Green Operators in Low Regularity Spacetimes and Quantum Field Theory

In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field $\phi$ on a globally hyperbolic spacetime $M$ with $C^{1,1}$ metric $g$. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both $\phi$ and $\square_g\phi$ in order to ensure that $\square_g \circ G^\pm$ and $G^\pm \circ \square_g$ are the identity maps on those spaces. The causal propagator $G=G^+-G^-$ is then used to define a symplectic form $\omega$ on a normed space $V(M)$ which is shown to be isomorphic to $\ker \square_g$. This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local $C^*$-algebras.


Introduction
This paper is concerned with developing the theory of quantum fields on low regularity spacetimes. We will follow the algebraic approach to quantisation as described in [55] and [27]. In particular we will draw heavily on the detailed scheme given in the book [1]. The starting point is a smooth Lorentzian manifold (M, g) and a field equation P φ = f where P is a normally hyperbolic differential operator P acting on a vector bundle F . In this paper we will consider the scalar operator P = g as there are no significant additional mathematical issues in dealing with the general case. The essence of the algebraic approach as outlined in [1] is to first construct the advanced and retarded Green operators for P and use these to construct the causal propagator G = G + − G − . Note that in order for the Green operators to be unique we require the spacetime to be globally hyperbolic. The causal propagator is then used to construct a skew symmetric bilinear map on the space of smooth functions of compact support byω(φ, ψ) := G(φ), ψ L 2 (M,g) . This form is degenerate but gives rise to a symplectic form ω on the quotient space D(M )/ker(G) of the test function space. The next step in the process is to use ω to construct representations of the canonical commutation relations (CCRs) on the space of quasi-local C * -algebras [1,Theorem 4.4.11] which satisfy the Haag-Kastler axioms [23]. Each of these steps may be described in terms of a functor so that we have a functorial description of how to go to from the category of globally hyperbolic manifolds equipped with (formally self-adjoint) normally hyperbolic operators to the space of quasi-local C * -algebras whose elements describe the observables of the field (see details below). Indeed this scheme gives rise to a locally covariant quantum field in the sense of [8]. Going from the rather abstract quantisation procedure described here to the more familiar Fock space representation requires one to pick out the physically relevant states. For the smooth case a mathematically appealing criterion (which corresponds to the standard answer in Minkowski space) is the micro-local spectrum condition of Radzikowski [46]. However this is not directly applicable in the low-regularity case. We return to this point and suggest a suitable modification in the discussion section at the end. In generalising the smooth results to the low regularity setting we need to choose a class of metrics that are sufficiently regular to establish the results we would like, while being sufficiently general to cover the cases of physical interest. From this point of view the choice of C 1,1 metrics is natural since it allows one to deal with spacetimes where the curvature remains finite while allowing for discontinuities in the energy-momentum tensor at, for example, an interface or the surface of a star. From the mathematical point of view C 1,1 is the minimal condition which ensures existence and uniqueness of geodesics and for which the standard results from smooth causality theory go through more or less unchanged [31]. It also ensures that the solutions to the wave equation are in H 2 loc (M ) (see Appendix B for details of the function spaces we use) as shown in Theorem 4.7 below, which ensures we have enough regularity to define the quantisation functors we need. Although one can define solutions to the wave equation for metrics of lower regularity [51,52] there are difficulties in defining the corresponding advanced and retarded Green operators for these cases. In Section 3 we establish the results we need to prove existence and uniqueness of solutions to the forward (and backward) initial value problem for the wave equation on R n+1 for C 1,1 metrics. Rather than rework the entire theory of the wave equation for metrics of low-regularity we use a method of regularising the coefficients [32], using the smooth theory to obtain the corresponding solutions of the Cauchy problem and then using a compactness argument to show that this converges to a weak H 2 loc (R n+1 ) solution of the original equation. This proceeds via the theory of Colombeau generalised functions [13] and is related to the work of [20] on very weak solutions. Furthermore by controlling the causal structure of the regularisation g ε by insisting that J + ε (U ) ⊂ J + (U ) we can ensure that the forward solution u + with zero initial data satisfies the causal support condition supp(u + ) ⊂ J + (supp(f )) (cf. [48,Theorem 2.6.4]). In Section 4 we introduce the notion of global hyperbolicity [49] and temporal functions for nonsmooth metrics [43] as well as the other results from C 1,1 causality theory that we require [31]. We use the fact that even for C 1,1 spacetimes the temporal function can be chosen to be smooth so we can write M as R × Σ, where Σ is a Cauchy surface, and define function spaces where we make a split between space and time. However, as far as possible we formulate our final results in a way that is independent of the choice of temporal function, so that the particular choice of space-time split is not important. The remainder of the section shows how to go from existence and uniqueness results on R n+1 to global results on a globally hyperbolic C 1,1 spacetime. Our approach to this closely follows Ringström [48] and the causality results for C 1,1 metrics [31] ensure that the existence proof remains similar to the smooth case. The next step is to define appropriate Green operators. In the smooth case the Green operator takes (compactly supported) smooth functions to smooth functions. However in the C 1,1 case it is crucial that the map is between suitable Sobolev type spaces. In fact, we find precise conditions on the regularity of the solutions and the causal support in order to define unique Green operators in globally hyperbolic spacetimes of limited differentiability. We need to control the (local) Sobolev norms of both φ and g φ in order to ensure that g • G ± and G ± • g are the identity maps on the corresponding spaces. The choice of function space is also relevant in the definition of ker(G) which is used to construct the factor space for the symplectic map ω. We end the section by considering the dependence of the construction on the choice of temporal function. The passage from the symplectic space defined by ω to the canonical commutation relations proceeds almost identically to that of the smooth case [1] so we only sketch out the details although the analogue of [1,Theorem 4.5.1] requires some work. We end the paper with a summary of the results we have obtained and a discussion of the outstanding issues, including the choice of a Sobolev micro-local spectrum condition to single out the physical states in the low-regularity setting. Appendix A briefly describes the basic properties of regularisation methods we use while Appendix B gives a brief description of various Sobolev spaces we use in the paper. Notation. We denote the derivative of a function u with respect to t by u t or ∂ t u and by u i or ∂ i u if it is with respect to the spatial x i -coordinate. The space of smooth functions of compact support on a manifold M will be denoted by D(M ). A function f on an open subset U of R n is said to be Lipschitz if there is some constant K such that for each pair of points p, q ∈ U, |f (p) − f (q)| ≤ K|p − q|. We denote by C k,1 those C k functions where the k th derivative is a Lipschitz continuous function. A function on a smooth manifold is said to be Lipschitz or C k,1 if it has this property upon composition with any smooth coordinate chart. We gather basic notions and results concerning Sobolev and related function spaces in Appendix B.

The smooth setting
In this section we briefly review the results in the smooth setting. The starting point is an existence and uniqueness result for solutions to a smooth second order hyperbolic equation on R n+1 (see e.g. [48,Theorem 8.6]). This is used to obtain a corresponding result showing the existence of a unique smooth solution u ∈ C ∞ (M ) to the Cauchy initial value problem for a smooth normally hyperbolic operator P on a smooth globally hyperbolic manifold with smooth spacelike Cauchy surface Σ and normal vector field n given by which satisfies the causal support condition supp(u) ⊂ J(supp(u 0 ) ∪ supp(u 1 ) ∪ supp(f )).
Note that the globally hyperbolic condition is essential in order to ensure that the solution is unique. The next step is to show the existence of advanced and retarded Green operators.  We want to use G to construct a symplectic vector space to which one can apply the CCR functor. We first define a skew symmetric bilinear form on D(M ) byω(φ, ψ) := G(φ), ψ L 2 (M,g) . Unfortunately the bilinear form is degenerate so it fails to provide the required symplectic form. However we can rectify this by passing to the quotient space V := D(M )/ker(G), which by the above is just D(M )/P (D(M )). Henceω induces a symplectic form ω on V . One can go on to use this to construct representations of the canonical commutation relations (CCRs) on the space of quasi-local C * -algebras [1,Theorem 4.4.11] which satisfy the Haag-Kastler axioms [1, Theorem 4.5.1].
3 The Cauchy problem on R n+1 for C 1,1 metrics In this section we establish Theorem 3.7 which gives the existence, uniqueness and causal support results we need concerning solutions to the Cauchy problem for the wave equation on R n+1 for a C 1,1 metric. The proof follows from Lemmas 3.1, 3.2, 3.3 below, which cover a slightly more general version of the Cauchy problem. The basic technique is to employ a Chruściel-Grant regularisation of the metric [12] (see details in Appendix A) to obtain a family of smooth metrics (g ε ) ε∈(0,1] which converge to g in the C 1 topology, have uniformly bounded second derivatives on compact sets and satisfy J + ε (K) ⊂ J + (K) for every compact subset K ⊂ M and ε > 0. This enables us to obtain the required solution of the wave equation by taking a suitable limit of solutions to the smooth equation as ε → 0 while preserving the causal support properties. Lemma 3.1 provides a detailed form of the energy estimates which proves crucial in the transition to the non-smooth setting. The causal support properties follow from Lemma 3.5. Note that although Lemma 3.2 shows the existence of a generalised Colombeau solution [13] to a corresponding generalised Cauchy problem the main result of this section, Theorem 3.7, concerns a classical weak solution and does not require explicitly referring to the Colombeau solution used in the proof. As a basic setup, we consider a Lorentzian metric g of signature (+, −, ..., −) on R n+1 with spatial components (−h ij ) 1≤i,j≤n and the corresponding wave operator where all coefficients are supposed to be real-valued, smooth, and bounded in all orders of derivatives. Moreover we assume that there exist positive constants τ min , τ max , λ min , λ max such that for all (t, x) ∈ R n+1 and for all ξ ∈ C n . It follows from Theorem 23.2.2 in [30] or Theorem 2.6 in [4] that for initial data u 0 , In the following lemma we provide an explicit energy estimate with s = 1.
Proof. We show how to derive the energy estimate in terms of application of P to any function u ∈ H ∞ (R n+1 ): We may write and Here we have used the notation f i,µ for ∂ x µ f i , where Greek indices range from 0 to n and x 0 := t.
Considering time t as a parameter and integrating equation (3.5) over the spatial domain R n , we obtain where we have used that R n ∂ xj X j (u(t, x); t, x)d(x 1 , ..., x n ) = 0 for all j = 1, ..., n and for all t ∈ [0, T ], since x → u(t, x) belongs to H 1 (R n ) and the latter possesses C ∞ c (R n ) as a dense subspace. Using the notation g µ · for the µ-th row vector and divg µ · for its divergence, we may estimate Y (u(t, x); t, x) as follows: We note that where C g ,a,b depends on the the metric g and its first derivatives (more precisely, only on g 0µ,µ and g ij,j ), as well as on a and b. (See Appendix B for the definition ofH 1 (M ).) Moreover we where C min := min(τ min , λ min ) and C max := max(τ max , λ max ). We observe that |2Re( Using the fundamental theorem of calculus we may write and integration over the spatial variables yields where we have deliberately multiplied by C min ≤ C max . Integrating equation (3.6) with respect to the time variable from 0 to t we obtain We will from now on writeũ 0 := u(0, ·) andũ 1 := ∂ t u(0, ·). Upon adding inequality (3.7) we get as well as the other estimates and rearranging terms, we arrive at Gronwall's inequality [14, Chapter XVIII, §5, Section 2.2] then yields the a-priori energy estimate To obtain similar estimates for spatial derivatives ∇u = (∂ x1 u, ..., ∂ xn u) T of u, we first divide the equation by g 00 : Differentiating this equation with respect to x k yields Multiplying by g 00 we may write this as where we have brought all terms which do not contain spatial derivatives ∂ x k u to the right-hand side. Multiplying by ∂ t ∂ x k u and summing over 1 ≤ k ≤ n, we get where the whole expression depends on (t, x) and X and Y are defined exactly as before. The additional term Z collects all terms from (3.10) which are of lower order with respect to ∂ x k u (that is, they contain spatial derivatives of order 2 at most). We have and it is easy to see that Z can be estimated as follows: We may now proceed as in the proof for the basic energy estimate (3.8), following the steps from (3.6) to (3.8). Upon integrating equation (3.11) over the spatial domain R n , we get |Z(∇u; s, ·))|dV ds + t 0 R n |F (∇u; s, ·))|dV ds and by employing the estimates for Y , Z, and F , we arrive at an energy estimate for ∇u (recall that which we may also write as where C g ,a ,b contains L ∞ (Ω T )-norms of at most first-order derivatives of the coefficients g, a, b. Applying Gronwall's inequality then results in an energy estimate for the spatial derivative vector ∇u, (3.12) where β 1 := C −1 min C g ,a ,b . Employing the basic energy estimate (3.8) to estimate u(s, ·) and ∂ s u(s, ·) 2 L 2 (R n ) in terms of the restrictionsũ 0 ,ũ 1 and P u, we arrive at To prepare for the extension to the Colombeau solutions, we proceed by discussing higher-order energy estimates. Applying the operator ∂ x l to equation (3.9) and afterwards multiplying by g 00 we essentially get a system of the form P (∇ 2 u) = Q (2) (Σ 2 u) + R (2) (Σ 2 P u), where Q 2 is a PDO of order 3, containing time derivatives of at most order 1, and R is a purely spatial PDO of order 2. Here we have used the notation Thus the terms produced by Q (2) (Σ 2 u) are either lower-order terms in ∇ 2 u or terms with less than two spatial derivatives of u. In the first case, they can be dealt with just as the generic lower-order terms appearing in P (∇ 2 u). In the second case, they can be interpreted as source terms on the right-hand side, just as the term R (2) (Σ 2 P u). More generally, we obtain where Q (r) is a PDO of order r + 1, containing time derivatives of at most order 1, and R (r) is a purely spatial PDO of order r. Moreover, the coefficients of Q (r) and R (r) depend on spatial derivatives of the metric g of at most order r. It is important to note that all principal coefficients of Q (r) depend only on g µν , a µ , b and on derivatives g µν,ρ (0 ≤ ρ ≤ n). In particular, they can be viewed as lower-order terms of an operator P r , which has the same principal symbol as P . The energy estimate following from (3.13) is of the form where β r depends only on g µν , a µ , b and on derivatives g µν,ρ . Summing up, we obtain energy estimates for ∇ r u and ∂ t ∇ r u for r ∈ N 0 . Equation (3.1) itself then immediately provides an estimate for ∂ 2 t u as well. Similarly, equation (3.9) allows us to estimate ∂ 2 t ∂ x k u in terms of (mixed) derivatives of order ≤ 1 in the time variable. More generally, equation (3.13) directly provides estimates for ∂ 2 t ∇ r u (r ∈ N) in terms of (mixed) derivatives of order ≤ 1 in the time variable. To get estimates for mixed derivatives ∂ m t ∇ r u of order m ≥ 3 in the time variable, we may apply the operator ∂ l t to equation (3.13), yielding where Q (r,l) is a PDO of order r + l with time derivatives only up to order l − 1 and R (r,l) is a PDO of order r + l − 1 with time derivatives up to order l − 1. The first term in P (∇ r ∂ l t u) is g 00 ∇ r ∂ l+2 t u. Thus, starting with l = 1, we immediately obtain energy estimates for all mixed derivatives of the form ∇ r ∂ 3 t u. In the next step, we can show energy estimates for ∇ r ∂ 4 t u, and proceed iteratively to get energy estimates for all mixed derivatives ∇ r ∂ m t u as well. These estimates are direct consequences of (3.15), once estimates for the terms with time derivatives of lower order have been established (there is no need for a Gronwall argument). However, a perhaps more elegant viewpoint is that for any r, l ∈ N, equation (3.15) also implies H 1 -estimates for ∇ r ∂ l t u in terms of initial data and source terms via the energy estimate (3.8) for the operator P . Again, it is important to note that the principal coefficients of Q (r,l) and R (r,l) only depend on g µν , a j , b and on first derivatives g µν,ρ . Higher-order derivatives of the coefficients will only enter in the lower order terms and thus do not show up in the exponent after applying Gronwall's inequality. In the following lemma we refer to Colombeau theoretic notions which are summarised in Appendix A.
Then, given initial data u 0 , u 1 ∈ G L 2 (R n ) and right-hand side f ∈ G L 2 (Ω T ), the Cauchy problem (3.4) has a unique solution u ∈ G L 2 (Ω T ).
Proof. We fix a symmetric representative of g and representatives of all lower-order coefficients, initial data, and right-hand side. As noted above, we have smooth solutions to the corresponding classical initial value problems for each ε < ε 0 . To show moderateness, we apply Lemma 3.1 and obtain where (e β ε ) is moderate thanks to the log-type condition on the coefficients and their first derivatives as well as the positivity condition (ii). Thus the estimate shows that ( u ε H 1 (Ω T ) ) and ( ∇u ε H 1 (Ω T ) ) is moderate. For r ≥ 2, the higher-order estimates (3.14) for the spatial derivative vector ∇ r u ε then imply that ( ∇ r u ε H 1 (Ω T ) ) is also moderate (for any r ∈ N), since all principal coefficients of the operators on the right-hand-side of equation (3.13) only depend on (g ε µν ), (a ε µ ), (b ε ) and on first derivatives (g ε µν,ρ ), all of which are of L ∞ −log-type. An iterative application of the higher-order estimates for spatial derivatives (3.14), starting with r ≥ 2 then establishes moderateness of ∇ r u ε H 1 (Ω T ) for all r ∈ N 0 , since 1/C ε min is of logarithmic growth in ε (and thus moderate), are of moderate growth and β ε r is of L ∞ -log-type (where β ε r depends on C ε min and (first derivatives of) g ε , a ε , and b ε ). Finally, the corresponding energy estimates for mixed derivatives following from equation (3.15) yield moderateness of ∇ r ∂ l t u ε H 1 (Ω T ) as well. Note that only the principal coefficients will be exponentiated in the Gronwall inequality. Thus the important observation in all these higherorder estimates is that the principal coefficients on both sides of equation (3.15) only depend on g ε µν , a ε µ , b ε and on derivatives g ε µν,ρ , all of which are of L ∞ -log-type. In total this implies that for all r, l ∈ N 0 there exists m ∈ N 0 such that ∇ r ∂ l t u ε 2 L 2 (Ω T ) = O(ε −m ) as ε → 0 and thus [(u ε ) ε>0 ] ∈ G L 2 (Ω T ). To show uniqueness of the generalised solution in G L 2 (Ω T ), we assume negligible initial data ( u 0ε ), ( u 1ε ) ∈ N L 2 (R n ) and right-hand side ( f ε ) ∈ N L 2 (Ω T ). Then the same energy estimates we used for moderateness yield negligibility of the solution [( u ε ) ε>0 ].
In the next lemma we want to identify conditions on the coefficients and data of a low-regularity Cauchy problem (3.4) such that the corresponding generalised Cauchy problem, obtained via regularisation, has a unique solution u ∈ G L 2 (Ω T ) and, moreover, this solution admits a distributional shadow.

Lemma 3.3.
We consider the Cauchy problem (3.4) where all coefficients belong to W 1,∞ (R n+1 ) and the bounds (3.2) and (3.3) are satisfied. Then, given initial data (u 0 , u 1 ) ∈ H 2 (R n ) × H 1 (R n ) and right-hand side f ∈ L 2 ([0, T ], H 1 (R n )), we consider the corresponding generalised Cauchy problem obtained via convolution regularisation of all coefficients and data. Then the unique generalised according to Lemma 3 Proof. We note that the regularised coefficients and data satisfy the assumptions in Lemma 3.2 and we obtain a unique generalised solution. We aim at showing that any representative (u ε ) of the solution is a Cauchy net. To this end we apply the variant of the energy estimate in Lemma 3.1, implicit in the proof (see also (3.8)), with order of spatial Sobolev norms reduced by one (except for the initial data): Applying the operator P ε to the difference u ε − uε, we obtain where β ε is bounded uniformly in ε thanks to the hypotheses on the non-smooth coefficients. We may write Considering the regularity of the initial data and right-hand side, it is easy to see that (u ε ) is a Cauchy net, if for all η > 0 there exists Since all coefficients belong to W 1,∞ and thus converge in the L ∞ -norm, it suffices to show that uε H 2 (Ω T ) is bounded uniformly in ε. First, observe that the energy estimate (3.8) implies that ). However, we can deduce even better regularity from the properties of the net (u ε ). For any t ∈ [0, T ] and any ϕ ∈ D(R n ), we have as ε → 0 and we already have uniform convergence of (u ε (t, ·)) in H 1 (R n ) by the Cauchy net estimate and hence as a distribution. It follows that ∂ xj ∂ x k u 0 (t, ·) ∈ L 2 (R n ) and therefore u 0 ∈ C 0 ([0, T ], H 2 (R n )). Moreover we have Summing up, we have obtained a distributional shadow u 0 of the generalised solution with In the last part of the proof we show that the distributional shadow u of the generalised solution is the unique weak solution to the Cauchy problem. The proof follows the line of arguments in the proof of [32,Corollary 4.6]. First note that both u and ∂ t u are continuous and thus, by construction of u, the initial conditions are satisfied. The Cauchy net estimate (3.16) implies that u ε → u as ε → 0 in the norm H 1 (Ω T ). Our aim is to prove that P u = f in a suitable weak sense. Since the coefficients belong to W 1,∞ (R n+1 ), we immediately get L 2 -convergence of all first-order terms and H 1 -convergence of all zero-order terms: We claim that . By the weak compactness theorem, this implies that there exists a weakly * convergent subsequence (u 1/n k ) k∈N with limit v ∈ H 2 (Ω T ) (cf. [47,Theorem 6.64]) and indeed v = u since we already know from the Cauchy net estimate that and we have for any ϕ ∈ L 2 (Ω T ): To see this, observe that the first term goes to zero because the coefficients of P 1/n k converge to those of P in L ∞ (Ω T ) as k → ∞ and (u 1/n k ) k∈N is bounded in H 2 (Ω T ); the second term vanishes as well in the limit k → ∞ since u 1/n k converges to u in H 2 (Ω T ) as k → ∞. We provide the explicit calculation for the g 00 -term (the others can be treated similarly): This shows that for any ϕ ∈ L 2 (Ω T ), and thus u is indeed a weak solution of the initial value problem.
To show uniqueness of the weak solution, we suppose that there exists another solution w ∈ We may regularise this solution so that The following estimate then shows that P ε w ε → P w = f in L 2 (Ω T ) as ε → 0: Here we have only used the H 2 (Ω T )-convergence of w ε to w as ε → 0. Denoting by (u ε ) ε a representative of generalised solution and applying the basic energy estimate (3.8) to the difference u ε − w ε then yields Letting Remark 3.4. The required conditions for the existence of a distributional shadow (and weak solution) are weaker than those that would be required in a similar result based on transforming the equation (3.1) into a first-order system as in [24], since the lower-order coefficients of this system would contain derivatives of the principal coefficients of equation (3.1) and therefore g µν ∈ W 2,∞ (R n+1 ) would be necessary (instead of g µν ∈ W 1,∞ (R n+1 )). However, for a wave equation derived from the Laplace-Beltrami operator of a Lorentzian metric g, where Γ ρ µν = 1 2 g ρσ (g σµ,ν + g σν,µ − g µν,σ ), the lower-order coefficients contain derivatives of the metric and thus the metric has to be W 2,∞ (C 1,1 ) anyway in order to obtain a distributional shadow of the generalised solution.
Proof. Let u + ε be the unique solution of the corresponding advanced problem for the regularised metric g ε where the right-hand side f is not necessarily smooth. Note that upon taking K slightly larger, we may assume that supp(f ) ⊂ J + ε (K) for ε sufficiently small. By Lemmas 3.1, 3.2 we can obtain u + ε as a limit α → 0 of u + ε,α where (u + ε,α ) α>0 is a Colombeau representative of the unique generalised solution with fixed smooth g ε -coefficients and right-hand side being the class of a convolution regularisation (f α ) α>0 . By the smooth theory we have for every ε and every α that At fixed ε and as α → 0 we have in terms of monotonically decreasing sets α>0 supp(f α ) = supp(f ), i.e. supp(f α ) supp(f ). By closedness of the causal relation [1, Lemma A.5.5] we obtain that Let ψ be a test function such that supp(ψ) ∩ J + ε (supp(f )) = ∅. By (3.17) and (3.18) there exists some α 0 > 0 such that supp(ψ) ∩ J + ε (supp(f α )) = ∅ for all α < α 0 . Therefore u + ε,α , ψ = 0 for every α < α 0 ; taking the limit α → 0 we obtain u + ε , ψ = 0 and therefore Now let ψ have support disjoint from J + (supp(f )). Then, by the causal properties of the Chruściel-Grant regularisation, we have that supp(ψ) is also disjoint from J + ε (supp(f )) for every ε > 0. Therefore, u + ε , ψ = 0 for every such ψ and for every ε > 0, showing that supp(u Remark 3.6. More generally for non-zero initial data one has The proof of this follows from the above by recasting the smoothed version of the problem as an equivalent inhomogeneous problem with non-zero initial data.
Proof. The existence and uniqueness of the weak solution follow from Lemma 3.3 and the causal support condition follows from Lemma 3.5 together with Remark 3.6 applied to both the past and future.
4 The Cauchy problem for C 1,1 globally hyperbolic spacetimes 4.1 Causality results for C 1,1, spacetimes 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 [39] 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 [1, 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 in the causal hierarchy of spacetimes [44] and recently there has been considerable interest in looking at the causal properties of low-regularity spacetimes [43] [31]. It was shown explicitly by Chrusćiel [10] that essentially all of causality theory for smooth spacetimes goes through to the C 2 case. However in the proofs of these results an important role is played by the existence of totally normal (convex) neighbourhoods and the Gauss Lemma whose existence is not automatic in the C 1,1 case. However it was shown in [31], [36] [42] 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 [31] and [42] for details).
For spacetimes with a C 2 metric there are four equivalent notions of global hyperbolicity (see for example [44, Section 3.11 p. 340ff.]). These are: 1. compactness of the causal diamonds and causality 1 , 2. compactness of the space of causal curves connecting two points and causality [39], 3. existence of a Cauchy hypersurface, 4. the metric splitting of the spacetime.
For C 1,1 spacetimes we will adopt the first definition. However for non-totally imprisoned [41] C 0 spacetimes (and hence in particular for globally hyperbolic C 1,1 spacetimes) these four definitions remain equivalent [49]. See also [43] 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 [43,Theorem 2.30] (see also [18]) 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 → 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 R × Σ where the first projection is t and the level surfaces 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 Sanchez [5] 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 [5] to the non-smooth case we need the concept of a stable Cauchy hypersurface introduced by Minguzzi in [43]. These are Cauchy hypersurfaces which are also Cauchy hypersurfaces for some metric g g with strictly wider lightcones than g. Bernard and Suhr [7,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) [7, Theorem 1]. A full discussion of this issue is given in the paper by Minguzzi [40]. The approach in [40] is complementary to that in [7] 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 Sanchez [5] to construct a smooth time function with Σ = t −1 (0) which as shown in [43] is a smooth temporal function for the original spacetime (M, g). See [40,Theorem 2.22] for details. Indeed given two smooth spacelike Cauchy hypersurfaces Σ 0 and Σ 1 with Σ 1 ⊂ J + (Σ 0 ) \ Σ 0 one can find a smooth temporal function t that interpolates between them so that Σ 0 ⊂ t −1 (0) and Σ 1 ⊂ t −1 (1) [40, Theorem 2.23].

Existence and Uniqueness
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 2 loc (M ) 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.
Proof. To establish the energy inequality we follow Hawking and Ellis by applying the divergence theorem to an enhanced energy-momentum tensor [25,Lemma 7.4.4]. Let (1) be the energy momentum tensor of the scalar field. Then (1) T αβ has vanishing divergence and satisfies the dominant energy condition. We now follow [25] and modify this by adding on the term (0) T αβ which still satisfies the dominant energy condition. Let ξ α = ∇ α t then we obtain the required inequality by applying the divergence theorem to S αβ ξ α over the region In order to do this we require that div(S αβ ξ α ) should be integrable with respect the volume form ν g , and this is guaranteed by the compactness ofŪ and the fact that our solution is in H 2 loc (M ). In fact it is enough that the weak solutions have two derivatives in L 2 loc (M, g) if the metric and the timelike vector field are in the space C 0,1 (see [11]). The boundary of U τ consists of three parts; the level surface Σ τ ∩Ū + , the level surface Σ 0 ∩Ū + and the remainder which we denote H. Because of the dominant energy condition and the fact that ∂U ∩Ū + is achronal, the contribution to the surface integral from H is positive. We therefore obtain the following inequality: where ν g is the volume form on U τ given by g, and µ τ is the volume form induced by g on the Σ τ . We now define an energy type integral Then onŪ this is equivalent [57] to the restricted Sobolev norm (B.3) Note that since the solution u is in H 2 loc (M ) we have well-defined traces inH 1 (Σ). In terms of the energy norm we may write (4.2) in the form Repeated application of the Cauchy-Schwartz inequality and g u = f then gives [57].
which on applying Gronwall's inequality gives In terms of the Sobolev type norms this gives 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 spacelike n-dimensional spacelike Cauchy 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 [31]. Now suppose there exist two solutions u andũ to the above initial value problem. Then applying Lemma 4.1 toû := u −ũ over the region U + gives Hence ||û||H1 (Στ ∩U + ) = 0 so thatû and ∇û 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).
Proof. We prove the result for J + . A similar proof holds for J − . Let V = J + (supp(u 0 ) ∪ supp(u 1 ) ∪ supp(f )) and suppose q ∈ M \ V . Then we may find a point p ∈ D + (Σ) such that q ∈ I − (p) ∩ I + (Σ) and u and ∇u vanish on J − (p) ∩ Σ and f vanishes on J − (p) ∩ J + (Σ). Now let U = I − (p) ∩ I + (Σ) and apply (4.3) on this region to obtain 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). So that 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 [48]. In both cases the proof given in [48] for the smooth case goes through to that of a C 1,1 metric unchanged. 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 = (x 0 , x 1 , . . . , x n ) such that q ∈ U ∩ Σ if and only if q ∈ U and x 0 (q) = 0. Furthermore we may choose x so that ∂ ∂x 0 is the future directed unit normal to Σ for q ∈ Σ ∩ U . If we fix > 0 and let g µν := g( ∂ ∂x µ , ∂ ∂x ν ), then we can assume U to be such that |g 0i | ≤ i = 1, . . . , n on U . If we let a = g 00 (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)ξ i ξ j > b|ξ| 2 ) then we may assume that g 00 (q) < a/2 and g ij (q) regarded as a positive definite matrix is bounded below by b/2 for q ∈ U . 3. there is a chart (U, φ) with φ = (x 0 , . . . , x n ) and x 0 = t, such that there exist a, b > 0 with g 00 (q) < −a and g ij (q)ξ i ξ j ≥ bδ ij ξ i ξ j for q ∈ U ; 4. for any compact K ⊂ U there is a C 1,1 matrix valued function h on R n+1 such the h µν = g µ,ν • φ −1 on φ −1 (K) and such that there are positive constants Note that that although the proof is identical to that in [48] it relies on the C 1,1 causality results of [31] and [7,Theorem 1]. Note also that in point (4) above the matrix valued function is only C 1,1 rather than smooth as it is in [48,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.
Let K 1 ⊂ Σ be a compact set such that supp(u 0 ) ∪ supp(u 1 ) ⊂ K 1 and K 2 ⊂ M a compact set such that supp(f ) ⊂ K 2 . Let t 1 > 0 and define R t1 to be the set of q such that 0 ≤ t(q) ≤ t 1 .
Then R t1 is closed and K 3 = K 2 ∩ R t1 is compact. The union of I + (p) for p ∈ I − (Σ) is an open cover of K 1 ∪ K 3 so there is a finite number of points p 1 , . . . , p such that the I + (p i ) are finite subcover of Wp i Let χ ∈ C ∞ 0 (Up k ) be such that χ(q) = 1 for all q ∈ K p . Then we use our solution up to time s 1 to define new initial data on Σ s1 given byũ 0 := (χu)| Σs 1 ∈ H 2 (Σ s1 ) andũ 1 := (χ∇ n u)| Ss 1 ∈ H 1 (Σ s1 ) and sourcef := χf . These all have their support within Up k so we may use the chart (Up k , φ) to regard these are data and source on the whole of R n and 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 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 R n+1 . The third condition of Lemma 4.5 and the fact that the solution is in C 0 (R, H 2 (Σ t )) ∩ C 1 (R, H 1 (Σ t )) ∩ H 2 loc (M ) ensures that we may apply Proposition 3.3 to obtain a solution on R n+1 which on φ(Up k ) may be transferred back to give a solution on K p . In the region V p := I − (p) ∩ J + (Σ s1 ) 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 O 1 to be the union of the V p for p ∈ F s , s ∈ [s 1 , τ + ] then the above construction defines a unique solution in O 1 . Note that the interior of O 1 contains F s for all s ∈ (s 1 , τ + ). Now define O 2 to be the set of points for which s 1 ≤ t(q) < τ + 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 O 1 and O 2 . If q ∈ O 2 ∩ O 1 with t(q) > s 1 then both u and ∇u vanish at J − (q) ∩ S s1 and f vanishes in J − (q) ∩ J + (Σ s1 ). Furthermore there is an r such that q ∈ V r ⊂ O 1 . So by uniqueness the solution defined on O 1 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 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 A is not empty. We have also shown that if τ ∈ 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 A is open in the relative topology of [0, ∞). Finally we note that by definition A is also closed in [0, ∞) because it contains its limit points. Then this set is open, closed and non-empty so must be the whole of [0, ∞) and we have a solution for all future times. By time reversal we also have a solution for all past times and hence on the whole of M .
Proof. Let p be any point to the future of Σ. We also want to show that the initial value problem is well-posed. For solutions in H 1 loc (M ) this follows immediately from the energy estimate (4.1). But well-posedness in H 2 loc (M ) requires a higher order estimate which we now establish. (4.5) Proof. We use a similar approach to that in the existence proof given in [1, 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 p 1 , · · · , p N such that the corresponding W pj cover K. Now let {χ j } N j=1 be a partition of unity of K subordinate to the W pj . We now define u 0,j := χ j u 0 , u 1,j := χ j u 1 , f j := χ j f.

So that
We also define Let u j be the (weak) solution of the IVP We will employ (an implicit choice of a temporal function in terms of) the diffeomorphism M ∼ = R × Σ 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 real interval, thus suppressing the transfers of functions via restrictions of the underlying global diffeomorphism. By point two of Lemma 4.5 there exists an pj > 0 such that Let δ = min{ p1 , · · · , p N }. 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 Since by (4.6) each of the u j lie entirely within some chart (U pj , φ pj ) we may regard the initial value problem as one on I ×R n where I is an interval chosen sufficiently large such that the images of (−2δ, 2δ) × Σ under all the φ pj are contained in I × 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 × R n to ones for u j on U j ⊂ M to give where we may replace f L 2 ([0,δ],H 1 (Σ)) by the larger value f H 1 (M ) , since f ∈ H 1 comp (M ).
We remark that in the above proof (and formulation of the result) we have replaced the norm f L 2 ([0,δ],H 1 (Σ)) by f H 1 (M ) , which is valid for f ∈ H 1 comp (M ), 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 (u 0 , u 1 ) ∈ H 2 comp (Σ) × H 1 comp (Σ) and source f ∈ H 1 comp (M )), then the (weak) Proof. We first use Lemma 4.8 to obtain an estimate for the dataû 0 := u| Σ δ andû 1 := ∇ n u| Σ δ induced by u on Σ δ . It follows from (4.5) and the fact that that u ∈ C 0 (R, Now applying Lemma 4.8 to the initial surface Σ δ we obtain aδ > δ such that

Combining the two energy inequalities on [0, δ] and [δ,δ] we have
This shows that we may extend the energy inequality from [0, δ] to the larger time interval [0,δ] by repeatedly applying Lemma 4.8.
We now use the above proposition to obtain a spacetime energy inequality for any compact K ⊂ M .
Proof. Without loss of generality we may assume that K ⊂ [0, T ] × Σ. Due to the regularity of u ∈ C 0 (R, H 2 (Σ t )) ∩ C 1 (R, H 1 (Σ t )) we have control of the second order spatial derivatives, the second order mixed derivatives and the lower order terms In order to obtain the required estimate we also need to control the ∂ tt u in the L 2 (K) norm. Using From (4.8),(4.9), the regularities of f and g, we obtain an L 2 estimate for ∂ tt u, Combining the above with Proposition 4.9 completes the proof. Equation (4.7) implies the following result.

Green operators for C 1,1, spacetimes
In this section we will define Green operators for g on globally hyperbolic manifolds M with C 1,1 metrics. We will show existence and uniqueness of Green operators via the existence of solutions to the wave equation with appropriate regularity and causal support. We define below the notion of generalised hyperbolicity which will give us the required conditions in this situation. Proof. Theorem 4.7 shows that a globally hyperbolic C 1,1 spacetime satisfies the condition of generalised hyperbolicity to the future by considering the forward initial value problem where f ∈ H 1 comp (M ) and Σ + is a smooth spacelike Cauchy hypersurface such that J + (supp(f )) ∩ Σ + = ∅. Note: If we were to choose some other smooth spacelike Cauchy hypersurfaceΣ + , which also satisfies J + (supp(f )) ∩Σ + = ∅, then the corresponding solution is the same, since the divergence theorem arguments used in Lemma 4.1 apply and yield that the solution must vanish in the region between Σ + andΣ + . Similarly, Theorem 4.7 shows it satisfies the condition of generalised hyperbolicity to the past by considering the backwards initial value problem where f ∈ H 1 comp (M ) and Σ − is a smooth spacelike Cauchy hypersurface such that J − (supp(f )) ∩ Σ − = ∅. Again the solution is independent of the choice of Cauchy surface as long as it satisfies the causal support condition.

Green operators
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: 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
satisfying the properties is called an advanced Green operator for g . A retarded Green operator G − is defined similarly.  [28,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) = τ 2 − |ξ| 2 , which is hyperbolic with respect to the directional vectors (±1, 0) and produces the temperate weight functionp(τ, ξ) := |α|≥0 |∂ α p(τ, ξ)| 2 = (τ 2 − ξ 2 ) 2 + 4(τ 2 + ξ 2 + 2) ≥ 1 + τ 2 + ξ 2 =: w 1 (τ, ξ). The unique fundamental solution E ± with support in the half space where ±t ≥ 0 belongs to B loc ∞,p , i.e., for every test function φ the Fourier transform F(φE ± ) times p is measurable and bounded. The advanced and retarded Green operators are then given by = H 2 loc (R n+1 ), sincep(τ, ξ)w 1 (τ, ξ) ≥ w 2 1 (τ, ξ) = 1 + τ 2 + ξ 2 . We next show that the advanced and retarded Green operators are adjoints of one another. To do this we use the following Lemma 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 L 2 inner product. We may now prove the following theorem.  g). Moreover, is compact by the global hyperbolicity condition. Hence, 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
Proof. We will only discuss the advanced Green operator, the existence and the properties of the retarded Green operator follow from time reversal.
Existence: We define the linear map 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 1 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 2 and 3 in Definition 5.4. Property 2: Let f ∈ V 0 and v ∈ D(M ), then where we have used Theorem 5.7 and Property 1 for the retarded Green operator G − . Thus the weak form of the required identity holds, which implies G + g (f ) = f for every f ∈ V 0 (see Remark 5.5(ii)). Property 3 follows because supp(G + ) = supp(u + ) ⊂ J + (supp(f )) by Lemma 4.3.
Uniqueness: LetG + be another linear operator satisfying Definition 5.4. Given f ∈ H 1 comp (M ) we have that v :=G + (f ) satisfies g v = f and supp(v) ⊂ J + (supp(f )). Since f ∈ H 1 comp (M ), 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 ∇ n v = 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 thatG We now show that the low-regularity Green operators satisfy an exact sequence result similar to that in the smooth case [1, 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
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 • Exactness at V 0 , i.e., injectivity of g : Let φ ∈ V 0 be such that g φ = 0. By compactness of the support there is a smooth spacelike Cauchy hypersurface Σ such that φ = 0 and ∇ n φ = 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.

Restrictions
We briefly discuss the restriction of Green operators to causally compatible subsets Ω ⊂ M , that is, sets such that We have the following theorem (cf. [1, Proposition 3.5.1]).
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 g . Notice that for all u ∈ H 2 loc (M ) we have g (u| Ω ) = g | Ω (u| Ω ) = ( g u)| Ω and for all u ∈ H 2 (Ω) with supp(u) ⊆ Ω we have ( g u) ext = g (u ext ).
Proof of Theorem 5.10:

6 Quantisation functors
In this section we discuss suitable categories and functors as in the smooth case that will allow us to construct the algebra of observables of the quantum theory.

The functor SYMPL and the categories GENHYP and SYMPLVECT
This subsection defines a category based on the analytic results in the previous sections and a functor assigning to each object a symplectic space. Before considering quantisation we prove the following result on compatibility of Green operators.
Proof. Theorem 5.10 shows thatG ± (φ) := G ± 2 (φ ext )| M1 is a Green operator. By uniqueness, this operator has to be equal to G ± 1 and the result follows. Remark 6.4. In the smooth setting [1] 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 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 : M 1 → M 2 is a time-orientation preserving isometric embedding such that ι(M 1 ) ⊂ M 2 is a causally compatible open subset [1]. Moreover, given the condition of globally hyperbolicity one can form the category GLOBHYP where objects are 3-tuples (M, F, P ), where M is a Lorentzian manifold, F is 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 : M 1 → M 2 is a time-orientation preserving isometric embedding such that ι(M 1 ) ⊂ M 2 is a causally compatible open subset. The existence and uniqueness of Green operators allow us to form a functor from GLOBHYP to LORFUND [1].
We now use the Green operators in order to construct a symplectic vector space. Let (M, G + , G − ) be an object of GENHYP and definẽ ω : Theorem 5.9). Thenω is bilinear and skewsymmetric by Theorem 5.7. However,ω is degenerate because ker(G) is nontrivial. Moreover, using Theorem 5.9 we have that ker(G) = g V 0 .
Therefore on the quotient space V (M ) = U 0 / ker(G) = U 0 / g V 0 the degenerate formω 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.

The functor CCR and the categories C * -ALG and QUASILOCALALG
In this section we closely follow [1] 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 the smooth case is that when considering the symplectic space (V, ω) in the smooth theory one has 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 A with unit and a map W : V → A such that for all ϕ, ψ ∈ V , A Weyl system (A, W ) of a symplectic vector space (V, ω) is called a CCR-representation of (V, ω) if A is generated as a C * -algebra by the elements W (ϕ), ϕ ∈ V . In this case we call A a CCRalgebra 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 [1, Example 4.2.2] ). Moreover, the construction is categorical in the sense that if (V 1 , ω 1 ) and (V 2 , ω 2 ) are two symplectic vector spaces and S : V 1 → V 2 is a symplectic linear map. Then, there exist a unique injective * -morphism CCR(S) : CCR(V 1 , ω 1 ) → CCR(V 2 , ω 2 ) The proof can be found in Corollary 4.2.11 in [1]. From uniqueness of the map CCR(S) it is possible to define a functor CCR : SYMPL → C * −ALG 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: 1. for all α, β ∈ I there exists a γ ∈ I with α ≤ γ and β ≤ γ, 2. for every α ∈ I there is a β ∈ I with α ⊥ β, 3. α ≤ β and β ⊥ γ, then α ⊥ γ, 4. 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 (U, {U α } α∈I ) of a C * -algebra U and a family {U α } α∈I of C * -subalgebras, where I is a directed set with orthogonality relation such that the following holds: 3. The algebras U α have a common unit 1, 4. If α ⊥ β, then the commutators of elements from U α with those of U β are trivial.
A morphism between two quasi-local C * -algebras (U, {U α } α∈I ) and (V, {V β } β∈J ) is defined as a pair (ϕ, Φ) where Φ : U → V is a unit-preserving C * -morphism and ϕ : I → J is a map such that In the remainder of this section we discuss a functor from GENHYP to QUASILOCALALG. Let 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 [12,49,31] 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.

The Haag-Kastler axioms
In this subsection we show that the functor QUANT given by Theorem 6.10 satisfies the Haag-Kastler axioms.
The proof will be based on the following two lemmas. The proof of Lemma 6.13 in the C 1,1 setting can be carried out following that of [1] with suitable modifications using results of low regularity causality theory [12,49,31]. In particular, the proof uses the facts that, the causal relation is closed, that if S, S t are Cauchy hypersurfaces of O and S is also a Cauchy hypersurface of M , then S t is a Cauchy hypersurface of M and the existence of Cauchy hypersurfaces in globally hyperbolic spacetimes. Set , 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 χ ± ∈ H 2 comp (M ). Moreover, which implies g χ ± ∈ H 1 loc (M ). Notice that g ρ ± is not smooth but C 0,1 . Now ψ is the difference of H 1 loc (M ) functions so it remains to show that supp(ψ) is compact and contained in O. By the first property of ρ ± , one has

Discussion
In this paper we have constructed Green operators for spacetime metrics of regularity C 1,1 . The function spaces for the domain and range of the Green operators play a fundamental role in low regularity spacetimes and our choices for these spaces were motivated by the following two requirements: Global well posedness of the Cauchy problem and employing Sobolev spaces, such as H k loc (M ) and H k comp (M ) (k ∈ N 0 ), that do not depend on a Riemannian background metric. We have shown that the quotient space V (M ) = U 0 / g V 0 can be used to construct quasi-local C * -algebras that satisfy the Haag-Kastler axioms, so that in a quantum theoretic setting the self-adjoint elements in these C * -algebras can be associated with the observables of the theory.

Topological Issues
Let us describe the quotient vector space U 0 / g V 0 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 U 0 / g V 0 = U 0 /ker(G) in this case. Recall that G is a linear map U 0 → V sc and let G 0 denote the associated map from the quotient U 0 /ker(G) to im(G) ⊆ V sc , defined by G 0 (φ + ker(G)) := Gφ for every φ ∈ U 0 . Therefore, G 0 is linear and bijective by construction and we arrive at the following chain of (algebraic) isomorphisms of vector spaces Recall that the analogue of (7.1) in the smooth globally hyperbolic case, as discussed in [1], is 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 G 0 , is topological, where the quotient U 0 /ker(G) is equipped with the finest topology such that the canonical surjection π : U 0 → U 0 /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 U 0 , hence U 0 /ker(G) is a normed space (in particular, Hausdorff). Furthermore, G 0 is continuous by construction and the continuity of G, thus it remains to be checked whether the inverse of G 0 is continuous, or, equivalently, whether G 0 is an open map. , respectively). We choose a finer topology σ on V sc to make g : (V sc , σ) → H 1 loc (M ) continuous by adding the seminorms p χ (φ) := χ · g φ H 1 (χ ∈ D(M )) to those on V sc inherited from H 2 loc (M ). 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 2 loc (M )-topology on V sc , which we denote by τ 2 , and renders g continuous as a map V sc → H 1 loc (M ), i.e., σ is the supremum (in the lattice of topologies on V sc ) of τ 2 and the initial (projective) topology τ 1 with respect to g . Therefore, we have continuity of G : U 0 → (V sc , σ), since G is continuous U 0 → (V sc , τ 2 ) by Corollary 4.11 and also continuous U 0 → (V sc , τ 1 ) due to the obvious continuity of g • G = 0 from U 0 into H 1 loc (M ). Proof. We will show that G −1 0 can be written as the composition G −1 0 = π•P •Z of three continuous linear maps. The map π : U 0 → U 0 /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) → U 0 and such that G 0 • π • P • Z = id im(G) and π • P • Z • G 0 = id U0/ker (G) . Let V ± sc := {φ ∈ V sc | supp(φ) ⊆ J ± (K) for some compact subset K ⊆ M } and define the subspace , which we equip with the trace of the product topology stemming from σ. Construction of P : We consider P : W → U 0 , given by P (φ − , φ + ) := ( g φ + − g φ − )/2. Note that a priori, P (φ − , φ + ) is only in H 1 loc (M ) and we have to show that P (φ − , φ + ) has compact support, thus belongs to U 0 = H 1 comp (M ). To prove this, observe that is compact by global hyperbolicity ( [1,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 ± : V sc → V ± sc , such that φ = S − φ + S + φ holds for every φ ∈ V sc and, moreover, 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 ± ∈ V ± sc for every φ ∈ V sc and Equation (7.2) is true.
is relatively compact by [1,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 + φ ∈ V + sc . The reasoning for S − φ ∈ V − sc is analogous. For the proof of (7.2) we start by noting that φ ∈ im(G) = ker( g ) implies 0 = g φ = g S − φ + g S + φ, so that the part with S − in (7.2) follows once the equation for S + is shown. Recall that we have g S + φ = − g S − φ ∈ H 1 comp (M ) 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 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 G 0 .
Remark 7.4. We are not using an inductive limit construction for the topology on V sc as, e.g., in [2], 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 U 0 /ker(G), since we decided coherently that U 0 should inherit the norm topology from H 1 (M ), thus rendering U 0 = H 1 comp (M ) normed, but incomplete. However, the basic constructions of quantisation for the associated symplectic (quotient) vector spaces do not require completeness.

An equivalent symplectic structure
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 [55]. In that context, one defines a symplectic structure Ξ on ker( g ) given by Ξ(φ, ψ) = Σ (u 1 v 0 − v 1 u 0 )µ h where (u 0 , u 1 ), (v 0 , v 1 ) are compactly supported smooth initial data induced by the smooth solutions φ, ψ respectively on the Cauchy hypersurface Σ. Moreover, the symplectic structure and the Weyl system generated by it is independent of the chosen Cauchy hypersurface and it is isomorphic to the Weyl system generated by (U 0 /ker(G), ω) [55,16]. In the C 1,1 setting the above construction 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 φ| Σ ∈ H 2 comp (Σ) and ∇ n φ| Σ ∈ H 1 comp (Σ). 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 (R, H 2 (Σ t ))∩ C 1 (R, H 1 (Σ t )). Therefore, given any smooth spacelike Cauchy hypersurface Σ, we define for φ, ψ ∈ ker g with u 0 := φ| Σ , 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, u 1 ) and (v 0 , 0), i.e., with u 0 = 0 and v 1 = 0 and employ uniqueness in the Cauchy problem (cf. [16]). We show that Ξ Σ does not depend on Σ: This follows from the divergence theorem in a region bounded by two Cauchy hypersurfaces Σ 1 , Σ 2 and the conservation of the current j µ (φ, ψ) = g µν (φ∇ ν ψ − ψ∇ ν φ). Explicitly we have for any φ, ψ ∈ ker( g ) Therefore, Ξ Σ1 (φ, ψ) = Ξ Σ2 (φ, ψ), so we will drop the Σ from the notation of Ξ. Notice that the H 2 loc regularity is required in order to make sense of the divergence of the current. Finally, we show that the linear bijective factor map G 0 of G, as defined before (7.1), provides a symplectic map from (U 0 /ker(G), ω) to (ker( g ), Ξ). Proposition 7.5. Let the symplectic vector spaces (ker( g ), Ξ), (U 0 / ker(G), ω) and the factor map G 0 be defined as above. Then we have for every Proof. Without loss of generality we may consider M ∼ = R × Σ and suppose that supp(f ) ⊂ (t 1 , t 2 ) × Σ for some real t 1 < t 2 . Then we have for every φ ∈ ker( g ) upon integrating by parts twice, Using the fact that g φ = 0 and that Σ t1 is disjoint 4 we obtain (t1,t2)×Σ Similarly, from the causal properties again we have that Σ t2 and supp(G − (f )) are disjoint. There- Recalling that ψ = G(f ) and t 1 < t < t 2 in supp(f ) we obtain Here, we use also the assumption G(f ) = φ to proceed with We have established a symplectomorphism between the spaces (ker( g ), Ξ) and (U 0 /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 U 0 / ker(G) or those of ker( g ) to construct the algebra of quantum observables.

The physical quantum states
Finally, in order to construct a full quantum field theory in a low regularity spacetime, a suitable choice of quantum states must be made. Usually, quantum states Λ are given by certain positive linear functionals on the quasi-local C * -algebra. A common candidate for the physical quantum states in the smooth case are the quasi-free states that satisfy the microlocal spectrum condition. as described below. To be precise, given a real scalar product µ : ker( g ) × ker( g ) → R satisfying |Ξ(φ, ψ)| 2 ≤ 1 4 µ(φ, ψ)µ(φ, ψ) for all φ, ψ ∈ ker( g ), we define a quasi-free state by Λ µ (W (φ)) = e 1 2 µ(φ,φ) [55]. 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 where (x 1 , η) ∼ (x 2 ,η) means that η,η are cotangent to the null geodesic γ at x 1 , x 2 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 Using the isomorphism between ker( g ) and V (M ) the two-point function can be seen to induce a bidistribution on spacetime, i.e., Λ 2 ∈ D (M × M ). 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 D (M × M ) and satisfies the following wavefront set condition The states that satisfy the microlocal spectrum condition are called Hadamard states and their class includes ground states and KMS states ( [19,46,35]).
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 [33]. 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 [15] and they can be described as adiabatic states [50]. 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 ∈ R if its two-point function Λ 2N is a bidistribution that satisfies for every s ≤ N + 3 Colombeau algebras contain the distributions as a linear subspace, though not every element of a Colombeau algebra is a regularisation of a distribution. Their elements are equivalence classes of nets of smooth functions, G(Ω) u = [(u ε ) ε ]. We say that a Colombeau function u is associated with a distribution w ∈ D (Ω) if some (and hence every) representative (u ε ) ε converges to w in D (Ω). The distribution w represents the macroscopic behaviour of u and is called the distributional shadow of u.
Logarithmic growth conditions on the coefficients of a differential equation are typical in statements on existence and uniqueness of generalised solutions. These results are usually derived from a detailed analysis of regularisation techniques and Colombeau solutions often lead to very weak solutions in the sense of [20]. A related methodology of regularisation is used in the approximation results of [37] which show how to approximate a globally hyperbolic C 1,1 metric by a smooth family of globally hyperbolic metrics while controlling the causal structure. We recall from [44, Sec. 3.8.2], [12, Sec. 1.2] that for two Lorentzian metrics g 1 , g 2 , we say that g 2 has strictly wider light cones than g 1 , denoted by g 1 ≺ g 2 , if for any tangent vector X = 0, g 1 (X, X) ≤ 0 implies that g 2 (X, X) < 0.
(iv) If g is C 1,1 and globally hyperbolic then theĝ ε (andǧ ε ) can be chosen to be globally hyperbolic.
(v) If g is C 1,1 then the regularisations can in addition be chosen such that they converge to g in the C 1 -topology and such that their second derivatives are bounded, uniformly in ε on compact sets.
Remark A.2. In our application the main point we will need compared to [12, Sec. 1.2] is property (iv) which guarantees that for globally hyperbolic metrics there exist approximations with strictly narrower (wider) lightcones that are themselves globally hyperbolic. Extending methods of [21], it was shown in [3] that global hyperbolicity is stable in the interval topology. Consequently, if g is a smooth, globally hyperbolic Lorentzian metric, then there exists some smooth globally hyperbolic metric g ≺ g (resp. g g). Constructingĝ ε resp.ĝ j as in (ii) then automatically gives globally hyperbolic metrics (cf. [3, Sec. II]).

B Function Spaces
The (real) Hilbert space L 2 (M, g) is used in the section on Green operators to formulate adjointness properties and is defined as follows: Recall that for any Lorentzian manifold (M, g) we have a unique positive density µ g on M [56, Proposition 2.1.15], which has the local coordinate expression | det(g ij )| |dx 0 ∧ . . . ∧ dx n |; in case of a Lipschitz continuous metric g the density µ g is continuous and induces a positive Borel measure on M , which we employ to define the corresponding L 2 space and denote it by L 2 (M, g). If M is orientable, then we have a global volume form ν g on M [45, Ch. 7] from which the density µ g can be obtained. We consider the (real) Sobolev spaces H m (M ) for a nonnegative integer m to be defined with respect to some chosen smooth Riemannian background metric on M as described in [26], i.e., by completion of the space of (real) smooth functions whose covariant derivatives up to order m are square integrable with respect to the positive Borel measure on M associated with the Riemannian metric (cf. where γ is a Riemannian metric on Σ. We will then often consider a function v ∈ L 2 ((0, T ) × Σ) as a map t → v(t) from the interval into the Hilbert space L 2 (Σ) in the sense that v(t)(x) = v(t, x) holds pointwise for continuous v. Thanks to Fubini's theorem, we may then write For general constructions with measurable functions valued in Banach spaces we refer to [38,34]; in particular we will make use of the isomorphism L 2 ((0, T ) × Σ) ∼ = L 2 ((0, T ), L 2 (Σ)) [38,Theorem 8.28]. If v is differentiable and interpreted as a function t → v(t), we will occasionally denote the partial derivative ∂ t v byv and write ∂ t for the corresponding vector field on (0, T ) × Σ.  In place of a bounded time interval we will occasionally consider the basic spacetime to be R × Σ and deal with function spaces of Bochner measurable maps from R to some of the Sobolev-type Hilbert spaces (cf. [38,Chapter 8]), in particular, L 2 (R, H 1 (Σ)). We will then use the notation L 2 loc (R, H 1 (Σ)) for the set of all Bochner-measurable functions v : R → H 1 (Σ) such that for every compact subinterval I ⊂ R the restriction v| I belongs to L 2 (I, H 1 (Σ)). In looking at energy estimates on R × Σ we will also need versions of the Sobolev norms where the derivatives are taken in both the space and time directions but the integration and volume form are confined to the t = τ level hypersurfaces S τ := {τ } × Σ. These norms will be denoted by where∇ is the covariant derivative with respect to the spatial background metric γ and µ τ is the Riemannian measure on S τ which is just that given by the spatial metric γ. Finally let us adapt the basic function space structures to the situation of a general globally hyperbolic C 1,1 spacetime (M, g) with Cauchy hypersurface Σ, where we suppose that-according to the discussion in the subsection on C 1,1 causality theory-we have chosen a smooth temporal function t : M → R such Σ = t −1 (0) and a corresponding diffeomorphism Φ : M → R × Σ. For τ ∈ R denote the corresponding level surface by Σ τ := t −1 (τ ) = Φ −1 ({τ } × Σ), hence Σ 0 = Σ, and consider again a background Riemannian metric of the form h = dt ⊗ dt + γ on the product manifold R × Σ, which in turn provides us with the convenient background metric Φ * h on M . In the sequel, all Sobolev spaces on submanifolds of M or R × Σ will be considered to be defined via Riemannian metrics induced by Φ * h or h, respectively. Let Φ τ denote the induced diffeomorphism Σ τ → Σ, i.e., Φ(x) := (τ, Φ τ (x)) for every x ∈ Σ τ . We will commit another abuse of notation and a somewhat naive simplification in defining now the spaces C k (I, H m (Σ t )) for the case of a compact interval I = [0, T ] or for I = R without using the full theory of more sophisticated constructions in terms of sections, e.g., as in [2]. Let B m (I) denote the set of all maps u : I → τ ∈I H m (Σ τ ) such that u(τ ) ∈ H m (Σ τ ) for every τ ∈ I. Then we have that u ∈ B m (I) implies (equivalently) u(τ ) • Φ −1 τ ∈ H m (Σ) for every τ ∈ I. We define C k (I, H m (Σ t )) to be the subset of those elements u ∈ B m (I) such that the map τ → u(τ ) • Φ −1 τ belongs to C k (I, H m (Σ)). For elements u ∈ C k (I, H m (Σ t )) we can then also define the norms over spatial domains, but involving derivatives in space and time directions, such as u Hm (Στ ) via the corresponding . Hm (Sτ ) -norm evaluated for the associated map τ → u(τ ) • Φ −1 τ in C k (I, H m (Σ)). Note that the definition of the spaces C k (I, H m (Σ t )) depends on the splitting M ∼ = R × Σ and on the choice of temporal function. However, the reasoning in the main text tries to use the temporal function only in intermediate calculations and afterwards gives formulations of results essentially in "pure" spacetime terms without recourse to the splitting.