BV formalism

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

arXiv:1306.1058v2 [math-ph] 9 Jul 2013

Romeo Brunetti(1) , Klaus Fredenhagen(2) , Katarzyna Rejzner(3)

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Department of Mathematics at the University of Trento, II Institute for Theoretical Physik, University of Hamburg, INdAM Cofund (Marie Curie) fellow, University of Rome Tor Vergata, [email protected], [email protected], [email protected] (2)

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Dedicated to Roberto Longo on the occasion of his 60th birthday Abstract 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.

Contents 2

1 Introduction 2 Classical theory 2.1 Configuration space of the classical theory 2.2 Functionals . . . . . . . . . . . . . . . . . 2.3 Dynamics . . . . . . . . . . . . . . . . . . 2.4 Diffeomorphism invariance . . . . . . . . . 2.5 BV complex . . . . . . . . . . . . . . . . . 3 Quantization 3.1 Outline of the approach 3.2 Perturbative formulation 3.3 Free quantum theory . . 3.4 Interacting theory . . .

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17 17 18 20 23

4 Background independence

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5 States 30 5.1 Perturbative construction of a state for on-shell backgrounds . . . . . . . . 30 5.2 Existence of states for globally hyperbolic on-shell spacetimes . . . . . . . 34 6 Conclusions and Outlook

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A Aspects of classical relativity seen as a locally covariant field theory

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B Time-slice axiom 39 B.1 Time-slice axiom for the free theory in the presence of sources . . . . . . . 39 B.2 Time-slice axiom for the interacting theory . . . . . . . . . . . . . . . . . . 40 C BRST charge

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D Existence of Hadamard states on ultrastatic spacetimes

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1

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 [65, 72] 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 [75]. 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 [74]. Results supporting this perspective have been obtained by Reuter et al. [63, 64]. 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 2

Hawking’s prediction of black hole evaporation [42], 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 [31].) 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 [27]. 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, 44]. 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 [66, 22, 69]. 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, 71]. 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 [32] two of us performed this construction for classical gravity, and in another paper [34] 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 [43] and of Brennecke and Dütsch on a general treatment of anomalies [10].

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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 [54, 56, 55] 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 BatalinVilkovisky formalism for classical theories as developed in [32] 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 [34]. A crucial role is played by the Mœller 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 [32], 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 [28] and [61, 32]). 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 [66, 22, 69]. 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 [33]). 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 |Ψi are considered to be functions on the diffeomorphism group, so to obtain expectation values we have to

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evaluate hΨ| A |Ψi at the identity. Construction of states on arbitrary globally hyperbolic on-shell backgrounds is possible with the use of the deformation argument of [35]. 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 [35] 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 2.1

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 [32], 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 [41] 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 [50]. 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: g′ ≺ g if g′ (X, X) ≥ 0 implies g(X, X) > 0 ,

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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 . E(M) = {g′ ∈ Lor(M )| g′ ≺ g} . (2)

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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 τW , given by open neighborhoods of the form Ug0 ,V = {g0 + h, h ∈ V open in Γc ((T ∗ M )⊗2 )}. It turns out that E(M) is an open subset of Γ((T ∗ M )⊗2 ) with respect to τW (for details, see the Appendix A and [15]). The topology τW induces on E(M) a structure of an infinite dimensional manifold modeled on the locally convex vector space Γc ((T ∗ M )⊗2 ), of compactly supported contravariant 2-tensors. The coordinate chart associated to Ug0 ,V is given by κg0 (g0 +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 τW induces on the configuration space also a smooth manifold structure, in the sense of the convenient calculus [50], so E becomes a contravariant functor from Loc to CnMfd where morphisms χ are mapped to pullbacks χ∗ .

2.2

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 [41, 57] (see Appendix A for details). In particular, the definition of smoothness which we use implies that for all g˜ ∈ E(M), n ∈ N, F (n) (˜ g ) ∈ Γ′ ((T ∗ M )n ), i.e. 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: T E(M) = E(M) × Γc ((T ∗ M )⊗2 ), T ∗ E(M) = E(M) × Γ′c ((T ∗ M )⊗2 ). By a slight abuse of notation we denote the space Γc ((T ∗ M )⊗2 ) by Ec (M). The assignment of Ec (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) = {x ∈ X|f (x) 6= g(x)} .

Now we can define the spacetime support of a functional on E(M): . supp F = {x ∈ M |∀ neighbourhoods U of x ∃h1 , h2 ∈ E(M ), rel supp(h1 , h2 ) ⊂ U

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such that F (h1 ) 6= F (h2 )} .

Another crucial property is additivity. Definition 2.1. Let h1 , h2 , h3 ∈ E(M), such that rel supp(h1 , h2 ) ∩ rel supp(h1 , h3 ) = ∅ and we define . . h = h3 ↾(rel supp(h1 ,h2 ))c , h = h2 ↾(rel supp(h1 ,h3))c , 6

where the superscript c denotes the complement in M . We say that F is additive if F (h1 ) = F (h2 ) + F (h3 ) − F (h)

holds.

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A smooth compactly supported functional is called local if it is additive and, for (k) (˜ each n, the wavefront set  of F (n) (˜ g ) satisfies: WF(F g )) ⊥ T Diagk (M ) with the . thin diagonal Diagk (M ) = (x, . . . , x) ∈ M k : x ∈ M . In particular F (1) (˜ g ) has to be a smooth section for each fixed g˜. 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 Floc (M). The algebraic completion of Floc (M) with respect to the pointwise product F · G(˜ g ) = F (˜ g )G(˜ g) (5) 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˜).

2.3

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 Floc satisfying supp(LM (f )) ⊆ supp(f ) ,

∀ M ∈ Obj(Loc), f ∈ D(M) ,

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LM (f1 + f2 + f3 ) = LM (f1 + f2 ) − LM (f2 ) + LM (f2 + f3 ) ,

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and the additivity rule

for f1 , f2 , f3 ∈ D(M) and supp f1 ∩ supp f3 = ∅. The action S(L) is defined as an equivalence class of Lagrangians [16], where two Lagrangians L1 , L2 are called equivalent L1 ∼ L2 if supp(L1,M − L2,M )(f ) ⊂ supp df , (8) for all spacetimes M and all f ∈ D(M). In general relativity the dynamics is given by the Einstein-Hilbert Lagrangian: Z . EH g ) = R[˜ g]f dµg˜ , g˜ ∈ E(M) , (9) LM (f )(˜ where we use the Planck units, so in particular the gravitational constant G is set to 1.

2.4

Diffeomorphism invariance

In this subsection we discuss the symmetries of (9).L As a natural transformation LEH . is an element of Nat(Tensc , F),1 where Tensc (M) = k Tenskc (M) and Tensc (M) is the

1 Both Tensc and F have to be treated as functors into the same category. In [14] this category is chosen to be Top, the category of topological spaces, but in the present context it is more natural to include some notion of smoothness. A possible choice is the category of convenient vector spaces [50].

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L ⊗m ⊗ space of smooth compactly supported sections of the vector bundle m,l (T M ) ∗ ⊗l (T M ) . The space Nat(Tensc , F) is quite large, so, to understand the motivation for such an abstract setting, let us now discuss the physical interpretation of Nat(Tensc , F). In [32] we argued that this space contains quantities which are identified with diffeomorphism invariant partial observables of general relativity, similar to the approach of [66, 22, 69]. Let Φ ∈ Nat(Tensc , F). A test tensor f ∈ Tensc (M) corresponds to a concrete geometrical setting of an experiment, so we obtain a functional ΦM (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 [58]. Example 2.2. Let S : [0, 1] → R4 , λ 7→ s(λ) be a spacelike curve in Minkowski space M = (R4 , η). For g˜ = η + h ∈ E(M) the curve is still spacelike, and its length is . Λg˜ (S) =

Z

0

1q

|˜ gµν (s)s˙ µ s˙ ν |dλ .

Here s˙ µ is the tangent vector of s. We write it as s˙ µ = se ˙ µ , with ηµν eµ eν = −1. Expanding the formula above in powers of h results in Λg˜ (S) =

∞ X

n

(−1)

n=0

1 Z 2

n

0

1

hµ1 ν1 (s) . . . hµn νn (s)se ˙ µ1 eν1 . . . eµn eνn dλ .

Now, if we want to measure the length up to the k-th order, we have to consider a field Z Z Z µν µν µ1 ν1 ...µk νk 4 4 hµ1 ν1 . . . hµk νk d4 x , Λ(M) (fS )(h) = fS,0 ηµν d x + fS,1 hµν d x + . . . + fS,k where the curve whose length we measure is specified by the test tensor fS = (fS,0 , . . . , fS,k ) ∈ Tensc (M), which depends on the parameters of the curve in the following way: 1 Z 1 µ1 ν1 ...µk νk (x) = (−1)k 2 fS,k δ(x − s(λ))se ˙ µ1 eν1 . . . eµk eνk dλ, k ≥ 1 , k 0 Z 1 µν δ(x − s(λ))se ˙ µ eν dλ . fS,0 (x) = − 0

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(Tensc , F), which are additive in test tensors (condition (7)) and satisfy the support condition (6), correspond to local measurements, i.e. ΦM (f ) ∈ Floc (M). The condition for a family (ΦM )M∈Obj(Loc) to be a natural transformation reads ΦM′ (χ∗ f )(h) = ΦM (f )(χ∗ h) ,

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where f ∈ Tensc (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 [32], where an associative, commutative product was defined as follows: (ΦΨ)M (f1 , ..., fp+q ) =

1 X ΦM (fπ(1) , ..., fπ(p) )ΨM (fπ(p+1) , ..., fπ(p+q) ) , p!q!

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π∈Pp+q

Note, however, that the dependence on test tensors fi physically corresponds to a geometrical setup of an experiment, so ΦM (f1 )ΨM (f2 ) means that, on a spacetime M, we measure the observable Φ in a region defined by f1 and Ψ in the region defined by f2 . 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 ∈ Tensc (M) we recover not only the functional ΦM (f ), but also the whole diffeomorphism class of functionals ΦM (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∈Diff c (M) . Therefore, for fixed M, Φ and f , a physically meaningful object is the function Diff c (M) ∋ α 7→ ΦM (α∗ f ). It was proven in [50] that Diff c (M) can be endowed with a structure of an infinite dimensional manifold modeled on Xc (M), where Xc (M) is the space of compactly supported vector fields on M . Assuming the notion of smoothness proposed in [50], evaluation and composition are smooth maps, so we can prove that a pair (M, f ) induces a mapping: evM,f : Nat(Tensc , F) →C∞ (Diff c (M), F(M)) , . (evM,f Φ)(α) = ΦM (α∗ f ) , where α ∈ Diff c (M). We will denote evM,f Φ by ΦMf , or simply by Φf 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: . (F G)(α) = F (α)G(α) . (11) Let us denote by Fdiff (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 evM,f (Φ), for some Φ ∈ Nat(Tensc , F), f ∈ Tensc (M). 9

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 ∈ Fdiff (M) at α = id and in the values of the derivatives. Therefore, in the next step, we replace elements of Fdiff (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 ⊗Pη − η ⊗ ζ − [ζ, η]. More explicitly, on fixed M, to Φ and f we associate a series Φf = n∈N0 Φnf , whose components are given by derivatives of ΦM ((eζ )∗ f ) with respect to ζ at 0, i.e. dn . Φnf (ζ1 ⊗· · ·⊗ζn ) = , ζ1 ⊗· · ·⊗ζn ∈ U X(M) . ΦM ((et1 ζ1 ◦· · ·◦etn ζn )∗ f ) dt1 . . . dtn t1 =···=tn =0

Instead of writing Φnf (ζ1 ⊗ · · · ⊗ ζn ), n ∈ N0 , we will use a shorthand notation Φf (eζ⊗ ). The first few components of Φf are given by 1 . Φf (eζ⊗ ) = ΦM (f ) + Φ′M (£ζ f ) + (Φ′′M (£ζ f ⊗ £ζ f ) + Φ′M (£ζ £ζ f )) + . . . , 2

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′

where the prime denotes the derivative along a test function. An elegant way to represent Φf , is to write it as a (possibly infinite dimensional) lower triangular matrix   ΦM (f ) 0 0 ...  ΦM (f ) 0 . . . Φ′M (£ζ f )   1 (13)  (Φ′′ (£ζ f ⊗ £ζ f ) + Φ′ (£ζ £ζ f )) Φ′ (£ζ f ) ΦM (f ) . . . . M M 2 M  .. .. . . Note that the product (11) induces a product

n   . X n F k Gn−k , (F G) = k n

(14)

k=0

P P 1 n ⊗n 1 n ⊗n where F (eζ⊗ ) = ∞ ) and G(eζ⊗ ) = ∞ ) and, using the matrix n=0 n! F (ζ n=0 n! G (ζ notation (13), this product is just the usual product of matrices. The naturality condition for Φ implies that Φf depends locally on ζ. Note that the expression (12) makes sense also for ζ, which is not compactly supported. This motivates the following definition: Definition 2.3. The space F(M) of multilocal evaluated fields on M is a subspace of ′ U X(M)′ ⊗F(M) ≡ C∞ ml (E(M), (U X(M)) ) defined as the algebraic completion of {evM,f Φ| Φ ∈ Nat(Tensc , F),

f ∈ D(M)}

under the product (14). Clearly, F(M) is an infinitesimal realization of Fdiff (M). 10

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 Fdiff (M) by (αF )(β)(˜ g ) = F (β ◦ α)(α∗ g˜) F is diffeomorphism invariant if αF = F .

(15)

Remark 2.4. To relate (15) to the invariance condition for natural transformations, g ) = ΦM (β ∗ f )(˜ g), then proposed in [32], we note that if F (β)(˜ (αF )(β)(˜ g ) = ΦM (αM ∗ (β ∗ f ))(α∗M g˜) . If we choose a whole family α = (αM )M∈Obj(Loc) and for each M we choose a test function fM , then we obtain (αM ΦfM )(β)(˜ g ) = ΦM (αM ∗ (β ∗ fM ))(α∗M g˜) , and, at β = id, the invariance condition (15) implies that ΦM (αM ∗ (fM ))(α∗M g˜) = ΦM (fM )(˜ g) , which is exactly the condition (αΦ)M = ΦM for natural transformations proposed in [32]. In other words, invariant natural transformations Φ induce diffeomorphism invariant observables Φf on any fixed spacetime M. . Now, we come back to the discussion of infinitesimal diffeomorphisms. The derived action ρ of Xc (M) on F(M) is given by E D ζ δ F (e )(˜ g ), £ g ˜ . (16) (ρ(ξ)F )(eζ⊗ )(˜ g ) = F (ξ ⊗ eζ⊗ )(˜ g ) + δ˜ ξ ⊗ g

To make contact with [32], note that, in degree 0 in ζ, we have for elements of the form (12) D E δ (ρ(ξ)Φf )0 (˜ g ) = δ˜ g) , Φ (f )(˜ g ), £ g ˜ + Φ′M (£ξ f )(˜ ξ g M Infinitesimal diffeomorphism invariance is the requirement that ρ(ξ)Φf = 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.

11

2.5

BV complex

(for the Einstein Hilbert Lagrangian LEH ) is Any element of F(M) of the form LEH f an example of a diffeomorphism invariant quantity. In [32] 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 Xc (M) on F(M). The Chevalley-Eilenberg differential is constructed by replacing the infinitesimal diffeomorphism in (16) by a ghost, i.e. a form on Xc (M). We consider the space U X(M)′ ⊗CE(M) of functions on the enveloping algebra of X(M), with values in . CE(M) = 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. Definition 2.5. CE(M), the underlying algebra of the Chevalley-Eilenberg complex, is a subspace of U X(M)′ ⊗CE(M) defined as the algebraic completion of {evM,f Φ| Φ ∈ Nat(Tensc , CE),

f ∈ D(M)}

under the product (14). 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 γCE defined by . ∗ γCE = γCE + Dc , where Dc is the insertion of a ghost defined by . (Dc F )(eζ⊗ ) = F (c ⊗ eζ⊗ ) , ∗ is a derivation of CE(M) given by3 and γCE ∗ γCE : CEq (M) → CEq+1 (M) , q D E . X ∗ δ ˆi , . . . , ξq )), £ξ g˜ + (−1)i+q δ˜ (ω(ξ , . . . , ξ (γCE ω)(ξ0 , . . . , ξq ) = 0 i g i=0

+

X

(−1)i+j+q (ω(−[ξi , ξj ], . . . , ξˆi , . . . , ξˆj , . . . , ξq ) ,

(17)

i 0. Interacting fields are formal power series in in ~, λ and the interaction S1 . For onshell backgrounds, S˜1 the #af = 0 term of the interaction, is of order λ1 and θ is of order λ0 . Therefore we obtain formal power series in λ, ~ and the antifield number #af. Let us assume that we have a representation π0 of A(M) on an indefinite product . space K0 (M) and we denote K(M) = K0 (M)[[~, λ]]. 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 . K (M) = U X(M)′ ⊗K(M) ,

(57)

and the state space on BVq (M) will be constructed as a certain subspace of K (M). It is convenient to write elements of BVq (M) in the matrix notation (13). A typical element of K (M) will be written as a column vector  0  ψ ψ 1 (ζ) ζ (58) Ψ(e⊗ ) =  . .. . The action of BVq (M) on K (M) is now defined by   0  π0 (ΦM (f )) 0 0 ... ψ . π0 (Φ′ (£ζ f )) π0 (ΦM (f )) 0 . . . ψ 1 (ζ) ζ M (π(Φf )Ψ)(e⊗ ) =    .. .. .. . . .

For the simplicity of notation we will omit π0 , i.e. we write ΦM (f ) instead of π0 (ΦM (f )), if no confusion arises. Next, we have to choose an indefinite product on K (M). A 31

pointwise product on the space of K-valued maps induces

Ψ, Ψ′



∞ n E D . XX k ′ n−k 1 (eζ⊗ ) = ψ , ψ (ζ ⊗n ) . n!(n−k)!

K

K

n=0 k=0

(59)

Performing the physical measurement means evaluating Φf on β = id, so when we interpret our theory we have to evaluate hΨ, Ψ′ iK at ζ = 0. The positivity of the product is, therefore, the requirement that hΨ, Ψ′ iK (1) > 0. Let Φf ∈ RS1 (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 [52, 53]. 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 [62] we conclude that sˆ∗ Φ(f ) =

i [F, Q]⋆V ~

holds on-shell for F ∈ BV(M). We can also use RS1 to map to the free algebra and then, observables are elements of the form RS1 (F ), F ∈ BV(M), which belong to the 0-th cohomology of ~i [., RS1 (Q)]⋆ . The full BV operator is obtained by adding Df c . Note that this term is just an insertion of f c, with an appropriate sign, so if we define . df c Ψ(eζ⊗ ) = Ψ(f c ⊗ eζ⊗ ) , where c has to be understood as an operator7 , then Df c G = [df c , G]⋆ , . ˜= Denoting Q

1 i~ RS1 (Q(η))

G ∈ BVq (M) .

+ dc , we obtain ˜ G]⋆ . sˆG = [Q,

1 Here i~ RS1 (Q(η)) ≡ Qint is the interacting BRST charge and its nilpotency can be shown ˜ is nilpotent. Using by arguments analogous to [43]. Next, we have to prove that also Q ∗ the fact that s acts from the right, we obtain

˜ 2 Ψ)n (ζ ⊗n ) = s∗ (QΨ) ˜ n (ζ ⊗n ) + (QΨ) ˜ n+1 (c ⊗ ζ ⊗n ) (Q

= ψn+1 (s∗ c ⊗ ζ ⊗n ) − (s∗ ψn+1 )(c ⊗ ζ ⊗n )+

(s∗ ψn+1 )(c ⊗ ζ ⊗n ) + ψn+2 (c ⊗ c ⊗ ζ ⊗n ) =

= ψn+1 (− 21 [c, c] ⊗ ζ ⊗n ) + ψn+2 (c ⊗ c ⊗ ζ ⊗n ) = 0 .

In the last step we used the fact that ΨM is a function on U X(M), so it has to vanish on ˜ the ideal generated by elements of the form ξ1 ⊗ ξ2 − ξ2 ⊗ ξ1 − [ξ1 , ξ2 ]. We can now use Q 7

More precisely, let us write the representation of c in the form π0 (c)ψ = . P R ζ M can now define ΨM (f c ⊗ eζ⊗ ) = µ f (x)π0 (cµ (x))( δζδΨ µ (x) (e⊗ ))dx.

32

P R µ

π0 (cµ (x))ψdx. We

˜ defines a space to characterize the physical states in K . Indeed, the 0 cohomology of Q 0 closed under the action of physical observables (i.e. under H (BV, sˆ)). To see that it is ˜ and Φf ∈ RS (BV(M)). Then consistent, let us take Ψ ∈ ker(Q) 1 ˜ Φf ]⋆ Ψ = QΦ ˜ fΨ (ˆ sΦf )Ψ = [Q, ˜ so it vanishes in the cohomology. We will now show how to holds, i.e. (ˆ sΦf )Ψ ∈ Im(Q), ˜ If Ψ is such a vector, then construct vectors belonging to ker(Q).  0    ψ Qint ψ 0 + ψ 1 (c) 1 2  1    (Q + dc ) ψ (ζ) = Qint ψ (ζ) + ψ (c ⊗ ζ) = 0 .. .. . .

The general solution of this equation is:

ψ n = (−1)n (Qζ )n ψ 0 ,

(60)

where Qζ is defined by Qζ = π0 (ιζ Qint ) , and by ιζ 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). ˜ but this space is not yet the subspace of K where We have characterized KerQ, the product (59) 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])8 : given an evaluated field Φf = Φ0f + Φ1f + . . ., such that Φ0f k is the lowest non-vanishing order of the λexpansion of Φ0f with #af = 0, we require that Φlf 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 ˜ as a power series in λ. Let Φf be an evaluated interacting field. Assume that the of Q lowest non-vanishing λ-order of Φ0f with #af = 0 is Φ0f k and Φlf m ≡ 0 for all l > 0, m < k holds. Then we have: ˜ [Q(λ), Φf (λ)]⋆ = λk [Q0 , Φ0f k ]⋆ + O(λk+1 ) . 8

This condition is not entirely physical, since it is related to the expansion in λ which is just a trick we use to avoid discussing the issues of convergence. It appears, because our notion of positivity of states is a rather weak one and it relies on the positivity of the λ = 0 order in the expansion of Ψ ∈ K0 [[~, λ]]. If we manage to formulate the theory in such a way that the λ expansion will be avoided, this condition would not appear.

33

The BRST invariance of Φf (λ) implies that, in particular, [Q0 , Φ0f k ]⋆ = 0 ,

(61)

so the additional condition we imposed on observables guarantees that the first nonvanishing term in the formal power series Φ0f (λ) commutes with Q0 . 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 ψk0 , the lowest λ-order of ψ 0 ∈ K (M), belongs to the cohomology of Q0 , i.e.: ψk0 ∈ H 0 (K (M), Q0 ) .

(62)

From (61) it follows that condition (62) is preserved under applying physical quantum ˜ consisting of elements fields. Let us denote by H (M) the subspace of H 0 (K (M), Q) ′ that satisfy (62) as well. Then, for Ψ, Ψ ∈ H (M), with lowest non-vanishing λ-orders k1 , k2 respectively, we have: D E

 0 Ψ, Ψ′ K (1) = λk1 +k2 ψk01 , ψ ′ k2 + O(λk1 +k2 +1 ) . K

hΨ, Ψ′ iK

It (1) > 0 in the sense of formal power series if and only if D follows Ethat 0 ψk01 , ψ ′ k2 > 0. Therefore a sufficient condition for the positivity of h., .iK on H (M) K

is that h., .iK of the linearized theory is positive on the space H 0 (K(M), Q0 ). 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.

5.2

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 [35]. We start with an arbitrary globally hyperbolic spacetime M1 = (M1 , g1 ), where g1 is a solution to Einstein’s equation and we consider a causally convex neighborhood N1 of a Cauchy surface Σ ⊂ M1 . We choose an ultrastatic auxiliary spacetime M2 = (Σ × R, g2 ) with a causally convex neighborhood N2 of a Cauchy surface. Now we take an intermediate spacetime M = (M, g) χ1 χ2 such that (N1 , g1 ) −→ M, (N2 , g2 ) −→ M, where χ1 , χ2 are morphisms in Loc. Let us denote the images of embeddings χ1 and χ2 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 S0 . Free equations of motion include now an external ′ . In appendix B we argue that, for off-shell backgrounds, in current J induced by SM 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 Φf ∈ H0 (BVS1 (N), sˆ), such that Φ0f k is the lowest non-vanishing order of the λ-expansion 34

with #af = 0 of Φ0f and Φlf m ≡ 0 for all l > 0, m < k (additional condition derived in subsection 5.1). Using the local covariance we can map Φf to Φχ∗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 BVS1 (N) to BVS1 (N′ ). Here we only give the final formula: α ˜ (Φχ∗1 f ) = RV−1

   ! ˜ 1b +θ ′ )/~ ˜ 1b +θ ′ )/~ −1 i( L i(L b b ⋆ α0 (RV Φχ∗1 f ) ⋆ eT = eT =



˜ 1b +θ ′ )/~ i(L b eT

−1

⋆V (RV Φχ∗1 f ) ⋆V



˜ 1b +θ ′ )/~ i(L b eT



+ I1 , (63)

where I1 = RV−1 (I0 ) is an element of the ideal IV generated by the interacting equations of motion. Test functions b, b′ do not intersect the future of the support of RµV (Φχ∗1 f ), b ≡ 1 on supp b′ , V ∈ VS1 (O) and both b and b′ are supported inside O. We will now show that if Φχ∗1 f is in H 0 (BV(M), sˆ), then the same holds for α ˜ (Φχ∗1 f ). −1 Recall that sˆ = RV ◦ ({., S0 }⋆ + Dc ) ◦ RV . Using the fact that δ0 is a resolution of the algebra of on-shell functionals, we know that there exists a element X of BV(M), such that I2 = δ0 X = {X, S0 } = {X, S0 }⋆ . It follows that RV−1 (I2 ) = sˆ(RV−1 X), so the second term in (63) drops out from the cohomology. Concerning the first term, we use the fact that {., S0 }⋆ and Dc are ⋆-derivations and we consider   ˜ 1b +θ ′ )/~ i(L b , L0f ′′ , eT ⋆

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 therefore   !  ˜ 1b +θ ′ )/~ ˜ 1b +θ ′ )/~ −1 i(L i(L b b −1 ⋆ {RV (Φχ∗1 f ), S0 }⋆ ⋆ eT = eT sˆ ◦ α ˜ (Φχ∗1 f ) = RV =α ˜ ◦ sˆ(Φχ∗1 f ) Additional condition, concerning the behavior of the λ-expansion of α ˜ (Φχ∗1 f ) under small changes of the coordinate system is also fulfilled. To see this, note that the lowest λ-order     ˜ 1b +θ ′ )/~ −1 ˜ 1b +θ ′ )/~ i(L i(L b b with #af = 0 of eT ⋆V (RV Φχ∗1 f ) ⋆V eT is the same as of Φχ∗1 f , 35

˜ 1 is of order λ. Hence that the lowest order of α because L ˜ (Φχ∗1 f ), is the same as that of Φχ∗1 f , modulo the equations of motion. We have just proven that if Φχ∗1 f is a physical observable, then so is α ˜ (Φχ∗1 f ), so a state on A(N1 ) can be transported to a state on A(N2 ).

6

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 [34]. In the spirit of algebraic quantum field theory [40] 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 [66], Dittrich [22] and Thiemann [69]. 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 [51, 52, 53] (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 Mœller 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 calcu36

lations could be compared. It would also be of great interest to adapt the renormalization approach of Reuter et al. (see, e.g., [63, 64]) to our framework. Further interesting problems are the validity of the semiclassical Einstein equation (for an older discussion see [73]) and the possible noncommutativity of the physical spacetime [24].

Acknowledgements We would like to thank Dorothea Bahns, Roberto Conti and Jochen Zahn for enlightening discussions and comments. K. R. is also grateful to INdAM (Instituto Nazionale di Alta Mathematica "Francesco Severi") for supporting her research.

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 τW , given by open neighborhoods of the form Ug0 ,V = {g0 + h, h ∈ V open in Γc ((T ∗ M )⊗2 )}, where Γc ((T ∗ M )⊗2 ) is equipped with the standard inductive limit topology. In our case, τW coincides with the Whitney C∞ topology, W O∞ , hence the notation. After [48], Whitney C∞ topology is the initial topology on C∞ (M, (T ∗ M )⊗2 ) induced by the graph topology on C∞ (M, J ∞ (M, (T ∗ M )⊗2 ) through maps Γ((T ∗ M )⊗2 ) ∋ h 7→ j ∞ h, where J ∞ (M, (T ∗ M )⊗2 ) is the jet space and j ∞ h is the infinite jet of h. On the space of all Lorentzian metrics we have also another nattural topology, namely the interval topology τI introduced by Geroch [37], which is given by intervals {g|g1 ≺ g ≺ g2 }, where the partial order relation ≺ is defined by (1), i.e. g′ ≺ g if g′ (X, X) ≥ 0 implies g(X, X) > 0 . The configuration space E(M), defined (2) is, by definition, an open subset of Lor(M ), with respect to τI . Moreover, if g ′ ∈ E(M), then we know that there exists λ ∈ R such that λg − g′ is positive definite, so we can find a neighborhood V ⊂ Γc ((T ∗ M )⊗2 ) of 0, such that g ′ + h ∈ Lor(M ) and λg − g′ − h is also positive definite. It follows that g′ + h < λg and g ′ + V ⊂ E(M). This shows that E(M) is open also with respect to τW . More generally, it was shown in [48] that the C0 Whitney topology, W O0 , on Lor(M ) conincides with the interval topology on the space of continuous Lorentz metrics. This result was than used in [15] to show that the space of smooth, time-oriented and globally hyperbolic Lorentzian metrics on M is an open subset of Lor(M ), with respect to W O∞ . Functionals on E(M) are required to be smooth in the sense of calculus on locally convex vector spaces, but the relevant topology is the compact open topology τCO not the Whitney topology τW . More precisely, let U be an open neighborhood of h0 in the compact open topology τCO . The derivative of F at h0 in the direction of h1 ∈ 37

Γ((T ∗ M )⊗2 ) is defined as E D 1 . F (1) (h0 ), h1 = lim (F (h0 + th1 ) − F (h0 )) t→0 t

(64)



whenever the limit exists. The function F is called differentiable at h0 if F (1) (h0 ), h1 exists for all h1 ∈ E(M). It is called continuously differentiable if it is differentiable at all

points of E(M) and dF : U × E(M) → R, (h0 , h1 ) 7→ F (1) (h0 ), h1 is a continuous map. It is called a C1 -map if it is continuous and continuously differentiable. Higher derivatives are defined in a similar way. Note that above definition means that F is smooth, in the sense of calculus on locally convex vector spaces, as a map U → R. It was shown in [15, Remark 2.3.9] that this fits also into the manifold structure on E(M) induced by τW . To see this, note that a compactly supported functional F , defined on a τW -open set Ug0 ,V can be extended to a functional F ◦ ιχ defined on an τCO -open neighborhood ∗ ⊗2 ∗ ⊗2 ι−1 χ (Ug0 ,V ) by means of a continuous map ιχ : (Γ((T M ) ), τCO ) → (Γ((T M ) ), τW ), . ′ ′ defined by ιχ (g ) = g0 + (g − g0 )χ. From the support properties of F follows that F ◦ ιχ is independent of χ. In particular, F (1) defines a kinematical vector field on E(M) in the sense of [50]. Moreover, since Ec (M) is reflexive and has the approximation property, it follows (theorem 28.7 of [50]) 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 BVµc (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 7→ F (1) (ϕ) is smooth as a map E(M) → E(M), where E(M) is equipped with its standard Fréchet topology. Since the lightcone of g˜ 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 BVµc (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 D E D E (⌊F, G⌋g˜ )(1) (ϕ) = F (2) (ϕ), ∆G(1) (ϕ) − ∆F (1) (ϕ), G(2) (ϕ) + D E D E − ∆A F (1) (ϕ), S ′′′ (ϕ)∆R G(1) (ϕ) + ∆R F (1) (ϕ), S ′′′ (ϕ)∆A G(1) (ϕ) , (65)

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 Diag3 (M ) and ∆R/A F (1) (ϕ), ∆R/A G(1) (ϕ) are smooth. The first term of (65) can be written as 38

d (1) (ϕ dt F

+ th)

t=0

, where h = ∆G(1) (ϕ) is smooth. By assumption, ϕ 7→ F (1) (ϕ) is

smooth, so the above derivative exists as a smooth section in E(M). The same argument can be applied to the second term in (65), so we can conclude that ⌊F, G⌋g˜ )(1) (ϕ) is a smooth section. From a similar reasoning follows also that ϕ 7→ (⌊F, G⌋g˜ )(1) (ϕ) is a smooth map. The antisymmetry of ⌊., .⌋g˜ is clear, so it remains to prove the Jacobi identity. In [47, 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.

B B.1

Time-slice axiom Time-slice axiom for the free theory in the presence of sources

δS0 In the algebra of free fields A(M), the equations of motion are δϕ(x) = 0 and we denote by I0 the ideal generated (in the algebraic and topological sense) by such elements with respect to the product ⋆. It coincides with the ideal of functionals vanishing on solutions. Let N ⊂ M be a causally convex neighborhood of a Cauchy surface in M. The time-slice axiom of the free theory requires that that there exists a map α0 : (A(M), ⋆) → (A(N), ⋆), which is an isomorphism modulo I0 . 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:

Oϕ(x) = J(x) , ′′ and J is an “external current” induced where O is the differential operator induced by SM ′ by SM . 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 Σ0 , Σ1 ⊂ N such that Σ0 lies in the past of Σ1 . Let χ be a function which is 1 in the future of Σ1 and 0 in the past of Σ0 . We use the left inverse (cf. Section 3.3) Gχ = ∆R χ + ∆A (1 − χ).

The map α0 is defined by . α0 (F )(ϕ) = F (ϕ − Gχ (Oϕ − J)) .

(66)

The support of α0 (F ) is contained in N. Namely, let h have a compact support which does not intersect N. Then Gχ Oh = h, hence α0 (F )(ϕ + h) = α0 (F )(ϕ). Moreover, α0 (F ) coincides with F on solutions of the equations of motion, thus the difference is an element of I0 . 39

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 I0 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 RV (F ). Therefore, we want to completely avoid discussing convergence issues at this stage and apply (66) 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 (66) 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 g0 which is a solution to Einstein’s equation S ′ (g0 ) = 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 = g0 + g1 (J) + g2 (J ⊗2 ) + . . .. Propagators are now also formal power series in J. . The map from equation (66) acts on observables as α0 (F )(λϕ) = F (λϕ − Gχ (λOϕ − J)). 2 We can apply α0 also to the formal S-matrix eiV /(λ ~) and to the interacting fields, so that α0 (RV (F )) = RV (F ) + I0 holds, where I0 ∈ I0 . The expression on the left hand side is a formal expression both in J and in λ and the ideal I0 is generated by λOϕ(x) − J(x).

B.2

Time-slice axiom for the interacting theory

In the interacting net (BVV (O), ⋆V ), the equations of motion are   δ(S0 + V ) δS0 −1 , = RV δϕ(x) δϕ(x) and they generate with respect to ⋆V an ideal which we denote by IV . To prove the timeslice axiom, we have to show that (BVV (M), ⋆V ) is isomorphic to (BVV (N), ⋆V ), modulo IV , where N is a causally convex neighborhood of a Cauchy surface Σ. Let us start with an arbitrary element F ∈ BVV (O), where O ⊂ M is relatively compact. Using RV , we map it to RV (F ) ∈ (AV (O), ⋆). By definition (AV (O), ⋆) ⊂ (A(M), ⋆). From the time-slice axiom of the free theory it follows that there exists a map α0 : (A(M), ⋆) → (A(N), ⋆), which is an isomorphism modulo I0 . Now we want to find a map from (A(N), ⋆) to (AV (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 α0 of the free theory. More concretely, one has to multiply the relative S-matrices corresponding to the interaction L1N , 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 α0 is applied and then the free fields are mapped back to interacting ones. In [18] such a map α : (A(N), ⋆) → (AV (N), ⋆) is constructed explicitly and, for 40

each relatively compact region O1 , there is an invertible element U such that α(G) = U −1 ⋆ G ⋆ U , where G is supported in O1 . The invertible element U can be realized as ˜ 1N (b)+θN (b′ ))/~ iT(L

˜ 1N (b) + θN (b′ ))/~) = eT S(i(L

,

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 ˜ 1 are evaluated on different test functions. Let f1 be a smooth function with that θ and L past compact support in the interior of the future of Σ1 , which is a Cauchy surface in the past of supp G. Let N1 be a causally convex neighborhood of Σ, which is contained in the interior of N. We choose a function f2 with supp f2 ⊂ N which coincides with f1 on N1 and a test function b2 with support in N which coincides with f2 on J− (K). K is a compact region, such that supp G ⊂ K o . Similarly, we can choose a test function b1 which coincides with f1 in J− (K). Let b1 − b2 = b + b, where supp b does not intersect the past and supp b not the future of supp G. In [18] fi , bi are used as smearing functions for L1M , 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 ˜ 1 , so it remains to construct test functions for θ. We consider N2 , N3 causally convex for L neighborhoods of Σ, such that the closure of N is contained in the interior of N2 the closure of N2 is contained in the interior of N3 . Let Σ2 be a Cauchy surface, which lies in the past of N3 and in the future of Σ1 . Now we essentially repeat the construction of test functions for the new choice of neighborhoods and Cauchy surfaces: f1′ is a smooth function with past compact support contained in the interior of the future of Σ2 , f2′ has supp f2′ ⊂ N3 and coincides with f1′ on N2 , b′2 has suppb′2 ⊂ N3 and coincides with f2′ on ′ J− (K), b′1 coincides with f1′ in J− (K), b′1 − b′2 = b + b′ and supp b′ does not intersect the future of supp G. Moreover, we choose our smearing objects in such a way that the region where b1 ≡ 1 has a nonzero intersection with the past of N3 and b ≡ 1 on the support of b′ . We consider a product of relative S-matrices ˜ 1N (b2 ) + θN (b′2 )))−1 ⋆ Sf1 ,f1′ ,f2 ,f2′ (G) = SL˜ 1M (f1 )+θM (f ′ ) (−(L 1

˜ 1N (b2 ) + θN (b′ )) + G) , ⋆ SL˜ 1M (f1 )+θM (f ′ ) (−(L 2 1

which is, by definition, an interacting field, but due to causal factorisation property, it is equal to a product of free fields, ˜ 1N (b) + θN (b′ ))/~). ˜ 1N (b) + θN (b′ ))/~)−1 ⋆ S(G) ⋆ S(i(L Sf1 ,f1′ ,f2 ,f2′ (G) = S(i(L In the last step we can use RV−1 to map (AV (N), ⋆) to (BVV (N), ⋆V ). Let us denote . ˜ 1N (b) + θN (b′ ))/~). Note that ˜ 1N (b) + θN (b′ ))/~)−1 ⋆ α0 (RV (F )) ⋆ S(i(L α ˜ (F ) = RV−1 (S(i(L . −1 −1 ′ ˜ 1N (b) + θN (b ))/~)−1 ⋆ RV (F ) ⋆ S(i(L ˜ 1N (b) + ˜ (F ) = RV (S(i(L I1 = RV (I0 ) ∈ IV , so α ′ θN (b ))/~) modulo interacting equations of motion. We can summarize this construction in the following diagram: 41

(BVV (M), ⋆V )

(BVV (N), ⋆V )

⊃

R−1 V

RV

(AV (O), ⋆)

C

⊂

α0

(A(M), ⋆)

α

(A(N), ⋆)

(AV (N), ⋆)

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: Z   ′ ˜λ Γ ˜ρ ˜ρ ˜λ L(M,g) (f )(h) = dvol(M,˜g) g˜µν Γ µρ νλ − Γµν Γρλ . M

It differs from the Einstein-Hilbert Lagrangian by a term

R

M

˜ µρσ − ˜gµν Γ ˜λ Dµ = ˜gρσ Γ νλ

f ∇µ Dµ , where

˜ are the Christoffel symbols relating the covariant derivative ∇ ˜ to ∇. Explicitly and Γ’s they are defined by ˜ αµν = g˜αβ (∇µ g˜αν + ∇ν g˜αµ + ∇α g˜µν ) . Γ

˜ µνρσ , corresponding to the full metric g˜, can be written The Riemann curvature tensor R as: ˜ σ = ∇ν Γ ˜ σ − ∇µ Γ ˜σ + Γ ˜α Γ ˜σ ˜α ˜σ R µνρ µρ νρ µρ αν − Γνρ Γαµ .

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:    . X µ α ∂LM (x) α ∂LM (x) µ α ∂LM (x) + 2∇ν γϕ − ∇ν γϕ −KM (x) , J (x) = γϕ α) α) α) ∂(∇ ϕ ∂(∇ ∇ ϕ ∂(∇ ∇ ϕ µ µ ν µ ν α µ where KM is the divergence term appearing after applying γ ∗ to LM (f ). Using this formula we obtain (compare with [59, 51, 56]):

J µ = ˜gµλ (bρ ∇λ cρ − (∇λ bρ )cρ ) + α(bρ + icα ∇α c¯ρ )(bρ + icα ∇α c¯ρ ) + i˜gµλ cα cρ Rλαρβ c¯β . (67) The free BRST current is given by: √ J0µ = −gg µλ (bρ ∇λ cρ − (∇λ bρ )cρ ) . For a spacetime M with compact Cauchy surface Σ, for Rany closed 3-form β there exists R 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: Z . η∧J Q= M

42

and analogously for the free BRST charge Q0 . 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 ηM 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 −1 0 metric g = , where gˆ is a Riemannian metric on the Cauchy surface Σ. The 0 gˆ ˆ k , where by ˆ we will only nonvanishing Christoffel symbols for this metric are Γkji = Γ ji always denote quantities that refer to metric gˆ. The only non-vanishing components of the curvature tensor are the ones corresponding to the intrinsic curvature of (Σ, gˆ), i.e. l ˆ l . Therefore if (M, g) is a an Einstein manifold, then so is (Σ, gˆ). Moreover, Rijk =R ijk if (Σ, gˆ) 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 [30]. To see this, let us first fix a global frame {ea }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 {eb }b=0,1,2,3 as a set of four global sections of T ∗ M by demanding that eb (x)(ea (x)) = δab holds at all points of M . Using these frames we can write the perturbation metric in the form: h = h00 e0 ⊗ e0 +

3 X i=1

3 X

hi0 ei ⊗ e0 +

i=1

h0i e0 ⊗ ei +

3 X

i,j=1

hij ei ⊗ ej .

(68)

This can be also expressed in the matrix form   h00 hT h= , (69) ˆ h h   h10 where h is the vector with components h = h20 . In this section we will adapt small h30

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 (68). It is now easy to check that the wave operator L acts on the components of h in a following way: L h00 = (−∂02 + ∆s )h00 , L h = (−∂02 + ∆v )h , ˆ, ˆ = (−∂ 2 + ∆t )h L h 0

43

where ∆s , ∆v , ∆t are the 3-dimensional Lichnerowicz Laplacians on (Σ, gˆ) acting on scalar, vector and symmetric contravariant 2 tensors respectively. If ξ ∈ E(M) doesn’t depend on the ultrastatic time parameter x0 ≡ t, then A (t, x) = e−iωt ξ(x) solves the equation L A = 0 if and only if −∆s ξ00 = ω 2 ξ00 ,

(70)

2

−∆v ξ = ω ξ , −∆t ξˆ = ω 2 ξˆ ,

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 ˜ containing O and the compact Cauchy surface Σ with a boundary ∂Σ. We have to O 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 metric gˆ 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 [68]. The non-negativity of the Lichnerowicz Laplacian was also proven for some Ricci flat metrics on compact manifolds [38], 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: −∆ξ00 = ω 2 ξ00 , 2

−∆ξ = ω ξ , −∆ξˆ = ω 2 ξˆ ,

ξ00 |∂Σ = 0 ,

(71)

ξ 00 |∂Σ = 0 , ξˆ00 |∂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 (69). On the space of symmetric contravariant 2 tensors on Σ we can now introduce an inner product by: 3 Z ab ab . X h h′ dµgˆ . (h, h ) = ′

(72)

a,b=0 Σ

We can complete Γ∞ (S 2 T ∗ Σ) with respect to the norm topology induced by this inner product to obtain a Hilbert space L2 (S 2 T ∗ Σ) of square integrable 2-tensors. In the similar way we define spaces of square integrable 1-forms L2 (T ∗ Σ) and functions L2 (Σ). The direct sum of these three spaces is the Hilbert space Ht = L2 (S 2 T ∗ Σ)⊕L2 (T ∗ Σ)⊕L2 (Σ), where the inner product is (., .). It can be identified with the space of static symmetric 44

rank 2 contravariant tensors, since the decomposition (69) is orthogonal with respect to (., .). On Ht 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 (72), we can also introduce on Ht a more geometrically natural structure, namely an indefinite inner product h., .i: Z

 ab cd ′ . (73) h, h = dvolgˆ h h′ Sabcd , Σ

where Sabcd is the deWitt supermetric Sabcd = gac gbd + gad gbc − gab gcd . Ht equipped with h., .i is a Krein space which we will denote by Kt . It is easy to see that h., .i 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 Ht are orthogonal also with respect to h., .i. Explicitely we have: Z Z

 ′ ′ ′ ˆ ˆ ˆ h, h = (h, Gh ) − 2 h · h dµgˆ − h00 h′00 dµgˆ , Σ

Σ

ˆ is the trace reversal operator on Σ. where G As proven in [30], in order to construct a Hadamard state we need a complete pseudoorthonormal basis of Kt consisting of eigenvectors of K. To analyze the eigenvalue problem (71) it is convenient to make a further decomposition of L2 (S 2 T ∗ Σ) into traceless and scalar component: 1 . ˆ ij = h Hij + gˆij φ , 3 . 1 ˆ where φ = tr(h). This decomposition is also orthogonal, since: 3

1 ˆ ′ + 1 gˆφ′ )) = (H + gˆφ, G(H 3 3 Z
1 1 ′ ′ ′ ′ ′ g φ, gˆφ ) = (H, H ) − φφ′ dµgˆ . = (H, H ) + (ˆ g φ, H ) − (H, gˆφ ) − (ˆ 3 9 Σ

This means that we can write the h., .i-product as:

 1 h, h′ = (H, H ′ ) − (φ, φ′ ) − 2(h, h′ ) − (h00 , h′00 ) . 3

The corresponding decomposition of Ht can be written as: Ht = HT ⊕ HT S ⊕ HS ⊕ HV , where HT is the space of traceless 2 tensors in L2T (S 2 T ∗ Σ), HT S contains the scalar components of tensors in L2T (S 2 T ∗ Σ) and HS , HV are L2 (Σ), L2 (T ∗ Σ) respectively. The operator K acts on all these direct summands simply as the bare Laplacian −∆, 45

since the curvature terms vanish. We shall now analyze the eigenvalue problem (71) 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 HS . They form a complete orthonormal basis and the scalar eigenfunctions can be written as ξ(S, j)(x) = u(j, x)e0 ⊗e0 . The corresponding solutions are A (S, j)(t, x) = e−iω(S,j)t u(j, x)e0 ⊗ e0 .

Similarly we find that eigenfunctions of ∆ in HT S can be written as ξ(T S, j)(x) = √1 u(j, x)gik ei ⊗ ek and therefore 3 A (T S, j)(t, x) = e−iω(S,j)t u(j, x)ˆ gik √13 ei ⊗ ek . Now we come to the Laplace equation on HV . Since we assummed Σ to be simply . connected, the Hodge decomposition gives HV = dΣ C∞ (Σ) ⊕ δΣ Λ2 (Σ) = HV L ⊕ HV T , where dΣ , δΣ are the de Rham differential and the codifferential on Σ and Λ2 (Σ) denotes the space of 2-forms. It is easy to calculate that the properly normalized VL and VT modes can be written as: 1 ∇i u(j, x)e−iω(V L,j)t (ei ⊗ e0 + e0 ⊗ ei ) , A (V L, j)(t, x) = √ 2ω(V L, j) 1 A (V T, j)(t, x) = √ vi (j, x)e−iω(V T,j)t (ei ⊗ e0 + e0 ⊗ ei ), ∇i v(j, .)i = 0 . 2 Finally we analyze the eigenvalue problem for traceless symmetric 2 tensors in HT . Using the decomposition theorem of Berger and Ebin [7] we can write HT as: HT = Im(div∗0 ) ⊕ ker(div) ,

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 HT L = Im(div∗0 ) to be the space of traceless longitudinal square . integrable symmetric 2 tensors and HT T = ker(div) the space of traceless transversal tensors. Using the Hodge decomposition for 1-forms we can decompose HT L further into components HT LS and HT LV . The corresponding modes are given by: 1 √ (∇i vk (j, x) + ∇k vi (j, x))e−iω(T LV,j)t ei ⊗ ek , ω(T LV, j) 2 r   1 1 3 ∇i ∇k u(j, x) − u(j, x)ˆ gik e−iω(T LS,j)t ei ⊗ ek , A (T LS, j)(t, x) = 2 ω(T LS, j)2 3

A (T LV, j)(t, x) =

A (T T, j)(t, x) = Hik (j, x)e−iω(T T,j)t ei ⊗ ek , 46

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 [30], so we only recall the results. Again we impose the Dirichlet boundary conditions on the Cauchy surface and we obtain the following families of modes: B(S, j)(t, x) = u(j, x)e−iω(S,j)t e0 , 1 ∇i u(j, x)e−iω(L,j)t ei , B(L, j)(t, x) = ω(L, j) B(T, j)(t, x) = vk (j, x)e−iω(T,j)t ek ,

∇k v(j, .)k = 0 .

These modes form an orthonormal basis for the Hilbert space of static one-forms Hv = L2 (T ∗ Σ) ⊕ L2 (Σ). We can now come to construction of the Hadamard state. Let IA denote the index set consisting of all the values of λ for A -modes (i.e. IA = {S, T S, . . . , T T }). Similarly, IB is the corresponding index set for B-modes. The bisolution for 2-tensors is defined as: s(λ) . X X hA (λ, j), h1 i hA (λ, j), h2 i , ωt (h1 , h2 ) = − 2ω(λ, j) λ j∈J(λ)

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: . X X ωv (f1 , f2 ) = −

λ j∈J(λ)

s(λ) hB(λ, j), f1 i hB(λ, j), f2 i , 2ω(λ, j)

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 ωt , ωv 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 Kt , Kv equipped with the indefinite product h., .i discussed above. Next we build a Fock space K which is a tensor product of the bosonic Fock space Fb of Kt and the fermionic Fock space Ff of Kv . Quantum fields are represented in the following way:   1 . X X p A (λ, j)(x)a+ π0 (h(x)) = λ,j + h.c. , 2ω(λ, j) λ∈IA j∈J(λ)   1 . X X p + B(λ, j)(x)c+ B(λ, j)(x)c π0 (C(x)) = λ,j,2 , λ,j,1 2ω(λ, j) λ∈IB j∈J(λ)   1 . X X ¯ p + B(λ, j)(x)c+ π0 (C(x)) = B(λ, j)(x)c λ,j,1 , λ,j,2 2ω(λ, j) λ∈I j∈J(λ) B

+ creation operators of Fb and c+ where a+ λ,j,1 , cλ,j,2 of Ff . Additionally λ,j are the standard . we define π0 (B(x)) = −π0 (div G h(x)). One can check by an explicit calculation that the

47

free BRST charge is represented as: Z X X † ← → π0 (Q0 ) = π0 (Bρ ∇0 C ρ )dvol(Σ,ˆg) = (aλ,j cλ,j,2 + aλ,j c†λ,j,1 ) . λ∈T / T j∈J(λ)

Σ

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 π0 (Q0 ) and so are ones corresponding to C and B excitations. The latter, however, belong to the image of π0 (Q0 ) as well, so the physical Hilbert space (defined as the cohomology of π0 (Q0 ) in ghost number 0) consists only of T T modes and the product h., .i is positive definite on its elements.

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