Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds

In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hyperbolic manifolds.


Introduction
Symmetric hyperbolic systems are an important class of first-order linear differential operators acting on sections of vector bundles on Lorentzian manifolds. The most prominent examples are the classical Dirac operator and the geometric wave operator, which can be understood by reducing a suitable second-order normally hyperbolic differential operator to a first-order differential operator. In the class of Lorentzian manifolds with empty boundary known as globally hyperbolic, the Cauchy problem of a symmetric hyperbolic system is well-posed. As a consequence the existence of advanced and retarded Green operators in guaranteed. These operators are of essential importance in the quantization of a classical field theory: Indeed they implement the canonical commutation relation for a bosonic field theory or the canonical anti-commutation relation for fermionic field theory. Moreover, their difference, dubbed causal propagator (or Pauli-Jordan commutator), can be used to construct quantum states. For further details, we recommend the recent reviews [4,10,35].
In this paper, we investigate the existence of a geometrical map connecting the space of solutions of different symmetric hyperbolic systems over (possibily different) globally hyperbolic manifolds. A summary of the main result obtained is the following (cf. Theorem 3.4): Theorem 1.1. Let α P t0, 1u and M α " pM, g α q be globally hyperbolic manifolds admitting the same Cauchy temporal function. Consider the symmetric hyperbolic systems S α over M α acting on sections of a real (or complex) vector bundle E α endowed with a non-degenerate sesquilinear fiber metric. If there exists a vector bundle isometry κ : E 0 Ñ E 1 and the set of timelike vectors for g 1 is contained in the one for g 0 , then the spaces of inhomogeneous solutions for S 0 and S 1 are isomorphic. Remark 1.2. As we shall see in more details in Theorem 3.4, actually there exists a oneparameter family of isomorphisms R ̺ : Sol pS 0 q Ñ Sol pS 1 q, where ̺ P C 8 pM, Rq is a strictly positive smooth function which plays a fundamental role in the conservation of Hermitian forms naturally defined on the space of spacially compact solutions (cf. Section 3.1). Unfortunately, none of these isomorphisms is canonical, in the sense that they depends on the choice of a smooth function χ P C 8 pM, r0, 1sq.
Let us briefly comment on the geometric setting. First of all, the globally hyperbolic manifolds and the vector bundles can coincide, as for the case of scalar wave equations propagating on the same manifold which differs by a smooth potential. In this case, our analysis incorporates the results of Dappiaggi and Drago in [23]. If the manifolds do not coincides but the vector bundles do, then a relation between the causal cones for the different metrics is need it in order to defined a suitable 'intertwining' symmetric hyperbolic systems (cf. Lemma 3.3). Finally, if also the vector bundles do not coincides, as for the case of spinor bundles, then an isometry is need it in order to compare the different symmetric hyperbolic systems. Let us recall that the existence of a spinor bundle isometry in the Lorentzian setting is guaranteed by the result of Bär, Gauduchon and Moroianu [3]. As a by-product of our analysis, we expect that the intertwining operator defined in Proposition 3.9 can be used to probe spectral properties of the Riemannian Dirac operator as follows: Consider two Riemannian Dirac operators D α acting on sections of spinors bundles SΣ α over a compact Riemannian manifold pΣ, h α q. By defining pM α " RˆΣ α , g α "´dt 2`h α q we immediately obtain a globally hyperbolic manifold. On M α consider the following symmetric hyperbolic system S α " B t´ı D α : where E α is also considered as a Hermitian vector bundle on M α via the pull-back along the projection π : M α Ñ Σ α . Since D α is essentially self-adjoint on L 2 pSΣ α q, any vector in the kernel of S α can be written employing the spectral calculus, namely Ψ α " expp´ı t D α qf α , "´ż σpDαq e´ı ω t dE ω¯fα where σpD α q and dE ω are respectively the spectral measure and the spectrum of D α and f α are initial data. It follows that intertwining operators should interplay between the spectral measure of D 0 and D 1 respectively. Very recently, Capoferri and Vassiliev gave an explicit formula for the Dirac evolution operator on any 3-dimensional oriented close Riemannian manifold in [22]. Combining their results with ours, we expect to get a better understanding of eigenvalues of the Dirac operators.
Our main result has a deep implication in free quantum field theory over generic globally hyperbolic spacetimes. Indeed, by denoting with A CCR α (resp. A CAR α ) the algebra of real scalar fields (resp. Dirac fields) over a globally hyperbolic spacetimes M α , the isomorphism R ̺ defined in Proposition 3.9 (resp. Proposition 3.13) can be lift to a˚-isomorphism A CCR ) -cf. Theorem 4.3 (resp. Theorem 4.6). Remarkably, the pullback of a quasifree state along this˚-isomorphism preserves the singular structure of the two-point distribution associated to the state (cf. Theorem 4.12). This result is used to provide a new geometrical proof of the existence of the so-called Hadamard states (cf. Corollary 4.14). This paper is structured as follows. In Section 2 we review well know-facts about symmetric hyperbolic systems. In particular in the Subsection 2.1.1 and 2.1.2 we introduce respectively the (classical) Dirac operator and the geometric wave operator as main examples of symmetric hyperbolic systems. In Section 3 we prove the main result of this paper and we investigate the conservation of Hermitian forms. Finally, Section 4 is devoted to analyzing the consequence of such isomorphism in the context of algebraic quantum field theory.
Notation and convention -The symbol K denotes on of the elements of the set tR, Cu; -M g :" pM, gq is a globally hyperbolic manifold (with empty boundary) -cf. Definition 2.1 -and g has the signature p´,`. . . ,`q; -GH M denotes the space of globally hyperbolic metrics on a smooth manifold M such that, for any g 0 , g 1 P GH M , M 0 " pM, g 0 q and M 1 " pM, g 1 q have the same Cauchy temporal function; -E is a K-vector bundle over M g with N -dimensional fibers, denoted by E p for p P M, and endowed with a nondegenerate sesquilinear fiber metric ă¨|¨ą: ΓpE p qˆΓpE p q Ñ K; -Γ c pEq, Γ pc pEq, Γ f c pEq, Γ tc pEq, Γ sc pEq resp. ΓpEq denote the spaces of compactly supported, past compactly supported, future compactly supported, timelike compactly supported, spacelike compactly supported resp. smooth sections of E; -S : ΓpEq Ñ ΓpEq is a symmetric hyperbolic system -cf. Definition 2.5 -and S : , S˚are respectively the formal adjoint and the formal dual operator of S -cf. Remark 2.7; -when M is spin, SM g denotes the spinor bundle -cf. Definition 2.13 -and D : ΓpSM g q Ñ ΓpSM g q denotes the classical Dirac operator -cf. Definition 2.14; -for a fixed, but arbitrary f P ΓpEq, Sol pSq denotes the space of inhomogeneous solutions while Sol sc pSq denotes the space of homogeneous solutions with spacially compact support Sol sc pSq " tΨ P Γ sc pEq | Ψ P ker Su .

Aknowledgements
We would like to thank Claudio Dappiaggi, Nicolò Drago and Valter Moretti for helpful discussions related to the topic of this paper. We are grateful to the referees for useful comments on the manuscript.

Symmetric hyperbolic systems
To goal of this section is to present a self-contained overview of symmetric hyperbolic systems and their properties on Lorentzian manifolds. For a detailed introduction, we recommend the lecture notes of Bär [2].
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 ourself to the class of globally hyperbolic manifolds.
Definition 2.1. A globally hyperbolic manifold is a pn`1q-dimensional, oriented, time-oriented, smooth Lorentzian manifold pM, gq such that (i) There are no closed causal curves; (ii) For every point p, q P M, J`ppq X J´pqq is compact; where J`pUq (resp. J´pUq) denotes the set of points of M that can be reached by future (resp. past) directed causal curves starting from U Ă M. Notation 2.2. For the rest of this section, M g :" pM, gq will always denote a globally hyperbolic manifold and we adopt the convention that the metric g has signature p´,`, . . . ,`q.
The class of globally hyperbolic manifolds contains many important spacetimes, e.g. Minkowski spacetime, Friedmann-Robertson-Walker models, the Schwarzschild blackhole and de Sitter space. In [41], Geroch established the equivalence for a Lorentzian manifold being global hyperbolic and the existence of a Cauchy hypersurface Σ (i.e. an achronal subset which is crossed exactly once by any inextendible timelike curve), which implies that M is homeomorphic to RˆΣ and all Cauchy hypersurfaces are homeomorphic. The proof was carried out by finding a Cauchy time function, namely a continuous function t : M Ñ R which increases strictly on any future-directed causal curve such that each level t´1pt 0 q, t 0 P R, is a Cauchy hypersurface. In [17] Bernal and Sánchez "smoothened" the result of Geroch by introducing the notion of Cauchy temporal function. Definition 2.3. We say that a smooth time function t : M Ñ R is a Cauchy temporal function if its gradient ∇t is past-directed timelike and its level set is a smooth Cauchy hypersurface. . Any globally hyperbolic manifold admits a Cauchy temporal function. In particular, it is isometric to the smooth product manifold RˆΣ with metric g "´β 2 dt 2`h t where t : RˆΣ Ñ R is Cauchy temporal function, β : RˆΣ Ñ R is a smooth positive function and h t is a Riemannian metric on each level set of t.
Let now E be a K-vector bundle over a globally hyperbolic manifold M g with finite rank N and endowed with a (possibly indefinite) non-degenerate sesquilinear fiber metric ă¨|¨ą: E pˆEp Ñ K .
(2.1) Definition 2.5 ( [1], Definition 5.1). A linear differential operator S : ΓpEq Ñ ΓpEq of first order is called a symmetric hyperbolic system over M g if (S) The principal symbol σ S pξq : E p Ñ E p is Hermitian with respect to ă¨|¨ą for every ξ P TpM and for every p P M; (H) For every future-directed timelike covector τ P TpM, the bilinear form ă σ S pτ q¨|¨ą is positive definite on E p .
Remark 2.6. Notice that Definition 2.5 depends on the fiber metric ă¨|¨ą and on the Lorentzian metric g which defined the set of future-directed timelike covectors.
Let us recall that for a first-order linear differential operator S : ΓpEq Ñ ΓpEq the principal symbol σ S : T˚M Ñ EndpEq can be characterized by where u P ΓpEq and f P C 8 pMq. If we choose local coordinates pt, x 1 , . . . , x n q on M, with x i local coordinates on Σ t , and a local trivialization of E, any linear differential operator S : ΓpEq Ñ ΓpEq of first order reads in a point p P M as where the coefficients A 0 , A j , B are NˆN matrices, with N being the rank of E, depending smoothly on p P M. In these coordinates, Condition (S) in Definition 2.5 reduces to ă A 0¨|¨ą "ă¨| A 0¨ą and ă A j¨|¨ą "ă¨| A j¨ą for j " 1, . . . , n. Condition (H) can be stated as follows: For any future directed, timelike covector τ " dt`ř j α j dx j , ă σ S pτ q¨|¨ą"ă pA 0`n ÿ j"1 α j A j q¨|¨ą defines a scalar product on E p .
Remark 2.7. Let S : be the formal adjoint operator of S, i.e. the unique linear differential operator S : : ΓpEq Ñ ΓpEq defined by ż for every Φ, Ψ P ΓpEq with supp Φ X supp Ψ compact. Its principal symbol σ S : satisfies for every ξ P T˚M, see e.g. [ . The Cauchy problem for a symmetric hyperbolic system S on a globally hyperbolic manifold is well-posed, i.e. for any f P Γ c pEq and h P Γ c pE| Σ 0 q there exists a unique smooth solution Ψ P Γ sc pEq with spatially compact support to the initial value problem which depends continuously on the data pf, hq.
As a byproduct of the well-posedness of the Cauchy problem, it follows the existence of Green operators. Proposition 2.9 ( [1], Theorem 5.9 and Theorem 3.8). A symmetric hyperbolic system is Green hyperbolic, i.e. there exist linear maps, dubbed advanced Green operator G`: Γ pc pEq Ñ Γ pc pEq and retarded Green operator G´: Remark 2.10. On account of Remark 2.7 it immediately follows that the formal dual operator S˚is Green hyperbolic for every Φ1 P dompG`˚q, Φ2 P dompG´˚q, Ψ 1 P dompG´q and Ψ 2 P dompG`q.
Definition 2.11. Let S be a Green hyperbolic operator and denote with G`and G´respectively the advanced and retared Green operator. We call causal propagator G : Γ f c pEq Ñ ΓpEq the operator defined by G :" G`´G´.
The causal propagator characterize the space of solutions to the homogeneous Cauchy problem, namely for any f P Γ tc pEq, Ψ :" Gf P ker S. The properties of the causal propagator are summarized in the following proposition.

Proposition 2.12 ( [4], Theorem 3.5).
Let G be the causal propagator for a Green hyperbolic operator S : ΓpEq Ñ ΓpEq. Then the following linear maps forms an exact sequence

Geometric examples
In this section, we shall review two of the most important examples of symmetric hyperbolic systems: the classical Dirac operator and the geometric wave operator. More examples can be found in [42,47], while for further details on spin geometry on Lorentzian spin manifold, we refer to [3,25,49].

The classical Dirac operator
Let M g be a globally hyperbolic manifold and assume to have a spin structure i.e. a twofold covering map from the Spin 0 p1, nq-principal bundle P Spin 0 to the bundle of positively-oriented tangent frames P SO`o f M such that the following diagram is commutative: The existence of spin structures is related to the topology of M g . A sufficient (but not necessary) condition for the existence of a spin structure is the parallelizability of the manifold. Therefore, since any 3-dimensional orientable manifold is parallelizable, it follows by Theorem 2.4 that any 4-dimensional globally hyperbolic manifold admits a spin structure. Given a fixed spin structure, one can use the spinor representation to construct the spinor bundle Definition 2.13. Let M g be a (globally hyperbolic) spin manifold. The (complex) spinor bundle is the complex vector bundle SM g :" Spin 0 p1, nqˆρ C N where ρ : Spin 0 p1, nq Ñ AutpC N q is the complex Spin 0 p1, nq representation and N :" 2 t n`1 2 u .
The spinor bundle is enriched with the following structure: -a natural Spin 0 p1, nq-invariant indefinite fiber metrics -a Clifford multiplication, i.e. a Clifford multiplication, i.e. a fiber-preserving map Using the spin product (2.1.1), we denote as adjunction map, the complex anti-linear vector bundle isomorphism by where SpM g is the so-called cospinor bundle, i.e. the dual bundle of S p M g .
Definition 2.14. The (classical) Dirac operator D is the operator defined as the composition of the metric connection ∇ SMg on SM g , obtained as a lift of the Levi-Civita connection on TM, and the Clifford multiplication: In local coordinates and with a trivialization of the spinor bundle SM g , the Dirac operator reads as where te µ u is a local Lorentzian-orthonormal frame of TM and ε µ " gpe µ , e µ q "˘1.
Remark 2.15. Note that unlike differential forms, the definition of spinors (and cospinors) requires the choice of a spin structure and it depends on the metric of the underlying manifold.
Proposition 2.16. The classical Dirac operator D on globally hyperbolic spin manifolds M g is a symmetric hyperbolic system.
Proof. The principal symbol σ D of the Dirac operator reads as where ξ P ΓpT˚Mq, ψ P ΓpSM g q and 7 : T˚M Ñ TM is the musical isomorphism implemented by the Lorentzian metric. Therefore, Property (S) of Definition 2.5 is verified on account of (2.5), while Property (H) follows by [30, Proposition 1.1], provided that the spin product (2.1.1) was chosen with the appropriate sign.
Remark 2.17. Noticed that in the literature it is often used the canonical positive-definite scalar product ă | ą C N despite the indefinite, non-degenerate spin product (2.1.1). As a consequence, D is no longer a symmetric hyperbolic system, but γp B Bt qD does satisfies the required properties, see e.g. [43,44].

The geometric wave operator
Let V be an Hermitian vector bundle of finite rank and consider a normally hyperbolic operator P : ΓpVq Ñ ΓpVq , i.e. a 2 nd -order linear differential operator with principal symbol σ P defined by σ P pξq "´gpξ, ξq¨Id V , for every ξ P T˚M. Following [2, Remark 3.7.11], we shall reduce P to a symmetric hyperbolic system, but first we assume, without loss of generality, that M g " pM, gq is given by there exists a unique (metric) connection ∇ V on V and a unique endomorphism field c P ΓpEndpVqq such that where ∇ Σ is defined by ∇ Σ X :" ∇ V X for all X P TΣ, while b 0 P C 8 pMq and b P ΓpTΣq are given by Equation (2.9) allows us to rewrite the Cauchy problem for P : as a Cauchy problem for S : ΓpEq Ñ ΓpEq, where E is the Hermitian vector bundle E :" V ' pT˚Σ b Vq ' V, B P ΓpEndpEqq and The Cauchy problem (2.11) should be read as follows: is more or less the Weingarten map put into the TΣ slot. The curvature tensor R is that of ∇ and is by convention given for all X, Y P TM by R X,Y " r∇ X , ∇ Y s´∇ rX,Y s . As in [2, Remark 3.7.11], Conditions (S) and (H) can be easily checked. Hence S is a symmetric hyperbolic system. Remark 2.19. Notice that, while any solution u of the Cauchy problem (2.10) gives a solution Ψ to the Cauchy problem (2.11), the contrary does not hold. Indeed, the space of initial data for Ψ is "too large" and some a suitable restriction has to be imposed. For further details we refer to [2, Remark 3.7.11].

Intertwining operators
This section aims to generalize the results of Dappiaggi and Drago in [23] by constructing a geometric map between the solutions space of symmetric hyperbolic systems defined on (possibly different) vector bundles over (possibly different) globally hyperbolic manifolds. Since the construction of a vector bundle can depend in general on the metric of the underlying Lorentzian manifold, as for the case of classical Dirac operator, it became necessary first to find a path connecting different metrics. Despite the space of Lorentzian metrics on a fixed smooth manifold is not path-connected, when we restrict our attention to globally hyperbolic manifolds, we get the following result.
Lemma 3.1. Let GH M be the space of globally hyperbolic metrics on a smooth manifold M such that, for any g 0 , g 1 P GH M , M 0 " pM, g 0 q and M 1 " pM, g 1 q have the same Cauchy temporal function. Then GH M is convex.
Proof. Let g 0 , g 1 P GH M . Since M 0 and M 1 admit the same Cauchy temporal function, the there exists a isometric splitting M " RˆΣ with metric g α "´β 2 α dt 2`h tα , for α P t0, 1u. This in particular, shows that the convex linear combination g λ :" λg 1`p 1´λq g 0 for any λ P r0, 1s, is a globally hyperbolic metric.
Keeping in mind Lemma 3.1 we introduce the following setup, which we shall use through this section: Setup 3.2. For α P t0, 1u, we have the following: -GH M denotes the space of globally hyperbolic metrics on a smooth manifold M such that, for any g 0 , g 1 P GH M , M 0 " pM, g 0 q and M 1 " pM, g 1 q have the same Cauchy temporal function; -M α :" pM, g α q, where g α P GH M and g 1 ď g 0 (i.e. the set of timelike vectors for g 1 is contained in the one for g 0 ); -E α is a K-vector bundle over M α with finite rank and endowed with a nondegenerate sesquilinear fiber metric ă¨|¨ą α ; κ 1,0 : E 0 Ñ E 1 is a fiberwise linear isometry of vector bundles and, for any ̺ P C 8 pM, Rq striclty positive, we set κ ̺ 1,0 :" ̺ κ 1,0 and κ ̺ 0,1 :" ̺´1 κ 0,1 ; -S ̺ 0,1 : ΓpE 1 q Ñ ΓpE 1 q is the operator defined by S ̺ 0,1 :" κ ̺ 1,0 S 0 κ ̺ 0,1 ; -Sol pS α q denotes the space of solutions for the symmetric hyperbolic system S α over M α To construct an intertwining operator, we need a preliminary lemma.
is a symmetric hyperbolic system over M 1 .
Proof. Using the characterization of the principal symbols given as in Equation (2.2), for every Ψ 1 P ΓpE 1 q and f P C 8 pMq we thus obtain We first show that S ̺ 0,1 is a symmetric hyperbolic system. By Equation (2.2), for every Ψ λ P ΓpE λ q and f P C 8 pMq we thus obtain where Ψ 0 " κ ̺ 0,1 Ψ 1 and κ ̺ 0,1 κ ̺ 1,0 " Id. Since κ 1,0 is a fiberwise linear isometry by assumption, it follows where ă¨|¨ą 0 is a fiberwise pairing on E 0 . Using that S 0 is a symmetric hyperbolic system, it follows immediately that S ̺ 0,1 satisfies property (S) in Definition 2.5. Furthermore, since g 1 ď g 0 , any timelike covector ξ for g 1 is also a timelike covector for g 0 . Therefore S ̺ 0,1 satisfies also property (H) as well. Hence it is a symmetric hyperbolic system. To conclude our proof, it is enough to notice that a convex linear combination of Hermitian operators is a Hermitian operator, and a convex linear combination of positive operators is a positive operator. Hence also S ̺ χ,1 is a symmetric hyperbolic system. Building on Lemma 3.3, we now prove the main result of this paper.
Proof. Since, for any f 1 P ΓpE 1 q , χf 1 P Γ pc pE 1 q and p1´χqf 1 P Γ f c pE 1 q then the intertwining operator R ̺ is well defined. Indeed, for any Ψ 0 P Sol pS 0 q, it turns out that The smoothness of R ̺ Ψ 0 is a by-product of the regularity properties of the advanced and retarded Green operators. By straightforward computation, we thus obtain where we used that S 1˝G1 " Id and S ̺ χ,1˝Gχ " Id (cf. Proposition 2.9). In particular, this implies that Therefore, for any Ψ 0 P Sol pS 0 q, it holds Hence R ̺ Ψ 0 P Sol pS 1 q.
To conclude our proof, it suffices to prove that R´and R`are invertible. Indeed, by defining it follows R ̺˝R´1 ̺ " Id : Sol pS 1 q Ñ Sol pS 1 q and R´1 ̺˝R ̺ " Id : Sol pS 0 q Ñ Sol pS 0 q. To this end, we make the following ansatz: where G0 ,1 is the advanced Green operator for S ̺ 0,1 . We begin by showing that R´1 is a right inverse for RŔ´˝R´1 Since R´1˝R´and R´1˝R`have analogous computations, we can conclude.
Remark 3.5. By uniqueness of solution, R ̺ implements the following geometric map: Let be Ψ 0 P Sol pS 0 q and denote with Ψ 1,0 " κ ̺ 1,0 Ψ 0 . Then consider the operator which maps the Cauchy data Ψ 1,0 | Σ´t o the corresponding Cauchy data on Σ`by evolving it via the evolution operator of S ̺ χ,1 , which exists on account of Lemma 3.3 and Theorem 2.8. Finally define R ̺ Ψ 0 as the solution for S 1 R ̺ Ψ 0 " f 1 and Cauchy data provided by those previously obtained on Σ`. If the inhomogeneity f 1 " 0, then the evolution operator is an unitary operator from L 2 pSM| Σα q Ñ L 2 pSM| Σ β q, where Σ α , Σ β Ă M are Cauchy hypersurfaces.

Conservation of Hermitian structures
In this section, we are going to show that the intertwining operator R ̺ preserves Hermitian structures on the spaces of homogeneous solutions with spacially compact support, once that ̺ P C 8 pMq is chosen suitably. Despite our result can be formulated abstractly for a generic symmetric hyperbolic system, we believe that analyzing the conservation of symplectic forms (for waves-like fields) and Hermitian scalar products (for Dirac fields) separately is more preparatory for Section 4.

Hermitian scalar products for Dirac fields
As already underlined in Remark 2.15, the space of spinors depends on the metric of the underlying manifold M α . Therefore, an identification between spaces of sections of spinor bundles for different metrics is needed to construct an intertwining operator. This can be achieved by following [3, Section 5].
Let be λ P R, g λ P GH M and consider a family of globally hyperbolic manifolds with the same Cauchy temporal function M λ :" pM, g λ q. Let Z be the Lorentzian manifold On Z there exists a globally defined vector field which we denote as e λ :" B Bλ . For any λ, the spin structures on Z and M λ are in one-to-one correspondence: Any spin structure on Z can be restricted to a spin structure on M λ and a spin structure on M λ it can be pulled back on Z -see [3,Section 3 and 5]. Actually, the spinor bundle SM λ on each globally hyperbolic spin manifold M λ can be identified with the restriction of the spinor bundle SZ on M λ , in particular By denoting with γ Z the Clifford multiplication on SZ, the family of Clifford multiplications γ λ satisfies satisfy γ λ pvqψ " γ Z pe λ qγ Z pvqψ if n is even, (3.8) γ λ pvqpψ``ψ´q " γ Z pe λ qγ Z pvqpψ`´ψ´q if n is odd, (3.9) where in the second case ψ " ψ``ψ´P SZ| M λ ' SZ| M λ and each component ψ˘is identified with an element in S˘Z| M λ .
Lemma 3.7. Let Z be the Lorentzian manifold given by where pM, g λ q :" M λ is a family of globally hyperbolic manifolds with g λ P GH M for any λ P R, and denote with SM λ be the spinor bundle over M λ . For any p P M λ , the map defined by the parallel translation on Z along the curve λ Þ Ñ pλ, pq is a linear isometry and preserves the Clifford multiplication.
Therefore the spin product on SZ is constant along the curve λ Þ Ñ pλ, pq which implies that κ is an isometry. We conclude by showing that the Clifford multiplication is preserved. But this follows from the fact that e λ and the Clifford multiplication γ Z are parallel along λ Þ Ñ pλ, pq. Indeed, on account of the relations (3.8)-(3.9), we get so we can conclude our proof.
Let now D α be the classical Dirac operator on a global hyperbolic manifold M α . With the bundle isomorphism κ λ 1 ,λ 0 we define the intertwining operator where Sol sc pD α q is the space of homogeneous solutions with spatially compact support, i.e.
Our next task is to show that, assigned M 0 and M 1 there exists a choice of ̺ such that R ̺ preserves the positive definite Hermitian scalar product (3.11) naturally defined on Sol sc pD α q. We begin by recasting the definition and an important property of the Hermitian scalar product. Lemma 3.17). Let Σ α Ă M α be a smooth spacelike Cauchy hypersurface with its future-oriented unit normal vector field n and its induced volume element vol Σα . Then yields a positive definite Hermitian scalar product which does not depend on the choice of Σ α . Proposition 3.9. Assume the setup of Theorem 3.4 and let ̺ P C 8 pMq be such that vol Σ 0 " ̺ 2 vol Σ 1 , where vol Σα is the volume induced by the metric g α on a Cauchy hypersurface Σ Ă J´pΣ´q. Then the intertwining operator R ̺ : Sol sc pD 0 q Ñ Sol sc pD 1 q preserves the scalar products (3.11).
Proof. On account of Proposition 2.16 and Theorem 3.4, for any r ψ :" R`κ ̺ 1,0 ψ 0 and r φ :" R`κ ̺ 1,0 φ 0 , by Equation (3.4), it holds R´r ψ, R´r φ P Sol sc pD 1 q. Moreover, by Lemma 3.8, the scalar product does not depend on the choice of Σ, therefore By choosing Σ 1 Ă J`pΣ`q we have χ " 1, therefore the Hermitian scalar product reads as On account of Equation (3.5), we get r ψ, r φ P Sol sc pD ̺ χ,1 q. This implies that latter Hermitian form can be read as the scalar product p¨|¨q χ,1 on the space of solution D ̺ χ,1 , i.e.
pR´r ψ | R´r φq 1 " This because on Σ 1 , we have χ " 1 which implies that D ̺ χ"1,1 " D 1 on small tubular neighborhood of Σ 1 contained in the future of Σ`. By Lemma 3.8, we thus obtain By choosing Σ 2 " Σ, which lies in the past of Σ´, we obtain where the latter forms is defined on the solution space of the operator D 0,1 . But, on account of Lemma 3.7, κ 1,0 is an isometry of spinor bundles which preserves the Clifford multplication. Therefore it follows Plugging all together we can conclude.

Symplectic structures for geometric wave operator
Let M α be globally hyperbolic manifolds with the same Cauchy temporal function and let P α be a normally hyperbolic operator acting on section of a vector bundle V α with finite rank and with a positive definite Hermitian form ă¨|¨ą α . On account of Section 2.1.2, P α can be reduced to a symmetric hyperbolic operator S α acting on sections of the vector bundle E α " V α ' pT˚Σ b V α q ' V α , and the map ι : Sol pP α q Ñ Sol pS α q is injective. We denote with Sol Pα pS α q the subspace of Sol pS α q where ι is bijective, i.e. Sol Pα pS α q :" ιpSol pP α qq.
Proposition 3.10. Let α P t0, 1u and consider P α and S α as above. If there exists a linear isometry κ 1,0 : V 0 Ñ V 1 , then there exists an intertwining operator R ̺ :" Sol pP 0 q Ñ Sol pP 1 q which implements a non-canonical isomorphism.
Proof. Since the tangent bundle does not depend on the underline metric, then we can define Since κ 1,0 is an isometry, then also κ 1,0 enjoys the same property. Hence the operator R ̺ which makes the following diagram commutative implements the desired non-canonical isomorphism where R ̺ is given as in Theorem 3.4. where R´:" Id´G1 p1´χqpP 1´P ̺ 0,1 q and R`:" Id´Gχ χpP 1´P ̺ 0,1 q where Gχ is the advanced Green operator for P ̺ χ,1 and G1 is the retared Green operator for P 1 . Proof. We begin by noticing that the principal symbols of P ̺ χ,1 :" p1´χq P ̺ 0,1`χ P 1 it satifies σ P ̺ χ,1 pdf qu 1 " 1 2 P ̺ χ,1 pf 2 qu 1`1 2 f 2 P ̺ χ,1 pu 1 q " p1´χqσ P ̺ 0,1 pdf qu 1`χ σ P 1 pdf qu 1 "´´p1´χqg 0 pdf, df q`χg 1 pdf, df q¯u 1 , where we used that σ P ̺ 0,1 " κ ̺ 1,0 σ P 0 κ ̺ 0,1 , see e.g. Equation 3.2. By defining h :" p1´χqg 0`χ g 1 , computations analogous to the one in Section 2.1.2, shows that P ̺ χ,1 can be reduced to a symmetric hyperbolic system. This is enough to guaranteed the existence of solutions and hence Green operators for P ̺ χ,1 as well. Using analogous computations to the ones performed in the proof of Theorem 3.4 we can already conclude.
As for Section 3.1.1, let Sol sc pP α q be the space of homogeneous solutions with spacially compact support, i.e.
Our next task is to show that, assigned M 0 and M 1 there exists a choice of ̺ such that R ̺ preserves the symplectic form naturally defined on Sol sc pP α q. As for the scalar product (3.11), also the symplectic form does not depends on the choice of the Cauchy hypersurface. . Let Σ α Ă M α be a smooth spacelike Cauchy hypersurface with its future-oriented unit normal vector field n and its induced volume element vol Σα . Then yields a symplectic form which does not depend on the choice of Σ α .
Using the same arguments as in the proof of Proposition 3.9 we can conclude an analog conservation of the symplectic form.
Proposition 3.13. Assume the setup of Corollary 3.11 and let ̺ P C 8 pMq be such that vol Σ 0 " ̺ 2 vol Σ 1 , where vol Σα is the volume induced by the metric g α on a Cauchy hypersurface Σ Ă J´pΣ´q. Then the intertwining operator R ̺ preserves the symplectic form (3.12).

Applications
We conclude this paper with an application inspired by [23,32]. In loc. cit., the quantization of a free field theory 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 the canonical commutation/anticommutation 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.
As explained in details by Benini, Dappiaggi and Schenkel in [12], the space of observables for the inhomogeneous solution coincides with the one of the homogeneous solution plus an extra observable assigned to the particular solution G`f, where f is the source term and G`is the advanced Green operator for the inhomogeneous Cauchy problem. Therefore, in what follows we shall only consider the quantization of classical fields which satisfies an homogeneous Cauchy problem.

Algebra of Dirac fields
As in Section 3.1.1, let Sol sc pDq be the space of homogeneous solutions with spatially compact support of the Dirac operator endowed with the positive definite Hermitian scalar product (3.11).
Definition 4.2. We call algebra of Dirac fields the unital, complex˚-algebra A freely generated by the abstract elements Ξpψq, with ψ P Sol sc pDq, together with the following relations for all ψ, φ P Sol sc pDq and α, β P C: (i) Linearity: Ξpαψ`βφq " αΞpψq`βΞpφq

*
for every ψ 1 , φ 1 , . . . ψ k , φ k P Sol sc pDq. As always,˚is extended to all elements of T by antilinearity, thus turning T into a unital complex˚-algebra. The canonical relations are implemented taking the quotient of T by the˚-ideal I generated by Ξpψq ‚ Ξpφq`Ξpφq ‚ Ξpψq and Ξpψq ‚ Ξpφq˚`Ξpφq˚‚ Ξpψq´pψ | φq Id A (4.1) for every ψ, φ P Sol sc pDq. If follows that A " T{I is a realization of the algebra of Dirac fields.
Keeping in mind this concrete realization, we can prove the following isomorphism. Proof. Theorem 3.4 and Proposition 3.6 establish via R ̺ and R̺ an isomorphism between Sol pD α q and Sol pDαq respectively. As a by-product, R ̺ 'R̺ extends first of all to an isomorphism between the tensor algebras TpSol ' α q by linearity. Finally, on account of Proposition 3.9 R̺ " Υ 1 R ̺ Υ 0 and R ̺ preserve the Hermitian scalar product, which implies that the ideals I α are˚-isomorphic.
Before concluding this subsection, we want to make the following remark: Remark 4.4. The algebra of Dirac fields cannot be considered an algebra of observables, since observables are required to commute at spacelike separations and A does not fulfil such requirement. However, the subalgebra A obs Ă A composed by even elements, i.e. Ξpψq "´Ξpψq, which are invariant under the action of Spin 0 p1, nq (extended to A) is a good candidate as algebra of observables. For further details we refer to [25,30].

Algebras of real scalar fields
Consider a normally hyperbolic operator P acting on a real line bundle V :" RˆM. As in Section 3.1.2, let Sol sc pPq be the space of homogeneous solutions with spatially compact support endowed with the symplectic form (3.11).
Definition 4.5. We call algebra of real scalar fields the unital, complex˚-algebra A freely generated by the abstract elements Φpuq, with u P Sol sc pPq, together with the following relations for all u, v P Sol sc pPq and α, β P C: (i) Linearity: Φpαu`βvq " αΦpuq`βΦpvq (ii) Hermiticity: Φpuq˚" Φpuq (iii) Canonical commutation relations (CCRs): where x¨|¨y is the symplectic form (3.11).
A more concrete construction can be obtained by mimicking the one for the Dirac fields. First, consider the tensor C-algebra T :"´À nPN Sol sc pPq bn , ‚¯, where the generators of T are given by Id T " t1, 0 . . . u Φpuq :" t0, u, 0 . . . u for any u P Sol sc pDq and the involution˚: T Ñ T is implemented by means of the antilinear isomorphism Υ t0, . . . , u 1 b¨¨¨b u k , 0 . . . u˚" t0, . . . , u k b¨¨¨b u 1 , 0 . . . u for every ψ 1 , φ 1 , . . . ψ k , φ k P Sol sc pDq. As always,˚is extended to all elements of T by antilinearity, thus turning T into a unital complex˚-algebra. The canonical commutation relations are implemented taking the quotient of T by the˚-ideal I generated by for every u, v P Sol sc pPq. If follows that A " T{I is a realization of the algebra of real scalar fields.
Keeping in mind this concrete realization, we can prove the following isomorphism.
Theorem 4.6. Assume the setup Proposition 3.13 and let A α be the algebra of real scalar fields on M α . Then there exists a˚-isomorphism R 1,0 : Proof. Corollary 3.11 establish via R ̺ an isomorphism between Sol pP α q. As a by-product, such result extends first of all to an isomorphism between the tensor algebras TpSol sc pP α qq by linearity. Finally, by Proposition 3.13 R ̺ preserves the Hermitian scalar product, which implies that the ideals I α are˚-isomorphic.

Hadamard states
We conclude this section by studying (algebraic) states and their interplay with the intertwining operator R.
Definition 4.7. Given a complex˚-algebra A we call (algebraic) state any linear functional from A into C that is positive, i.e. ωpa˚aq ě 0 for any a P A, and normalized, i.e. ωpId A q " 1.
Let V be a vector bundle and consider a Green hyperbolic operator Q : V Ñ V. Let Sol sc pQq be the space of homogeneous solutions with spacially compact support. where I is a suitable˚-ideal which encodes CCR-or CAR-relations.
Remark 4.9. Notice that if we set Q " P : C 8 pMq Ñ C 8 pMq and we consider the ideal generated by the CCR relations (4.2), we thus obtain the algebra of real scalar fields. While if we set Q " D ' D˚: ΓpSMq ' ΓpS˚Mq Ñ ΓpSMq ' ΓpS˚Mq and we consider the ideal generated by the CAR relations (4.1), we thus obtain the algebra of Dirac fields.
Due to the natural grading on the algebra of fields A, it suffices to ω on the monomials. This gives rise to the n-points distributions ω pnq P`Γ c pVq˘1 by means of the relations ω pnq pf 1 , . . . , f n q " ω pf 1 b¨¨¨b f n q : where u j " Gf j , with j " 1, . . . , n, and G is the causal Green propagators for Q. This leads us to the following definition.
Definition 4.10. A state ω on the algebra of fields is quasifree if its n-point functions ω pnq vanish for odd n, while for even n, they are defined as where S 1 n denotes the set of ordered permutations of n elements.
Remark 4.11. It is widely accepted that, among all possible (quasifree) states, the physical ones are required to satisfy the so-called the 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 following a covariant scheme, see [46] or [48] for recent reviews. This requirement is conveniently translated in the language of microlocal analysis, in particular into a microlocal characterization of the two-points distribution of the state. Since a full characterization is out of the scope of the paper, for further details we refer to [21,[37][38][39] for scalar fields and to [25,33] for Dirac fields -see also [9,11,40,51] for gauge theory.
With the next theorem, we show that the pull-back of a quasifree state along the isomorphism R 1,0 : A 0 Ñ A 1 induced by the intertwining operator R ̺ for Q (see e.g. Theorem 4.3 or Theorem 4.6) preserves singularity structure of the two-point distribution ω p2q , i.e. it preserves the wavefront set.
Theorem 4.12. Assume the Setup 3.2 and denote with A α the algebra of fields on M α . Finally let ω α : A α Ñ C be quasifree states satisfying ω 0 " ω 1˝R1,0 : A 0 Ñ C with R 1,0 is the isomorphism induced by R ̺ . Then the bi-distributions of the associated two-point distributions ω p2q α have the same singularity structure.
Proof. Since R 1,0 is˚-isomorphism, ω 0 inherits the property of being a quasifree state from ω 1 . In particular two-point function ω Consider now the restriction of ω p2q 1 to a neighborhood N 1 of a Cauchy hypersurface Σ 1 Ă J`pΣ`q. On account of Theorem 3.4, χ " 1 and we have R ̺ | N 1 " R`κ ̺ 1,0 . In particular, ω 0 reads as The latter two-points distribution can also be read as the restriction to N of the two-point distribution ω χ of a state on the algebra of Q χ -fields (cf. Proof of Theorem 3.4), namely On account of the Theorem on the propagation of singularites, see e.g. [50], ω p2q χ has the same singularities in the whole manifold. By restricting ω χ to a neighborhood N 2 of a Cauchy hypersurface Σ 2 Ă J´pΣ´q we thus obtain ω p2q Since the composition with a smooth map does not change the wavefront set, ω p2q 0 has the same singularity structure of ω p2q 1 .
Remark 4.13. The main drawback of the definition of the intertwining operator R, used in Theorem 4.12, is the lack of any control on the action of the group of˚-automorphism induced by the isometry group of M on ω 2 . Let us remark, that the study of invariant states is a wellestablished 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: Under which conditions it is possible to perform an adiabatic limit, namely when lim χÑ1 ω 1 is 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. A partial answer is given in [23,31] for the case of scalar field theory on globally hyperbolic manifolds. In those papers, it is investigated how to relate normally hyperbolic operators which differ from a smooth potential, e.g. massive and massless wave operators.
We conclude this paper with the following corollary, which is a new proof of the existence of Hadamard states on every globally hyperbolic manifold.