Hilbert spaces built over metrics of fixed signature

We construct two Hilbert spaces over the set of all metrics of arbitrary but fixed signature, defined on a manifold. Every state in one of the Hilbert spaces is built of an uncountable number of wave functions representing some elementary quantum degrees of freedom, while every state in the other space is built of a countable number of them. Each Hilbert space is unique up to natural isomorphisms and carries a unitary representation of the diffeomorphism group of the underlying manifold. The Hilbert spaces constructed in the case of signature (3, 0) may be possibly used for canonical quantization of the ADM formulation of general relativity.


Introduction
In [1] we constructed a space of quantum states and an algebra of quantum observables over the set of all metrics of arbitrary but fixed signature, defined on a manifold. This space and this algebra were obtained by means of the Kijowski's projective method [2,3,4,5,6]. The motivation for this construction was a desire to find a space of quantum states, which could be used for quantization of the ADM formulation [7] of general relativity (GR), as the "position" part of the ADM phase space is the set of all Riemannian metrics defined on a three-dimensional manifold.
The space of quantum states built in [1] is not a Hilbert space, but a convex set of mixed states. It turns out, however, that a structural component of that space can be used to obtain two distinct Hilbert spaces related to metrics of arbitrary signature.
To outline the construction of these Hilbert spaces, which will be denoted by H and K, let us first describe that structural component. To this end consider a manifold M and fix a metric signature (p, p ′ ) such that p + p ′ = dim M. Given point x ∈ M, denote by Γ x the set of values at x of all metrics of signature (p, p ′ ) defined on M. The structural component is a diffeomorphism invariant assignment where dµ x is a natural measure on Γ x .
This assignment allows us to define for every x ∈ M a Hilbert space H x being the space of all complex functions on Γ x square integrable with respect to the measure dµ x . Each Hilbert space H x thus defined will be treated as an elementary quantum degree of freedom (d.o.f.).
To arrange the Hilbert spaces {H x } x∈M into the Hilbert space H, we will proceed as follows. First, we will associate with every point x the setH x of all half-densities over the tangent space T x M valued in H x -a section of the bundle-like set x∈MH x will be a half-density on M valued in the Hilbert spaces {H x }. Given two such half-densitiesΨ andΨ ′ , we can pair their valuesΨ(x) andΨ ′ (x) using the inner product on H x , obtaining as a result a complex-valued density over T x M. Doing this point by point gives us a scalar density on M, which can be naturally integrated over the manifold. This procedure, that is, the pairing and the integration, defines an inner product on a set of sections of x∈MH x . This set equipped with the product will form a Hilbert space H 1 .
The Hilbert space H 1 alone will not be suitable for quantization of physical theories since it is more like an uncountable orthogonal sum of the Hilbert spaces {H x } and therefore it does not contain any tensor product of them. But these Hilbert spaces represent independent d.o.f. and a physically acceptable Hilbert space should contain tensor products of {H x }. We will include these tensor products as follows.
We will fix a natural number N ≥ 2 and will consider the set M N of all N -element subsets of M. Next we will equip M N with a differential structure and will associate with each element {x 1 , . . . , x N } ∈ M N the Hilbert space H x 1 ⊗ . . . ⊗ H x N . Then, following the construction of the Hilbert space H 1 outlined above, we will build a Hilbert space H N .
Finally, the Hilbert space H will be defined as an orthogonal sum of all the spaces {H N }.
Regarding the Hilbert space K, its construction will be similar to that of H, but simpler: we will consider sections of x∈M H x , which are non-zero merely on countable subsets of M, and will construct a Hilbert space K 1 using these sections and the inner products on {H x }. In an analogous way, we will obtain a Hilbert space K N for N ≥ 2 using the manifold M N and the tensor products {H x 1 ⊗ . . . ⊗ H x N } assigned to points of the manifold. All the spaces {K N } will be then merged into K by means of an orthogonal sum.
Thus both Hilbert spaces H and K, will be constructed of the same elementary quantum d.o.f. being the Hilbert spaces {H x }. A difference between the spaces will be that a state in H will be built of an uncountable number of wave functions belonging to the spaces {H x } (and their finite tensor products), while a state in K will be built of a countable number of them.
The diffeomorphism invariance of the assignment x → dµ x used to build both H and K will allow us to define unitary representations of the diffeomorphism group of M on the Hilbert spaces. Moreover, as shown in [1], the assignment is unique up to a positive multiplicative constant. This will imply that each Hilbert space H and K is unique up to distinguished unitary maps. We will also show that the two Hilbert spaces {H} built over M = R (for signature (1,0) and (0, 1)) are separable and that all the Hilbert spaces {K} are non-separable.
With regard to the original motivation underlying this research, that is, to quantization of the ADM formalism: let us note that in the case of signature (3,0) both Hilbert spaces H and K are constructed over the "position" part of the ADM phase space. Therefore one may try to apply these spaces to canonical quantization of the formalism. More precisely, since the formalism describes a constrained Hamiltonian system, an appropriate method to use here is the Dirac procedure of quantization of such systems. This procedure (see e.g. [8,9]) requires to construct so-called kinematical Hilbert space, which corresponds to the unconstrained phase space of the system, and the spaces H and K could possibly serve as such a space for the ADM formalism.
According to the Dirac procedure, constraints on the phase space are to be taken into account by (i ) defining corresponding constraint operators on the kinematical Hilbert space and (ii ) by finding states, which are annihilated by the operators. The so-called vector constraints on the ADM phase space generate gauge transformations which coincide with the action of spatial diffeomorphisms on the canonical variables. On the other hand, in the case of signature (3,0), the manifold M can be interpreted as a three-dimensional spatial slice of a spacetime and then the unitary representations of the diffeomorphism group of M on H and K, are representations of spatial diffeomorphisms. As it is in loop quantum gravity (see e.g. [10,11]), these representations may be helpful in taking into account the vector constraints at the quantum level.
A quantum model resulting from canonical quantization of the ADM formalism is called quantum geometrodynamics (see e.g. [12] and references therein). To the best of our knowledge, so far no kinematical Hilbert space for quantum geometrodynamics equipped with a (non-trivial) unitary representation of the diffeomorphism group, has been known [12,13,14]. Thus the spaces H and K obtained in the case of signature (3, 0) seem to be the first known spaces of this sort. It has to be however emphasized that although the constructions of H and K appear to be fairly natural, neither space consists of square integrable functions on the set of metrics of signature (3,0). Therefore it is not obvious whether H or K can be actually useful for quantum geometrodynamics-a further research is needed to answer this question. As the first step towards this goal, in the forthcoming paper [15] we will define some operators on H and K related to the ADM canonical variables.
The paper is organized as follows: Section 2 contains preliminaries-there we recall first of all some necessary notions and facts from [1]. In Section 3 we construct the Hilbert space H 1 , and in Section 4 we define a unitary representation of diffeomorphisms of M on H 1 . Then, in Section 5, we build the Hilbert space H, and in Section 6 the Hilbert space K. Section 7 contains a summary and an outlook for future research. In Appendix A we define the differential structure on the set M N , and in Appendices B and C we present proofs of some lemmas.

Manifold of scalar products of fixed signature
Suppose that V is a real vector space of non-zero finite dimension. Let us fix a pair (p, p ′ ) of non-negative integers such that p + p ′ = dim V and denote by Γ the set of all scalar products of signature (p, p ′ ) defined on V . As shown in [1], Γ is a noncompact connected real-analytic manifold of dimension dim V (dim V + 1)/2. If (e i ) i=1,2,...,dim V is a basis of V , then the following map Γ ∋ γ → χ(γ) := γ(e i , e j ) i≤j ∈ R dim Γ (2.1) defines 1 a global coordinate system on Γ. This system will be called here linear coordinate system on Γ and denoted (γ i≤j ), where Γ ∋ γ → γ i≤j (γ) := γ(e i , e j ) ∈ R.
Thus Γ is diffeomorphic to the open subset χ(Γ) of R dim Γ , which since now will be denoted by Γ R : In practice one often expresses a function on a manifold in terms of a coordinate system on the manifold. Here we will often express a function f on Γ in terms of a linear coordinate system i.e. we will use the pull-back χ −1 * f : Γ R → C instead of f . However, using coordinates (γ i≤j ) with the restriction i ≤ j is a bit cumbersome and therefore we would like to express functions on Γ R in terms of all components (γ ij ). Formally this can be achieved in the following way.
Let (t ij ) i,j=1,...,dim V be an element of R (dim V ) 2 . The following map restricted to the set is a bijection onto Γ R . In this paper we will not distinguish between a function f : Γ R → C and the pull-back (S| A ) ⋆ f : A → C, denoting at the same time elements (t ij ) of A by (γ ij ). Moreover, we will usually abuse slightly the notation of elements of Γ R and the notation of a linear coordinate system on Γ by dropping the restriction i ≤ j in (γ i≤j ) and will write simply (γ ij ).

Invariant measure on the homogeneous space of scalar products
The group GL(V ) of all linear automorphisms of V acts naturally on the set Γ via pull-back: The pair (GL(V ), Γ) together with the action (2.4) is a homogeneous space [1] isomorphic is the (pseudo-)orthogonal group, consisting of all those elements of GL(dim V, R), which preserve the matrix diag( Γ is a locally compact Hausdorff (l.c.H.) space being homeomorphic to the open set Γ R ⊂ R dim Γ . Since now, unless stated otherwise, a measure will mean a regular Borel measure on a l.c.H. space (see e.g. [16]). The symbol C c (Y ) will denote here the linear space of all real-valued continuous functions of compact support defined on a l.c.H. space Y . If Y ′ is another such a space, α : Y → Y ′ a homeomorphism and dµ a (regular Borel) measure on Y , then there exists 2 a unique (regular Borel) measure α ⋆ dµ on Y ′ called push-forward measure such that for every h ∈ C c (Y ′ ), where α ⋆ h denotes the pull-back of the function h: given by g ∈ GL(V ) and the action (2.4). We say that a measure dµ on Γ is invariant if for every g ∈ GL(V )ḡ In [1] we showed that on Γ there exists a (non-zero) invariant measure and that it is unique up to a positive multiplicative constant. Γ is second countable (i.e. Γ has a countable base for its topology) being homeomorphic to the open subset Γ R of R dim Γ . On the other hand, each regular measure on second countable l.c.H. space is σ-finite 3 [16], which means that every invariant measure on Γ is σ-finite.
Consider now real vector spaces V 0 , V 1 and V 2 of the same dimension, and suppose that Γ i (i = 0, 1, 2) is the homogeneous space of all scalar products of signature (p, p ′ ) on V i (the signature is fixed and does not depend on i). Every linear isomorphism l ij : V j → V i defines a pull-back l * ij : Γ i → Γ j , being a diffeomorphism between the manifolds.
Lemma 2.1. If dµ 0 is an invariant measure on Γ 0 , then 1. (l * 01 ) ⋆ dµ 0 is an invariant measure on Γ 1 , which is independent of the choice of linear isomorphism l 01 ; 2. for every triplet of linear isomorphisms l 01 , l 02 and l 12 For a proof of these statements see [1]. We have shown in [1] that on every homogeneous space Γ of scalar products of signature (p, p ′ ), there exists a special 4 metric Q called in [1] natural metric on Γ. Let us denote by (Q αβ ) components of the metric in a linear coordinate system (γ ij ) ≡ (γ α ) on Γ given by a map χ : Γ → Γ R (see (2.1)). It turns out that the natural metric looks the same in every linear coordinate system [1]. More precisely, there exist smooth functions such that for every space Γ as above and for every linear coordinate system on Γ the pull-back χ −1⋆ Q αβ = ∆ αβ (note that the components (Q αβ ) are functions on Γ). Moreover, the natural metric Q defines a measure dµ Q on Γ-for every continuous (real or complex) function Ψ of compact support on Γ where dµ L is the Lebesgue measure on R dim Γ ⊃ Γ R and is a positive function on Γ R . Consider two real vector spaces V andV of the same dimension and the corresponding spaces Γ andΓ of scalar products of the same signature (p, p ′ ). Let dµ Q and dµQ be measures on, respectively, Γ andΓ defined by the corresponding natural metrics Q andQ. If l :V → V is a linear isomorphism, then the pull-back l * : Γ →Γ is a diffeomorphism. It was shown in [1] that the push-forward measure then Ψ = 0.
Proof. Suppose that Ψ(γ 0 ) = 0 for some γ 0 ∈ Γ. It follows from continuity of Ψ that there exists a non-negative compactly supported continuous function h on Γ such that h(γ 0 ) > 0 andΨΨ ≥ h. Then -here the second equality holds by virtue of (2.9), and the last inequality does by virtue of properties of the Lebesgue measure dµ L and the fact that the function ∆ is positive everywhere on Γ.
Clearly, the inequality above shows that the only continuous function, which satisfies (2.10), is the constant function of zero value.

Diffeomorphism invariant field of invariant measures
Let M be a smooth connected paracompact manifold. We fix a pair of non-negative integers (p, p ′ ) such that p + p ′ = dim M and denote by Q(M) the set of all (smooth) metrics of signature (p, p ′ ) defined on M. Let us denote by Γ x the space of all scalar products on T x M, of signature (p, p ′ ). Obviously, for every x the pair (GL(T x M), Γ x ) is a homogeneous space. Moreover, as shown in [1], if Q(M) is non-empty, then where q x denotes the value of the metric q at x ∈ M.
An assignment x → dµ x , where dµ x is a measure on Γ x , will be called a field of measures or a measure field on the manifold M.
Let x 0 be any point of M and dµ x 0 an invariant measure on Γ x 0 . In [1] we introduced the following measure field on M: where l x 0 x : T x M → T x 0 M is a linear isomorphism and l * x 0 x : Γ x 0 → Γ x the corresponding pull-back. By virtue of Lemma 2.1, (i ) dµ x is an invariant measure on Γ x , which does not depend on the choice of the map l x 0 x , and (ii ) for every two points x, x ′ ∈ M and for every linear isomorphism l xx ′ : The latter property means that the measure field (2.11) is diffeomorphism invariant since l xx ′ above can be the tangent map θ t defined by any diffeomorphism θ on M, which maps x ′ to x. Thus (2.11) is a diffeomorphism invariant field of invariant measures.
In [1] we showed moreover, that the measure field (2.11) is unique up to a positive multiplicative constant, i.e., for any two measure fields x → dµ x and x → dμ x constructed according to (2.11), there exists a number c > 0 such that for every x ∈ M dμ x = c dµ x . (2.13) Let dµ Qx be the invariant measure on Γ x given by the natural metric on the homogeneous space. It follows from (2.8) that the measure field x → dµ Qx can be obtained via the formula (2.11). Taking into account (2.13), we see that every measure field (2.11) is of the form x → c dµ Qx (2.14) for some (independent of x) positive number c.

Pseudo Hilbert space of half-densities
Let W be a (possibly infinite dimensional) complex vector space, V a finite dimensional real vector space and α a real number. Denote by B the set of all bases of V . If e = (e i ) i=1,...,dim V is a basis of V and Λ = (Λ j i ) i,j=1,...,dim V a non-singular real matrix, then the symbol Λe will represent the basis (Λ j i e j ). An α-density over V valued in W is a mapw : B → W of the following property: for every two bases e and Λe of V ,w where det Λ ≡ det(Λ j i ) = 0. We will denote byW the set of all α-densities over V valued in W . This set possesses a natural complex vector space structure: if z ∈ C andw,w ′ ∈W , then (zw)(e) := zw(e), (w +w ′ )(e) :=w(e) +w ′ (e).
Denote byC the vector space of one-densities over V valued in complex numbers. The complex conjugatew ofw ∈C is an element ofC such that for a basis e of V (if the equality above holds for e, then it does for every basis of V ).
Letw,w ′ ∈C be real-valued. We will say thatw ′ is greater than or equal tow and writew ′ ≥w ifw ′ (e) ≥w ( This map satisfies what follows: where, abusing slightly the notation, we used the symbol 0 to denote both the zero ofC and the zero ofH. Therefore the map (2.17) will be called density product onH.
The spaceH equipped with the density product (2.17) will be called pseudo-Hilbert space of half-densities over V valued in H.

Construction of the Hilbert space H 1
Let us recall that M is a smooth connected paracompact manifold. For the sake of the construction of the Hilbert space H 1 , let us fix a pair (p, p ′ ) of non-negative integers such that p + p ′ = dim M and treat it as a metric signature-all objects used to construct H 1 , which need a metric signature to be chosen, will be given by this (p, p ′ ).
3.1 Hilbert half-densities and scalar densities on M

Definition
Let x → dµ x be a diffeomorphism invariant field of invariant measures given by (2.11). It allows to define a separable [1] Hilbert space for every x ∈ M: (3.1) We will use the symbol ·|· x to represent the inner product on H x . Given a point x ∈ M, letC x stands for the vector space of all one-densities over the tangent space T x M valued in C. Denote byH x the pseudo-Hilbert space of half-densities over T x M valued in H x , and by (·|·) x the density product (2.17) A mapΨ : M →H such thatΨ(x) ∈H x for every x ∈ M, will be called Hilbert halfdensity on M (this name comes from a shortening of the precise but inconvenient term "half-density on M valued in the Hilbert spaces {H x }"). In other words, a Hilbert halfdensity is a section of the bundle-like setH. All such half-densities form a complex vector space with multiplication by complex numbers and addition defined point by point: where z ∈ C, andΨ,Ψ ′ are two Hilbert half-densities. The M-support of a Hilbert half-densityΨ on M is the closure of the following set: LetC := x∈MC x . A mapF : M →C, such thatF (x) ∈C x for every x ∈ M, is a complex scalar density on M. The set of all such densities is a complex vector space with multiplication by complex numbers and addition defined point by point by formulas analogous to (3.3). Complex conjugateF of a densityF is defined naturally point by point: The support of a scalar densityF on M is the closure of

Regularity conditions
As it is said in the introduction to this paper, we are going to define an inner product on a set of sections ofH, that is, on a set of Hilbert half-densities. To this end we will pair the values of two such half-densities point by point using the density products {(·|·) x } x∈M , obtaining thereby a scalar density on M. This density, once integrated over the manifold, will yield a complex number, which by definition will be the value of the inner product of these two half-densities. An important question is how to choose that set of Hilbert half-densities, which together with this inner product, will form the desired Hilbert space. It seems that an obvious answer to this question is that one should choose the set of all half-densities of finite norm with respect to the inner product. However, it has to be proven that this set equipped with the product is indeed a Hilbert space.
Moreover, in the case of signature (3, 0), we would like to use the resulting Hilbert space for quantization of GR and this means in particular that we will try to define on the space some operators, which will represent physical observables. To define such operators it is often very convenient to have a dense linear subspace of sufficiently regular (continuous, smooth, of compact support etc.) wave functions. Thus if we defined the desired Hilbert space using all the half-densities of finite norm, then we would have to find within it a linear subspace of sufficiently regular half-densities and prove that this subspace is dense.
To avoid having to carry out these two proofs, we will choose a linear space of sufficiently regular Hilbert half-densities and will simply define the desired Hilbert space as a completion of the former space in the norm defined by the inner product. The issue of a relation between this Hilbert space and that space given by all the half-densities of finite norm, will be postponed for the future, being not very essential at this moment.
There is, however, a related issue: is this space of sufficiently regular Hilbert halfdensities "large enough" from a physical point of view? Again, we will postpone this question in its generality for the future and will limit ourselves to examining only a simple particular case-the result of this examination (see Section 5.7) will suggest that the answer to this (general) question is in affirmative.
Below we will introduce a notion of continuous Hilbert half-densities (which can be easily modified to a notion of smooth half-densities). Moreover, we will impose on the continuous half-densities an additional regularity condition, which will guarantee that the half-densities, once paired point by point by means of the density products {(·|·) x }, yield continuous densities on M-continuity in combination with a compact support of a density, will ensure that the density is integrable over the manifold.
Let us emphasize finally that the present section is quite technical and it may be skipped on the first reading.
Continuous scalar densities Let us begin by recalling the notion of continuous scalar density, which will be a model for introducing the notion of continuous Hilbert half-density.
Let U be an open subset of M and ϕ : U → R dim M a map defining a coordinate system (x i ) on U . Given a scalar densityF and a point x ∈ U , the valueF (x) is a one-density over T x M valued in C. Since (∂ x k ) is a basis of the tangent space, thenF x, (∂ x k ) is a complex number. Every x ∈ U can be expressed in terms of the coordinate system (x i ), which allows us to define coordinate representation ofF in the system (x i ) as a function (3.4) If (x ′i ) is an other coordinate system of the domain U , then the corresponding coordinate representation f ′ satisfies This property means that the function f is continuous if and only if f ′ is continuous. We will say that the scalar densityF is continuous if for every local coordinate system (x i ) the corresponding coordinate representation (3.4) is continuous.
Coordinate representation of a Hilbert half-density Let U be again an open subset of M and ϕ : U → R dim M a map defining a coordinate system (x i ) on U . Given Hilbert half-densityΨ and a point x ∈ U , the valueΨ(x) is a half-density over T x M valued in H x . ThusΨ(x, (∂ x k )) is an element of H x being an equivalence class 5 of a function (3.6) Now, every x ∈ U can be expressed in terms of the coordinate system (x i ) and the scalar product γ can be expressed in terms of components (γ ij ) given by the basis This allows us to define coordinate representation ofΨ in the system (x i ) as the following function: Note that sinceΨ(x, (∂ x k )) is an equivalence class of functions on Γ x , the coordinate representation above is not unique, even if the system (x i ) is fixed.
To reconstruct the Hilbert half densityΨ on U from its coordinate representative ψ, it is enough to observe that the function is a representative of the equivalence classΨ x, Continuous Hilbert half-densities Definition 3.1. Let (x i ) be a coordinate system on an open set U ⊂ M. We will say that a Hilbert half-densityΨ is continuous on U in the coordinate system (x i ) if for every x ∈ U the representative (3.6) ofΨ(x, (∂ x k )) ∈ H x can be chosen in such a way that the coordinate representation (3.7) is a continuous map.
It follows from Lemma 2.2 that if a Hilbert half-densityΨ is continuous on U in the coordinate system (x i ), then the choice of the representatives (3.6), which give the continuous function (3.7), is unique. In other words, given a coordinate system (x i ), a continuous coordinate representation ofΨ in the system is unique (provided it exists). Proof. Suppose that for every x ∈ U , a representative ofΨ(x, (∂ x j )) ∈ H x is chosen, which results in a coordinate representative ψ ofΨ in the coordinates (x j ).
For every x ∈ UΨ Therefore if γ →Ψ(x, (∂ x j ), γ) is the selected representative ofΨ(x, (∂ x j )), then the function ) we obtain from the formula above where ψ is the coordinate representation ofΨ in the system (x i ) introduced at the very beginning of the proof, and the derivatives ∂x k /∂x ′l and ∂x ′l /∂x k are treated as functions of (x ′i ). We conclude that if ψ is a coordinate representative ofΨ in the coordinates (x i ), then ψ ′ given by (3.9) is a representative ofΨ in the coordinates (x ′i ).
Taking into account that the transition map (x ′i ) → x j (x ′i ) is smooth, we see from (3.9) that the coordinate representation ψ ′ is continuous if and only if the coordinate representation ψ is continuous (the "only if" part of these statement comes from the fact that the dependence of ψ ′ on ψ given by (3.9) can be inverted to a dependence of ψ on ψ ′ of an analogous form).
We will say that a Hilbert half-densityΨ is continuous if for every local coordinate system, there exists a continuous coordinate representation (3.7).
Since now in the case of a continuous Hilbert half-density we will use exclusively its continuous coordinate representations.
Hilbert half-densities of compact and slowly changing Γ R -support Let us consider again the map ϕ : U → R dim M and the corresponding coordinate system (x i ). Definition 3.3. Suppose that a Hilbert half-densityΨ is continuous, and ψ is its coordinate representation in the system (x i ). We will say that the Γ R -support ofΨ around x 0 ∈ U is compact and slowly changing in the coordinate system (x i ), if there exist an open neighborhood U 0 ⊂ U of x 0 , and a compact set K ⊂ Γ R , such that for every value (x i ) ∈ ϕ(U 0 ), the support of the function Let us emphasize that if the support of ψ (x i ) is contained in a compact set, then the support is compact itself (since each closed subset of a compact set is compact). Thus the definition above implies that for every ( and (x ′i ) be coordinate systems on an open set U ⊂ M, and letΨ be a continuous Hilbert half-density. The Γ R -support ofΨ around x 0 ∈ U is compact and slowly changing in the coordinate system (x i ) if and only if the Γ R -support ofΨ around x 0 is compact and slowly changing in the coordinate system (x ′i ).
Suppose that the Γ R -support ofΨ around x 0 ∈ U is compact and slowly changing in the coordinate system (x i ). Let U 0 and K be the sets introduced in Definition 3.3 for the system (x i ). Choose a compact set U 1 ⊂ U 0 of non-empty interior Int U 1 such that is continuous (here we treat the derivative ∂x i /∂x ′k as a function of (x i )). Therefore the set is compact. Consider now the following function where ψ ′ (x ′j , γ ′ kl ) represents the half-densityΨ in the coordinate system (x ′j ) (see (3.7)). Let us fix a value (x ′j ) ∈ ϕ ′ (U ) of the coordinates and the corresponding value x i (x ′j ) ∈ ϕ(U ). By virtue of Equation (3.9), (γ ′ kl ) belongs to the support of ψ ′ (x ′j ) if and only if is an element of K ′ by definition of the latter set. We thus see that for every value (x ′j ) ∈ ϕ ′ (Int U 1 ) the support of ψ ′ (x ′j ) is contained in the compact set K ′ . Therefore, if the Γ R -support ofΨ around x 0 is compact and slowly changing in the coordinate system (x i ), then it is the same in the coordinate system (x ′i ). But these coordinate systems are arbitrary and can be swapped in the previous statement. Thus the lemma follows.
Since now we will say that the Γ R -support around x 0 ∈ M of a continuous Hilbert half-densityΨ, is compact and slowly changing if it is compact and slowly changing in every coordinate system defined on a neighborhood of x 0 . Finally, we will say that the Γ R -support ofΨ is compact and slowly changing if the Γ R -support is such around every point of M.
Lemma 3.5. IfΨ andΨ ′ are continuous Hilbert half-densities of compact and slowly changing Γ R -support, then any (finite) linear combination of them is a continuous Hilbert half-density of compact and slowly changing Γ R -support.
Proof. Suppose that U is an open subset of M and that a map ϕ : Consider now a linear combination and suppose that ψ and ψ ′ are (continuous) coordinate representations of, respectively,Ψ andΨ ′ in the system (x i ). Then, for every x ∈ U , the function substituted to the reconstruction formula (3.8) gives us an appropriate linear combination of functions on Γ x representing the equivalence class zΨ x, ξ is thus a continuous coordinate representation ofΞ in the system (x i ). Since U is an arbitrary open subset of M, thenΞ is continuous. Fix an arbitrary point x 0 ∈ U . Let K be a compact subset of Γ R and U 0 ⊂ U be an open neighborhood of x 0 such that for every value (x i ) ∈ ϕ(U 0 ), the support of the function ψ (x i ) related toΨ via the formulas (3.10) and (3.7), is contained in K. In the same way, let K ′ be a compact subset of Γ R and U ′ 0 ⊂ U be an open neighborhood of x 0 such that for every value (x i ) ∈ ϕ(U ′ 0 ), the support of the function ψ ′ (x i ) related toΨ ′ via the formulas (3.10) and (3.7), is contained in K ′ .
Obviously, if ξ (x i ) is related via (3.10) to ξ, then Therefore for every value ( 0 is open and K ∪ K ′ compact, the Γ R -support ofΞ around x 0 is compact and slowly changing in the coordinate system (x i ). But x 0 is an arbitrary point in M, and (x i ) an arbitrary local coordinate system. Therefore the Γ R -support ofΞ is compact and slowly changing.

Pairing of Hilbert half-densities into scalar densities
Consider now two Hilbert half-densitiesΨ andΨ ′ on M. Then the map is a scalar density on M. The lemma below ensures that if Hilbert half-densities satisfy the regularity conditions introduced in the previous section, then the resulting density is sufficiently regular for our purposes.
Lemma 3.6. Suppose that Hilbert half-densitiesΨ andΨ ′ are continuous and that the Γ Rsupport ofΨ is compact and slowly changing. Then the scalar density (Ψ ′ |Ψ) is continuous.
Proof. Let us consider the scalar density (Ψ ′ |Ψ) ≡F and a map ϕ : where ·|· x is the inner product on the Hilbert space H x . Consequently, (3.14) where in the second step we used the fact that every measure field (2.11) is of the form (2.14).
For further transformation ofF x, (∂ x i ) we would like to use Equation (2.7). In order to do this we have to show that (for fixed x) the integrand in (3.14) is a continuous function of compact support. To this end let us note that if χ : where ψ ′ (x i ) and ψ (x i ) are continuous functions related to, respectively,Ψ ′ andΨ via the formulas (3.10) and (3.7). Thus the integrand in (3.14) is continuous.
Let us fix a point x 0 ∈ U and suppose that U 0 ⊂ U is an open neighborhood of x 0 introduced in Definition 3.3 for the half-densityΨ. Then it follows from the assumptions imposed onΨ that for every (x i ) ∈ ϕ(U 0 ), the support of ψ ′ (x i ) ψ (x i ) is contained in a compact set K ⊂ Γ R and thereby the support is compact as well. This together with (3.15) mean that, indeed, the integrand in (3.14) is of compact support.
Since the integrand in (3.14) is a continuous compactly supported function on Γ x , we can use (2.7) to get Using (3.15) once again we obtain for every value (x i ) ∈ ϕ(U 0 ). Now, it follows from assumed continuity ofΨ ′ andΨ that the function is continuous (note that the function ∆ is continuous and independent of (x i ), which follows from the properties of the natural metrics described in Section 2.2). Therefore the function (3.17) becomes a bounded one once restricted to a compact set ϕ( and Since the set K is compact and the Lebesgue measure dµ L is regular [16], the function h is integrable over Γ R with respect to dµ L . Moreover, for every value ( . These three facts allows us to apply the Lebesgue's dominated convergence theorem to conclude that f given by (3.16), is continuous at ( . But x 0 is an arbitrary point, and (x i ) an arbitrary local coordinate system. Thus (Ψ ′ |Ψ) is a continuous scalar density on M.

The Hilbert space H 1
Let H c 1 be the set of all continuous Hilbert half-densities on M of compact M-support and of compact and slowly changing Γ R -support. Any (finite) linear combination of elements of H c 1 is again a Hilbert half-density of compact M-support. This fact and Lemma 3.5 guarantee that H c 1 is a complex vector space. By virtue of Lemma 3.6 for any two elementsΨ andΨ ′ of H c 1 , the scalar density (Ψ ′ |Ψ) on M is continuous. This density is also compactly supported and therefore it can be naturally integrated 6 over M being a paracompact manifold. The following map where the integral at the r.h.s. is the integral of the scalar density (Ψ ′ |Ψ), is an inner product on H c 1 . Indeed, it is clear that the map (3.18) is linear in the second argument and that it satisfies the Hermitian (or conjugate) symmetry condition. For everyΨ ∈ H c 1 , the scalar density (Ψ|Ψ) is continuous and non-negative i.e., for every x ∈ M, (Ψ|Ψ)(x) ≥ 0 (see the formula (2.16)). Therefore Ψ |Ψ ≥ 0. Suppose thatΨ(x) = 0 for a point x ∈ M. Then, by continuity of (Ψ|Ψ) its support contains a non-empty open set. Consequently, Ψ |Ψ > 0 and (3.18) is positive definite. 6 Let us emphasize that the assumption that the Γ R -support of each element of H c 1 is slowly changing, is essential here-without it one can get a non-integrable scalar density as seen in the following example. In the case M = R and the signature (1, 0) the set Γ R is the set R+ of all positive real numbers. Let x 1 be the canonical coordinate on M = R and A : Thus the Hilbert half-densityΨ on M obtained from ψ by means of the reconstruction formula (3.8), is continuous and of compact Γ R -support everywhere on M (and of compact M-support), but the Γ R -support is not slowly changing around x 1 = 0. The measure ∆dµL on Γ R is here of the form (γ11) −1 dγ11 [1]. Using this fact and (3.16) it is not difficult to realize that the coordinate representation of (Ψ|Ψ) in the coordinate x 1 , diverges to infinity as (x 1 ) −1 , as x 1 goes to zero from the right. This means that the density (Ψ|Ψ) is not continuous at the point x 1 = 0 and is also non-integrable over M.
The completion H 1 of H c 1 in the norm induced by the inner product (3.18), is a Hilbert space built over the set Q(M).
Let us end this section by a remark concerning the standard Hilbert space H QM of the ordinary quantum mechanics. This space is usually defined as L 2 (R 3 , dµ L ), where dµ L is the Lebesgue measure. Note, however, that wave functions in H QM can be viewed as Hilbert half-densities of special sort defined on R 3 . Indeed, the set C of complex numbers equipped with the map , is the set of all halfdensities over T x R 3 valued in the Hilbert space C, then a section of x∈R 3C 1/2 x is a sort of a Hilbert half-density. It is clear that each wave function in H QM , if expressed in a Cartesian coordinate system on R 3 , can be understood as a coordinate representation of such a half-density. It is also not difficult to convince oneself that the inner product of two wave functions in H QM , can be expressed in terms of (i ) pairing of corresponding Hilbert half-densities to a scalar density and (ii ) integrating of the resulting scalar density over R 3 . Thus, counterintuitively, the Hilbert spaces H 1 and H QM are fairly similar.

Uniqueness of H 1
Given manifold M and signature (p, p ′ ), the only choice we have to make in order to obtain the Hilbert space H 1 , is the choice of a diffeomorphism invariant field (2.11) of invariant measures. However, since all such fields are unique up to a positive multiplicative constant (see Equation (2.13)), the freedom to choose the measure field is actually not relevant.
Indeed, if x → dµ x and x → dμ x are two such measure fields on M, and H 1 andȞ 1 the resulting Hilbert spaces, then it follows from (2.13) that is a unitary map. Thus all Hilbert spaces constructed according to the prescription presented in Section 3 are isomorphic. Moreover, there exists a distinguished or natural isomorphism (3.19) between each pair of such Hilbert spaces.
We conclude then that the Hilbert space H 1 is unique up to natural isomorphisms.
4 Action of diffeomorphisms on the Hilbert space H 1

Action of diffeomorphisms on scalar densities
LetF be a scalar density on M, and θ : M → M a diffeomorphism. The diffeomorphism θ acts on the densityF by means of the following pull-back: where is a basis of T θ(x) M.
The pull-back θ * F is again a scalar density on M. To see this let us calculate where Λ ≡ (Λ j i ) is any non-singular matrix. If e x = (e xi ), then Λe x = (Λ j i e xj ). Hence by virtue of linearity of θ t Consequently, LetF be a scalar density integrable over M. Then for every diffeomorphism θ of M (4.3)

Action of diffeomorphisms on H c 1
Let us consider a Hilbert half-densityΨ being an element of H c is an element of the Hilbert space H θ(x) , that is, an equivalence class of functions on Γ θ(x) . SinceΨ ∈ H c 1 , the equivalence class (4.4) can be represented by a unique continuous compactly supported function on Γ θ(x) (see Lemma 2.2).
To pull-back the equivalence class (4.4) of functions on Γ θ(x) to an equivalence class of functions on Γ x being an element of H x , we will proceed as follows. First we will pullback by means of (θ −1 ) t * the unique continuous representative of (4.4), obtaining thereby a continuous compactly supported function on Γ x . Then we will find an element of H x defined by this resulting function, denote it by 8 and treat it as the desired pull-back 9 of (4.4). Using (4.2) and linearity of the pull-back (θ −1 ) t * ⋆ one easily shows that which means that the following map defined on the set of all bases of T x M and valued in H x is a half-density.
We thus see that on the manifold M there exists a Hilbert half-density θ * Ψ , such that for every x ∈ M and for every basis e x of T x M, (4.6) We will say that θ * Ψ is pull-back ofΨ under the diffeomorphisms θ. Proof. We have to show that θ * Ψ is (i ) of compact M-support, (ii ) continuous and (iii ) of compact and slowly changing Γ R -support. Let us recall that defining θ * Ψ we assumed thatΨ ∈ H c 1 .
Now let us find a relation between the (continuous) coordinate representation ψ ofΨ in the system (x i ) (see (3.7)), and a coordinate representation Let e x be a basis of T x M. It follows from (4.6) and the procedure, which defines (4.5), that (θ * Ψ )(x, e x ) ∈ H x contains a continuous representative such that for every γ x ∈ Γ x its value reads where the number at the r.h.s. is a value of the continuous representative ofΨ(θ(x), θ t e x ).
Using all these results we can transform (4.7) obtaining This equality together with (3.7) mean that on Note that ψ at the r.h.s. of the equation above, is the continuous coordinate representation ofΨ in the system (x i )-to be continuous, for every value (x i ) ∈ ϕ(U ) the representation must come from the unique continuous representative ofΨ(ϕ −1 (x i ), (∂ x i )) and, indeed, ψ on ϕ(U ) × Γ R does come from these representatives (see the remark just below Equation (4.7)). Now an immediate implication of Equation (4.8) is that the coordinate representation θ * ψ of θ * Ψ in the system (x i ) is continuous-its continuity follows obviously from continuity of ψ. This is sufficient for θ * Ψ to be continuous, since (x i ) is an arbitrary local coordinate system on M.
Regarding (iii ): let us consider now a point θ(x 0 ) ∈ U . Let U 0 ⊂ U be an open neighborhood of θ(x 0 ) and K a compact subset of Γ R such that for every value (x i ) ∈ ϕ(U 0 ), the support of the function ψ (x i ) related to ψ via the formula (3.10), is contained in K-the existence of U 0 and K follows from the assumption, thatΨ is an element of H c 1 . By virtue of (4.8), for every value (x i ) ∈ (ϕ • θ)(θ −1 (U 0 )), the support of the function (θ * ψ) (x i ) related to θ * ψ via the formula (3.10), is contained in K. This means that the Γ Rsupport of the pull-back θ * Ψ around x 0 is compact and slowly changing in the coordinate system (x i ). But x 0 is an arbitrary point of M and (x i ) an arbitrary local coordinate system on the manifold. Therefore θ * Ψ is a Hilbert half-density on M of compact and slowly changing Γ R -support.

Unitary representation of diffeomorphisms on H 1
Lemma 4.2. Suppose thatΨ,Ψ ′ ∈ H c 1 and θ is a diffeomorphism of M. Then the pull-back θ * preserves the inner product (3.18) on H c 1 : Proof. Consider first the scalar density (θ * Ψ′ |θ * Ψ ) (see (3.13) for the definition). If e x is a basis of T x M, then -here we used in turn: in the first step the definition (2.17), in the second step Equation (4.6), in the third step we chosen the continuous (compactly supported) representatives of Ψ ′ (θ(x), θ t (e x )) andΨ(θ(x), θ t (e x )), in the forth step we applied (2.5), finally in the last step we used the diffeomorphism invariance of the measure field x → dµ x (see Equation (2.12)). Thus Comparing this with (4.1) we see that (θ * Ψ′ |θ * Ψ ) = θ * (Ψ ′ |Ψ).

By virtue of this result and Equation (4.3)
Let us define an operator on H c 1 : It follows from (4.7) that u θ is linear. Manifestly, for everyΨ ∈ H c which means that u θ is surjective. Lemma 4.2 guarantees that u θ preserves the inner product on H c 1 being a dense linear subspace of H 1 . Taking into account all these facts, we see that the operator u θ can be uniquely extended to a unitary operator U 1 (θ) on H 1 .
It is not difficult to check that for two diffeomorphisms θ 1 and θ 2 , is a unitary representation of the group Diff(M) of all diffeomorphisms of M on the Hilbert space H 1 .

Motivation
Suppose that the set Q(M) is non-empty. Then the following surjective [1] map where q x is the value of the metric q at x ∈ M, can be treated as a degree of freedom (d.o.f.) on Q(M). Consequently, the Hilbert space H x given by (3.1) can be treated as a quantum counterpart of κ x . Evidently, for each pair x = x ′ , κ x and κ x ′ are independent d.o.f.. It seems therefore, that a Hilbert space being a quantum counterpart of the configuration space Q(M) should contain tensor products of the Hilbert spaces {H x }.
It is easy to realize that the structure of H 1 does not meet this expectation. Indeed, the inner product (3.18) is an integral, that is, an "uncountable sum" of values of inner products on the Hilbert spaces {H x }. Therefore H 1 is more like a direct integral of Hilbert spaces [18]: This fact suggests that H 1 (in the case of signature (3, 0)) is not well suited for quantization of the ADM formalism. Fortunately, it is relatively easy to find a way around this problem.
Namely, consider the set M N of all N -element subsets of M: It is possible to define on M N a differential structure in such a way that the set becomes a smooth paracompact manifold locally diffeomorphic to M N -for details see Appendix A.1. We will associate with each point y = {x K } of M N a Hilbert space H ⊗ y naturally isomorphic to H x 1 ⊗ . . . ⊗ H x N (for every ordering of the factors in the tensor product), Note also that the structure of this H will resemble to a certain degree the structure of Fock spaces. Obviously, the Hilbert space H will contain all finite tensor products of the Hilbert spaces {H x }. Therefore the space H constructed in the case of signature (3, 0), seems to be better suited for quantization of the ADM formalism than H 1 alone.

Remarks on the definition (5.4) of H
A rigorous construction of H N , we are going to present below, is fairly long and technically involved (this concerns in particular the construction of the smooth atlas on the set M N ). Let us note that an other method to take into account the tensor products of {H x }, is to use in (5.4) the tensor product N H 1 of N copies of H 1 , instead of H N . This may seem to be the simplest way to achieve the goal, which does not require any extra effort, but a closer look at this construction makes it clear that it is not the case.
Namely, it is easy to realize that, given {x K } ∈ M N , one can find N ! tensor products of Hilbert spaces {H x 1 , . . . , H x N } in the space N H 1 -these tensor products differ from each other by the ordering of the factors and are given by all possible orderings. From a physical point of view the ordering of the factors in H x 1 ⊗ . . . ⊗ H x N is irrelevant-each such tensor product describes a space of quantum states of the same quantum system, obtained by quantization of a classical system, whose configuration space is given by the collection {κ x 1 , . . . , κ x N } of independent classical d.o.f..
Consequently, a generic state in N H 1 seems to correspond to N ! distinct states in H N . It can be then expected that working with such generic states would be rather cumbersome. To avoid this, one would have to impose on elements of N H 1 some restrictions in order to isolate those states, which would correspond to single states in H N .
Moreover, for many purposes it would be convenient or even necessary to describe elements of N H 1 as fields on M N (preferably, as half-densities on M N valued in the tensor products of {H x }) and to impose some regularity conditions on these fields. Therefore, in order to put the space N H 1 into practice, one would have to elaborate such a description. Thus the application of N H 1 does require some additional effort, which makes this space not as attractive as it seemed to be at the very beginning.
In our opinion the advantage of H N over N H 1 is conceptual simplicity of H N : the idea of its construction is simple and natural, and complies with the fact that the ordering of factors in H x 1 ⊗ . . . ⊗ H x N is physically irrelevant. Consequently, each state in H N fits well our need to take into account the tensor products of {H x }, without the necessity to impose on it any extra conditions. Moreover, once the space H N is rigorously built, one can work with it without the need to refer to many technical details of its construction like e.g. the construction of the smooth atlas on M N . For convenience of the reader not interested in such details, we will place a considerable part of these technicalities in the appendix to this paper.

Construction of the Hilbert space H N
The construction of the Hilbert space H N will follow as closely as possible the construction of the Hilbert space H 1 described in Section 3.
Let (p, p ′ ) be the signature of metrics on M fixed at the very beginning of Section 3 for the sake of the construction of the Hilbert space H 1 . Let us fix additionally a natural number N ≥ 2.

Preliminaries
Suppose that V ⊕ is a real vector space of dimension N (p + p ′ ) and that a decomposition is given, such that each V I is a linear subspace of V ⊕ of dimension p + p ′ . Denote by Γ I the homogeneous space of all scalar products on V I of signature (p, p ′ ). For each I ∈ {1, . . . , N } let us choose γ I ∈ Γ I and define where Clearly, γ ⊕ is a scalar product on V ⊕ of signature (N p, N p ′ ). We will use the symbol Γ ⊕ to represent the set of all scalar products on V ⊕ of the form (5.6) (with the fixed decomposition (5.5)). Let us emphasize that Γ ⊕ is a proper subset of the set of all scalar products on V ⊕ of signature (N p, N p ′ ).
Note that one can assign to γ ⊕ a sequence of the scalar products used to define γ ⊕ via (5.6). Obviously, this assignment, is a bijection, which can be used to induce some structures on Γ ⊕ . First, the bijection together with charts {(Γ I , χ I )} given by (2.1), allow us to construct a map which define a global coordinate system on Γ ⊕ . This map can be used to "pull-back" the topology from R N dim Γ I onto Γ ⊕ . Obviously, coordinate systems given by all maps (5.8) form an analytic atlas on Γ ⊕ . Suppose that dµ I is an invariant measure on Γ I . Since dµ I is σ-finite (see Section 2.2), the product This product is also a regular Borel measure on the Cartesian product Γ 1 × . . . × Γ N -this is because [16] each dµ I is a regular Borel measure on a second countable l.c.H. space Γ I . We can use the bijection (5.7) to push-forward this measure obtaining thereby a (regular Borel) measure on Γ ⊕ , which will be denoted by dµ × : Recall that each L 2 (Γ I , dµ I ) is separable [1] and each measure dµ I is σ-finite. These properties of the space and the measure guarantee [19] that Let us emphasize that the Cartesian product Γ 1 × . . . × Γ N , unlike the construction of Γ ⊕ , does require the spaces {Γ I } to be ordered. However, neither the topology nor the differential structure nor the measure dµ × induced on Γ ⊕ depends on the ordering. Thus the Hilbert space H ⊗ is also independent of the ordering.
Finally, let us state a fact, which concerns a push-forward of a product of measures: Lemma 5.1. Let X, Y , X ′ and Y ′ be second countable l.c.H. spaces and let α : X ′ → X, and β : Y ′ → Y be homeomorphisms. If dµ and dν are regular Borel measures on, respectively, X ′ and Y ′ , then This lemma may seem to be obvious, but a strict proof of it, done in line with the definition (2.5) of push-forward measure, requires some effort. Therefore we relegate the proof to Appendix B.
The lemma above can be easily generalized to a product of any finite number of suitable measures since [16] (i ) a product X × Y of second countable l.c.H. spaces X and Y is such a space again and (ii ) if dµ and dν are regular Borel measures on, respectively, X and Y , then dµ × dν is such a measure on X × Y .
Let us now apply Lemma 5.1 to express an integral over Γ ⊕ with respect to dµ × in terms of an integral over Γ N R . Recall first that each Γ I in (5.7) and Γ R are second countable l.c.H. spaces. Each invariant measure dµ I on Γ I used to define dµ × is regular and Borel. Moreover, dµ I = c I dµ Q I , where c I > 0 and dµ Q I is the invariant measure on Γ I given by the natural metric on this space (see (2.9)). On the other hand, we can treat the r.h.s. of (2.7) as the definition of a positive functional on C c (Γ R ), which (by virtue of the Riesz representation theorem) allows us to regard ∆ dµ L as a regular Borel measure on Γ R . Taking into account all these facts, Equation (2.7) and Lemma 5.1, it is straightforward to make the following transformations: is given by the maps appearing in (5.8), and Ś ∆ dµ L is the product of N copies of ∆ dµ L . Equation (5.11) can be now used to prove the following generalization of Lemma 2.2: then Ψ = 0.

Hilbert half-densities on M N
Here we will apply the construction of the Hilbert space H ⊗ just presented to associate the Hilbert space H ⊗ y with every point y ≡ {x K } of the manifold M N (let us recall that the set M N is defined by the formula (5.2), and the smooth atlas on M N is introduced in Appendix A.1).
As shown in Appendix A.3, for every y ∈ M N , there exists a distinguished decomposition of the tangent space T y M N into a direct sum of (linear subspaces naturally isomorphic to) the tangent spaces {T x I M} x I ∈ y : Suppose that {γ x I } x I ∈ y is a collection of scalar products of signature (p, p ′ ) on, respectively, {T x I M}-in other words, each γ x I ∈ Γ x I . This collection together with the decomposition (5.12) define a scalar product γ ⊕ y on T y M N of signature (N p, N p ′ ) according to the prescription (5.6). We will denote by Γ ⊕ y the set of all such scalar products on T y M N .
Let x → dµ x be the diffeomorphism invariant field (2.11) of invariant measures on M, used in Section 3.1.1 to construct the Hilbert space H 1 . Denote by dµ × y the measure on Γ ⊕ y defined as the push-forward of the measure given by the inverse of the natural bijection (see (5.7)) This allows us to associate with the point y the following Hilbert space (see (5.9)): Let us emphasize that the measure dµ × y does not depend on the choice of the ordering of the spaces {Γ x K } in (5.13). Consequently, the Hilbert space H ⊗ y does not distinguish any ordering of the Hilbert spaces {H x K } in (5.14).
Definition LetH ⊗ y denote the pseudo-Hilbert space of all half-densities over T y M N valued in H ⊗ y . We will use the symbol (·|·) y to represent the density product onH ⊗ y . A mapΨ from M N toH such thatΨ(y) ∈H ⊗ y , will be called Hilbert half-density on M N . Equivalently, one can think ofΨ as of a section of the bundle-like setH ⊗ .
Regularity conditions Again, we would like to impose on the Hilbert half-densities just defined, some regularity conditions, which (i ) would be helpful while defining physical operators on H N and (ii ) will ensure that the half-densities paired by means of the density products {(·|·) y } y∈M N , give integrable scalar densities on M N .
The regularity conditions, we are going to introduce here, will be analogous to those presented in Section 3.1.2. However, some differences will be unavoidable. The reason is that each space Γ ⊕ y does not consist of all scalar products on T y M N of signature (N p, N p ′ ), but contains only some special ones. Therefore, introducing and working with these new regularity conditions, we will restrict ourselves to some coordinate systems on M N , which, in a sense, are compatible with decompositions (5.12) and, thereby, with the special form of the elements of Γ ⊕ y . We showed in Appendix A.1 that for every y ∈ M N , there exist local charts {(U K , ϕ K )}  11)). All charts of this sort constitute a smooth atlas on the manifold denoted in the appendix by A. Here we will extend this atlas by admitting all charts obtained by restricting domains of charts in A. The extended atlas will be denoted by A ′ .
Let Σ be the set of all permutation of the sequence (1, . . . , N ). Every chart in A ′ defines a local coordinate system 16) on M N , which is compatible with the decomposition (5.12) in the following sense: there exists σ ∈ Σ such that the tangent vectors (∂ x i I ) (with fixed I) form a basis of T x σ(I) M in (5.12) (see Appendix A.3 for a justification of this claim). Then each element of Γ ⊕ y , is of the sort of the map (5.8).
The description (5.17) of γ ⊕ y in terms of the components (γ xσ(K) ) i K j K is more explicit than that in terms of (γ ⊕ y ) ab . However, the symbols (γ xσ(K) ) i K j K are fairly complex and thereby somewhat unreadable. Therefore we would like to use the components (γ ⊕ y ) ab instead of them. To this end we will neglect each zero component (γ ⊕ y ) ab for which a and b refer to coordinates, respectively, x i I I and x j J J with I = J. This will allow us to treat the set (γ ⊕ y ) ab (being in fact an element of R (N dim M) 2 ) as an element of Γ N R and identify the two sets of components under consideration: Let us now introduce a coordinate representation of a Hilbert half-densityΨ on M N . To this end consider a chart (Z, Φ) ∈ A ′ and the corresponding coordinate system (5.16). Given y ∈ Z,Ψ(y) is a half-density over T y M N valued in H ⊗ y , which means that the value ofΨ(y) on the basis (∂ ThusΨ y, (∂ x a ) is an equivalence class of a function Expressing y by means of the coordinates and γ ⊕ as in (5.17) we obtain a coordinate representation ofΨ being the map Now we can introduce the notion of continuous Hilbert half-densities on M N of compact and slowly changing Γ N R -support exactly as we did in Section 3.1.2 in the case of Hilbert half-densities on M with only three exceptions: 1. the scalar product components (γ ⊕ ab ) in (5.18) do not describe arbitrary scalar products on T y M N , y = Φ −1 (x a ) ∈ Z, of signature (N p, N p ′ ), but exclusively those in Γ ⊕ y . Therefore the components are restricted to be elements of Γ N R .
2. we do not allow ourselves to use arbitrary local coordinate systems on M N , but only those given by charts in A ′ .

Lemma 5.2 should be used instead of Lemma 2.2.
This means in particular that (i ) given an admissible coordinate system (x a ), a continuous coordinate representation of every Hilbert half-density on M N in the system is unique (provided it exists) and (ii ) appropriate counterparts of Lemmas 3.2, 3.4 and 3.5 can be proven in the same way without any essential changes.

Pairing of Hilbert half-densities into scalar densities
Suppose thatΨ andΨ ′ are Hilbert half-densities on M N . Clearly, the map is a scalar density on M N (C here is defined analogously toC in Section 3.1.1).
As before, if we assume that both half-densitiesΨ andΨ ′ are continuous and that the Γ N R -support of one of them is compact and slowly changing, then the scalar density (5.19) is continuous. This fact can be proven analogously 10 to the proof of Lemma 3.6. The only essential difference is a bit more complicated passage from the counterpart of Equation (3.14) to the counterpart of Equation (3.16), where now Equation (5.11) should be used.

The Hilbert space H N
Let H c N be the vector space of all continuous Hilbert half-densities on M N of compact M N -support and of compact and slowly changing Γ N R -support. For any two elementsΨ andΨ ′ of H c N , the scalar density (Ψ ′ |Ψ) on M N is continuous and of compact support and therefore the density can be naturally integrated over this paracompact manifold. The following map 20) where the integral at the r.h.s. is the integral of the scalar density (Ψ ′ |Ψ), is an inner product on H c N . By definition, the Hilbert space H N is the completion of H c N in the norm induced by the inner product (5.20).

Uniqueness of H N
Let us recall that to construct the Hilbert spaces H 1 and H N we used the same diffeomorphism invariant field x → dµ x of invariant measures. If for this purpose we used an other such field x → dμ x instead, then we would obtain an other Hilbert spaceȞ N . It is not difficult to realize (see Equation (2.13)) that there exists a positive number c such that is a unitary map. Thus we conclude that the Hilbert space H N is unique up to natural isomorphisms (5.21). 10 The regularity conditions imposed on elements of H c N are expressed in terms of coordinate systems defined by charts in A ′ . Therefore the result of this analogous proof will be a conclusion that a coordinate representation of the density (5.19) in every coordinate system of this special sort, is continuous. But this is sufficient to claim that all coordinate representations of the density are continuous (see Equation (3.5)) and thereby the density is continuous. To this latter end recall that if y = {x K } ∈ M N , then the measure dµ × y is given by the push-forward of the measure Ś dµ x K under the bijection b −1 y (see (5.13)). Let

Action of diffeomorphisms of
, which means that y = Θ(y ′ ). As shown in Appendix A.4, the bijections b y and b y ′ intertwine the pull-back Θ t * : Γ ⊕ y → Γ ⊕ y ′ and the pull-back Ś θ t * : A.29) for definition of Ś θ t * and Equation (A.30) for the relation between the bijections and the pull-backs). Consequently, -here in the third step we used Lemma 5.1, and in the forth step the diffeomorphism invariance of the measure field x → dµ x on M (see Equation (2.12)). We thus conclude that, indeed, the measure field (5.23) is invariant with respect to the action of elements of Diff M (M N ).
As in the case of the Hilbert space H 1 , the pull-back (5.22) given by a diffeomorphism Θ ∈ Diff M (M N ) corresponding to θ ∈ Diff(M), can be unambiguously extended to a unitary operator on the Hilbert space H N . It is convenient to denote this operator by is a unitary representation of Diff(M) on H N .

Uniqueness of H
We are then allowed to state that the Hilbert space H is unique up to natural (or distinguished) isomorphisms.

Unitary representation of Diff(M) on H
The unitary representations (4.9) and (5.24) of Diff(M) on, respectively, H 1 and H N , N ≥ 2, can be used to define the following unitary representation of the diffeomorphism group on the Hilbert space H: where θ ∈ Diff(M).

Hilbert spaces {H} built over M = R
In this section we will consider the Hilbert space H built in the case M = R for signature either (1, 0) or (0, 1) and will show that this space is separable (regardless of signature). To this end we will show first that for every N ≥ 1 the Hilbert space H N constructed over M = R is separable.
In Appendix A.2 we considered the manifold M N , N ≥ 2, constituted of points of M = R and constructed a bijection ι : is an open subset of R N . We have further demonstrated that ι defines a global coordinate system on M N . Setting M ≡ M 1 , R 1 > ≡ R and ι ≡ id on M 1 ≡ R 1 > will allow us to treat the case N = 1 together with all the cases N ≥ 2 in the considerations below.
Let us then fix N ≥ 1 and a Hilbert half-densityΨ ∈ H c N . As shown in Appendix A.2, for every y ∈ M N there exists its open neighborhood Z such that the chart (Z, ι| Z ) belongs to the atlas A on M N and thereby to the atlas A ′ . This together with continuity ofΨ mean that if a coordinate system is given by (Z, ι| Z ), then there exists a continuous coordinate representation ofΨ in this system. Moreover, this continuous representation is unique (see the last sentence of Section 5.3.2). This uniqueness allows us to merge all such continuous coordinate representations ofΨ into one continuous representation ofΨ in the global coordinate system (x a ) defined by the map ι (see (3.7) and (5.18)) 11 . Furthermore,Ψ under consideration is of compact and slowly changing Γ N R -support. Using this fact we can conclude in an analogous way that for every y ∈ M N there exists its open neighborhood Z y and a compact set K y ∈ Γ N R such that for every (x a ) ∈ ι(Z y ) the support of ψ (x a ) is contained in K y -ψ (x a ) is related to the map (5.25) by an obvious generalization of (3.10).
It is easy to convince oneself that, the other way round, if a Hilbert half-densityΨ of compact M N -support is (i ) continuous in the coordinate system (x a ) and (ii ) of compact and slowly changing Γ N R -support in the same system, thenΨ ∈ H c N .
where ψ is the function (5.25), is a linear bijection from H c N onto the linear space C c (R N > × Γ N R , C) of all complex compactly supported continuous functions on R N > × Γ N R . Proof. By reasoning similar to that used in the proof of Lemma 3.5 one can show that the map (5.26) is linear.
Let us fixΨ and ψ related by the map (5.26). We know already that ψ is continuous. Let us then show that ψ is compactly supported.
If suppΨ denotes the M N -support ofΨ, then obviously On the other hand, if ψ(x a , γ ⊕ ab ) = 0, then (γ ⊕ ab ) ∈ supp ψ (x a ) . But since (x a ) ∈ ι(suppΨ), by virtue of (5.27) there exists n ∈ {1, . . . , m} such that (x a ) ∈ ι(Z yn ). Then supp ψ (x a ) ⊂ K yn (the sets {K y } are introduced just above the lemma). Consequently, (γ ⊕ ab ) ∈ K yn . We are then allowed to state that if ψ(x a , γ ⊕ ab ) = 0, then Note now that ι(suppΨ) is compact, because it is the image of a compact set under a continuous map. The set m n=1 K yn is compact being a union of a finite number of compact sets. The Cartesian product of ι(suppΨ) and m n=1 K yn is then a compact subset of R N > × Γ N R and therefore it is closed. Because it is closed and ψ(x a , γ ⊕ ab ) = 0 implies (5.28), then We thus see that supp ψ is a closed subset of a compact set. Therefore supp ψ is compact. We just proved that the map (5.26) is valued in C c (R N > × Γ N R , C). To show that the map is injective note that if ψ is the value of the map atΨ, thenΨ can be unambiguously reconstructed from ψ by means of the obvious generalization of the formula (3.8).
To finish the proof it remains to show that the map (5.26) is surjective. To this end assume that ψ ∈ C c (R N > × Γ N R , C). If are canonical projections, then both sets π 1 (supp ψ) and π 2 (supp ψ) are compact being images of a compact set under continuous maps. It is not difficult to show that for every (x a ) ∈ R N > , supp ψ (x a ) is a subset of π 2 (supp ψ)-see the reasoning concerning the support of a function h x in Appendix B. This means that the Hilbert half-densityΨ defined on M N by ψ with the help of the generalization of (3.8), is of compact and slowly changing Γ N R -support.
Thus every ψ ∈ C c (R N > × Γ N R , C) defines by means of the generalization of (3.8) an elementΨ ∈ H c N . The map (5.26) is then surjective.
By definition of H N , the space H c N is dense in H N . We will show now that H N is isomorphic to a Hilbert space, which contains the space C c (R N > × Γ N R , C) as its dense subset.
Let us begin by expressing the inner product (5.20) in terms of an iterated integral over R N > ×Γ N R . To this end consider an integrable scalar densityF on M N . If f is its coordinate representation in the global coordinate system (x a ) defined by ι (see (3.4)), then where dµ N L is the Lebesgue measure on R N . IfF = (Ψ ′ |Ψ), whereΨ ′ ,Ψ ∈ H c N , then combining Equations (3.15), (3.16) and (5.11) we get Here ψ ′ (x a ) is the function on Γ N R related by the obvious generalization of (3.10) to the coordinate representation ψ ′ ofΨ ′ in the coordinates (x a ), and Ś ∆ dµ L is the product of N copies of ∆ dµ L . Taking into account the last two equations and the definition (5.20), we see that the inner product turns out to be a dense subset [16] 12 of this Hilbert space.
We conclude that the map (5.26) (i ) is a linear bijection between linear dense subspaces of H N and L 2 (R N > × Γ N R , dν) and (ii ) preserves the inner products. Therefore the map can be unambiguously extended to a unitary map from H N onto L 2 (R N > × Γ N R , dν). R N > × Γ N R is a second countable l.c.H. space being an open subset of R 2N . As each regular measure on such a space is σ-finite [16], so is dν. On the other hand, each σ-finite Borel measure on a second-countable space defines a separable L 2 space [19]. This means that L 2 (R N > × Γ N R , dν) is separable. H N being isomorphic to the former Hilbert space, is separable as well. Consequently, H built over M = R is separable, since it is defined as the countable orthogonal sum (5.4) of separable Hilbert spaces.
There is also another important conclusion, which can be drawn from the results just obtained. Let us recall that on elements of H c N there are imposed seemingly strong conditions of compact M N -support and of compact and slowly changing Γ N R -support. This fact may raise concerns about whether the resulting Hilbert space H N is "large enough" from a physical point of view. But since H N built over M = R turned out to be isomorphic to L 2 (R N > × Γ N R , dν), then at least in this case we can regard these concerns to be unfounded.

Construction of the Hilbert space K
Let us fix a manifold M, a metric signature (p, p ′ ) such that p + p ′ = dim M and a diffeomorphism invariant field (2.11) of invariant measures on M. This field defines via (5.14) the Hilbert space H ⊗ y for every y ∈ M N , where N ≥ 2. Note however that the definition of M N applied to the case N = 1, gives the original manifold M (provided we identify y = {x} with x ∈ M). Then H ⊗ y coincides with H y defined by (3.1) (under the same identification). This observation allows us to simplify the presentation below by considering spaces {M N } and corresponding Hilbert spaces for all N ≥ 1.
Let us then fix an integer N ≥ 1 and consider a bundle-like set Let K N be a set, which consists of some special sections of H ⊗ : a section Ψ of H ⊗ belongs to K N if where || · || y is the norm on H ⊗ y , is finite (note that by virtue of the previous assumption, the uncountable sum above reduces to a sum of countable number of positive terms).
Lemma 6.1. The map is well-defined. K N equipped with this map is a Hilbert space.
Let us recall that ·|· y in (6.4) is the inner product on H ⊗ y . Note also that the map (6.4) can be expressed alternatively as where dµ 0 is the counting measure on M N , being a diffeomorphism invariant measure on the manifold.
A proof of Lemma 6.1, as following the well-known case of the Hilbert sequence space l 2 , is relegated to Appendix C.
For many practical purposes it would be convenient to have a dense linear subspace of K N , which would contain sufficiently regular elements of the Hilbert space. Denote by K c N a set, which consists of all elements {Ψ} of K N of the following property: for every y ∈ M N , the value Ψ(y) ∈ H ⊗ y is (an equivalence class of) an element of C c (Γ ⊕ y , C), i.e., a complex continuous function on Γ ⊕ y of compact support. Let K cf N be the set consisting of all elements of K c N , for which the set (6.2) is finite.
Proof. Since every set C c (Γ ⊕ y , C) is a linear space, then both K c N and K cf N are linear subspaces of K N . It remains then to prove that K cf N is dense in K N . We will first show that K c N is dense in K N . Let us fix Ψ ∈ K N . Then the set (6.2) is countable and all its elements can be ordered to form a sequence (y n ).
We know that every Γ ⊕ y ∼ = Γ N R is l.c.H. space, and the measure dµ × y is regular and Borel. These imply that the space C c (Γ ⊕ y , C) is a dense subset of H ⊗ y = L 2 (Γ ⊕ y , dµ × y ) [16] 13 . This means that if ψ is a non-zero element of H ⊗ y , then there exists a sequence (ψ ′ m ) of non-zero elements of C c (Γ ⊕ y , C), which converges to ψ. Then the sequence (||ψ ′ m || y ), where || · || y is the norm on H ⊗ y , converges to ||ψ|| y . Consequently, functions form a sequence, which converges to ψ and for every m, ||ψ m || y = ||ψ|| y (in other words, all elements of the sequence (ψ m ) belong to the sphere of radius ||ψ|| y centered at zero of H ⊗ y ). Let us fix a natural number m > 0. The conclusions above allow us to choose for every y n a function ψ nm ∈ C c (Γ ⊕ yn , C) such that Define now a section Ψ m of H ⊗ : Obviously, for every Ψ m the set (6.2) is countable and Thus Ψ m is an element of K c N . On the other hand, the norm of Ψ m − Ψ in the Hilbert space K N , can be bounded from above as follows: This result means that K c N is dense in K N . Note now that for every Ψ ∈ K c N and for every ǫ > 0, there exists Ψ f ∈ K cf N such that ||Ψ − Ψ f || < ǫ-if the set (6.2) for Ψ is infinite, then to obtain the desired Ψ f it is enough to zero out values of Ψ at appropriately chosen points in M N . Consequently, K cf N is also dense in K N .
Taking into account experiences gained from the study of the spaces {H N }, it is easy to realize that 1. if the Hilbert spaces K N andǨ N are constructed as above, starting from two distinct diffeomorphism invariant measure fields, then there exists a number c > 0 such that is a unitary map. Each Hilbert space H ⊗ y is separable, because every Hilbert space H x is separable [1]. Let {ψ yn } n∈N be a basis of H ⊗ y and let Ψ yn be an element of K N such that It is not difficult to demonstrate that {Ψ yn } y∈M N , n∈N is an orthonormal basis 14 of K N , which thereby is a non-separable Hilbert space. Now we are able to merge the spaces {K N } into the Hilbert space K in the same way, the spaces {H N } were merged into H, that is, by means of an orthogonal sum: Let us recall that all the Hilbert spaces {K N } above stem from the same diffeomorphism invariant field (2.11) of invariant measures.
It is now a simple exercise 1. to show that K is unique up to natural isomorphisms built of the unitary maps (6.5); 2. to construct a unitary representation of Diff(M) on K from the representations {U N } just defined.
Note also that K is a non-separable Hilbert space being built from non-separable Hilbert spaces {K N }.

Summary and outlook
In this paper we constructed two Hilbert spaces H and K over the set Q(M) of all metrics of arbitrary signature (p, p ′ ), defined on a (smooth connected paracompact) manifold M 15 . Each space was obtained by merging the tensor products Let us now present an outlook to future research. The most important question is whether either H or K constructed in the case of signature (3, 0), can be used for the quantization of the ADM formalism. As emphasized in the introduction to this paper, there is no guarantee that the answer to this question is in affirmative. The first step to be done to clarify this issue, is an attempt to define on the Hilbert spaces operators [15] related to the ADM canonical variables. As it seems, the fact that each measure dµ x is an invariant measure on the homogeneous space Γ x , should result in self-adjointness of operators related to the momentum variable.
Other issues we left open here are: (i ) the relation between each Hilbert space H N (being a building block of H) and the set of all "square integrable" Hilbert half-densities on M N , (ii ) the question whether each space H N generated by the space H c N of special Hilbert half-densities, is "large enough" from a physical point of view 16 and (iii ) the question whether all the Hilbert spaces {H} are separable.
In this paper we considered the bundle-like setsH,H ⊗ and H ⊗ defined, respectively, by the formulas (3.2), (5.15) and (6.1). It is interesting, at least from a mathematical point of view, whether these spaces can be endowed with local trivializations, which would make them genuine bundles. In particular, it is interesting, whether the set H ⊗ is a Hilbert bundle (see e.g. [20]) over M N .
Let us emphasize that the Hilbert spaces H and K are distinctly different from the space S of quantum states built in [1] by means of the Kijowski's projective method over the same set Q(M) of metrics. To construct S, we extended each Hilbert space in {H x 1 ⊗ . . . ⊗ H x N } N =1,2,... to a larger space S λ , where λ ≡ {x 1 , . . . , x N }. Namely, the space S λ was defined as the set of all algebraic states on the C * -algebra B λ of all bounded operators on H λ ≡ H x 1 ⊗ . . . ⊗ H x N . Since all the sets {S λ } form naturally a projective family, the space S were obtained as the projective limit of the family. As a result, the space S is not a Hilbert space, but it is rather a convex set of all algebraic states on a "large" C * -algebra, obtained by merging all the algebras {B λ } [6].
Despite these differences, the spaces of quantum states: H, K and S, are constructed of the same building blocks (being the Hilbert spaces {H x 1 ⊗ . . . ⊗ H x N } N =1,2,... ) and in our opinion it is worthwhile to explore more closely the relations between these three spaces.

A The set M N as a manifold
Let us fix an integer N ≥ 2 and a smooth connected paracompact manifold M and define

A.1 Smooth atlas on M N
Our goal in this section is to define a smooth atlas on M N , which will allow us to treat this set a smooth manifold.  (1, 2, 3, . . . , N ). Given σ ∈ Σ, the following map is a diffeomorphism on M N 0 . To see this let us fix a point (x 0 K ) ∈ M N 0 and for every x 0 I ∈ (x 0 K ) choose its open neighborhood U I is such a way that (i ) the neighborhoods {U I } satisfy (A.1) and (ii ) each U I is a domain of an R dim M -valued map ϕ I , which defines a coordinate system between appropriate open subsets of is a natural surjection (projection) from M N 0 onto M N , which "forgets" about the ordering of points in (x K ). This map will be used to define the smooth atlas on M N .
It follows immediately 17 from (A.5) that for every {x K } ∈ M N and for every subset U of M N 0 , To show that the topology just introduced is Hausdorff, consider two distinct elements It is clear that π Ś U x K and π Ś U under the surjection π are also not disjoint: , keeping in mind that for I = J the sets U I and U J are disjoint. It turns out that the map π| Ś U K , that is, the map π restricted to Ś U K , is a bijection onto its image. Indeed, it follows from the first of Equations (A.6) that for some σ ∈ Σ. But since the sets {U I } are pairwise disjoint, the intersection ą for every σ ∈ Σ except the identity permutation. This means that if (A.8) holds for two elements of Ś U K , then the elements coincide.  6)). This means that π −1 Ś U K is a continuous map. Since the map π is also continuous, π −1 Ś U K is a homeomorphism. This property of π −1 Ś U K allows us to define a local coordinate system on M N : given a map (A.3), the composition is a homeomorphism onto its image and defines thereby a A is smooth Let us show now that A is also smooth. To this end consider the map (A.11) and an other one of this sort: 12) and suppose that the domains of Φ and Φ ′ are not disjoint: Our goal now is to show that the transition function Φ ′ • Φ −1 related to Z is smooth.
or, equivalently, if and only if This fact together with the inclusion Z ⊂ π(M N 0 ) (see (A.13)) mean that Applying (A.10) we see that if σ = σ ′ , then But π restricted to Ś U ′ K is injective and therefore the sets {π(U σ )} appearing at the r.h.s. of (A.14) are pairwise disjoint. Now it is enough to find the transition function Φ ′ • Φ −1 on each non-empty set π(U σ ).

Consider then
Thus (x ′i K K ) and (x i K K ) are values of coordinates defined by, respectively, Φ ′ and Φ, of the same point {x ′ K } = {x K }. Using (A.16) we obtain the following relation between the values: 17) Obviously, this relation is nothing else but the value of the transition function Φ ′ • Φ −1 at (x i K K ). We thus conclude that Φ ′ • Φ −1 on the set Φ π(U σ ) is smooth, since it coincides with the coordinate expression (A.17) of the diffeomorphismσ-given I, Thus the transition map is smooth on its whole domain Φ(Z) = σ∈Σ Φ π(U σ ) .
is a diffeomorphism. Indeed, let us choose a collection of charts {(U K , ϕ K )} K=1,...,N on M such that U K ⊂ U ′ K . Using the maps (A.3) and (A.11) we see that the following coordinate expression for π −1 This fact allows us to state that (i ) M N is locally diffeomorphic to M N and (ii ) the projection π is smooth. M N is paracompact A manifold is paracompact if and only if each connected component of the manifold is second countable [21]. We assumed that M is paracompact and connected, which means that M is second countable. Thus M N and M N 0 are second countable as well. Let

A.2 An example of M N
Here we will find an explicit description of M N for M = R by means of a global coordinate system on M N . Let us fix an integer N ≥ 2. If M = R, then for every y ≡ {x K } ∈ M N it is possible to form a decreasing sequence from all elements of y i.e. there exists a permutation σ ∈ Σ such that This observation allows us to define the following map It is obvious that the map is a bijection onto its image such that where the last equation holds on ι(M N ) ⊂ M N 0 . Let It is clear that ι(M N ) = R N > . Fix an arbitrary z ≡ (x 1 , . . . , x N ) ∈ R N > and define  To demonstrate that this map is smooth consider the following smooth map Clearly, ξ y I = π • ξ 0 y I , (A. 21) which means that ξ y I is a composition of two smooth maps. If ξ t y I is the tangent map defined by ξ y I , then ξ t y I (T x I M) is a linear subspace of T y M N . This subspace is generated by all curves in M N of the following form: It is evident that where ξ 0t y K denotes the tangent map given by ξ 0 y K . Let us now act on both sides of this equation by the tangent map π t defined by π. Since π restricted to Ś U K is a (local) diffeomorphism 18 to M N , we thus obtain where in the last step we used (A. 21). Suppose now that the product Ś U K is the domain of the map (A.3), which is used to define the map Φ via (A.11). Denote by (x i K K ) the value at y = {x K } of the coordinates defined by Φ. Then the following curve Consider now maps (A.11) and (A.12) assuming that U ′ K = θ(U K ) and ϕ ′ K = ϕ K • θ −1 . Then using (A.26) we obtain This means that for every diffeomorphism θ of M, the map Θ is smooth. But Θ −1 exists and is smooth, since it is given via (A.25) by θ −1 . Θ is thus a diffeomorphism of M N .
18 This is because π −1 Ś U K is a local diffeomorphism as proven at the end of Appendix A.1.
If Θ is induced by θ via (A.25), then is a homomorphism. Its image is a subgroup of the diffeomorphism group of M N . This subgroup will be denoted by Diff M (M N ). Note also that it follows from (A.27) that Φ • Θ −1 coincides with Φ ′ . This means that Θ −1 maps a chart in A to an other one in A. In other words, the atlas A is preserved by all diffeomorphisms in Diff M (M N ). Let y = {x K } ∈ M N and let x ′ K = θ −1 (x K ). Then y ′ = {x ′ K } = Θ −1 (y). Suppose that {γ x K } are scalar products (of the same signature) such that γ x I ∈ Γ x I and that γ ⊕ y ∈ Γ ⊕ y is constructed of these scalar products according to Equation (5.6). Consider now the pull-back Θ t * γ ⊕ y :

Diffeomorphisms in Diff
where v,v ∈ T y ′ M N . We see that the pull-back Θ t * γ ⊕ y is a scalar product on T y ′ M N constructed of scalar products {θ t * γ x I } via (5.6). Therefore Θ t * γ ⊕ y ∈ Γ ⊕ y ′ . We are then allowed to conclude that diffeomorphisms in Diff M (M N ) preserve the spaces {Γ ⊕ y } y∈M N . Moreover, regarding the correspondence given by the bijection (5.13), we see that the pull-back Θ t * : Γ ⊕ y → Γ ⊕ y ′ corresponds to the pull-back ą θ t * ≡ θ t * × . . . × θ t * : provided the ordering of the spaces of scalar products {Γ x K } and {Γ x ′ K } is chosen as in (A.29).

B Proof of Lemma 5.1
Since X, Y , X ′ and Y ′ are second countable l.c.H. spaces, so are the products X × Y and X ′ × Y ′ [16]. If two regular Borel measures are defined on second countable l.c.H. spaces, then their product is well defined and again is a regular Borel measure [16]. Therefore both product measures, which appear in (5.10) are regular and Borel. That being the case, by virtue of the Riesz representation theorem, it is enough to show that for every h ∈ C c (X × Y ), (B.1) Each second countable l.c.H. space is metrizable [16]. Let then δ X and δ Y be corresponding metrics on X and Y . Then δ (x, y), (x,y) := δ 2 X (x,x) + δ 2 Y (y,y) is a metric on X × Y compatible with the product topology. The canonical projections π X : X × Y → X, (x, y) → x, and π Y : X × Y → Y , (x, y) → y, are continuous maps. Given x ∈ X, let us define Y ∋ y → h x (y) := h(x, y) ∈ R.
The support of h x , if non-empty, can be characterized as follows: y ∈ supp h x if and only if for every ǫ > 0 there existsy ∈ Y such that h x (y) = 0 and δ Y (y,y) < ǫ, or, equivalently, if and only if for every ǫ > 0 there exists a pair (x,y) ∈ X × Y such that h(x,y) = 0 and δ (x, y), (x,y) < ǫ. This last statement implies that (x, y) ∈ supp h. We thus showed that if y ∈ supp h x , then (x, y) ∈ supp h. But if (x, y) ∈ supp h, then y = π Y (x, y) ∈ π Y (supp h). Thus supp h x ⊂ π Y (supp h). This inclusion holds also if supp h x is empty. Note now that π Y (supp h) is compact being the image of the compact set supp h under the continuous map π Y . We see that supp h x is a closed subset of a compact set and therefore is compact as well 19 .
On the other hand, continuity of h implies continuity of h x . We conclude that for every x ∈ X, h x ∈ C c (Y ) and accordingly to (2.5) Consider now the following function Suppose that x ∈ π X (supp h). This means that for every y ∈ Y , h x (y) = h(x, y) = 0 and consequentlyȟ(x) = 0. Thus ifȟ(x) = 0, then x ∈ π X (supp h). Therefore suppȟ ⊂ π X (supp h), since π X (supp h) is compact and thereby closed. We conclude then that suppȟ is compact being a closed subset of a compact set. Let s = sup (x,y)∈X×Y |h(x, y)| and Y ∋ y → h(y) := s if y ∈ π Y (supp h), 0 otherwise .
Since π Y (supp h) is compact and the measure β ⋆ dν is regular, h is integrable with respect to the measure. Moreover, for every x ∈ X, |h x | ≤ h. These two facts allow us to use the Lebesgue's dominated convergence theorem to conclude that the functionȟ is continuous 20 . We thus showed thatȟ ∈ C c (X) and consequently by virtue of (2.5) (note also that α × β is a homeomorphism and therefore (α × β) ⋆ h is continuous and compactly supported.) C Proof of Lemma 6.1 To prove Lemma 6.1 we have to show that (i ) K N is a linear space, (ii ) the map (6.4) is an inner product on K N and (iii ) K N is complete in the norm defined by the inner product (6.4). The proof of the lemma we are going to present here, follows a proof of an analogous lemma concerning the Hilbert sequence space l 2 (see e.g. [22,23]).
Thus Ψ + Ψ ′ ∈ K N and K N is a linear space.
The map (6.4) is an inner product To prove that the map (6.4) is well defined, consider again the same elements Ψ, Ψ ′ ∈ K N and the same sequence (y n ) of points in (C.1). Then where ·|· y is the inner product on H ⊗ y . Let us show now that the series above is absolutely convergent-then its sum does not depend on the ordering of points in (C.1) into a sequence and, consequently, Ψ ′ |Ψ is well defined.
To this end we will apply the Schwarz inequality to every term in the following series: ||Ψ ′ (y n )|| 2 yn + ||Ψ(y n )|| 2 yn = 1 2 ||Ψ ′ || 2 + ||Ψ|| 2 < ∞ (here in the second step we again used the inequality 2ab ≤ a 2 + b 2 ). Thus the map (6.4) is well-defined. It is now an easy exercise to show that it is an inner product on K N and that the norm defined on the set by the inner product coincides with (6.3).
K N is complete It remains to show that K N equipped with the norm is a complete space. Let us then suppose that (Ψ m ) m=1,2,... is a Cauchy sequence of elements of K N . Then the set { y ∈ M N | ∃ m such that Ψ m (y) = 0 } is countable and we can form a sequence (y n ) using all elements of this set. Thus for every ǫ > 0, there exists m 0 such that for every m, m ′ > m 0 , ||Ψ m (y n ) − Ψ m ′ (y n )|| 2 yn < ǫ 2 . (C.2) Consequently, for every n and every ǫ > 0, there exists m 0 such that for each m, m ′ > m 0 , This implies that for every (fixed) n, the sequence Ψ m (y n ) has a limit in (the complete space) H ⊗ yn -this limit will be denoted by ψ n . Let then Ψ be a section of H ⊗ such that Ψ(y) := ψ n if y = y n 0 otherwise .
Therefore for every l and for every m > m 0 , Consequently, for m > m 0 , passing to the limit as l tends to the infinity, we obtain We conclude then that Ψ is the limit of (Ψ m ). Evidently, the set (6.2) for Ψ − Ψ m is countable. It follows from (C.3) that Ψ − Ψ m is of finite norm (6.3). Thus Ψ − Ψ m is an element of K N . But because Ψ m ∈ K N and K N is a linear space (as proven above), Ψ belongs to K N .
We thus showed that every Cauchy sequence of elements of K N converges to an element of this space. The space is then complete.