Green operators in low regularity spacetimes and quantum field theory

In this paper we will be considering solutions of the wave equation on orientable spacetimes (M, g) endowed with a C^1,1 metric. Note that although the metric is only C^1,1 we will always assume that the manifold has a smooth structure. The concept of global hyperbolicity (for smooth metrics) was introduced by Leray [25] as a condition to ensure the existence of unique solutions to hyperbolic equations and in particular the Cauchy problem for the wave equation is well-posed for smooth globally hyperbolic spacetimes [3, theorem 3.2.11 p 84ff]. For our situation it is therefore natural to consider globally hyperbolic spacetimes with C^1,1 metrics. Global hyperbolicity is the strongest of the conditions on (M, g) in the causal hierarchy of spacetimes [26] and recently there has been considerable interest in looking at the causal properties of low-regularity spacetimes [8, 18]. It was shown explicitly by Chrusćiel [27] that essentially all of causality theory for smooth spacetimes goes through to the C² case. However in the proofs of these results an important role is played by the Gauss lemma and the existence of totally normal (convex) neighbourhoods whose existence is not at all obvious in the C^1,1 case. However it was shown in [8, 28, 29] that such neighbourhoods do exist and the exponential map gives a local lipeomorphism. This enables essentially the whole of the results of standard smooth causality theory to go through to the C^1,1 case (see [8, 29] for details).

For spacetimes with a C² metric there are four equivalent notions of global hyperbolicity (see for example [26, section 3.11 p 340ff.]). These are:

• (a)  
  Compactness of the causal diamonds and causality⁴,
• (b)  
  Compactness of the space of causal curves connecting two points and causality [25],
• (c)  
  Existence of a Cauchy hypersurface,
• (d)  
  The metric splitting of the spacetime.

For C^1,1 spacetimes we will adopt the first definition. However for non-totally imprisoned [31] C⁰ spacetimes (and hence in particular for globally hyperbolic C^1,1 spacetimes) these four definitions remain equivalent [17]. See also [18, theorem 2.45] for a more general notion formulated in terms of closed cone structures.

In our constructions below we will make use of time functions and temporal functions. A time function is a function that is strictly increasing along every causal curve while a temporal function has the additional property that its gradient is everywhere past-directed and timelike. It is shown by Minguzzi [18 theorem 2.30] (see also [32]) that for a stably causal closed cone structure (and hence in particular for a C^1,1 globally hyperbolic spacetime) there exists a smooth temporal function $t:M\to \mathbb{R}$. Furthermore every globally hyperbolic closed cone structure is the domain of dependence of a stable Cauchy surface (see definition below) Σ, so that M = D(Σ) and that M is the topological product $\mathbb{R}{\times}{\Sigma}$ where the first projection is t and the level surfaces Σ_τ = {x ∈ M : t(x) = τ} are diffeomorphic to Σ [18, theorem 2.42]. So that although the metric is only C^1,1 the topological splitting remains smooth.

In the case of a smooth metric Bernal and Sánchez [33] show that given a smooth spacelike Cauchy hypersurface Σ there exists a smooth temporal function t such that Σ = t⁻¹(0). However in the case of a non-smooth metric the temporal function they construct will not be smooth. To generalise the results of [33] to the non-smooth case we need the concept of a stable Cauchy hypersurface introduced by Minguzzi in [18]. These are Cauchy hypersurfaces which are also Cauchy hypersurfaces for some metric g' ≻ g with strictly wider lightcones than g.

Bernard and Suhr [34, corollary 2.4] show that a smooth spacelike Cauchy hypersurface is a stable Cauchy hypersurface and that furthermore one can construct a smooth temporal function such that Σ = t⁻¹(0) [34, theorem 1]. A full discussion of this issue is given in the paper by Minguzzi [35]. The approach in [35] is complementary to that in [34] and consists of using topological arguments to show that the causal cones can be widened while preserving the Cauchy property of the hypersurface. One may then use the methods of Bernal and Sánchez [33] to construct a smooth time function with Σ = t⁻¹(0) which as shown in [18] is a smooth temporal function for the original spacetime (M, g). See [35, theorem 2.22] for details. Indeed given two smooth spacelike Cauchy hypersurfaces Σ₀ and Σ₁ with Σ₁ ⊂ J⁺(Σ₀)\Σ₀ one can find a smooth temporal function t that interpolates between them so that Σ₀ ⊂ t⁻¹(0) and Σ₁ ⊂ t⁻¹(1) [35, theorem 2.23].

In this section we extend the results of section 3 to a globally hyperbolic C^1,1 spacetime (M, g). The main result is theorem 4.7 which establishes the existence and uniqueness of ${H}_{\mathrm{loc}}^{2}\left(M\right)$ solutions to the wave equation for a globally hyperbolic C^1,1 spacetime. We start by obtaining an energy inequality which we use to establish uniqueness and the causal support properties of solutions to the wave equation.

Lemma 4.1. (energy inequality). [4, lemma 7.4.4]. Let (M, g) be an (n + 1)-dimensional C^1,1 globally hyperbolic spacetime with Σ a smooth spacelike n-dimensional Cauchy surface and t a smooth temporal function such that Σ = Σ₀ := t⁻¹(0). Let U ⊂ M be an open set with compact closure and let U⁺ := U ∩ J⁺(Σ) be such that ∂U ∩ Ū⁺ is achronal. Then if u is a (weak) ${H}_{\mathrm{loc}}^{2}\left(M\right)$ solution of □_gu = f where $f\in {L}_{\mathrm{loc}}^{2}\left(M\right)$ then

$$\begin{equation}{{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {U}^{+}\right)}{\leqslant}K\left({{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{0}\cap {U}^{+}\right)}+{{\Vert}f{\Vert}}_{{L}^{2}\left({U}_{\tau }\right)}\right)\end{equation} \tag{ 4.1 }$$

where ${U}_{\tau }=\left\{q\in U:0{\leqslant}t\left(q\right){\leqslant}\tau \right\}$.

Proof. To establish the energy inequality we follow Hawking and Ellis by applying the divergence theorem to an enhanced energy–momentum tensor [4, lemma 7.4.4]. Let

$$\begin{equation*}{\left(1\right){T}}_{\alpha \beta }{:=}{\nabla }_{\alpha }u{\nabla }_{\beta }u-\frac{1}{2}\left({g}^{\rho \sigma }{\nabla }_{\rho }u{\nabla }_{\sigma }u\right){g}_{\alpha \beta }.\end{equation*}$$

be the energy momentum tensor of the scalar field. Then ${\left(1\right){T}}_{\alpha \beta }$ has vanishing divergence and satisfies the dominant energy condition. We now follow [4] and modify this by adding on the term

$$\begin{align}\hfill {\left(0\right){T}}_{\alpha \beta }{:=}-\frac{1}{2}{g}_{\alpha \beta }{u}^{2}\end{align}$$

to obtain

$$\begin{equation*}{S}_{\alpha \beta }={\left(0\right){T}}_{\alpha \beta }+{\left(1\right){T}}_{\alpha \beta }\end{equation*}$$

which still satisfies the dominant energy condition. Let ξ_α = ∇_αt, then we obtain the required inequality by applying the divergence theorem to S^αβξ_α over the region ${U}_{\tau }=\left\{q\in U:0{\leqslant}t\left(q\right){\leqslant}\tau \right\}$. In order to do this we require that div(S^αβξ_α) should be integrable with respect to the volume form ν_g, and this is guaranteed by the compactness of Ū and the fact that our solution is in ${H}_{\mathrm{loc}}^{2}\left(M\right)$. In fact it is enough that the weak solutions have two derivatives in ${L}_{\mathrm{loc}}^{2}\left(M,g\right)$ if the metric and the timelike vector field are in the space C^0,1 (see [36]). The boundary of U_τ consists of three parts; the level surface Σ_τ ∩ Ū⁺, the level surface Σ₀ ∩ Ū⁺ and the remainder which we denote $\mathcal{H}$. Because of the dominant energy condition and the fact that ∂U ∩ Ū⁺ is achronal, the contribution to the surface integral from $\mathcal{H}$ is positive. We therefore obtain the following inequality:

$$\begin{equation}{\int }_{{{\Sigma}}_{\tau }\cap {\bar{U}}^{+}}{S}^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }{\mu }_{\tau }-{\int }_{{{\Sigma}}_{0}\cap {\bar{U}}^{+}}{S}^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }{\mu }_{0}{\leqslant}{\int }_{{U}_{\tau }}{\nabla }_{\alpha }\left({S}^{\alpha \beta }{\xi }_{\beta }\right){\nu }_{g}\end{equation} \tag{ 4.2 }$$

where ν_g is the volume form on U_τ given by g, and μ_τ is the volume form induced by g on Σ_τ.

We now define an energy type integral

$$\begin{equation*}E\left(\tau \right)={\int }_{{{\Sigma}}_{\tau }\cap {\bar{U}}^{+}}{S}^{\alpha \beta }{\xi }_{\alpha }{\xi }_{\beta }{\mu }_{\tau }.\end{equation*}$$

Then on Ū this is equivalent [37] to the restricted Sobolev norm (B.3)

$$\begin{equation*}{C}_{1}{{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {\bar{U}}^{+}\right)}{\leqslant}E\left(\tau \right){\leqslant}{C}_{2}{{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {\bar{U}}^{+}\right)}.\end{equation*}$$

Note that since the solution u is in ${H}_{\mathrm{loc}}^{2}\left(M\right)$ we have well-defined traces in ${\tilde {H}}^{1}\left({\Sigma}\right)$. In terms of the energy norm we may write (4.2) in the form

$$\begin{equation*}E\left(\tau \right){\leqslant}E\left(0\right)+{\int }_{{U}_{\tau }}\left(\left({\nabla }_{\alpha }{S}^{\alpha \beta }\right){\xi }_{\beta }+{S}^{\alpha \beta }{\nabla }_{\alpha }{\xi }_{\beta }\right){\nu }_{g}.\end{equation*}$$

Repeated application of the Cauchy–Schwarz inequality and □_gu = f then gives [37]

$$\begin{equation}E\left(\tau \right){\leqslant}E\left(0\right)+{C}_{1}{{\Vert}f{\Vert}}_{{L}^{2}\left({U}_{\tau }\right)}^{2}+{C}_{2}{\int }_{0}^{\tau }E\left(s\right)\mathrm{d}s\end{equation} \tag{ 4.3 }$$

which on applying Gronwall's inequality gives

$$\begin{equation*}E\left(\tau \right){\leqslant}\left(E\left(0\right)+{C}_{1}{{\Vert}f{\Vert}}_{{L}^{2}\left({U}_{\tau }\right)}^{2}\right){\mathrm{e}}^{{C}_{2}\tau }.\end{equation*}$$

In terms of the Sobolev type norms this gives

$$\begin{equation*}{{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {U}^{+}\right)}{\leqslant}K\left({{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{0}\cap {U}^{+}\right)}+{{\Vert}f{\Vert}}_{{L}^{2}\left({U}_{\tau }\right)}\right).\end{equation*}$$

□

We now use the energy inequality (4.1) to prove uniqueness of the solution as well as the causal support properties of the solution to the Cauchy problem.

Proposition 4.2. (uniqueness).Let (M, g) be a (connected, oriented, time oriented,) globally hyperbolic (n + 1)-dimensional Lorentzian manifold with C^1,1 metric and Σ a smooth n-dimensional spacelike Cauchy surface. Let u be a (weak) ${H}_{\mathrm{loc}}^{2}\left(M\right)$ solution of

$$\begin{equation*}{\square }_{g}u=f\end{equation*}$$

with $f\in {H}_{\mathrm{loc}}^{1}\left(M\right)$, which satisfies the initial conditions

$$\begin{align*}\hfill u{\vert }_{{\Sigma}}& ={u}_{0},\hfill \\ \hfill {\nabla }_{n}u{\vert }_{{\Sigma}}& ={u}_{1},\quad \mathrm{where}\,n\,\mathrm{is}\,\mathrm{the}\,\mathrm{unit}\,\mathrm{normal}\,\mathrm{to}\,{\Sigma}\hfill \end{align*}$$

with (u₀, u₁) ∈ H²(Σ) × H¹(Σ). Then u is unique.

Proof. Let q ∈ M and without loss of generality suppose that q ∈ I⁺(Σ). Then since our spacetime is globally hyperbolic we may find a p such that q ∈ I⁻(p) ∩ I⁺(Σ) := U where by C^1,1 causality theory U has compact closure [8]. Now suppose there exist two solutions u and to the above initial value problem. Then applying lemma 4.1 to := u − over the region U⁺ gives

$$\begin{equation*}{{\Vert}\hat{u}{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {U}^{+}\right)}{\leqslant}K{{\Vert}\hat{u}{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{0}\cap {U}^{+}\right)}=0.\end{equation*}$$

Hence ${{\Vert}\hat{u}{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {U}^{+}\right)}=0$ so that and ∇ vanish in U⁺. Since q is arbitrary the solution is unique in D⁺(Σ). A similar result applies to D⁻(Σ), so we have uniqueness in the whole of M = D(Σ). □

Proposition 4.3. (causal support). Let (M, g) be a connected, oriented, time oriented, globally hyperbolic (n + 1)-dimensional Lorentzian manifold with C^1,1 metric, Σ₀ a smooth spacelike n-dimensional Cauchy surface and t a smooth temporal function such that t⁻¹(0) = Σ₀. Let $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ be a (weak) solution to the initial value problem

$$\begin{align*}\hfill {\square }_{g}u& =f\quad \text{on}\;M\text{,}\hfill \\ \hfill u& ={u}_{0}\quad \text{on}\;{{\Sigma}}_{0}\text{,}\hfill \\ \hfill {\nabla }_{n}& ={u}_{1}\quad \text{on}\;{{\Sigma}}_{0}\text{.}\hfill \end{align*}$$

Then $\mathrm{supp}\left(u\right)\subset J\left(\mathrm{supp}\left({u}_{0}\right)\cup \mathrm{supp}\left({u}_{1}\right)\cup \mathrm{supp}\left(f\right)\right)$.

Proof. We prove the result for J⁺. A similar proof holds for J⁻. Let $V={J}^{+}\left(\mathrm{supp}\left({u}_{0}\right)\cup \mathrm{supp}\left({u}_{1}\right)\cup \mathrm{supp}\left(f\right)\right)$ and suppose q ∈ M\V. Then we may find a point p ∈ D⁺(Σ₀) such that q ∈ I⁻(p) ∩ I⁺(Σ₀) and u and ∇u vanish on J⁻(p) ∩ Σ₀ and f vanishes on J⁻(p) ∩ J⁺(Σ₀). Now let U = I⁻(p) ∩ I⁺(Σ₀) and apply (4.3) on this region to obtain

$$\begin{equation*}{{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{\tau }\cap {J}^{+}\left({{\Sigma}}_{0}\right)\cap {I}^{-}\left(p\right)\right)}{\leqslant}K\left({{\Vert}u{\Vert}}_{{\tilde {H}}^{1}\left({{\Sigma}}_{0}\cap {J}^{-}\left(p\right)\right)}+{{\Vert}f{\Vert}}_{{L}^{2}\left({J}^{+}\left({{\Sigma}}_{0}\right)\cap {J}^{-}\left(p\right)\right)}\right),\quad \text{for}\enspace 0{\leqslant}\tau {\leqslant}t\left(p\right).\end{equation*}$$

But by the choice of p the right-hand side vanishes, so u must also vanish in the region U with τ in the range 0 ⩽ τ ⩽ t(p). Hence u vanishes on a neighbourhood of q. Since q was arbitrary, u vanishes on M\V which proves the result. □

To establish existence on M, we need the following two lemmas from Ringström [5]. In both cases the proof given in [5] for the smooth case goes through to that of a C^1,1 metric unchanged.

Lemma 4.4. (Ringström [5, lemma 12.5]). Let (M, g) be an (n + 1)-dimensional Lorentzian manifold with C^1,1 metric and Σ a smooth spacelike n-dimensional submanifold. If p ∈ S there is a chart (U, x) with p ∈ U and x = (x⁰, x¹, ..., xⁿ) such that q ∈ U ∩ Σ if and only if q ∈ U and x⁰(q) = 0. Furthermore we may choose x so that $\frac{\partial }{\partial {x}^{0}}$ is the future directed unit normal to Σ for q ∈ Σ ∩ U.

If we fix > 0 and let ${g}_{\mu \nu }{:=}g\left(\frac{\partial }{\partial {x}^{\mu }},\frac{\partial }{\partial {x}^{\nu }}\right)$, then we can assume U to be such that |g_0i| ⩽ , i = 1, ..., n on U. If we let a = g₀₀(p) and b > 0 be such that g_ij(p) regarded as a positive definite matrix is bounded below by b (i.e. g_ij(p)ξⁱξ^j > b|ξ|²) then we may assume that g₀₀(q) < a/2 and g_ij(q) regarded as a positive definite matrix is bounded below by b/2 for q ∈ U.

Lemma 4.5. (Ringström [5, lemma 12.16]). Let (M, g) be a (connected, oriented, time oriented,) globally hyperbolic (n + 1)-dimensional Lorentzian manifold with C^1,1 metric and Σ a smooth spacelike n-dimensional Cauchy surface. Let t be a (smooth) temporal function (as given by [34, 35]) with ${t}^{-1}\left(0\right)={{\Sigma}}_{{t}_{0}}$. If $p\in {{\Sigma}}_{{t}_{0}}$ there is an > 0 and open neighbourhoods U, W of p such that

• (a)  
  The closure of W is compact and contained in U;
• (b)  
  If q ∈ W and τ ∈ [t₀ − , t₀ + ], then J⁺(Σ_τ) ∩ J⁻(q) is compact and contained in U;
• (c)  
  There is a chart (U, ϕ) with ϕ = (x⁰, ..., xⁿ) and x⁰ = t, such that there exist a, b > 0 with g₀₀(q) < − a and g_ij(q)ξⁱξ^j ⩾ bδ_ijξⁱξ^j for q ∈ U;
• (d)  
  For any compact K ⊂ U there is a C^1,1 matrix valued function h on ${\mathbb{R}}^{n+1}$ such that h_μν = g_μ,ν◦ϕ⁻¹ on ϕ⁻¹(K) and such that there are positive constants a₁, b₁ c₁ with h₀₀ ⩽ −a₁, h_ijξⁱξ^j ⩾ b₁δ_ijξⁱξ^j and |h_μν| ⩽ c₁ on all of ${\mathbb{R}}^{n+1}$.

Note that although the proof is identical to that in [5] it relies on the C^1,1 causality results of [8] and [34, theorem 1]. Note also that in point (d) above the matrix valued function is only C^1,1 rather than smooth as it is in [5, lemma 12.16].

We are now in a position to establish existence.

Proposition 4.6. (existence for compactly supported source and initial data). Let (M, g) be a time oriented (n + 1)-dimensional Lorentzian manifold with C^1,1 metric and Σ a smooth spacelike n-dimensional hypersurface. Let t be a smooth temporal function with t⁻¹(0) = Σ and let n be the future directed timelike unit normal to Σ. Given initial data $\left({u}_{0},{u}_{1}\right)\in {H}_{\mathrm{comp}}^{2}\left({\Sigma}\right){\times}{H}_{\mathrm{comp}}^{1}\left({\Sigma}\right)$ and source $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$, then there exists a (weak) solution $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ to the initial value problem

$$\begin{align*}\hfill {\square }_{g}u& =f\quad \text{on}\;M\text{,}\hfill \\ \hfill u& ={u}_{0}\quad \text{on}\;{\Sigma}\text{,}\hfill \\ \hfill {\nabla }_{n}u& ={u}_{1}\quad \text{on}\;{\Sigma}\text{.}\hfill \end{align*}$$

Proof. We again closely follow Ringström [5, theorem 12.17]. Let K₁ ⊂ Σ be a compact set such that supp(u₀) ∪ supp(u₁) ⊂ K₁ and K₂ ⊂ M a compact set such that supp(f) ⊂ K₂. Let t₁ > 0 and define ${R}_{{t}_{1}}$ to be the set of q such that 0 ⩽ t(q) ⩽ t₁. Then ${R}_{{t}_{1}}$ is closed and ${K}_{3}={K}_{2}\cap {R}_{{t}_{1}}$ is compact. The union of I⁺(p) for p ∈ I⁻(Σ) is an open cover of K₁ ∪ K₃, so there is a finite number of points p₁, ..., p_ℓ such that the I⁺(p_i) are a finite subcover of K₁ ∪ K₃. Note that the set $F={\bigcup }_{i=1}^{\ell }{J}^{+}\left({p}_{i}\right)\cap {J}^{-}\left({{\Sigma}}_{{t}_{1}}\right)$ is compact and that if there is a solution in ${R}_{{t}_{1}}$ it has to be zero in ${R}_{{t}_{1}}{\backslash}F$ by proposition 4.3. We now show that there is a solution in the compact set ${R}_{{t}_{1}}\cap F$.

Let F_τ := F ∩ Σ_τ. Let 0 ⩽ τ < t₁ and assume we have a solution in the function space specified in the proposition up to time τ, i.e. on R_τ or for every R_s with 0 ⩽ s < τ. We now extend the solution to the future of τ. For every p ∈ F_τ there are neighbourhoods U_p, W_p and _p with the properties of lemma 4.5. By compactness there is a finite number of points ${\tilde {p}}_{i},\dots ,\;{\tilde {p}}_{N}$ such that the ${W}_{{\tilde {p}}_{i}}$ cover F_τ. Let $0{< }{\epsilon}{\leqslant}\mathrm{min}\left\{{{\epsilon}}_{{\tilde {p}}_{1}},\dots ,\;{{\epsilon}}_{{\tilde {p}}_{N}}\right\}$ be such that

$$\begin{equation}{F}_{s}\subset \bigcup _{i=1}^{N}{W}_{{\tilde {p}}_{i}}\end{equation} \tag{ 4.4 }$$

for all s ∈ [τ − , τ + ]. Now let s₁ ∈ [τ − , τ] be such that there is a solution, in the function space specified in the proposition up to and including s₁ and let p ∈ F_s for any s ∈ [s₁, τ + ]. Then ${K}_{p}{:=}{J}^{-}\left(p\right)\cap {J}^{+}\left({{\Sigma}}_{{s}_{1}}\right)$ is compact and contained in one of the charts, say $\left({U}_{{\tilde {p}}_{k}},\phi \right)$. Let $\chi \in {C}_{0}^{\infty }\left({U}_{{\tilde {p}}_{k}}\right)$ be such that χ(q) = 1 for all q ∈ K_p. Then we use our solution up to time s₁ to define new initial data on ${{\Sigma}}_{{s}_{1}}$ given by ${\tilde{u}}_{0}{:=}\left(\chi u\right){\vert }_{{{\Sigma}}_{{s}_{1}}}\in {H}^{2}\left({{\Sigma}}_{{s}_{1}}\right)$ and ${\tilde{u}}_{1}{:=}\left(\chi {\nabla }_{n}u\right){\vert }_{{S}_{{s}_{1}}}\in {H}^{1}\left({{\Sigma}}_{{s}_{1}}\right)$ and source $\tilde {f}{:=}\chi f$. These all have their support within ${U}_{{\tilde {p}}_{k}}$ so we may use the chart $\left({U}_{{\tilde {p}}_{k}},\phi \right)$ to regard these as data and source on the whole of ${\mathbb{R}}^{n}$ and ${\mathbb{R}}^{n+1}$ respectively. We may also extend the Lorentz metric g_μν◦ϕ⁻¹ to a Lorentz matrix-valued function h_μν on the whole of ${\mathbb{R}}^{n+1}$ which coincides with g_μν◦ϕ⁻¹ on K_p. We may therefore regard the tildered version as an initial value problem on ${\mathbb{R}}^{n+1}$. The third condition of lemma 4.5 and the fact that the solution is in ${C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ ensures that we may apply lemma 3.3 to obtain a solution on ${\mathbb{R}}^{n+1}$ which on $\phi \left({U}_{{\tilde {p}}_{k}}\right)$ may be transferred back to give a solution on K_p. In the region ${V}_{p}{:=}{I}^{-}\left(p\right)\cap {J}^{+}\left({{\Sigma}}_{{s}_{1}}\right)$ we define u to be this solution. In the region V_p ∩ V_q then uniqueness ensures that the two potential solutions coincide. We now define O₁ to be the union of the V_p for p ∈ F_s, s ∈ [s₁, τ + ] then the above construction defines a unique solution in O₁. Note that the interior of O₁ contains F_s for all s ∈ (s₁, τ + ). Now define O₂ to be the set of points for which s₁ ⩽ t(q) < τ + and for which q ∉ F. We want to define the solution to be zero in this set, however we need to check that there is no contradiction for points in both O₁ and O₂. If q ∈ O₂ ∩ O₁ with t(q) > s₁ then both u and ∇u vanish at ${J}^{-}\left(q\right)\cap {S}_{{s}_{1}}$ and f vanishes in ${J}^{-}\left(q\right)\cap {J}^{+}\left({{\Sigma}}_{{s}_{1}}\right)$. Furthermore there is an r such that q ∈ V_r ⊂ O₁. So by uniqueness the solution defined on O₁ has to vanish for a sufficiently small neighbourhood at q.

In summary we have shown that if there exists a solution for all s < τ or up to time τ in the required function space, we get a solution in the same space on the larger region R_τ+ for some > 0. Let $\mathcal{A}$ be the set of s ∈ [0, ∞) such that there is a solution up to time s. Taking τ = 0 in the above we have a solution on R so $\mathcal{A}$ is not empty. We have also shown that if $\tau \in \mathcal{A}$ then a solution exists for [0, τ + ), so that for any τ > 0 we may find an open interval containing τ in which a solution exists. Thus $\mathcal{A}$ is open in the relative topology of [0, ∞). Finally we note that by definition $\mathcal{A}$ is also closed in [0, ∞) because it contains its limit points. Then this set is open, closed and non-empty, so it must be the whole of [0, ∞) and we have a solution for all future times. By an analogous argument interchanging past and future we also have a solution for all past times and hence on the whole of M. □

Theorem 4.7. (global existence and uniqueness). Let (M, g) be a connected, oriented, time oriented (n + 1)-dimensional Lorentzian globally hyperbolic manifold with C^1,1 metric and Σ a smooth spacelike n-dimensional Cauchy hypersurface. Let t be a smooth temporal function with t⁻¹(0) = Σ and let n be the future directed timelike unit normal to Σ.

Given initial data (u₀, u₁) ∈ H²(Σ) × H¹(Σ) and source $f\in {C}^{0}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{1}\left(M\right)$ then there exists a unique (weak) solution $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ to the initial value problem

$$\begin{align*}\hfill {\square }_{g}u& =f\quad \text{on}\;M\text{,}\hfill \\ \hfill u& ={u}_{0}\quad \text{on}\;{\Sigma}\text{,}\hfill \\ \hfill {\nabla }_{n}u& ={u}_{1}\quad \text{on}\;{\Sigma}\text{.}\hfill \end{align*}$$

Moreover $\mathrm{supp}\left(u\right)\subset J\left(\mathrm{supp}\left({u}_{0}\right)\cup \mathrm{supp}\left({u}_{1}\right)\cup \mathrm{supp}\left(f\right)\right)$.

Proof. Let p be any point to the future of Σ. Then K_p = J⁻(p) ∩ J⁺(Σ) is a compact set. Let $\chi \in {C}_{0}^{\infty }\left(M\right)$ be such that χ(q) = 1 for all q ∈ K_p. Now define f' = χf, u₀' = χu₀ and u₁' = χu₁. Then by proposition 4.6 there is a unique solution u' to the primed initial value problem. Now set u = u' in V_p := I⁻(p) ∩ J⁺(Σ). Similarly given some other point $\tilde {p}$ to the future of Σ there is a unique solution ' to the initial value problem in ${V}_{\tilde {p}}{:=}{I}^{-}\left(\tilde {p}\right)\cap {J}^{+}\left({\Sigma}\right)$ and we may set u = ' in ${V}_{\tilde {p}}$. If we now consider some further point $q\in {V}_{p}\cap {V}_{\tilde {p}}$ then by uniqueness the two potential solutions, given by u' and ' respectively, agree in ${V}_{p}\cap {V}_{\tilde {p}}$, so there is no contradiction in setting u = u' in V_p and u = ' in ${V}_{\tilde {p}}$ since u' = ' in the intersection. Thus we have a solution on the whole of D⁺(Σ). A similar argument gives us a solution on D⁻(Σ) and hence on the whole of M. The solution is unique by proposition (4.2) and satisfies the causal support condition by proposition (4.3).

□

We also want to show that the initial value problem is well-posed in the sense of corollary 4.11 below. For solutions in ${H}_{\mathrm{loc}}^{1}\left(M\right)$ this follows immediately from the energy estimate (4.1). But well-posedness in ${H}_{\mathrm{loc}}^{2}\left(M\right)$ requires a higher order estimate which we now establish.

Lemma 4.8. For each compact subset K ⊂ M there exists a δ > 0 with the following property: if (u₀, u₁) ∈ H²(Σ) × H¹(Σ) with supp(u_j) ⊂ K ∩ Σ (j = 1, 2) and f ∈ H¹(M) with supp(f) ⊂ K, then the (weak) solution $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ of □_gu = f with initial data (u₀, u₁) satisfies the energy inequality

$$\begin{equation}{{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,\delta \right],{H}^{2}\left({{\Sigma}}_{t}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,\delta \right],{H}^{1}\left({{\Sigma}}_{t}\right)\right)}{\leqslant}C\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(M\right)}\right).\end{equation} \tag{ 4.5 }$$

Proof. We use a similar approach to that in the existence proof given in [3, theorem 3.2.11].

For every p ∈ K there are neighbourhoods U_p, W_p and _p > 0 with the properties of lemma 4.5. By compactness there is a finite number of points p₁, ..., p_N such that the corresponding ${W}_{{p}_{j}}$ cover K. Now let ${\left\{{\chi }_{j}\right\}}_{j=1}^{N}$ be a partition of unity of K subordinate to the ${W}_{{p}_{j}}$.

We now define

$$\begin{equation*}{u}_{0,j}{:=}{\chi }_{j}{u}_{0},\quad {u}_{1,j}{:=}{\chi }_{j}{u}_{1},\quad {f}_{j}{:=}{\chi }_{j}f,\end{equation*}$$

so that

$$\begin{equation*}{u}_{0}=\sum _{j=1}^{N}{u}_{0,j},\quad {u}_{1}=\sum _{j=1}^{N}{u}_{1,j},\quad f=\sum _{j=1}^{N}{f}_{j}.\end{equation*}$$

We also define

$$\begin{equation*}{K}_{j}{:=}\mathrm{supp}\left({u}_{0,j}\right)\cup \mathrm{supp}\left({u}_{1,j}\right)\cup \mathrm{supp}\left({f}_{j}\right)\subset K\cap \mathrm{supp}\left({\chi }_{j}\right)\subset {W}_{{p}_{j}}.\end{equation*}$$

Let u_j be the (weak) solution of the initial value problem

$$\begin{equation*}\square {u}_{j}={f}_{j},\quad {u}_{j}{\vert }_{{\Sigma}}={u}_{0,j},\quad {\nabla }_{n}{u}_{j}{\vert }_{{\Sigma}}={u}_{1,j}.\end{equation*}$$

We will employ (an implicit choice of a temporal function in terms of) the diffeomorphism $M\cong \mathbb{R}{\times}{\Sigma}$ and slightly abuse notation from now on by considering all functions u, u_j etc to be defined already on products I × Σ, where I is some open interval, thus suppressing the transfers of functions via restrictions of the underlying global diffeomorphism.

By point two of lemma 4.5 there exists an ${{\epsilon}}_{{p}_{j}}{ >}0$ such that

$$\begin{equation}\left(\left(-2{{\epsilon}}_{{p}_{j}},2{{\epsilon}}_{{p}_{j}}\right){\times}{\Sigma}\right)\cap J\left({K}_{j}\right)\subset {U}_{{p}_{j}}.\end{equation} \tag{ 4.6 }$$

Let $\delta =\mathrm{min}\left\{{{\epsilon}}_{{p}_{1}},\dots ,\;{{\epsilon}}_{{p}_{N}}\right\}$. Given the solutions u_j of the local problem we may extend them by zero on all of (−2δ, 2δ) × Σ and sum them to give our unique solution

$$\begin{equation*}u=\sum _{j=1}^{N}{u}_{j},\quad \text{on}\;\left(-2\delta ,2\delta \right){\times}{\Sigma}.\end{equation*}$$

Since by (4.6) each of the u_j lie entirely within some chart $\left({U}_{{p}_{j}},{\phi }_{{p}_{j}}\right)$ we may regard the initial value problem as one on $I{\times}{\mathbb{R}}^{n}$ where I is an interval chosen sufficiently large such that the images of (−2δ, 2δ) × Σ under all the ${\phi }_{{p}_{j}}$ are contained in $I{\times}{\mathbb{R}}^{n}$.

Then the third condition of lemma 4.5 enables us to transfer the basic energy estimate according to lemma 3.1 from $I{\times}{\mathbb{R}}^{n}$ to ones for u_j on U_j ⊂ M to give

$$\begin{equation*}{{\Vert}{u}_{j}{\Vert}}_{{C}^{0}\left(\left[0,\delta \right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}{u}_{j}{\Vert}}_{{C}^{1}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}{\leqslant}{C}_{j}\left({{\Vert}{u}_{0,j}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1,j}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}{f}_{j}{\Vert}}_{{L}^{2}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}\right).\end{equation*}$$

Now u = ∑_ju_j so that

$$\begin{align*}\hfill {{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,\delta \right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}& {\leqslant}\sum _{j=1}^{N}\left({{\Vert}{u}_{j}{\Vert}}_{{C}^{0}\left(\left[0,\delta \right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}{u}_{j}{\Vert}}_{{C}^{1}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}\right)\hfill \\ \hfill & {\leqslant}\sum _{j=1}^{N}{C}_{j}\left({{\Vert}{u}_{0,j}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1,j}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}{f}_{j}{\Vert}}_{{L}^{2}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}\right)\hfill \\ \hfill & {\leqslant}\sum _{j=1}^{N}{C}_{j}\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{L}^{2}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}\right)\hfill \\ \hfill & {\leqslant}C\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{L}^{2}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}\right),\hfill \end{align*}$$

where we may replace ${{\Vert}f{\Vert}}_{{L}^{2}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}$ by the larger value ${{\Vert}f{\Vert}}_{{H}^{1}\left(M\right)}$, since $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$. □

We remark that in the above proof (and formulation of the result) we have replaced the norm ${{\Vert}f{\Vert}}_{{L}^{2}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}$ by ${{\Vert}f{\Vert}}_{{H}^{1}\left(M\right)}$, which is valid for $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$, to avoid the need to specify a particular choice of a temporal function.

Proposition 4.9. (global higher energy estimates). Let (M, g) be a time oriented (n + 1)-dimensional Lorentzian manifold with C^1,1 metric and Σ a smooth spacelike n-dimensional hypersurface. Let t be a smooth temporal function with Σ = t⁻¹(0) and let n be the future directed timelike unit normal to Σ.

Given initial data $\left({u}_{0},{u}_{1}\right)\in {H}_{\mathrm{comp}}^{2}\left({\Sigma}\right){\times}{H}_{\mathrm{comp}}^{1}\left({\Sigma}\right)$ and source $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$, then the (weak) solution $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ satisfies

$$\begin{equation*}{{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,T\right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,T\right],{H}^{1}\left({\Sigma}\right)\right)}{\leqslant}C\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(M\right)}\right)\end{equation*}$$

for any interval [0, T].

Proof. We first use lemma 4.8 to obtain an estimate for the data ${\hat{u}}_{0}{:=}u{\vert }_{{{\Sigma}}_{\delta }}$ and ${\hat{u}}_{1}{:=}{\nabla }_{n}u{\vert }_{{{\Sigma}}_{\delta }}$ induced by u on Σ_δ. It follows from (4.5) and the fact that $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)\cap {H}_{\mathrm{loc}}^{2}\left(M\right)$ that

$$\begin{equation*}{{\Vert}{\hat{u}}_{0}{\Vert}}_{{H}^{2}\left({{\Sigma}}_{\delta }\right)}+{{\Vert}{\hat{u}}_{1}{\Vert}}_{{H}^{1}\left({{\Sigma}}_{\delta }\right)}{\leqslant}\tilde {C}\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[0,\delta \right]{\times}{\Sigma}\right)}\right).\end{equation*}$$

Now applying lemma 4.8 to the initial surface Σ_δ we obtain a $\hat{\delta }{ >}\delta$ such that

$$\begin{align*}\hfill & {{\Vert}u{\Vert}}_{{C}^{0}\left(\left[\delta ,\hat{\delta }\right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[\delta ,\hat{\delta }\right],{H}^{1}\left({\Sigma}\right)\right)}{\leqslant}\hat{C}\left({{\Vert}{\hat{u}}_{0}{\Vert}}_{{H}^{2}\left({{\Sigma}}_{\delta }\right)}+{{\Vert}{\hat{u}}_{1}{\Vert}}_{{H}^{1}\left({{\Sigma}}_{\delta }\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[\delta ,\hat{\delta }\right]{\times}{\Sigma}\right)}\right)\hfill \\ \hfill & {\leqslant}\hat{C}\left(\tilde {C}\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[0,\delta \right]{\times}{\Sigma}\right)}\right)+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[\delta ,\hat{\delta }\right]{\times}{\Sigma}\right)}\right)\hfill \\ \hfill & {\leqslant}{C}_{1}\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[0,\hat{\delta }\right]{\times}{\Sigma}\right)}\right).\hfill \end{align*}$$

Combining the two energy inequalities on [0, δ] and $\left[\delta ,\hat{\delta }\right]$ we have

$$\begin{align*}\hfill & {{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,\hat{\delta }\right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,\hat{\delta }\right],{H}^{1}\left({\Sigma}\right)\right)}\hfill \\ \hfill & {\leqslant}{{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,\delta \right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,\delta \right],{H}^{1}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{0}\left(\left[\delta ,\hat{\delta }\right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[\delta ,\hat{\delta }\right],{H}^{1}\left({\Sigma}\right)\right)}\hfill \\ \hfill & {\leqslant}C\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{\tilde{u}}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[0,\delta \right]{\times}{\Sigma}\right)}\right)+{C}_{1}\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[0,\hat{\delta }\right]{\times}{\Sigma}\right)}\right)\hfill \\ \hfill & {\leqslant}{C}_{2}\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{\tilde{u}}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(\left[0,\hat{\delta }\right]{\times}{\Sigma}\right)}\right).\hfill \end{align*}$$

This shows that we may extend the energy inequality from [0, δ] to the larger time interval $\left[0,\hat{\delta }\right]$ by repeatedly applying lemma 4.8.

Similarly, for every τ ∈ [0, T] we may find a δ(τ) > 0 such that lemma 4.8 applies to the time interval (τ − δ(τ), τ + δ(τ)) with initial data given on Σ_τ. By compactness, finitely many intervals (τ_k − δ(τ_k), τ_k + δ(τ_k)) (k = 1, ..., m) cover [0, T] and the energy inequalities on these may be combined. □

We now use the above proposition to obtain a spacetime energy inequality.

Proposition 4.10. (higher order spacetime energy estimates). Let (M, g) be a time oriented (n + 1)-dimensional Lorentzian manifold with C^1,1 metric and Σ a smooth spacelike n-dimensional hypersurface. Given initial data $\left({u}_{0},{u}_{1}\right)\in {H}_{\mathrm{comp}}^{2}\left({\Sigma}\right){\times}{H}_{\mathrm{comp}}^{1}\left({\Sigma}\right)$ and source $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$, then the (weak) solution u satisfies the energy-inequality

$$\begin{equation}{{\Vert}u{\Vert}}_{{H}^{2}\left(K\right)}{\leqslant}C\left({{\Vert}{u}_{0}{\Vert}}_{{H}^{2}\left({\Sigma}\right)}+{{\Vert}{u}_{1}{\Vert}}_{{H}^{1}\left({\Sigma}\right)}+{{\Vert}f{\Vert}}_{{H}^{1}\left(M\right)}\right).\end{equation} \tag{ 4.7 }$$

for any compact K ⊂ M.

Proof. Without loss of generality we may assume that K ⊂ [0, T] × Σ. Due to the regularity of $u\in {C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)$ we have control of the second order spatial derivatives, the second order mixed derivatives and the lower order terms

$$\begin{equation}\mathrm{max}\left({{\Vert}{\partial }_{i}{\partial }_{j}u{\Vert}}_{{L}^{2}\left(K\right)},{{\Vert}{\partial }_{j}u{\Vert}}_{{L}^{2}\left(K\right)}\right){\leqslant}C\left({{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,T\right],{H}^{2}\left({\Sigma}\right)\right)}\right),\end{equation} \tag{ 4.8 }$$

$$\begin{equation}\mathrm{max}\left({{\Vert}{\partial }_{i}{\partial }_{t}u{\Vert}}_{{L}^{2}\left(K\right)},{{\Vert}{\partial }_{t}u{\Vert}}_{{L}^{2}\left(K\right)}\right){\leqslant}C\left({{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,T\right],{H}^{1}\left({\Sigma}\right)\right)}\right).\end{equation} \tag{ 4.9 }$$

In order to obtain the required estimate we also need to control ∂_ttu in the L²(K) norm. Using □_gu = f we have

$$\begin{equation*}{{\Vert}{g}^{00}{\partial }_{tt}u{\Vert}}_{{L}^{2}\left(K\right)}={{\Vert}f+\left(-{g}^{ti}{\partial }_{t}{\partial }_{i}-{g}^{ij}{\partial }_{j}{\partial }_{i}+{g}^{\alpha \beta }{{\Gamma}}_{\alpha \beta }^{\gamma }{\partial }_{\gamma }\right)u{\Vert}}_{{L}^{2}\left(K\right)}\end{equation*}$$

From (4.8) and (4.9), the regularities of f and g, we obtain an L² estimate for ∂_ttu,

$$\begin{align*}\hfill {{\Vert}{\partial }_{tt}u{\Vert}}_{{L}^{2}\left(K\right)}& ={C}_{1}{{\Vert}f+\left(-{g}^{ti}{\partial }_{t}{\partial }_{i}-{g}^{ij}{\partial }_{j}{\partial }_{i}+{g}^{\alpha \beta }{{\Gamma}}_{\alpha \beta }^{\gamma }{\partial }_{\gamma }\right)u{\Vert}}_{{L}^{2}\left(K\right)}\hfill \\ \hfill & {\leqslant}{C}_{2}\left({{\Vert}u{\Vert}}_{{C}^{0}\left(\left[0,T\right],{H}^{2}\left({\Sigma}\right)\right)}+{{\Vert}u{\Vert}}_{{C}^{1}\left(\left[0,T\right],{H}^{1}\left({\Sigma}\right)\right)}+{{\Vert}f{\Vert}}_{{L}^{2}\left(K\right)}\right).\hfill \end{align*}$$

Combining the above with proposition 4.9 completes the proof. □

Equation (4.7) implies the following result.

Corollary 4.11. The solution to the Cauchy problem described in theorem 4.7 is well-posed in the sense that the solution map

$$\begin{equation*}\text{Sol}:{H}^{2}\left({\Sigma}\right){\times}{H}^{1}\left({\Sigma}\right){\times}{H}_{\mathrm{comp}}^{1}\left(M\right)\to {H}_{\mathrm{loc}}^{2}\left(M\right),\left({u}_{0},{u}_{1},f\right){\mapsto}u,\end{equation*}$$

is continuous in the topologies coming from the respective Sobolev spaces (see appendix B).

The definition of the Green operators in the non-smooth setting will require us to choose suitable spaces of functions as domain and range (see theorem 5.9). We therefore define the following spaces:

$$\begin{align}\hfill {V}_{0}=& \left\{\phi \in {H}_{\mathrm{comp}}^{2}\left(M\right)\;\text{s.t.}\;{\square }_{g}\phi \in {H}_{\mathrm{comp}}^{1}\left(M\right)\right\}\hfill \\ \hfill {U}_{0}=& {H}_{\mathrm{comp}}^{1}\left(M\right)\hfill \\ \hfill {V}_{sc}=& \left\{\phi \in {H}_{\mathrm{loc}}^{2}\left(M\right)\;\text{s.t.}\;{\square }_{g}\phi \in {H}_{\mathrm{loc}}^{1}\left(M\right)\right.\hfill \\ \hfill & \left.\;\text{and}\;\mathrm{supp}\left(\phi \right)\subset J\left(K\right)\;\text{where}\;K\;\text{is}\;\text{a}\;\text{compact}\;\text{subset}\;\text{of}\;M\right\}\hfill \end{align} \tag{ 5.1 }$$

Remark 5.3. Note that none of the spaces defined above depend upon the choice of background metric used in the definition of the Sobolev spaces.

Definition 5.4. A linear map

$$\begin{equation*}{G}^{+}:{H}_{\mathrm{comp}}^{1}\left(M\right)\to {H}_{\mathrm{loc}}^{2}\left(M\right)\end{equation*}$$

satisfying the properties

• (a)  
  ${\square }_{g}{G}^{+}={\text{id}}_{{H}_{\mathrm{comp}}^{1}\left(M\right)}$,
• (b)  
  ${G}^{+}{\square }_{g}{\vert }_{{V}_{0}}={\text{id}}_{{V}_{0}}$,
• (c)  
  supp(G⁺(f)) ⊂ J⁺(supp(f)) for all $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$,

is called an advanced Green operator for □_g. A retarded Green operator G⁻ is defined similarly.

Remark 5.5. 

• (a)  
  Clearly, the regularity condition in the definition of the space V₀ was chosen to guarantee that □_gf, given f ∈ V₀ belongs to the domain ${H}_{\mathrm{comp}}^{1}\left(M\right)$ of the Green operators.
• (b)  
  In several proofs below we will show the identities in properties (a) and (b) to hold weakly, i.e., when evaluated on test functions in $\mathcal{D}\left(M\right)$. However it can be shown using the results of Hörmander [38, theorem 1.25] that if two such Sobolev functions have the same effect on test functions then they are actually equal as Sobolev functions.
• (c)  
  The function spaces ${H}_{\mathrm{loc}}^{2}\left(M\right)$ and ${H}_{\mathrm{comp}}^{1}\left(M\right)$ used as target space and domain for the Green operators are in perfect accordance with the theory of so-called regular fundamental solutions for hyperbolic operators with constant coefficients⁵ (cf [39, section 12.5]) as we sketch briefly in the following: let M be (n + 1)-dimensional Minkowski space so that □ has the symbol p(τ, ξ) = τ² − |ξ|², which is hyperbolic with respect to the directional vectors (±1, 0) and produces the temperate weight function $\tilde {p}\left(\tau ,\xi \right){:=}\sqrt{{\sum }_{\vert \alpha \vert {\geqslant}0}{\vert {\partial }^{\alpha }p\left(\tau ,\xi \right)\vert }^{2}}=\sqrt{{\left({\tau }^{2}-{\xi }^{2}\right)}^{2}+4\left({\tau }^{2}+{\xi }^{2}+2\right)}{\geqslant}\sqrt{1+{\tau }^{2}+{\xi }^{2}}=:{w}_{1}\left(\tau ,\xi \right)$. The unique fundamental solution E_± with support in the half space where ±t ⩾ 0 belongs to ${B}_{\infty ,\tilde {p}}^{\mathrm{loc}}$, i.e., for every test function ϕ the Fourier transform $\mathcal{F}\left(\phi {E}_{{\pm}}\right)$ times $\tilde {p}$ is measurable and bounded. The advanced and retarded Green operators are then given by convolution G^±f = E_±*f for every $f\in {H}_{\mathrm{comp}}^{1}\left({\mathbb{R}}^{n+1}\right)={\mathcal{E}}^{\prime }\left({\mathbb{R}}^{n+1}\right)\cap {B}_{2,{w}_{1}}$, where ${B}_{2,{w}_{1}}=\left\{u\in {\mathcal{S}}^{\prime }\vert {w}_{1}\cdot \mathcal{F}u\in {L}^{2}\right\}={H}^{1}\left({\mathbb{R}}^{n+1}\right)$. Finally, we may apply [39, theorem 12.5.3 or theorem 10.1.24] to obtain ${G}^{{\pm}}f\in {B}_{2,\tilde {p}\cdot {w}_{1}}^{\mathrm{loc}}\subseteq {B}_{2,{w}_{1}^{2}}^{\mathrm{loc}}={H}_{\mathrm{loc}}^{2}\left({\mathbb{R}}^{n+1}\right)$, since $\tilde {p}\left(\tau ,\xi \right){w}_{1}\left(\tau ,\xi \right){\geqslant}{w}_{1}^{2}\left(\tau ,\xi \right)=1+{\tau }^{2}+{\xi }^{2}$.

We next show that the advanced and retarded Green operators (if they exist) are adjoints of one another. To do this we use the following lemma.

Lemma 5.6. Given $\chi ,\varphi \in {H}_{\mathrm{loc}}^{2}\left(M,g\right)$ and supp(χ) ∩ supp(φ) compact. Then, we have

$$\begin{equation}{\int }_{M}{\square }_{g}\chi \varphi {\nu }_{g}={\int }_{M}\chi {\square }_{g}\varphi {\nu }_{g}.\end{equation} \tag{ 5.2 }$$

The proof of the lemma follows from using integration by parts twice and the support properties given in the hypothesis. Note that the specified regularity of the metric g and of the functions is needed in order to use the L² inner product. We may now prove the following theorem.

Theorem 5.7. Given Green operators satisfying conditions (a) and (c) of definition 5.4 and $\chi ,\varphi \in {H}_{\mathrm{comp}}^{1}\left(M\right)$ we have that

$$\begin{equation*}{\int }_{M}{G}^{+}\left(\chi \right)\varphi {\nu }_{g}={\int }_{M}\chi {G}^{-}\left(\varphi \right){\nu }_{g}.\end{equation*}$$

Proof. First, notice that if $\chi ,\varphi \in {H}_{\mathrm{comp}}^{1}\left(M\right)$ we have that ${G}^{+}\left(\chi \right),{G}^{-}\left(\varphi \right)\in {H}_{\mathrm{loc}}^{2}\left(M,g\right)$. Moreover, G⁺(χ) ∩ G⁻(φ) ⊂ J⁺(supp(χ)) ∩ J⁻(supp(φ)) is compact by the global hyperbolicity condition.

Hence,

$$\begin{align*}\hfill {\int }_{M}{G}^{+}\left(\chi \right)\varphi {\nu }_{g}& ={\left({G}^{+}\left(\chi \right),\varphi \right)}_{{L}^{2}\left(M,g\right)}={\left({G}^{+}\left(\chi \right),{\square }_{g}{G}^{-}\left(\varphi \right)\right)}_{{L}^{2}\left(M,g\right)}\hfill \\ \hfill & ={\left({\square }_{g}{G}^{+}\left(\chi \right),{G}^{-}\left(\varphi \right)\right)}_{{L}^{2}\left(M,g\right)}={\left(\chi ,{G}^{-}\left(\varphi \right)\right)}_{{L}^{2}\left(M,g\right)}={\int }_{M}\chi {G}^{-}\left(\varphi \right){\nu }_{g}.\hfill \end{align*}$$

□

We are now in a position to prove the main result about existence of Green operators.

Theorem 5.8. Let (M, g) be a spacetime that satisfies the definition of generalised hyperbolicity (definition 5.1). Then there exist unique continuous advanced and retarded Green operators for □_g on M.

Proof. We will only discuss the advanced Green operator, the existence and the properties of the retarded Green operator follow from an analogous argument with the roles of future and past interchanged.

Existence: we define the linear map

$$\begin{equation*}{G}^{+}:{H}_{\mathrm{comp}}^{1}\left(M\right)\to {H}_{\mathrm{loc}}^{2}\left(M\right)\end{equation*}$$

which sends a source function f to the (unique) advanced weak solution u⁺. That such a u⁺ exists and is unique is a consequence of generalised hyperbolicity. Property (a) in definition 5.4 is immediate. In addition, the energy estimate (4.7) shows that G⁺ is a continuous operator. It remains to prove properties (b) and (c) in definition 5.4.

Property (b): let f ∈ V₀ and $v\in \mathcal{D}\left(M\right)$, then

$$\begin{equation*}{\left({G}^{+}\left({\square }_{g}f\right),v\right)}_{{L}^{2}\left(M,g\right)}={\left({\square }_{g}f,{G}^{-}\left(v\right)\right)}_{{L}^{2}\left(M,g\right)}={\left(f,{\square }_{g}{G}^{-}\left(v\right)\right)}_{{L}^{2}\left(M,g\right)}={\left(f,v\right)}_{{L}^{2}\left(M,g\right)},\end{equation*}$$

where we have used theorem 5.7 and property (a) for the retarded Green operator G⁻. Thus the weak form of the required identity holds, which implies G⁺□_g(f) = f for every f ∈ V₀ (see remark 5.5)(b).

Property (c) follows because supp(G^+(f))= supp(u⁺) = supp(u⁺) ⊂ J⁺(supp(f)) by proposition 4.3.

Uniqueness: let ${\tilde {G}}^{+}$ be another linear operator satisfying definition 5.4. Given $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$ we have that $v{:=}{\tilde {G}}^{+}\left(f\right)$ satisfies □_gv = f and supp(v) ⊂ J⁺(supp(f)). Since $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$, supp(v) ⊂ J⁺(supp(f)) and M is globally hyperbolic, there is a smooth timelike Cauchy surface Σ to the past of the support of f where the Cauchy data vanishes, i.e., v = 0 and ∇_nv = 0. Hence, v is a solution to the zero initial data forward Cauchy problem on Σ. By uniqueness we must have v = u⁺ so we can conclude that ${\tilde {G}}^{+}\left(f\right)={G}^{+}\left(f\right)$ for all $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$. □

We now show that the low-regularity Green operators satisfy an exact sequence result similar to that in the smooth case [3, theorem 3.4.7].

Theorem 5.9. Let M be a connected time-oriented globally hyperbolic Lorentzian manifold that satisfies definition 5.1. Define the causal propagator as

$$\begin{equation*}G={G}^{+}-{G}^{-}:{H}_{\mathrm{comp}}^{1}\left(M\right)\to {H}_{\mathrm{loc}}^{2}\left(M\right)\end{equation*}$$

Then the image of G is contained in V_sc and the following complex is exact:

Proof of theorem 5.9. First we show that the sequence is a complex: we have from the definitions ${G}^{+}{\square }_{g}{\vert }_{{V}_{0}}={G}^{-}{\square }_{g}{\vert }_{{V}_{0}}={\text{id}}_{{V}_{0}}$ and ${\square }_{g}{G}^{+}={\square }_{g}{G}^{-}={\text{id}}_{{H}_{\mathrm{comp}}^{1}\left(M\right)}$, therefore G□_gϕ = 0 for all ϕ ∈ V₀ and □_gGψ = 0 for all ψ ∈ U₀.

• Exactness at V₀, i.e., injectivity of □_g: let ϕ ∈ V₀ be such that □_gϕ = 0. By compactness of the support there is a smooth spacelike Cauchy hypersurface Σ such that ϕ = 0 and ∇_nϕ = 0 on Σ. Therefore, ϕ is a solution to the Cauchy problem with vanishing initial data and source. Uniqueness of the solution implies ϕ = 0.
• Exactness at U₀: let ϕ ∈ U₀ be such that ϕ ∈ ker(G), i.e., G⁺(ϕ) = G⁻(ϕ). Define ψ := G⁺(ϕ) = G⁻(ϕ), hence $\psi \in {H}_{\mathrm{loc}}^{2}\left(M\right)$ and □_gψ = ϕ. Moreover, ψ is compactly supported in M because supp(ψ) ⊂ supp(G⁺(ϕ)) ∩ supp(G⁻(ϕ)) ⊂ J⁺(supp(ϕ)) ∩ J⁻(supp(ϕ)) and the latter is compact due to global hyperbolicity. Thus there exists $\psi \in {H}_{\mathrm{loc}}^{2}\left(M\right)$ such that □_gψ = ϕ ∈ U₀ and supp(ψ) is compact. Hence, ψ ∈ V₀ and ϕ ∈ Im(G).
• Exactness at V_sc: let ϕ ∈ ker(□_g) and ϕ ∈ V_sc. Without loss of generality we may assume that supp(ϕ) ⊂ I⁺(K) ∪ I⁻(K) for some compact set⁶ K of M. Using a partition of unity {χ₋, χ₊} subordinate to {I⁻(K), I⁺(K)} we let ϕ₁ = χ₋ϕ and ϕ₂ = χ₊ϕ, thus ϕ = ϕ₁ + ϕ₂. Then supp(ϕ₁) ⊂ J⁻(K) and supp(ϕ₂) ⊂ J⁺(K), hence ϕ₁, ϕ₂ ∈ V_sc.

Define ψ := −□_gϕ₁ = □_gϕ₂. Then supp(ψ) is compact because supp(ψ) ⊂ J⁻(K) ∩ J⁺(K). Moreover, $\psi \in {H}_{\mathrm{loc}}^{1}\left(M\right)$ since ϕ ∈ V_sc. Combining these two observations, we conclude that $\psi \in {H}_{\mathrm{comp}}^{1}\left(M\right)$ and therefore G⁺(ψ) is defined.

For arbitrary $\chi \in \mathcal{D}\left(M\right)$ we have

$$\begin{equation*}{\left(\chi ,{G}^{+}\left(\psi \right)\right)}_{{L}^{2}\left(M,g\right)}={\left(\chi ,{G}^{+}\left({\square }_{g}{\phi }_{2}\right)\right)}_{{L}^{2}\left(M,g\right)}={\left({\square }_{g}{G}^{-}\chi ,{\phi }_{2}\right)}_{{L}^{2}\left(M,g\right)}={\left(\chi ,{\phi }_{2}\right)}_{{L}^{2}\left(M,g\right)}\end{equation*}$$

which shows G⁺(ψ) = ϕ₂, where we have made use of the fact that the supports of G⁻χ and ϕ₂ intersect in a compact set due to global hyperbolicity. Similarly, G⁻(ψ) = −ϕ₁ and therefore G(ψ) = G⁺(ψ) − G⁻(ψ) = ϕ₂ + ϕ₁ = ϕ. In summary, we may conclude that there exists ψ ∈ U₀ satisfying G(ψ) = ϕ.

□

We briefly discuss the restriction of Green operators to causally compatible subsets Ω ⊂ M, that is, sets such that

$$\begin{equation*}{J}_{{\Omega}}\left(x\right)={J}_{M}\left(x\right)\cap {\Omega}\quad \forall x\in {\Omega}.\end{equation*}$$

We have the following theorem (cf [3, proposition 3.5.1]).

Theorem 5.10. Let M be a time oriented connected globally hyperbolic manifold with a C^1,1 Lorentzian metric, G⁺ be the advanced Green operator for □_g and Ω ⊂ M be a causally compatible open subset. Then we may define an advanced Green operator for the restriction of □_g to Ω by

$$\begin{equation*}{\tilde {G}}^{+}\left(\varphi \right){:=}{G}^{+}\left({\varphi }_{\mathrm{ext}}\right){\vert }_{{\Omega}},\quad \text{for}\enspace \varphi \in {H}_{\mathrm{comp}}^{1}\left(M\right)\;\text{with}\;\mathrm{supp}\left(\varphi \right)\subseteq {\Omega}\end{equation*}$$

where φ_ext denotes the extension of ϕ by zero. Similar results hold for G⁻.

Remark 5.11. We denote the restriction of □_g to Ω by ${\tilde {\square }}_{g}$. Notice that for all $u\in {H}_{\mathrm{loc}}^{2}\left(M\right)$ we have ${\tilde {\square }}_{g}\left(u{\vert }_{{\Omega}}\right)={\square }_{g}{\vert }_{{\Omega}}\left(u{\vert }_{{\Omega}}\right)=\left({\square }_{g}u\right){\vert }_{{\Omega}}$ and for all u ∈ H²(Ω) with supp(u) ⊆ Ω we have ${\left({\tilde {\square }}_{g}u\right)}_{\text{ext}}={\square }_{g}\left({u}_{\text{ext}}\right)$.

Proof of theorem 5.10. Property (a): let $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$ with supp(f) ⊆ Ω, then

$$\begin{equation*}{\tilde {\square }}_{g}{\tilde {G}}^{+}\left(f\right)={\tilde {\square }}_{g}\left({G}^{+}\left({f}_{\text{ext}}\right){\vert }_{{\Omega}}\right)={\square }_{g}\left({G}^{+}\left({f}_{\text{ext}}\right)\right){\vert }_{{\Omega}}={f}_{\text{ext}}{\vert }_{{\Omega}}=f.\end{equation*}$$

Property (b): let f ∈ V₀ with supp(f) ⊂ Ω, then

$$\begin{equation*}{\tilde {G}}^{+}\left({\tilde {\square }}_{g}f\right)=\left({G}^{+}\left({\left({\tilde {\square }}_{g}f\right)}_{\text{ext}}\right){\vert }_{{\Omega}}\right)=\left({G}^{+}\left({\square }_{g}{f}_{\text{ext}}\right)\right){\vert }_{{\Omega}}={f}_{\text{ext}}{\vert }_{{\Omega}}=f.\end{equation*}$$

Property (c): for $f\in {H}_{\mathrm{comp}}^{1}\left(M\right)$ with supp(f) ⊆ Ω we have

$$\begin{align*}\hfill \mathrm{supp}\left({\tilde {G}}^{+}\left(f\right)\right)& =\mathrm{supp}\left({G}^{+}\left({f}_{\text{ext}}\right){\vert }_{{\Omega}}\right)=\mathrm{supp}\left({G}^{+}\left({f}_{\text{ext}}\right)\right)\cap {\Omega}\subset {J}_{M}^{+}\left(\mathrm{supp}\left({f}_{\text{ext}}\right)\right)\cap {\Omega}\hfill \\ \hfill & ={J}_{M}^{+}\left(\mathrm{supp}\left(f\right)\right)\cap {\Omega}={J}_{{\Omega}}^{+}\left(\mathrm{supp}\left(f\right)\right).\hfill \end{align*}$$

□

This subsection defines a category based on the analytic results in the previous sections and a functor assigning to each object a symplectic space.

Definition 6.1. Let GENHYP denote the category whose objects are 3-tuples (M, G⁺, G⁻) where M is a time oriented connected generalised globally hyperbolic manifold as in definition 5.1 and G⁺, G⁻ are the unique Green operators of □_g. Let $X=\left({M}_{1},{G}_{1}^{+},{G}_{1}^{-}\right)$ and $Y=\left({M}_{2},{G}_{2}^{+},{G}_{2}^{-}\right)$ be two objects in GENHYP, then Mor(X, Y) consists of all smooth maps ι : M₁ → M₂ which are time-orientation preserving isometric embeddings such that ι(M₁) ⊂ M₂ is a causally compatible open subset.

Remark 6.2. Theorem 5.2 shows that time oriented globally hyperbolic spacetimes with C^1,1 metrics are objects in this category.

Before considering quantisation we prove the following result on compatibility of Green operators.

Theorem 6.3. Let M₁ and M₂ be as in definition 6.1, then the following diagram commutes:

Proof. Theorem 5.10 shows that ${\tilde {G}}^{{\pm}}\left(\phi \right){:=}{G}_{2}^{{\pm}}\left({\phi }_{\text{ext}}\right){\vert }_{{M}_{1}}$ is a Green operator. By uniqueness, this operator has to be equal to ${G}_{1}^{{\pm}}$ and the result follows. □

Remark 6.4. In the smooth setting [3] the category LORFUND is defined as the category with objects being 5-tuples (M, F, G⁺, G⁻, P), where M is a Lorentzian manifold, F is a real vector bundle over M with non-degenerate inner product, P is a formally self-adjoint normally hyperbolic operator acting on sections in F and G⁺, G⁻ are the advanced and retarded Green operators for P. The morphisms consist of maps ι such that ι : M₁ → M₂ is a time-orientation preserving isometric embedding such that ι(M₁) ⊂ M₂ is a causally compatible open subset [3]. Moreover, given the condition of global hyperbolicity one can form the category GLOBHYP where objects are 3-tuples (M, F, P), where M is a Lorentzian manifold, F is a real vector bundle over M with non-degenerate inner product, P is a formally self-adjoint normally hyperbolic operator acting on sections in F. The morphisms are then given by maps ι such that ι : M₁ → M₂ is a time-orientation preserving isometric embedding such that ι(M₁) ⊂ M₂ is a causally compatible open subset. The existence and uniqueness of Green operators allow us to form a functor from GLOBHYP to LORFUND [3].

We now use the Green operators in order to construct a symplectic vector space. Let (M, G⁺, G⁻) be an object of GENHYP and define

$$\begin{equation*}\tilde {\omega }:{H}_{\mathrm{comp}}^{1}\left(M\right){\times}{H}_{\mathrm{comp}}^{1}\left(M\right)\to \mathbb{R}\end{equation*}$$

by

$$\begin{equation*}\tilde {\omega }\left(\phi ,\psi \right)={\int }_{M}G\left(\phi \right)\psi {\nu }_{g}\end{equation*}$$

where G = G⁺ − G⁻ is the causal propagator (see theorem 5.9). Then $\tilde {\omega }$ is bilinear and skew-symmetric by theorem 5.7. However, $\tilde {\omega }$ is degenerate because ker(G) is nontrivial. Moreover, using theorem 5.9 we have that

$$\begin{equation*}\mathrm{ker}\left(G\right)={\square }_{g}{V}_{0}.\end{equation*}$$

Therefore on the quotient space V(M) = U₀/ker(G) = U₀/□_gV₀ the degenerate form $\tilde {\omega }$ induces a symplectic form which we denote by ω.

Remark 6.5. It follows from corollary 4.11 that G is continuous so that ker G is a closed subspace and hence V(M) is a normed space (and in particular, Hausdorff). See the discussion section for more details on this point.

Finally, we need a functor SYMPL : GENHYP → SYMPLVECT, where SYMPLVECT is the category whose objects are symplectic vector spaces with morphisms given by symplectic maps, i.e., linear maps A such that ω₁(f, g) = ω₂(Af, Ag). The following theorem shows the existence of such a functor.

Theorem 6.6. Let $X=\left({M}_{1},{G}_{1}^{+},{G}_{1}^{-}\right)$ and $Y=\left({M}_{2},{G}_{2}^{+},{G}_{2}^{-}\right)$ be two objects in GENHYP and f ∈ Mor(X, Y) be a morphism. Then $\text{ext}:{H}_{\mathrm{comp}}^{1}\left({M}_{1}\right)\to {H}_{\mathrm{comp}}^{1}\left({M}_{2}\right)$ maps the null space ker(G₁) into the null space ker(G₂) and hence induces a continuous symplectic linear map V(M₁) → V(M₂).

Proof. Let ϕ ∈ ker(G₁) then $\phi ={\square }_{{g}_{1}}\psi$ for some ψ ∈ V₀(M₁) where we have used theorem 5.9.

From the fact that ${G}_{2}\left({\phi }_{\text{ext}}\right)={G}_{2}\left({\left({\square }_{{g}_{1}}\psi \right)}_{\text{ext}}\right)={G}_{2}{\square }_{{g}_{2}}{\psi }_{\text{ext}}=0$ we see that ext( ker(G₁)) ⊂ ker(G₂). Hence, ext induces a linear map from V(M₁) → V(M₂). Moreover, for $\phi ,\psi \in {H}_{\mathrm{comp}}^{1}\left({M}_{1}\right)$ we have on taking representatives

$$\begin{equation*}{\omega }_{1}\left(\phi ,\psi \right)={\int }_{{M}_{1}}{G}_{1}\left(\phi \right)\psi {\nu }_{{g}_{1}}={\int }_{{M}_{1}}{G}_{2}\left({\phi }_{\text{ext}}\right){\vert }_{{M}_{1}}\psi {\nu }_{{g}_{1}}={\int }_{{M}_{2}}{G}_{2}\left({\phi }_{\text{ext}}\right){\psi }_{\text{ext}}{\nu }_{{g}_{2}}={\omega }_{2}\left({\phi }_{\text{ext}},{\psi }_{\text{ext}}\right).\end{equation*}$$

Therefore, ext induces a symplectic map from V(M₁) to V(M₂). This induced map is also continuous (with the respective quotient topologies), because it is the composition π₂◦E of two continuous maps, namely $E:V\left({M}_{1}\right)={H}_{\mathrm{comp}}^{1}\left({M}_{1}\right)/\mathrm{ker}\left({G}_{1}\right)\to {H}_{\mathrm{comp}}^{1}\left({M}_{2}\right)$, the factor map of ext, and π₂, the quotient map ${H}_{\mathrm{comp}}^{1}\left({M}_{2}\right)\to {H}_{\mathrm{comp}}^{1}\left({M}_{2}\right)/\mathrm{ker}\left({G}_{2}\right)=V\left({M}_{2}\right)$. □

In this section we closely follow [3] and define the algebraic structures that will be required to represent the observables of the quantum theory. The definitions are algebraic in nature and do not require any further analytical considerations with respect to the regularity of solutions to the Cauchy problem. Nevertheless, the C^1,1 causality theory is required and will be mentioned below when it is used. Another modification with respect to the smooth case is that when considering the symplectic space (V, ω) in the smooth theory one has $V\left(M\right)=\mathcal{D}\left(M\right)/\mathrm{ker}\;G$ where G = G⁺ − G⁻ is a map $G:\mathcal{D}\left(M\right)\to {C}_{\mathrm{s}\mathrm{c}}^{\infty }\left(M\right)$. Employing the short-hand notation [f] for the class f + ker(G) in U₀/ker(G), the symplectic form is given by $\omega \left(\left[f\right],\left[h\right]\right)={\left(f,Gh\right)}_{{L}^{2}\left(M,g\right)}$ whereas in this section we will have $V\left(M\right)={H}_{\mathrm{comp}}^{1}\left(M\right)/\mathrm{ker}\;G$ where now $G:{H}_{\mathrm{comp}}^{1}\left(M\right)\to {V}_{\text{sc}}$ and the symplectic form is given by $\omega \left(\left[f\right],\left[h\right]\right)={\left(f,Gh\right)}_{{L}^{2}\left(M,g\right)}$.

We now introduce the definition of a Weyl system and a CCR-representation of (V, ω).

Definition 6.7. A Weyl system of the symplectic vector space (V, ω) consists of a C*-algebra $\mathcal{A}$ with unit and a map $W:V\to \mathcal{A}$ such that for all φ, ψ ∈ V,

• (a)  
  W(0) = 1,
• (b)  
  W(−φ) = W(φ)*,
• (c)  
  W(φ) ⋅ W(ψ) = e^{−iω(φ,ψ)/2}W(φ + ψ).

A Weyl system $\left(\mathcal{A},W\right)$ of a symplectic vector space (V, ω) is called a CCR-representation of (V, ω) if $\mathcal{A}$ is generated as a C*-algebra by the elements W(φ), φ ∈ V. In this case we call $\mathcal{A}$ a CCR-algebra of (V, ω) and write it as CCR(V, ω).

It is always possible to construct a CCR-representation (CCR(V, ω), W) for any symplectic vector space (V, ω) (see [3, example 4.2.2]). Moreover, the construction is categorical in the sense that if (V₁, ω₁) and (V₂, ω₂) are two symplectic vector spaces and S : V₁ → V₂ is a symplectic linear map, then there exists a unique injective ∗-morphism CCR(S) : CCR(V₁, ω₁) → CCR(V₂, ω₂).

The proof can be found in corollary 4.2.11 in [3].

From uniqueness of the map CCR(S) it is possible to define a functor

$$\begin{equation*}\mathrm{C}\mathrm{C}\mathrm{R}:\mathrm{S}\mathrm{Y}\mathrm{M}\mathrm{P}\mathrm{L}\to {C}^{{\ast}}-\mathrm{A}\mathrm{L}\mathrm{G}\end{equation*}$$

where C*-ALG is the category whose objects are C*-algebras and whose morphisms are injective unit preserving ∗-morphisms.

A set I is called a directed set with orthogonality relation, if it carries a partial order ⩽ and a symmetric relation ⊥ between its elements such that:

• (a)  
  For all α, β ∈ I there exists a γ ∈ I with α ⩽ γ and β ⩽ γ,
• (b)  
  For every α ∈ I there is a β ∈ I with α ⊥ β,
• (c)  
  If α ⩽ β and β ⊥ γ, then α ⊥ γ,
• (d)  
  If α ⊥ β and α ⊥ γ, then there exists a δ ∈ I such that β ⩽ δ, γ ⩽ δ and α ⊥ δ.

Sets of this type allow to define the objects and morphisms of the category QUASILOCALALG.

Definition 6.8. The objects of the category QUASILOCALALG are bosonic quasi-local C*-algebras which are pairs $\left(\mathcal{U},{\left\{{\mathcal{U}}_{\alpha }\right\}}_{\alpha \in I}\right)$ of a C*-algebra $\mathcal{U}$ and a family ${\left\{{\mathcal{U}}_{\alpha }\right\}}_{\alpha \in I}$ of C*-subalgebras, where I is a directed set with orthogonality relation such that the following holds:

• (a)  
  ${\mathcal{U}}_{\alpha }\subset {\mathcal{U}}_{\beta }$ whenever α ⩽ β,
• (b)  
  $\mathcal{U}=\overline{{\bigcup }_{\alpha }{\mathcal{U}}_{\alpha }}$,
• (c)  
  The algebras ${\mathcal{U}}_{\alpha }$ have a common unit 1,
• (d)  
  If α ⊥ β, then the commutators of elements from ${\mathcal{U}}_{\alpha }$ with those of ${\mathcal{U}}_{\beta }$ are trivial.

A morphism between two quasi-local C*-algebras $\left(\mathcal{U},{\left\{{\mathcal{U}}_{\alpha }\right\}}_{\alpha \in I}\right)$ and $\left(\mathcal{V},{\left\{{\mathcal{V}}_{\beta }\right\}}_{\beta \in J}\right)$ is defined as a pair (φ, Φ) where ${\Phi}:\mathcal{U}\to \mathcal{V}$ is a unit-preserving C*-morphism and φ : I → J is a map such that

• (a)  
  φ is monic, i.e., if α₁ ⩽ α₂ in I then φ(α₁) ⩽ φ(α₂) in J,
• (b)  
  φ preserves orthogonality, i.e., if α₁ ⊥ α₂ in I, then φ(α₁) ⊥ φ(α₂),
• (c)  
  ${\Phi}\left(\mathcal{U}\right)\subset {\mathcal{V}}_{\varphi \left(\alpha \right)}$ for all α ∈ I.

In the remainder of this section we discuss a functor from GENHYP to QUASILOCALALG. Let (M, G⁺, G⁻) be an object in GENHYP and

$$\begin{equation*}I=:\left\{O\subset M\vert O\;\text{is}\;\text{open,}\;\text{rel.}\;\text{compact,}\;\text{causally}\;\text{compatible,}\;\text{glob.}\;\text{hyperbolic}\right\}\cup \left\{\varnothing ,M\right\}.\end{equation*}$$

The relation O ⊥ O' means that O and O' are causally independent, i.e., there is no causal curve connecting a point in $\overline{O}$ to a point in $\overline{{O}^{\prime }}$.

Remark 6.9. The proof that the set I is a directed set with orthogonality relation requires results from causality theory in a low regularity setting [8, 17, 19] to obtain lemma A.5.11 in [3] and the existence of smooth time functions [34, 35] to obtain proposition A.5.13 in [3]. Properties (a) and (b) follow upon taking α = M, β = ∅. Property (c) follows from the observation that O ⊂ O' implies J(O) ⊂ J(O'), and Property (d) is implied by lemma 4.4.8 in [3] with the appropriate modifications of lemma A.5.11 and proposition A.5.13 therein.

For any non-empty set O ∈ I take the restriction of the operator □_g to the region O. Due to causal compatibility of O ⊂ M the restriction of Green operators G⁺, G⁻ to the region O yields Green operators ${G}_{O}^{+},{G}_{O}^{-}$. Therefore, we get an object $\left(O,{G}_{O}^{+},{G}_{O}^{-}\right)$ for each O ≠ ∅ ∈ I. For ∅ ≠ O₁ ⊂ O₂ the inclusion induces a morphism ${\iota }_{{O}_{2},{O}_{1}}$ in the category GENHYP. This morphism is given by the embedding O₁ → O₂. Let ${\alpha }_{{O}_{2},{O}_{1}}$ denote the morphism $\text{CCR}{\circ}\text{SYMPL}\left({\iota }_{{O}_{2},{O}_{1}}\right)$ in C*-ALG and recall that ${\alpha }_{{O}_{2},{O}_{1}}$ is an injective unit preserving ∗-morphism.

We set for ∅ ≠ O ∈ I,

$$\begin{equation*}\left({V}_{O},{\omega }_{O}\right){:=}\mathrm{S}\mathrm{Y}\mathrm{M}\mathrm{P}\mathrm{L}\left(O,{G}_{O}^{+},{G}_{O}^{-}\right),\end{equation*}$$

and for O ∈ I, O ≠ ∅, M,

$$\begin{equation*}{\mathcal{U}}_{O}{:=}{\alpha }_{M,O}\left(\text{CCR}\left({V}_{O},{\omega }_{O}\right)\right),\end{equation*}$$

for O = M define

$$\begin{equation*}{\mathcal{U}}_{M}{:=}{C}^{{\ast}}\left({\mathcal{U}}_{O\in I,O\ne \varnothing ,M}\right)\end{equation*}$$

which is the algebra of CCR(V_M, ω_M) generated by all the ${\mathcal{U}}_{O}$; for O = ∅, set ${\mathcal{U}}_{\varnothing }=\mathbb{C}$.

Now we assign to any morphism in GENHYP a morphism between quasi-local algebras in QUASILOCALALG: consider a morphism $\iota :\left(M,{G}^{+},{G}^{-}\right)\to \left(N,{\tilde {G}}^{+},{\tilde {G}}^{-}\right)$ in GENHYP. Let I₁, I₂ denote the index sets associated to M, N respectively. We define a map φ : I₁ → I₂ by M → N and O₁ → ι(O₁) if O₁ ≠ M. Since ι is an embedding such that ι(M) ⊂ N is causally compatible, the map φ is monotonic and preserves causal independence. Therefore, (φ, Φ) with Φ = CCR◦SYMPL(ι) is the required morphism. To be precise we have the following result.

Theorem 6.10. The assignment $\left(M,{G}^{+},{G}^{-}\right)\to \left({\mathcal{U}}_{M},{\left\{{\mathcal{U}}_{O}\right\}}_{O\in I}\right)$ and ι → (φ, Φ) yields a functor QUANT from GENHYP to QUASILOCALALG.

Proof. A detailed proof can be found in [3, lemma 4.4.10, theorem 4.4.11 and lemma 4.4.13]. □

Remark 6.11. In the low regularity setting the proof above requires one to consider elements $\phi \in {H}_{\mathrm{comp}}^{1}\left(O\right)$ rather than $\phi \in \mathcal{D}\left(O\right)$ in lemma 4.4.10 and the low regularity quotient ${H}_{\mathrm{comp}}^{1}\left(M\right)/\mathrm{ker}\left(G\right)$ instead of $\mathcal{D}\left(M\right)/\mathrm{ker}\left(G\right)$ with C^1,1 causality theory in lemma 4.4.13 in [3].

In this subsection we show that the functor QUANT given by theorem 6.10 satisfies the Haag–Kastler axioms.

Theorem 6.12. The functor QUANT : GENHYP → QUASILOCALALG satisfies the Haag–Kastler axioms, i.e., for every object (M, G⁺, G⁻) in GENHYP the corresponding quasi-local C*-algebra $\left({\mathcal{U}}_{M},{\left\{{\mathcal{U}}_{\mathcal{O}}\right\}}_{\mathcal{O}\in I}\right)$ satisfies:

• (a)  
  If O₁ ⊂ O₂ then ${\mathcal{U}}_{{\mathcal{O}}_{1}}\subset {\mathcal{U}}_{{O}_{2}}$ for all O₁, O₂ ∈ I.
• (b)  
  ${\mathcal{U}}_{M}=\overline{{\bigcup }_{O\in I,O\ne M,\varnothing }{\mathcal{U}}_{\mathcal{O}}}$.
• (c)  
  ${\mathcal{U}}_{M}$ is simple.
• (d)  
  The ${\mathcal{U}}_{O}$'s have a common unit 1.
• (e)  
  For all O₁, O₂ ∈ I with $J\left(\overline{{O}_{1}}\right)\cap \overline{{O}_{2}}=\varnothing$ the subalgebras ${\mathcal{U}}_{{O}_{1}},{\mathcal{U}}_{{O}_{2}}$ commute.
• (f)  
  (Time-slice axiom) let O₁ ⊂ O₂ be nonempty elements of I admitting a common smooth spacelike Cauchy hypersurface, then ${\mathcal{U}}_{{O}_{1}}={\mathcal{U}}_{{O}_{2}}$.
• (g)  
  Let O₁, O₂ ∈ I and let the Cauchy development D(O₂) be relatively compact in M. If O₁ ⊂ D(O₂), then ${\mathcal{U}}_{{O}_{1}}\subset {\mathcal{U}}_{{O}_{2}}$.

The proof will be based on the following two lemmas.

Lemma 6.13. Let O be a causally compatible globally hyperbolic open subset of a C^1,1 globally hyperbolic manifold M. Assume there exists a smooth spacelike Cauchy hypersurface Σ of O which is also a Cauchy hypersurface of M, let h be a smooth Cauchy time function on O and K ⊂ M be compact. Assume that there exists $t\in \mathbb{R}$ with K ⊂ I⁺(h⁻¹(t)). Then there is a smooth function ρ : M → [0, 1] such that

• (a)  
  ρ = 1 on a neighbourhood of K,
• (b)  
  supp(ρ) ∩ J⁻(K) ⊂ M is compact, and
• (c)  
  $\overline{\left\{x\in M\vert 0{< }\rho \left(x\right){< }1\right\}}\cap {J}^{-}\left(K\right)$ is compact and contained in O.

The proof of lemma 6.13 in the C^1,1 setting can be carried out following that of [3] with suitable modifications using results of low regularity causality theory [8, 17, 19].

We reproduce the main argument of the proof as done in [3].

Proof. By assumption there are real numbers t₋, t₊ in the range of h such that $K\subset {I}^{{\pm}}\left({S}_{{t}_{{\pm}}}\right)$ where S_t = h⁻¹(t). Since S_t is a Cauchy surface of O and since O and M admit a common Cauchy hypersurface, from C^1,1 causality theory it follows that S_± are also Cauchy hypersurfaces of M. Since J^±(S_±) are disjoint closed subsets of M there exists a smooth function ρ : M → [0, 1] such that $\rho {\vert }_{{J}^{+}\left({S}_{{t}_{+}}\right)}=1$ and $\rho {\vert }_{{J}^{-}\left({S}_{{t}_{-}}\right)}=0$.

The function ρ satisfies the first property stated in the lemma because $K\subset {I}^{+}\left({S}_{{t}_{+}}\right)$.

Since $\rho {\vert }_{{J}^{-}\left({S}_{{t}_{-}}\right)}=0$, we have $\mathrm{supp}\left(\rho \right)\subset {J}^{+}\left({S}_{{t}_{-}}\right)$. Moreover, in the C^1,1 setting the causal relationship is still closed and implies that ${J}^{+}\left({S}_{{t}_{-}}\right)\cap {J}^{-}\left(K\right)$ is compact and therefore supp(ρ) ∩ J⁻(K) ⊂ M is compact. This shows that the second property is satisfied.

The last property follows from two observations. The first one is that the closed set $\overline{\left\{x\in M\vert 0{< }\rho \left(x\right){< }1\right\}}\cap {J}^{-}\left(K\right)$ is contained in the compact set supp(ρ) ∩ J⁻(K) ⊂ M. The second observation is that ${J}^{+}\left({S}_{{t}_{-}}\right)\cap {J}^{-}\left({S}_{{t}_{+}}\right)\subset O$ which implies $\overline{\left\{x\in M\vert 0{< }\rho \left(x\right){< }1\right\}}\cap {J}^{-}\left(K\right)$ is contained in O. The statement that ${J}^{+}\left({S}_{{t}_{-}}\right)\cap {J}^{-}\left({S}_{{t}_{+}}\right)\subset O$ follows from the characterisation of Cauchy hypersurfaces as surfaces that are met exactly once by every inextendible causal curve. This characterisation also holds in the C^1,1 setting. □

Lemma 6.14. Let (M, G⁺, G⁻) be an object of GENHYP and O be a causally compatible globally hyperbolic open subset of M. Assume that there exists a Cauchy hypersurface Σ which is also a Cauchy hypersurface of M. Let φ ∈ U₀, then there exist χ ∈ V₀ and ψ ∈ U₀ such that supp(ψ) ⊂ O and φ = ψ + □_gχ.

Proof. Let h be a Cauchy time function on O. Fix t₋ ⩽ t₊ in the range of h. Then the subsets Σ₋ = h⁻¹(t₋), Σ₊ = h⁻¹(t₊) are Cauchy hypersurfaces of M. Hence every inextendible timelike curve in M meets Σ₋, Σ₊. Since t₋ ⩽ t₊, the set {I⁺(Σ₋), I⁻(Σ₊)} is a finite open cover of M. Let {f₊, f₋} be a smooth partition of unity subordinated to this cover. In particular, supp(f_±) ⊂ I^±(Σ_∓). Set K_± := supp(f_±φ) = supp(φ) ∩ supp(f_±). Then K_± is a compact subset of M satisfying K_± ⊂ I^±(Σ_∓). Applying lemma 6.13 we obtain two smooth functions ρ_± : M → [0, 1] satisfying

• (a)  
  ρ_± = 1 on a neighbourhood of K_±,
• (b)  
  supp(ρ_±) ∩ J^∓(K_±) ⊂ M is compact, and
• (c)  
  $\overline{\left\{x\in M\vert 0{< }{\rho }_{{\pm}}\left(x\right){< }1\right\}}\cap {J}^{\mp }\left({K}_{{\pm}}\right)$ is compact and contained in O.

Set χ_± := ρ_±G^∓(f_±φ), χ := χ₊ − χ₋ and ψ := φ − □_gχ. Since supp(G^∓(f_±φ)) ⊂ J^∓(K_∓), the support of χ_± is contained in supp(ρ_±) ∩ J^∓(K_±) which is compact by the second property of ρ_±. Since ρ_± and f_± are smooth by construction, we have ${\chi }_{{\pm}}\in {H}_{\mathrm{comp}}^{2}\left(M\right)$. Moreover,

$$\begin{align*}\hfill {\square }_{g}{\chi }_{{\pm}}& ={G}^{\mp }\left({f}_{{\pm}}\varphi \right){\square }_{g}{\rho }^{{\pm}}+{g}^{\alpha \beta }{\partial }_{\alpha }{\rho }^{{\pm}}{\partial }_{\beta }{G}^{\mp }\left({f}_{{\pm}}\varphi \right)+{\rho }^{{\pm}}{\square }_{g}{G}^{\mp }\left({f}_{{\pm}}\varphi \right)\hfill \\ \hfill & ={G}^{\mp }\left({f}_{{\pm}}\varphi \right){\square }_{g}{\rho }^{{\pm}}+{g}^{\alpha \beta }{\partial }_{\alpha }{\rho }^{{\pm}}{\partial }_{\beta }{G}^{\mp }\left({f}_{{\pm}}\varphi \right)+{\rho }^{{\pm}}\left({f}_{{\pm}}\varphi \right),\hfill \end{align*}$$

which implies ${\square }_{g}{\chi }_{{\pm}}\in {H}_{\mathrm{loc}}^{1}\left(M\right)$. Notice that □_gρ^± is not smooth but C^0,1.

Now ψ is the difference of ${H}_{\mathrm{loc}}^{1}\left(M\right)$ functions so it remains to show that supp(ψ) is compact and contained in O. By the first property of ρ_±, one has χ_± := G^∓(f_±φ) in a neighbourhood of K_±. Moreover, f_±φ = 0 on {ρ_± = 0}. Hence, □_gχ_± = f_±φ on {ρ_± = 0} ∪ {ρ_± = 1}. Therefore, f_±φ − □_gχ_± vanishes outside $\overline{\left\{x\in M\vert 0{< }{\rho }_{{\pm}}\left(x\right){< }1\right\}}$, i.e., $\mathrm{supp}\left({f}_{{\pm}}-{\square }_{g}{\chi }_{{\pm}}\right)\subset \overline{\left\{0{< }{\rho }_{{\pm}}\left(x\right){< }1\right\}}$. By the definitions of χ_±, f_± one also has supp(f_±φ − □_gχ_±) ⊂ J^∓(K_±), hence $\mathrm{supp}\left({f}_{{\pm}}\varphi -{\square }_{g}{\chi }_{{\pm}}\right)\subset {J}^{\mp }\left({K}_{{\pm}}\right)\cap \overline{\left\{0{< }{\rho }_{{\pm}}\left(x\right){< }1\right\}}$ which is compact and contained in O by the third property of ρ_±. Therefore, ψ ∈ U₀ with supp(ψ) ⊂ O. Moreover, φ − ψ = □_gχ ∈ U₀ which gives χ ∈ V₀. □

Proof of theorem 6.12. The first, fourth and fifth axiom follow from the definition of the quasi-local C*-algebra, the definition of the set I and [3, lemma 4.4.10]. The second axiom follows from [3, lemma 4.4.13 ] and the third axiom follows from [3, remark 4.5.3]. Remark 6.11 mentions the necessary modifications of those lemmas in the C^1,1 setting.

It therefore remains to prove the time-slice axiom. Let O₁ ⊂ O₂ be nonempty causally compatible globally hyperbolic subsets of M admitting a common smooth spacelike Cauchy hypersurface Σ. Let [ϕ] ∈ V(O₂). Then lemma 6.14 applied to M := O₂ and O = O₁ yields χ ∈ V₀, ψ ∈ U₀ such that ϕ = ψ_ext + □_gχ with supp(ψ) ⊂ O₁. Since, ${\square }_{g}\chi \in \mathrm{ker}\left({G}_{{O}_{2}}\right)$ we have [ϕ] = [ψ_ext], that is, [ϕ] is the image of the symplectic linear map V(O₁) → V(O₂) induced by the inclusion ι : O₁ → O₂. Therefore, the map is surjective, and hence an isomorphism of symplectic topological vector spaces. This isomorphism functorially induces an isomorphism of C*-algebras, hence ${\mathcal{U}}_{{O}_{1}}={\mathcal{U}}_{{O}_{2}}$. This proves the time-slice axiom. Finally, the seventh axiom can be deduced from the first and the sixth axiom [3, theorem 4.5.1]. □

Let us describe the quotient vector space U₀/□_gV₀ in some more detail for the globally hyperbolic case, where we have ker(G) = Im(□_g) as a consequence of the spectral sequence given in theorem 5.9, thus U₀/□_gV₀ = U₀/ker(G) in this case. Recall that G is a linear map U₀ → V_sc and let G₀ denote the associated map from the quotient U₀/ker(G) to Im(G) ⊆ V_sc, defined by G₀(ϕ + ker(G)) := Gϕ for every ϕ ∈ U₀. Therefore, G₀ is linear and bijective by construction and we arrive at the following chain of (algebraic) isomorphisms of vector spaces

$$\begin{equation}{U}_{0}/{\square }_{g}{V}_{0}={U}_{0}/\mathrm{ker}\left(G\right)\cong \text{Im}\left(G\right)=\mathrm{ker}\left({\square }_{g}\right)\subseteq {V}_{\text{sc}}.\end{equation} \tag{ 7.1 }$$

Recall that the analogue of (7.1) in the smooth globally hyperbolic case, as discussed in [3], is

$$\begin{equation*}\mathcal{D}\left(M\right)/{\square }_{g}\mathcal{D}\left(M\right)=\mathcal{D}\left(M\right)/\mathrm{ker}\left(G\right)\cong \text{im}\left(G\right)=\mathrm{ker}\left({\square }_{g}\right)\subseteq {C}_{\text{sc}}^{\infty },\end{equation*}$$

showing also that the quotient is isomorphic to the space of solutions to the homogeneous wave equation.

The question arises whether the isomorphism in the middle part of (7.1), obtained via the factored map G₀, is topological, where the quotient U₀/ker(G) is equipped with the finest topology such that the canonical surjection π : U₀ → U₀/ker(G), ϕ ↦ ϕ + ker(G) is continuous. Note that by continuity of G we have that ker(G) is closed in the normed space U₀, hence U₀/ker(G) is a normed space (in particular, Hausdorff). Furthermore, G₀ is continuous by construction and the continuity of G, thus it remains to be checked whether the inverse of G₀ is continuous, or, equivalently, whether G₀ is an open map.

Remark 7.1. We note that by [40, chapter III, proposition 1.2], the factored map G₀ is a topological isomorphism if and only if G is open as a map from U₀ to Im(G) (with the relative topology on the latter). In case of Fréchet spaces such a property for G could be deduced conveniently via an open mapping principle or from a closed image criterion, but observe that neither ${U}_{0}={H}_{\mathrm{comp}}^{1}\left(M\right)$ nor Im(G) is complete (with respect to the metric inherited from the Banach space H¹(M) and the Fréchet space ${H}_{\mathrm{loc}}^{2}\left(M\right)$, respectively).

We choose a finer topology σ on V_sc to make ${\square }_{g}:\left({V}_{\text{sc}},\sigma \right)\to {H}_{\mathrm{loc}}^{1}\left(M\right)$ continuous by adding the semi-norms ${p}_{\chi }\left(\phi \right){:=}{{\Vert}\chi \cdot {\square }_{g}\phi {\Vert}}_{{H}^{1}}$ ($\chi \in \mathcal{D}\left(M\right)$) to those on V_sc inherited from ${H}_{\mathrm{loc}}^{2}\left(M\right)$. Note that this has no effect on the subspace Im(G) ⊆ V_sc, since Im(G) ⊆ ker(□_g) in the complex of maps in theorem 5.9 (even equality holds due to global hyperbolicity). In fact, σ is precisely the coarsest topology that is finer than the ${H}_{\mathrm{loc}}^{2}\left(M\right)$-topology on V_sc, which we denote by τ₂, and renders □_g continuous as a map ${V}_{\text{sc}}\to {H}_{\mathrm{loc}}^{1}\left(M\right)$, i.e., σ is the supremum (in the lattice of topologies on V_sc) of τ₂ and the initial (projective) topology τ₁ with respect to □_g. Therefore, we have continuity of G : U₀ → (V_sc, σ), since G is continuous U₀ → (V_sc, τ₂) by corollary 4.11 and also continuous U₀ → (V_sc, τ₁) due to the obvious continuity of □_g◦G = 0 from U₀ into ${H}_{\mathrm{loc}}^{1}\left(M\right)$.

Lemma 7.2. The inverse of G₀ : U₀/ker(G) → Im(G), ϕ + ker(G) ↦ Gϕ, is continuous.

Proof. We will show that ${G}_{0}^{-1}$ can be written as the composition ${G}_{0}^{-1}=\pi {\circ}P{\circ}Z$ of three continuous linear maps. The map π : U₀ → U₀/ker(G) is the canonical surjection, which is continuous by construction. It remains to construct suitable continuous maps P and Z with P◦Z : Im(G) → U₀ and such that G₀◦π◦P◦Z = id_Im(G) and $\pi {\circ}P{\circ}Z{\circ}{G}_{0}={\text{id}}_{{U}_{0}/\mathrm{ker}\left(G\right)}$.

Let ${V}_{\text{sc}}^{{\pm}}{:=}\left\{\phi \in {V}_{\text{sc}}\vert \mathrm{supp}\left(\phi \right)\subseteq {J}^{{\pm}}\left(K\right)\;\text{for}\;\text{some}\;\text{compact}\;\text{subset}\;K\subseteq M\right\}$ and define the subspace

$$\begin{equation*}W{:=}\left\{\left({\phi }_{-},{\phi }_{+}\right)\in {V}_{\text{sc}}^{-}{\times}{V}_{\text{sc}}^{+}\vert {\phi }_{-}+{\phi }_{+}\in \mathrm{ker}\left({\square }_{g}\right)\right\}\subseteq {V}_{\text{sc}}{\times}{V}_{\text{sc}},\end{equation*}$$

which we equip with the trace of the product topology stemming from σ.

Construction of P: we consider P : W → U₀, given by P(ϕ₋, ϕ₊) := (□_gϕ₊ − □_gϕ₋)/2. Note that a priori, P(ϕ₋, ϕ₊) is only in ${H}_{\mathrm{loc}}^{1}\left(M\right)$ and we have to show that P(ϕ₋, ϕ₊) has compact support, thus belongs to ${U}_{0}={H}_{\mathrm{comp}}^{1}\left(M\right)$. To prove this, observe that ϕ = ϕ₋ + ϕ₊ ∈ ker(□_g) implies □_gϕ₋ = −□_gϕ₊, hence Pϕ = □_gϕ₊ = −□_gϕ₋. Let K₋ and K₊ be compact subsets of M with supp(ϕ_±) ⊆ J^±(K_±), then we have supp(Pϕ) ⊆ supp(ϕ₋) ∩ supp(ϕ₊) ⊆ J⁻(K₋) ∩ J⁺(K₊), where J⁻(K₋) ∩ J⁺(K₊) is compact by global hyperbolicity ([3], lemma A.5.7]). The continuity of P is clear by construction of the topology σ.

Construction of Z: as a preparation we will first construct two continuous maps ${S}^{{\pm}}:{V}_{\text{sc}}\to {V}_{\text{sc}}^{{\pm}}$, such that ϕ = S⁻ϕ + S⁺ϕ holds for every ϕ ∈ V_sc and, moreover,

$$\begin{equation}G\;{\square }_{g}{S}^{{\pm}}\phi ={\pm}\phi \quad \forall \phi \in \text{Im}\left(G\right).\end{equation} \tag{ 7.2 }$$

Choose a smooth spacelike Cauchy surface Σ ⊆ M and let $t:M\to \mathbb{R}$ be a smooth temporal function such that Σ = t⁻¹(0) (compare the earlier discussion on causality in C^1,1-spacetimes). We obtain an open covering of M by the two sets O₋ := {x ∈ M|t(x) < 1} and O₊ := {x ∈ M|t(x) > −1} and choose a subordinate partition of unity χ₋, χ₊ ∈ C^∞(M), i.e., supp(χ_±) ⊆ O_± and χ₋ + χ₊ = 1. We define S^±ϕ := χ_±ϕ, then the relation ϕ = S⁻ϕ + S⁺ϕ holds by construction and the continuity of S^± is clear from continuity of multiplication by fixed smooth functions with respect to (localised) Sobolev norms. It remains to show that ${S}^{{\pm}}\in {V}_{\text{sc}}^{{\pm}}$ for every ϕ ∈ V_sc and equation (7.2) is true.

Let ϕ ∈ V_sc and K ⊆ M be compact such that supp(ϕ) ⊆ J⁻(K) ∪ J⁺(K). Then supp(χ₊ϕ) ⊆ O₊ ∩ (J⁻(K) ∪ J⁺(K)) ⊆ (O₊ ∩ J⁻(K)) ∪ J⁺(K). Note that O₊ ∩ J⁻(K) is relatively compact by [3, corollary A.5.4], since O₊ ⊆ J⁺(Σ₋) holds with Σ₋ := t⁻¹(−1) (note that the time function is strictly increasing along causal curves). Therefore, with some compact set K₊ containing K as well as O₊ ∩ J⁻(K) we obtain supp(χ₊ϕ) ⊆ J⁺(K₊), thus ${S}^{+}\phi \in {V}_{\text{sc}}^{+}$. The reasoning for ${S}^{-}\phi \in {V}_{\text{sc}}^{-}$ is analogous.

For the proof of (7.2) we start by noting that ϕ ∈ Im(G) = ker(□_g) implies 0 = □_gϕ = □_gS⁻ϕ + □_gS⁺ϕ, so that the part with S⁻ in (7.2) follows once the equation for S⁺ is shown. Recall that we have ${\square }_{g}{S}^{+}\phi =-{\square }_{g}{S}^{-}\phi \in {H}_{\mathrm{comp}}^{1}\left(M\right)$ from the reasoning in the construction of P above. Moreover, for every test function ψ on M we have that supp(G^∓ψ) ∩ supp(S^±ϕ) ⊆ J^∓(supp(ψ)) ∩ J^±(K) for some compact set K, hence global hyperbolicity guarantees that the supports of G⁻ψ and S⁺ϕ as well as those of G⁺ψ and S⁻ϕ always have compact intersection. To summarise, we may apply theorem 5.7 and lemma 5.6 to obtain the following chain of weak equalities

$$\begin{align*}\hfill & {\left(\psi ,{G}^{{\pm}}{\square }_{g}{S}^{+}\phi \right)}_{{L}^{2}\left(M,g\right)}={\left({G}^{\mp }\psi ,{\square }_{g}{S}^{+}\phi \right)}_{{L}^{2}\left(M,g\right)}={\left({G}^{\mp }\psi ,{\pm}{\square }_{g}{S}^{{\pm}}\phi \right)}_{{L}^{2}\left(M,g\right)}\hfill \\ \hfill & \quad ={\left({\square }_{g}{G}^{\mp }\psi ,{\pm}{S}^{{\pm}}\phi \right)}_{{L}^{2}\left(M,g\right)}={\left(\psi ,{\pm}{S}^{{\pm}}\phi \right)}_{{L}^{2}\left(M,g\right)},\hfill \end{align*}$$

which implies G^±□_gS⁺ϕ = ±S^±ϕ and therefore G□_gS⁺ϕ = G⁺□_gS⁺ϕ − G⁻□_gS⁺ϕ = S⁺ϕ − (−S⁻ϕ) = S⁺ϕ + S⁻ϕ = ϕ. Thus, equation (7.2) is proved and concludes the preparatory construction of S^±.

Finally, we turn to the definition of the map Z. Observe that ϕ ∈ Im(G) = ker(□_g) ⊆ V_sc implies (S⁻ϕ, S⁺ϕ) ∈ W, which allows to set Zϕ := (S⁻ϕ, S⁺ϕ) for every ϕ ∈ Im(G) and obtain a continuous linear map Z : Im(G) → W.

We complete the proof by showing that π◦P◦Z is the inverse of G₀.

• The relation G₀◦π◦P◦Z = id_Im(G) holds, since for every ϕ ∈ Im(G) we have
  $$\begin{align*}\hfill & {G}_{0}\left(\pi \left(P\left(Z\phi \right)\right)\right)={G}_{0}\left(\pi \left(P\left({S}^{-}\phi ,{S}^{+}\phi \right)\right)\right)=\frac{1}{2}{G}_{0}\left(\pi \left({\square }_{g}{S}^{+}\phi -{\square }_{g}{S}^{-}\phi \right)\right)\hfill \\ \hfill & \quad =\frac{1}{2}{G}_{0}\left({\square }_{g}{S}^{+}\phi -{\square }_{g}{S}^{-}\phi +\mathrm{ker}\left(G\right)\right)=\frac{1}{2}\left(G\;{\square }_{g}{S}^{+}\phi -G\;{\square }_{g}{S}^{-}\phi \right),\hfill \end{align*}$$
  where we may apply (7.2) to rewrite the last term as $\frac{1}{2}\left(\phi +\phi \right)=\phi$.
• We finally show that the equation $\pi {\circ}P{\circ}Z{\circ}{G}_{0}={\text{id}}_{{U}_{0}/\mathrm{ker}\left(G\right)}$ is true. Let f ∈ U₀, then
  $$\begin{align*}\hfill & \pi \left(P\left(Z\left({G}_{0}\left(f+\mathrm{ker}\left(G\right)\right)\right)\right)\right)=\pi \left(P\left(Z\left(Gf\right)\right)\right)=\pi \left(P\left({S}^{-}Gf,{S}^{+}Gf\right)\right)\hfill \\ \hfill & \quad =\pi \left(\frac{1}{2}\left({\square }_{g}{S}^{+}Gf-{\square }_{g}{S}^{-}Gf\right)\right)=\frac{1}{2}\left({\square }_{g}{S}^{+}Gf-{\square }_{g}{S}^{-}Gf\right)+\mathrm{ker}\left(G\right)\hfill \end{align*}$$
  and in the last term we may replace ${f}_{1}{:=}\frac{1}{2}\left({\square }_{g}{S}^{+}Gf-{\square }_{g}{S}^{-}Gf\right)$ by f, since thanks to (7.2) the difference is in the kernel: $G\left({f}_{1}-f\right)=\frac{1}{2}\left(Gf+Gf\right)-Gf=Gf-Gf=0$. □

Proposition 7.3. For a globally hyperbolic C^1,1 spacetime (M, g), we obtain a topological isomorphism U₀/ker(G) ≅ Im(G) according to (7.1), where V_sc carries the topology σ.

Remark 7.4. We are not using an inductive limit construction for the topology on V_sc as, e.g., in [41], because we preferred to stay with questions of convergence and continuity in the simpler realm of local Sobolev norms. Moreover, in the above context, we would otherwise not have a topological isomorphism of Im(G) with U₀/ker(G), since we decided coherently that U₀ should inherit the norm topology from H¹(M), thus rendering ${U}_{0}={H}_{\mathrm{comp}}^{1}\left(M\right)$ normed, but incomplete. However, the basic constructions of quantisation for the associated symplectic (quotient) vector spaces do not require completeness.

An analogous construction of the CCR representation can be achieved using a symplectic structure on the vector space of solutions to the homogeneous problem parametrised by their initial data [1]. In that context, one defines a symplectic structure Ξ on ker(□_g) given by Ξ(ϕ, ψ) = ∫_Σ(u₁v₀ − v₁u₀)μ_h where (u₀, u₁), (v₀, v₁) are compactly supported smooth initial data induced by the smooth solutions ϕ, ψ respectively on the Cauchy hypersurface Σ. Moreover, the Weyl system generated by the symplectic space (ker(□_g), Ξ) is isomorphic to the Weyl system generated by (U₀/ker(G), ω) [1, 42]. In the C^1,1 setting this isomorphism remains true with suitable modifications.

To be precise, using theorem 5.9 we know that ker(□_g) = Im(G). Moreover, for any smooth spacelike Cauchy hypersurface Σ, if ϕ = G(f) then $\phi {\vert }_{{\Sigma}}\in {H}_{\mathrm{comp}}^{2}\left({\Sigma}\right)$ and ${\nabla }_{n}\phi {\vert }_{{\Sigma}}\in {H}_{\mathrm{comp}}^{1}\left({\Sigma}\right)$. This follows from the observation that ϕ ∈ V_sc and is the difference of two solutions to the Cauchy problem with zero initial data, which by theorem 4.7 belong to the space ${C}^{0}\left(\mathbb{R},{H}^{2}\left({{\Sigma}}_{t}\right)\right)\cap {C}^{1}\left(\mathbb{R},{H}^{1}\left({{\Sigma}}_{t}\right)\right)$. Therefore, given any smooth spacelike Cauchy hypersurface Σ, we define for ϕ, ψ ∈ ker □_g with u₀ := ϕ|_Σ, u₁ := ∇_nϕ|_Σ, v₀ := ψ|_Σ, v₁ := ∇_nψ|_Σ, hence $\left({u}_{0},{u}_{1}\right),\left({v}_{0},{v}_{1}\right)\in {H}_{\mathrm{comp}}^{2}\left({\Sigma}\right){\times}{H}_{\mathrm{comp}}^{1}\left({\Sigma}\right)$, the skew-symmetric bilinear form

$$\begin{equation*}{{\Xi}}_{{\Sigma}}\left(\phi ,\psi \right)={\int }_{{\Sigma}}\left({u}_{1}{v}_{0}-{v}_{1}{u}_{0}\right){\mu }_{h}.\end{equation*}$$

It follows from linearity, the uniqueness of solutions to the Cauchy problem, and direct computations that Ξ is symplectic where to show non-degeneracy one tests with elements of the form (0, u₁) and (v₀, 0), i.e., with u₀ = 0 and v₁ = 0 and employs uniqueness in the Cauchy problem (cf [42]).

We show that Ξ_Σ does not depend on Σ: this follows from the divergence theorem in a region bounded by two Cauchy hypersurfaces Σ₁, Σ₂ and the conservation of the current ${j}^{\mu }\left(\phi ,\psi \right)={g}^{\mu \nu }\left(\phi {\nabla }_{\nu }\psi -\psi {\nabla }_{\nu }\phi \right)$. Explicitly we have for any ϕ, ψ ∈ ker(□_g)

$$\begin{equation*}{\int }_{{J}^{-}\left({{\Sigma}}_{1}\right)\cap {J}^{+}\left({{\Sigma}}_{2}\right)}\mathrm{div}\left({j}^{\mu }\left(\phi ,\psi \right)\right){\nu }_{g}=0\end{equation*}$$

and

$$\begin{equation*}{\int }_{{J}^{-}\left({{\Sigma}}_{1}\right)\cap {J}^{+}\left({{\Sigma}}_{2}\right)}\mathrm{div}\left({j}^{\mu }\left(\phi ,\psi \right)\right){\nu }_{g}={\int }_{{{\Sigma}}_{2}}{j}^{\mu }\left(\phi ,\psi \right){n}_{\mu }{\mu }_{{h}_{1}}-{\int }_{{{\Sigma}}_{1}}{j}^{\mu }\left(\phi ,\psi \right){n}_{\mu }{\mu }_{{h}_{2}}=0.\end{equation*}$$

Therefore,

$$\begin{equation*}{{\Xi}}_{{{\Sigma}}_{1}}\left(\phi ,\psi \right)={{\Xi}}_{{{\Sigma}}_{2}}\left(\phi ,\psi \right),\end{equation*}$$

so we will drop the Σ from the notation of Ξ. Notice that the ${H}_{\mathrm{loc}}^{2}$ regularity is required in order to make sense of the divergence of the current.

Finally, we show that the linear bijective factor map G₀ of G, as defined before (7.1), provides a symplectic map from (U₀/ker(G), ω) to (ker(□_g), Ξ).

Proposition 7.5. Let the symplectic vector spaces (ker(□_g), Ξ), (U₀/ker(G), ω) and the factor map G₀ be defined as above. Then we have for every f, f' ∈ U₀ with ϕ = G₀([f']) = Gf', ψ = G₀([f]) = Gf ∈ ker(□_g),

$$\begin{equation*}{\Xi}\left(\phi ,\psi \right)=\omega \left(\left[{f}^{\prime }\right],\left[f\right]\right).\end{equation*}$$

Proof. Without loss of generality we may consider $M\cong \mathbb{R}{\times}{\Sigma}$ and suppose that supp(f) ⊂ (t₁, t₂) × Σ for some real t₁ < t₂. Then we have for every ϕ ∈ ker(□_g) upon integrating by parts twice,

$$\begin{align*}\hfill & {\int }_{\left({t}_{1},{t}_{2}\right){\times}{\Sigma}}\phi {\square }_{g}{G}^{+}\left(f\right){\nu }_{g}={\int }_{\left({t}_{1},{t}_{2}\right){\times}{\Sigma}}{\square }_{g}\phi {G}^{+}\left(f\right){\nu }_{g}\hfill \\ \hfill & \quad \quad -{\int }_{{{\Sigma}}_{{t}_{2}}}\left(\phi {\nabla }_{n}{G}^{+}\left(f\right)-{G}^{+}\left(f\right){\nabla }_{n}\phi \right){\mu }_{{h}_{2}}+{\int }_{{{\Sigma}}_{{t}_{1}}}\left(\phi {\nabla }_{n}{G}^{+}\left(f\right)-{G}^{+}\left(f\right){\nabla }_{n}\phi \right){\mu }_{{h}_{1}}.\hfill \end{align*}$$

Using the fact that □_gϕ = 0 and that ${{\Sigma}}_{{t}_{1}}$ is disjoint⁷ from supp(G⁺f) we obtain

$$\begin{equation*}{\int }_{\left({t}_{1},{t}_{2}\right){\times}{\Sigma}}\phi {\square }_{g}{G}^{+}\left(f\right){\nu }_{g}=-{\int }_{{{\Sigma}}_{{t}_{2}}}\left(\phi {\nabla }_{n}{G}^{+}\left(f\right)-{G}^{+}\left(f\right){\nabla }_{n}\phi \right){\mu }_{{h}_{2}}.\end{equation*}$$

Similarly, from the causal properties again we have that ${{\Sigma}}_{{t}_{2}}$ and supp(G⁻(f)) are disjoint. Therefore ${G}^{+}f{\vert }_{{{\Sigma}}_{{t}_{2}}}=Gf{\vert }_{{{\Sigma}}_{{t}_{2}}}$ and ${\nabla }_{n}{G}^{+}f{\vert }_{{{\Sigma}}_{{t}_{2}}}={\nabla }_{n}Gf{\vert }_{{{\Sigma}}_{{t}_{2}}}$ which gives

$$\begin{equation*}{\int }_{\left({t}_{1},{t}_{2}\right){\times}{\Sigma}}\phi {\square }_{g}{G}^{+}\left(f\right){\nu }_{g}=-{\int }_{{{\Sigma}}_{{t}_{2}}}\left(\phi {\nabla }_{n}G\left(f\right)-G\left(f\right){\nabla }_{n}\phi \right){\mu }_{{h}_{2}}.\end{equation*}$$

Recalling that ψ = G(f) and t₁ < t < t₂ in supp(f) we obtain

$$\begin{equation*}{\Xi}\left(\phi ,\psi \right)={\int }_{\left({t}_{1},{t}_{2}\right){\times}{\Sigma}}\phi f{\nu }_{g}={\int }_{M}\phi f{\nu }_{g}.\end{equation*}$$

Here, we use also the assumption G(f') = ϕ to proceed with

$$\begin{equation*}{\int }_{M}\phi f{\nu }_{g}={\int }_{M}G\left({f}^{\prime }\right)f{\nu }_{g}=\tilde {\omega }\left({f}^{\prime },f\right)=\omega \left(\left[{f}^{\prime }\right],\left[f\right]\right).\end{equation*}$$

□

We have established a symplectomorphism between the spaces (ker(□_g), Ξ) and (U₀/ker(G), ω). This implies that the functor CCR will give isomorphic C*-algebras in the quantisation. Therefore, the result shows that one can use either the elements of U₀/ker(G) or those of ker(□_g) to construct the algebra of quantum observables.

Finally, in order to construct a full quantum field theory in a low regularity spacetime, a suitable choice of quantum states must be made. In the algebraic quantisation method, a quantum state Λ is a normalised positive linear functional on the quasi-local C*-algebra. In particular, given a real scalar product $\mu :\mathrm{ker}\left({\square }_{g}\right){\times}\mathrm{ker}\left({\square }_{g}\right)\to \mathbb{R}$ satisfying ${\vert {\Xi}\left(\phi ,\psi \right)\vert }^{2}{\leqslant}\frac{1}{4}\mu \left(\phi ,\psi \right)\mu \left(\phi ,\psi \right)$ for all ϕ, ψ ∈ ker(□_g), we define a quasi-free state by ${{\Lambda}}_{\mu }\left(W\left(\phi \right)\right)={\mathrm{e}}^{\frac{1}{2}\mu \left(\phi ,\phi \right)}$. As a consequence of the GNS construction, a quasi-free state on a C*-algebra possesses a natural Fock space structure [1]. Moreover, if additional symmetries exist such as the existence of a timelike Killing vector field one can define ground states and thermal equilibrium states at finite temperature as quasi-free states that can be represented as elements of a suitable Fock space.

In the absence of symmetries a common candidate for the physical quantum states in the smooth case are the quasi-free states that satisfy the microlocal spectrum condition. This condition allows via a renormalization procedure to define the energy momentum tensor of the state [1].

To specify the microlocal spectrum condition, we need to define appropriate subsets of T*(M × M)\0, i.e., the cotangent bundle with the zero section removed, and the two-point function of the state Λ_μ, which is a distribution on M × M. Let

$$\begin{align*}\hfill & C=\left\{\left({x}_{1},\eta ,{x}_{2},\tilde {\eta }\right)\in {T}^{{\ast}}\left(M{\times}M\right){\backslash}0;{g}^{ab}\left({x}_{1}\right){\eta }_{a}{\eta }_{b}=0,{g}^{ab}\left({x}_{2}\right){\tilde {\eta }}_{a}{\tilde {\eta }}_{b}=0,\left({x}_{1},\eta \right)\sim \left({x}_{2},\tilde {\eta }\right)\right\},\hfill \\ \hfill & \text{and}\;{C}^{+}=\left\{\left({x}_{1},\eta ,{x}_{2},\tilde {\eta }\right)\in C;{\eta }^{0}{\geqslant}0,{\tilde {\eta }}^{0}{\geqslant}0\right\},\hfill \end{align*}$$

where $\left({x}_{1},\eta \right)\sim \left({x}_{2},\tilde {\eta }\right)$ means that η, $\tilde {\eta }$ are cotangent to the null geodesic γ at x₁, x₂ respectively, and parallel transports of each other along γ. The value of the two-point function of a state Λ_μ acting on the elements of the algebra defined by ϕ and ψ is

$$\begin{equation*}\langle {{\Lambda}}_{2},\phi \otimes \psi \rangle {:=}-\frac{{\partial }^{2}}{\partial s\partial t}{{\Lambda}}_{\mu }\left(W\left(t\phi \right)W\left(s\psi \right)\right){\vert }_{s=t=0}=-\frac{{\partial }^{2}}{\partial s\partial t}\left({{\Lambda}}_{\mu }\left[W\left(s\phi +t\phi \right)\right]{\mathrm{e}}^{\frac{\mathrm{i}st{\Xi}\left(\phi ,\psi \right)}{2}}\right){\vert }_{s=t=0}.\end{equation*}$$

Using the isomorphism between ker(□_g) and V(M) the two-point function can be seen to induce a bidistribution on spacetime, i.e., ${{\Lambda}}_{2}\in {\mathcal{D}}^{\prime }\left(M{\times}M\right)$.

Definition 7.6. A quasi-free state Λ_H on the algebra of observables satisfies the microlocal spectrum condition if its two-point function Λ_2H is a distribution ${\mathcal{D}}^{\prime }\left(M{\times}M\right)$ and satisfies the following wavefront set condition

$$\begin{equation*}W{F}^{\prime }\left({{\Lambda}}_{2H}\right)={C}^{+},\end{equation*}$$

where $W{F}^{\prime }\left({{\Lambda}}_{2H}\right){:=}\left\{\left({x}_{1},\eta ;{x}_{2},-\tilde {\eta }\right)\in {T}^{{\ast}}\left(M{\times}M\right);\left({x}_{1},\eta ;{x}_{2},\tilde {\eta }\right)\in WF\left({\omega }_{2H}\right)\right\}$.

The states that satisfy the microlocal spectrum condition⁸ are called Hadamard states and their class includes ground states and KMS states ([13, 44, 45]). An alternative method of constructing a Fock space makes use of the S–J vacuum states. These are quasi-free states that in general do not satisfy this condition [46].

In the low regularity setting we require a generalisation of Hadamard states. A larger class of states, called adiabatic states of order N and characterised in terms of their Sobolev wavefront set, has been obtained by Junker and Schrohe [47]. These states are natural candidates to replace the Hadamard states in spacetimes with limited regularity. In particular, quantum ground states have been constructed in static spacetimes using semigroup techniques [48] and they can be described as adiabatic states [49]. We briefly recall the definition of this class of states and of the Sobolev wavefront set.

Definition 7.7. A quasi-free state Λ_N on the algebra of observables is called an adiabatic state of order $N\in \mathbb{R}$ if its two-point function Λ_2N is a bidistribution that satisfies for every $s{\leqslant}N+\frac{3}{2}$ the H^s-wavefront set condition

$$\begin{equation*}{W{F}^{s}}^{\prime }\left({{\Lambda}}_{2N}\right)\subset {C}^{+},\end{equation*}$$

where WF^s denotes the refinement of the notion of the wavefront set in terms of Sobolev regularity ([43]), i.e., (x, ξ) ∉ WF^s(u) if and only if u = u₁ + u₂ with u₁ ∈ H^s and (x, ξ) ∉ WF(u₂).