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 is 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 Simone Murro and Daniele Volpe acknowledge the support of the INFN-TIFPA project "Bell".  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. 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 [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 define 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 et al. [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 acting on sections of spinors bundles Σ over a compact Riemannian manifold (Σ, h ) . By defining ( = ℝ × Σ , g = −dt 2 + h ) , we immediately obtain a globally hyperbolic manifold. On consider the following symmetric hyperbolic system where is also considered as a Hermitian vector bundle on via the pull-back along the projection ∶ → Σ . Since is essentially self-adjoint on L 2 ( Σ ) , any vector in the kernel of can be written employing the spectral calculus, namely where ( ) and dE are, respectively, the spectral measure and the spectrum of and are initial data. It follows that intertwining operators should interplay between the spectral measure of 0 and 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 CCR (resp. CAR ) the algebra of real scalar fields (resp. Dirac fields) over a globally hyperbolic spacetimes , the isomorphism defined in Proposition 3.9 (resp. Proposition 3.13) can be lift to a * -isomorphism 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 Sect. 2, we review well-known facts about symmetric hyperbolic systems. In particular, in Sects. 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 Sect. 3, we prove the main result of this paper, and we investigate the conservation of Hermitian forms. Finally, Sect. 4 is devoted to analyzing the consequence of such isomorphism in the context of algebraic quantum field theory.

Symmetric hyperbolic systems
The 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 (n + 1)-dimensional, oriented, timeoriented, smooth Lorentzian manifold ( , g) such that (i) There are no closed causal curves; (ii) For every point p, q ∈ , J + (p) ∩ J − (q) is compact; where J + ( ) (resp. J − ( ) ) denotes the set of points of that can be reached by future (resp. past) directed causal curves starting from ⊂ .

Notation 2.2
For the rest of this section, g ∶= ( , g) will always denote a globally hyperbolic manifold, and we adopt the convention that the metric g has signature (−, +, … , +).
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 is homeomorphic to × Σ and all Cauchy hypersurfaces are homeomorphic. The proof was carried out by finding a Cauchy time function, namely a continuous function t ∶ → which increases strictly on any future-directed causal curve such that each level t −1 (t 0 ) , t 0 ∈ , is a Cauchy hypersurface. In Bernal and Sànchez [17] "smoothened" the result of Geroch by introducing the notion of Cauchy temporal function.

Definition 2.3
We say that a smooth time function t ∶ → is a Cauchy temporal function if its gradient ∇t is past-directed timelike, and its level set is a smooth Cauchy hypersurface. Theorem 2.4 ([17], Theorem 1.1 and Theorem 1.2) Any globally hyperbolic manifold admits a Cauchy temporal function. In particular, it is isometric to the smooth product manifold ℝ × Σ with metric where t ∶ ℝ × Σ → ℝ is Cauchy temporal function, ∶ ℝ × Σ → ℝ is a smooth positive function and h t is a Riemannian metric on each level set of t.
Let now be a -vector bundle over a globally hyperbolic manifold g with finite rank N and endowed with a (possibly indefinite) non-degenerate sesquilinear fiber metric Definition 2.5 ([1], Definition 5.1) A linear differential operator ∶ Γ( ) → Γ( ) of first order is called a symmetric hyperbolic system over g if (S) The principal symbol ( ) ∶ p → p is Hermitian with respect to ≺ ⋅ | ⋅ ≻ for every ∈ * p and for every p ∈ ; (H) For every future-directed timelike covector ∈ * p , the bilinear form ≺ ( )⋅ | ⋅ ≻ is positive definite on p .

Remark 2.6
Notice that Definition 2.5 depends on the fiber metric ≺ ⋅ | ⋅ ≻ 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 ∶ Γ( ) → Γ( ) the principal symbol ∶ * → End( ) can be characterized by where u ∈ Γ( ) and f ∈ C ∞ ( ) . If we choose local coordinates (t, x 1 , … , x n ) on , with x i local coordinates on Σ t , and a local trivialization of , any linear differential operator ∶ Γ( ) → Γ( ) of first order reads in a point p ∈ as where the coefficients A 0 , A j , B are N × N matrices, with N being the rank of , depending smoothly on p ∈ . In these coordinates, Condition (S) in Definition 2.5 reduces to for j = 1, … , n . Condition (H) can be stated as follows: For any future-directed, timelike covector = dt + ∑ j j dx j , defines a scalar product on p .

Remark 2.10
On account of Remark 2.7, it immediately follows that the formal dual operator * is Green hyperbolic A straightforward computation shows that

Definition 2.11
Let be a Green hyperbolic operator and denote with + and − , respectively, the advanced and retarded Green operator. We call causal propagator ∶ Γ fc ( ) → Γ( ) the operator defined by ∶= + − − .
The causal propagator characterize the space of solutions to the homogeneous Cauchy problem, namely for any ∈ Γ tc ( ) , Ψ ∶= ∈ ker . The properties of the causal propagator are summarized in the following proposition. ([4], Theorem 3.5) Let be the causal propagator for a Green hyperbolic operator ∶ Γ( ) → Γ( ) . 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 g be a globally hyperbolic manifold and assume to have a spin structure i.e., a twofold covering map from the Spin 0 (1, n)-principal bundle Spin 0 to the bundle of positively oriented tangent frames SO + of such that the following diagram is commutative: The existence of spin structures is related to the topology of 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 g be a (globally hyperbolic) spin manifold. The (complex) spinor bundle is the complex vector bundle The spinor bundle is enriched with the following structure: • a natural Spin 0 (1, n)-invariant indefinite fiber metrics • a Clifford multiplication, i.e., a Clifford multiplication, i.e., a fiber-preserving map which satisfies for all p ∈ g , u, v ∈ p and , ∈ p g Using the spin product (2.1.1), we denote as adjunction map, the complex anti-linear vector bundle isomorphism by where * p g is the so-called cospinor bundle, i.e., the dual bundle of p g .
Definition 2.14 The (classical) Dirac operator is the operator defined as the composition of the metric connection ∇ g on g , obtained as a lift of the Levi-Civita connection on , and the Clifford multiplication: In local coordinates and with a trivialization of the spinor bundle g , the Dirac operator reads as where {e } is a local Lorentzian-orthonormal frame of and = g(e , e ) = ±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 on globally hyperbolic spin manifolds g is a symmetric hyperbolic system.

Proof
The principal symbol of the Dirac operator reads as where ∈ Γ( * ) , ∈ Γ( g ) and ♯ ∶ * → 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 ≺ | ≻ ℂ N despite the indefinite, non-degenerate spin product (2.1.1). As a consequence, is no longer a symmetric hyperbolic system, but ( t ) does satisfies the required properties, see e.g. [43,44].

The geometric wave operator
Let be an Hermitian vector bundle of finite rank and consider a normally hyperbolic operator ∶ Γ( ) → Γ( ) , i.e., a 2 nd -order linear differential operator with principal symbol defined by for every ∈ * . Following [2, Remark 3.7.11], we shall reduce to a symmetric hyperbolic system, but first we assume, without loss of generality, that g = ( , g) is given by see Theorem 2.4. By [6, Lemma 1.5.5], there exists a unique (metric) connection ∇ on and a unique endomorphism field c ∈ Γ(End( )) such that where ∇ Σ is defined by ∇ Σ X ∶= ∇ X for all X ∈ Σ , while b 0 ∈ C ∞ ( ) and b ∈ Γ( Σ) are given by Equation (2.9) allows us to rewrite the Cauchy problem for ∶ Γ( ) → Γ( ) where is the Hermitian vector bundle ∶= ⊕ ( * Σ ⊗ ) ⊕ , B ∈ Γ(End( )) and (2.10) The Cauchy problem (2.11) should be read as follows: t h t⌟ is more or less the Weingarten map put into the Σ slot. The curvature tensor R is that of ∇ and is by convention given for all As in [2, Remark 3.7.11], Conditions (S) and (H) can be easily checked. Hence, 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 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 [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 be the space of globally hyperbolic metrics on a smooth manifold such that, for any g 0 , g 1 ∈ GH , 0 = ( , g 0 ) and 1 = ( , g 1 ) have the same Cauchy temporal function. Then, GH is convex.
Proof Let g 0 , g 1 ∈ GH . Since 0 and 1 admit the same Cauchy temporal function, there exists a isometric splitting = ℝ × Σ with metric g = − 2 dt 2 + h t , for ∈ {0, 1} . This, in particular, shows that the convex linear combination g ∶= g 1 + (1 − )g 0 for any ∈ [0, 1] , is a globally hyperbolic metric. ◻ Keeping in mind Lemma 3.1, we introduce the following setup, which we shall use through this section: , 1} , we have the following: • GH denotes the space of globally hyperbolic metrics on a smooth manifold such that, for any g 0 , g 1 ∈ GH , 0 = ( , g 0 ) and 1 = ( , g 1 ) have the same Cauchy temporal function; • ∶= ( , g ) , where g ∈ GH and g 1 ≤ g 0 (i.e., the set of timelike vectors for g 1 is contained in the one for g 0 ); • is a -vector bundle over with finite rank and endowed with a nondegenerate sesquilinear fiber metric ≺ ⋅ | ⋅ ≻ ; • 1,0 ∶ 0 → 1 is a fiberwise linear isometry of vector bundles and, for any ∈ C ∞ ( , ℝ) strictly positive, we set 1,0 ∶= 1,0 and 0,1 ∶= −1 0,1 ; • ( ) denotes the space of solutions for the symmetric hyperbolic system over To construct an intertwining operator, we need a preliminary lemma. Proof Using the characterization of the principal symbols given as in Eq. (2.2), for every Ψ 1 ∈ Γ( 1 ) and f ∈ C ∞ ( ) , we thus obtain We first show that 0,1 is a symmetric hyperbolic system. By Eq. (2.2), for every Ψ ∈ Γ( ) and f ∈ C ∞ ( ) , we thus obtain where Ψ 0 = 0,1 Ψ 1 and 0,1 1,0 = Id . Since 1,0 is a fiberwise linear isometry by assumption, it follows where ≺ ⋅ | ⋅ ≻ 0 is a fiberwise pairing on 0 . Using that 0 is a symmetric hyperbolic system, it follows immediately that 0,1 satisfies property (S) in Definition 2.5. Furthermore, since g 1 ≤ g 0 , any timelike covector for g 1 is also a timelike covector for g 0 . Therefore, 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 ,1 is a symmetric hyperbolic system. ◻ Building on Lemma 3.3, we now prove the main result of this paper. Proof Since, for any 1 ∈ Γ( 1 ) , 1 ∈ Γ pc ( 1 ) and (1 − ) 1 ∈ Γ fc ( 1 ) then the intertwining operator is well defined. Indeed, for any Ψ 0 ∈ ( 0 ) , it turns out that The smoothness of 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 1 • − 1 = Id and ,1 • + = Id (cf. Proposition 2.9). In particular, this implies that Therefore, for any Ψ 0 ∈ ( 0 ) , it holds Hence, Ψ 0 ∈ ( 1 ). To conclude our proof, it suffices to prove that − and + are invertible. Indeed, by defining . To this end, we make the following ansatz: where + 0,1 is the advanced Green operator for 0,1 . We begin by showing that −1 − is a right inverse for − where we used ( 1 − 0,1 ) = ,1 − 0,1 together with + ( ,1 − 0,1 ) + 0,1 = + 0,1 − + Since −1 − • − and −1 + • + have analogous computations, we can conclude. ◻

Remark 3.5 By uniqueness of solution, implements the following geometric map: Let
be Ψ 0 ∈ ( 0 ) and denote with Ψ 1,0 = 1,0 Ψ 0 . Then, consider the operator which maps the Cauchy data Ψ 1,0 | Σ − to the corresponding Cauchy data on Σ + by evolving it via the evolution operator of ,1 , which exists on account of Lemma 3.3 and Theorem 2.8. Finally define Ψ 0 as the solution for 1 Ψ 0 = 1 and Cauchy data provided by those previously obtained on Σ + . If the inhomogeneity 1 = 0 , then the evolution operator is an unitary operator from L 2 ( | Σ ) → L 2 ( | Σ ) , where Σ , Σ ⊂ are Cauchy hypersurfaces.
On account of Remark 2.7, the formally adjoint operator † ∶ Γ( ) → Γ( ) is a symmetric hyperbolic system, clearly up to a sign. Therefore, the results of Theorem 3.4 applies immediately to − † , and we denote its intertwining operator by † . By denoting Υ ∶ → * , we immediately get the following result. Proof As shown in Remark 2.7, it holds * = Υ † Υ −1 , which should be reads as for every Φ ∈ ( † ) and Ψ ∈ Γ c ( ) . Therefore, Eq. (3.6) rewrite as

Conservation of Hermitian structures
In this section, we are going to show that the intertwining operator preserves Hermitian structures on the spaces of homogeneous solutions with spacially compact support, once that ∈ C ∞ ( ) 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 Sect. 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 . 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 ∈ ℝ , g ∈ GH and consider a family of globally hyperbolic manifolds with the same Cauchy temporal function ∶= ( , g ) . Let be the Lorentzian manifold On , there exists a globally defined vector field which we denote as e ∶= . For any , the spin structures on and are in one-to-one correspondence: Any spin structure on can be restricted to a spin structure on and a spin structure on it can be pulled back on -see [3,Section 3 and 5]. Actually, the spinor bundle on each globally hyperbolic spin manifold can be identified with the restriction of the spinor bundle on , in particular ≃ | if n is even, while ≃ + | ≃ − | if n is odd. Equivalently, we may identify By denoting with the Clifford multiplication on , the family of Clifford multiplications satisfies where in the second case = + + − ∈ S | ⊕ S | and each component ± is identified with an element in ± | .

Lemma 3.7 Let be the Lorentzian manifold given by
where ( , g ) ∶= is a family of globally hyperbolic manifolds with g ∈ GH for any ∈ ℝ , and denote with S be the spinor bundle over . For any p ∈ , the map if n is even, if n is odd,

defined by the parallel translation on along the curve ↦ ( , p) is a linear isometry and preserves the Clifford multiplication.
Proof Let ∈ Γ( ) for = 1, 2 be parallel transported along the curve ↦ ( , p) , i.e., ∇ e = 0 . Since the spin connection preserves the spin product, it follows that Therefore, the spin product on is constant along the curve ↦ ( , p) which implies that is an isometry. We conclude by showing that the Clifford multiplication is preserved. But this follows from the fact that e and the Clifford multiplication are parallel along ↦ ( , p) . Indeed, on account of the relations (3.8)-(3.9), we get so we can conclude our proof. ◻ Let now be the classical Dirac operator on a global hyperbolic manifold . With the bundle isomorphism 1 , 0 , we define the intertwining operator where sc ( ) is the space of homogeneous solutions with spatially compact support, i.e., Our next task is to show that assigned 0 and 1 there exists a choice of such that preserves the positive definite Hermitian scalar product (3.11) naturally defined on sc ( ) . We begin by recasting the definition and an important property of the Hermitian scalar product.
Proof On account of Proposition 2.16 and Theorem 3.4, for any ̃∶= + 1,0 0 and ̃∶ = + 1,0 0 , by Eq. (3.4), it holds −̃, −̃∈ sc ( 1 ) . Moreover, by Lemma 3.8, the scalar product does not depend on the choice of Σ , therefore, By choosing Σ � ⊂ J + (Σ + ) , we have = 1 , therefore, the Hermitian scalar product reads as On account of Eq. (3.43.5), we get ̃,̃∈ sc ( ,1 ) . This implies that latter Hermitian form can be read as the scalar product (⋅ | ⋅) ,1 on the space of solution ,1 , i.e., This because on Σ � , we have = 1 which implies that =1,1 = 1 on small tubular neighborhood of Σ � contained in the future of Σ + . By Lemma 3.8, we thus obtain By choosing Σ �� = Σ , which lies in the past of Σ − , we obtain where the latter forms is defined on the solution space of the operator 0,1 . But, on account of Lemma 3.7, 1,0 is an isometry of spinor bundles which preserves the Clifford multiplication. Therefore, it follows Plugging all together, we can conclude. ◻

Symplectic structures for geometric wave operator
Let be globally hyperbolic manifolds with the same Cauchy temporal function and let be a normally hyperbolic operator acting on section of a vector bundle with finite rank and with a positive definite Hermitian form ≺ ⋅ | ⋅ ≻ . On account of Sect. 2.1.2, can be reduced to a symmetric hyperbolic operator acting on sections of the vector bundle = ⊕ ( * Σ ⊗ ) ⊕ , and the map ∶ ( ) → ( ) is injective. We denote with ( ) the subspace of ( ) where is bijective, i.e. ( ) ∶= ( ( )).
Proof Since the tangent bundle does not depend on the underline metric, then we can define 1,0 ∶ 0 → 1 as Since 1,0 is an isometry, then also 1,0 enjoys the same property. Hence, the operator which makes the following diagram commutative implements the desired non-canonical isomorphism where is given as in Theorem 3.4. ◻ where where + is the advanced Green operator for ,1 and − 1 is the retared Green operator for 1 .

Proof
We begin by noticing that the principal symbols of ,1 ∶= (1 − ) 0,1 + 1 it satisfies where we used that 0,1 = 1,0 0 0,1 , see e.g., Eq. 3.2. By defining h ∶= (1 − )g 0 + g 1 , computations analogous to the one in Sect. 2.1.2, shows that ,1 can be reduced to a symmetric hyperbolic system. This is enough to guaranteed the existence of solutions and hence Green operators for ,1 as well. Using analogous computations to the ones performed in the proof of Theorem 3.4, we can already conclude. ◻ As for Sect. 3.1.1, let sc ( ) be the space of homogeneous solutions with spacially compact support, i.e., Our next task is to show that assigned 0 and 1 there exists a choice of such that preserves the symplectic form naturally defined on sc ( ) . As for the scalar product (3.11), also the symplectic form does not depends on the choice of the Cauchy hypersurface.

3
Lemma 3.12 ([4], Lemma 3.17) Let Σ ⊂ be a smooth spacelike Cauchy hypersurface with its future-oriented unit normal vector field and its induced volume element vol Σ . Then, yields a symplectic form which does not depend on the choice of Σ .
Using the same arguments as in the proof of Proposition 3.9, we can conclude an analog conservation of the symplectic form.

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/anti-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.
As explained in details by Benini et al. [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 + , where is the source term and + 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 Sect. 3.1.1, let sc ( ) be the space of homogeneous solutions with spatially compact support of the Dirac operator endowed with the positive definite Hermitian scalar product (3.11).
A more concrete construction can be obtained as follow. Denote with and consider the tensor ℂ-algebra ∶= � ⨁ . Notice that, on account of Remark 2.18, sc ( * ) = Υ sc ( ) . The generators of are given by for any , ∈ sc ( ) and the involution * ∶ → is implemented by means of the antilinear isomorphism Υ for every 1 , 1 , … k , k ∈ sc ( ) . As always, * is extended to all elements of by anti-linearity, thus turning into a unital complex * -algebra. The canonical relations are implemented taking the quotient of by the * -ideal ℑ generated by for every , ∈ sc ( ) . If follows that = ∕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 establish via and * an isomorphism between ( ) and ( * ) , respectively. As a by-product, ⊕ * extends first of all to an isomorphism between the tensor algebras ( ⊕ ) by linearity. Finally, on account of Proposition 3.9 * = Υ 1 Υ 0 and preserve the Hermitian scalar product, which implies that the ideals ℑ 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 does not fulfill such requirement. However, the subalgebra obs ⊂ composed by even elements, i.e., Ξ( ) = −Ξ( ) , which are invariant under the action of Spin 0 (1, n) (extended to ) 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 acting on a real line bundle ∶= ℝ × . As in Sect. 3.1.2, let sc ( ) 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 freely generated by the abstract elements Φ(u) , with u ∈ sc ( ) , together with the following relations for all u, v ∈ sc ( ) and , ∈ ℂ : where ⟨⋅ � ⋅⟩ 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 ∶= � ⨁ n∈ℕ sc ( ) ⊗ n , • � , where the generators of are given by for any u ∈ sc ( ) and the involution * ∶ → is implemented by means of the antilinear isomorphism Υ for every 1 , 1 , … k , k ∈ sc ( ) . As always, * is extended to all elements of by antilinearity, thus turning into a unital complex * -algebra. The canonical commutation relations are implemented taking the quotient of by the * -ideal ℑ generated by for every u, v ∈ sc ( ) . If follows that = ∕ℑ is a realization of the algebra of real scalar fields.
Keeping in mind this concrete realization, we can prove the following isomorphism. Proof Corollary 3.11 establish via an isomorphism between ( ) . As a by-product, such result extends first of all to an isomorphism between the tensor algebras ( sc ( )) by linearity. Finally, by Proposition 3. 13 preserves the Hermitian scalar product, which implies that the ideals ℑ are * -isomorphic. ◻

Hadamard states
We conclude this section by studying (algebraic) states and their interplay with the intertwining operator ℜ.
Let be a vector bundle and consider a Green hyperbolic operator ∶ → . Let sc ( ) be the space of homogeneous solutions with spacially compact support.

Definition 4.8 Following Sects. 4.1 and 4.2 we call algebra of fields
where ℑ is a suitable * -ideal which encodes CCR-or CAR-relations.

Remark 4.9
Notice that if we set = ∶ C ∞ ( ) → C ∞ ( ) 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 = ⊕ * ∶ Γ( ) ⊕ Γ( * ) → Γ( ) ⊕ Γ( * ) , 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 , it suffices to on the monomials. This gives rise to the n-points distributions (n) ∈ Γ c ( ) � by means of the relations where u j = f j , with j = 1, … , n , and is the causal Green propagators for . This leads us to the following definition.

Definition 4.10
A state on the algebra of fields is quasifree if its n-point functions (n) vanish for odd n, while for even n, they are defined as where S ′ 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 (2) f (2i−1) , f (2i) , conveniently translated in the language of microlocal analysis, in particular into a microlocal characterization of the two-point 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 ∶ 0 → 1 induced by the intertwining operator for (see e.g. Theorems 4.3 or 4.6) preserve singularity structure of the two-point distribution (2) , i.e., it preserves the wavefront set.
instead, invariant states can represent equilibrium states in statistical mechanics e.g., KMSstates 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 →1 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.