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Quantum gravity from the point of view of locally covariant quantum field theory

We construct perturbative quantum gravity in a generally covariant way. In particular our construction is background independent. It is based on the locally covariant approach to quantum field theory and the renormalized Batalin-Vilkovisky formalism. We do not touch the problem of nonrenormalizability and interpret the theory as an effective theory at large length scales.


Introduction
The incorporation of gravity into quantum theory is one of the great challenges of physics. The last decades were dominated by attempts to reach this goal by rather radical new concepts, the best known being string theory and loop quantum gravity. A more conservative approach via quantum field theory was originally considered to be hopeless because of severe conceptual and technical problems. In the meantime it became clear that also the other attempts meet enormous problems, and it might be worthwhile to reconsider the quantum field theoretical approach. Actually, there are indications that the obstacles in this approach are less heavy than originally expected.
One of these obstacles is perturbative non-renormalisability [66,73] which actually means that the counter terms arising in higher order of perturbation theory cannot be taken into account by readjusting the parameters in the Lagrangian. Nevertheless, theories with this property can be considered as effective theories with the property that only finitely many parameters have to be considered below a fixed energy scale [76]. Moreover, it may be that the theory is actually asymptotically safe in the sense that there is an ultraviolet fixed point of the renormalisation group flow with only finitely many relevant directions [75]. Results supporting this perspective have been obtained by Reuter et al. [64,65].
Another obstacle is the incorporation of the principle of general covariance. Quantum field theory is traditionally based on the symmetry group of Minkowski space, the Poincaré group. In particular, the concept of particles with the associated notions of a vacuum (absence of particles) and scattering states heavily relies on Poincaré symmetry. Quantum field theory on curved spacetime which might be considered as an intermediate step towards quantum gravity already has no distinguished particle interpretation. In fact, one of the most spectacular results of quantum field theory on curved spacetimes is Hawking's prediction of black hole evaporation [43], a result which may be understood as a consequence of different particle interpretations in different regions of spacetime. (For a field theoretical derivation of the Hawking effect see [32]. ) Quantum field theory on curved spacetime is nowadays well understood. This success is based on a consequent use of appropriate concepts. First of all, one has to base the theory on the principles of algebraic quantum field theory since there does not exist a distinguished Hilbert space of states. In particular, all structures are formulated in terms of local quantities. Global properties of spacetime do not enter the construction of the algebra of observables. They become relevant in the analysis of the space of states whose interpretation up to now is less well understood. It is at this point where the concept of particles becomes important if the spacetime under consideration has asymptotic regions similar to Minkowski space. Renormalization can be done without invoking any regularization by the methods of causal perturbation theory [28]. Originally these methods made use of properties of a Fock space representation, but could be generalized to a formalism based on algebraic structures on a space of functionals of classical field configurations where the problem of singularities can be treated by methods of microlocal analysis [13,11,45]. The lack of isometries in the generic case could be a problem for a comparison of renormalisation conditions at different points of spacetime. But this problem could be overcome by requiring local covariance, a principle, which relates theories at different spacetimes. The arising theory is already generally covariant and includes all typical quantum field theoretical models with the exception of supersymmetric theories (since supersymmetry implies the existence of a large group of isometries (Poincaré group or Anti de Sitter group)). See [14,16] for more details.
It is the aim of this paper to extend this approach to gravity. But here there seems to be a conceptual obstacle. As discussed above, a successful treatment of quantum field theory on generic spacetimes requires the use of local observables, but unfortunately there are no diffeomorphism invariant localized functionals of the dynamical degrees of freedom (the metric in pure gravity). The way out is to replace the requirement of invariance by covariance which amounts to consider partial observables in the sense of [67,22,70].
Because of its huge group of symmetries the quantization of gravity is plagued by problems known from gauge theories, and a construction seems to require the introduction of redundant quantities which at the end have to be removed. In perturbation theory the Batalin-Vilkovisky (BV) approach [3,4] has turned out to be the most systematic method, generalizing the BRST approach [5,6,72]. In the BV approach one constructs at the end the algebra of observables as a cohomology of a certain differential. But here the absence of local observables shows up in the triviality of the corresponding cohomology, as long as one restricts the formalism to a fixed spacetime. A nontrivial cohomology class arises on the level of locally covariant fields which are defined simultaneously on all spacetimes.
In a previous paper [33] two of us performed this construction for classical gravity, and in another paper [35] we developed a general scheme for a renormalized BV formalism for quantum physics, based on previous work of Hollands on Yang-Mills theories on curved spacetimes [44] and of Brennecke and Dütsch on a general treatment of anomalies [10].
In the present paper it therefore suffices to check whether the assumptions used in the general formalism are satisfied in gravity.
It turns out that we can rely to a large extent on older heuristic work by Nakanishi [55,57,56] and by DeWitt [21]. In particular the work of Nakanishi is very near to our stand point with the exception that he formulates quantum gravity with Minkowski space as a background whereas we admit arbitrary backgrounds.
The paper is organized as follows. We first describe the functional framework for classical field theory adapted to gravity. This framework was in detail developed in [15] but many ideas may already be found in the work of DeWitt [21], and an earlier version is [26]. In this framework, many aspects of quantum gravity can be studied, in particular the gauge symmetry induced by general covariance. We use the Batalin-Vilkovisky formalism for classical theories as developed in [33] where we take serious the fact that the configuration spaces in field theories are infinite dimensional manifolds.
Then we construct the quantum theory by deformation quantization. For this purpose we split the action around some background metric into a free part and an interacting one and quantize the free part by choosing a Hadamard solution of the linearized Einstein equation. We then can apply the renormalized BV formalism as developed in [35]. A crucial role is played by the Moeller map which maps interacting fields to free ones. In particular it also intertwines the free BV differential with that of the interacting theory, in spite of the fact that the symmetries are different.
As already discussed in [33], the candidates for local observables are locally covariant fields which act simultaneously on all spacetimes in a coherent way. Mathematically, they can be defined as natural transformations between suitable functors (see [14]). It seems, however, difficult to use them directly as generators of an algebra of observables for quantum theory (for attempts see [29] and [62,33]). We therefore choose a different way: due to the naturality condition, the smeared locally covariant field induce, on every spacetime, an action of the diffeomorphism group. We interpret the obtained algebra valued functions on the diffeomorphism group as relative observables, similar to concepts developed in loop quantum gravity [67,22,70]. The diffeomorphism may be interpreted as the choice of a coordinate system in terms of which the observable is specified. For the purposes of perturbation theory we replace the diffeomorphism group by the enveloping algebra of the Lie algebra of vector fields.
We then show that the theory is background independent, in the sense that a localized change in the background which formally yields an automorphism on the algebra of observables (called relative Cauchy evolution in [14]) is actually trivial, in agreement with the proposal made in [12] (see also [34]).
We finally construct states on the algebra of observables, as linear positive normalized functionals yielding the expectation values. In the first step we consider functions on the diffeomorphism group with values in a Krein space, which provides a representation of the extended algebra of fields. Among these diffeomorphism dependent vectors we can distinguish those, which correspond to physical states by considering the 0-th cohomology of the BRST charge. Both observables A and states |Ψ are considered to be functions on the diffeomorphism group, so to obtain expectation values we have to evaluate Ψ| A |Ψ at the identity. Construction of states on arbitrary globally hyperbolic on-shell backgrounds is possible with the use of the deformation argument of [36]. First, we show that one can construct states for the free theory explicitly on ultrastatic, on-shell backgrounds. Next, we construct from them states of the interacting theory by the perturbative method proposed in [25]. Then, we use the time-slice axiom of the interacting theory to identify the algebra associated to the full spacetime with the algebra associated to a causally convex neighborhood of a Cauchy surface. The deformation argument of [36] says that we can then deform the spacetime in the past of this neighborhood in such a way that it becomes ultrastatic in a neighborhood of some Cauchy surface. In section 5.2 we show that the isomorphism which connects the algebras associated to neighborhoods of different Cauchy surfaces commutes with the full BV operator, so we can transport states constructed on some special class of spacetimes to arbitrary on-shell backgrounds.
2 Classical theory

Configuration space of the classical theory
We start with defining the kinematical structure which we will use to describe the gravitational field. We follow [33], where the classical theory was formulated in the locally covariant framework. To follow this approach we need to define some categories. Let Loc be the category of time-oriented globally hyperbolic spacetimes with causal isometric embeddings as morphisms. The configuration space of classical gravity is a subset of the space of Lorentzian metrics, which can be equipped with an infinite dimensional manifold structure. To formulate this in the locally covariant framework we need to introduce a category, whose objects are infinite dimensional manifolds and whose arrows are smooth injective linear maps. There are various possibilities to define this category. One can follow [42] and use the category LcMfd of differentiable manifolds modeled on locally convex vector spaces or use the more general setting of convenient calculus, proposed in [51]. The second of these possibilities allows one to define a notion of smoothness, where a map is smooth if it maps smooth curves into smooth curves. We will denote by CnMfd, the category of smooth manifolds that arises in the convenient setting. Actually, as far as the definition of the configuration space goes, these two approaches are equivalent. This was already discussed in details in [15], for the case of a scalar field and the generalization to higher rank tensor is straightforward. Let Lor(M ) denote the space of Lorentzian metrics on M . We can equip it with a partial order relation ≺ defined by: i.e. the closed lightcone of g ′ is contained in the lighcone of g. Note that, if g is globally hyperbolic, then so is g ′ . We are now ready to define a functor E : Loc → LcMfd that assigns to a spacetime, the classical configuration space. To an object M = (M, g) ∈ Obj(Loc) we assign Note that, if g is globally hyperbolic, then so is g ′ ∈ E(M, g). The spacetime (M, g ′ ) is also an object of Loc, since it inherits the orientation and time-orientation from (M, g). A subtle point is the choice of a topology on E(M). Let Γ((T * M ) ⊗2 ) be the space of smooth contravariant 2-tensors. We equip it with the topology τ W , given by open neighborhoods of the form is an open subset of Γ((T * M ) ⊗2 ) with respect to τ W (for details, see the Appendix A and [15]). The topology τ W induces on E(M) a structure of an infinite dimensional manifold modeled on the locally convex vector space Γ c ((T * M ) ⊗2 ), of compactly supported contravariant 2-tensors. The coordinate chart associated to U g 0 ,V is given by κ g 0 (g 0 +h) = h. Clearly, the coordinate change map between two charts is affine, so E(M) is an affine manifold. It was shown in [15] that τ W induces on the configuration space also a smooth manifold structure, in the sense of the convenient calculus [51], so E becomes a contravariant functor from Loc to CnMfd where morphisms χ are mapped to pullbacks χ * .

Functionals
Let us now proceed to the problem of defining observables of the theory. We first introduce functionals F : E(M) → R, which are smooth in the sense of the calculus on locally convex vector spaces [42,58] (see Appendix A for details). In particular, the definition of smoothness which we use implies that for allg ∈ E(M), n ∈ N, it is a distributional section with compact support. Later, beside functionals, we will also need vector fields on E(M). Since the manifold structure of E(M) is affine, the tangent and cotangent bundles are trivial and are given by . By a slight abuse of notation we denote the space Γ c ((T * M ) ⊗2 ) by E c (M). The assignment of E c (M) to M is a covariant functor from Loc to Vec where morphisms χ are mapped to pushforwards χ * . Another covariant functor between these categories is the functor D which associates to a manifold the space D(M) . = C ∞ 0 (M, R) of compactly supported functions. An important property of a functional F is its spacetime support. Here we introduce a more general definition than the one used in our previous works, since we don't want to rely on an additive structure of the space of configurations. To this end we need to introduce the notion of relative support. Let f, g be arbitrary functions between two sets X and Y , then rel supp(f, g) .
Now we can define the spacetime support of a functional on E(M): Another crucial property is additivity.
where the superscript c denotes the complement in M . We say that F is additive if A smooth compactly supported functional is called local if it is additive and, for each n, the wavefront set of F (n) (g) satisfies: In particular F (1) (g) has to be a smooth section for each fixedg. From the additivity property follows that F (n) (g) is supported on the thin diagonal. The space of compactly supported smooth local functions F : E(M) → R is denoted by F loc (M). The algebraic completion of F loc (M) with respect to the pointwise product is a commutative algebra F(M) consisting of sums of finite products of local functionals. We call it the algebra of multilocal functionals. F becomes a (covariant) functor by setting Fχ(F ) = F • Eχ, i.e. Fχ(F )(g) = F (χ * g ).

Dynamics
Dynamics is introduced by means of a generalized Lagrangian L which is a natural transformation between the functor of test function spaces D and the functor F loc satisfying and the additivity rule for f 1 , f 2 , f 3 ∈ D(M) and supp f 1 ∩ supp f 3 = ∅. The action S(L) is defined as an equivalence class of Lagrangians [16], where two Lagrangians L 1 , L 2 are called equivalent for all spacetimes M and all f ∈ D(M). In general relativity the dynamics is given by the Einstein-Hilbert Lagrangian: where we use the Planck units, so in particular the gravitational constant G is set to 1.

Diffeomorphism invariance
In this subsection we discuss the symmetries of (9). As a natural transformation L EH is an element of Nat(Tens c , F), 1 where Tens c (M) . = k Tens k c (M) and Tens c (M) is the space of smooth compactly supported sections of the vector bundle m,l (T M ) ⊗m ⊗ (T * M ) ⊗l . The space Nat(Tens c , F) is quite large, so, to understand the motivation for such an abstract setting, let us now discuss the physical interpretation of Nat(Tens c , F). In [33] we argued that this space contains quantities which are identified with diffeomorphism invariant partial observables of general relativity, similar to the approach of [67,22,70]. Let Φ ∈ Nat(Tens c , F). A test tensor f ∈ Tens c (M) corresponds to a concrete geometrical setting of an experiment, so we obtain a functional Φ M (f ), which depends covariantly on the geometrical data provided by f . We allow arbitrary tensors to be test objects, because we don't want to restrict a priori possible experimental settings. A simple example of an experiment is the length measurement, studied in detail in [59].
the curve is still spacelike, and its length is Hereṡ µ is the tangent vector of s. We write it asṡ µ =ṡe µ , with η µν e µ e ν = −1.
Expanding the formula above in powers of h results in Now, if we want to measure the length up to the k-th order, we have to consider a field where the curve whose length we measure is specified by the test tensor f S = (f S,0 , . . . , f S,k ) ∈ Tens c (M), which depends on the parameters of the curve in the following way: The framework of category theory, which we are using, allows us also to formulate the notion of locality in a simple manner. It was shown in [15] that natural transformations Φ ∈ Nat(Tens c , F), which are additive in test tensors (condition (7)) and satisfy the support condition (6), correspond to local measurements, i.e. Φ M (f ) ∈ F loc (M). The condition for a family (Φ M ) M∈Obj(Loc) to be a natural transformation reads where f ∈ Tens c (M), h ∈ E(M ′ ), χ : M → M ′ . Now we want to introduce a BV structure on natural transformations defined above. One possibility was proposed in [33], where an associative, commutative product was defined as follows: Note, however, that the dependence on test tensors f i physically corresponds to a geometrical setup of an experiment, so Φ M (f 1 )Ψ M (f 2 ) means that, on a spacetime M, we measure the observable Φ in a region defined by f 1 and Ψ in the region defined by f 2 . From this point of view, there is no a priori reason to consider products of fields which are symmetric in test functions. Therefore, we take here a different approach and replace the collection of natural transformations with another structure. Let us fix M. Physically, a natural transformation Φ tells us how to identify functionals in F(M) localized in different regions. Given a test tensor f ∈ Tens c (M) we recover not only the functional Φ M (f ), but also the whole diffeomorphism class of functionals Φ M (α * f ), where α ∈ Diff c (M). We have already mentioned that the test function specifies the geometrical setup for an experiment, but a concrete choice of f ∈ D(M) can be made only if we fix some coordinate system. This is related to the fact that, physically, points of spacetime have no meaning. To realize this in our formalism we have to allow for a freedom of changing the labeling of the points of spacetime, which corresponds to a diffeomorphism transformation. Keeping this in mind, it is clear that physical meaning should be assigned not to f , but rather to a class of test functions, obtained from f by composing with diffeomorphisms. In our formalism, it means that the full information about the dependence of the measurement on the geometrical setup should be contained in the family (α * f ) α∈Diffc(M) . Therefore, for fixed M, Φ and f , a physically meaningful object is the function It was proven in [51] that Diff c (M) can be endowed with a structure of an infinite dimensional manifold modeled on X c (M), where X c (M) is the space of compactly supported vector fields on M . Assuming the notion of smoothness proposed in [51], evaluation and composition are smooth maps, so we can prove that a pair (M, f ) induces a mapping: where α ∈ Diff c (M). We will denote ev M,f Φ by Φ Mf , or simply by Φ f if M is fixed and no ambiguity arises. The reformulation we made here has an important advantage; on C ∞ (Diff c (M), F(M)) one has a natural notion of a product, namely the pointwise product: Let us denote by F diff (M) the subspace of C ∞ (Diff c (M), F(M)) consisting of sums of finite products of elements that can be written in the form ev M,f (Φ), for some Φ ∈ Nat(Tens c , F), f ∈ Tens c (M).
Physically, the choice of a diffeomorphism α ∈ Diff c (M) corresponds to a choice of the coordinate system in which we parametrize the geometrical setup. When we interpret the results of a classical measurement, we have to fix a coordinate system, and we have to know what happens, if we change it. Since we restrict ourselves to infinitesimal transformations, we are interested only in the values F (α) of F ∈ F diff (M) at α = id and in the values of the derivatives. Therefore, in the next step, we replace elements of F diff (M) by their Taylor expansions at α = id. As such, they are elements of an appropriate completion of U X(M) ′ ⊗ F(M), where U X(M) is the universal enveloping algebra of X(M), i.e. the quotient of the tensor algebra over X(M), by the ideal generated by ζ ⊗ η − η ⊗ ζ − [ζ, η]. More explicitly, on fixed M, to Φ and f we associate a series Φ f = n∈N 0 Φ n f , whose components are given by derivatives of Φ M ((e ζ ) * f ) with respect to ζ at 0, i.e.
Instead of writing Φ n f (ζ 1 ⊗ · · · ⊗ ζ n ), n ∈ N 0 , we will use a shorthand notation Φ f (e ζ ⊗ ). The first few components of Φ f are given by where the prime ′ denotes the derivative along a test function. An elegant way to represent Φ f , is to write it as a (possibly infinite dimensional) lower triangular matrix  Note that the product (11) induces a product where F (e ζ ⊗ ) = ∞ n=0 1 n! F n (ζ ⊗n ) and G(e ζ ⊗ ) = ∞ n=0 1 n! G n (ζ ⊗n ) and, using the matrix notation (13), this product is just the usual product of matrices. The naturality condition for Φ implies that Φ f depends locally on ζ. Note that the expression (12) makes sense also for ζ, which is not compactly supported. This motivates the following definition: Among elements of F(M) we have to identify those, which correspond to physical observables, so, in the next step, we need to formulate the notion of diffeomorphism invariance for elements of F(M). A diffeomorphism α in a sufficiently small neighborhood of id acts on elements of F diff (M) by Remark 2.4. To relate (15) to the invariance condition for natural transformations, proposed in [33], we note that if If we choose a whole family α = (α M ) M∈Obj(Loc) and for each M we choose a test function f M , then we obtain and, at β = id, the invariance condition (15) implies that which is exactly the condition (αΦ) M = Φ M for natural transformations proposed in [33]. In other words, invariant natural transformations Φ induce diffeomorphism invariant observables Φ f on any fixed spacetime M. . Now, we come back to the discussion of infinitesimal diffeomorphisms. The derived action ρ of X c (M) on F(M) is given by To make contact with [33], note that, in degree 0 in ζ, we have for elements of the form Infinitesimal diffeomorphism invariance is the requirement that ρ(ξ)Φ f = 0. Note that the property of diffeomorphism invariance is a notion depending on a physical observable we measure, not on the concrete geometrical setup. For example, the length measurement described in Example 2.2 is an invariant quantity, independent of the choice of a curve, whose length we measure.

BV complex
Any element of F(M) of the form L EH f (for the Einstein Hilbert Lagrangian L EH ) is an example of a diffeomorphism invariant quantity. In [33] we proposed how to quantize theories with such invariant Lagrangians, using an appropriate extension of the Batalin-Vilkovisky (BV) formalism. Here we have to modify that formalism a little bit, since our algebra of observables is F(M), rather than F(M). In the first step we construct the Chevalley-Eilenberg complex corresponding to the componentwise action ρ of X c (M) on F(M). The Chevalley-Eilenberg differential is constructed by replacing the infinitesimal diffeomorphism in (16)  . = F(M)⊗ΛX ′ (M). The notation ⊗ is clarified in Definition 2.3. Note that CE is a functor from Loc to Vec and also a functor to graded algebras, since CE is equipped with a natural grading (called the pure ghost number #pg) 2 We can now define the underlying algebra of the Chevalley-Eilenberg complex. CE(M) inherits the #pg grading from CE(M) and we can make it into a differential graded algebra, by equipping it with a differential γ CE defined by where D c is the insertion of a ghost defined by and γ * CE is a derivation of CE(M) given by 3 where ξ 0 , . . . , ξ q ∈ X(M). We name γ * CE , the partial Chevalley-Eilenberg differential. We have already pointed out in [33] that D c is the part of the Chevalley-Eilenberg we assign grade k and by CE k (M) we denote the subspace of CE(M) arising from natural transformations from Tensc to CE k . 3 We have adjusted the sign convention in such a way that γ * acts from the right and we used the differential that appears in gauge theories, where the local symmetry involves a coordinate transformation. To see that γ CE maps CE(M) to itself, we have to use the fact that symmetries act locally, so γ * CE can be lifted to a map on natural transformations 4 . The same happens with D c , since elements of the form Φ f depend locally on ζ and inserting c is a local operation. For the nilpotency of γ CE , note that γ * CE is nilpotent by definition and, on elements of the form (13) Comparing with (16), we see that the 0-th cohomology of γ CE is the space of diffeomorphism invariant elements of F(M).
Now we construct the Batalin-Vilkovisky complex, following the ideas of [33]. Note that CE(M) can be formally seen as the space of multilocal, compactly supported functions on a graded manifold E(M) = E(M)[0] ⊕ X(M) [1]. BV(M), the underlying space of the BV complex, is the graded symmetric tensor algebra of graded derivations of CE(M). We require that elements of BV(M), considered as smooth maps on E(M) with values in a certain graded algebra, are multilocal and compactly spacetime supported, i.e.
where ⊗ is the sequentially completed tensor product. Elements of the above space can be seen as functionals on is induced by the topology τ W introduced in section 2.1. We denote a field multiplet in E by ϕ and its components by ϕ α , where the index α runs through all the metric and ghost indices. "Monomial" elements 5 of BV(M) can be written formally as where we keep the summation over the indices α implicit and the product denoted by juxtaposition is the graded associative product of BV(M). Polynomials are sums of elements of the form (19). We identify the functional derivatives δ δϕ α (x) with the so called antifields, Φ ‡ α (x). Here we treat them as densities and f F is a density-valued distribution with a compact support, which is contained in the product of partial diagonals. The WF set of f F has to be chosen in such a way, that F is multilocal. In the appropriate topology (more details may be found in [33]) polynomials (19) are dense in BV(M). Functional derivatives with respect to odd variables and antifields are defined on polynomials as left derivatives and are extended to BV(M) by continuity. We always assume that are left derivatives, unless stated otherwise. The assignment (18) induces a functor BV from Loc to Vec. Finally we can define the underling algebra of the BV complex.
To simplify the notation, we write from now on δ EH F = {F, S EH } instead of (20). In a similar manner, one can find a natural transformation θ CE , that implements γ * CE , i.e. γ * CE = {., θ CE }. This definition allows, in a natural way, to extend γ * CE to antifields. With a slight abuse of notation we can also have {., . This definition extends D c to a differential on BV(M). Note, however that D c itself is not an element of BV(M). The total BV differential is the sum of the Koszul-Tate and the Chevalley-Eilenberg differentials: The nilpotency of s BV is guaranteed by the so called classical master equation (CME). In [33] it was formulated as a condition on the level of natural transformations. With an appropriate choice of a natural Lagrangian θ CE which generates γ * CE , we can impose a stronger condition. Let then the CME for evaluated fields on M can be formulated as If (21) holds, we have s 2 BV = 0 and the 0-th cohomology of s BV on BV(M) is the space of gauge invariant on-shell observables BV ph (M) = H 0 (BV(M), s BV ). We write s BV as s * BV + D c and call s * BV the partial BV differential. This is similar to the distinction between the full and the "intrinsic" BRST differential made in [56]. Note that s BV can act also on natural Lagrangians. We have already mentioned that γ * BV can be lifted to the level of natural transformations. The same holds for δ EH , so there exists a natural Lagrangian so s BV has a well defined action on equivalence classes of natural Lagrangians.
In the next step we introduce the gauge fixing along the lines of [33]. For the specific choice of gauge we need, we have to extend the BV complex by adding auxiliary fields: antighostsc and Nakanishi-Lautrup fields b. The new extended configuration space is given by: and the extended space of multilocal evaluated fields is again denoted by BV(M). Next, we perform an automorphism α Ψ of (BV(M), {., .}) such that the part of the transformed action which doesn't contain antifields has a well posed Cauchy problem. For details on the construction of α Ψ , see [33]. The full transformed Lagrangian is given by: where L GF , L F P are respectively the gauge-fixing and Fadeev-Popov terms and L AF is the term containing antifields. The variables of the theory (i.e. the components ϕ α of the multiplet ϕ ∈ E(M)) are now: the metricg, the Nakanishi-Lautrup fields b ∈ X(M)[0], the ghosts c ∈ X(M) [1] and antighostsc ∈ X(M)[−1]. We introduce a new grading, called the total antifields number #ta. It is equal to 0 for functions on E(M) and equal to 1 for all the vector fields on E(M). The Lagrangian terms listed in (22) are explicitely given by: where we introduced a product u, v g = M u # , v dµ g for u, v tensors of the same rank and # is the isomorphism between T * M and T M induced by g. Also the covariant derivative ∇ is the one induced by g. We denoteg := √ −gg, K(g) = 1 √ −g ∇ νg νµ and K (1) is the derivative of K with respect tog. We suppressed all the indices and wrote the graded product of the BV algebra simply by the juxtaposition. New field equations are now equations for the full multiplet ϕ = (g, b, c,c) and are derived from the #ta = 0 term of L ext , denoted by L. The corresponding action S(L) is called the gauge fixed action.
We denote {., S ext } by s * and write it as The differential δ is the Koszul operator for the field equations derived from S and γ * is related to the the gauge-fixed BRST operator γ by means of The action of γ * on F(M) and the evaluation functionals b, c,c is summarized in the table below: The equations of motion are: andg, b, c,c have to be understood as evaluation functionals and not as field configurations. The equation for b can be derived from the first equation by means of the Bianchi identity. One can already see that, after linearization, we obtain a normally hyperbolic system of equations. As shown in [33], also the full system becomes normally hyperbolic, after a suitable change of variables. Since s = δ + γ and (BV(M), δ) is a resolution of the algebra of on-shell evaluated fields, the space of gauge invariant on-shell fields is recovered as the cohomology BV ph (M) = H 0 (s, BV(M)) = H 0 (γ, H 0 (δ, BV(M))). Since the system of linearized field equations is normally hyperbolic, there exist retarded and advanced propagators for spacetime the M = (M, g) (if we linearize around g) as well as for (M,g) (if we linearize aroundg). The latter are denoted by ∆ R/Ã g and we define a Poisson (Peierls) bracket on BV(M) by: where, to simplify the sign convention, we use both the right and the left derivative. Note that the support of ⌊A, B⌋g is contained in the support of ⌊A, B⌋ g . Hence, ⌊., .⌋g is a well defined operation on BV(M), taking values in the space of compactly supported smooth functionals. BV(M) is not closed under ⌊., .⌋g, but one can extend this space to a suitable completion in such a way that (BV(M), ⌊., .⌋g ) becomes a Poisson algebra (more details in Appendix A and in [15]). A Poisson bracket on BV(M) induces a Poisson bracket on BV (a suitable completion of BV), where the product between δ l F δϕ α and δ r G δϕ β is the product (14). Now we want to see if ⌊., .⌋g is compatible with s. First, note that the image of δ is a Poisson ideal, so ⌊., .⌋g is well defined on H 0 (δ, BV(M)). It remains to show that, on H 0 (δ, BV(M)), γ is a derivation with respect to ⌊., .⌋g. To prove it, we have to show that After a short calculation, we obtain the following condition (compare with Prop. 2.3. of [63]): where Kg is defined by and γ 0g is the linearization of γ aroundg. The condition (26) is fulfilled if γ 0g commutes with δ 0g (the linearization of δ), i.e. if the action linearized aroundg is invariant under γ 0g . This is the case ifg is a solution of the full equations of motion (see [63] for more details). We conclude that γ is a derivation with respect to ⌊., .⌋g modulo terms that vanish on the ideal generated by the full equations of motion, i.e. modulo the image of δ. It follows that γ is a derivation on H 0 (δ, BV(M)), hence ⌊., .⌋g induces a Poisson bracket ). This way we obtain a Poisson algebra (BV ph (M), ⌊., .⌋g ), which we interpret as a classical algebra of observables in general relativity.

Outline of the approach
In the previous section we defined the classical theory, now we want to quantize this structure. The usual prescription involving the star product cannot be applied to {., .}g, because acting iteratively with the functional differential operator ∆g αβ , δ l δϕ α ⊗ δ r ϕ β involves also derivatives of ∆g. Therefore, from the point of view of quantization, it is convenient to split the gauge fixed action S into a free part and the rest and quantize the free theory first. Interaction is introduced in the second step, with the use of timeordered products. One can make this split by writing the Taylor expansion of L(g) around a reference metric g, so h =g − g is the perturbation. Later on h will be interpreted as a quantum fluctuation around a classical background. We denote L 0 and consider it to be the free action. We also introduce the notation S 1 = S ext − S 0 and θ = S ext − S. We also expand θ around g. The first nontrivial term in the expansion is linear in configuration fields and we denote it by θ 0 . It generates the free gauge-fixed BRST differential γ * 0 . The Taylor expansion of the classical master equation (21) yields in particular: The first two terms of this identity correspond to the classical master equation for the free Lagrangian S (2) (0) + θ 0 . The third term vanishes only for on-shell backgrounds, so γ * 0 is a symmetry of S 0 only if g is a solution of Einstein's equations. Consequences of this fact are discussed in detail in [63].
Let us now summarize the general strategy for the perturbative quantization of gravity, which we will follow in this work. We start with the full classical theory, described by the gauge-fixed action S which is invariant under the BRST operator γ. Then, we linearize the action and the BRST differential around a fixed background metric g. This way, the "gauge" invariance of the theory is broken and the linearized classical theory doesn't posses the full symmetry anymore. If we linearize around g which is a solution of the full Einstein's equations, then part of the symmetry remains and S 0 is invariant under γ * 0 . This, however, is not needed for performing a deformation quantization of the linearized theory along the lines of [35], which works for arbitrary (M, g) ∈ Obj(Loc). The free theory, quantized this way, still contains non-physical fields and is not invariant under the full BRST symmetry. This is to be expected, since the linearization breaks this symmetry in an explicit way. To restore the symmetry we have to include the interaction. This can be done with the use of time-ordered products and relative S-matrices. The full interacting theory is again invariant under the full BRST symmetry γ. This is guaranteed by the so called quantum master equation (QME), which is a renormalization condition for the time-ordered products (see [35] for more details). A crucial step in our construction is to prove that the quantized interacting theory which we obtain in the end doesn't depend on the choice of the background g. This will be done in section 4.

Perturbative formulation of classical theory
The starting point for the construction of the linearized classical theory is the gauge-fixed free action S 0 . For simplicity we choose from now on the gauge with α = −1. To write S 0 in a more convenient way, we introduce some notation. Let us define the divergence operator div : where ., . is the pairing between tangent and cotangent spaces. The formal adjoint of div with respect to the product ., . g , introduced in the previous section, is denoted by div * : In local coordinates we have: Another important operation is the trace reversal operator G : We have tr(Gh) = −trh and G 2 = id. Using this notation we can write the quadratic part of the gauge fixed Lagrangian on a generic background M = (M, g) ∈ Obj(Loc) in the form: denotes the linearization of the Ricci scalar around the background g and we treat b, c,c as elements of Γ(T * M ). Now we calculate the variation of L 0 , to obtain S ′′ M (x, y). We write it here in a block matrix form: = * −1 d * is the codifferential and L is given in local coordinates by where we denoted . = ∇ µ ∇ µ . In the literature, L it is called the Lichnerowicz Laplacian [50] and provides a generalization of the Hodge Laplacian to the space of symmetric contravariant 2 tensors. Note that L commutes with G on E(M). It is now easy to check that the retarded and advanced propagators for S 0 are given by: where δ 4 denotes the Dirac delta in 4 dimensions and subscript y in div * y means that the operator should be applied on the second variable. In the above formula ∆ A/R t are the advanced/retarded propagators for the operator L acting on symmetric tensor fields with compact support Using the above formula we can write down the expression for the causal propagator and use this propagator to define the classical linearized theory, by introducing the Peierls bracket: An appropriate extension of the space of multilocal functionals, which is closed under ⌊., .⌋ g is ∈ BV µc (M), the space of microcausal functionals. They are defined as smooth, compactly supported functionals whose derivatives (with respect to both ϕ and ϕ ‡ ) satisfy the WF set condition: where Ξ n is an open cone defined as where V ± is the closed future/past lightcone with respect to the metric g. The space of microcausal functionals is equipped with the Hörmander topology τ Ξ , which allows to control properties of functional derivatives (see [33] for a precise definition). The Peierls bracket on BV µc (M) is next extended to BV µc (M).

Free quantum theory
In the next step we want to construct the quantized algebra of free fields by means of deformation quantization of the classical algebra (BV µc (M), ⌊., .⌋ g ). To this end, we equip the space of formal power series BV µc (M)[[ ]] with a noncommutative star product. In this construction one needs Hadamard parametrices, i.e. a set of distributions in Here O α β are the coefficients of the differential operator induced by S ′′ M , written in the basis {ϕ α }. They can be easily read off from (28). By C + we denoted the following subset of the cotangent bundle T * M 2 : if there is a lightlike geodesic from x 1 to x 2 and k 2 is a parallel transport of k 1 along this geodesics. These are the properties which we will require for a Hadamard parametrix on the general background M ∈ Obj(Loc). If we replace the condition (32b) by a stronger one then the Hadamard parametrix becomes a Hadamard 2-point function. We will now show that such a distribution can be constructed on generic backgrounds. Assume that ω is of the form: In this case, the conditions for ω to be a Hadamard 2-point function reduce to: The existence of a Hadamard parametrix is already clear, since one just needs to pick arbitrary parametrices ω t , ω v of L and H respectively. Their existence was already proven in [68] (the paper actually discusses general wave operators acting on vectorvalued field configurations). Now, from a parametrix, one can construct a bisolution using a following argument: let ω be a Hadamard parametrix and by O we denote the hyperbolic operator from (32b), so O x ω = h, O y ω = k, hold for some smooth functions h and k. Let χ be a smooth function such that suppχ is past-compact and Clearly G χ is a left inverse for O. A Hadamard bisolution ω χ can be now obtained as can be extended to a larger space BV µc (M) [[ ]] and the star product on it can be defined as: The resulting algebra is denoted by A H (M). As there is no preferred two-point function ω, and hence no preferred H, we have to consider all of them simultaneously. An elegant, though abstract construction of the quantum algebra was provided in [16]. It can be see as a where Φ x i are evaluation functionals, f ∈ E ′ Ξn (M n , V ) and we suppress all the indices. Let us now discuss the covariance properties of Wick powers. The assignment of A(M) to a spacetime M can be made into a functor A from the category Loc of spacetimes to the category of topological *-algebras Obs and, by composing with a forgetful functor, to the category Vec of topological vector spaces. Admissible embeddings are mapped to pullbacks, i.e. for χ : M → M ′ we set AχF (ϕ) . = F (χ * ϕ). Locally covariant quantum fields are natural transformations between D and A. We require Wick products to be locally covarint in the above sense. On each object M we have to construct the map T 1M from the classical algebra BV loc (M) to the quantum algebra A(M) in such a way that As we noted above, classical functionals can be mapped to A H (M) by identification (36). This, however, doesn't have right covariance properties. A detailed discussion is presented in section 5 of [14], where it is shown that redefining Wick products to become covariant amounts to solving a certain cohomological problem. The result reproduces the solution, which was proposed earlier in [45]. One has to define T 1 as α −1 H+w , where w is the smooth part of the Hadamard 2-point function ω = u σ + v ln σ + w with σ(x, y) denoting the square of the length of the geodesic connecting x and y and with geometrically determined smooth functions u and v.
Analogously to the classical case, the space BV q (M) of evaluated quantum fields is defined as a subspace of A(M)⊗S • X ′ (M) generated by sums of ⋆-products of elements that can be written in the form Φ f for some natural transformation Φ ∈ Nat(Tens c , A). Note that a family of mappings T 1M : BV loc (M) → A(M) satisfying (37) induces a map T 1 : BV(M) → BV q (M). The operator D c is extended to the space of quantum fields by requirement that it is a derivation with respect to ⋆ and commutes with T 1M .

Interacting theory
Following [35], we introduce the interaction by means of renormalized time-ordered products. Let us define operators T n : BV loc (M) ⊗n → A(M) by means of H+w is fixed up to the choice of finitely many parameters (see [45] for more details). Maps T n have to be extended to functionals with coinciding supports and are required to satisfy the standard conditions given in [16,44]. In particular, we require graded symmetry, unitarity, scaling properties, suppT n (F 1 , . . . , F n ) ⊂ suppF i and causal factorization property: if the supports of F 1 . . . F i are later than the supports of F i+1 , . . . F n , then we have Maps satisfying the conditions above are constructed inductively, and T n is uniquely fixed by the lower order maps T k , k < n, up to the addition of an n-linear map which describes possible finite renormalizations. In [35] it was shown that the renormalized time ordered product can be extended to an associative, commutative binary product defined on the domain D T (M) is the inverse of the multiplication, as defined in [35,62]. D T (M) contains in particular A loc (M) and is invariant under the renormalization group action. Renormalized time ordered products are defined by Time ordered products on different spacetimes have to be defined in a covariant way. This means that if Φ f is an evaluated classical field, then we have to define the families T nM in such a way that (TΦ) f .
is an evaluated quantum field. In other words, we need to prove that T can be lifted to the level of natural transformations. A straightforward generalization of a result of [44] shows that this indeed can be done and the family (T M ) M∈Obj(Loc) induces for each M a map T : BV(M) → BV q (M).
Using covariant time-ordered products we can now introduce the interaction. As indicated in section 3.2, we split the action into L ext = L 0 + L 1 . L 1 is the interaction term and the corresponding quantum field is TL 1 . We can split L 1 into two terms: L 1 =L 1 + θ, whereL 1 has the antifield number equal to 0 and θ has #af = 1. Choosing test functions f and f ′ we obtain elements TL 1f and Tθ f ′ of BV q (M). T(L 1f + θ f ′ ) is interpreted as an interaction with a spacetime cutoff. Later on we will see that, within the freedom allowed for by our formalism, we can choose f ′ such that f ′ ≡ 1 on supp f .
The formal S-matrix S is a map BV loc (M) → BV q (M) defined as the time-ordered exponential. In particular, we have Now we want to construct a local net of * -algebras corresponding to the interacting theory on a fixed spacetime M. This is done along the lines of [16], We interpret this set as a class of interaction terms that locally represent the action S 1 .
The relative S-matrix in the algebraic adiabatic limit is constructed from a family of Since all the operations are local, this definition is independent of the choice of V ∈ V S 1 (O). The relative S-matrix defined this way is a formal power series in V and . It satisfies the following property: for any V 1 , V 2 ∈ V S 1 (O), there exists a formal automorphism β of A(M) such that Interacting quantum fields R O S 1 (F ), associated to a region O are generated by S O V (F ), in the sense that for given V ∈ V S 1 (O), we have so an interacting field R O S 1 (F ) is a covariantly constant section of a bundle of algebras over V S 1 (O). More explicitly, the intertwining map R V can be written as The interacting net is constructed by assigning to each relatively compact region O, so the interacting fields can be constructed from the free ones, but they have different localization properties.
We can now define the space of interacting fields associated to M as the space of sequences The map that associates to an element of BV(M) the corresponding interacting, evaluated field will be denoted by R S 1 and its image by A S 1 (M).
The local algebra of interacting fields associated to a region O is given by . This is a rather abstract way of formulating the problem, but it has some advantages, since working with ⋆ V is sometimes more convenient conceptually. On the other hand, elements of BV V (O) can no longer be understood as functionals, so one has to be careful with manipulating them.
Let us now come back to quantization of structures appearing in the BV formalism. Following the approach proposed in [35], we define the renormalized time-ordered antibracket on T(BV(M)) by We can also write it as: The above formula has to be understood as: where D * denotes the pullback by the diagonal map and T −1 δF δϕ (ϕ) is a compactly supported distribution (i.e. an element of E ′ (M)) defined by In the second step we used the field independence of time ordered products. Since F ∈ T(BV(M)), the distribution T −1 δF δϕ (ϕ) defined by the above equation is actually a smooth function and the pullback in (47) is well defined. Similarly, we define the antibracket with the ⋆-product: whenever it exists. Clearly, it is well defined if one of the arguments is regular or equal to S 0 . The antibracket {., S 0 } ⋆ with the free action defines a ⋆-derivation and, similarly, {., S 0 } T is a ·T -derivation. A relation between these two is provided by the Master Ward Identity [10,44]: Both (46) and (48) can be lifted to the level of natural transformations so they are well defined operations on T(BV(M)). Now we can use the BV formalism to discuss the gauge invariance in the quantum theory. In the framework of [35], the S-matrix is independent of the gauge fixing-fermion if the quantum master equation (QME) is fulfilled on the level of natural transformations. Here, as in the classical case (compare with (21)), we replace the condition used in [35] with a stronger one, namely: where f ′′ ≡ 1 on suppf and suppf ′ . Since we chose f ≡ 1 on suppf ′ , terms of the extended action with #af = 0 (i.e.L 1 and L 0 ) are treated on the equal footing. Using the Master Ward Identity [10,44] and our choice of f , f ′ , we can rewrite the above condition as: where ∆L 1f +θ f ′ is the anomaly term, which in the formalism of [35], is interpreted as the renormalized version of the BV Laplacian. If we redefine time-ordered products in such a way that the anomaly is equal to 0, the above condition is fulfilled, because the first term vanishes due to the invariance of S under γ and the second term vanishes due to the nilpotency of γ. To show that such a redefinition of time-ordered products is possible, one has to analyze the cohomology H 1 (γ * |d) on the space of density-valued local functionals (called also local forms). For the case of gravity in the metric formulation, this cohomology was computed in [17]. We can apply this result to conclude that the QME can be fulfilled, by an argument used in [44] (see section 4.4.1, proof of identity T12a).
The quantum BV operatorŝ is a map on BV defined bŷ where supp X ⊂ O and f ≡ 1 on O. Note that, because of locality, the above expression is independent of the choice of f and f ′ . If (50) holds, thenŝ has the following property: and can be written asŝ where ∆L 1f +θ f ′ (X) ∆L 1f +θ f ′ +λX . Note that, by definition,ŝ is a derivation with respect to ⋆ V . To analyze the cohomology ofŝ, it is convenient to extend BV(M) to a larger space BV S 1 (M) by a completion procedure described in Remark 3.1. BV S 1 can be equipped with a product ⋆ S 1 , defined as follows: where V ∈ V S 1 (O) and, O is chosen in such a way that it contains J − (supp f 1 ) ∩ J + (supp f 2 ) and J + (supp f 1 ) ∩ J − (supp f 2 ). The above product is well defined, since ⋆ V depends only on the behavior of V between the supports of the arguments. We define the space of gauge invariant fields as the 0th cohomology of (ŝ, BV S 1 (M)). From the intertwining relation (52) follows that this cohomology is equivalent to the 0th cohomology of (R −1 S 1 {., S 0 } ⋆ , R S 1 (BV(M)), ⋆). This concludes the construction of the algebra of diffeomorphism invariant quantum fields for general relativity.

Background independence
In the previous section we constructed the algebra of interacting evaluated fields of quantum gravity, by choosing a background and splitting the action into a free and interacting part. Now we prove that the result is independent of that split. In [12] it was proposed that a condition of background independence can be formulated by means of the relative Cauchy evolution. Let us fix a spacetime M 1 = (M, g 1 ) ∈ Obj(Loc) and choose Σ − and Σ + , two Cauchy surfaces in M, such that Σ + is in the future of Σ − . Consider another globally hyperbolic metric g 2 on M , such that g . = g 2 − g 1 is compactly supported and its support K lies between Σ − and Σ + . Let us take N ± ∈ Obj(Loc) that embed into M 1 , M 2 , via χ 1± , χ 2± and χ i± (N ± ), i = 1, 2 are causally convex neighborhoods of Σ ± in M i . We can then use the time-slice axiom to define isomorphisms α χ i± . = Aχ i± and the free relative Cauchy evolution is an automorphism of A(M 1 ) given by It was shown in [14] that the functional derivative of β with respect to g is the commutator with the free stress-energy tensor. Let us recall briefly that argument, using a different formulation. We can apply β to the S-matrix, which works as the generating function for free fields, and calculate the functional derivative using an explicit formula for relative Cauchy evolution. To this end we use the perturbative agreement condition introduced by Hollands and Wald in [46]. They construct a map τ ret : the support of f lies outside the causal future of K. Physically it means that free algebras A(M 1 ) and A(M 2 ) are identified in the past of K. Analogously, one defines a map τ adv , which identifies the free algebras in the future. The free relative Cauchy evolution is then given by Let us briefly recall the definition of τ ret . In [46] this map was defined on the on-shell algebra of fields, i.e. on H 0 (δ, BV). Following [46], we choose two Cauchy surfaces Σ 0 and Σ 1 , inside χ 1− (N − ), such that Σ 0 is in the past of Σ − and Σ 1 in its future. Let ψ be a function which is 0 in the future of Σ 1 and 1 in the past of Σ 0 . The map τ ret g 1 g 2 is defined on linear on-shell fields (i.e. equivalence classes in H 0 (δ, BV)) by where O g 2 is the differential operator induced by S ′′ M 2 and ∆ g 2 , the corresponding causal propagator. It extends to an algebra homomorphism by continuity. The map τ ret g 1 g 2 can also be extended to an off shell mapping in such a way that τ ret δϕ(x) (this was done in [27]). The perturbative agreement is now a condition that, on shell, Here S S 0M 1 −S 0M 2 denotes the relative S-matrix constructed with the interaction S 0M 1 − S 0M 2 and the background metric g 1 . More explicitly, we have where o.s.
= means "holds on-shell with respect to free equations of motion" and f ≡ 1 on the support of f ′ . An analogous condition for τ adv g 1 g 2 reads: Conditions (55) and (56) were proven in [46] for the case of the free scalar field, but the same argument can be used also for pure gravity.
To fulfill the perturbative agreement condition, one fixes the time-ordered product T M 1 and shows that there exists a definition of T M 2 on the background M 2 compatible with other axioms, such that also (54) can be fulfilled. In particular, the quantum master equation holds automatically for T M 2 if it holds for T M 1 . To prove this, we use the off-shell definition of τ ret g 1 g 2 , given in [27], and from (54) it follows that τ ret . From the nilpotency of {., S 0M 1 } ⋆g 1 and the fact that τ ret g 1 g 2 is an algebra morphism, follows that τ ret Now we use the fact that S 0M 2 − S 0M 1 doesn't depend on antifields and that i(S 0M 2 − S 0M 1 )/ + V 2 = V 1 . This yields so the qme holds for T M 2 . Let us go back to the relative Cauchy evolution. The functional derivative of β 0g can now be easily calculated, yielding where T 0µν is the stress-energy tensor of the linearized theory.
Let us now discuss a corresponding construction in the interacting theory. It was conjectured in [12] that, for the full interacting theory of quantum gravity, the relative Cauchy evolution should be trivial (equal to the identity map), hence the derivative with respect to g should vanish. To map the interacting fields in (BV S 1 (M i ), ⋆ V ) to (A(M i ), ⋆), i = 1, 2, we can use retarded and advanced maps . The interacting relative Cauchy evolution can be written as: We can now formulate the condition of background independence as: Note that we can avoid potential problems with domains of definition of R −1 V 1 and A −1 V 1 , by rewritting the above condition as Using formulas for τ ret g 1 g 2 and τ adv g 1 g 2 and the fact that L 0M 2 (f ) + V 2 = L ext M 2 (f ), we obtain: Differentiating with respect to g µν yields a condition is the full stress-energy tensor. We can write the above condition in a more elegant way, using the formal notation with ⋆ V 1 , namely = means "holds on-shell with respect to the equations of motion of the full interacting theory". To prove that the infinitesimal background independence is fulfilled, we have to show that T µν = 0 in the cohomology ofŝ. Recall that L ext = L EH + L AF + L F P + L GF , where L EH is the Einstein-Hilbert Lagrangian and the remaining terms are given by the formulas (23)- (25). Both L EH and L AF don't depend explicitly on g, so they do not contribute to T µν . The remaining terms are given by γΨ, where Ψ is the gauge-fixing Fermion. Since γ commutes with δ δg , we obtain where in the last step we used the fact that Ψ doesn't depend on antifields. Since the gauge-fixing term has to depend explicitly on g, to conclude the proof of background independence, we need to show that the contribution from sΨ vanishes in the cohomology of the quantum BV operator, i.e. that s , or in other words, This holds for the non-renormalized BV Laplacian △, because Ψ doesn't contain antifields. For the renormalized △ V 1 we know, from the Master Ward Identity, that and n factors of V 1 , n > 0. We only have to check if this doesn't stay in conflict with other properties, which we require for time-ordered products. In particular, we have to prove that the renormalization group transformation which we are performing acts on Ψ M 1 itself in a desirable way. To have the independence of the theory of the gauge fixing we need △ V 1 (Ψ M 1 ) to vanish, so we need to check if the redefinition of time-ordered products containing one factor of is independent of the redefinition of the products containing one factor of {Ψ M 1 , V 1 }. This is indeed the case, since {Ψ M 1 , V 1 } contains the terms with derivatives of the form M b ν ∂ µg νµ dµ g , Mc ν ∂ µ £ c (g νµ )dµ g , which are not present in and in accordance with the Action Ward Identity we can renormalize them independently. As a consequence, the anomaly can be removed from both Ψ M 1 and , so the theory can be made both gauge fixing and background independent.

Perturbative construction of a state for on-shell backgrounds
Finally we come to the discussion of states. We start with the construction of states of the full interacting theory for on-shell backgrounds (i.e. backgrounds for which the metric is a solution to Einstein's equations). We will use the method proposed in [25] which relies on the gauge invariance of the linearized theory under the free BV transformation s 0 . We have already indicated that this requires the background metric g to be a solution of the Einstein's equation, so throughout this subsection we assume that this is indeed the case. The construction we perform is only formal, since we don't control the convergence of interacting fields and we treat them as formal power series in the interaction. In subsection 5.2 we construct, also formally, states for arbitrary on-shell backgrounds. The construction presented in subsection 5.2 makes use of a deformation argument of [36], so it reduces the problem of existence of states on arbitrary globally hyperbolic on-shell spacetimes to construction of states on a distinguished class of backgrounds. This last step is performed in Appendix D, where we show the existence of states for the linearized theory on ultrastatic, on-shell spacetimes. For a fixed spacetime M, we define the quantum algebra A(M) of the free theory as in section 3.2. Since we assumed in this subsection that g is a solution of Einstein's equation, the free action L 0 contains only the term quadratic in h. To keep track of the order in h it is convenient to introduce a formal parameter λ (identified with the square of the gravitational coupling constant, i.e. λ = √ κ) and the field multiplet (g + λh, λb, λc, λc), together with corresponding antifields (λh † , λb † , λc † , λc † ). We denote (h, b, c,c) collectively by ϕ. It is convenient to use the natural units, where κ is not put to 1, but has a dimension of length squared, so h has a dimension of 1/length. The action used in quantization must be dimensionless, so, as in path integral approach, we use L/λ 2 , where L is the full extended action defined before. The free action is now defined as L 0M (ϕ) . 0, 0, 0), ϕ , analogously forL 1 and θ. Observables are formal power series in λ obtained by expanding functions on E(M) around (g, 0, 0, 0) and the ideal of free equations of motion is generated by λOϕ, where O is the differential operator induced by S ′′ M . Note that we can write elements of this ideal as {X, S 0 }, where X ∈ BV(M) has an antifield number #af > 0.
Interacting fields are formal power series in in , λ and the interaction S 1 . For onshell backgrounds,S 1 the #af = 0 term of the interaction, is of order λ 1 and θ is of order λ 0 . Therefore we obtain formal power series in λ, and the antifield number #af.
Let us assume that we have a representation π 0 of A(M) on an indefinite product space K 0 (M) and we denote K(M) .
The difficulty in constructing a representation of the field algebra BV lies in the fact that its elements are not the local observables assigned to a spacetime. To treat such objects, we have to extend our formalism a little bit. In section 3.2 we defined the space of evaluated fields as the space of smooth maps on the diffeomorphism group with values in A(M). In the infinitesimal version (in a neighborhood of the identity), this was replaced by the space of formal power series U X(M) ′ ⊗A(M). In a similar manner, we consider the space of smooth maps on Diff c (M) with values in K(M) which, in the neighborhood of identity, gets replaced by and the state space on BV q (M) will be constructed as a certain subspace of K (M). It is convenient to write elements of BV q (M) in the matrix notation (13). A typical element of K (M) will be written as a column vector The action of BV q (M) on K (M) is now defined by

  
For the simplicity of notation we will omit π 0 , i.e. we write Φ M (f ) instead of π 0 (Φ M (f )), if no confusion arises. Next, we have to choose an indefinite product on K (M). A pointwise product on the space of K-valued maps induces Performing the physical measurement means evaluating Φ f on β = id, so when we interpret our theory we have to evaluate Ψ, Ψ ′ K at ζ = 0. The positivity of the product is, therefore, the requirement that Ψ, Ψ ′ K (1) > 0. Let Φ f ∈ R S 1 (BV(M)) be an interacting evaluated field.
In order to distinguish a subspace of K (M) that corresponds to physical states, we will apply the Kugo-Ojima formalism [53,54]. For this we need the BRST charge Q. It is defined as the Noether charge corresponding to the BRST transformation. A concrete formula is provided in Appendix C. Using the result of [63] we conclude that holds on-shell for F ∈ BV(M). We can also use R S 1 to map to the free algebra and then, observables are elements of the form R S 1 (F ), F ∈ BV(M), which belong to the 0-th cohomology of i [., R S 1 (Q)] ⋆ . The full BV operator is obtained by adding D f c . Note that this term is just an insertion of f c, with an appropriate sign, so if we define .
where c has to be understood as an operator 6 , then Here 1 i R S 1 (Q(η)) ≡ Q int is the interacting BRST charge and its nilpotency can be shown by arguments analogous to [44]. Next, we have to prove that alsoQ is nilpotent. Using the fact that s * acts from the right, we obtain In the last step we used the fact that Ψ M is a function on U X(M), so it has to vanish on the ideal generated by elements of the form We can now useQ to characterize the physical states in K . Indeed, the 0 cohomology ofQ defines a space closed under the action of physical observables (i.e. under H 0 (BV,ŝ)). To see that it is consistent, let us take Ψ ∈ ker(Q) and Φ f ∈ R S 1 (BV(M)). Then More precisely, let us write the representation of c in the form π0(c)ψ = µ π0(c µ (x))ψdx. We .
holds, i.e. (ŝΦ f )Ψ ∈ Im(Q), so it vanishes in the cohomology. We will now show how to construct vectors belonging to ker(Q). If Ψ is such a vector, then . . .
The general solution of this equation is: where Q ζ is defined by and by ι ζ we mean the insertion of a vector field ζ ∈ X(M) to a functional of ghosts, which is, by definition, an alternating multilinear form on X(M). We have characterized KerQ, but this space is not yet the subspace of K where the product (61) is positive. This is related to the fact that the condition we imposed on observables is not sufficient to have a physical interpretation of the theory. Note that we work perturbatively, so in addition to usual requirements, we need a condition that guarantees the stability of the perturbative expansion under small changes of the coordinate system. This means that a quantity which we consider as "free" has to stay this way if we change the coordinate system by a small perturbation. Let us formalize this idea by imposing a following condition (first proposed in [2]) 7 : given an evaluated . ., such that Φ 0 f k is the lowest non-vanishing order of the λexpansion of Φ 0 f with #af = 0, we require that Φ l f m ≡ 0 for all l > 0, m < k. We can see it as a condition on the choice of a test tensor f for a given natural transformation Φ and a background M. Now we can check, how this additional condition is compatible with the expansion ofQ as a power series in λ. Let Φ f be an evaluated interacting field. Assume that the lowest non-vanishing λ-order of Φ 0 f with #af = 0 is Φ 0 f k and Φ l f m ≡ 0 for all l > 0, m < k holds. Then we have: The BRST invariance of Φ f (λ) implies that, in particular, so the additional condition we imposed on observables guarantees that the first nonvanishing term in the formal power series Φ 0 f (λ) commutes with Q 0 . We can now impose a corresponding condition on elements of K (M), which suffices to distinguish physical states. Let Ψ = ψ 0 + ψ 1 (ζ) + . . .. We require that ψ 0 k , the lowest λ-order of ψ 0 ∈ K (M), belongs to the cohomology of Q 0 , i.e.: From (63) it follows that condition (64) is preserved under applying physical quantum fields. Let us denote by H (M) the subspace of H 0 (K (M),Q) consisting of elements that satisfy (64) as well. Then, for Ψ, Ψ ′ ∈ H (M), with lowest non-vanishing λ-orders k 1 , k 2 respectively, we have: It follows that Ψ, Ψ ′ K (1) > 0 in the sense of formal power series if and only if ψ 0 k 1 , ψ ′ 0 Therefore a sufficient condition for the positivity of ., . K on H (M) is that ., . K of the linearized theory is positive on the space H 0 (K(M), Q 0 ). This is a condition similar to the one obtained in the case of electrodynamics in [25] and is fulfilled also for linearized gravity, assuming that the background metric is on-shell.

Existence of states for globally hyperbolic on-shell spacetimes
In this section we show that the existence of states on globally hyperbolic spacetimes which are solution to Einstein's equations, can be reduced to showing their existence on some particularly symmetric ones (for example ultrastatic). To construct states, we use a modification of the deformation argument applied in [36]. We start with an arbitrary globally hyperbolic spacetime M 1 = (M 1 , g 1 ), where g 1 is a solution to Einstein's equation and we consider a causally convex neighborhood N 1 of a Cauchy surface Σ ⊂ M 1 . We choose an ultrastatic auxiliary spacetime M 2 = (Σ × R, g 2 ) with a causally convex neighborhood N 2 of a Cauchy surface. Now we take an intermediate spacetime M = (M, g) such that (N 1 , g 1 ) where χ 1 , χ 2 are morphisms in Loc. Let us denote the images of embeddings χ 1 and χ 2 as N and N ′ respectively. We assume that N ′ is in the past of N . Since the intermediate spacetime M is not on-shell we have to include a linear term into the free action S 0 . Free equations of motion include now an external current J induced by S ′ M . In appendix B we argue that, for off-shell backgrounds, in order to have the time-slice axiom in the theory which involves formal expressions in λ, it is necessary to extend the free algebra with formal power series in the current J. Hence A(M) is a space of formal power series in and J and Laurent series in λ.
In the representation which we are using, an on-shell observable is an element Φ f ∈ H 0 (BV S 1 (N),ŝ), such that Φ 0 f k is the lowest non-vanishing order of the λ-expansion with #af = 0 of Φ 0 f and Φ l f m ≡ 0 for all l > 0, m < k (additional condition derived in subsection 5.1). Using the local covariance we can map Φ f to Φ χ * 1 f . Next we apply the time-slice axiom for the interacting theory to transport observables supported in N to N ′ . A general proof of the time-slice axiom for interacting quantum field theories constructed in the framework of perturbative algebraic quantum field theory was provided in [18].
There are two differences between the situation described in [18] and the present case. Firstly, we want to work off-shell, so we have to keep track of elements belonging to the ideal generated by the equations of motion. Secondly, the extended Lagrangian has a term with #af > 0 and we evaluate this term on a different test function from the test function used in the #af = 0 part. In Appendix B we repeat the argument of [18], taking into account these two differences and we construct explicitely the isomorphism which maps BV S 1 (N) to BV S 1 (N ′ ). Here we only give the final formula: where is an element of the ideal I V generated by the interacting equations of motion. Test functions b, b ′ do not intersect the future of the support of We will now show that if Φ χ * 1 f is in H 0 (BV(M),ŝ), then the same holds forα(Φ χ * 1 f ).
Using the fact that δ 0 is a resolution of the algebra of on-shell functionals, we know that there exists a element X of BV(M), such that , so the second term in (65) drops out from the cohomology. Concerning the first term, we use the fact that {., S 0 } ⋆ and D c are ⋆-derivations and we consider where f ′′ ≡ 1 on supports of f , b and b ′ . By assumption b ≡ 1 on supp b ′ , so using the quantum master equation (50) we conclude that the above expression vanishes and thereforê Additional condition, concerning the behavior of the λ-expansion ofα(Φ χ * 1 f ) under small changes of the coordinate system is also fulfilled. To see this, note that the lowest λ-order is the same as of Φ χ * 1 f , becauseL 1 is of order λ. Hence that the lowest order ofα(Φ χ * 1 f ), is the same as that of Φ χ * 1 f , modulo the equations of motion. We have just proven that if Φ χ * 1 f is a physical observable, then so isα(Φ χ * 1 f ), so a state on A(N 1 ) can be transported to a state on A(N 2 ).

Conclusions and Outlook
We showed in this paper how the conceptual problems of a theory of quantum gravity can be solved, on the level of formal power series. The crucial new ingredient was the concept of local covariance [14] by which a theory is formulated simultaneously on a large class of spacetimes. Based on this concept, older ideas dating back to Nakanishi and De Witt could be extended and made rigorous. The construction uses the renormalized Batalin Vilkovisky formalism as recently developed in [35].
In the spirit of algebraic quantum field theory [41] we first constructed the algebras of local observables. In a theory of gravity, this is a subtle point, since on a first sight one might think that in view of general covariance local observables do not exist. We approached this problem in the following way. Locally covariant fields are, by definition, simultaneously declared on all spacetimes. On each spacetime, they give rise to an action of the diffeomorphism group on their spacetime averages. One obtains algebra valued functions on the diffeomorphism group which may be compared to the partial observables used by Rovelli [67], Dittrich [22] and Thiemann [70]. The algebra of local observables can be understood in terms of these objects.
The states in the algebraic approach are linear functionals on the algebra of observables interpreted as expectation values. In gauge theories the algebra of observables is obtained as the cohomology of the BRST differential on an extended algebra. The usual construction first described by Kugo and Ojima [52,53,54] (for an earlier attempt see [19]) starts from a representation of the extended algebra on some Krein space and an implementation of the BRST differential as the graded commutator with a nilpotent (of order 2) operator (the BRST charge). The cohomology of this operator is then a representation space for the algebra of observables. This construction can be explicitly performed in the linearized theory on an ultra static spacetime; moreover the physical Krein space turns out to be a pre Hilbert space. One then transports this state to a generic on-shell spacetime by a deformation argument (here it is crucial that deformed backgrounds are admitted which are not solutions of the classical Einstein equation) and goes to the full theory by the retarded Moeller map which maps interacting fields to free ones.
In this paper we treated pure gravity. It is, however, to be expected that the procedure can be easily extended to include matter fields (scalar, Dirac, Majorana, gauge). It is less clear whether supergravity can be treated in an analogous way.
On the basis of the formalism developed in this paper one should be able to perform reliable calculations for quantum corrections to classical gravity, under the assumption that these corrections are small and allow a perturbative treatment. There exist already some calculations of corrections, e.g. for the Newton potential [8] with which these calculations could be compared. It would also be of great interest to adapt the renormalization approach of Reuter et al. (see, e.g., [64,65]) to our framework. Further interesting problems are the validity of the semiclassical Einstein equation (for an older discussion see [74]) and the possible noncommutativity of the physical spacetime [24].
A Aspects of classical relativity seen as a locally covariant field theory In this appendix we discuss some details concerning the formulation of classical relativity in the framework of locally covariant quantum field theory. The first issue concerns the choice of a topology on the configuration space E(M). In section 2.1 we already indicated that a natural choice of such topology is τ W , given by open neighborhoods of the form is equipped with the standard inductive limit topology. In our case, τ W coincides with the Whitney C ∞ topology, W O ∞ , hence the notation. After [49], Whitney C ∞ topology is the initial topology on On the space of all Lorentzian metrics we have also another nattural topology, namely the interval topology τ I introduced by Geroch [38], which is given by where the partial order relation ≺ is defined by (1), i.e.
From the support properties of F follows that F • ι χ is independent of χ.
In particular, F (1) defines a kinematical vector field on E(M) in the sense of [51]. Moreover, since E c (M) is reflexive and has the approximation property, it follows (theorem 28.7 of [51]) that kinematical vector fields are also operational i.e., they are derivations of the space of smooth functionals on E(M).
At the end of section 2.5 we have indicated that the space of multilocal functionals can be extended to a space BV(M) which is closed under ⌊., .⌋g. Here we give a possible choice for this space. We define BV(M) to be a subspace of BV µc (M) (defined in section 3.2) consisting of functionals F , such that the first derivative F (1) (ϕ) is a smooth section for all ϕ ∈ E(M) and ϕ → F (1) is equipped with its standard Fréchet topology. Since the lightcone ofg is contained in the interior of the lightcone of g, the WF set condition (30) guarantees that ⌊., .⌋g is well defined on BV(M). Using arguments similar to [15] we can prove the following proposition: Proposition A.1. The space BV(M) together with ⌊., .⌋g is a Poisson algebra.
Proof. First we have to show that BV(M) is closed under ⌊., .⌋g. It was already shown in [15] that BV µc (M) is closed under the Peierls bracket. It remains to show that the additional condition we imposed on the first derivative is also preserved under ⌊., .⌋g. Consider where S ′′′ (ϕ) denotes the third derivative of the action. The last two terms in the above formula are smooth sections, since the wavefront set of S ′′′ (ϕ) is orthogonal to T Diag 3 (M ) and ∆ R/A F (1) (ϕ), ∆ R/A G (1) (ϕ) are smooth. The first term of (67) can be written as smooth, so the above derivative exists as a smooth section in E(M). The same argument can be applied to the second term in (67), so we can conclude that ⌊F, G⌋g) (1) (ϕ) is a smooth section. From a similar reasoning follows also that ϕ → (⌊F, G⌋g) (1) (ϕ) is a smooth map.
The antisymmetry of ⌊., .⌋g is clear, so it remains to prove the Jacobi identity. In [48,15] it was shown that this identity follows from the symmetry of the third derivative of the action, as long as products of the form ∆ R/A F (1) (ϕ) are well defined. With our definition of BV(M) this is of course true, since F (1) (ϕ) is required to be a smooth section. which is an isomorphism modulo I 0 . An explicit construction can be found in [18] for the case where the free interaction contains only a quadratic term. For the Einstein-Hilbert Lagrangian expanded around an arbitrary background metric g we also have to include a linear term and the field equation can be written in the form: where O is the differential operator induced by S ′′ M and J is an "external current" induced by S ′ M . The modification of the argument used in [18] is rather straightforward for a linear configuration space.
Let N be a causally convex neighborhod of a Cauchy surface Σ. We choose two Cauchy surfaces Σ 0 , Σ 1 ⊂ N such that Σ 0 lies in the past of Σ 1 . Let χ be a function which is 1 in the future of Σ 1 and 0 in the past of Σ 0 . We use the left inverse (cf. Section 3.3) The map α 0 is defined by The support of α 0 (F ) is contained in N. Namely, let h have a compact support which does not intersect N. Then G χ Oh = h, hence α 0 (F )(ϕ + h) = α 0 (F )(ϕ). Moreover, α 0 (F ) coincides with F on solutions of the equations of motion, thus the difference is an element of I 0 .
In case of a configuration space which is an affine manifold as in gravity, one has to check that the argument of the functionals remains inside the allowed domain. But this is always the case if ϕ is sufficiently near to a solution. To solve the problem exactly one has to check that an appropriate neighborhood exists. If this is the case, then the quotients by the ideal I 0 fulfill the time slice axiom.
The above reasoning cannot, however, be applied if we include also formal power series in λ into our treatment, as it was done in section 5.1. In the present state of knowledge this is necessary, since we do not have control over the convergence of interacting fields R V (F ). Therefore, we want to completely avoid discussing convergence issues at this stage and apply (68) also to formal expressions in λ with coefficients in functionals. This can be done, if we treat the external current J as a formal generator and regard the expression (68) as a Laurent series in λ and a formal power series in J. To make it consistent, we extend A(M) to a space of Laurent series in and λ and formal power series in J. We start with a metric g 0 which is a solution to Einstein's equation S ′ (g 0 ) = 0. Next we construct, perturbatively, a metric g which is solution to Einstein's equation with an external source, i.e. S ′ (g) = J. As mentioned before, if J is small enough, this problem can be solved exactly, but formally we can always obtain g as a formal power series g = g 0 + g 1 (J) + g 2 (J ⊗2 ) + . . .. Propagators are now also formal power series in J. The map from equation (68) acts on observables as α 0 (F )(λϕ) . = F (λϕ − G χ (λOϕ − J)). We can apply α 0 also to the formal S-matrix e iV /(λ 2 ) and to the interacting fields, so that α 0 (R V (F )) = R V (F ) + I 0 holds, where I 0 ∈ I 0 . The expression on the left hand side is a formal expression both in J and in λ and the ideal I 0 is generated by λOϕ(x) − J(x).

B.2 Time-slice axiom for the interacting theory
In the interacting net (BV V (O), ⋆ V ), the equations of motion are  N), ⋆) to show that free fields can be constructed from the interacting ones and obtain the time-slice axiom for the interacting theory using the map α 0 of the free theory. More concretely, one has to multiply the relative S-matrices corresponding to the interaction L 1N , choosing the supports of test functions in such a way, that the interacting fields are mapped to free ones in a certain region, next the map α 0 is applied and then the free fields are mapped back to interacting ones. In [18] such a map α : (A(N), ⋆) → (A V (N), ⋆) is constructed explicitly and, for each relatively compact region O 1 , there is an invertible element U such that α(G) = U −1 ⋆ G ⋆ U , where G is supported in O 1 . The invertible element U can be realized as for suitably chosen test functions b and b ′ , with the property that supp b and supp b ′ do not intersect the future of supp G and b ≡ 1 on supp b ′ . Let us briefly recall the argument of [18] with a slightly different notation and a minor modification arising from the fact that θ andL 1 are evaluated on different test functions. Let f 1 be a smooth function with past compact support in the interior of the future of Σ 1 , which is a Cauchy surface in the past of supp G. Let N 1 be a causally convex neighborhood of Σ, which is contained in the interior of N. We choose a function f 2 with supp f 2 ⊂ N which coincides with f 1 on N 1 and a test function b 2 with support in N which coincides with f 2 on J − (K). K is a compact region, such that supp G ⊂ K o . Similarly, we can choose a test function b 1 which coincides with f 1 in J − (K). Let b 1 − b 2 = b + b, where supp b does not intersect the past and supp b not the future of supp G. In [18] f i , b i are used as smearing functions for L 1M , in construction of free fields from interacting ones. Here the situation is slightly different, since we use different test functions for the antifield number 0 and for the antifield number 1 term. Test functions we have introduced will be used as test objects forL 1 , so it remains to construct test functions for θ. We consider N 2 , N 3 causally convex neighborhoods of Σ, such that the closure of N is contained in the interior of N 2 the closure of N 2 is contained in the interior of N 3 . Let Σ 2 be a Cauchy surface, which lies in the past of N 3 and in the future of Σ 1 . Now we essentially repeat the construction of test functions for the new choice of neighborhoods and Cauchy surfaces: f ′ 1 is a smooth function with past compact support contained in the interior of the future of the future of supp G. Moreover, we choose our smearing objects in such a way that the region where b 1 ≡ 1 has a nonzero intersection with the past of N 3 and b ≡ 1 on the support of b ′ . We consider a product of relative S-matrices which is, by definition, an interacting field, but due to causal factorisation property, it is equal to a product of free fields, )/ ) modulo interacting equations of motion. We can summarize this construction in the following diagram:

C BRST charge
In this section we construct the BRST charge that generates the gauge-fixed BRST transformation s * . It is convenient to pass from the original Einstein-Hilbert Lagrangian to an equivalent one given by: It differs from the Einstein-Hilbert Lagrangian by a term andΓ's are the Christoffel symbols relating the covariant derivative∇ to ∇. Explicitly they are defined byΓ α µν =g αβ (∇ µgαν + ∇ νgαµ + ∇ αgµν ) .
The Riemann curvature tensorR σ µνρ , corresponding to the full metricg, can be written as: αµ . Let L be the gauge-fixed Lagrangian, where the Einstein-Hilbert term is replaced by L ′ . The full BRST current corresponding to γ * is given by the formula: where K µ M is the divergence term appearing after applying γ * to L M (f ). Using this formula we obtain (compare with [60,52,57]): The free BRST current is given by: For a spacetime M with compact Cauchy surface Σ, for any closed 3-form β there exists a closed compactly supported 1-form η on M such that M η ∧ β = Σ β. In this case we can define the BRST charge on a fixed spacetime as: and analogously for the free BRST charge Q 0 . Note that the BRST current depends locally on field configurations, so we can view the BRST charge as an evaluated quantum field Q(η) ∈ BV, where each η M is chosen as indicated above.

D Existence of Hadamard states on ultrastatic spacetimes
We will here consider an ultrastatic spacetime, i.e. M = Σ × R equipped with the metric g = −1 0 0ĝ , whereĝ is a Riemannian metric on the Cauchy surface Σ. The only nonvanishing Christoffel symbols for this metric are Γ k ji =Γ k ji , where byˆwe will always denote quantities that refer to metricĝ. The only non-vanishing components of the curvature tensor are the ones corresponding to the intrinsic curvature of (Σ,ĝ), i.e. R l ijk =R l ijk . Therefore if (M, g) is a an Einstein manifold, then so is (Σ,ĝ). Moreover, if (Σ,ĝ) is Ricci flat it is also flat, since it's 3 dimensional. For now we do not require (M, g) to be on-shell. We will see later on how this condition arises in our construction.
As we know, on ultrastatic spacetimes the wave equation can be reduced to the analysis of an elliptic eigenvalue problem [31]. To see this, let us first fix a global frame {e a } a=0,1,2,3 . as a set of four global sections of T M . This is always possible since a four-dimensional, globally hyperbolic spacetime is parallelisable. Similarly, we define a dual Lorentz frame {e b } b=0,1,2,3 as a set of four global sections of T * M by demanding that e b (x)(e a (x)) = δ b a holds at all points of M . Using these frames we can write the perturbation metric in the form: This can be also expressed in the matrix form where h is the vector with components h = h 10 h 20 h 30 . In this section we will adapt small latin letters from the beginning of alphabet a, b, .. to denote the components 0, ..., 3 of a tensor in the fixed frame. For components 1, ..., 3 we will use small latin letters from the middle of the alphabet, i.e. i, j, k, .... Wa also adapt the Einstein summation convention so we will omit the sum symbols appearing in (70). It is now easy to check that the wave operator L acts on the components of h in a following way: where ∆ s , ∆ v , ∆ t are the 3-dimensional Lichnerowicz Laplacians on (Σ,ĝ) acting on scalar, vector and symmetric contravariant 2 tensors respectively. If ξ ∈ E(M) doesn't depend on the ultrastatic time parameter x 0 ≡ t, then A (t, x) = e −iωt ξ(x) solves the equation L A = 0 if and only if −∆ s ξ 00 = ω 2 ξ 00 , At this point we have to make further assumptions on the topology of Σ. Firstly, since we want the spectrum of ∆ s/v/t to be discrete, we need Σ to be compact. On the other hand we know that for compact spaces there are no interesting embeddings. We solve this problem in a similar way as in [25]. Let O be a region of M with the local algebra A(M) on which we want to define a state. We consider a compact causally convex set O containing O and the compact Cauchy surface Σ with a boundary ∂Σ. We have to choose the boundary conditions for the Laplacian −∆ s/v/t in such a way that it has only nonnegative eigenvalues. For −∆ s/v this problem was already discussed in [26]. For a metricĝ with vanishing curvature, −∆ t is just the "bare" Laplacian −∆ and with the assumed boundary conditions, it is positive semidefinite. For compact surfaces without boundary the positivity problem for −∆ t is also well studied, since it is related to the stability of the Ricci flow, see for example [69]. The non-negativity of the Lichnerowicz Laplacian was also proven for some Ricci flat metrics on compact manifolds [39], but it is not known if it holds for all of them. For our application it is, however, enough to study the vanishing curvature case, since the Cauchy surface is 3 dimensional and in this case the Ricci flatness implies also vanishing of the Riemann tensor. To summarize, our problem reduces to looking for solutions of the equations: −∆ξ 00 = ω 2 ξ 00 , ξ 00 | ∂Σ = 0 , (73) −∆ξ = ω 2 ξ , ξ 00 | ∂Σ = 0 , −∆ξ = ω 2ξ ,ξ 00 | ∂Σ = 0 . Now we want to analyze the space of solutions in more detail. As shown above we can write any time independent (static) ξ ∈ E(M) in the form (71). On the space of symmetric contravariant 2 tensors on Σ we can now introduce an inner product by: We can complete Γ ∞ (S 2 T * Σ) with respect to the norm topology induced by this inner product to obtain a Hilbert space L 2 (S 2 T * Σ) of square integrable 2-tensors. In the similar way we define spaces of square integrable 1-forms L 2 (T * Σ) and functions L 2 (Σ). The direct sum of these three spaces is the Hilbert space H t = L 2 (S 2 T * Σ)⊕L 2 (T * Σ)⊕L 2 (Σ), where the inner product is (., .). It can be identified with the space of static symmetric rank 2 contravariant tensors, since the decomposition (71) is orthogonal with respect to (., .). On H t we can now define a positive semidefinite self-adjoint operator K by forming the Friedrichs extension of −∆ t ⊕ −∆ v ⊕ −∆ s . As discussed before, K has a purely discrete spectrum.
Beside the auxiliary inner product (74), we can also introduce on H t a more geometrically natural structure, namely an indefinite inner product ., . : where S abcd is the deWitt supermetric S abcd = g ac g bd + g ad g bc − g ab g cd . H t equipped with ., . is a Krein space which we will denote by K t . It is easy to see that ., . can be also written as h, h ′ = 2(h, Gh ′ ) = 2(Gh, h ′ ) , where G is the trace reversal operator defined in (27). Writing h and h ′ in components we can see that the direct summands of H t are orthogonal also with respect to ., . . Explicitely we have: whereĜ is the trace reversal operator on Σ.
As proven in [31], in order to construct a Hadamard state we need a complete pseudoorthonormal basis of K t consisting of eigenvectors of K. To analyze the eigenvalue problem (73) it is convenient to make a further decomposition of L 2 (S 2 T * Σ) into traceless and scalar component:ĥ where φ . = 1 3 tr(ĥ). This decomposition is also orthogonal, since: This means that we can write the ., . -product as: The corresponding decomposition of H t can be written as: where H T is the space of traceless 2 tensors in L 2 T (S 2 T * Σ), H T S contains the scalar components of tensors in L 2 T (S 2 T * Σ) and H S , H V are L 2 (Σ), L 2 (T * Σ) respectively. The operator K acts on all these direct summands simply as the bare Laplacian −∆, since the curvature terms vanish. We shall now analyze the eigenvalue problem (73) by identifying different families of eigenfunctions. We will denote the eigenfunctions by ξ(λ, j), where λ labels the family to which it belongs and j (which labels eigenfunctions within families) takes values in a labeling set J(λ), so we have: Kξ(λ, j) = ω(λ, j) 2 ξ(λ, j) , and the corresponding solution of the wave equation is A (λ, j)(t, x) = e −iω(λ,j)t ξ(λ, j)(x) .
We start with scalar modes of ∆ in H S . They form a complete orthonormal basis and the scalar eigenfunctions can be written as ξ(S, j)(x) = u(j, x)e 0 ⊗e 0 . The corresponding solutions are A (S, j)(t, x) = e −iω(S,j)t u(j, x)e 0 ⊗ e 0 .
Similarly we find that eigenfunctions of ∆ in H T S can be written as ξ(T S, j)(x) = Now we come to the Laplace equation on H V . Since we assummed Σ to be simply connected, the Hodge decomposition gives H V = d Σ C ∞ (Σ) ⊕ δ Σ Λ 2 (Σ) .
= H V L ⊕ H V T , where d Σ , δ Σ are the de Rham differential and the codifferential on Σ and Λ 2 (Σ) denotes the space of 2-forms. It is easy to calculate that the properly normalized VL and VT modes can be written as: Finally we analyze the eigenvalue problem for traceless symmetric 2 tensors in H T . Using the decomposition theorem of Berger and Ebin [7] we can write H T as: where div * 0 is the traceless component of div * . In other words, we can identify Im(div * 0 ) acting on 1-forms Γ ∞ (T * M ) with the image of div * acting on the space of divergence free 1-forms. Let us denote H T L . = Im(div * 0 ) to be the space of traceless longitudinal square integrable symmetric 2 tensors and H T T . = ker(div) the space of traceless transversal tensors. Using the Hodge decomposition for 1-forms we can decompose H T L further into components H T LS and H T LV . The corresponding modes are given by: A (T T, j)(t, x) = H ik (j, x)e −iω(T T,j)t e i ⊗ e k , div(A (T T, j)) = 0 .
In the same way as the equation L A = 0 for the 2-tensors, one can also analyze the solutions of H B = 0 for 1-forms. This was done in [31], so we only recall the results.
These modes form an orthonormal basis for the Hilbert space of static one-forms H v = L 2 (T * Σ) ⊕ L 2 (Σ).
We can now come to construction of the Hadamard state. Let I A denote the index set consisting of all the values of λ for A -modes (i.e. I A = {S, T S, . . . , T T }). Similarly, I B is the corresponding index set for B-modes. The bisolution for 2-tensors is defined as: where λ runs through all the families of modes described above and s(λ) is 1 for λ = V L, T LS, T T and −1 otherwise. Similarly the bisolution for 1-forms is given by: where λ = S, L, T and s(S) = −1, s(L) = −1, s(T ) = 1. Explicit calculation shows that the 2-point function ω, constructed from ω t , ω v defined above, satisfies also the property (26). Finally we turn to the Hilbert space representation of the free on-shell algebra of observables. We start from the Krein spaces K t , K v equipped with the indefinite product ., . discussed above. Next we build a Fock space K which is a tensor product of the bosonic Fock space F b of K t and the fermionic Fock space F f of K v . Quantum fields are represented in the following way: π 0 (h(x)) . = λ∈I A j∈J(λ) 1 2ω(λ, j) A (λ, j)(x)a + λ,j + h.c. , π 0 (C(x)) . = λ∈I B j∈J(λ) 1 2ω(λ, j) B(λ, j)(x)c + λ,j,1 + B(λ, j)(x)c λ,j,2 , π 0 (C(x)) . = λ∈I B j∈J(λ) 1 2ω(λ, j) B(λ, j)(x)c + λ,j,2 + B(λ, j)(x)c λ,j,1 , where a + λ,j are the standard creation operators of F b and c + λ,j,1 , c + λ,j,2 of F f . Additionally we define π 0 (B(x)) . = −π 0 (div G h(x)). One can check by an explicit calculation that the free BRST charge is represented as: (a † λ,j c λ,j,2 + a λ,j c † λ,j,1 ) .
where we identified frequencies of B modes which coincide with frequencies of A modes. It is now easy to see that the transversal modes T T are in the kernel of π 0 (Q 0 ) and so are ones corresponding to C and B excitations. The latter, however, belong to the image of π 0 (Q 0 ) as well, so the physical Hilbert space (defined as the cohomology of π 0 (Q 0 ) in ghost number 0) consists only of T T modes and the product ., . is positive definite on its elements.