# Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

Nicoló Drago, Nicolas Ginoux, Simone Murro

► To cite this version:

Nicoló Drago, Nicolas Ginoux, Simone Murro. Møller operators and Hadamard states for Dirac fields with MIT boundary conditions. Documenta Mathematica, 2022, 27, pp.1643-1687. 10.48550/arXiv.2109.01375 . hal-03333715v2

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Submitted on 17 Nov 2021

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<table><tr><td>2.6</td><td>Møller operators on manifolds with timelike boundary</td><td>12</td></tr><tr><td>2.7</td><td>Conservation of positive definite Hermitian scalar products</td><td>18</td></tr><tr><td><b>3</b></td><td><b>The algebraic approach to quantum Dirac fields</b></td><td><b>20</b></td></tr><tr><td>3.1</td><td>The Dirac operator</td><td>21</td></tr><tr><td>3.2</td><td>Self-adjoint admissible boundary conditions</td><td>22</td></tr><tr><td>3.3</td><td>Linear isometry between spinor bundles</td><td>23</td></tr><tr><td>3.4</td><td>The algebra of Dirac fields with MIT boundary condition</td><td>26</td></tr><tr><td>3.5</td><td>Hadamard states</td><td>27</td></tr></table>

# 1 Introduction

The initial value problem for a symmetric hyperbolic system on a Lorentzian manifold $M$ is a classical problem which has been exhaustively studied in many contexts. If the underlying background is *globally hyperbolic*, a complete answer is known: In [3] it has been shown that the Cauchy problem is well-posed for any smooth initial data. Even if there exists a plethora of models in physics where globally hyperbolic spacetimes have been used as a background, there also exist many applications which require a manifold with non-empty boundary. Indeed, recent developments in quantum field theory focused their attention on manifolds with timelike boundary [12, 75], e.g. anti-de Sitter spacetime [28, 29] and BTZ spacetime [15]. Moreover, experimental setups for studying the Casimir effect enclose (quantum) fields between walls, which may be mathematically described by introducing timelike boundaries [32, 71]. Also moving walls in a spatial set-up correspond to a timelike boundary in the Lorentzian manifold. For the class of globally hyperbolic manifolds with timelike boundary, the Cauchy problem was investigated by the last two named authors. In particular, we showed in [49] that the Cauchy problem for any symmetric hyperbolic system coupled with an admissible boundary condition is well-posed and the unique solution propagates with at most the speed of light. As a byproduct the existence of a causal propagator is guaranteed. This operator plays a pivotal role in the algebraic approach to quantum field theory since it allows to construct an algebra of observables in a covariant manner, see e.g. [13, 47] for textbooks, [5, 43] for recent reviews and [14, 16, 25–28, 30–32, 39, 40] for some applications. In order to complete the quantization of a free field theory, it is necessary to define an (algebraic) state, i.e. a positive functional on the algebra of the observables. Clearly not any state can be considered physically meaningful and, on globally hyperbolic spacetimes with empty boundary, only those satisfying the so-called Hadamard condition are regarded as states of physical interest.

Indeed, within this setting, Hadamard states guarantee the possibility of constructing Wick polynomials following a local and covariant scheme [52, 55], granting also the finiteness of the quantum fluctuations of such Wick polynomials, see e.g. [38]. Let us remark that in globally hyperbolic stationary spacetimes with empty boundary, the ground state and any KMS state satisfy the Hadamard condition, see e.g. [65, 69]. In close analogy, in the presence of a timelike boundary a generalization of the Hadamard condition has been proposed in [74]. Once the Hadamard condition is assumed, several natural questions arise. The most important one concerns the existence of such states, a problem which was answered positively for free field theories on globally hyperbolic spacetimes with empty boundary in [45] (except linearised gravity for which the method cannot be applied – see for example [11]) by means of a spacetime deformation technique. In detail, once a globally hyperbolic manifold $(M, g)$ is assigned, the key point is to find a (ultra-)static globally hyperbolic metric $g_0$ on $M$ has well as a Lorentzian metric $g_\chi$ interpolating between $g$ and $g_0$ which is globally hyperbolic. This is not an easy task, since the convex combination of two given globally hyperbolic metric is not in general globally hyperbolic. If the boundary is not empty, the situation gets worse. This is first due to the need for a boundary condition for Cauchy problem. Furthermore, if the interpolating metric is not explicit, then it is not clear how to construct that
boundary condition. Hence the arguments used in [45] cannot be applied in a straightforward
manner and a new proof has to be thought out.

The aim of this paper is to provide a geometric proof of the existence of Hadamard states for Dirac field with a boundary condition dubbed *MIT boundary condition*. Let us recall that the MIT boundary condition is a local boundary condition which was introduced for the first time in [21] in order to reproduce the confinement of quark in a finite region of space: “Dirac waves” are indeed reflected on the boundary, see also [20,51] for the description of hadronic states, like baryons and mesons. The MIT boundary condition has been used more recently for many other applications, like the computation of the Casimir energy in a three-dimensional rectangular box [41, 42, 71] in order to construct an integral representation for the Dirac propagator in Kerr Black Hole Geometry and finally also in [56] to prove the asymptotic completeness for linear massive Dirac fields on the Schwarzschild Anti-de Sitter spacetime. A summary of our main result is the following:

**Theorem 1.1.** Let $(M, g := -\beta^2 dt^2 + h(t))$ be a globally hyperbolic spin spacetime with timelike boundary and let $D$ be the Dirac operator coupled with MIT boundary condition —cf. Equation (3.3). If for any $u \in \text{Sol}_{\text{MIT}}(D)$, the b-wave front set $WF_b(u)$ is the union of maximally extended generalized broken bicharacteristics, then there exists a state for the algebra of Dirac fields with MIT boundary conditions that satisfies the Hadamard condition as per Definition 3.15.

**Remark 1.2.** The requirement in Theorem 1.1 is also known as “propagation of singularity theorem” and it has been used for scalar wave equation in many different settings e.g. [9, 29, 46, 57, 60, 61, 72, 73]. We expect a similar result to hold since the propagation of singularity for Dirac operators reduces to the propagation of singularity for scalar wave operator. This is because any element in the kernel of the Dirac operator is also in the kernel of the spinorial wave operator, whose principal symbol is equal to the principal symbol of the scalar wave equation times an identity matrix. What is left to check is how the boundary condition for the Dirac operator and the spinorial wave equation are related. In globally hyperbolic asymptotically anti-de Sitter spacetimes $^1$, Dappiaggi and Marta in [29] proved the propagation of singularity for the scalar wave equation for a very large class of boundary conditions, which contains all self-adjoint boundary conditions. For these reasons, we expect that the Dirac operator coupled with MIT boundary condition should also enjoy a propagation of singularity in this class of spacetimes.

Our strategy to prove the existence of Hadamard states is as follows. In Section 2.5 we introduce the class of symmetric weakly-hyperbolic operators (cf. Definition 2.14) extending that of symmetric hyperbolic ones. We show that the Cauchy problem is well-posed for that generalized category of operators (cf. Theorem 2.19). Then in Section 2.6 we construct a Møller operator, i.e. a geometric map which compares the space of solutions of two given symmetric weakly-hyperbolic systems coupled with admissible boundary conditions on (possibly different) globally hyperbolic manifolds with timelike boundary (cf. Theorem 2.27). In particular Section 2.7, we show that this geometric map can be constructed to preserve the natural scalar product defined on the space of solution (cf. Proposition 2.33). In Section 3 we specialize ourself to the case of Dirac operators: after introducing the classical Dirac operator and the MIT boundary conditions 3.1 and 3.2 respectively, we construct an isomorphism between spinor bundles defined on different Lorentzian manifolds, see Section 3.3. In Section 3.4 we construct the algebras of Dirac fields and we promote the unitary map between the spaces of solutions of the Dirac equation to a $*$-isomorphism between the algebras of Dirac fields (cf. Theorem 3.11). Finally in Section 3.5 we discuss and prove the existence of Hadamard states for Dirac fields with MIT boundary conditions.

<footnote>$^1$A globally hyperbolic manifold $(M, g)$ is called asymptotically anti-de Sitter spacetime if it holds the following: (1) for any boundary function $x$ the metric $\hat{g} = x^2 g$ extends smoothly to a Lorentzian metric on $M$; (2) the pullback $\iota_{\partial M} \hat{g}$ is a smooth Lorentzian metric on $\partial M$; (3) $\hat{g}^\sharp(dx, dx) = 1$ on $\partial M$.</footnote>
Acknowledgements

We would like to thank Matteo Capoferri, Claudio Dappiaggi, Christian Gérard, Valter Moretti, Miguel Sánchez, Daniele Volpe and Michał Wrochna for helpful discussions related to the topic of this paper.

**Funding** S.M. is supported by the DFG research grant MU 4559/1-1 “Hadamard States in Linearized Quantum Gravity”.

Notation and convention

- The symbol $\mathbb{K}$ denotes one of the elements of the set $\{\mathbb{R}, \mathbb{C}\}$.

- $M := (M, g)$ is a globally hyperbolic manifold with timelike boundary $\partial M$ and we adopt the convention that $g$ has the signature $(-, +, \dots, +)$. If $g$ is a Lorentzian metric such that $(M, g)$ is globally hyperbolic, then we shall write $g \in \mathcal{G}\mathcal{H}_M$.

- For two Lorentzian metrics $g, g', g \le g'$ means that any causal tangent vector for $g$ is causal for $g'$ or equivalently $J_{g'} \subset J_g$.

- $t: M \to \mathbb{R}$ is a Cauchy temporal function and $M_T := t^{-1}(t_0, t_1)$ is a time strip.

- $n$ is the outward unit normal vector to $\partial M$.

- $b: TM \to T^*M$ and $\sharp: T^*M \to TM$ are the musical isomorphisms.

- $E$ is a $\mathbb{K}$-vector bundle over $M$ with $N$-dimensional fibers, denoted by $E_p$ for $p \in M$, and endowed with a Hermitian fiber metric $\langle \cdot, \cdot \rangle_{p}$.

- $\Gamma_c(E), \Gamma_{sc}(E) \text{ resp. } \Gamma(E)$ denote the spaces of compactly supported, spacelike compactly supported resp. smooth sections of $E$.

- $S$ is a symmetric weakly-hyperbolic system of constant characteristic coupled with principal symbol denoted by $\sigma_S$ and $B$ is admissible boundary space for $S$.

- When $M$ is a Lorentzian spin manifold, we denote with $\overline{SM}$ the spinor bundle over $M$ and with $D$ the classical Dirac operator.

2 Møller operators for symmetric weakly-hyperbolic systems

The aim of this section is to construct a geometric map, named Møller operator, to compare the
solution spaces of two symmetric weakly-hyperbolic operators coupled with admissible bound-
ary conditions on possibly different (though related) globally hyperbolic manifolds with timelike
boundary. To this end, we shall first recall the theory of symmetric hyperbolic systems on glob-
ally hyperbolic manifolds with timelike boundary. Then, after showing the well-posedness of the
Cauchy problem for weakly hyperbolic systems, we shall construct a family of Møller operators
depending on the choice of an arbitrary smooth function $f$. By setting suitably such a function,
we shall show that the resulting Møller operator is actually a unitary map between the spaces of
initial data endowed with a naturally defined positive scalar product. Our goal is achieved with
the help of [49, 63].
## 2.1 Globally hyperbolic manifolds

Let $M$ be a connected oriented smooth manifold with boundary. We assume $M$ to be endowed with a smooth Lorentzian metric $g$ for which $M$ becomes time-oriented. Here and in the following we shall assume that the boundary is timelike, i.e. the pullback of $g$ with respect to the natural inclusion $ι: ∂M → M$ defines a Lorentzian metric $ι^*g$ on the boundary. In the class of Lorentzian manifolds with timelike boundary, those called globally hyperbolic provide a suitable background where to analyze the Cauchy problem for hyperbolic operators.

**Definition 2.1.** [1, Definition 2.14] A globally hyperbolic manifold with timelike boundary is an $(n+1)$-dimensional, oriented, time-oriented, smooth Lorentzian manifold $M$ with timelike boundary $∂M$ such that

(i) $M$ is causal, i.e. there are no closed causal curves;

(ii) for all points $p, q ∈ M$, the subset $J^+(p) ∩ J^-(q)$ of $M$ is compact, where $J^+(p)$ (resp. $J^-(p)$) denotes the causal future (resp. past) of $p$ (resp. $q$) in $M$.

**Remark 2.2.** In case of an empty boundary, this definition agrees with the standard one, see e.g. [10, Section 3.2] or [6, Section 1.3].

Recently, Aké, Flores and Sánchez gave a characterization of globally hyperbolic manifolds with timelike boundary:

**Theorem 2.3** ([1], Theorem 1.1). Any globally hyperbolic manifold with timelike boundary admits a Cauchy temporal function $t: M → R$ with gradient tangent to $∂M$. This implies that $M$ splits into $R × Σ$ with metric

$$g = -β^2 dt^2 ⊕ h(t),$$

where $β: R × Σ → R$ is a smooth positive function, $h(t)$ is a Riemannian metric on each slice $Σ_t := \{t\} × Σ$ varying smoothly with $t$, and these slices are spacelike Cauchy hypersurfaces with boundary $∂Σ_t := \{t\} × ∂Σ$, namely achronal sets intersected exactly once by every inextensible timelike curve.

## 2.2 Symmetric hyperbolic systems of constant characteristic

Let now $E → M$ be a Hermitian vector bundle over a globally hyperbolic manifold with timelike boundary $M$, namely a $K$-vector bundle with finite rank $N$ endowed with a positive definite Riemannian or Hermitian fiber metric $<·|·>_{p}: E_p × E_p → K$.

**Definition 2.4.** A linear differential operator $S: Γ(E) → Γ(E)$ of first order is called a *symmetric hyperbolic system* over $M$ if

(S) The principal symbol $σ_S(ξ): E_p → E_p$ is Hermitian with respect to $<·|·>_{p}$ for every $ξ ∈ T_p^*M$ and for every $p ∈ M$;

(H) For every future-directed timelike covector $τ ∈ T_p^*M$, the bilinear form $<σ_S(τ) |·>_{p}$ is positive definite on $E_p$ for every $p ∈ M$.

Furthermore, we say that $S$ is *of constant characteristic* if $dim ker σ_S(n^p)$ is constant on the boundary. In particular, if $σ_S(n^p)$ has maximal rank equal to $rk(E) = N$ everywhere on $∂M$ we say that $S$ is *nowhere characteristic*.

**Remark 2.5.** Notice that, if a system $S$ is hyperbolic with respect to a metric $g$ then it is also hyperbolic with respect to any metric in the conformal class of $g$. Indeed, conformal changes preserve each type of covector. Furthermore, Condition (H) implies that for any spacelike covector $ξ ∈ T_p^*M$ such that $τ := dt + ξ$ is timelike future-directed,

$$<σ_S(dt) · |·>_{p} + <σ_S(ξ) · |·>_{p} = <σ_S(dt + ξ) · |·>_{p} > 0$$

Therefore, a symmetric system is hyperbolic if and only if:
(H') For every spacelike covector $\xi \in T_p^*M$ such that $dt + \xi$ is a future-directed timelike covector, the bilinear form verifies

$$ \prec \sigma_S(\xi) \cdot |\cdot \succ_p > - \prec \sigma_S(dt) \cdot |\cdot \succ_p . $$

## 2.3 Admissible boundary conditions

In order to discuss the Cauchy problem for a symmetric hyperbolic system, we have to impose suitable boundary conditions, depending of course if we want to solve the forward or the backward Cauchy problem. We begin by fixing a Cauchy surface $\Sigma_0 := t^{-1}(\{0\})$ where we shall assign the initial data. In this paper we shall focus on a class introduced by Friedrichs and Lax-Phillips respectively in [44, 58], dubbed admissible boundary conditions.

**Definition 2.6.** A smooth linear bundle map $\pi_{B_+}: E_{|_{\partial M}} \to E_{|_{\partial M}}$ is said to be a *future admissible boundary condition* for a first-order Friedrichs system S if

(i-f) the pointwise kernel $B_+$ of $\pi_{B_+}$ is a smooth subbundle of $E_{|_{\partial M}}$;

(ii-f) the quadratic form $\Psi \mapsto \prec \sigma_S(n^b)\Psi | \Psi \succ_p$ is positive semi-definite on $B_+$;

(iii-f) the rank of $B_+$ is equal to the number of pointwise non-negative eigenvalues of $\sigma_S(n^b)$ counting multiplicity.

Similarly we say that $\pi_{B_-}: E_{|_{\partial M}} \to E_{|_{\partial M}}$ is *past admissible* if

(i-p) the pointwise kernel $B_-$ of $\pi_{B_-}$ is a smooth subbundle of $E_{|_{\partial M}}$;

(ii-p) the quadratic form $\Psi \mapsto \prec \sigma_S(n^b)\Psi | \Psi \succ_p$ is negative semi-definite on $B_-$;

(iii-p) the rank of $B_-$ is equal to the number of pointwise non-positive eigenvalues of $\sigma_S(n^b)$ counting multiplicity.

The pair $B = (B_+, B_-)$ is called the *admissible boundary space* or *admissible boundary condition* for S.

**Remark 2.7.** The role of $B_+$ and $B_-$ will become apparent when looking for energy estimates for symmetric hyperbolic S. It turns out that $B_+$ (resp. $B_-$) is only needed in the future (resp. past) of the chosen Cauchy hypersurface $\Sigma_0$.

Conditions (ii-f) and (ii-p) are equivalent to require that the boundary conditions are *maximal* with respect to properties (iii-f) and (iii-p) respectively, namely no smooth vector subbundles $(B')_{\pm}$ of E exist that properly contains $B_{\pm}$ and such that for all $\Phi' \in (B')_+$ and $\Phi'' \in (B')_-$

$$ \prec \sigma_S(n^b)\Phi' | \Phi' \succ \ge 0 \quad \prec \sigma_S(n^b)\Phi'' | \Phi'' \succ \le 0 $$

holds. For further details we refer to [49, Section 2.2].

With the next lemma, we shall see that admissible boundary conditions are “stable” under conformal transformations, namely if B is a future/past admissible boundary space for a system on a globally hyperbolic manifold $(M, g)$, then it is also future/past admissible for the same system on $(M, \Omega^2 g)$, where $\Omega$ is a positive smooth function on M.

**Lemma 2.8.** Let $(M, g)$ be a globally hyperbolic spacetime with timelike boundary and let $B_{\pm}$ be a future/past admissible boundary space for a hyperbolic Friedrichs system of constant characteristic S. Then $B_{\pm}$ is future/past admissible w.r.t $g$ if and only if it is future/past admissible w.r.t. $\Omega^2 g$, for any positive $\Omega \in C^{\infty}(M)$.

*Proof.* We only prove the case of a future admissible boundary condition, since the other case is analogous. Let denote with $n$ and $\tilde{n}$ the normal vector w.r.t $g$ and $\Omega^2 g$. Since $\tilde{n} = \Omega^{-1}n$, we get $\sigma_S(n) = \Omega\sigma_S(\tilde{n})$. This guarantees conditions(i-f)-(ii-f) in Definition 2.6 to be satisfied. $\square$
smooth spacelike Cauchy hypersurface in M. Then, for every $f \in \Gamma_c(E)$ and $h \in \Gamma_c(E|_{\Sigma_{t_0}})$ satisfying the compatibility conditions (2.2) up to any order, there exists a unique $\Psi \in \Gamma(E)$ satisfying the Cauchy problem

$$ 
\left\{
\begin{array}{l}
S\Psi = f \\
\Psi|_{\Sigma_{t_0}} = h \\
\Psi|_{\partial M \cap J^+(\Sigma_{t_0})} \in B_+ \\
\Psi|_{\partial M \cap J^-(\Sigma_{t_0})} \in B_-
\end{array}
\right.
\qquad (2.3) $$

and the map $(f, h) \mapsto \Psi$ sending a pair $(f, h) \in \Gamma_c(E) \times \Gamma_c(E|_{\Sigma_{t_0}})$ to the solution $\Psi \in \Gamma_{sc}(E)$ of (2.3) is continuous.

The assignment $U_{S,t}: D(U_{S,t}) \subset \Gamma_c(E|_{\Sigma_t}) \ni h \to U_{S,t}h := \Psi \in \Gamma_{sc}(E)$ of a (unique) solution $\Psi$ to any smooth initial data $h \in D(U_{S,t})$ is known as a Cauchy evolution operator —here $D(U_{S,t})$ is made by sections $h \in \Gamma_c(E|_{\Sigma_t})$ fulfilling the compatibility conditions (2.2) with $f = 0$. For later convenience we shall denote by $\rho_t: \Gamma(E) \to \Gamma(E|_{\Sigma_t})$ the restriction map for smooth sections: Notice that, $\rho_t$ is a right-inverse for $U_{S,t}$. As shown in [16–19], on globally hyperbolic manifolds with empty boundary and compact Cauchy hypersurfaces, the evolution operator can be realized as a Fourier integral operator. As a matter of fact, the Fourier integral representation of the propagator contains the information on how singularities propagates in the manifold. As we shall see in Section 3.5, this is of fundamental importance in proving the existence of Hadamard states for a free quantum field theory on a curved spacetime.

We conclude this section with the following result:

**Corollary 2.13.** Let M be a globally hyperbolic manifold with timelike boundary and let B be an admissible boundary space for a symmetric hyperbolic system of constant characteristic S. Then the Cauchy problem for S on (M, g) is well-posed if and only if it is well-posed on (M, $\Omega^2g$) for any positive $\Omega \in C^\infty(M)$.

*Proof.* Our claim follows immediately by Remark 2.5 and Lemma 2.8. $\square$

## 2.5 Symmetric weakly-hyperbolic systems

We conclude this section by showing that the Cauchy problem for a symmetric system S is well-posed also if we assume that the principal symbol $\sigma_S(\xi)$ acts pointwise in a positive definite way only for a subset of future-directed timelike covectors $\xi$. We begin with the following definition.

**Definition 2.14.** A symmetric system of constant characteristic S over M is weakly-hyperbolic if there exists a positive smooth function $C \in C^\infty(M)$

(gh) The metric $g_C := -\beta^2 dt^2 \oplus C^2h(t)$ is globally hyperbolic, where $t$ is a Cauchy temporal function for $g$;

(wH) For any future-directed timelike covector $\tau$ of the form $\tau = dt + \xi \in T_p^*M$ it holds

$$ < \sigma_S(dt + C\xi) \cdot | \cdot >_p > 0. $$

## Remarks 2.15.

1. The idea behind the Definition 2.14 is to ‘shrink’ the light cone of the dual metric $g^\sharp$ in the cotangent bundle, so that the condition (H) in Definition 2.4 has to be checked for a smaller class of future-directed timelike covectors (cf. Figure 1). Mind that, in the cotangent bundle, the causal future/past of $g_C$ is not allowed to shrink too much along any $\Sigma_t$. Clearly any symmetric hyperbolic system is a symmetric weakly-hyperbolic system, just take $C = 1$ in Definition 2.14 (cf. Remark 2.5).
<figure><img src="image_2.png" /><figcaption>Figure 1: The future light cones of $g^\sharp$ and $g_C^\sharp$ in the cotangent bundle $T^*M$.</figcaption></figure>

2. Assuming the quadratic form $\prec \sigma_S(dt) \cdot | \cdot \succ$ to be pointwise positive definite, there exists at each $p \in M$ an $C(p) > 0$ such that $\prec \sigma_S(\xi) \cdot | \cdot \succ$ is positive definite for every $\xi \in J_{g_C}^+(p)$, where $J_{g_C}^+(p) \subset T_p^*M$ is the causal future in $T_p^*M$ w.r.t. the metric $g_C := -\beta^2 dt^2 \oplus C(p)^2 h(t)$ defined on $T_pM$, which is hence only defined at $p$.

The following lemma shows that symmetric weakly-hyperbolic systems are not so far from
symmetric hyperbolic systems.

**Lemma 2.16.** Let $S$ be any symmetric weakly-hyperbolic system on a globally hyperbolic manifold $(M, g) = (\mathbb{R} \times \Sigma, -\beta^2 dt^2 \oplus h(t))$ with or without any timelike boundary. Let $C \in C^\infty(\mathbb{R}, (0, \infty))$ be a function depending only on time which satisfies (wH) in Definition 2.14. Then $(M, g_C) := (\mathbb{R} \times \Sigma, -\beta^2 dt^2 \oplus C^2 h(t))$ is globally hyperbolic and $S$ is symmetric hyperbolic on $(M, g_C)$.

*Proof.* Let $p \in M$ and let $\xi \in T_p^*M$ be $g_C$-timelike, that is, $g_C^\sharp(\xi, \xi) < 0$ where $g_C^\sharp = -\beta^{-2}\partial_t^{\otimes 2} \oplus C^{-2}h(t)^\sharp$. Then there exists unique $\lambda > 0$ and $\tilde{\xi} \in T_{\pi_\Sigma(p)}^*\Sigma$ such that $\xi = \lambda \cdot (dt + C\tilde{\xi})$ —here $\pi_\Sigma: M = \mathbb{R} \times \Sigma \to \Sigma$ is the standard projection. Condition $g_C^\sharp(\xi, \xi) < 0$ is then equivalent to $g^\sharp(dt + \tilde{\xi}, dt + \tilde{\xi}) = g_C^\sharp(dt + C\tilde{\xi}, dt + C\tilde{\xi}) < 0$, that is, to $dt + \tilde{\xi}$ being $g$-timelike. Then condition (wH) implies that $\sigma_S(dt + C\tilde{\xi}) = \lambda^{-1}\sigma_S(\xi)$ is positive definite. This shows that $S$ is symmetric hyperbolic on $(M, g_C)$. Therefore, only the global hyperbolicity of $g_C$ on $M$ remains to be proven. For this, it suffices to show that $\beta^{-2}g_C = -dt^2 \oplus \beta^{-2}C^2h(t)$ is globally hyperbolic when restricted to any subset of the form $(a, b) \times \Sigma$, with real $a < b$. But since for all such $a, b$ there exists a positive constant $C_0$ such that $C(t) \ge C_0 > 0$ for all $t \in [a, b]$, we have $\beta^{-2}g_C \le \beta^{-2}g_{C_0}$ on $[a, b] \times \Sigma$, where $g_{C_0} := -\beta^2 dt^2 \oplus C_0^2 h(t)$. Therefore, it suffices to show that $\beta^{-2}g_{C_0} = -dt^2 \oplus \beta^{-2}C_0^2 h(t)$ is globally hyperbolic on $(a, b) \times \Sigma$. But fixing any $t_0 \in (a, b)$ and any inextensible $\beta^{-2}g_{C_0}$-timelike curve (which is $C^0$ and piecewise $C^1$) $\gamma = (\gamma_0, \tilde{\gamma})$ in $(a, b) \times \Sigma$, the curve $\tilde{\gamma} := (C_0^{-1}\gamma_0, \tilde{\gamma})$ is $\beta^{-2}g$-timelike and still inextensible, therefore it meets $\{t_0\} \times \Sigma$ exactly once. This shows $\beta^{-2}g_{C_0}$ and therefore $\beta^{-2}g_C$ (hence $g_C$) to be globally hyperbolic on $(a, b) \times \Sigma$. This finishes the proof. $\square$

**Example 2.17.** Let $M = \mathbb{R} \times \Sigma$ be a globally hyperbolic manifolds and let $\mathbb{R}$ be trivial line bundle over $M$. Then any future directed timelike vector field $X \in \Gamma(\mathrm{TM})$ defines an operator $S := \nabla_X$ which is a symmetric weakly-hyperbolic system if and only if the projection $g(X, \partial_t)^{-1} X_{\Sigma_t}$ is bounded along $\Sigma_t$, where $X_{\Sigma_t}$ denotes the projection of $X$ on $\Sigma_t$. This applies in particular for $X = \partial_t + v$, $v \in \Gamma(\Sigma)$, the resulting transport equation being known with the name of Vlasov equation once applied in kinetic theory.

The definition 2.6 of admissible boundary condition can be straightforwardly generalized for a symmetric weakly-hyperbolic system $S$. The resulting connection with standard hyperbolic systems is described by the following lemma.
**Lemma 2.18.** Let B be a future/past admissible boundary space for a symmetric weakly-hyperbolic system S over a globally hyperbolic manifold $(M, g)$. Then B is future/past admissible for S over $(M, g_C)$. Furthermore, if S is of constant characteristic on $(M, g)$ then it is also of constant characteristic of $(M, g_C)$.

*Proof.* To verify our claim is it enough to notice that the unit normal vectors to $\partial M$ are the same up to a strictly positive smooth function. This is due to the choice of a Cauchy temporal function whose gradient is tangent to the boundary. □

Combining Lemmas 2.16 and 2.18, we can conclude that the Cauchy problem for a symmetric weakly-hyperbolic system S is well-posed. Indeed, these two lemmas guarantee that any smooth solution propagates with at most speed of light (w.r.t. $g_C$). Therefore, the Cauchy problem can be equivalently reformulated in terms of a Cauchy problem for a symmetric positive system with $\sigma_S(dt) > 0$ in a globally hyperbolic manifold with compact Cauchy surfaces. We summarize our results in the following theorem and we leave the details to the reader.

**Theorem 2.19** (Smooth solutions for symmetric weakly-hyperbolic systems). Let M be a globally hyperbolic spacetime with timelike boundary and let S be a symmetric weakly-hyperbolic system of constant characteristic. Assume $\pi_{B_+}, \pi_{B_-}$ to be future and past admissible boundary conditions for S. Let $\Sigma_{t_0}$ be any smooth spacelike Cauchy hypersurface in M. Then, for every $f \in \Gamma_c(E)$ and $h \in \Gamma_c(E|_{\Sigma_{t_0}})$ satisfying the compatibility conditions (2.2) up to any order, there exists a unique $\Psi \in \Gamma(E)$ satisfying the Cauchy problem

$$ egin{cases} S\Psi = f \\ \Psi|_{\Sigma_{t_0}} = h \\ \Psi|_{\partial M \cap J^+(\Sigma_{t_0})} \in B_+ \\ \Psi|_{\partial M \cap J^-(\Sigma_{t_0})} \in B_- \end{cases} \qquad (2.4) $$

and the map $(f, h) \mapsto \Psi$ sending a pair $(f, h) \in \Gamma_c(E) \times \Gamma_c(E|_{\partial M})$ to the solution $\Psi \in \Gamma_{sc}(E)$ of (2.4), is continuous.

As usual, as a byproduct of the well-posedness of the Cauchy problem, we get the existence of Green operators.

**Proposition 2.20.** A symmetric weakly-hyperbolic system S of constant characteristic on a globally hyperbolic manifold with timelike boundary coupled with an admissible boundary condition $B = (B_+, B_-)$ is Green-hyperbolic, i.e., there exist linear maps, called advanced/retarded Green operator respectively, $G^\pm: \Gamma_c(E) \to \Gamma_{sc,B_\pm}(E)$ satisfying

(i) $S \circ G^\pm f = f$ for all $f \in \Gamma_c(E)$ and $G^\pm \circ Sf = f$ for all $f \in \Gamma_{c,B_\pm}(E)$;

(ii) $\text{supp}(G^\pm f) \subset J_{gC}^\pm(\text{supp} f)$ for all $f \in \Gamma_c(E)$,

where $J_{gC}^\pm$ denote the causal future (+) and past (-) w.r.t. $g_C$ and $\Gamma_{\sharp,B_\pm}(E) \subset \Gamma_{\sharp}(E)$, $\sharp \in \{sc, c\}$ denotes the space of smooth sections on E (with $\sharp$ support property) which fulfil the $B_\pm$-boundary condition.

Moreover, let $G := G^+ - G^-$: $\Gamma_c(E) \to \Gamma_{sc,B_+ + B_-}(E)$ be the causal propagator associated with S and B. Then the following sequence is a complex

$$ 0 \to \Gamma_{c,B_+ \cap B_-}(E) \xrightarrow{S} \Gamma_c(E) \xrightarrow{G} \Gamma_{sc,B_+ + B_-}(E) \xrightarrow{S} \Gamma_{sc}(E) \to 0 $$

which satisfies $\ker(S|_{\Gamma_{c,B_+ \cap B_-}(E)}) = \{0\}$, $\ker(G) = S\Gamma_{c,B_+ \cap B_-}(E)$ and $S\Gamma_{sc,B_+ + B_-}(E) = \Gamma_{sc}(E)$. Moreover, if B is self-adjoint, i.e. $B_+ = B_-$, then $\ker(S|_{\Gamma_{sc,B_+}(E)}) = G\Gamma_c(E)$ and $S: \Gamma_{sc,B_+}(E) \to \Gamma_{sc}(E)$ is surjective, so that the complex is exact everywhere. In that case, the solution space $\text{Sol}_{sc,B}(S) := \Gamma_{sc,B_+}(E) \cap \ker(S)$ is characterized as

$$ \text{Sol}_{sc,B}(S) = G\Gamma_c(E) \simeq \Gamma_c(E)/S\Gamma_{c,B_+}(E). \qquad (2.5) $$
Proof. Properties (i) and (ii) are satisfied by definition of $G^{\pm}$. As a straightforward consequence, $GS = 0$ on $\Gamma_{c,B_+ \cap B_-}(E)$ and $SG = 0$ on $\Gamma_c(E)$, therefore the sequence is a complex. Note that, since $(G^{\pm}u)|_{\partial M} \in B_{\pm}$, we have $(Gu)|_{\partial M} \in \Gamma_{sc,B_+ + B_-}(E)$ but beware that there is no reason why $B_+ + B_- = E$ in general.

The injectivity of $S|_{\Gamma_{c,B_{\pm}}(E)}$ immediately follows from property (i) since $Su = 0$ for a $u \in \Gamma_{c,B_{\pm}}(E)$ yields $u = G^{\pm}Su = 0$. As a consequence, $S|_{\Gamma_{c,B_+ \cap B_-}(E)}$ is injective.

To show that $\ker(G) \subset S(\Gamma_{c,B_+ \cap B_-}(E))$, let $u \in \Gamma_c(E)$ with $Gu = 0$. Then $G^+u = -G^-u$, so that $\text{supp} G^+u \subset J_{gC}^+(\text{supp} u) \cap J_{gC}^-(\text{supp} u)$ must be compact by property (ii). Moreover, because $(G^{\pm}u)|_{\partial M} \in B_{\pm}$, we have $G^+u \in \Gamma_{c,B_+ \cap B_-}(E)$. Therefore $G^+u \in \Gamma_{c,B_+ \cap B_-}(E)$ and satisfies $SG^+u = u$ by property (i), from which $u \in S(\Gamma_{c,B_+ \cap B_-}(E))$ follows.

From now on let us assume $B_+ = B_-$ and prove that $\ker(S|_{\Gamma_{sc,B_+}(E)}) \subset G(\Gamma_c(E))$. Let $u \in \Gamma_{sc,B_+}(E)$ be such that $Su = 0$. By definition, there exists a compact subset $K$ of $M$ such that $\text{supp} u \subset J_{gC}^+(K) \cup J_{gC}^-(K)$. Up to possibly enlarging $K$, we may assume that $\text{supp} u \subset I_{gC}^+(K) \cup I_{gC}^-(K)$, where $I_{gC}^+$ and $I_{gC}^-$ denote the chronological future and past w.r.t. $gC$ respectively. Let $\{\chi_+, \chi_-\}$ be a partition of unity subordinated to the open covering $\{I_{gC}^+(K), I_{gC}^-(K)\}$ of $I_{gC}^+(K) \cup I_{gC}^-(K)$. Let $u_{\pm} := \chi_{\pm}u$. Then $u = u_+ + u_-$, where each $u_{\pm}$ is smooth with supp $u_{\pm} \subset I_{gC}^{\pm}(K)$. Furthermore, because $u_{\pm}$ is obtained by pointwise multiplication of $u$ by a real number, we have $u_{\pm}|_{\partial M} \in B_+$. Let $v := Su_+(=-Su_-)$. Then $v$ is smooth with support contained in $J_{gC}^+(K) \cap J_{gC}^-(K)$, therefore $\text{supp} v$ is compact. We would like to check that $Gv = u$ in the weak – and therefore also in the strong – sense. For that, we need the following fact: if $(G^{\pm})^*$ denotes the formal adjoint of $G^{\pm}$, then actually

$$(G^{\pm})^* = G_{\mp}^{\mp}$$

holds, where $G_{\mp}^{+}$ and $G_{\mp}^{-}$ are the Green operators for $S^{\dagger}$ with boundary condition $B^{\dagger} := (B_{+}^{\dagger}, B_{-}^{\dagger})$. Recall that, if $B$ is a future/past admissible boundary condition for $S$, then $B^{\dagger}$ is a future/past admissible boundary condition for $S^{\dagger}$. Moreover, $S^{\dagger}$ becomes a symmetric weakly hyperbolic system on $M$ with reversed time orientation, in particular $S^{\dagger}$ has unique advanced and retarded Green operators as well.

To check that $(G^{\pm})^* = G_{\mp}^{\mp}$, let $\varphi, \psi$ be arbitrary in $\Gamma_c(E)$. Because of $\text{supp} G^{\pm}\varphi \cap \text{supp} G_{\mp}^{\mp}\psi$ being compact, we may perform the following partial integration on $M$:

$$ \begin{aligned} \int_M \langle G^{\pm} \varphi | \psi \succ \text{vol}_M &= \int_M \langle G^{\pm} \varphi | S^{\dagger} G_{\mp}^{\mp} \psi \succ \text{vol}_M \\ &= \int_M \langle S G^{\pm} \varphi | G_{\mp}^{\mp} \psi \succ \text{vol}_M - \int_{\partial M} \langle \sigma_S(n^{\flat}) G^{\pm} \varphi | G_{\mp}^{\mp} \psi \succ \text{vol}_{\partial M} \\ &= \int_M \langle \varphi | G_{\mp}^{\mp} \psi \succ \text{vol}_M, \end{aligned} $$

where the boundary term vanishes by $G^{\pm}\varphi|_{\partial M} \in B_{\pm}$ and $G_{\mp}^{\pm}\psi|_{\partial M} \in B_{\pm}^{\dagger}$. This shows $(G^{\pm})^* = G_{\mp}^{\mp}$. Now given any $\psi \in \Gamma_c(E)$, we have

$$ \begin{aligned} \int_M \langle G^{\pm} v | \psi \succ \text{vol}_M &= \int_M \langle v | (G^{\pm})^* \psi \succ \text{vol}_M \\ &= \int_M \langle v | G_{\mp}^{\mp} \psi \succ \text{vol}_M \\ &= \pm \int_M \langle S u_{\pm} | G_{\mp}^{\mp} \psi \succ \text{vol}_M \\ &= \pm \int_M \langle u_{\pm} | S^{\dagger} G_{\mp}^{\mp} \psi \succ \text{vol}_M \quad \text{because } u_{\pm}|_{\partial M} \in B_{+} = B_{+}^{\dagger} \\ &= \pm \int_M \langle u_{\pm} | \psi \succ \text{vol}_M, \end{aligned} $$
where we have used in a crucial way that $G_{\perp}^{\pm}\psi_{|_{\partial M}} \in B_{\pm}^{\perp}$ and that $u_{\pm|_{\partial M}} \in B_{\pm}$ as well as $B_{\pm}^{\perp} = B_{\pm}$.
Therefore, $G^{\pm}v = \pm u_{\pm}$ and $Gv = u_{+} + u_{-} = u$, as we claimed.

It remains to look at the surjectivity of $S: \Gamma_{sc,B_{+}}(E) \to \Gamma_{sc}(E)$, still with the assumption that $B_{+} = B_{-}$. Let $f \in \Gamma_{sc}(E)$ be given and $K \subset M$ be compact such that $\text{supp} f \subset J_{gc}^{+}(K) \cup J_{gc}^{-}(K)$. As above, up to enlarging $K$ we may assume that $f = f_{+} + f_{-}$, where $f_{\pm} \in \Gamma_{sc}(E)$ with $\text{supp} f_{\pm} \subset J_{gc}^{\pm}(K)$. By Theorem 2.3 the spacetime $M$ can be identified with $\mathbb{R} \times \Sigma$, where $\Sigma$ a smooth spacelike Cauchy hypersurface of $M$. For each $n \in \mathbb{N}$ we let $M_{(-n,n)} := (-n,n) \times \Sigma$, where $\Sigma \simeq \{0\} \times \Sigma$. Note that $M_{(-n,n)}$ is still a globally hyperbolic spacetime with timelike boundary. Let $\chi_n$ be a smooth function with timelike compact support such that $\chi_{n|_{M_{(-n,n)}}} = 1$. Then $\chi_n f_{+}$ lies in $\Gamma_c(E)$ and we may consider $u_n := G^{+}\chi_n f_{+} \in \Gamma_{sc,B_{+}}(E)$. Now letting $u^{+}(x) := u_n^{+}(x)$ for every $x \in M_{(-n,n)}$ defines a smooth section of $E$ on $M$ with $Su^{+} = f_{+}$, for if e.g. $m > n$ then $v := u_m^{+} - u_n^{+}$ is a smooth spacelike compact section of $E$ satisfying $Sv = 0$ on $M_{(-n,n)}$ as well as $v_{|_{M\setminus J^{+}(\text{supp } f_{+})}} = 0$ and $v_{|_{\partial M_{(-n,n)}}} \in B_{+}$, so that $v = 0$ on $M_{(-n,n)}$ by uniqueness of the solution of the forward Cauchy problem. The support of $u^{+}$ must be contained in $J_{gc}^{+}(K)$ since this is the case for the support of each $u_n^{+}$. Analogously, there exists a $u^{-} \in \Gamma_{sc,B_{-}}(E) = \Gamma_{sc,B_{+}}(E)$ with $Su^{-} = f^{-}$ and therefore $S(u^{+} + u^{-}) = f$. This proves the surjectivity of $S: \Gamma_{sc,B_{+}}(E) \to \Gamma_{sc}(E)$ and concludes the proof of Proposition 2.20. $\square$

For further details we refer to [49, Proposition 5.1], [24, Proposition 20] and [23, Proposition 36].

## 2.6 Møller operators on manifolds with timelike boundary

In [63] a geometric process was realized to compare solutions of symmetric hyperbolic systems on different globally hyperbolic manifolds $M_0 := (M, g_0)$ and $M := (M, g_1)$ with empty boundary, provided that $M_0$ and $M_1$ admit the same Cauchy temporal function and $g_1 \le g_0$, namely the set of timelike vectors for $g_1$ is contained in the one for $g_0$. This was achieved via the construction of a family of so-called Møller operators [22, 34, 53]. The aim of this paper is to generalize that construction on manifolds with timelike boundary, where the assumption $g_1 \le g_0$ is adapted to the situation.

Let us introduce the following setup:

### Setup 2.21.

(i) $M_0 = (M, g_0)$ and $M_1 = (M, g_1)$ are globally hyperbolic manifolds with timelike boundary and with the same Cauchy temporal function $t: M \to \mathbb{R}$. Moreover, by realizing $(M, g_i) = (\mathbb{R} \times \Sigma, -\beta_i^2 dt^2 \oplus h_i(t))$ for $i = 0, 1$—cf. Theorem 2.3— we shall assume that there exists a smooth positive function $C > 0$ such that

$$C^2 \beta_1^{-2} h_1(t) \le \beta_0^{-2} h_0(t)$$

holds for every $p \in M$ and $g_C := -\beta_1^2 dt^2 \oplus C^2 h_1(t)$ is globally hyperbolic;

(ii) $E_1$ (resp. $E_0$) is a $\mathbb{K}$-vector bundle over $M_1$ (resp. $M_0$) with finite rank and endowed with a nondegenerate bilinear or sesquilinear fiber metric $\prec \cdot | \cdot \succ_1$ (resp. $\prec \cdot | \cdot \succ_0$);

(iii) $\kappa_{1,0}: E_0 \to E_1$ is a fiberwise linear isometry of vector bundles with inverse $\kappa_{0,1}: E_1 \to E_0$. With a slight abuse of notation we shall also denote by $\kappa_{1,0}: \Gamma(E_0) \to \Gamma(E_1)$ the corresponding linear map between sections defined by $[\kappa_{1,0}u](x) = \kappa_{1,0}u(x)$ for all $u \in \Gamma(E_0)$ and $x \in M$. Finally for any positive $f \in C^{\infty}(M)$, we set $\kappa_{1,0}^f := f \kappa_{1,0}: \Gamma(E_0) \to \Gamma(E_1)$ with inverse $\kappa_{0,1}^f := f^{-1} \kappa_{0,1}$;

(iv) $S_1$ (resp. $S_0$) is a symmetric weakly-hyperbolic system with self-adjoint admissible boundary space we shall denote by $B_1$ (resp. $B_0$). Moreover we shall assume that $\dim \ker \sigma_{S_1}(\xi)$ is constant for any nonzero spacelike covector $\xi \in T^*M_1$;
(v) Let $S_{0,1}^f: \Gamma(E_1) \to \Gamma(E_1)$ be the operator defined by $S_{0,1}^f := \kappa_{1,0}^f S_0 \kappa_{0,1}^f$. We assume there exists a linear isometry $\wp_{1,0}: T^*M_0 \to T^*M_1$ such that $\sigma_{S_{0,1}^f}(\xi) = \sigma_{S_1}(\wp_{1,0}\xi)$ for every $\xi \in T^*M_1$.

**Remarks 2.22.**

1. The assumption (i) in the Setup 2.21 can be equivalently rephrased as follows. Consider the metric $g_C := -\beta_1^2 dt^2 \oplus C^2 h_1(t)$ which is globally hyperbolic on account of Definition 2.14. Then we get the following two situations: for any vector $v \in TM$, we get

$$ (C \le 1) \quad g_C(v, v) \le g_0(v, v) \quad \text{and} \quad g_C(v, v) \le g_1(v, v), \text{ which implies that } J_{g_0}^{\pm} \cup J_{g_1}^{\pm} \subset J_{g_C}^{\pm}. $$

$$ (C \ge 1) \quad g_1(v, v) \le g_C(v, v) \le g_0(v, v), \text{ which implies that } J_{g_0}^{\pm} \subset J_{g_C}^{\pm} \subset J_{g_1}^{\pm}. $$

<figure><img src="image_3.png" /><figcaption>Figure 2: Future light cones of $g_0$ and $g_1$ satisfying $\beta_1^{-2}h_1(t) \le C^2\beta_0^{-2}h_0(t)$.</figcaption></figure>

2. Using assumption (iv), assumption (v) implies both that dim ker $\sigma_{S_0}(\xi)$ is constant for any nonzero spacelike covector $\xi$ and that $\wp_{1,0}$ is time-orientation preserving.

**Remark 2.23** (The principal symbol of $S_{0,1} := S_{0,1}^1$). For later convenience we shall compute the principal symbol $\sigma_{S_{0,1}}$ of $S_{0,1}$ in a slightly more general framework that the one depicted above. (Or maybe this is not at all necessary and we set $\zeta = \mathrm{id}_M$ from the beginning.) Let $E_0 \to M_0$ and $E_1 \to M_1$ be two vector bundles such that there exists a diffeomorphism $\zeta_{1,0}: M_0 \to M_1$ with (inverse $\zeta_{0,1}$) which is lifted to a vector bundle isomorphism $\kappa_{1,0}: E_0 \to E_1$. With a slight abuse of notation we shall denote with $\kappa_{1,0}: \Gamma(E_0) \to \Gamma(E_1)$ the associated map of vector bundles defined by

$$ (\kappa_{1,0}u_0)(x_1) := \kappa_{1,0}(u_0(\zeta_{0,1}x_1)). $$

for all $u_0 \in \Gamma(E_0)$ and $x_1 \in M_1$. Notice that $\kappa_{1,0}(fu_0) = (\zeta_{0,1}^*f)\kappa_{1,0}u_0$ for all $u_0 \in \Gamma(E_0)$ and $f \in C^\infty(M_0)$ where $\zeta_{0,1}^*: C^\infty(M_0) \to C^\infty(M_1)$. Moreover, $\kappa_{1,0}: \Gamma(E_0) \to \Gamma(E_1)$ is invertible with inverse $\kappa_{0,1}$.

The principal symbol of $S_{0,1} := \kappa_{1,0}S_0\kappa_{0,1}$ is obtained as follow. For all $u_1 \in E_1|_{x_1}$ and $\xi_1 \in T^*M_1$, let $\tilde{u}_1 \in \Gamma(E_1)$ and $f \in C^\infty(M_1)$ be such that $\tilde{u}_1(x_1) = u_1$ and $df(x_1) = \xi_1$. Then
we have

$$
\begin{align*}
\sigma_{S_{0,1}}(\xi_1)u_1 &= [\kappa_{1,0}S_0\kappa_{0,1}, f_1]\tilde{u}_1|_{x_1} \\
&= \kappa_{1,0}S_0\kappa_{0,1}f_1\tilde{u}_1 - f_1\kappa_{1,0}S_0\kappa_{0,1}\tilde{u}_1 \\
&= \kappa_{1,0}[S_0, \zeta_{1,0}^*f_1]\kappa_{0,1}\tilde{u}_1 \\
&= \kappa_{0,1}\sigma_{S_0}(d(\zeta_{1,0}^*f_1))\kappa_{0,1}\tilde{u}_1|_{x_1} \\
&= \kappa_{0,1}\sigma_{S_0}((d\zeta_{1,0}^*d f_1)\kappa_{0,1}\tilde{u}_1|_{x_1} \\
&= \kappa_{0,1}\sigma_{S_0}((d\zeta_{1,0}^*\xi_1)\kappa_{0,1}\tilde{u}_1|_{x_1},
\end{align*}
$$

where $(d\zeta_{1,0})^*: T^*M_1 \to T^*M_0$. Overall we have

$$
\sigma_{S_{0,1}}(\xi_1) = \kappa_{1,0}\sigma_{S_0}[(d\zeta_{1,0})^*\xi_1]\kappa_{0,1}.
$$

Similarly to the case of an empty boundary, the construction of a family of Møller operators requires to control the Cauchy problem for the operator $S_{0,1}^f$. With the next proposition, we shall show that the $S_{0,1}^f$ is actually symmetric weakly-hyperbolic over $M_1$.

**Proposition 2.24.** *Assume the Setup 2.21. Then the operator $S_{0,1}^f$ is a symmetric weakly-hyperbolic system of constant characteristic on $M_1$ and $\kappa_{1,0}^f(B_0) = \kappa_{1,0}(B_0)$ is a self-adjoint admissible boundary space for $S_{0,1}^f$.*

*Proof.* On account of Remark 2.23 — with $\zeta_{0,1} = \text{id}_M$ — we have that, for every $\xi \in T^*M$, $\sigma_{S_{0,1}^f}(\xi) = \kappa_{1,0}^f\sigma_{S_0}(\xi)\kappa_{0,1}^f = \kappa_{1,0}\sigma_{S_0}(\xi)\kappa_{0,1}$. Since $S_0$ is symmetric and $\kappa_{1,0}$ is a fiberwise linear isometry by assumption, $S_{0,1}^f$ clearly satisfies property (S) in Definition 2.4. Moreover, because $S_0$ has constant characteristic and $n_1^b$ is a pointwise positive scalar multiple of $n_0^b$, the operator $S_{0,1}^f$ has constant characteristic. Because $B_0$ is an admissible boundary condition for $S_0$, the subbundle $\kappa_{1,0}(B_0) = \kappa_{1,0}^f(B_0)$ must be an admissible boundary space for $S_{0,1}^f$, and it remains self-adjoint. We shall next verify property (wH) in Definition 2.14. To this end let $g_{0,C_0} = -\beta_0^2 dt^2 \oplus C_0^2 h_0(t)$ be the globally hyperbolic metric chosen accordingly with Lemma 2.16. Then $S_0$ is a symmetric hyperbolic system and $\sigma_{S_0}(\tau)$ is fiberwise positive definite for any future-directed (w.r.t. $g_{0,C_0}$) covector $\tau$. Since any conformal transformation does not change the set of future-directed covectors, the operator $S_0$ is hyperbolic w.r.t. $\bar{g}_0 := \beta_0^{-2} g_{0,C_0} = -dt^2 \oplus C_0^2 \beta_0^{-2} h_0(t)$. We shall now prove that $S_{0,1}^f$ is symmetric hyperbolic with respect to the metric $\bar{g}_1 := -dt^2 \oplus C_0^2 C(t)^{-2} \beta_1^{-2} h_1(t)$. For that, let $\tau = dt + \xi$ be $\bar{g}_1$-timelike future directed: On account of the assumption $\beta_1^{-2} h_1(t) \le C^2(t) \beta_0^{-2} h_0(t)$ we find

$$
\bar{g}_0^\sharp(dt + \xi, dt + \xi) \le \bar{g}_1^\sharp(dt + \xi, dt + \xi) < 0,
$$

so that $dt + \xi$ is $\bar{g}_0$-timelike future directed —notice that $\bar{g}_0^\sharp = \partial_t^{\otimes 2} \oplus C_0^{-2} C(t)^2 \beta_0^2 h_0(t)^\sharp$ and similarly for $\bar{g}_1$ (cf. Figure 3).

It follows that $\sigma_{S_0}(dt + \xi) > 0$ and therefore $\sigma_{S_{0,1}^f}(dt + \xi) > 0$ as well. This shows that $S_{0,1}^f$ is symmetric hyperbolic with respect to $\bar{g}_1$ and therefore the same holds true for $g_{1,C_1} := \beta_1^2 \bar{g}_1 = -\beta_1^2 dt^2 \oplus C_1^2 h_1(t)$, where $C_1^2 := C_0^2 C(t)^{-2} > 0$ on account of the hypothesis on $C$ and $C_0$. This proves that $S_{0,1}^f$ is weakly-hyperbolic with respect to $g_1$ as per Definition 2.14. $\square$

Note that the existence of a linear isometry $\wp_{1,0}$ from assumption (v) is not required in the proof of Proposition 2.24.

So far, we considered a setting where the operators $S_0, S_1$, though being defined on different bundles, can be compared through $\kappa_{1,0}$. As a matter of fact the next step in the construction
<figure><img src="image_4.png" /><figcaption>Figure 3: The future light cones of $g_1^\#$ and $g_0^\#$ in the cotangent bundle $T^*M$.</figcaption></figure>

of a Møller operator intertwining $S_0$ and $S_1$ is to build an intertwining operator between $S_{0,1}^f$ and $S_1$. To this avail, we shall first consider an interpolating operator $S_\chi^f$ defined by $S_{\chi,1}^f := (1-\chi)S_{0,1}^f + \chi S_1$, where $\chi \in C^\infty(M)$ is an arbitrary smooth function with $0 \le \chi \le 1$.

The following proposition ensures that $S_{\chi,1}^f$ is a symmetric weakly-hyperbolic system of constant characteristic as soon as $\wp_{1,0}(\mathbf{n}_1^b) \neq \mu\mathbf{n}_1^b$ for any $\mu < 0$. Then for any $\chi \in C^\infty(M, [0, 1])$, the operator defined by

$$S_{\chi,1}^f := (1-\chi)S_{0,1}^f + \chi S_1 + \frac{1}{2}(\sigma_{S_1} - \sigma_{S_{0,1}^f}) (d\chi) : \Gamma(E_1) \to \Gamma(E_1) \quad (2.6)$$

is a symmetric weakly-hyperbolic system of constant characteristic over $M_1$.

*Proof.* By Remark 2.23 it follows that

$$\sigma_{S_{\chi,1}^f}(\xi) = (1-\chi)\sigma_{S_{0,1}^f}(\xi) + \chi\sigma_{S_1}(\xi).$$

Therefore, $S_{\chi,1}^f$ is a symmetric system. We shall next notice that a convex combination of weakly-hyperbolic system is still weakly-hyperbolic. As a matter of fact if $C_{0,1}, C_1 \in (0, 1]$ denotes the positive functions of Definition 2.14 associated with $S_{0,1}^f$ and $S_1$, it follows that

$$\prec \sigma_{S_{\chi,1}^f} (dt + C\xi) \cdot | \succ_p > 0,$$

where $C = \min\{C_{0,1}, C_1\}$ for every future-directed $g_1$-timelike covector $\tau = dt + \xi$.

To conclude our proof, we shall show that $S_{0,1}^f$ is of constant characteristic. To this end, we consider

$$\sigma_{S_{\chi,1}^f}(\mathbf{n}_1^b) = (1-\chi)\sigma_{S_{0,1}^f}(\mathbf{n}_1^b) + \chi\sigma_{S_1}(\mathbf{n}_1^b) = \sigma_{S_1}((1-\chi)\wp_{1,0}\mathbf{n}_1^b + \chi\mathbf{n}_1^b).$$

Since we have by assumption that $\wp_{1,0}\mathbf{n}_1^b \neq \mu\mathbf{n}_1^b$ for any $\mu < 0$, the covector $(1-\chi)\wp_{1,0}\mathbf{n}_1^b + \chi\mathbf{n}_1^b$ is non-vanishing along $\partial M$, which implies that $(1-\chi)\wp_{1,0}\mathbf{n}_1^b + \chi\mathbf{n}_1^b$ is a nonzero spacelike covector. In particular, by assumption (iv) in Setup 2.21, $S_{\chi,1}^f$ is of constant characteristic. $\square$

**Remarks 2.26.**

1. Note that the zero-order operator $V := \frac{1}{2}(\sigma_{S_1} - \sigma_{S_{0,1}^f}) (d\chi)$ is a Hermitian operator which vanishes on every open subset where $\chi$ is constant – in particular on both the chronological past of $\Sigma_-$ and the chronological future of $\Sigma_+$. The zero-order operator $V$ does not play any role in the proof of Theorem 2.27. However, whenever $S_1, S_0$ are formally skew-adjoint, the presence of $V$ ensures that $S_{\chi,1}^f$ is formally skew-adjoint provided a suitable choice of $f$ is made —cf. Proposition 2.32 for the precise statement.
2. The Assumption (iv) in Setup 2.21 is needed in order to ensure that $S_{\chi,1}^f$ is of constant characteristic. It can be dropped if $\wp_{1,0}\mathbf{n}_1^b = \mathbf{n}_1^b = \mathbf{n}_0^b$.

Building on Proposition 2.25, we now prove the main result of this paper.

**Theorem 2.27.** Assume the Setup 2.21 and that $\wp_{1,0}(\mathbf{n}_1^b) \neq \mu\mathbf{n}_1^b$ for any $\mu < 0$. Consider two Cauchy hypersurfaces $\Sigma^\pm \subset M_1$ such that $\Sigma_+ \subset J_{g_1}^+(\Sigma_-)$ — where $J_{g_1}^\pm$ denote the causal cones w.r.t. $g_1$ — and let $\chi \in C^\infty(M_1, [0, 1])$ be non-decreasing along any future-oriented timelike curve such that

$$ \chi|_{J_{g_1}^+(\Sigma_+)} = 1, \quad \text{and} \quad \chi|_{J_{g_1}^-(\Sigma_-)} = 0. $$

Finally let $B_\chi$ be a self-adjoint admissible boundary space for $S_{\chi,1}^f$ such that

$$ B_{\chi} = \begin{cases} B_{0,1} := \kappa_{1,0}^f(B_0) & \text{where } \chi = 0 \\ B_1 & \text{where } \chi = 1 \end{cases}. $$

Then the Cauchy problem for $S_{\chi,1}^f$ with $B_\chi$-boundary conditions is well-posed. Moreover, let $U_{S_{\chi,1}^f, \pm}: D(U_{S_{\chi,1}^f, \pm}) \subset \Gamma_c(E_1|_{\Sigma_\pm}) \to \Gamma_{sc}(E_1)$ be the Cauchy evolution operator associated with $S_{\chi,1}^f$ and initial data on $\Sigma_\pm$ and let $\rho_\pm: \Gamma_{sc}(E_1) \to \Gamma_c(E_1|_{\Sigma_\pm})$ be the standard restriction maps.

Then the Møller operator $R_{0,1} = U_{S_{1,+}} \circ \rho_+ \circ U_{S_{\chi,1,-}} \circ \rho_- \circ \kappa_{1,0}^f$, implements an isomorphism between the spaces of solutions of $S_0$ and $S_1$ given by

$$ \mathrm{Sol}_{B_c}(S_c) := \{ \Psi_c \in \Gamma(E_c) \mid S_c \Psi_c = 0 \text{ and } \Psi_c|_{\partial M} \in B_c \} \quad \text{for } C = 0, 1. $$

*Proof.* Since $B_\chi$ is a self-adjoint admissible boundary space, the Cauchy evolution operators and the Cauchy data map are well-defined on account of Theorem 2.19. Furthermore, for any $\Psi_0 \in \mathrm{Sol}_{B_0}(S_0)$ we have $\rho_-\kappa_{1,0}^f\Psi_0 \in D(U_{S_{\chi,1,-}})$ because of $B_\chi$ coincide with $\kappa_{1,0}^f(B_0)$ on $\Sigma_-$. Therefore $U_{S_{\chi,1,-}}\rho_-\kappa_{1,0}^f\Psi_0$ is well defined. For a similar reason, for any $\Psi \in \mathrm{Sol}(S_{\chi,1}^f)$ we have $\rho_+\Psi \in D(U_{S_{1,+}})$. It follows that $R$ is well-defined.

To conclude our proof, it is enough to notice that the Møller operator is a composition of isomorphisms. As such the inverse $R_{1,0}^{-1}$ of $R_{1,0}$ can be computed explicitly as $R_{1,0}^{-1} = \kappa_{0,1}^f \circ U_{S_{0,1},-} \circ \rho_- \circ U_{S_{\chi,1,+}} \circ \rho_+$.

**Example 2.28.** Let $(M, g)$ be a globally hyperbolic spacetime with timelike boundary and let $S$ and $\bar{S}$ be symmetric weakly-hyperbolic systems of constant characteristic which differ by a zero order term, i.e. $S - \bar{S} = V$, for $V \in \Gamma(\mathrm{End}(E))$. It follows that any self-adjoint admissible boundary condition $B$ for $S$ is also a self-adjoint admissible boundary condition for $\bar{S}$. Therefore we can choose $B_\chi = B$.

We conclude this section by showing that for any pair of admissible boundary conditions $B, B'$ for a given symmetric weakly-hyperbolic system there exists an interpolating admissible boundary condition $B_\chi$. In case $B$ and $B'$ are self-adjoint and the interpolating admissible boundary condition can be constructed to be self-adjoint, then this applies in particular to Theorem 2.27 for the choices $V = E_1$, $W_0 = \kappa_{1,0}^f(B_0)$, $W_1 = B_1$.

**Lemma 2.29.** Let $V \to M$ be any smooth vector bundle over any smooth manifold $M$. Let $q$ be any smooth quadratic form on $V$. Assume $k := \dim \ker(q)$, the numbers $n_+$ of positive and $n_-$ of negative pointwise eigenvalues of $q$ to be constant on $M$. Let $W_0, W_1 \to M$ be any $n_+ + k$-dimensional subbundles of $V$ such that $q|_{W_i} \ge 0$ holds pointwise for both $i = 0, 1$. Then there exists a smooth map $\phi: [0, 1] \times W_0 \to V$ such that, for every $t \in [0, 1]$, $\phi_t := \phi(t, \cdot)$ is a linear and injective vector-bundle-map, $q|_{\phi_t(W_0)} \ge 0$ holds pointwise, and $\phi_0 = \mathrm{Id}_{W_0}$ as well as $\phi_1(W_0) = W_1$ are satisfied.
**Proof.** If we can find a smooth subbundle $W'_0$ of $V$ such that $W_0 \oplus W'_0 = W_1 \oplus W'_0 = V$ and $q_{|W'_0} \le 0$ pointwise, then the map $\phi$ can be constructed as follows. Let $\pi_{W_0}$ (resp. $\pi_{W'_0}$) be the pointwise linear projection onto $W_0$ with kernel $W'_0$ (resp. onto $W'_0$ with kernel $W_0$). Then $\pi_{W_0|_{W_1}}: W_1 \to W_0$ is an isomorphism by $W_1 \cap W'_0 = \{0\}$ and equality of space dimensions. Let $F := \pi_{W'_0} \circ (\pi_{W_0|_{W_1}})^{-1}: W_0 \to W'_0$. Observe that, for every $v \in W_0$, we can write

$$v + F(v) = \pi_{W_0} \left( (\pi_{W_0|_{W_1}})^{-1}(v) \right) + \pi_{W'_0} \left( (\pi_{W_0|_{W_1}})^{-1}(v) \right) = (\pi_{W_0|_{W_1}})^{-1}(v),$$

so that $v + F(v) \in W_1$. Now define $\phi: [0, 1] \times W_0 \to V$ by $\phi(t, v) := v + tF(v)$ for all $(t, v) \in [0, 1] \times W_0$. Clearly $\phi$ is smooth, $\phi_t = \phi(t, \cdot)$ is a linear injective vector-bundle-map for every $t \in [0, 1]$ because of $W_0 \cap W'_0 = \{0\}$ and obviously $\phi_0 = \text{Id}_{W_0}$ and $\phi_1(W_0) = W_1$ hold by the above observation. Moreover, for any $(t, v) \in [0, 1] \times W_0$,

$$q(v + tF(v), v + tF(v)) = q(v, v) + 2q(v, F(v))t + q(F(v), F(v))t^2.$$

Since the r.h.s of the last identity is a degree-2-polynomial in $t$, it is non-negative on $[0, 1]$ as soon as it is for $t = 0$ and $t = 1$ and $q(F(v), F(v)) \le 0$. Therefore, if $q_{|W'_0} \le 0$, then $q_{|\phi_t(W_0)} \ge 0$.

To construct $W'_0$, we make use of the following fact:

**Lemma:** Let $A$ be any smooth section of $\text{End}(V)$. If $x \mapsto \text{dim ker}(A(x))$ is constant on $M$, then $\text{ker}(A) \to M$ defines a smooth vector subbundle of $V$.

**Proof:** Fix any Euclidean resp. Hermitian inner product on $V$ and let $k := \text{dim ker}(A(x))$ for all $x \in U$. For any $x \in U$, we have $\text{ker}(A(x)) = \text{ran}(A(x)^\perp)$, where $A(x)^\perp$ is the adjoint of $A(x)$ w.r.t. the chosen inner product on $V$. Now $\text{ran}(A^*) \to M$ defines a smooth subbundle of $V$. Namely it defines an $(n-k)$-dimensional vector subspace of $V$ at each point of $M$; moreover, for any $x_0 \in M$, there exists an open neighbourhood $U$ of $x_0$ in $M$ and a family of smooth sections $v_1, \dots, v_{n-k}$ of $V|_U$ such that $\{A(x)^*v_1(x), \dots, A(x)^*v_{n-k}(x)\}$ is a family of linearly independent vectors and therefore a basis of $\text{ran}(A(x)^*)$ for any $x \in U$. This shows $\text{ran}(A^*) \to M$ to be a smooth subbundle of $V$. As a straightforward consequence, its pointwise orthogonal complement must be a smooth subbundle as well. This proves our claim. ✓

It can be deduced from the claim that $\text{ker}(q) \to M$ defines a smooth subbundle of $V$. Therefore there exists a smooth supplementary subbundle $W$ to $\text{ker}(q)$. The restriction of $q$ to $W$ defines a smooth nondegenerate quadratic form. Its signature is also constant, namely it is $(n_+, n_-)$. By e.g. [64, Theorem C.1.4], the bundle $W$ can therefore be split as $W = W_+ \oplus W_-$, where $W_\pm$ are smooth subbundles of $W$ of rank $n_\pm$ and on which $q$ restricts pointwise as a positive- resp. negative-definite quadratic form. On the whole, we obtain the smooth splitting $V = \text{ker}(q) \oplus W_+ \oplus W_-$. Now $W'_0 := W_-$ does the job since automatically $W_0 \cap W_- = W_1 \cap W_- = \{0\}$ by the fact that $q_{|W_-}$ is pointwise negative definite. This concludes the proof of Lemma 2.29. □

To apply Lemma 2.29, consider $q := \sigma_{S_1}(n^\flat) \cdot | \succcurlyeq$ on $V := E_{1|_{\partial M}}$ as well as $W_0 := B_{0,1}$ and $W_1 := B_1$. Then the map $\hat{\phi}$ realizing the interpolation of the boundary conditions is defined by

$$\hat{\phi}: B_{0,1} \to E_{1|_{\partial M}}, \quad v \mapsto \phi(\chi(\pi(v)), v),$$

where $\pi: E_{1|_{\partial M}} \to \partial M$ is the footpoint map.

**Remarks 2.30.**

1. In case $W_0$ and $W_1$ are null spaces for $q$, then unless $q$ vanishes identically on $V$ and thus $W_0 = W_1 = V$ the space $\phi_t(W_0)$ as constructed in the proof of Lemma 2.29 is not null for almost every $t \in [0, 1]$. This does not prevent the existence of a path of null subspaces connecting $W_0$ and $W_1$: namely the question is only whether the Grassmannian of $n_+ + n_0$-dimensional $q$-nonnegative subspaces in an $n_+ + n_0 + n_-$-dimensional one is connected or not, where $n_0 = \text{dim ker } \sigma_S(n^\flat)$.
2. Note also that Lemma 2.29 can be applied to the situation where a stronger condition as condition (iv) on the operator $S_1$ is assumed, namely that the numbers $n_0, n_+, n_-$ of vanishing, positive resp. negative eigenvalues of $\sigma_{S_1}(\xi)$ are constant along $\partial M$ whenever $\xi$ is a nonvanishing covector in $T^*\Sigma|_{\partial\Sigma}$; then we need not assume any longer that $S_{0,1} = S_1$. This applies for instance to the Dirac operators associated to two different globally hyperbolic metrics $g_0, g_1$ and where the boundary condition is the MIT one, see Section 3.2 below.

## 2.7 Conservation of positive definite Hermitian scalar products

Consider now the pre-Hilbert space given by

$$ \mathrm{Sol}_{sc,B}(S) = \{ \Psi \in \Gamma_{sc}(E) | S\Psi = 0, \Psi|_{\partial M} \in \mathbb{B} \} $$

where $(\cdot|\cdot)$ is the positive definite Hermitian form defined by

$$ (\cdot|\cdot) = \int_{\Sigma} \prec \cdot | \sigma_S(n^b) \succ \mathrm{vol}_{\Sigma}, \qquad (2.7) $$

where $n = -\frac{1}{\beta}\partial_t$ is the past-directed unit normal vector to $\Sigma$ while $n^b = g(n, \cdot) = \beta dt$. In the next lemma, we shall show that if $S$ is skew-adjoint then the scalar product (2.7) does not depend on the choice of the Cauchy hypersurface $\Sigma \subset M$.

**Lemma 2.31.** Let $\Sigma \subset M$ be a smooth spacelike Cauchy hypersurface with its past-oriented unit normal vector field $n$ and its induced volume element $\mathrm{vol}_{\Sigma}$. Furthermore, let $S$ be a formally skew-adjoint, symmetric weakly-hyperbolic system of constant characteristic with self-adjoint admissible boundary condition, i.e. $B_+ = B_-$, see Definition 2.9. Then

$$ (\cdot|\cdot): \mathrm{Sol}_{sc,B}(S) \times \mathrm{Sol}_{sc,B}(S) \to \mathbb{C} \qquad (\Psi|\Phi) = \int_{\Sigma} \prec \Psi | \sigma_S(n^b) \Phi \succ \mathrm{vol}_{\Sigma}, $$

where $n^b$ denotes here the future-directed unit normal, yields a positive definite Hermitian scalar product which does not depend on the choice of $\Sigma$.

*Proof.* The proof virtually coincides with the one of [5, Lemma 3.17]. First note that $\mathrm{supp}(\Psi) \cap \Sigma$ is compact since $\mathrm{supp}(\Psi)$ is spacelike compact, so that the integral is well-defined. Let $\Sigma'$ be any other smooth spacelike Cauchy hypersurface. Without loss of generality we may assume that $\Sigma \cap \Sigma' = \emptyset$, otherwise a third Cauchy hypersurface lying in the common pasts of $\Sigma$ and $\Sigma'$ has to be chosen, see proof of [5, Lemma 3.17]. Let $M_T = t^{-1}(\tau, \tau')$ be the time strip such that $t^{-1}(\tau) = \Sigma$ and $t^{-1}(\tau') = \Sigma'$. Its boundary is $\partial M_T = (\partial M \cap M_T) \cup \Sigma \cup \Sigma'$. By the Green identity [49, Lemma 2.11] we have

$$ \int_{M_T} (\prec S\Psi | \Phi \succ - \prec \Psi | S^\dagger\Phi \succ) \mathrm{vol}_{M_T} = \int_{\partial M_T} \prec \Psi | \sigma_S(n^b) \Phi \succ \mathrm{vol}_{\partial M_T} $$

for any $\Psi, \Phi \in \mathrm{Sol}_{sc,B}(S)$. Since $S$ is assumed to be skew-adjoint, the left-hand side of the latter equality vanishes identically. Moreover, since $B = B^\dagger$ also $\prec \Psi | \sigma_S(n^b) \Phi \succ$ vanishes identically at $\partial M \cap M_T$. Therefore we can conclude

$$ 0 = \int_{\Sigma'} \prec \Psi | \sigma_S(n^b) \Phi \succ \mathrm{vol}_{\Sigma'} - \int_{\Sigma} \prec \Psi | \sigma_S(n^b) \Phi \succ \mathrm{vol}_{\Sigma}. $$

This concludes our proof. $\square$

With the next lemma, we shall show that there exists a choice of $f$ which makes the operator $S_{\chi,1}^f: \Gamma_{sc,B_\chi}(E_1) \to \Gamma_{sc}(E_1)$ formally skew-adjoint on $\Gamma_{sc,B_\chi}(E_1)$, provided that $B_\chi$ is a self-adjoint boundary condition and $S_0$ (resp. $S_1$) are formally skew-adjoint with respect to the pairing $(\cdot|\cdot)_0$ (resp. $(\cdot|\cdot)_1$).
**Proposition 2.32.** Assume the setup of Theorem 2.27, that $S_0$ and $S_1$ are formally skew-adjoint with respect to the pairings $(\cdot|\cdot)_0$ and $(\cdot|\cdot)_1$ respectively. Finally assume that $B_\chi$ is self-adjoint boundary condition for $S_{\chi,1}^f$. If $f \in C^\infty(M)$ is the positive smooth function such that

$$ \mathrm{vol}_{M_0} = f^2 \mathrm{vol}_{M_1} $$

on $M$, where $\mathrm{vol}_{M_0}$ (resp. $\mathrm{vol}_{M_1}$) is the volume form of the metric $g_0$ (resp. $g_1$) on $M$, then $S_{\chi,1}^f$ is a skew-adjoint operator in the Hilbert space $\mathcal{H}_1$.

*Proof.* First we compute the formal adjoint of $S_{0,1}^f$ on $(M, g_1)$. Let $\Psi_1, \Phi_1 \in \Gamma_c(E_1)$ be such that their supports do not meet $\partial M$. Since by assumption $f^2 \mathrm{vol}_{M_1} = \mathrm{vol}_{M_0}$ and $S_0^\dagger = -S_0$, we have

$$
\begin{align*}
\int_M \prec S_{0,1}^f \Psi_1 |\Phi_1\succ_1 \mathrm{vol}_{M_1} &= \int_M \prec \kappa_{1,0}^f S_0 \kappa_{0,1}^f \Psi_1 |\Phi_1\succ_1 \mathrm{vol}_{M_1} \\
&= \int_M \prec \kappa_{1,0}^f S_0 \kappa_{0,1}^f \Psi_1 |\kappa_{1,0}^f \kappa_{0,1}^f \Phi_1\succ_1 \mathrm{vol}_{M_1} \\
&= \int_M f^2 \prec \kappa_{1,0} S_0 \kappa_{0,1}^f \Psi_1 |\kappa_{1,0} \kappa_{0,1}^f \Phi_1\succ_1 \mathrm{vol}_{M_1} \\
&= \int_M f^2 \prec S_0 \kappa_{0,1}^f \Psi_1 |\kappa_{0,1}^f \Phi_1\succ_0 \mathrm{vol}_{M_1} \\
&= \int_M \prec S_0 \kappa_{0,1}^f \Psi_1 |\kappa_{0,1}^f \Phi_1\succ_0 \mathrm{vol}_{M_0} \\
&= \int_M \prec \kappa_{0,1}^f \Psi_1 | S_0^\dagger \kappa_{0,1}^f \Phi_1\succ_0 \mathrm{vol}_{M_0} \\
&= \int_M \prec \kappa_{0,1}^f \Psi_1 | \kappa_{0,1}^f \kappa_{1,0} S_0^\dagger \kappa_{0,1}^f \Phi_1\succ_0 \mathrm{vol}_{M_0} \\
&= \int_M f^{-2} \prec \Psi_1 | \kappa_{1,0}^f S_0^\dagger \kappa_{0,1}^f \Phi_1\succ_1 \mathrm{vol}_{M_0} \\
&= \int_M \prec \Psi_1 | \kappa_{1,0}^f S_0^\dagger \kappa_{0,1}^f \Phi_1\succ_1 \mathrm{vol}_{M_1} \\
&= -\int_M \prec \Psi_1 | S_{0,1}^f \Phi_1\succ_1 \mathrm{vol}_{M_1},
\end{align*}
$$

that is, $(S_{0,1}^f)^\dagger = -S_{0,1}^f$ on $(M, g_1)$. As a consequence,

$$
\begin{align*}
((1-\chi)S_{0,1}^f + \chi S_1)^\dagger &= (1-\chi)(S_{0,1}^f)^\dagger - \sigma_{S_{0,1}^f}(d(1-\chi)) + \chi S_1^\dagger - \sigma_{S_1}(d\chi) \\
&= -(1-\chi)S_{0,1}^f + \sigma_{S_{0,1}^f}(d\chi) - \chi S_1 - \sigma_{S_1}(d\chi) \\
&= -(1-\chi)S_{0,1}^f - \chi S_1 - 2V,
\end{align*}
$$

where $V$ is the zero-order operator defined as above by $V := \frac{1}{2}[\sigma_{S_1}(d\chi) - \sigma_{S_{0,1}^f}(d\chi)]$. Since $V$ is a Hermitian operator it follows that $S_{\chi,1}^f = (1-\chi)S_{0,1}^f + \chi S_1 + V$ is formally skew-adjoint. $\square$

Building on Lemma 2.31 and Proposition 2.32, we can show that $R_{1,0}$ is a unitary map between $\mathrm{Sol}_{sc,B_0}(S_0)$ and $\mathrm{Sol}_{sc,B_1}(S_1)$.

**Proposition 2.33.** Assume the setup of Theorem 2.27, that $S_0$ and $S_1$ are formally skew-adjoint and that $B_\chi$ is a self-adjoint boundary condition for $S_{\chi,1}^f$. Let $\Sigma_1 \subset J^+(\Sigma_+)$ and $\Sigma_0 \subset J^-(\Sigma_-)$ be fixed spacelike Cauchy hypersurfaces of $M$ (w.r.t. $g_0$ or $g_1$, it makes no difference). Let $f \in C^\infty(M)$ be the positive smooth function such that

$$ \mathrm{vol}_{M_0} = f^2 \mathrm{vol}_{M_1} $$
on M, where vol$_{M_0}$ (resp. vol$_{M_1}$) is the volume form of the metric g_0 (resp. g_1) on M. Then the Møller operator R$_{1,0}$ : Sol$_{sc,B_0}$(S_0) → Sol$_{sc,B_1}$(S_1) is a unitary map once Sol$_{sc,B_0}$(S_0) (resp. Sol$_{sc,B_1}$(S_1)) is equipped with the scalar product defined in Equation (2.7) associated with S_0 (resp. with S_1).

Proof. Let be Ψ₀, Φ₀ ∈ Sol<sub>sc,B₀</sub>(S₀) and Ψ₁ := R₁,₀(Ψ₀), Φ₁ := R₁,₀(Φ₀) ∈ Sol<sub>sc,B₁</sub>(S₁), where the Møller operator R₁,₀ is defined using the interpolating operator S<sub>χ,1</sub><sup>f</sup> instead of S<sub>χ,1</sub><sup>f</sup>. We also denote by Ψ<sub>χ,1</sub> (resp. Φ<sub>χ,1</sub>) the smooth section with spacelike compact support in ker (S<sub>χ,1</sub><sup>f</sup>) on M with Ψ<sub>χ,1|<sub>Σ_−</sub></sub> = κ<sub>1,0</sub><sup>f</sup>Ψ<sub>0|<sub>Σ_−</sub></sub> (resp. Φ<sub>χ,1|<sub>Σ_−</sub></sub> = κ<sub>1,0</sub><sup>f</sup>Φ<sub>0|<sub>Σ_−</sub></sub>). By Lemma 2.31 and definition of f, we have n<sub>0</sub><sup>b</sup> ⊗ vol<sub>Σ_−,g_0</sub> = f<sup>2</sup>n<sub>1</sub><sup>b</sup> ⊗ vol<sub>Σ_−,g_1</sub> along Σ_− and therefore

$$
\begin{align*}
\int_{\Sigma_0} \prec \sigma_{S_0}(\mathbf{n}_0^b) \Psi_0 | \Phi_0 \succ_0 \mathrm{vol}_{\Sigma_0, g_0} &= \int_{\Sigma_0} \prec \sigma_{S_0}(\mathbf{n}_0^b) \Psi_0 | \Phi_0 \succ_0 \mathrm{vol}_{\Sigma_-, g_0} \\
&= \int_{\Sigma_0} f^{-2} \prec \kappa_{1,0}^f \sigma_{S_0}(\mathbf{n}_0^b) \kappa_{0,1}^f \kappa_{1,0}^f \Psi_0 | \kappa_{1,0}^f \Phi_0 \succ_1 \mathrm{vol}_{\Sigma_-, g_0} \\
&= \int_{\Sigma_0} f^{-2} \prec \sigma_{S_{0,1}^f}(\mathbf{n}_0^b) \Psi_{\chi,1} | \Phi_{\chi,1} \succ_1 \mathrm{vol}_{\Sigma_-, g_0} \\
&= \int_{\Sigma_0} \prec \sigma_{S_{0,1}^f}(\mathbf{n}_1^b) \Psi_{\chi,1} | \Phi_{\chi,1} \succ_1 \mathrm{vol}_{\Sigma_-, g_1} \\
&= \int_{\Sigma_0} \prec \sigma_{S_{\chi,1}^f}(\mathbf{n}_1^b) \Psi_{\chi,1} | \Phi_{\chi,1} \succ_1 \mathrm{vol}_{\Sigma_-, g_1} \\
&= \int_{\Sigma_+} \prec \sigma_{S_{\chi,1}^f}(\mathbf{n}_1^b) \Psi_{\chi,1} | \Phi_{\chi,1} \succ_1 \mathrm{vol}_{\Sigma_+, g_1} \\
&= \int_{\Sigma_+} \prec \sigma_{S_1}(\mathbf{n}_1^b) \Psi_1 | \Phi_1 \succ_1 \mathrm{vol}_{\Sigma_+, g_1},
\end{align*}
$$

which concludes the proof of Proposition 2.33.

<figure><img src="image_5.png" /></figure>

**Definition 2.34.** We call *unitary Møller operator* the operator R<sub>1,0</sub> defined in accordance with Proposition 2.33.

We conclude this section with the following Remark.

**Remark 2.35.** The unitary Møller operator $R_{1,0} : \text{Sol}_{sc,B_0}(S_0) \to \text{Sol}_{sc,B_1}(S_1)$ can be seen as the composition of two unitary Møller operators

$$
R_{\chi,0}: \mathrm{Sol}_{sc,B_0}(S_0) \to \mathrm{Sol}_{sc,B_\chi}(S_{\chi,1}^f) \qquad R_{\chi,0} := U_{S_{\chi,1}^f, -} \circ \rho_- \circ \kappa_{1,0}^f,
$$

$$
R_{1,\chi}: \mathrm{Sol}_{sc,B_{\chi}}(S_{\chi,1}^{f}) \to \mathrm{Sol}_{sc,B_{1}}(S_{1}) \qquad R_{1,\chi} := U_{S_{1,+}} \circ \rho_{+}.
$$

3 The algebraic approach to quantum Dirac fields

In this section we shall compare the quantization of the Dirac field on two different (yet related)
globally hyperbolic spacetimes with timelike boundary. To this end, we shall benefit from [22,
37,63], where a class of Møller operator was introduced in order to construct unitary equivalent
quantum field theories, together of the results of the previous Sections 2.5-2.6.

As a first step we introduce the relevant geometrical objects, showing how they fit within the
framework introduced in Section 2.6. In particular we shall apply Theorem 2.27 and Proposition
2.33 for the case of the Dirac operator with MIT boundary conditions — cf. Equation (3.3).
## 3.1 The 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 $Spin_0(1, n)$-principal bundle $P_{Spin_0}$ to the bundle of positively-oriented tangent frames $P_{SO+}$ of M such that the following diagram is commutative:

<figure><img src="image_6.png" /></figure>

**Remark 3.1.** Note that unlike differential forms, the definition of a spin structure depends on the metric of the underlying manifold.

The existence of spin structures is related to the topology of M. 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 2.3 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, i.e. the complex vector bundle

$$ \mathcal{SM} := \mathrm{Spin}_0(1, n) \times_{\rho} \mathbb{C}^N $$

where $\rho: \mathrm{Spin}_0(1, n) \to \mathrm{Aut}(\mathbb{C}^N)$ is the complex $\mathrm{Spin}_0(1, n)$ representation and $N := 2^{\lfloor \frac{n+1}{2} \rfloor}$. The spinor bundle comes together with the following structures:

- a natural $\mathrm{Spin}_0(1, n)$-invariant indefinite fiber metrics

$$ \prec \cdot | \cdot \succ_p : \mathcal{S}_p M \times \mathcal{S}_p M \to \mathbb{C}; $$

- a Clifford multiplication, i.e. a fiber-preserving map

$$ \gamma: \mathrm{TM} \to \mathrm{End}(\mathcal{SM}); $$

which satisfies for all $p \in M$, $u, v \in T_p M$ and $\psi, \phi \in \mathcal{S}_p M$

$$ \gamma(u)\gamma(v) + \gamma(v)\gamma(u) = -2g(u, v)\mathrm{Id}_{\mathcal{S}_p M} \quad \text{and} \quad \prec\gamma(u)\psi|\phi\succ_p = \prec\psi|\gamma(u)\phi\succ_p. $$

Using the spin product (3.1), we denote as *adjunction map*, the complex anti-linear vector bundle isomorphism by

$$ \Upsilon_p : \mathcal{S}_p M_g \to \mathcal{S}_p^* M_g \quad \psi \mapsto \prec\psi|\cdot\succ, \qquad (3.1) $$

where $\mathcal{S}_p^* M_g$ is the so-called *cospinor bundle*, i.e. the dual bundle of $\mathcal{S}_p M_g$.

**Definition 3.2.** The (classical) Dirac operator D is the operator defined as the composition of the metric connection $\nabla^S$ on $\mathcal{SM}$, obtained as a lift of the Levi-Civita connection on $\mathrm{TM}$, and the Clifford multiplication:

$$ D = \gamma \circ \nabla^{\mathcal{SM}} : \Gamma(\mathcal{SM}) \to \Gamma(\mathcal{SM}). $$

In local coordinates and with a trivialization of the spinor bundle $\mathcal{SM}$, the Dirac operator reads as

$$ D\psi = \sum_{\mu=0}^{n} \varepsilon_{\mu} \gamma(e_{\mu}) \nabla_{e_{\mu}}^{\mathcal{SM}} \psi, $$

where $\{e_{\mu}\}$ is a local Lorentzian-orthonormal frame of $\mathrm{TM}$ and $\varepsilon_{\mu} = g(e_{\mu}, e_{\mu}) = \pm 1$.
**Proposition 3.3** ([49, Proposition 6.2]). *The classical Dirac operator D on globally hyperbolic spin manifolds M with timelike boundary is a nowhere characteristic symmetric hyperbolic system.*

It follows, by Theorem 2.12, that the Cauchy problem for the Dirac operator on globally hyperbolic spacetimes with empty boundary is well-posed, therefore, it admits a Cauchy evolution operator $U_t: \Gamma_c(\mathcal{S}M|_{\Sigma_t}) \to \Gamma_{sc}(\mathcal{S}M)$. Remarkably, as shown by Capoferri and Vassilliev [18], the Cauchy evolution operator for Dirac fields on Cauchy-compact ultrastatic manifolds (with empty boundary) can be realized as a Fourier integral operator. As a matter of fact, the Fourier integral representation of the propagator contains the information on how singularities propagates in the manifolds. For this reason, it would be desirable to extend their techniques to more general globally hyperbolic manifolds with possibly not empty boundary.

## 3.2 Self-adjoint admissible boundary conditions

The aim of this section is to recast the boundary conditions for the Dirac operator which are self-adjoint and admissible in the sense of Definition 2.6. Let us remark that not all physical interesting boundary conditions for Dirac fields enter in this class of boundary condition. Indeed there exists physically interesting non-local boundary conditions, like the so-called APS boundary condition, which guarantees that the Cauchy problem is well-posed [36], but they are not admissible (since any admissible boundary condition is a local condition). For further details on self-adjoint admissible boundary conditions for the Dirac fields we refer to [49, Section 6.1.1] and [50, Remark 3.19].

The first example of self-adjoint admissible boundary conditions are the so-called *chiral boundary conditions*. They are defined as follows: let $\mathcal{G}$ be a chirality operator on $\mathcal{S}M$, i.e. a parallel involutive antiunitary (with respect to $\prec \cdot | \cdot \succ$) endomorphism-field of $\mathcal{S}M$ that anti-commutes with Clifford multiplication by vectors. Notice that chirality operators exist only in even-dimensional manifolds. Then the so-called *chirality* boundary spaces $B_{\text{CHI}}$ are defined as the range of the maps

$$ \pi_{\text{CHI}}^{\pm} := \frac{1}{2} (\text{Id} \pm \gamma(\mathbf{n})\mathcal{G}), \qquad (3.2) $$

where $\gamma(\mathbf{n})$ denotes Clifford multiplication for the outward-pointing unit normal along $\partial M$. It is not difficult to check that the range of $\pi_{\text{CHI}}$ has dimension $2^{\lfloor \frac{n}{2} \rfloor - 1}$, which is the number of nonnegative eigenvalues of the endomorphism $\sigma_D(\mathbf{n}^b)$, and

$$ \prec \sigma_D(\mathbf{n}^b) \pi_{\text{CHI}}^{\pm}(\psi) | \pi_{\text{CHI}}^{\pm} \succ = 0. $$

Furthermore, since

$$ \pi_{\text{CHI}}^{\pm} \pi_{\text{CHI}}^{\pm} = \pi_{\text{CHI}}^{\pm}, \quad \pi_{\text{CHI}}^{\mp} \pi_{\text{CHI}}^{\pm} = 0, \quad \pi_{\text{CHI}}^{+} + \pi_{\text{CHI}}^{-} = \text{Id}, $$

it can be easily verified that the boundary conditions are self-adjoint.

The second example of self-adjoint boundary conditions is the so-called *MIT boundary conditions*. The boundary space $B_{\text{MIT}}$ is defined as the range of

$$ \pi_{\text{MIT}}^{\pm} := \frac{1}{2} (\text{Id} \pm i\gamma(\mathbf{n})), \qquad (3.3) $$

where $\gamma(\mathbf{n})$ is again the Lorentzian Clifford multiplication for the outward-pointing unit normal vector along $\partial M$. Similarly to the chiral boundary conditions, the range of $\pi_{\text{CHI}}$ has dimension $2^{\lfloor \frac{n}{2} \rfloor - 1}$,

$$ \prec \sigma_D(\mathbf{n}^b) \pi_{\text{CHI}}^{\pm}(\psi) | \pi_{\text{CHI}}^{\pm} \succ = 0 $$

and we have

$$ \pi_{\text{MIT}}^{\pm} \pi_{\text{MIT}}^{\pm} = \pi_{\text{MIT}}^{\pm}, \quad \pi_{\text{MIT}}^{\mp} \pi_{\text{MIT}}^{\pm} = 0, \quad \pi_{\text{MIT}}^{+} + \pi_{\text{MIT}}^{-} = \text{Id}. $$
### 3.3 Linear isometry between spinor bundles

We shall now apply the results obtained in the Sections 2.6 to compare the solution spaces associated with pairs of Dirac operators $D_0, D_1$ defined using different metrics $g_0, g_1 \in \mathcal{GH}_M$ and equipped with admissible self-adjoint boundary conditions. In what follows $g_0, g_1 \in \mathcal{GH}_M$ are assumed to fulfil assumption (i) of Setup 2.21.

As already underlined in Remark 3.1, the space of spinors depends on the metric of the underlying manifold $M_\lambda$. Therefore, an identification between spaces of sections of spinor bundles for different metrics is needed to construct a unitary Møller operator. This can be achieved by following [4, Section 5].

Consider a family of Lorentzian spin manifolds $M_\lambda := (M, g_\lambda)$ with a common Cauchy temporal function, where $g_\lambda \in \mathcal{GH}_M$ for any $\lambda \in \mathbb{R}$. For a given nonempty interval $I$ in $\mathbb{R}$ let $Z$ be the Lorentzian manifold

$$Z = I \times M \qquad g_Z = d\lambda^2 + g_\lambda.$$

On $Z$ there exists a globally defined vector field which we denote as $e_\lambda := \frac{\partial}{\partial \lambda}$. For any $\lambda$, the spin structures on $Z$ and $M_\lambda \simeq \{\lambda\} \times M$ are in one-to-one correspondence: Any spin structure on $Z$ can be restricted to a spin structure on $M_\lambda$ and a spin structure on $M_\lambda$ it can be pulled back on $Z$ – see [4, Section 3 and 5]. Actually, the spinor bundle $\mathcal{SM}_\lambda$ on each globally hyperbolic spin manifold $M_\lambda$ can be identified with the restriction of the spinor bundle $\mathcal{SZ}$ on $M_\lambda$, in particular $\mathcal{SM}_\lambda \simeq \mathcal{SZ}|_{M_\lambda}$ if $n$ is even, while $\mathcal{SM}_\lambda \simeq \mathcal{S}^+Z|_{M_\lambda} \simeq \mathcal{S}^-Z|_{M_\lambda}$ if $n$ is odd. Equivalently we may identify

$$\mathcal{SZ}|_{M_\lambda} = \begin{cases} \mathcal{SM}_\lambda & \text{if } n \text{ is even,} \\ \mathcal{SZ}|_{M_\lambda} \oplus \mathcal{SZ}|_{M_\lambda} & \text{if } n \text{ is odd.} \end{cases} \qquad (3.4)$$

By denoting with $\gamma_Z$ (resp. $\gamma_\lambda$) the Clifford multiplication on $\mathcal{SZ}$ (resp. on $\mathcal{SM}_\lambda$), the family of Clifford multiplications $\gamma_\lambda$ satisfies

$$\gamma_{\lambda}(v)\psi = \gamma_{Z}(e_{\lambda})\gamma_{Z}(v)\psi \qquad \text{if } n \text{ is even,} \qquad (3.5)$$

$$\gamma_{\lambda}(v)(\psi_{+} + \psi_{-}) = \gamma_{Z}(e_{\lambda})\gamma_{Z}(v)(\psi_{+} - \psi_{-}) \qquad \text{if } n \text{ is odd,} \qquad (3.6)$$

where in the second case $\psi = \psi_+ + \psi_- \in \mathcal{SZ}|_{M_\lambda} \oplus \mathcal{SZ}|_{M_\lambda}$ and each component $\psi_\pm$ is identified with an element in $\mathcal{S}^\pm Z|_{M_\lambda}$.

**Lemma 3.4** ([63, Lemma 3.7]). Let $Z$ be the Lorentzian spin manifold given by

$$Z = I \times M \qquad g_Z = d\lambda^2 + g_\lambda,$$

where $(M, g_\lambda) := M_\lambda$ is a family of Lorentzian spin manifolds with a common Cauchy temporal function, and denote with $\mathcal{SM}_\lambda$ be the spinor bundle over $M_\lambda$. For any $p \in M_\lambda$, the map

$$\kappa_{1,0}: \mathcal{S}_p M_0 \to \mathcal{S}_p M_1. \qquad (3.7)$$

defined by the parallel translation on $Z$ along the curve $\lambda \mapsto (\lambda, p)$ is a linear isometry and preserves the Clifford multiplication, i.e. for any $v \in \Gamma(\mathrm{TM})$ and any $\Psi_0 \in \Gamma(\mathcal{SM}_0)$ it holds

$$\gamma_1(\wp_{1,0}v)(\kappa_{1,0}\Psi_0) = \kappa_{1,0}(\gamma_0(v)\Psi_0),$$

where $\wp_{1,0}: \mathrm{TM}_0 \to \mathrm{TM}_1$ is the parallel transport along the curve $\lambda \mapsto (\lambda, p)$.

**Remark 3.5.** Let us remark, that for any couple of Lorentzian metric $g_0$ and $g_1$ admitting a common Cauchy temporal function, there always exists a path of Lorentzian metric $g_\lambda$ connecting $g_0$ to $g_1$, e.g. $g_\lambda = \lambda g_1 + (1 - \lambda)g_0$ where $\lambda \in [0, 1]$. For more details we refer to [62, 63].
Lemma 3.4 provides an isomorphism $\kappa_{1,0}: \mathbb{SM} \to \mathbb{SM}$ with the same properties introduced in the Setup 2.21. We shall denote by $D_{0,1}^f$ the intertwining Dirac operator as in Proposition 2.24. Similarly $D_{\chi,1}^f$ shall denote the operator interpolating between $D_{0,1}^f$ and $D_1$. Here and in what follows $f$ is chosen as per Proposition 2.33.

**Remark 3.6.** Keeping the notation of Remark 2.23 and Lemma 3.4, the diffeomorphism $\zeta: M \to M$ is simply the identity Id. Since $\sigma_{D_0}(\xi) = \gamma_0(\xi^{\sharp_0})$, where $\sharp_0$ denotes the musical isomorphism with respect to $g_0$, we find

$$ \sigma_{D_{0,1}^f}(\xi_1) = \kappa_{0,1}^f \sigma_{D_0}(\xi_1) \kappa_{0,1}^f = \kappa_{1,0}^f \gamma_0(\xi_1^{\sharp_0}) \kappa_{0,1}^f = \gamma_1(\wp_{1,0}\xi_1^{\sharp_0}) = \sigma_{D_1}((\wp_{1,0}\xi_1^{\sharp_0})^{\flat_1}) = \sigma_{D_1}(\wp_{1,0}\xi_1), $$

where $\sharp_1 := b_1^{-1}$ is the musical isomorphism associated with $g_1$. In the last equality we used that, for $\xi \in T_x^*M$ and $X \in T_x M$ we have

$$ \begin{aligned} (\wp_{1,0}\xi^{\sharp_0})^{\flat_1}(X)|_x &= g_1(\wp_{1,0}\xi^{\sharp_0}, X)|_x = g_Z(\wp_{1,0}\xi^{\sharp_0}, X)|_{(1,x)} = g_Z(\xi^{\sharp_0}, \wp_{0,1}X)|_{(0,x)} \\ &= g_0(\xi^{\sharp_0}, \wp_{0,1}X)|_x = \xi(\wp_{0,1}X)|_x = [\wp_{0,1}\xi](X)|_x, \end{aligned} $$

where, with a slight abuse of notation, we denoted $\wp_{0,1}\xi$ the parallel transport of the 1-form $\xi$ along the curve $\lambda \to (\lambda, x)$ within $Z$: The latter coincides with $\wp_{0,1}^*\xi$, being $\wp_{0,1}: TM_1 \to TM_0$.

We are almost in position to apply Theorem 2.27 and Proposition 2.33. In the next lemma we shall prove that the assumption in Theorem 2.27 that the parallel transport of $n_1^b$ is not proportional to $\mu n_1^b$ for any $\mu < 0$, is always satisfied, provided $g_\lambda = (1 - \lambda)g_0 + \lambda g_1$ for all $\lambda \in [0, 1]$.

**Lemma 3.7.** Let $(M, g_0)$ and $(M, g_1)$ be globally hyperbolic manifolds with timelike boundary split as $(M, g_i) = (\mathbb{R} \times \Sigma, -\beta_i^2 dt^2 \oplus h_i(t))$ for both $i = 0, 1$. Consider the manifold $Z := [0, 1] \times M$ endowed with the metric $g_Z := d\lambda^2 \oplus g_\lambda$, where

$$ g_\lambda := (1 - \lambda)g_0 + \lambda g_1 = -\beta_\lambda dt^2 \oplus h_\lambda(t), $$

where $\beta_\lambda := (1 - \lambda)\beta_0 + \lambda\beta_1$ and $h_\lambda(t) = (1 - \lambda)h_0(t) + \lambda h_1(t)$. Then $h_1(\wp_{1,0}(n_1), n_1) > 0$ along $\partial M$, where $\wp_{1,0}$ is the parallel transport in $(Z, \hat{g})$ along $[0, 1] \to Z$, $\lambda \mapsto (\lambda, p)$, for any $p \in \partial M$.

*Proof.* Note that, by definition of both $g_i$ and of $g_Z$, we have $\nabla_{\partial\lambda}^Z \partial_\lambda = \nabla_{\partial\lambda}^Z \beta_\lambda^{-1} \partial_t = 0$, so that, for any $\lambda_0 \in [0, 1]$, the parallel transport along $[0, \lambda_0] \to Z$, $\lambda \mapsto (\lambda, p)$ preserves $T\Sigma$. Writing $p = (t, x)$, we fix a pointwise $h_0$-o.n.b. of $T_x\Sigma$ in which $h_1 = h_1(t)$ is diagonal i.e. , there exist $\mu_1, \dots, \mu_n > 0$ such that $h_1(e_i, e_j) = \mu_i\delta_{ij}$ for all $1 \le i, j \le n$. This basis $(e_i)_{1 \le i \le n}$ is extended constantly in $\lambda$ along $\lambda \mapsto (\lambda, p)$. Splitting $\wp_{\lambda,0}n_1 = \sum_{j=1}^n \alpha_j e_j$, where $\alpha_j = h_0(\wp_{\lambda,0}n_1, e_j)$, we have

$$ \begin{aligned} 0 &= \nabla_{\partial\lambda}^Z (\wp_{\lambda,0} n_1) \\ &= \sum_{j=1}^n (\partial_\lambda \alpha_j) e_j + \alpha_j \nabla_{\partial\lambda}^Z e_j \\ &= \sum_{j=1}^n (\partial_\lambda \alpha_j) e_j + \alpha_j \left( \underbrace{[\partial_\lambda, e_j]}_0 + \frac{1}{2} h_\lambda^{-1} \partial_\lambda h_\lambda(e_j, \cdot) \right), \end{aligned} $$

so that, denoting by $Y(\lambda) := \begin{pmatrix} \alpha_1(\lambda) \\ \vdots \\ \alpha_n(\lambda) \end{pmatrix}$ and identifying $h_\lambda$ (as a homomorphism $T\Sigma \to T^*\Sigma$) and $\partial_\lambda h_\lambda$ (as symmetric 2-tensor on $T\Sigma$) with their respective matrices $H_\lambda$ and $\partial_\lambda H_\lambda$ in the
bases $(e_j)_{1 \le j \le n}$ and $(e_j^*)_{1 \le j \le n}$ respectively, the vector-valued function $Y$ must satisfy the linear first-order ODE

$$ Y'(\lambda) + \frac{1}{2} H_{\lambda}^{-1} \partial_{\lambda} H_{\lambda} \cdot Y(\lambda) = 0 \qquad (3.8) $$

on $[0, 1]$. In case $[H_{\lambda}, \partial_{\lambda}H_{\lambda}] = 0$ is fulfilled for all $\lambda$, equation (3.8) can be solved explicitly, namely $Y(\lambda) = H_{\lambda}^{-\frac{1}{2}} \cdot Y(0)$ for all $\lambda \in [0, 1]$ is the solution with initial condition $Y(0) \in \mathbb{R}^n$. But with $h_{\lambda} = (1 - \lambda)h_0 + \lambda h_1$, we have $H_{\lambda} = (1 - \lambda)I_n + \lambda \text{diag}(\mu_1, \dots, \mu_n)$, so that $\partial_{\lambda}H_{\lambda} = \text{diag}(\mu_1, \dots, \mu_n) - I_n$ and therefore $[H_{\lambda}, \partial_{\lambda}H_{\lambda}] = 0$ holds for all $\lambda \in [0, 1]$. This implies that

$$ Y(\lambda) = H_{\lambda}^{-\frac{1}{2}} \cdot Y(0) = \text{diag} \left( (1 - \lambda + \lambda \mu_1)^{-\frac{1}{2}}, \dots, (1 - \lambda + \lambda \mu_n)^{-\frac{1}{2}} \right) \cdot Y(0) $$

holds for all $\lambda \in [0, 1]$. As a consequence, $Y(1) = \text{diag} \left( \mu_1^{-\frac{1}{2}}, \dots, \mu_n^{-\frac{1}{2}} \right) \cdot Y(0)$, from which

$$ h_1(\wp_{1,0}\mathbf{n}_1, \mathbf{n}_1) = H_1(Y(1), Y(0)) = \sum_{j=1}^{n} \mu_j^{\frac{1}{2}} \alpha_j(0)^2 > 0 $$

and the claim follows.

□

We conclude this section by stating Theorem 2.27 and Proposition 2.33 for the particular case of MIT boundary conditions.

**Proposition 3.8.** Let assume $g_0, g_1 \in \mathcal{GH}_M$ fulfils (i) in Setup 2.21. Let $M_0$ (resp. $M_1$) be a globally hyperbolic spin manifold with timelike boundary and let $D_0$ (resp. $D_1$) be a classical Dirac operator coupled with MIT boundary condition $B_{\text{MIT}_0}$ (resp. $B_{\text{MIT}_1}$). Then the boundary space defined by

$$ B_{\chi} = \ker M_{\chi} := \ker (\gamma_1(v) - \iota \|v\|_1) $$

is a self-adjoint boundary space for the operator

$$ D_{\chi,1}^f := (1 - \chi) \kappa_{1,0}^f D_0 \kappa_{0,1}^f + \chi D_1 + \frac{1}{2} (\sigma_{D_1} + \sigma_{D_{0,1}^f}) (d\chi), $$

where $v = \chi \mathbf{n}_1 + (1 - \chi) \wp_{1,0} \mathbf{n}_1$ and $\|v\|_1 = \sqrt{g_1(v, v)}$.

Therefore, letting $\text{Sol}_{\text{sc,MIT}}(D_i) := \{\Psi \in \mathcal{SM}_i | D_i \Psi = 0, \Psi|_{\partial M} \in B_{\text{MIT}}\}$, there exists a unitary isomorphism (Møller operator) $R_{1,0}: \text{Sol}_{\text{sc,MIT}}(D_0) \to \text{Sol}_{\text{sc,MIT}}(D_1)$ where $\text{Sol}_{\text{sc,MIT}}(D_0)$ (resp. $\text{Sol}_{\text{sc,MIT}}(D_1)$) is equipped with the scalar product defined in Equation (2.7) associated with $D_0$ (resp. with $D_1$).

**Proof.** The last part of the statement is nothing but Theorem 2.27 together with Proposition 2.33. Thus, it remains to prove the first part. We begin by noticing that when $\chi = 0$ then $B_{\chi}$ reduces to $\ker(\gamma_1(\wp_{1,0}\mathbf{n}_1) - \iota) = \kappa_{1,0}(B_{\text{MIT}_0})$ on account of Remark 3.6, while when $\chi = 1$ $B_{\chi}$ reads as $B_{\text{MIT}_1}$. To conclude we need to show that $B_{\chi}$ is a self-adjoint admissible boundary condition for $D_{\chi,1}^f$. To this end, we first notice that the bilinear form $\prec \gamma_1(v)\Psi | \Psi \succ_q$ satisfies simultaneously

$$ \prec \gamma_1(v) \Psi | \Psi \succ_q = \prec \Psi | \gamma_1(v) \Psi \succ_q = \prec \gamma_1(v) \Psi | \Psi \succ_q $$

$$ \prec \gamma_1(v) \Psi | \Psi \succ_q = \prec \iota \|v\| \Psi | \Psi \succ_q = \prec \Psi | \iota \|v\| \Psi \succ_q = - \prec \Psi | \gamma_1(v) \Psi \succ_q = - \prec \gamma_1(v) \Psi | \Psi \succ_q $$

which implies that $\prec \gamma_1(v)\Psi | \Psi \succ_q = \prec \gamma_{D_{\chi,1}^f}(\mathbf{n}_1^b)\Psi | \Psi \succ_q = 0$. Furthermore the range of the projector $\pi = \frac{1}{2}(\text{Id} - \iota \|v\|_1^{-1}\gamma_1(v))$ which is equal to $B_{\chi}$, has dimension $2^{\lfloor \frac{n}{2} - 1 \rfloor}$, which is exactly the number of nonnegative eigenvalues of $\gamma_{D_{\chi,1}^f}(\mathbf{n}^b)$. This concludes our proof. □

**Remark 3.9.** Since, for any nonzero spacelike covector $v$ on $M$, the operator $\sigma_D(v)$ has vanishing kernel and $\pm|v|$ as nonvanishing eigenvalues, each with multiplicity $2^{\lfloor \frac{n+1}{2} \rfloor - 1}$, the existence of an interpolating $B_{\chi}$ between $B_{\text{MIT}_0}$ and $B_{\text{MIT}_1}$ for $D_0$ and $D_1$ respectively follows from Lemma 2.29, see Remark 2.30 above. Note however that the interpolating $B_{\chi}$ from Lemma 2.29 is not self-adjoint.
### 3.4 The algebra of Dirac fields with MIT boundary condition

In this section we shall exploit Proposition 3.8 to compare the quantization of the Dirac field with MIT boundary condition on $M_0$ and $M_1$. With this purpose we shall briefly recall the quantization procedure from the algebraic point of view.

In [28,37,63], the quantization of a free field theory is realized as a two-step procedure. On the one hand, the physical system classically described by $Sol_{sc,MIT}(D)$ is quantized by introducing a unital $*$-algebra $\mathfrak{A}$, whose elements are interpreted as observables for the system under investigation. In a second stage, the description of possible physical states of the system is described through the choice of a suitable subclass of linear, positive and normalized functionals $\omega: \mathfrak{A} \to \mathbb{C}$.

By extending the analogous definition for a spacetime without boundary, we shall now introduce the $*$-algebra $\mathfrak{A}$ associated with the space $Sol_{sc,MIT}(D)$ of solutions with spatially compact support of the Dirac operator $D$ coupled with MIT boundary conditions and endowed with the positive definite Hermitian scalar product (2.7).

To this avail we shall profit of the results and definition already present in the literature, see [2]. For later convenience let $Sol_{sc,MIT}^{\oplus}(D)$ be the Hilbert space obtained by completion of

$$ Sol_{sc,MIT}(D) \oplus \Upsilon Sol_{sc,MIT}(D), $$

equipped with the natural scalar product $(, )_{Sol_{sc,MIT}^{\oplus}}$ induced by $Sol_{sc,MIT}(D)$ —cf. Equation (2.7)— in particular $(\psi_1 | \psi_2) = \int_{\Sigma} <\psi_1 | \gamma(-\beta^{-1}\partial_t)\psi_2> \geq \text{vol}_{\Sigma}$. Moreover, let $\Gamma: Sol_{sc,MIT}^{\oplus} \to Sol_{sc,MIT}^{\oplus}$ be the antilinear involution defined by $\Gamma(\psi_1 \oplus \Upsilon\psi_2) := (-\psi_2) \oplus \Upsilon\psi_1$ where $\Upsilon: SM \to S^*M$ has been defined in Equation (3.1).

**Definition 3.10.** The algebra of Dirac fields with MIT boundary condition is the unital, complex $*$-algebra $\mathfrak{A}$ freely generated by the abstract elements $\Xi(\psi)$, $1_{\mathfrak{A}}$, with $\psi \in Sol_{sc,MIT}^{\oplus}$, together with the following relations for all $\psi, \phi \in Sol_{sc,MIT}^{\oplus}$ and $\alpha, \beta \in \mathbb{C}$:

$$ (i) \text{ Linearity: } \Xi(\alpha\psi + \beta\phi) = \alpha\Xi(\psi) + \beta\Xi(\phi) $$

$$ (ii) \text{ Hermiticity: } \Xi(\psi)^* = \Xi(\Gamma\psi) $$

(iii) Canonical anti-commutation relations (CARs):

$$ \Xi(\psi) \cdot \Xi(\phi) + \Xi(\phi) \cdot \Xi(\psi) = 0 \quad \text{and} \quad \Xi(\psi) \cdot \Xi(\phi)^* + \Xi(\phi)^* \cdot \Xi(\psi) = (\psi | \phi) 1_{\mathfrak{A}}. $$

As a matter of fact $\mathfrak{A}$ can be completed in a unique way into a $C^*$-algebra [2] the $C^*$-norm being induced by the natural Hilbert structure of $Sol_{sc,MIT}^{\oplus}$. Occasionally we shall implicitly regard $\mathfrak{A}$ as a $C^*$-algebra.

Recollecting the results of the previous sections we have the following:

**Theorem 3.11.** Assume that $g_0, g_1 \in \mathcal{G}\mathcal{H}_M$ fulfils (i) in the Setup 2.21 and let $\mathfrak{A}_\alpha$ be the algebra of Dirac fields with MIT boundary condition on $M_\alpha$. Then the unitary Møller operator $R_{1,0}: Sol(D_0) \to Sol(D_1)$ lifts to a $*$-isomorphism $\mathfrak{R}_{1,0}: \mathfrak{A}_0 \to \mathfrak{A}_1$.

*Proof.* Let $\Upsilon_\alpha: SM_\alpha \to S^*M_\alpha$ the adjunction map defined in (3.1) between the spinor and cospinor bundle over $M_\alpha$ and set $R_{1,0}^\Upsilon := \Upsilon_1 R_{1,0} \Upsilon_0^{-1}$. Then $R_{1,0}^\Upsilon$ implements an isomorphism between $\Upsilon_0 Sol_{sc,MIT}(D_0)$ and $\Upsilon_1 Sol_{sc,MIT}(D_1)$. On account of Proposition 3.8 $R_{1,0}^{\oplus} := R_{1,0} \oplus R_{1,0}^\Upsilon: Sol_{sc,MIT}^{\oplus} \to Sol_{sc,MIT}^{\oplus}$ is a unitary isomorphism. By direct inspection, the linear map $\mathfrak{R}_{1,0}: \mathfrak{A}_0 \to \mathfrak{A}_1$ defined by $\mathfrak{R}_{1,0} \Xi(\psi) := \Xi(R_{1,0}^{\oplus} \psi)$ extends to the seen $*$-isomorphism. $\square$

**Remark 3.12.** The algebra of Dirac fields with MIT boundary condition cannot be considered as an algebra of observables, since observables are required to commute at spacelike separations and $\mathfrak{A}$ does not fulfil such requirement. A good candidate as algebra of observables is the subalgebra $\mathfrak{A}_{\text{obs}} \subset \mathfrak{A}$ consisting of elements which are even, i.e. invariant by replacement $\Xi(\psi) \mapsto -\Xi(\psi)$, and invariant under the action of $\text{Spin}_0(1, n)$ (extended to $\mathfrak{A}$). For further details we refer to [26].
## 3.5 Hadamard states

In this section we study (algebraic) states and their interplay with the *-isomorphism $\mathfrak{A}_{1,0}$.

**Definition 3.13.** Given a complex *-algebra $\mathfrak{A}$ we call (algebraic) state any linear functional from $\mathfrak{A}$ into $\mathbb{C}$ that is positive, i.e. $\omega(\mathfrak{a}^*\mathfrak{a}) \ge 0$ for any $\mathfrak{a} \in \mathfrak{A}$, and normalized, i.e. $\omega(1_{\mathfrak{A}}) = 1$.

Due to the natural grading on the algebra of Dirac fields with MIT boundary conditions $\mathfrak{A}$, it suffices to define $\omega$ on the monomials. Among all states, the so-called quasi-free states play a distinguished role.

**Definition 3.14.** A state $\omega$ on $\mathfrak{A}$ is *quasifree* if it satisfies

$$ \omega(\Xi(\psi_1) \cdots \Xi(\psi_n)) = \begin{cases} 0 & n \text{ odd} \\ \sum_{\sigma \in S'_n} (-1)^{\text{sign}(\sigma)} \prod_{i=1}^{n/2} \omega(\Xi(\psi_{\sigma(2i-1)})\Xi(\psi_{\sigma(2i)})) & n \text{ even} \end{cases}, $$

where $S'_n$ denotes the set of ordered permutations of $n$ elements.

As shown in [2, Lemma 3.2], for any quasi-free state $\omega$ on the $C^*$-algebra $\mathfrak{A}$ there exists a bounded operator $Q_\omega \in B(\text{Sol}_{sc,MIT}^\oplus)$ on $\text{Sol}_{sc,MIT}^\oplus$ such that $0 \le Q_\omega = Q_\omega^* \le 1$, $Q_\omega + \Gamma Q_\omega \Gamma = \text{Id}_{\text{Sol}_{sc,MIT}^\oplus}$ and

$$ \omega(\Xi(\psi_1)^*\Xi(\psi_2)) = (\psi_1, Q_\omega\psi_2)_{\text{Sol}_{sc,MIT}^\oplus}. \qquad (3.9) $$

From a different perspective, we can realize $\omega(\Xi(\psi_1)^*\Xi(\psi_2))$ in terms of distributions. This turns out to be quite useful when looking for physically relevant states. To this avail we observe that, an application of Proposition 2.20 leads to $\text{Sol}_{sc,MIT}^\oplus \simeq (\Gamma_c(\mathfrak{SM})/\mathfrak{D}\Gamma_{c,MIT}(\mathfrak{SM}))^{\oplus 2}$ —cf. Equation (2.5)— the isomorphism being given by $(\Gamma_c(\mathfrak{SM})/\mathfrak{D}\Gamma_{c,MIT}(\mathfrak{SM}))^{\oplus 2} \ni ([f_1], [f_2]) \to \mathfrak{G}f_1 \oplus \Gamma\mathfrak{G}f_2 \in \text{Sol}_{sc,MIT}^\oplus$. In particular we can endow $\Gamma_c(\mathfrak{SM})$ with the standard locally convex topology which induces a locally convex topology on the quotient $\Gamma_c(\mathfrak{SM})/\mathfrak{D}\Gamma_{c,MIT}(\mathfrak{SM})$. With this choices the map $(\Gamma_c(\mathfrak{SM})/\mathfrak{D}\Gamma_{c,MIT}(\mathfrak{SM}))^{\oplus 2} \to \text{Sol}_{sc,MIT}^\oplus$ turns out to be continuous, so that to any quasi-free state we may associate its 2-point distributions $\omega^{(2)} \in \Gamma_c((\mathfrak{SM} \oplus \mathfrak{SM})')$ defined by

$$ \omega^{(2)}(f_1, f_2) := \omega(\Xi(\psi_{f_1})^*\Xi(\psi_{f_2})). $$

where $\psi_f \in \text{Sol}_{sc,MIT}^\oplus$ is the element associated with $[f] \in [\Gamma_c(\mathfrak{SM})/\mathfrak{D}\Gamma_{c,MIT}(\mathfrak{SM})]^{\oplus 2}$. In particular, we have that the 2-point distribution is a solution of the Dirac equation with MIT boundary conditions, meaning that

$$ \omega^{(2)}(f_1, (D \oplus D)f_1) = 0 \quad \forall f_1, f_2 \in \Gamma_{c,MIT}(\mathfrak{SM} \oplus \mathfrak{SM}). \qquad (3.10) $$

Notice that, due to the CAR relations, Equation (3.10) cannot be strengthened to hold true for all $f_1, \dots, f_n \in \Gamma_c(\mathfrak{SM} \oplus \mathfrak{SM})$.

A widely accepted criterion to select physically relevant states is the renowned Hadamard condition [54,66–68]. On a globally hyperbolic spacetime with empty boundary, the latter allows to construct Wick polynomials in a local and covariant fashion, moreover, it guarantees the finiteness of the fluctuations of such Wick polynomials [38].

At a technical level, the Hadamard condition characterizes the wave front set WF($\omega^{(2)}$) $\subseteq \mathfrak{T}^*\mathfrak{M}^2$ of the 2-point function of a quasi-free state —generalization to non-quasi free states are possible [70]. Such a microlocal characterization is also possible for the case of a globally hyperbolic manifold with timelike boundary: therein the Hadamard condition has been formulated in [74] for the case of asymptotically Anti-de Sitter spacetimes and then exploited in [29] for a wider class of boundary conditions. In these situations the proper replacement for WF($\omega^{(2)}$) is given by WF$_b(\omega^{(2)}) \subset {}^b\mathfrak{T}^*\mathfrak{M}^2 \setminus \{0\}$, where WF$_b$ stands for the b-wave front set [59].
**Definition 3.15.** Let $(\mathbf{M}, g)$ be a globally hyperbolic spin manifold with timelike boundary. A bidistribution $\omega^{(2)} \in \Gamma_c(\mathbf{SM} \oplus \mathbf{SM})'$ is called of *Hadamard form* if it has the following b-wave front set

$$WF_b(\omega^{(2)}) = \{(x, y, k_x, -k_y) \in \mathbf{T}^*(\mathbf{M} \times \mathbf{M}) \setminus \{0\} | (x, k_x) \sim (y, k_y), k_x \succ 0\},$$

where $\sim$ entails that $(x, k_x)$ and $(y, k_y)$ are connected by a generalized broken bicharacteristic while that $k_x \succ 0$ means that the covector $k_x$ at $x \in \mathbf{M}$ is future pointing. Since we deal with vector-valued distributions, the standard convention for the wave front set is to take the union of the wave front set of its components in an arbitrary but fixed local frame.

For further details on Hadamard states on globally hyperbolic manifolds with empty boundary we refer to [47, 48, 55], while on globally hyperbolic manifolds with timelike boundary, we refer to [29, 46, 74].

With the next theorem, we show that the pull-back of a quasifree state along the isomorphism $\mathfrak{R}_{1,0} : \mathfrak{A}_0 \to \mathfrak{A}_1$ induced by the unitary Møller operator $\mathbf{R}$ for $\mathbf{D}$ preserves the singularity structure of the two-point distribution $\omega^{(2)}$.

**Theorem 3.16.** Assume that $g_0, g_1 \in \mathcal{GH}_M$ fulfil (i) in the Setup 2.21. Assume furthermore that a propagation of singularities theorem holds true for $\mathbf{D}$ with MIT boundary condition, namely for any $u \in \mathbf{Sol}_{\text{MIT}}(\mathbf{D})$, $WF_b(u)$ is the union of maximally extended generalized broken bicharacteristics. Denote with $\mathfrak{A}_\alpha$, $\alpha = 0, 1$, the algebras of Dirac fields with MIT boundary condition on $\mathbf{M}_\alpha$ and let $\omega_\alpha : \mathfrak{A}_\alpha \to \mathcal{C}$ be quasifree states satisfying

$$\omega_0 = \omega_1 \circ \mathfrak{R}_{1,0} : \mathfrak{A}_0 \to \mathcal{C}$$

with $\mathfrak{R}_{1,0}$ is the isomorphism induced by $\mathbf{R}$ as per Theorem 3.11. Then, if $\omega_1$ is a Hadamard state as per Definition 3.15, then so is $\omega_0$.

*Proof.* Since $\mathfrak{R}_{1,0}$ preserves the grading of $\mathfrak{A}_0$, $\mathfrak{A}_1$, $\omega_0$ inherits the property of being a quasifree state from $\omega_1$. In particular the two-point function $\omega_0^{(2)}$ satisfies

$$\omega_0^{(2)}(f_0, g_0) = \omega_0(\Xi(\psi_{f_0})^*\Xi(\psi_{g_0})) = \omega_1^{(2)}(\Xi(\mathbf{R}_{1,0}^{\oplus} \psi_{f_0})^*\Xi(\mathbf{R}_{1,0}^{\oplus} \psi_{g_0})).$$

We shall now prove that $\omega_1$ fulfils the Hadamard condition. To this avail we first observe that $\mathbf{R}_{1,0}$ can in fact be decomposed as $\mathbf{R}_{1,0} = \mathbf{R}_{1,\chi} \circ \mathbf{R}_{\chi,0}$ (cf. Remark 2.35). With reference to Theorem 2.27, we have $\mathbf{R}_{\chi,0} := U_{\mathbf{D}_{\chi,1,-}} \circ \rho_{-} \circ \kappa_{1,0}^f$ whereas $\mathbf{R}_{1,\chi} := U_{\mathbf{D}_{1,+}} \circ \rho_{+}$. Let us consider $\mathfrak{R}_{1,\chi} : \mathfrak{A}_\chi \to \mathfrak{A}_1$, where $\mathfrak{R}_{1,\chi}$ is the $*$-isomorphism defined as per Theorem 3.11 with $\mathfrak{A}_0$ replaced with $\mathfrak{A}_\chi$. Moreover let $\omega_\chi := \omega_1 \circ \mathfrak{R}_{1,\chi}$. With reference to Theorem 2.27, let $f_1, f_2 \in \Gamma_c(\mathbf{SM} \oplus \mathbf{SM})$ be with support contained in a neighbourhood of $\Sigma_+$. Then

$$
\begin{align*}
\omega_{\chi}^{(2)}(f_1, f_2) &= \omega_{\chi}(\Xi(\mathbf{G}_{\chi}f_1)^*\Xi(\mathbf{G}_{\chi}f_2)) \quad \text{def. } \omega_{\chi}^{(2)} \\
&= \omega_1(\Xi(\mathbf{R}_{1,\chi}^{\oplus}\mathbf{G}_{\chi}f_1)^*\Xi(\mathbf{R}_{1,\chi}^{\oplus}\mathbf{G}_{\chi}f_2)) \quad \text{def. } \mathfrak{R}_{1,\chi} \\
&= (\mathbf{R}_{1,\chi}\mathbf{G}_{\chi}f_1, Q_{\omega_1}\mathbf{R}_{1,\chi}\mathbf{G}_{\chi}f_2)_{\mathbf{Sol}_{sc,MIT}(\mathbf{D}_1)} \quad \text{Eq. (3.9)} \\
&= (\rho_{-}\mathbf{R}_{1,\chi}^{\oplus}\mathbf{G}_{\chi}f_1, Q_{\omega_1}\rho_{-}\mathbf{R}_{1,\chi}^{\oplus}\mathbf{G}_{\chi}f_2)_{\Sigma_+} \quad \text{choice of } \Sigma_+ \\
&= (\rho_{-}\mathbf{G}_{\chi}f_1, Q_{\omega_1}\rho_{-}\mathbf{G}_{\chi}f_2)_{\Sigma_+} \quad \rho_{-}\mathbf{R}_{1,\chi} = \rho_{-} \\
&= (\rho_{-}\mathbf{G}_1f_1, Q_{\omega_1}\rho_{-}\mathbf{G}_1f_2)_{\Sigma_+} \\
&= \omega_1^{(2)}(f_1, f_2),
\end{align*}
$$

where we exploited the fact that, when computing $(, )_{\mathbf{Sol}_{sc,MIT}^{\oplus}}$, we may choose $\Sigma$ arbitrarily. In the second to last equation we used that $\mathbf{G}_{\chi}f|_{\Sigma_+} = \mathbf{G}_1f|_{\Sigma_+}$ for $f$ supported in a small enough neighbourhood of $\Sigma_+$. This shows that $\omega_\chi^{(2)}$ coincides with $\omega_1^{(2)}$ in a neighbourhood of $\Sigma_+$ and
therefore fulfils the Hadamard condition therein. By the assumed propagation of singularities, it follows that $\omega_{\chi}^{(2)}$ fulfils the Hadamard condition on M.

By observing that $\omega_1 = \omega_\chi \circ \mathfrak{R}_{\chi,0}$ and proceeding with a similar argument we have that $\omega_1$ fulfils the Hadamard condition. $\square$

**Remark 3.17.** We expect the propagation of singularities to hold true, as there are already positive results in this direction, see e.g. [29, 46, 60, 61, 73] for the scalar wave equation, [57, 72] for first order systems, and [9] for the Dirac-Coulomb system. We postpone its investigation to a forthcoming paper.

We have finally all the tools to prove the existence of Hadamard states.

*Proof of Theorem 1.1.* Let $t$ be a Cauchy temporal function for $g$ and define $g_u := -dt^2+h$, where $h$ is a complete Riemannian metric on $t^{-1}(s)$ for every $s \in \mathbb{R}$. On account of [62, Proposition 2.23], there exists a globally hyperbolic metric $\bar{g}$ such that $J_{\bar{g}}^+ \subset J_{g_u}^+ \cap J_g^+$. Denote with $\mathbb{SM}_{\bar{g}}$ the spinor bundle over $(M, \bar{g})$ and consider the linear isometries

$$ \kappa_{\bar{g},g}^{f'} : \mathbb{SM}_g \to \mathbb{SM}_{\bar{g}} \qquad \kappa_{\bar{g},g_u}^{f''} : \mathbb{SM}_{g_u} \to \mathbb{SM}_{\bar{g}} $$

defined as in Section 3.3. It is easy to see that the operators

$$ D_{g,\bar{g}}^{f'} := \kappa_{g,\bar{g}}^{f'} D_{\bar{g}} \kappa_{g,\bar{g}}^{f'} : \Gamma(\mathbb{SM}_g) \to \Gamma(\mathbb{SM}_g) \quad \text{and} \quad D_{\bar{g},g_u}^{f''} := \kappa_{g_u,\bar{g}}^{f''} D_{\bar{g}} \kappa_{\bar{g},g_u}^{f''} : \Gamma(\mathbb{SM}_{g_u}) \to \Gamma(\mathbb{SM}_{g_u}) $$

are weakly-hyperbolic on $(M, g)$ and $(M, g_u)$ respectively, so we can construct a unitary Møller operator $R_{g_u, g} : \text{Sol}(D_g) \to \text{Sol}(D_{g_u})$, composing the unitary Møller operators $R_{\bar{g}, g_u} : \text{Sol}(D_{g_u}) \to \text{Sol}(D_{\bar{g}})$ and $R_{g, \bar{g}} : \text{Sol}(D_{\bar{g}}) \to \text{Sol}(D_g)$ obtained using the same arguments as in Sections 2.6 and 2.7. In particular, we can lift the action of the unitary Møller operator to a $*-\text{isomorphism}$ between the algebra of Dirac fields on $(M, g)$ and $(M, g_u)$ respectively. Hence for any Hadamard state $\omega_H$ on $\mathfrak{A}_u$, the state defined by

$$ \omega = \omega_H \circ \mathfrak{R}_{1,0} : \mathfrak{A} \to \mathbb{C}, $$

is also a Hadamard state on account of Theorem 3.16.

It remains to show that there exists a Hadamard state $\omega_H$ for $\mathfrak{A}_u$. For that, we shall define a quasi-free state by identifying a suitable operator $Q_\omega$ and then exploiting Equation (3.9). In order to construct the desired $Q_\omega$, let us write the Dirac equation as $D = \sigma(dt)\partial_t + L$, where $L$ differentiates only in the tangential part of $\Sigma$. Since we coupled $D$ with self-adjoint boundary condition, it follows that $L$ is skew-adjoint. As a consequence we may define the self-adjoint operator $H = iL$. To obtain a pure, quasi-free state it is enough to define the operator

$$ Q_{\omega} := P_{+}(H) \oplus \text{Id}_{\mathcal{H}} - \Upsilon P_{+}(H) \Upsilon^{-1} $$

where $P_+(H)$ is the spectral projection in the positive spectrum of $H$. It is not difficult to see, that on globally hyperbolic ultrastatic manifolds with empty boundary, the associated quasi-free state is of the Hadamard form, since it provides the canonical frequencies splitting. This concludes our proof. $\square$

**Remark 3.18.** The main drawback of the definition of the Møller $*-\text{isomorphism} \mathfrak{R}$, used in Theorem 3.16, is the lack of any control on the action of the group of $*-\text{automorphism}$ induced by the isometry group of $M$ on $\omega_2$. 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: Under which conditions it is
possible to perform an adiabatic limit, namely when is $\lim_{\omega_1} \omega$ well-defined?

$x \to 1$

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. Note that this is not a no-go Theorem, but at least an indication that, in these
situation, the map $\omega \to \omega \circ \mathcal{R}$ cannot be expected to preserve the ground state property. A partial
investigation in this direction has been carried on in [22,33] 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.

Since our results depends only on the principal symbol of the Dirac operator and on the
chosen boundary condition, we conclude our paper with the following corollary.

**Corollary 3.19.** Let $(M, g)$ be a globally hyperbolic spin spacetime with timelike boundary and let $D_V = D + V$ be the Dirac operator coupled with an external skew-symmetric potential $V \in \text{End}(\mathbb{S}M)$ and with the MIT boundary condition. Then there exists a state for the algebra of Dirac fields with MIT boundary conditions which satisfies the Hadamard condition.

References

[1] L. Aké, J. L. Flores and M. Sánchez, *Structure of globally hyperbolic spacetimes with timelike boundary*. Rev. Mat. Iberoam. **37**, 45–94 (2021).

[2] H. Araki, *On quasifree states of CAR and Bogoliubov automorphisms*. Publ. Res. Inst. Math. Sci. Kyoto **6**, 385 (1971).

[3] C. Bär, *Green-hyperbolic operators on globally hyperbolic spacetimes*. Comm. Math. Phys. **333**, 1585–1615 (2015).

[4] C. Bär, P. Gauduchon and A. Moroianu, *Generalized cylinders in semi-Riemannian and spin geometry*. Math. Z. **249**, 545–580 (2005).

[5] C. Bär and N. Ginoux, *Classical and quantum fields on Lorentzian manifolds*. in: C. Bär, J. Lohkamp and M. Schwarz (eds.), Global Differential Geometry, 359-400, Springer-Verlag Berlin Heidelberg (2012).

[6] C. Bär, N. Ginoux and F. Pfäffle, *Wave equations on Lorentzian Manifolds and Quantization*. ESI Lectures in Mathematics and Physics (2007).

[7] F. Bambozzi and S. Murro, *On the uniqueness of invariant states*. Adv. Math. **376** 107445 (2021).

[8] F. Bambozzi, S. Murro and N. Pinamonti, *Invariant states on noncommutative tori*. Int. Math. Res. Not. **2021**, 3299–3313 (2021).

[9] D. Baskin and J. Wunsch, *Diffraction for the Dirac-Coulomb propagator*. arXiv:2011.08890 [math.AP] (2020).

[10] J.K. Beem, P.E. Ehrlich and K.L. Easley, *Global Lorentzian geometry*. Marcel Dekker, New York (1996).

[11] M. Benini, C. Dappiaggi and S. Murro, *Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states*. J. Math. Phys. **55**, 082301 (2014).
[12] M. Benini, C. Dappiaggi and A. Schenkel, *Algebraic quantum field theory on spacetimes with timelike boundary*. Ann. Henri Poincaré **19**, 2401–2433 (2018).

[13] R. Brunetti, C. Dappiaggi, K. Fredenhagen and J. Yngvason, *Advances in algebraic quantum field theory*. Springer (2015)

[14] R. Brunetti, K. Fredenhagen and N. Pinamonti, *Algebraic approach to Bose Einstein Condenration in relativistic Quantum Field Theory. Spontaneous symmetry breaking and the Goldstone Theorem*. Ann. Henri Poincaré volume **22**, 951–1000 (2021).

[15] F. Bussola, C. Dappiaggi, H.R.C. Ferreira and I. Khavkine, *Ground state for a massive scalar field in the BTZ spacetime with Robin boundary conditions*. Phys. Rev. D 96, 105016 (2017).

[16] M. Capoferri, C. Dappiaggi and N. Drago, *Global wave parametrices on globally hyperbolic spacetimes*. J. Math. Anal. App. **490**, 124316 (2020).

[17] M. Capoferri, M. Levitin and D. Vassiliev, *Geometric wave propagator on Riemannian manifolds*. to appear in Comm. Anal. Geom, arXiv:1902.06982 [math.AP] (2019).

[18] M. Capoferri and D. Vassiliev, *Global propagator for the massless Dirac operator and spectral asymptotics*. arXiv:2004.06351 [math.AP], (2020).

[19] M. Capoferri and D. Vassiliev, *Invariant subspaces of elliptic systems II: spectral theory*. to appear in J. Spectr. Theory, arXiv:2103.14334 [math.AP] (2020).

[20] A. Chodos, R.L. Jaffe, K. Johnson and C.B. Thorn, *Baryon Structure in the Bag Theory*. Phys. Rev. D **10**, 2599 (1974).

[21] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, *New extended model of hadrons*. Phys. Rev. D **9**, 3471 (1974).

[22] C. Dappiaggi and N. Drago, *Constructing Hadamard States via an Extended Møller Operator*. Lett. Math. Phys. **106**, 1587–1615 (2016).

[23] C. Dappiaggi, N. Drago, H. Ferreira, *Fundamental solutions for the wave operator on static Lorentzian manifolds with timelike boundary*. Lett Math Phys **109**, 2157-2186 (2019).

[24] C. Dappiaggi, N. Drago, R. Longhi, *On Maxwell's Equations on Globally Hyperbolic Space-times with Timelike Boundary*. Ann. Henri Poincaré **21**, 2367-2409 (2020).

[25] C. Dappiaggi, F. Finster, S. Murro and E. Radici, *The Fermionic Signature Operator in De Sitter Spacetime*. J. Math. Anal. Appl. **485**, 123808 (2020).

[26] C. Dappiaggi, T.-P. Hack and N. Pinamonti, *The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor*. Rev. Math. Phys. **21**, 1241–1312 (2009).

[27] C. Dappiaggi, T.-P. Hack and K. Sanders, *Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss' law*. Commun. Math. Phys. **328**, 625 (2014).

[28] C. Dappiaggi and A. Marta, *A generalization of the propagation of singularities theorem on asymptotically anti-de Sitter spacetimes*. arXiv:2006.00560 (2020).

[29] C. Dappiaggi and A. Marta, *Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes*. Math. Phys., Anal. Geom. **24**, 28 (2021).

[30] C. Dappiaggi, V. Moretti and N. Pinamonti, *Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime*. Adv. Theor. Math. Phys. **15**, 355 (2011).
[31] C. Dappiaggi, S. Murro and A. Schenkel, *Non-existence of natural states for Abelian Chern-Simons theory*. J. Geom. Phys. **116**, 119-123 (2017).

[32] C. Dappiaggi, G. Nosari and N. Pinamonti, *The Casimir effect from the point of view of algebraic quantum field theory*. Math. Phys. Anal. Geom. **19**, 12 (2016).

[33] N. Drago and C. Gérard, *On the adiabatic limit of Hadamard states*. Lett. Math. Phys. **107**, 1409-1438 (2017).

[34] N. Drago, T-P. Hack and N. Pinamonti, *The generalized principle of Perturbative Agreement and the thermal mass*. Ann. Henri Poincaré **18**, 807 (2017).

[35] N. Drago, N. Ginoux and S. Murro, *On the Cauchy problem for hyperbolic Friedrichs systems with additional constraints*. in preparation.

[36] N. Drago, N. Große and S. Murro, *The Cauchy problem of the Lorentzian Dirac operator with APS boundary conditions*. arXiv:2104.00585 [math.AP] (2021).

[37] N. Drago and S. Murro, *A new class of Fermionic Projectors: Møller operators and mass oscillation properties*. Lett. Math. Phys. **107**, 2433-2451 (2017).

[38] C. J. Fewster, R. Verch, *The necessity of the Hadamard condition*. Class. Quantum Grav. **30**, 235027 (2013).

[39] F. Finster, S. Murro and C. Röken, *The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States*. J. Math. Phys. **57**, 072303 (2016).

[40] F. Finster, S. Murro and C. Röken, *The Fermionic Signature Operator and Quantum States in Rindler Space-Time*. J. Math. Anal. Appl. **454**, 385 (2017).

[41] F. Finster and C. Röken, *Self-adjointness of the Dirac Hamiltonian for a class of non-uniformly elliptic boundary value problems*. Annals of Mathematical Sciences and Applications **1**, 301 (2016).

[42] F. Finster and C. Röken, *An integral spectral representation for the massive Dirac propagator in Kerr geometry in Eddington-Finkelstein-type coordinates*. Adv. Theor. Math. Phys. **22**, no. 1, 47-92 (2018).

[43] K. Fredenhagen and K. Rejzner, *Quantum field theory on curved spacetimes: Axiomatic framework and examples*. J. Math. Phys. **57**, 031101 (2016).

[44] K.O. Friedrichs, *Symmetric positive linear differential equations*. Comm. Pure Appl. Math. **11**, 333-418 (1958).

[45] S.A. Fulling, F.J. Narcowich and R.M. Wald, *Singularity structure of the two-point function in quantum field theory in curved spacetime, II*. Annals of Physics **136** (1981), 243.

[46] O. Gannot and M. Wrochna, *Propagation of singularities on AdS spacetimes for general boundary conditions and the holographic Hadamard condition*. to appear in J. Inst. Math. Jussieu.

[47] C. Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes*. ESI Lectures in Mathematics and Physics (2019).

[48] C. Gérard, T. Stoskopf, *Hadamard states for quantized Dirac fields on Lorentzian manifolds of bounded geometry*, arXiv:2108.11630 [math.AP] (2021).
[49] N. Ginoux and S. Murro, *On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary*. arXiv:2007.02544 [math.AP] (2020).

[50] N. Große and S. Murro, *The well-posedness of the Cauchy problem for the Dirac operator on globally hyperbolic manifolds with timelike boundary*. Documenta Math. **25**, 737–765 (2020).

[51] R.W. Haymaker and T. Goldman, *Bag boundary conditions for confinement in the $q\bar{q}$ relative coordinate*. Phys. Rev. D **24**, 743 (1981).

[52] S. Hollands and R. M. Wald, *Existence of local covariant time ordered products of quantum fields in curved spacetime*. Commun. Math. Phys. **231**, 309 (2002).

[53] S. Hollands and R. M. Wald, *Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes*. Rev. Math. Phys. **17**, 227 (2005).

[54] B.S. Kay, R.M.Wald, *Theorems on the uniqueness and thermal properties of stationary, non-singular, quasifree states on spacetimes with a bifurcate Killing horizon*. Phys. Rep. 207(2), 49-136 (1991).

[55] I. Khavkine and V. Moretti, *Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction*. Advances in Algebraic Quantum Field Theory. Springer International Publishing, 2015.

[56] G. Idelon-Riton, *Scattering theory for the Dirac equation on the Schwarzschild-anti-de Sitter space-time*. Adv. Theor. Math. Phys. **22**, no.4, 1007–1069 (2018).

[57] V.Ya. Ivrii, *Microlocal Analysis, Sharp Spectral Asymptotics and Applications I*. Springer (2019).

[58] P.D. Lax and R.S. Phillips, *Local Boundary Conditions for Dissipative Symmetric Linear Differential Operators*. Comm. Pure Appl. Math. **13**, 427–455, (1960).

[59] R.B. Melrose, *The Atiyah-Patodi-Singer Index Theorem*. volume 4 of Research Notes in Mathematics. A K Peters Ltd., Wellesley, MA, (1993).

[60] R.B. Melrose and J. Sjöstrand, *Singularities of boundary value problems. i*. Comm. Pure Appl. Math. **31** (5), 593–617 (1978).

[61] R.B. Melrose and J. Sjöstrand, *Singularities of boundary value problems. ii*. Comm. Pure Appl. Math. **35** (2), 129–168 (1982).

[62] V. Moretti, S. Murro and D. Volpe, *Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theories*. arXiv:2109.06685 [math-ph] (2021).

[63] S. Murro and D. Volpe, *Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds*. Ann. Glob. Anal. Geom. **59**, 1–25 (2021).

[64] M. Nardmann, *Pseudo-Riemannian metrics with prescribed scalar curvature*. PhD thesis, Universität Leipzig, 2004.

[65] W. Pusz and S.L. Woronowicz, *Passive States and KMS States for General Quantum Systems*. Comm. Math. Phys. **58**, 273–290 (1978).

[66] M. J. Radzikowski, *Microlocal approach to the Hadamard condition in quantum field theory on curved space-time*. Commun. Math. Phys. **179**, 529–553 (1996).

[67] M. J. Radzikowski (with an Appendix by Rainer Verch), *A Local-to-Global Singularity Theorem for Quantum Field Theory on Curved Space-Time*. Commun. Math. Phys. **180**, 1-22 (1996).
[68] H. Sahlmann and R. Verch, *Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved space-time*. Rev. Math. Phys. **13**, 1203-1246 (2001).

[69] H. Sahlmann and R. Verch, *Passivity and Microlocal Spectrum Condition*. Comm. Math. Phys. **241**, 705-731 (2000).

[70] K. Sanders, *Equivalence of the (Generalised) Hadamard and Microlocal Spectrum Condition for (Generalised) Free Fields in Curved Spacetime*. Commun. Math. Phys. 295, 485-501 (2010).

[71] A. Seyedzahedi, R. Saghian and S.S. Gousheh, *Fermionic Casimir energy in a three-dimensional box*. Phys. Rev. A **82**, 032517 (2010).

[72] M.E. Taylor, *Reflection of Singularities of Solutions to Systems of Differential Equations*. Comm. Pure Appl. Math. **23** 457-478 (1975).

[73] A. Vasy, *Propagation of singularities for the wave equation on manifolds with corners*. Ann. Math. **168** 749-812, (2008).

[74] M. Wrochna, *The holographic Hadamard condition on asymptotically Anti-de Sitter spacetimes*. Lett. Math. Phys. **107**, 2291 (2017).

[75] J. Zahn, *Generalized Wentzell Boundary Conditions and Quantum Field Theory*. Ann. Henri Poincaré, **19**, 163187 (2018).