interplay concerning their light-cone structures which generalizes g ĺ g 1 in the sense of paracausal deformations.

General approach to construct Møller maps when g 0 ĺ g 1
Let us consider a globally hyperbolic spacetime pM, gq equipped with a vector bundle E Ñ M as before. If P : ΓpEq Ñ ΓpEq is a linear differential operator, a family of physically relevant solutions of the inhomogeneous equation Pf " h is the linear vector space of spacelike compact smooth solutions with compactly supported source: Sol g sc,c pPq :" tf P Γ g sc pEq | Pf P Γ c pEqu .
Its subspace corresponding to the solutions of the homogeneous equation Pf " 0 is denoted by Ker g sc pPq :" tf P Γ g sc pEq | Pf " 0u and it will play a pivotal role in the formulation of linear QFT. We now specialize the operators P to 2nd-order normally-hyperbolic linear operators N 1 , N 0 , N χ (3.8) over ΓpEq associated to globally hyperbolic metrics g 0 ĺ g 1 and g χ on the common spacetime manifold M. Our goal is to construct several families of Møller maps, namely linear operators such that (a) they are linear space isomorphisms between Sol g 0 sc,c pN 0 q, Sol g 1 sc,c pN 1 q, Sol gχ sc,c pN χ q; (b) they restrict to isomorphisms to the subspaces Ker g 0 sc pN 0 q, Ker g 1 sc pN 1 q, Ker gχ sc pN χ q.
For later convenience, we shall additionally require that the Møller maps preserve also the symplectic forms, which are of interest in applications to linear QFT.
The overall idea is inspired by the scattering theory. We start with two "free theories", described by the space of solutions of normally hyperbolic operators N 0 and N 1 in corresponding spacetimes pM, g 0 q and pM, g 1 q, respectively, and we intend to connect 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 .

Møller maps for metrics satisfying
The first step consists of comparing N 0 and N 1 with N χ separately to construct the Møller map. As usual, we denote with E the K-vector bundle over a spacetime pM, gq.
We first start with operators denoted by R˘defined on the whole space of smooth sections ΓpEq which is in common for the three metrics on M and next we will restrict these operators to the special spaces of solutions with spatially compact support and compactly supported sources, proving that these restrictions Ω˘are still linear space isomorphisms. Proposition 4.1. Let g 0 , g 1 P GM M be such that g 0 ĺ g 1 and V g 0x Ă V g 1x for all x P M . Let E be a vector bundle over M and N 0 , N 1 : ΓpEq Ñ ΓpEq be normally hyperbolic operators associated to g 0 and g 1 respectively. Choose (a) a smooth Cauchy time g 1 -function t : M Ñ R and χ P C 8 pM; r0, 1sq such that χppq " 0 if tppq ă t 0 and χppq " 1 if tppq ą t 1 for given t 0 ă t 1 ; (b) a pair of smooth functions ρ, ρ 1 : M Ñ p0,`8q such that ρppq " 1 for tppq ă t 0 and ρ 1 ppq " ρppq " 1 if tppq ą t 1 . (Notice that ρ " ρ 1 " 1 constantly is allowed.) The following facts are true.
The proof of the well definiteness of R´1 and R´1 on ΓpEq is analogous to the previous one for R´. We prove that R´defined as in (4.4) inverts R´from the right by direct computation:

Now, by exploiting the identity
we can prove our claim The proof that R´1 is also a left inverse is the same with obvious changes and analogous calculations show that R´1 is a left and right inverse of R`.
(2) Taking advantage of (ia)-(iib) in Definition 3.4 and the definition of N χ and the one of R˘, a direct computation establishes (4.5).
Lemma 4.3. Let P : ΓpEq Ñ ΓpEq be a 2nd order normally hyperbolic differential operator on the vector bundle E Ñ M on the globally hyperbolic spacetime pM, gq. Let Ψ P ΓpEq be such that PΨ P Γ c pEq. Then the following facts are equivalent.
(a) Ψ P Γ g sc pEq; (b) there is a spacelike Cauchy hypersurface of pM, gq such that Ψ have compactly supported Cauchy data thereon.
Proof. If Ψ P Γ g sc pEq then, by definition, (b) is true. Suppose that (b) is true for Σ 0 . According to Theorem 3.3, Ψ is the unique solution of the Cauchy problem whose equation is PΨ " f, where f P Γ c pEq. As a consequence the support of Ψ completely lies in Jpsupppfqq Y Jpsuppph 0 qq Y Jpsuppph 1 qq Ă JpKq where h 0 and h 1 are the Cauchy data of Ψ on Σ 0 and K :" supppfq Y suppph 0 q Y Jpsuppph 1 q. In particular K is compact. In view of well known properties of globally hyperbolic spacetimes (see e.g. [3, Proposition 1.2.56]), since K is compact JpKq X Σ is compact for every Cauchy hypersurface Σ of pM, gq so that Ψ P Γ g sc pEq.
Proof of Proposition 4.2. R˘and R´1 are bijective on ΓpEq. As a consequence (4.8) and thesis for Ω˘, including (4.10) which is a specialization of (4.5), immediately arise when proving that R`pSol g 0 sc,c pN 0 qq Ă Sol gχ sc,c pN χ q , R´1 pSol gχ sc,c pN χ qq Ă Sol g 0 sc,c pN 0 q (4.13) and R´pSol gχ sc,c pN χ qq Ă Sol g 1 sc,c pN 1 q , R´1 pSol g 1 sc,c pN 1 qq Ă Sol gχ sc,c pN χ q The identities in (4.9) and the thesis for Ω 0 immediately arise from the bijectivity of the linear maps Ω˘and (4.10) where we know that ρ, ρ 1 ą 0. To conclude, let us establish the first inclusion in (4.13), the remaining three inclusions have a strictly analogous proof. Suppose that f P Sol g 0 sc,c pN 0 q. Hence ρN χ R`f " N 0 f P Γ g 0 c pEq " Γ gχ c pEq and N χ R`f " ρ´1N 0 f P Γ gχ c pEq. Next pass to consider the Cauchy hypersurfaces of t which are in common with the three considered metrics g 0 , g 1 , g χ and choose t 1 ă t 0 . (3) in Proposition 4.1 yields pR`fqpt 1 , xq " fpt 1 , xq where x P Σ t 1 . The Cauchy data of f on Σ t 1 have compact support because f P Sol g 0 sc,c pN 0 q. On the ground of Lemma 4.3, noticing that N χ is normally hyperbolic in pM, g χ q, referring to the Cauchy problem on Σ t 1 for the equation N χ R`f " ρ´1N 0 f P Γ gχ c pEq in the spacetime pM, g χ q, we conclude that R`f P Γ gχ sc,c pEq because its Cauchy data on Σ t 1 (now interpreted as a Cauchy hypersurface for g χ q have compact support as they coincide with the ones of f itself.

General Møller maps for paracausally related metrics
We are now in a position to state a result regarding the existence of Møller maps between two normally hyperbolic operators N 0 and N 1 on respective globally hyperbolic spacetimes over the same manifold (and vector bundle) whose metrics are ĺ comparable. The final goal is to extend the results to pairs of paracausally related metrics.
Proposition 4.4. Let g 0 , g 1 P GM M be such that either g 0 ĺ g 1 or g 1 ĺ g 0 with, respectively, for all x P M or V g 1x Ă V g 0x for all x P M . Let E be a vector bundle over M and N 0 , N 1 : ΓpEq Ñ ΓpEq be normally hyperbolic operators associated to g 0 and g 1 respectively. There exist (infinitely many) vector space isomorphisms, S : Sol g 0 sc,c pN 0 q Ñ Sol g 1 sc,c pN 1 q such that, for some smooth function µ : M Ñ p0,`8q depending on S (which can be chosen µ " 1), (1) referring to the said domains, (2) the restriction S 0 :" S| Ker g 0 sc pN 0 q defines a vector space isomorphism Proof. First consider the case g 0 ĺ g 1 . Referring to a smooth Cauchy time function t of pM, g 1 q and a smoothing function χ, S :" Ω´Ω`constructed as in Proposition 4.2 satisfies all the requirements trivially for µ :" ρ 1 . The previous result is also valid for g 1 ĺ g 0 . It is sufficient to construct Ω˘as in Proposition 4.2, but using g 1 as the initial metric and g 0 as the final one, and eventually defining µ :" ρ´1, S :" pΩ´Ω`q´1 " Ω´1 Ω´1 , and S 0 :" pΩ 0 Ω 0 q´1 " pΩ 0 q´1pΩ 0 q´1.
We can pass to the generic case g » g 1 , obtaining the first main result of this work.
Theorem 4.5. Let pM, gq and pM, g 1 q be globally hyperbolic spacetimes, E a vector bundle over M and N, N 1 : ΓpEq Ñ ΓpEq normally hyperbolic operators associated to g and g 1 respectively.
If g » g 1 , then there exist (infinitely many) vector space isomorphisms, called Møller maps of g, g 1 (with this order), S : Sol g sc,c pNq Ñ Sol g 1 sc,c pN 1 q such that (1) referring to the said domains, µN 1 S " N for some smooth µ : M Ñ p0,`8q (which can be always taken µ " 1 constantly in particular), (2) the restriction S 0 :" S| Ker g sc pNq (also called Møller map of g 1 , g 1 ) defines a vector space isomorphism S 0 : Ker g sc pNq Ñ Ker g 1 sc pN 1 q .
Proof. First of all we notice that there always exists a normally hyperbolic operator N on E associated to every g P GM M : For instance the connection-d'Alembert operator in [3, Example 2.1.5] referred to a generic connection ∇ on E, which always exists, and the Levi-Civita connection on pM, gq. Let us consider a sequence g 0 " g, g 1 , . . . , g N " g 1 of globally hyperbolic metrics on M satisfying Definition 2.19 and a corresponding sequence of formally selfadjoint normally hyperbolic operators N k with N 0 :" N and N N :" N 1 . We can apply Proposition 4.4 for each pair g k , g k`1 for k " 0, 1, . . . , N´1. It turns immediately out that, with an obvious notation, satisfy the thesis of the theorem, where either S k :" Ω k´Ωk`, µ k :" ρ k or S k :" pΩ k`q´1 pΩ k´q´1 , µ k :" ρ´1 k according to g k ĺ g k`1 or g k`1 ĺ g k respectively. With the same convention it results that S 0 " S 0 0 S 1 1¨¨¨S 0 N´1 where either S 0 k " Ω 0 k´Ω 0 k`o r S k " pΩ 0 k`q´1 pΩ 0 k´q´1 according to the discussed cases.

Preservation of symplectic forms
The Møller maps S 0 as in Theorem 4.5 preserve the symplectic forms of the normal operators they relate when these operators are formally selfadjoint. Theorem 4.6. Consider g, g 1 P GH M with respectively associated normally hyperbolic operators N, N 1 on the K-vector bundle E over M. If g 1 » g and N and N 1 are formally selfadjoint with respect to a non-degenerate, Hermitian fiber metric x¨|¨y, then there are Møller maps S 0 satisfying the thesis of Theorem 4.5 such that where we used the notation σ N g in place of σ N pM,gq .
Proof. It is sufficient to prove the thesis for the maps Ω 0 referred to two metrics g 0 ĺ g 1 , which immediately implies the thesis also for the inverse maps pΩ 0 q´1 they being isomorphisms. Indeed, according the proof of Theorem 4.5, the isomorphisms S 0 are compositions of various copies of Ω 0 and their inverses. Let us consider Ω 0 : Ker sc pN 0 q Ñ Ker sc pN χ q and we prove the thesis for it, the other case being very similar. Consider a smooth Cauchy time function t for g 1 and the associated foliation made of spacelike Cauchy hypersurfaces Σ t in common for g 0 , g 1 , and g χ . If the smoothing function χ used to build up g χ and N χ vanishes before t 0 and we use Σ t with t ă t 0 to compute the relevant symplectic forms, due to (4.11), σ Nχ gχ pΩ 0 Ψ, Ω 0 Φq " σ N 0 g 0 pΨ, Φq for every Ψ, Φ P Ker g 0 sc pN 0 q.
Above, we have used the definition of the symplectic form, we have noticed that g χ " g 0 around Σ t and that the N 0 and N χ compatible connections must coincide there as they are locally defined and uniquely determined by N 0 Ψ " N χ Ψ " p´tr g p∇∇q`cqΨ for every smooth Ψ compactly supported around a point p with tppq ă t 0 . Thinking of σ Nχ gχ pΩ 0 Ψ, Ω 0 Φq as defined in pM, g χ q and of σ N 0 g 0 pΨ, Φq as defined in pM, g 0 q, though both computed on Σ t with t ă t 0 , Proposition 3.12 concludes the proof.
Definition 4.7. We call symplectic Møller map any linear isomorphism defined in accordance with Theorem 4.6.

Causal propagators and paracausally related metrics
In this section, we prove how is possible to choose the functions ρ and ρ 1 affecting the definitions (4.1)-(4.2) of R˘in order to satisfy a further requirement with some crucial implications in QFT: the preservation of the causal propagator of two operators N and N 1 when the associated metrics are paracausally related. Essentially speaking, a Møller map satisfying this further requirement will be named Møller operator.

Adjoint operators
To study the relation between Møller maps and the causal propagator of normally hyperbolic operators defined on a vector bundle equipped with a non-degenerate (Hermitian) fiber metric, we need a suitable notion of adjoint operator which generalizes the notion of formal adjoint of differential operators.
Let E be a K-vector bundle on the manifold M equipped with a non-degenerate, symmetric if K " R or Hermitian if K " C, fiber metric x¨|¨y. Suppose that g and g 1 (possibly g ‰ g 1 ) are Lorentzian metrics on M. Consider a K-linear operator where DompTq Ă ΓpEq is a K-linear subspace and DompTq Ą Γ c pEq. ż M x´T : gg 1 h¯pxq | fpxqyvol g pxq @f P DompTq , @h P Γ c pEq.
Notation 4.9. If g " g 1 then we shall denote the adjoint of T with respect to g simply as T :g .
We prove below that T : gg 1 is unique if exists so that calling it "the" adjoint operator of T is appropriate. xh | Tf n yvol g 1 Ñ 0 @h P Γ c pEq as Γ c pEq Q f n Ñ 0 for n Ñ`8 in the topology of test sections [3].
Vice versa, this only condition is not sufficient to guarantee the existence of T : gg 1 as a Γ c pEqvalued operator. Using a straightforward extension of the Schwartz kernel theorem, the condition above just implies the existence of a weaker version of T : gg 1 which is distribution-valued.
In the rest of the paper if T : DompTq Ñ ΓpEq and T 1 : DompT 1 q Ñ ΓpEq, we define the standard domains of their compositions as follows, where a P K. (b) DompT`T 1 q :" DompTq X DompT 1 q is the domain of aT`bT 1 defined pointwise; (c) DompT 1˝T q :" tf P DompTq | Tpfq P DompT 1 qu is the domain of T 1˝T . Proposition 4.11. Referring to the notion of adjoint in Definition 4.8, the following facts are valid.
(1) If the adjoint T : gg 1 of T exists, then it is unique.
(2) If T : ΓpEq Ñ ΓpEq 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 ΓpEq, T 1 : DompT 1 q Ñ ΓpEq and a, b P K. Then paT`bT 1 q : gg 1 " aT : gg 1`b T 1: gg 1 provided T : gg 1 and T 1: gg 1 exist.
Proof. We write below : in place of : gg 1 if it is not strictly necessary to specify the metrics. To prove (1)  Since Γ c pEq Ă DompTq, the thesis follows by reducing to every fixed local trivialization over every arbitrarily fixed coordinate patch U on M. Restricting to U , the equation above can be recast to ż U N ÿ a"1 pT : 1 h´T : 2 hq a ppqf a ppq vol g ppq " 0.
where f a ppq is a fiber component of x¨| fy p P Ep with p P U . Since U Q p Þ Ñ pT : 1 h´T : 2 hq a ppq is continuous and U Q p Þ Ñ f a ppq is smooth, compactly supported (with support in U ) and arbitrary (because x¨|¨y is non-degenerate), the fundamental lemma of calculus of variations implies that U Q p Þ Ñ pT : 1 h´T : 2 hq a ppq is the zero function for a " 1, . . . , N . Since U can be fixed as a neighborhood of every point of M, (1) follows.
The proof of (2) and (3) is obvious: (2) follows by comparing definitions 4.8 and 3.9, while (3) follows by direct computation checking that aT :`b T 1: satisfies the definition of paT`bT 1 q : (notice that Γ c pEq Ă DompaT :`b T 1: q if T : and T 1: exist).
To prove (4), since the composition is well defined on a suitable domain, we can just use twice the definition 4.8 for all f P DompT 1˝T q and all h P Γ c pEq: notice that using the definition of the adjoint in the second equality is possible because T 1: g 1 g 2 : Γ c pEq Ñ Γ c pEq. The found identity proves that T : gg 1˝T1: g 1 g 2 satisfies the definition of pT 1˝T q : gg 2 ending the proof of (4).
Finally, (6) arises by taking the :gg adjoint of both sides of the identity T˝T´1 " I and the : g 1 g 1 adjoint of both sides of the identity T´1˝T " I and taking (4) into account.

Møller operators and causal propagators
We are in a position to state one of the most important results of this work by specializing the isomorphisms introduced in Theorem 4.5 by means of a suitable choice of the function µ. As a matter of fact (1) and (3) have been already established in Theorem 4.5.
Theorem 4.12. Let E be K-vector bundle over the smooth manifold M with a non-degenerate, real or Hermitian depending on K, fiber metric x¨|¨y. Consider g, g 1 P GH M with respectively associated normally hyperbolic formally-selfadjoint operators N, N 1 . If g » g 1 , then it is possible to define (in infinite ways) a K-vector space isomorphism R : ΓpEq Ñ ΓpEq, called Møller operator of g, g 1 (with this order), such that the following facts are true.
(1) The restrictions to the relevant subspaces of ΓpEq respectively define Møller maps (hence linear isomorphisms) as in Theorem 4.5.
(2) The causal propagators G N 1 and G N , respectively of N 1 and N, satisfy (4.14) (3) By denoting c 1 the smooth function such that vol g 1 " c 1 vol g , we have (5) The maps R : gg 1 : Γ c pEq Ñ Γ c pEq and pR : gg 1 q´1 " pR´1q : g 1 g : Γ c pEq Ñ Γ c pEq are continuous with respect to the natural topologies of Γ c pEq in the domain and in the co-domain.
Remarks 4.13. Before proving our claim, we want to underline the following: (1) Any Møller operator defines a symplectic Møller map (cf. Definition 4.7). Indeed, the preservation of the causal propagator (cf. (2) in Theorem 4.12) implies that the symplectic forms are preserved in view of Proposition 3.13. However, the converse is false since the preservation of the causal propagator relies upon a suitable choice of the function ρ, whereas this choice is immaterial for the preservation of the symplectic forms.
(2) Møller operators can be explicitly constructed as follows. If g 1 » g, and referring to a finite sequence of metrics g 0 :" g, g 1 , . . . , g N :" g 1 P GH M as in Definition 2.19, then there exists a corresponding sequence of formally selfadjoint g k -normally hyperbolic operators N 0 :" N, N 1 , . . . , N N :" N 1 : ΓpEq Ñ ΓpEq such that is a Møller operator of g, g 1 where Above, for every given k, R pkq are defined as R˘as in Equations (4.1) and (4.2) where (i) N 0 is replaced by N k and N 1 is replaced by N k`1 if g k ĺ g k`1 , (ii) N 0 is replaced by N k`1 and N 1 is replaced by N k if g k`1 ĺ g k , (iii) ρ :" c χ 0 , and ρ 1 :" c 1 0 (assuming vol gχ " c χ 0 vol g 0 and vol g 1 " c 1 χ vol gχ ).
The smooth Cauchy time function χ in (4.1) and (4.2) can be chosen arbitrarily and depending on k in general. The final Møller operator R of g, g 1 also depends on all the made choices.
Proof of Theorem 4.12. We divide the proof into several steps.
(1)-(3) Let us first prove the thesis for the special case of g " g 0 ĺ g 1 " g 1 , with V g 0x Ă V g 1x for all x P M, and specialize the definition of the isomorphisms (4.1) and (4.2) to where vol gχ " c χ 0 vol g 0 and vol g 1 " c 1 0 vol g 0 It is easy to see that pc χ 0 N χ q :g 0 " c χ 0 N χ and pc 1 0 N 1 q :g 0 " c 1 0 N 1 . (4.20) Our goal is to prove that the isomorphism R :" R´R`: ΓpEq Ñ ΓpEq satisfies the thesis. Per direct inspection, applying the definition of adjoint operator and taking advantage of (4.20), Proposition 3.6, and (3.4), we almost immediately have that R :g 0 " Id´pc χ 0 N χ´N0 qGćχ 0 Nχ | ΓcpEq and R Again per direct inspection we see that and thus c 1 0 N 1 R " c 1 0 N 1 R´R`" N 0 as wanted. As we prove below, the following identities are valid On the other hand, we have A :g 0 c 1 0 " A :g 0 g 1 so that RG N 0 R :g 0 g 1 " R´R`G N 0 pR´R`q :g 0 g 1 " G N 1 .
To conclude the proof of (1)-(3) for the case g " g 0 ĺ g 1 " g 1 we prove (4.22). Since G N 0 is defined as the difference of the advanced and retarded Green operators restricted to compact sections, we perform the computation separately for the two operators. We start from R`G N 0 | ΓcpEq R :g 0 : the adjoint of the Møller operator is defined over Γ c pEq and gives back compactly supported sections, then the advanced Green operator maps a compactly supported section f P Γ c pEq to a solution such that supp pGǸ 0 fq Ă J0 psupp pfq Ă Jχ psupp pfqq, where the last inclusion is due to the crucial hypothesis g 0 ĺ g χ ĺ g 1 . Now since supp pfq is compact the smooth Cauchy time function t attains a minimum t 0 P R therein, so we choose a common smooth Cauchy hypersurface Σ t 1 of the foliation induced by t such that t 1 ă t 0 and deduce that supp pGǸ 0 fq Ă Jχ psupp pfqq Ă Jχ pΣ t 1 q which implies by [3, Lemma 1.2.61] that GǸ 0 f P Γ χ pc pEq. Omitting the restriction of the domain of the causal propagators from the notation for sake of clarity, but having in mind that it is crucial for the validity of the argument, we obtain: where now the restriction of the domains of the causal propagators to compactly supported sections is assumed from the definition of the adjoint. Collecting together the two identities found, we have R`G N 0 R :g 0 " pR`GǸ 0´GŃ0 R :g 0 q`M " G c χ 0 Nχ`M , with, where both sides have to be computed on compactly supported sections, A lengthy direct evaluation of M using (4.18) and the former in (4.21) shows that M " 0. All that establishes the first identity in (4.22), while the latter follows by almost identical facts. Let us pass to prove (1)-(3) for the case g 1 ĺ g 0 , with V g 1x Ă V g 0x for all x P M. First of all we observe that from the previously treated case (g 0 ĺ g 1 ) we have c 0 1 N 0 R´1 " N 1 where c 0 1 " pc 0 1 q´1 and vol g 0 " c 0 1 vol g 1 . Interchanging the names of g 0 and g 1 , this result implies that (4.15) is true for g 1 ĺ g 0 when using R´1 in place of R. An analogous procedure proves (4.14) for the case g 1 ĺ g 0 from the same equation, already established, valid when g 0 ĺ g 1 . Also in this case the relevant Møller operator is R´1. To this end, we have only to prove that pR´1q :g 1 g 0 exists and coincides to pR :g 0 g 1 q´1. Indeed, under these assumptions (4.14) implies which is our thesis when interchanging g 0 and g 1 . This fact that pR´1q :g 1 g 0 " pR :g 0 g 1 q´1 actually can be established exploiting (6) in 4.11: R is bijective over ΓpEq, and admits the adjoint R :g 0 g 1 , so if the inverse R´1 admits the adjoint pR´1q :g 1 g 0 , then R :g 0 g 1 is bijective and its inverse is such that pR :g 0 g 1 q´1 " pR´1q :g 1 g 0 . Let us prove that R´1 admits adjoint (with respect to any metric among g 0 , g χ , g 1 since the existence of the adjoint with respect to one of them trivially implies the existence of the adjoint with respect to the other metrics) to end the proof for the case g 1 ĺ g 0 . By recalling that R´1 " R´1˝R´1 it suffices to show that R´1 and R´1 both admit adjoints. We explicitly give the g 0 -adjoint of R´1 the other case being analogous, Let us pass to the proof of (1)-(3) for the general case g » g 1 also establishing the last part of the thesis. In this case there is a sequence g 0 " g, g 1 , . . . , g N " g 1 of globally hyperbolic metrics on M satisfying Definition 2.19 and a corresponding sequence of selfadjoint normally hyperbolic operators N k with N 0 :" N and N N :" N 1 . (This sequence always exists because, for every globally hyperbolic metric g, there is a normally hyperbolic operator N as proved in the proof of Theorem 4.5. The operator r N :" 1 2 pN`N :g q is simultaneously formally selfadjoint with respect to x¨|¨y and normally hyperbolic.) Taking advantage of the validity of the thesis in the cases g ĺ g 1 and g 1 ĺ g, using in particular (4) and (6) in Proposition 4.11, one immediately shows that we can build a Møller map for a paracausal deformation of metrics just by defining R as the composition of the various similar operators defined for each copy g k , g k`1 as in (4.16) and (4.17).
(4) If f P Γ c pEq, (5) It is sufficient to prove the thesis for the case g " g 0 ĺ g 1 " g 1 and for R :g 0 . The case of R :g 0 is analogous. In the case g 1 ĺ g 0 one uses the inverses of the operators above, and all remaining cases are proved just by observing that the considered Møller operators are compositions of the elementary operators R :g 0 and/or their inverses and smooth functions used as multiplicative operators. We know that The identity operator has already the requested continuity property, so we have only to focus on the second addend using the fact that a linear combination of continuous maps is continuous as well. The map Gćχ 0 Nχ | ΓcpEq : Γ c pEq Ñ ΓpEq is continuous with respect to the natural topologies of the domain and co-domain (see e.g. [3, Corollary 3.6.19 ]). Since pc χ 0 N χ´N0 q is a smooth differential operator pc χ 0 N χ´N0 qGćχ 0 Nχ | ΓcpEq : Γ c pEq Ñ ΓpEq is still continuous. To conclude the proof it is sufficient to prove that if Γ c pEqf n Ñ 0 in the topology of Γ c pEq and K Ą supppf n q for all n P N is a compact set, then there is a compact set K 1 such that K 1 Ą suppppc χ 0 N χ´N0 qGćχ 0 Nχ f n q for all n P N. If t : M Ñ R is the Cauchy temporal function of g 1 used to construct R`and R´, whose level sets Σ τ :" t´1pτ q are Cauchy hypersurfaces for g 0 , g χ , g 1 and g χ " g 0 in the past of Σ t 0 , then the set J pM,gχq pKq X D pM,gχq pΣ t 0 q, which is compact for known properties of globally hyperbolic spacetimes, includes all supports of pc χ 0 N χ´N0 qGćχ 0 Nχ f n from the very definition of retarded Green operator also using the fact that pc χ 0 N χ´N0 q vanishes in the past of Σ t 0 .
As a byproduct of Theorem 4.12 we get a technical, but important, corollary.
Corollary 4.14. Consider g, g 1 , g 2 P GH M , corresponding formally selfadjoint and normally hyperbolic operators N, N 1 , N 2 on the K-vector bundle E on M equipped with a non-degenerate, Hermitian, fiberwise metric. Assume that g » g 1 and g 1 » g 2 and suppose that R gg 1 is a Møller operator of g, g 1 and R g 1 g is a Møller operator of g 1 , g 2 according to (4.16). The following facts are true.
Proof. It is immediate form the construction of R described at the end of Theorem 4.12 relying on (4.16).

Remark 4.15.
Observe that the construction of the Møller operator R of g 0 , g 1 , for g 0 ĺ g 1 , as R " R´R`we used several times in this work is nothing but an elementary case of (2). Indeed, in that case, g 0 ĺ g χ ĺ g 1 and R`, R´are, respectively, a Møller operator of g 0 , g χ and g χ , g 1 .

Møller˚-isomorphisms in algebraic quantum field theory
The aim of this section is to investigate the role of the Møller operators at the quantum level. In order to achieve our goal, we will follow the so-called algebraic approach to quantum field theory, see e.g. [4,5,11,36,58]. In loc. cit. the quantization of a free field theory on a (curved) spacetime is interpreted as a two-step procedure: 1. The first consists of the assignment to a physical system of a˚-algebra of observables which encodes structural properties such as causality, dynamics and canonical commutation relations.
2. The second step calls for the identification of an algebraic state, which is a positive, linear and normalized functional on the algebra of observables.
Using this framework, in this section we shall lift the action of the Møller operators on the algebras of the free quantum fields and then we will pull-back the action of the Møller operators on quantum states, showing that the maps preserve the Hadamard condition with quite weak hypotheses which, in principle, permit an extension of the theory to a perturbative approach. For a more detailed introduction to the algebraic approach to quantum field theory we refer to [15,41] for textbook and to [10-12, 17-27, 43-50] for some recent applications.
We begin first by recalling the construction of the free quantum field theories in curved spacetime for the general class of Green hyperbolic operators, which, as we have seen contains the class of the normally hyperbolic operators which are the ones under exam.
Notation 5.1. Through this section, E will always denote an R-vector bundle over a globally hyperbolic spacetime pM, gq. In particular, we denote the non-degenerate, symmetric, fiberwise metric by x¨|¨y.

Algebras of free quantum fields and the Møller˚-isomorphism
Given a formally-selfadjoint normally hyperbolic operator N : ΓpEq Ñ ΓpEq and its causal propagator G, we first define the unital complex˚-algebra A f as the free complex unital˚-algebra with abstract (distinct) generators φpfq for all f P Γ c pEq, identity 1, and involution˚as discussed in [58]. (As a matter of fact A f is made of finite linear complex combinations of 1 and finite products of generic elements φpfq and φphq˚.) Then we define a refined complex unital˚-algebras by imposing the following relations by the quotient A " A f {I where I is the two sided˚-ideal generated by the following elements of A: where we have used the notation We have the further possibility to enrich the ideal with the generators: • φpNfq , @f P Γ c pEq.
Notation 5.2. The equivalence classes rφpfqs will be denoted by Φpfq and they will be called field operators (on-shell if the ideal is enlarged by including the generators φpNfq), and we use the notation I for the identity r1s of A f {I. Definition 5.3. Given a formally-selfadjoint normally hyperbolic operator N : ΓpEq Ñ ΓpEq and its causal propagator G, we call CCR algebra of the quantum fields Φ, the unital˚-algebra defined by A :" A f {I. The algebra is said to be on-shell in case the ideal is enlarged by including the generators φpNfq. Furthermore, we call observables of A any Hermitian element of it.
With the above notation, the following properties are valid The on-shell field operators also satisfy • Equation of motion. ΦpNfq " 0 , @f P Γ c pEq.
Remark 5.4. The idea behind the notation Φpfq is a formal smearing procedure that uses the scalar product From this perspective, since N is formally selfadjoint, the identity ΦpNfq " 0 for all f P Γ c pEq has the distributional meaning NΦ " 0. Alternatively, as explained in [69], one may use a different representation where Φ is viewed as a "generalized section" of the dual bundle E˚. In that case the formal identity NΦ " 0 corresponding to the equation of motion has to be replaced by N˚Φ " 0.
Given different normally hyperbolic operators N, N 1 all the information about causality and dynamics is encoded in the ideal I, I 1 . In that case we have two corresponding initial unital -algebras A f and A 1 f with respective generators φpfq and φ 1 pfq. Though the freely generated algebras are canonically isomorphic, under the unique unital˚-isomorphism such that φpfq Ñ φ 1 pfq for all f P Γ c pEq, the quotient algebras are intrinsically different because the CCR are different depending on the choice of the causal propagator G N or G N 1 . However there is an isomorphism between them as soon as a Møller operator exists. Indeed, the existence of the Møller operator discussed in the previous sections can be exploited to define first an isomorphism of the free algebras A f and A 1 f since the operator R : gg 1 : Γ c pEq Ñ Γ c pEq is an isomorphism.
Definition 5.5. Let N, N 1 : ΓpEq Ñ ΓpEq be two formally-selfadjoint (with respect to a fiber metric x¨|¨y) normally hyperbolic operators globally hyperbolic spacetimes pM, gq and pM, g 1 q. If g » g 1 , we define an isomorphism R f : A 1 f Ñ A f as the unique unital˚-algebra isomorphism between the said free unital˚-algebras such that R f pφ 1 pfqq " φpR : gg 1 fq @f P Γ c pEq. where R is a Møller operator of g, g 1 (in this order) satisfying Theorem 4.12 and Equation (4.16).
As we shall see in the next proposition, the isomorphism between freely generated algebras induces an isomorphism of the quotient algebras.
Proposition 5.6. Let N and N 1 be two formally-selfadjoint normally hyperbolic operators acting on the sections of the R-vector bundle E over M, and referred to respective g, g 1 P GM M . If g » g 1 and R is a Møller operator of g , g 1 in the sense of Theorem 4.12 and Equation (4.16), then the CCR algebras A and A 1 (possibly both on-shell) respectively associated to N and N 1 are isomorphic under the quotient isomorphism R : the unique unital˚-algebra isomorphism satisfying RpΦ 1 pfqq " ΦpR : gg 1 fq @f P Γ c pEq.
Proof. To prove the statement it suffices to show that the operator R f maps the ideal I 1 to the ideal I. Each ideal I and I 1 is the intersection of three (four) ideals corresponding to the requirements of linearity, Hermiticity, CCR (and equation of motion). The fact that R f preserves the ideals due to linearity and hermiticity is an immediate consequence of the fact that R f is a ˚-algebra homomorphism of the involved freely generates algebras. The ideal arising from the equation of motion condition is preserved due to the first statements of Theorem 4.12 and item (4) therein. The situation is more delicate regarding the ideal generated by the CCR. Preservation of that ideal is actually an immediate consequence of R f pI 1 q " I (R f is unital by hypothesis) and the structure of CCR together with (4.14): This concludes our proof.
Definition 5.7. A unital˚-isomorphism R : A 1 Ñ A defined in Proposition 5.6 out of the Møller operator R of g, g 1 as in Theorem 4.12 and (4.16) is called Møller˚-isomorphism of the CCR algebras A,A 1 (in this order)

Pull-back of algebraic states through the Møller˚-isomorphism
As explained in the beginning of this section, the subsequent step in the quantization of a field theory consists in identifying a distinguished state on the˚-algebra of the quantum fields. The GNS construction then guarantees the existence of a representation of the quantum field algebra through, in general unbounded, operators defined over a common dense subspace of some Hilbert space. We will not care about the explicit representation and recall some definitions (see [32] for a general discussion also pointing out several not completely solved standing issues).
Definition 5.8. We call an (algebraic) state over a unital˚-algebra B a C-linear functional ω : B Ñ C which is (i) Positive ωpa˚aq ě 0 @a P B, (ii) Normalized ωpIq " 1 A generic element of the CCR algebras A of a quantum field Φ associated to the normally hyperbolic operators discussed before can be written as a finite polynomial of the generators Φpf q, where the zero grade term is proportional to I, to specify the action of a state it's sufficient to know its action on the monomials, i.e its n-point functions ω n pf 1 , .., f n q :" ωpΦpf 1 q...Φpf n qq (5.1) The map Γ c pEqˆ¨¨¨ˆΓ c pEq Q pf 1 , . . . , f n q Þ Ñ ω n pf 1 , .., f n q can be extended by linearity to the space of finite linear combinations of sections f 1 b¨¨¨b f n P Γ c pE nb q, where E nb is n-times exterior tensor product of the vector bundle E with itself. If we impose continuity with respect to the usual topology on the space of compactly supported test sections, since the said linear combinations are dense, we can uniquely extend the n-point functions to distributions in Γ 1 c pE nb q we shall hereafter indicate by the same symbol ω n . It has a formal integral kernel, ω n pf 1 , .., f n q " ż M n r ω n px 1 , ..., x n qf 1 px 1 q...f n px n qvol M n px 1 , . . . , x n q, where vol M n px 1 , . . . , x n q :" vol g px 1 q b¨¨¨(n times)¨¨¨b vol g px n q henceforth. Notice that if more strongly ω n P Γ 1 c pE nb q, then ω n phq " ż M n r ω n px 1 , ..., x n qhpx 1 , . . . , x n qvol M n px 1 , . . . , x n q is also defined for h P Γ c pE nb q. The case n " 2 is the easiest one. The Schwartz kernel theorem implies Γ c pEq Q f Þ Ñ ω 2 ph, fq is (sequentially) continuous at f " 0 for every fixed h P Γ c pEq if and only if ω 2 continuously extends to a unique distribution we hereafter indicate with the same symbol ω 2 P Γ 1 c pE b Eq. An important fact (see the comment after [69,Proposition 5.6]) is that, if the CCR algebra A admits states, then the fiberwise metric x¨|¨y must be positive. In other words, x¨|¨y is a real symmetric positive scalar product. We shall assume it henceforth.
Differently from a free quantum field theory on Minkowski spacetime, where the Poincaré invariant state -known as Minkowski vacuum -might be a natural choice, on a general curved spacetime there might be no choice of a natural state. However there is a class of states, known as quasifree (or Gaussian) states, whose GNS representation mimics the Fock representation of Minkowski vacuum (see e.g. [58]). Definition 5.9. Let A be the CCR algebra. A state ω : A Ñ C is called quasifree, or equivalently Gaussian, if the following properties for its n-point functions hold (i) ω n pf 1 , ..., f n q " 0, if n P N is odd, where "partitions" for even n refers to the class of all possible decompositions of the set t1, 2, . . . , nu into n{2 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{2.
For these states all the information is encoded in the two-point distribution, as one can expect in a free theory. It is not difficult to prove that, for a quasifree state in view of the definition above, ω 2 P Γ 1 c pEq entails that ω n continuously extends to ω n P Γ 1 c pE nb q obtained, for n " 2k, as a linear combination of tensor products of distributions ω 2 and trivial if n " 2k`1.
as a consequence of (5.1) and ΦpNfq " 0. However it may happen that these identities are valid (for some n) even if the algebra is not on-shell.
In the next proposition, we shall see that the action of the Møller isomorphism R between CCR-algebras can be pull-backed on the quantum states. Furthermore, the pull-back of a quasifree state is again a quasifree state.
Proposition 5.11. Let be g, g 1 P GH M , consider the algebras A, A 1 respectively associated to formally-selfadjoint normally hyperbolic operators N, N 1 : ΓpEq Ñ ΓpEq constructed out of g and g 1 and let ω : A Ñ C be a state. Assuming that g » g 1 , we define a functional ω 1 : A 1 Ñ C by pull-back through a Møller˚-isomorphism R : A 1 Ñ A of A,A 1 as in Definition 5.7, i.e. ω 1 " ω˝R.
Then the following statements hold true: (1) ω 1 is a state on A 1 ; (2) ω 1 2 P Γ 1 c pE b Eq if and only if ω 2 P Γ 1 c pE b Eq; (3) ω 1 is quasifree if and only if ω is. Proof.
(1) Linearity is obvious since we are composing linear maps. Normalization follows from 1 in 5.6 and from the fact that ω is normalized. Positivity follows from positivity of ω and the fact that R preserves the involutions, the products, and is surjective. (2) Since ω 2 P Γ 1 c pEˆEq, then it is Γ c pEq-continuous in the right entry (taking values in Γ 1 c pEq and with respect to the corresponding topology). As a consequence, by composition of continuous functions, if h P Γ c pEq is given, Γ c pEq Q f Þ Ñ ω 1 2 ph, fq " ω 2 pR : gg 1 h, R : gg 1 fq is Γ c pEq-continuous as well because R : gg 1 : Γ c pEq Ñ Γ c pEq is continuous in the Γ c pEq topology in domain and co-domain for (5) of Theorem 4.12. In other words Γ c pEq Q f Þ Ñ ω 1 2 p¨, fq P Γ 1 c pEq is continuous. We conclude that ω 1 2 P Γ 1 c pE b Eq due to the Schwartz kernel theorem. The result can be reversed swapping the role of the states and the metrics, noticing that ω " ω 1˝R´1 where R´1 is also a Møller˚-isomorphism, the one constructed out of R´1 which is, in turn, a Møller operator associated to the pair g 1 , g in this order in view of Corollary 4.14.
(3) The proof is immediate and follows by construction.

Møller preservation of the microlocal spectrum condition for off-shell algebras
It is widely accepted that, among all possible (quasifree) states, the physical ones are required to satisfy the so-called Hadamard condition. The reasons for this choice are manifold: For example, it implies the finiteness of the quantum fluctuations of the expectation value of every observable and it allows us to construct Wick polynomials [55,57] and other observables, as the stress energy tensor, relevant in semi-classical quantum gravity following a covariant scheme [54,61], encompassing a locally covariant ultraviolet renormalization [56] (see also [58] for a recent pedagogical review). These states have been also employed, e.g. (the following list is far from being exhaustive) in the study of the Blackhole radiation [24,42,59,64], in cosmological models [21,23] and other applications to spacetime models [34,35,62], and to study energy quantum inequalities [33]. For later convenience, we decided to present the Hadamard condition as a microlocal condition on the wave-front set of the two-point distribution [67,68] instead of the equivalent geometric version based on the Hadamard parametrix [1,60,63]. Let's briefly sketch what they are and why they are useful.
From now on we adopt the definitions of wave-front set W F pψq of distribution ψ on R-vector bundles equipped with a non-degenerate, symmetric, fiberwise metric 3 as in [69].
We shall use some very known definitions and results of microlocal analysis applied to distributions of Γ 1 c pFq where F is a K-vector bundle, F " E b E for instance (see [69] for details). In particular, • ψ P Γ 1 c pFq is a smooth section of the dual bundle F˚, indicated with the same symbol ψ P ΓpF˚q, if and only if W F pψq " H.
• Let us assume that F " E b E where E is equipped with a non-degenerate, symmetric (Hermitian if K " C), fiberwise metric and let P : Γ c pEq Ñ Γ c pEq be a formally selfadjoint smooth differential operator. We say that ν P Γ 1 c pE b Eq is a bi-solution Pf " 0 mod C 8 , if there exist ϕ, ϕ 1 P ΓpF˚q such that We are in a position to state the definition of micro local spectrum condition and Hadamard state. Below, " is the equivalence relation in T˚M 2 zt0u such that px, k x q " py, k y q if there is a null geodesic passing through x, y P M and the geodesic parallely transports the co-tangent vector to that geodesic k x P Tx M into the co-tangent vector to that geodesic k y P Tẙ M. Finally, k x Ź 0 means that the covector k x is future directed.
Definition 5.12. With A as in Definition 5.9, a state ω : A Ñ C is called a Hadamard state if ω 2 P Γ 1 c pE b Eq and the following microlocal spectrum condition is valid W F pω 2 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 .
More generally, a distribution ν P Γ 1 c pE b Eq is said to be of Hadamard type if its wave-front set W F pνq is the right-hand side of (5.2).
(1) Notice that px, k x ; x,´k x q P W F pνq for every future directed lightlike covector k x P Tx M if ν P Γ 1 c pE b Eq is of Hadamard type.
(2) It is possible to prove that a fiberwise scalar product x¨|¨y must be necessarily positive if A admits quasifree Hadamard states as proved in the comment after [69, Proposition 5.6]. We henceforth assume that x¨|¨y is positive.
The theorem below, which is the second main result of this paper, shows that the Hadamard condition is preserved under the pull-back along the Møller isomorphism.
Theorem 5.14. Let E be an R-vector bundle over smooth manifold M and denote with x¨|¨y positive, symmetric, fiberwise metric. Let be g, g 1 P GH M , consider the corresponding formallyselfadjoint normally hyperbolic operators N, N 1 : ΓpEq Ñ ΓpEq and refer to the associated CCR algebras A and A 1 (off-shell in general). Finally, suppose that g » g 1 . ω : A Ñ C is a quasifree Hadamard state, if and only if ω 1 :" ω˝R : A 1 Ñ C , constructed out of a Møller˚-isomorphism R of A, A 1 , is a quasifree Hadamard state of A 1 .
Remark 5.15. We stress that it is not required that the algebras are on-shell nor that the relevant two-point functions satisfy the equation of motion with respect to the corresponding normally hyperbolic operators.
The rest of this section is devoted to prove Theorem 5.14, a refinement of it stated in the last Theorem 5.19, and a proof of existence of Hadamard states based on our formalism.
Our first observation is the following. If tf z u zPU Ă ΓpEq is such that MˆU Q px, zq Þ Ñ f z pxq is jointly smooth, then MˆU Q px, zq Þ Ñ pSf z qpxq is jointly smooth as well.
Proof. We consider the case of R`, the remaining three instances having a similar proof. What we have to prove is that MˆU Q px, zq Þ Ñ´Gρ Nχ pρN χ´N0 qf z¯p xq is smooth under the said hypotheses. Let us first consider the case where there is compact K Ă M such that supppf z q Ă K for all z P U . In this case, defining F px, zq :" pρN χ´N0 qf z pxq, the projection π : supppF q Q px, zq Þ Ñ z P U is proper 4 and this fact will be used shortly. Interpreting Gρ Nχ : Γ c pEq Ñ Γ c pEq and thus as a Schwartz kernel, we can compute the wavefront set of the map MˆU Q px, zq Þ Ñ´Gρ Nχ pρN χ´N0 qf z¯p xq viewed as the distributional kernel of the composition of the kernel Gρ Nχ px, yq and the smooth kernel F py, zq. We know that (see, e.g. [58] for the scalar case, the vector case being analogous) W F pGρ Nχ q " px, k x ; y,´k y q P T˚M 2 zt0u | px, k x q " py, k y q, x P J`pyq or k x " k y , x " y ( whereas, since F is jointly smooth, W F pF q " H . The known composition rules of wavefront sets of Schwartz kernels, which use in particular the fact that the projection π above is proper (in [58,Theorem 5.3.14] which is valid also in the vector field case), immediately yields W F pGρ Nχ˝F q Ă H .
It being W F pGρ Nχ˝F q " H, we conclude that MˆU Q px, zq Þ Ñ´Gρ Nχ pρN χ´N0 qf z¯p xq is a smooth function as desired.
Let us pass to consider the generic jointly smooth family tf z u zPU Ă ΓpEq without restrictions on the supports. First of all, we observe that f 1 z pxq :" ppρN χ´N0 qfqpxq is past compact by construction for every z P U , because its support is contained in the future of Σ t 0 referring to the construction of N χ . According to the proof of [3, Theorem 3.6.7], if h is past compact, x 0 P M, and A Ą suppphq X J´px 0 q is an open relatively compact set, for every compactly supported smooth function s A P C 8 c pM; r0, 1sq such that s A pxq " 1 if x P A, it holds pGρ Nχ hqpx 0 q " pGρ Nχ s A hqpx 0 q .
We want to apply this identity for h " f z . Take t 1 ă t 0 . Given x 0 P M we can always define A :" I´pr x 0 q X I`pΣ t 1 q where r x 0 P I`px 0 q 5 . With this choice, A does not depend on z P U and the same A can be used for x varying in an open neighborhood A 1 of x 0 , since I´pr x 0 q is open. We conclude that, if px, zq P A 1ˆU , theń Gρ Nχ pρN χ´N0 qf z¯p xq " pGρ Nχ˝F qpx, zq where F px, zq " s A pxqpρN χ´N0 qf z pxq . (5.3) In this case K :" suppps A q includes all the supports of the maps M Q x Þ Ñ F px, zq for every z P U . The first part of the proof is therefore valid for the map MˆU Q px, zq Þ Ñ pGρ Nχ˝F q which must be jointly smooth as a consequence. In particular, its restriction A 1ˆU Q px, zq Þ Ñ Gρ Nχ pρN χ´N0 qf z¯p xq is jointly smooth as well. Since A 1 can be taken as a neighborhood of every point in M and z P U is arbitrary, from (5.3) the whole function MˆU Q px, zq Þ Ñ Gρ Nχ pρN χ´N0 qf z¯p xq is jointly smooth.
Relying on Lemma 5.16, we can notice the following. Proof. We start by stressing that, as already noticed, in view of the known continuity properties of R :g 0 gχ and its inverse and using Schwartz' kernel theorem, ν 0 P Γ 1 c pE b Eq if and only if ν χ P Γ 1 c pE b Eq. We pass to prove that if ν 0 is a bisolution mod C 8 , then ν χ is a bisolution mod C 8 , referring to the corresponding operators. Let us hence suppose that ν 0 pN 0 f, hq " ψpfbhq and ν 0 pf, N 0 hq " ψ 1 pfbhq for some smooth sections ψ, ψ 1 P ΓppE b Eq˚q and all f, h P Γ c pEq. The identity immediately implies that, if ϕpx, yq :" c χ 0 pxqc χ 0 pyqψpx, yq, ϕ 1 px, yq :" c χ 0 pxqc χ 0 pyqψ 1 px, yq, The proof ends if proving that there are smooth sections ϕ 1 , ϕ 1 1 P ΓppE b Eq˚q, such that ż where we have locally defined ϕ 1 a yb pxq :" ξ ac pxqϕ cb px, yq, with ξ ab pxq being the inverse fiber metric at x P M in any considered local trivialization. Above, ϕ 1 ab px, yq :" c χ 0 pxqpR`ϕ 1 yb q a pxq is the candidate section of pE b Eq˚we were looking for, represented in local coordinates of the atlas of the said trivialization. That section is smooth, i.e., ϕ 1 P ΓppE b Eq˚q as desired. Indeed, the maps MˆU k Q px, yq Þ Ñ ϕ 1 yb pxq define a family of sections of ΓpEq parametrized by y P U k for every given b P t1, . . . , N u, where U k Ă M is the projection onto M of the domain of the considered local trivialization. This family is jointly smooth in x, y as established in Lemma 5.16. The converse statement, that ν 0 is a bisolution mod C 8 if ν χ is, can be proved with the same procedure simply replacing R`with pR`q´1 and using Lemma 5.16 again.
Before giving the proof of Theorem 5.14, we need a final lemma, which shows that any Hadamard distribution whose antisymmetric part is given by the causal propagator of a normally hyperbolic system N is actually a bisolution of N itself modulo smooth errors.
Lemma 5.18. Let N : ΓpEq Ñ ΓpEq be a formally selfadjoint normally hyperbolic operator and suppose that ν P Γ 1 c pE b Eq is of Hadamard type and satisfies νpx, yq´νpy, xq " iG N px, yq mod C 8 where G N px, yq is the distributional kernel of the causal propagator G N . In this case ν is a bisolution of Nf " 0 mod C 8 .
Proof. The proof is a straightforward re-adaptation of the proof appearing in the Note added in proof of [67].
We are finally in a position to prove Theorem 5.14.
Proof of Theorem 5.14. We have only to prove that ω 1 is Hadamard if and only if ω is, since the other preservation property has been already proved in (4) of Proposition 5.11. If g 0 » g 1 , there is a sequence of globally hyperbolic metrics g 1 0 " g 0 , g 1 1 , . . . , g 1 N " g 1 such that either g 1 k ĺ g 1 k`1 or g 1 k`1 ĺ g 1 k and the future cones satisfy a corresponding inclusion. The Møller operator R of A, A 1 is obtained as the composition of the Møller operators R k of the formally-selfadjoint normally hyperbolic operators N 1 k , N 1 k`1 associated to the pairs g 1 k , g 1 k`1 : as in the proof of Theorems 4.5, 4.12 and (4.16). The thesis is demonstrated if we prove that, with obvious notation, ω k`1 is Hadamard if and only if ω k is. So in principle we have to prove the thesis only for a pair of metrics g 0 , g 1 with the two cases g 0 ĺ g 1 and g 1 ĺ g 0 . Actually the latter is a consequence of the former, using the fact that a Møller˚-isomorphisms are bijective and that a Møller operator of the second case is the inverse operator of a Møller operator of the first case in accordance to Corollary 4.14. In summary, the proof is over if establishing the thesis for the case g " g 0 ĺ g 1 " g 1 and we shall concentrate on that case only in the rest of the proof. Recalling by (4.12) and (4.11) that R :g 0 g 1 " R , we write ω 1 2 pf 1 , f 2 q " ω 0 2 pR :g 0 g 1 f 1 , R :g 0 g 1 f 2 q " ω 0 2 pR To analyze the wave-front set of this bidistribution, we split again the operation in two steps. First we define a pull-back state on the algebra A χ of quantum fields defined for the formallyselfadjoint normally hyperbolic operator N χ , i.e a normally hyperbolic operator on pM, g χ q. This intermediate pull-back states reads We intend to prove that ω χ 2 P Γ 1 c pEq is of Hadamard type if and only ω 0 2 is. Notice that both two-point functions have antisymmetric parts that coincide with iG Nχ and iG N 0 , respectively, in view of the CCRs of the respective algebras. If ω 0 2 P Γ 1 c pEq is of Hadamard type, then it is a a bisolution of N 0 f " 0 mod C 8 in view of Lemma 5.18. The same argument proves that, if ω χ 2 P Γ 1 c pEq is of Hadamard type, then it is a bisolution of N χ f " 0 mod C 8 due to 5.18. Applying Lemma 5.17 to both cases we have that, (a) ω 0 2 P Γ 1 c pEq of Hadamard type implies that ω 0 2 is a bisolution N 0 f " 0 mod C 8 and ω χ 2 is a a bisolution of N χ f " 0 mod C 8 ; (b) ω χ 2 P Γ 1 c pEq of Hadamard type implies that ω χ 2 is a bisolution N χ f " 0 mod C 8 and ω 0 2 is a a bisolution of N 0 f " 0 mod C 8 .
We are now in a position to apply the Hadamard singularity propagation theorem. Consider the smooth Cauchy time function t in common with g 0 and g χ , such that χpxq " 0 if tpxq ă t 0 .
As a preparatory remark we notice that R :g 0 gχ f " f from (4.21) when the support of f stays in the past of the Cauchy surface Σ t 0 " t´1pt 0 q. In that region g 0 " g χ by definition of g χ . Finally due to (5.4), ω χ 2 pf, hq " ω 0 2 pf, hq if tpsupppfqq ă t 0 , tpsuppphqq ă t 0 Hence, in particular, ω χ 2 is of Hadamard type when the supports of the test functions are taken in that region if and only if ω 0 2 is of Hadamard type when the supports of the test functions are taken there. More precisely, it happens when the supports of the arguments f, h are taken in a (globally hyperbolic) neighborhood of a Cauchy surface (for both metrics!) Σ τ :" t´1pτ q with τ ă t 0 between two similar slices. Since both distributions are bisolutions of the respective equation of motion mod C 8 and the operators are normally hyperbolic, the theorem of propagation of Hadamard singularity (see, e.g., Theorem 5.3.17 in [58] 6 ) implies that ω χ 2 and ω 0 2 are of Hadamard type everywhere in pM, g χ q and pM, g 0 q, respectively. A similar reasoning shows that ω 1 2 P Γ 1 c pE b Eq, with ω 1 2 pf 1 , f 2 q " ω χ 2 pR :g χg1 is Hadamard on pM, g 1 q if and only if ω χ 2 is on pM, g χ q. Combining the two results we have that ω 1 " ω 1 is Hadamard on pM, g 1 " g 1 q if and only if ω " ω 0 is Hadamard on pM, g " g 0 q concluding the proof.
We are now in the position to prove our last result.
Theorem 5. 19. Let E be an R-vector bundle on a smooth manifold M equipped with a positive, symmetric, fiberwise metric x¨|¨y. Let g, g 1 P GH M , consider the corresponding formallyselfadjoint normally hyperbolic operators N, N 1 : ΓpEq Ñ ΓpEq and refer to the associated CCR algebras A and A 1 . Let ν P Γ 1 c pE b Eq be of Hadamard type and satisfy νpx, yq´νpy, xq " iG N px, yq mod C 8 , G N px, yq being the distributional Kernel of G N . Assuming g » g 1 , let us define ν 1 :" ν˝R : gg 1 b R : gg 1 , for a Møller operator R : ΓpEq Ñ ΓpEq of g, g 1 . Then the following facts are true.
Proof. Since we never exploited the fact that ω is positive, nor the fact that the antisymmetric part of its two points function is exactly the causal propagator, nor the fact that the relevant algebras of fields are on-shell (i.e., the equation of motion are satisfied by the two-point functions), we can use the same arguments as in the proof of the previous theorem to conclude.
We conclude this section with the following straightforward result of existence of Hadamard quasifree states which apparently does not use the Hadamard singularity propagation argument (actually this argument was used in the proof of Theorem 5.14).
Corollary 5.20. Let pM, gq be a globally hyperbolic spacetime, N be a formally-selfadjoint normally hyperbolic operator acting on the sections of the R-vector bundle E over M and refer to the associated CCR algebras A. Then there exists a Hadamard state on A.
Proof. It is well-known [38] that, in a globally hyperbolic ultrastatic spacetime, the (unique) CCR quasifree ground state which is invariant under the preferred Killing time is Hadamard. Hence, combining Corollary 2.23 with Theorem 5.14 we can conclude.

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.
Paracausally related metrics. One of the key ingredients in the realization of the Møller operator is the introduction of the new geometric notion that we have called paracausal deformation. In particular, we have seen in Section 4, that for any given two paracausally deformed metrics, there exists a Møller operator which realizes an algebraic equivalence between corresponding free quantum field theories defined on different curved spacetimes (on a given manifold). Therefore it seems very natural and important to classify the class of metrics which are not paracausally related in relation to the existence of inequivalent quantum field theories. We have already seen in Section 2.3, that if pM, gq and pM, g 1 q admit a common foliation of Cauchy hypersurfaces, then g and g 1 are paracausally related. As suggested by part (3) of the Example 2.21, the claim that if pM, gq and pM, g 1 q are Cauchy-compact globally hyperbolic spacetimes then g and g 1 are paracausally related, does not sound reasonable. The idea behind is that g and g 1 in part (3) of the Example 2.21 have somehow 'different time-orientation'. Since the time-orientation depends on the metric on M, we have to provide a criteria to translate the requirement that g and g 1 are both 'future-directed' in some sense. Keeping in mind what said above, a conjecture which urges to be proved or disproved is the following one (maybe adding some further hypothesis). Conjecture 6.1. Let t and t 1 be Cauchy temporal functions for globally hyperbolic spacetimes pM, gq and pM, g 1 q. Finally denote with x¨,¨y the natural paring between T˚M and TM. Then g » g 1 if and only if xB t , dt 1 y ą 0 and xB t 1 , dty ą 0 , where B t (resp. B t 1 ) is the dual of dt (resp. dt 1 ) with respect to g (resp. g 1 ).
Remark 6.2. The requirement xB t , dt 1 y ą 0 implies that the integral curve γ " γptq of B t on pM, g 1 q satisfies t 1 pγpt 2 qq ą t 1 pγpt 1 qq if t 2 ą t 1 . This requirement is weaker than assuming the B t is timelike and future-directed for g 1 . The reason why we also impose xB t 1 , dty ą 0 is that being paracausally related is an equivalence relation in GH M .
A similar conjecture has to be established or disproved in the class of asymptotically flat spacetimes. Conjecture 6.3. If pM, gq and pM, g 1 ) are asymptotically flat globally hyperbolic spacetimes then g and g 1 are paracausally related if xB t , dt 1 y ą 0 and xB t 1 , dty ą 0. Remark 6.4. Differently from Conjecture 6.1, the conjecture above would not provide a characterization of asymptotically flat spacetimes, since part (3) of the Example 2.21 provide a counterexample to the 'necessary' part of the statement. Indeed, the Minkowski metrics presented in Example 2.21 do not satisfy xdt 1 , B t y ą 0.
If the conjectures are proved to hold, the results would suggest that free quantum field theories on globally hyperbolic spacetimes are more sensitive to the topology of the manifold M with respect to the metric g endowing M. In particular, the physics encoded in the quasifree states ω for a quantum field propagating on curved spacetime (R 4 , gq can be found in the pull-backed state ω 1 " ω˝R on Minkowski spacetime. Loosely speaking, the quantum effects due to the interaction between the quantum field and the classical gravitational potential can be thought as special observable in spacetimes without gravitational interaction.
Homotopical properties of Møller operators Another issue which deserves to be investigated is the dependence of the Møller operator R of g, g 1 and the associated˚-isomorphism from the finite sequence of globally hyperbolic metrics joining g, g 1 in the sense of paracausal relationship. It is clear from the construction of R that the natural "composition" of sequences corresponds to the composition of operators. Question 6.5. Is there some homotopical notion in the space of globally hyperbolic metrics which is reflected in the space of Møller operators?
Pull-back of ground and KMS states through the Møller˚-isomorphism. Another issue we have faced is the lack of control on the action of the group of˚-automorphism induced by the isometry group of the spacetime M on ω. Let us remark, that the study of invariant states is a well-established research topic (cf. [7,8]). Indeed, the type of factor can be inferred by analyzing which and how many states are invariant. From a more physical perspective instead, invariant states can represent equilibrium states in statistical mechanics e.g. KMS-states or ground states. The previous remark leads us to the following open question: Question 6.6. Under which conditions it is possible to perform an adiabatic limit, namely when is lim χÑ1 ω 1 well-defined?
A priori we expect that there is no positive answer in all possible scenarios, since it is known that certain free-field theories, e.g., the massless and minimally coupled (scalar or Dirac) field on four-dimensional de Sitter spacetime, do not possess a ground state, even though their massive counterpart does. (Notice that this is not a no-go Theorem, but at least an indication that, in these situations, the map ω Ñ ω˝R cannot be expected to preserve the ground state property.) A partial investigation in this direction has been carried on in [20,28] for the case of a scalar field theory on globally hyperbolic spacetimes with empty boundary. In this situation it has been shown that, under suitable hypotheses the adiabatic limit can be performed preserving the invariance property under time translation but spoiling in general the ground state or KMS property.
Møller˚-isomorphism in perturbative AQFT. We conclude with the following observation. The results of Section 5 are valid also for off-shell algebras as well as for distribution of Hadamard type. Therefore, it could be possible to extend the action of the Møller operator also on the algebra of extended observables in a perspective of deformation product quantization (see for instance Section 2 of [27]), which include, e.g., the Wick polynomials of the underlying fields. Wick polynomials and time-ordered products of Wick polynomials are the building blocks for perturbative renormalization of quantum fields, both in Minkowski spacetime and in curved spacetime, where the metric is considered as a given external classical field. Although of utmost physical relevance, these formal operators as the stress-energy operator do not belong to the algebra of observables generated by the smoothly smeared field operators (operator-valued distributions). This is because they correspond to products of distributions at a given point and this notion is not well-defined in general. The popular and perhaps most effective procedure to eliminate the short-distance divergences consists of simply subtracting a suitable Hadamard distribution. This procedure is systematically embodied in a product deformation quantization procedure which relies on a suitable set of functionals with a specific wavefront set. The following observation leads to the following conjecture: Conjecture 6.7. Let A 0 , A 1 0 be the algebra of observables of the globally hyperbolic spacetimes pM, gq and pM, g 1 q and R 0 a Møller˚-isomorphism of them. If A, A 1 are corresponding extended algebras of observable (which include the Wick polynomials etc.) and g » g 1 , then R 0 extends to a (Møller)˚-isomorphism R : A Ñ A 1 .
Møller operators and gauge theory. Last but not least, all our results concern vector-valued fields of a vector bundle E over M. The fiber metric on the bundle does not depend on the globally hyperbolic metrics g chosen on M. A natural extension of the formalism would be the inclusion of that g-dependence in the fiber metric. This extension would allow to encompass the case of a Proca field and, possibly the case of the electromagnetic field, though issues with gauge invariance and gauge fixing are expected to pop out.