aa r X i v : . [ m a t h - ph ] M a r Hilbert spaces built over metrics of fixed signature
Andrzej OkołówMarch 2, 2021
Institute of Theoretical Physics, Warsaw Universityul. Pasteura 5, 02-093 Warsaw, [email protected]
Abstract
We construct two Hilbert spaces over the set of all metrics of arbitrary but fixedsignature, defined on a manifold. Every state in one of the Hilbert spaces is builtof an uncountable number of wave functions representing some elementary quantumdegrees of freedom, while every state in the other space is built of a countable numberof them. Each Hilbert space is unique up to natural isomorphisms and carries a unitaryrepresentation of the diffeomorphism group of the underlying manifold.
In [1] we constructed a space of quantum states and an algebra of quantum observables overthe set of all metrics of arbitrary but fixed signature, defined on a manifold. This spaceand 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, whichcould be used for quantization of the ADM formulation [7] of general relativity (GR), asthe “position” part of the ADM phase space is the set of all Riemannian metrics definedon a three-dimensional manifold.The space of quantum states built in [1] is not a Hilbert space, but a convex set ofmixed states. It turns out, however, that a structural component of that space can be usedto 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 andfix 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 structuralcomponent is a diffeomorphism invariant assignment x dµ x , 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 thespace 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 offreedom (d.o.f.). 1o arrange the Hilbert spaces { H x } x ∈M into the Hilbert space H , we will proceed asfollows. First, we will associate with every point x the set ˜ H x of all half-densities overthe tangent space T x M valued in H x —a section of the bundle-like set S x ∈M ˜ H x will bea half-density on M valued in the Hilbert spaces { H x } . Given two such half-densities ˜Ψ and ˜Ψ ′ , we can pair their values ˜Ψ( x ) and ˜Ψ ′ ( x ) using the inner product on H x , obtainingas a result a complex-valued density over T x M . Doing this point by point gives us ascalar 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 S x ∈M ˜ H x . This set equipped with the product will form a Hilbert space H .The Hilbert space H alone will not be suitable for quantization of physical theoriessince it is more like an uncountable orthogonal sum of the Hilbert spaces { H x } and there-fore it does not contain any tensor product of them. But these Hilbert spaces representindependent d.o.f. and a physically acceptable Hilbert space should contain tensor productsof { H x } . We will include these tensor products as follows.We will fix a natural number N ≥ and will consider the set N N of all N -elementsubsets of M . Next we will equip N N with a differential structure and will associate witheach element { x , . . . , x N } ∈ N N the Hilbert space H x ⊗ . . . ⊗ H x N . Then, following theconstruction of the Hilbert space H 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 S x ∈M H x , which are non-zero merely on countable subsets of M , and will construct a Hilbert space K using these sections and the inner productson { H x } . In an analogous way, we will obtain a Hilbert space K N for N ≥ using themanifold N N and the tensor products { H x ⊗ . . . ⊗ 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 quantumd.o.f. being the Hilbert spaces { H x } . A difference between the spaces will be that a statein 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 ofthem.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 onthe Hilbert spaces. Moreover, as shown in [1], the assignment is unique up to a positivemultiplicative constant. This will imply that each Hilbert space H and K is unique up todistinguished unitary maps. We will also show that the two Hilbert spaces { H } built over M = R (for signature (1 , and (0 , ) are separable and that all the Hilbert spaces { K } are non-separable.With regard to the original motivation underlying this research: although the construc-tions of both Hilbert spaces H and K seem to be fairly natural, neither space consists ofsquare integrable functions on the set of metrics of fixed signature. Therefore it is notobvious whether H or K (in the case of signature (3 , ) can be applied to quantization ofthe ADM version of GR—a further research is needed to answer this question. As the firststep towards this goal, in the forthcoming paper [8] we will define some operators on H and K related to the canonical variables on the ADM phase space.The paper is organized as follows: Section 2 contains preliminaries—there we recallfirst of all some necessary notions and facts from [1]. In Section 3 we construct the Hilbert2pace H , and in Section 4 we define a unitary representation of diffeomorphisms of M on H . 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 wedefine a smooth structure on the set N N , and in Appendices B and C we present proofs ofsome lemmas. Suppose that V is a real vector space such that < dim V < ∞ . 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 scalarproducts of signature ( p, p ′ ) defined on V . As shown in [1], Γ is a noncompact connectedreal-analytic manifold of dimension dim V (dim V + 1) / . If ( e i ) i =1 , ,..., dim V is a basis of V ,then the following map Γ ∋ γ χ ( γ ) := (cid:0) γ ( e i , e j ) (cid:1) i ≤ j ∈ R dim Γ (2.1)defines a global coordinate system on Γ . This system will be called here linear coordinatesystem on Γ and denoted ( γ i ≤ j ) , where Γ ∋ γ γ i ≤ j ( γ ) := γ ( e i , e j ) ∈ R . Occasionally we will use a single index α to label the coordinates: ( γ i ≤ j ) ≡ ( γ α ) α =1 ,..., dim Γ .Consider now another real vector space ˇ V such that dim ˇ V = dim V and the set ˇΓ of all scalar products of signature ( p, p ′ ) defined on ˇ V . Obviously, dim ˇΓ = dim Γ . Let ˇ χ : ˇΓ → R dim Γ be a map given by a basis (ˇ e i ) of ˇ V via (2.1). Then χ (Γ) = ˇ χ (ˇΓ) . (2.2)To see that the equality above holds, we will use dual bases ( ω i ) and (ˇ ω i ) to, respec-tively, ( e i ) and (ˇ e i ) . Assume that ( γ i ≤ j ) ∈ χ (Γ) —this holds if and only if γ = X i γ ii ω i ⊗ ω i + X i 4n [1] we showed that on Γ there exists a (non-zero) invariant measure and that it is uniqueup to a positive multiplicative constant. Γ is second countable (i.e. Γ has a countable base for its topology) being homeomorphicto the open subset Γ R of R dim Γ . On the other hand, each regular measure on secondcountable l.c.H. space is σ -finite [9], which means that every invariant measure on Γ is σ -finite.Consider now real vector spaces V , V and V of the same dimension, and suppose that Γ i ( i = 0 , , ) 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 is an invariant measure on Γ , then1. ( l ∗ ) ⋆ dµ is an invariant measure on Γ , which is independent of the choice of linearisomorphism l ;2. for every triplet of linear isomorphisms l , l and l ( l ∗ ) ⋆ ( l ∗ ) ⋆ dµ = ( l ∗ ) ⋆ dµ . 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 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 bya map χ : Γ → Γ R (see (2.1)). It turns out that the natural metric looks the same in everylinear coordinate system [1]. More precisely, there exist smooth functions ∆ αβ : Γ R → R , α, β = 1 , . . . , dim Γ , such that for every space Γ as above and for every linear coordinate system on Γ thepull-back χ − ⋆ 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 [1] Z Γ Ψ dµ Q := Z Γ R χ − ⋆ (cid:16) Ψ q | det Q αβ | (cid:17) dµ L = Z Γ R ( χ − ⋆ Ψ) ∆ dµ L , (2.7)where dµ L is the Lebesgue measure on R dim Γ ⊃ Γ R and ∆ ≡ q | det ∆ αβ | = χ − ⋆ (cid:16)q | det Q αβ | (cid:17) is a positive function on Γ R .Consider two real vector spaces V and ˇ V of the same dimension and the correspondingspaces Γ and ˇΓ of scalar products of the same signature ( p, p ′ ) . Let dµ Q and dµ ˇ Q be A measure dν on Y is σ -finite if Y is a union of a sequence ( Y n ) of its subsets such that each Y n hasa finite measure under dν . The metric is invariant with respect to the group action on Γ . Γ and ˇΓ defined by the corresponding natural metrics Q and ˇ Q .If l : ˇ V → V is a linear isomorphism, then the pull-back l ∗ : Γ → ˇΓ is a diffeomorphism. Itwas shown in [1] that the push-forward measure l ∗ ⋆ dµ Q = dµ ˇ Q , (2.8)which means in particular (i.e. in the case V = ˇ V ) that dµ Q is an invariant measure onthe homogeneous space Γ .Since an invariant measure on Γ is unique up to a positive multiplicative constant, forevery invariant measure dµ on the homogeneous space, there exist a number c > suchthat dµ = c dµ Q . (2.9) Lemma 2.2. Let Ψ : Γ → C be continuous and dµ be an invariant measure on Γ . If Z Γ ¯ΨΨ dµ = 0 , (2.10) then Ψ = 0 .Proof. Suppose that Ψ( γ ) = 0 for some γ ∈ Γ . It follows from continuity of Ψ that thereexists a non-negative compactly supported continuous function h on Γ such that h ( γ ) > and ¯ΨΨ ≥ h . Then Z Γ ¯ΨΨ dµ ≥ Z Γ h dµ = c Z Γ h dµ Q = c Z Γ R ( χ − ⋆ h ) ∆ dµ L > —here the second equality holds by virtue of (2.9), and the last inequality does by virtueof properties of the Lebesgue measure dµ L and the fact that the function ∆ is positiveeverywhere on Γ .Clearly, the inequality above shows that the only continuous function, which satisfies(2.10), is the constant function of zero value. Let M be a smooth connected paracompact manifold. We fix a pair of non-negativeintegers ( 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 scalarproducts on T x M , of signature ( p, p ′ ) . Obviously, for every x the pair ( GL ( T x M ) , Γ x ) isa homogeneous space. Moreover, as shown in [1], if Q ( M ) is non-empty, then Γ x = { q x | q ∈ Q ( M ) } , 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 be any point of M and dµ x an invariant measure on Γ x . In [1] we introducedthe following measure field on M : x dµ x := ( l ∗ x x ) ⋆ dµ x , (2.11)6here l x x : T x M → T x M is a linear isomorphism and l ∗ x x : Γ x → Γ x the correspondingpull-back. By virtue of Lemma 2.1, ( i ) dµ x is an invariant measure on Γ x , which doesnot depend on the choice of the map l x x , and ( ii ) for every two points x, x ′ ∈ M and forevery linear isomorphism l xx ′ : T x ′ M → T x M dµ x ′ = ( l ∗ xx ′ ) ⋆ dµ x . (2.12)The latter property means that the measure field (2.11) is diffeomorphism invariant since l xx ′ above can be the tangent map θ ′ 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 positivemultiplicative constant, i.e., for any two measure fields x dµ x and x d ˇ µ x constructedaccording to (2.11), there exists a number c > such that for every x ∈ M d ˇ µ x = c dµ x . (2.13)Let dµ Qx be the invariant measure on Γ x given by the natural metric on the homoge-neous 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 everymeasure field (2.11) is of the form x c dµ Qx (2.14)for some (independent of x ) positive number c . Let W be a (possibly infinite dimensional) complex vector space, V a finite dimensionalreal 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, thenthe symbol Λ e will represent the basis (Λ j i e j ) .An α -density over V valued in W is a map ˜ w : B → W of the following property: forevery two bases e and Λ e of V , ˜ w (Λ e ) = | det Λ | α ˜ w ( e ) , (2.15)where det Λ ≡ det(Λ j i ) = 0 .We will denote by ˜ W the set of all α -densities over V valued in W . This set possessesa natural complex vector space structure: if z ∈ C and ˜ w, ˜ w ′ ∈ ˜ W then ( z ˜ w )( e ) := z ˜ w ( e ) , ( ˜ w + ˜ w ′ )( e ) := ˜ w ( e ) + ˜ w ′ ( e ) . Denote by ˜ C the vector space of one-densities over V valued in complex numbers. Thecomplex conjugate ˜ w of ˜ w ∈ ˜ C is an element of ˜ C such that ˜ w ( e ) = ˜ w ( e ) for a basis e of V (if the equality above holds for e , then it does for every basis of V ).7et ˜ w, ˜ w ′ ∈ ˜ C be real-valued. We will say that ˜ w ′ is greater than or equal to ˜ w andwrite ˜ w ′ ≥ ˜ w if ˜ w ′ ( e ) ≥ ˜ w ( e ) (2.16)for a basis e of V (if (2.16) holds for e , then it does for every basis of V ).Let H be a (complex) Hilbert space with an inner product h·|·i . Consider a vector space ˜ H of half-densities (that is, -densities) over V valued in H . Let us define the followingmap: ˜ H × ˜ H ∋ ( ˜ w ′ , ˜ w ) ( ˜ w ′ | ˜ w ) ∈ ˜ C , ( ˜ w ′ | ˜ w )( e ) := h ˜ w ′ ( e ) | ˜ w ( e ) i . (2.17)This map satisfies what follows: ∀ z , z ∈ C , ∀ ˜ w, ˜ w , ˜ w ∈ ˜ H ( ˜ w | z ˜ w + z ˜ w ) = z ( ˜ w | ˜ w ) + z ( ˜ w | ˜ w ) , ∀ ˜ w, ˜ w ′ ∈ ˜ H ( ˜ w ′ | ˜ w ) = ( ˜ w | ˜ w ′ ) , ∀ ˜ w ∈ ˜ H ( ˜ w | ˜ w ) ≥ , ( ˜ w | ˜ w ) = 0 ⇒ ˜ w = 0 , (2.18)where, abusing slightly the notation, we used the symbol to denote both the zero of ˜ C and the zero of ˜ H . Therefore the map (2.17) will be called density product on ˜ H .The space ˜ H equipped with the density product (2.17) will be called pseudo-Hilbertspace of half-densities over V valued in H . H Let us recall that M is a smooth connected paracompact manifold. For the sake of theconstruction of the Hilbert space H , let us fix a pair ( p, p ′ ) of non-negative integers suchthat p + p ′ = dim M and treat it as a metric signature—all objects used to construct H ,which need a metric signature to be chosen, will be given by this ( p, p ′ ) . M Let x dµ x be a diffeomorphism invariant field of invariant measures given by (2.11). Itallows to define a separable [1] Hilbert space for every x ∈ M : H x := L (Γ x , dµ x ) . (3.1)We will use the symbol h·|·i x to represent the inner product on H x .Given a point x ∈ M , let ˜ C x stands for the vector space of all one-densities over thetangent space T x M valued in C . Denote by ˜ H x the pseudo-Hilbert space of half-densitiesover T x M valued in H x , and by ( ·|· ) x the density product (2.17) on ˜ H x valued in ˜ C x .Let ˜ H := [ x ∈M ˜ H x (3.2)A map ˜Ψ : M → ˜ H such that ˜Ψ( x ) ∈ ˜ H x for every x ∈ M , will be called Hilbert half-density on M (this name comes from a shortening of the precise but inconvenient term8half-density on M valued in the Hilbert spaces { H x } ”). In other words, a Hilbert half-density is a section of the bundle-like set ˜ H . All such half-densities form a complex vectorspace with multiplication by complex numbers and addition defined point by point: ( z ˜Ψ)( x ) := z ˜Ψ( x ) , ( ˜Ψ + ˜Ψ ′ )( x ) := ˜Ψ( x ) + ˜Ψ ′ ( x ) , (3.3)where z ∈ C , and ˜Ψ , ˜Ψ ′ are two Hilbert half-densities.The M -support of a Hilbert half-density ˜Ψ on M is the closure of the following set: { x ∈ M | ˜Ψ( x ) = 0 } . Let ˜ C := S x ∈M ˜ C x . A map ˜ F : M → ˜ C , such that ˜ F ( x ) ∈ ˜ C x for every x ∈ M , isa complex scalar density on M . The set of all such densities is a complex vector spacewith multiplication by complex numbers and addition defined point by point by formulasanalogous to (3.3). Complex conjugate ˜ F of a density ˜ F is defined naturally point bypoint: ˜ F ( x ) := ˜ F ( x ) . The support of a scalar density ˜ F on M is the closure of { x ∈ M | ˜ F ( x ) = 0 } . As it is said in the introduction to this paper, we are going to define an inner product ona set of sections of ˜ H , that is, on a set of Hilbert half-densities. To this end we will pairthe 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 ofthese two half-densities.An important question is how to choose that set of Hilbert half-densities, which togetherwith this inner product, will form the desired Hilbert space. It seems that an obviousanswer to this question is that one should choose the set of all half-densities of finite normwith respect to the inner product. However, it has to be proven that this set equippedwith the product is indeed a Hilbert space.Moreover, in the case of signature (3 , , we would like to use the resulting Hilbert spacefor quantization of GR and this means in particular that we will try to define on the spacesome operators, which will represent physical observables. To define such operators it isoften 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 Hilbertspace using all the half-densities of finite norm, then we would have to find within it alinear 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 suffi-ciently regular Hilbert half-densities and will simply define the desired Hilbert space as acompletion of the former space in the norm defined by the inner product. The issue of arelation between this Hilbert space and that space given by all the half-densities of finitenorm 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 half-densities “large enough” from a physical point of view? Again, we will postpone this9uestion in its generality for the future and will limit ourselves to examining only a simpleparticular case—the result of this examination (see Section 5.7) will suggest that the answerto this (general) question is in affirmative.Below we will introduce a notion of continuous Hilbert half-densities (which can beeasily modified to a notion of smooth half-densities). Moreover, we will impose on thecontinuous half-densities an additional regularity condition, which will guarantee that thehalf-densities, once paired point by point by means of the density products { ( ·|· ) x } , yieldcontinuous densities on M —continuity in combination with a compact support of a densitywill ensure that the density is integrable over the manifold.Let us emphasize finally that the present section is quite technical and it may be skippedon the first reading. Continuous scalar densities Let us begin by recalling the notion of continuous scalardensity, 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 density ˜ F and a point x ∈ U , the value ˜ F ( x ) is a one-densityover T x M valued in C . Since ( ∂ x k ) is a basis of the tangent space, then ˜ F (cid:0) x, ( ∂ x k ) (cid:1) is acomplex number. Every x ∈ U can be expressed in terms of the coordinate system ( x i ) ,which allows us to define coordinate representation of ˜ F in the system ( x i ) as a function ϕ ( U ) ∋ ( x i ) f ( x i ) := ˜ F (cid:0) ϕ − ( x i ) , ( ∂ x k ) (cid:1) ∈ C . (3.4)If ( x ′ i ) is an other coordinate system of the domain U , then the corresponding coordinaterepresentation f ′ satisfies f ′ ( x ′ i ) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ∂x k ∂x ′ l (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ( x ′ i ) f (cid:0) x k ( x ′ i ) (cid:1) . (3.5)This property means that the function f is continuous if and only if f ′ is continuous.We will say that the scalar density ˜ F 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 subsetof M and ϕ : U → R dim M a map defining a coordinate system ( x i ) on U . Given Hilberthalf-density ˜Ψ and a point x ∈ U , the value ˜Ψ( x ) is a half-density over T x M valued in H x .Thus ˜Ψ( x, ( ∂ x k )) is an element of H x being an equivalence class of a function Γ x ∋ γ Ψ (cid:0) x, ( ∂ x k ) , γ (cid:1) ∈ C . (3.6)Now, every x ∈ U can be expressed in terms of the coordinate system ( x i ) and the scalarproduct γ can be expressed in terms of components ( γ ij ) given by the basis ( ∂ x i ) of T x M .This allows us to define coordinate representation of ˜Ψ in the system ( x i ) as the followingfunction: ϕ ( U ) × Γ R ∋ ( x i , γ ij ) ψ ( x i , γ ij ) := Ψ (cid:0) ϕ − ( x i ) , ( ∂ x k ) , γ ij dx i ⊗ dx j (cid:1) ∈ C . (3.7)Note that since ˜Ψ( x, ( ∂ x k )) is an equivalence class of functions on Γ x , the coordinaterepresentation above is not unique, even if the system ( x i ) is fixed.10o reconstruct the Hilbert half density ˜Ψ on U from its coordinate representative ψ itis enough to observe that the function Γ x ∋ γ Ψ (cid:0) x, ( ∂ x k ) , γ (cid:1) = ψ (cid:0) ϕ ( x ) , γ ( ∂ x i , ∂ x j ) (cid:1) ∈ C (3.8)is a representative of the equivalence class ˜Ψ (cid:0) x, ( ∂ x i ) (cid:1) ∈ H x . Continuous Hilbert half-densitiesDefinition 3.1. We will say that a Hilbert half-density ˜Ψ is continuous on U in the coor-dinate system ( x i ) if for every x ∈ U the representative (3.6) of ˜Ψ( x, ( ∂ x k )) ∈ H x can bechosen 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-density ˜Ψ is continuous on U inthe coordinate system ( x i ) , then the choice of the representatives (3.6), which give thecontinuous function (3.7), is unique. In other words, given a coordinate system ( x i ) , acontinuous coordinate representation of ˜Ψ in the system is unique (provided it exists). Lemma 3.2. Let ( x i ) and ( x ′ i ) be coordinate systems on an open set U ⊂ M . ˜Ψ iscontinuous on U in the coordinate system ( x i ) if and only if it is continuous on U in thecoordinate system ( x ′ i ) .Proof. For every x ∈ U ˜Ψ( x, ( ∂ x ′ i )) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ∂x k ∂x ′ l (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) / ( x ) ˜Ψ( x, ( ∂ x j )) . Therefore if γ ˜Ψ( x, ( ∂ x j ) , γ ) is a representative of ˜Ψ( x, ( ∂ x j )) ∈ H x , then the function γ → (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ∂x k ∂x ′ l (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) / ˜Ψ( x, ( ∂ x j ) , γ ) is a representative of ˜Ψ( x, ( ∂ x ′ i )) ∈ H x . Hence ( x ′ i , γ ′ ij ) ψ ′ ( x ′ i , γ ′ ij ) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ∂x k ∂x ′ l (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) / ( x ′ i ) ˜Ψ (cid:0) ϕ ′− ( x ′ i ) , ( ∂ x j ) , γ ′ ij dx ′ i ⊗ dx ′ j (cid:1) , where ϕ ′ : U → R dim M is the map defining the system ( x ′ i ) , is a coordinate representationof ˜Ψ in the system ( x ′ i ) . Denoting ( ϕ ◦ ϕ ′− )( x ′ i ) ≡ ( x j ( x ′ i )) we obtain from the formulaabove ψ ′ ( x ′ i , γ ′ ij ) = (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ∂x k ∂x ′ l (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) / ˜Ψ (cid:16) ϕ − ( x j ( x ′ i )) , ( ∂ x j ) , γ ′ ij ∂x ′ i ∂x k ∂x ′ j ∂x l dx k ⊗ dx l (cid:17) == (cid:12)(cid:12)(cid:12)(cid:12) det (cid:16) ∂x k ∂x ′ l (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) / ψ (cid:16) x j ( x ′ i ) , γ ′ ij ∂x ′ i ∂x k ∂x ′ j ∂x l (cid:17) , (3.9)where ψ is a coordinate representation of ˜Ψ in the system ( x i ) , and the derivatives ∂x k /∂x ′ l and ∂x ′ l /∂x k are treated as functions of ( x ′ i ) .Taking into account that the transition map ( x ′ i ) (cid:0) x j ( x ′ i ) (cid:1) is smooth, we see from(3.9) that the coordinate representation ψ ′ is continuous if and only if the coordinaterepresentation ψ is continuous (the “only if” part of these statement comes from the factthat the dependence ψ ′ of ψ given by (3.9) can be inverted to a dependence of ψ of ψ ′ ofan analogous form). 11e will say that a Hilbert half-density ˜Ψ is continuous if for every local coordinatesystem, there exists a continuous coordinate representation (3.7).Since now in the case of a continuous Hilbert half-density we will use exclusively itscontinuous coordinate representations. Hilbert half-densities of compact and slowly changing Γ R -support Let us con-sider again the map ϕ : U → R dim M and the corresponding coordinate system ( x i ) . Definition 3.3. Suppose that a Hilbert half-density ˜Ψ is continuous, and ψ is its coordinaterepresentation in the system ( x i ) . We will say that the Γ R -support of ˜Ψ around x ∈ U is compact and slowly changing in the coordinate system ( x i ) , if there exist an openneighborhood U ⊂ U of x , and a compact set K ⊂ Γ R , such that for every value ( x i ) ∈ ϕ ( U ) , the support of the function Γ R ∋ ( γ ij ) ψ ( x i ) ( γ ij ) := ψ ( x i , γ ij ) ∈ C (3.10) is contained in K . Let us emphasize that if the support of ψ ( x i ) is contained in a compact set, then the supportis compact itself (since each closed subset of a compact set is compact). Thus the definitionabove implies that for every ( x i ) ∈ ϕ ( U ) the support of ψ ( x i ) is compact. Lemma 3.4. Let ( x i ) and ( x ′ i ) be coordinate systems on an open set U ⊂ M , and let ˜Ψ bea continuous Hilbert half-density on M . The Γ R -support of ˜Ψ around x ∈ U is compactand slowly changing in the coordinate system ( x i ) if and only if the Γ R -support of ˜Ψ around x is compact and slowly changing in the coordinate system ( x ′ i ) .Proof. Let ϕ : U → R dim M and ϕ ′ : U → R dim M be maps defining the coordinate systems,respectively, ( x i ) and ( x ′ i ) . Denote (cid:0) x i ( x ′ j ) (cid:1) ≡ ϕ ( ϕ ′− ( x ′ j )) .Suppose that the Γ R -support of ˜Ψ around x ∈ U is compact and slowly changing inthe coordinate system ( x i ) . Let U and K be the sets introduced in Definition 3.3 forthe system ( x i ) . Choose a compact set U ⊂ U of non-empty interior Int U such that x ∈ Int U . The map ϕ ( U ) × Γ R ∋ ( x i , γ ij ) ∂x i ∂x ′ k ∂x j ∂x ′ l γ ij ∈ Γ R (3.11)is continuous (here we treat the derivative ∂x i /∂x ′ k as a function of ( x i ) ). Therefore theset K ′ := the image of ϕ ( U ) × K under the map (3.11)is compact.Consider now the following function Γ R ∋ ( γ ′ kl ) ψ ′ ( x ′ j ) ( γ ′ kl ) := ψ ′ ( x ′ j , γ ′ kl ) ∈ C , where ψ ′ ( x ′ j , γ ′ kl ) represents the half-density ˜Ψ 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 (cid:0) x i ( x ′ j ) (cid:1) ∈ ϕ ( U ) . By virtue of Equation (3.9), ( γ ′ kl ) belongs to the support of ψ ′ ( x ′ j ) if and only if γ ij = ∂x ′ k ∂x i ∂x ′ l ∂x j γ ′ kl (3.12)12elongs to the support of ψ ( x i ) (here ∂x ′ k /∂x i is the value of the derivative at ( x i ) ).Suppose now that the value ( x ′ j ) ∈ ϕ ′ (Int U ) . Then the corresponding value ( x i ) ∈ ϕ (Int U ) ⊂ ϕ ( U ) . In this situation, if ( γ ′ kl ) belongs to the support of ψ ′ ( x ′ j ) , then ( γ ij ) given by (3.12) belongs to supp ψ ( x i ) ⊂ K . Moreover, ( γ ′ kl ) is the value of the map (3.11)at ( x i , γ ij ) ∈ ϕ ( U ) × K . Thus ( γ ′ kl ) is an element of K ′ by definition of the latter set.We thus see that for every value ( x ′ j ) ∈ ϕ ′ (Int U ) the support of ψ ′ ( x ′ j ) is contained inthe compact set K ′ . Therefore, if the Γ R -support of ˜Ψ around x is compact and slowlychanging 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 ∈ M of a continuous Hilberthalf-density ˜Ψ , is compact and slowly changing if it is compact and slowly changing inevery coordinate system defined on a neighborhood of x . Finally, we will say that the Γ R -support of ˜Ψ is compact and slowly changing if the Γ R -support is such around everypoint of M . Lemma 3.5. If ˜Ψ and ˜Ψ ′ are continuous Hilbert half-densities of compact and slowlychanging Γ R -support, then any (finite) linear combination of them is a continuous Hilberthalf-density of compact and slowly changing Γ R -support.Proof. Suppose that U is an open subset of M and that a map ϕ : U R dim M defines acoordinate system ( x i ) .Consider now a linear combination z ˜Ψ + z ′ ˜Ψ ′ ≡ ˜Ξ , z, z ′ ∈ C and suppose that ψ and ψ ′ are (continuous) coordinate representations of, respectively, ˜Ψ and ˜Ψ ′ in the system ( x i ) . Then, for every x ∈ U , the function zψ + z ′ ψ ′ ≡ ξ substituted to the reconstruction formula (3.8) gives us an appropriate linear combinationof functions on Γ x representing the equivalence class z ˜Ψ (cid:0) x, ( ∂ x i ) (cid:1) + z ′ ˜Ψ ′ (cid:0) x, ( ∂ x i ) (cid:1) ∈ H x . ξ is thus a continuous coordinate representation of Ξ in the system ( x i ) . Since U is anarbitrary open subset of M , then ˜Ξ is continuous.Fix an arbitrary point x ∈ U . Let K be a compact subset of Γ R and U ⊂ U bean open neighborhood of x such that for every value ( x i ) ∈ ϕ ( U ) , the support of thefunction ψ ( x i ) related to ˜Ψ via the formulas (3.10) and (3.7), is contained in K . In thesame way, let K ′ be a compact subset of Γ R and U ′ ⊂ U be an open neighborhood of x such that for every value ( x i ) ∈ ϕ ( U ′ ) , the support of the function ψ ′ ( x i ) related to ˜Ψ ′ viathe formulas (3.10) and (3.7), is contained in K ′ .Obviously, if ξ ( x i ) is related via (3.10) to ξ , then ξ ( x i ) = zψ ( x i ) + z ′ ψ ′ ( x i ) . Therefore for every value ( x i ) ∈ ϕ ( U ∩ U ′ ) the support of ξ ( x i ) is contained in K ∪ K ′ .Since U ∩ U ′ is open and K ∪ K ′ compact, the Γ R -support of ˜Ξ around x is compact andslowly changing in the coordinate system ( x i ) . But x is an arbitrary point in M , and ( x i ) an arbitrary local coordinate system. Therefore the Γ R -support of ˜Ξ is compact andslowly changing. 13 .2 Pairing of Hilbert half-densities into scalar densities Consider now two Hilbert half-densities ˜Ψ and ˜Ψ ′ on M . Then the map M ∋ x ( ˜Ψ ′ | ˜Ψ)( x ) := ( ˜Ψ ′ ( x ) | ˜Ψ( x )) x ∈ ˜ C (3.13)is a scalar density on M . The lemma below ensures that if Hilbert half-densities satisfythe regularity conditions introduced in the previous section, then the resulting density issufficiently regular for our purposes. Lemma 3.6. Suppose that Hilbert half-densities ˜Ψ and ˜Ψ ′ are continuous and that the Γ R -support of ˜Ψ 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 : U → R dim M defininga coordinate system ( x i ) on U ⊂ M . Using (3.13) and (2.17) we obtain ˜ F (cid:0) x, ( ∂ x i ) (cid:1) = ( ˜Ψ ′ ( x ) | ˜Ψ( x )) x ( ∂ x i ) = (cid:10) ˜Ψ ′ (cid:0) x, ( ∂ x i ) (cid:1)(cid:12)(cid:12) ˜Ψ (cid:0) x, ( ∂ x i ) (cid:1)(cid:11) x , where h·|·i x is the inner product on the Hilbert space H x . Consequently, ˜ F (cid:0) x, ( ∂ x i ) (cid:1) = Z Γ x ˜Ψ ′ (cid:0) x, ( ∂ x i ) (cid:1) ˜Ψ (cid:0) x, ( ∂ x i ) (cid:1) dµ x = c Z Γ x ˜Ψ ′ (cid:0) x, ( ∂ x i ) (cid:1) ˜Ψ (cid:0) x, ( ∂ x i ) (cid:1) dµ Qx , (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 of ˜ F (cid:0) x, ( ∂ x i ) (cid:1) we would like to use Equation (2.7). In orderto do this we have to show that (for fixed x ) the integrand in (3.14) is a continuous functionof compact support. To this end let us note that if χ : Γ x → Γ R is a map (2.1) given bythe basis ( ∂ x i ) of T x M , and if ϕ ( x ) = ( x i ) , then χ − ⋆ h ˜Ψ ′ (cid:0) ϕ − ( x i ) , ( ∂ x i ) (cid:1) ˜Ψ (cid:0) ϕ − ( x i ) , ( ∂ x i ) (cid:1)i = ψ ′ ( x i ) ψ ( x i ) , (3.15)where ψ ′ ( x i ) and ψ ( x i ) are continuous functions related to, respectively, ˜Ψ ′ and ˜Ψ via theformulas (3.10) and (3.7). Thus the integrand in (3.14) is continuous.Let us fix a point x ∈ U and suppose that U ⊂ U is an open neighborhood of x introduced in Definition 3.3 for the half-density ˜Ψ . Then it follows from the assumptionsimposed on ˜Ψ that for every ( x i ) ∈ ϕ ( U ) , 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 , wecan use (2.7) to get ˜ F (cid:0) x, ( ∂ x i ) (cid:1) = c Z Γ R χ − ⋆ h ˜Ψ ′ (cid:0) x, ( ∂ x i ) (cid:1) ˜Ψ (cid:0) x, ( ∂ x i ) (cid:1)i ∆ dµ L . Using (3.15) once again we obtain f ( x i ) = ˜ F (cid:0) ϕ − ( x i ) , ( ∂ x i ) (cid:1) = c Z Γ R ψ ′ ( x i ) ψ ( x i ) ∆ dµ L (3.16)for every value ( x i ) ∈ ϕ ( U ) . 14ow, it follows from assumed continuity of ˜Ψ ′ and ˜Ψ that the function ϕ ( U ) × Γ R ∋ ( x i , γ ij ) ψ ′ ( x i ) ( γ ij ) ψ ( x i ) ( γ ij )∆( γ ij ) ∈ C (3.17)is continuous (note that the function ∆ is continuous and independent of ( x i ) , which followsfrom 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 ( U ) × K , where U is acompact set of non-empty interior Int U ∋ x .Let then s ≡ sup ( x i ,γ ij ) ∈ ϕ ( U ) × K (cid:12)(cid:12) ψ ′ ( x i ) ( γ ij ) ψ ( x i ) ( γ ij )∆( γ ij ) (cid:12)(cid:12) and Γ R ∋ ( γ ij ) h ( γ ij ) := ( s if ( γ ij ) ∈ K , otherwise . Since the set K is compact and the Lebesgue measure dµ L is regular [9], the function h is integrable over Γ R with respect to dµ L . Moreover, for every value ( x i ) ∈ ϕ ( U ) , | ψ ′ ( x i ) ψ ( x i ) ∆ | ≤ h (recall that the support of ψ ′ ( x i ) ψ ( x i ) is contained in K ). On the otherhand, ψ ′ ( x i ) ψ ( x i ) converges pointwise to ψ ′ ( x i ) ψ ( x i ) as ( x i ) → ( x i ) = ϕ ( x ) .These three facts allows us to apply the Lebesgue’s dominated convergence theoremto conclude that f given by (3.16), is continuous at ( x i ) = ϕ ( x ) . But x is an arbitrarypoint, and ( x i ) an arbitrary local coordinate system. Thus ( ˜Ψ ′ | ˜Ψ) is a continuous scalardensity on M . H Let H c be the set of all continuous Hilbert half-densities on M of compact M -supportand of compact and slowly changing Γ R -support. Any (finite) linear combination of ele-ments of H c is again a Hilbert half-density of compact M -support. This fact and Lemma3.5 guarantee that H c is a complex vector space. By virtue of Lemma 3.6 for any twoelements ˜Ψ and ˜Ψ ′ of H c , the scalar density ( ˜Ψ ′ | ˜Ψ) on M is continuous. This density isalso compactly supported and therefore it can be naturally integrated over M being aparacompact manifold. The following map H c × H c ∋ ( ˜Ψ ′ , ˜Ψ) 7→ h ˜Ψ ′ | ˜Ψ i := Z M ( ˜Ψ ′ | ˜Ψ) ∈ C , (3.18)where the integral at the r.h.s. is the integral of the scalar density ( ˜Ψ ′ | ˜Ψ) , is an innerproduct on H c . Let us emphasize that the assumption that the Γ R -support of each element of H c 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 , the set Γ R is the set R + of all positive real numbers. Let x bethe canonical coordinate on M = R and A := { ( x , γ ) ∈ R × Γ R | x ≥ , γ − ≥ , − x γ ≥ } .Define a function ψ on R × Γ R as follows: ψ is zero outside A and ψ ( x , γ ) := √ x ( γ − − x γ ) on A . It is clear that ψ is continuous and for every x ∈ R , supp ψ ( x ) is compact. But for every openneighborhood U of x = 0 , S x i ∈ U supp ψ ( x ) = [1 , ∞ [ . Thus the Hilbert half-density ˜Ψ on M obtainedfrom ψ by means of the reconstruction formula (3.8), is continuous and of compact Γ R -support everywhereon M (and of compact M -support), but the Γ R -support is not slowly changing around x = 0 . Themeasure ∆ dµ L on Γ R is here of the form ( γ ) − dγ [1]. Using this fact and (3.16) it is not difficult torealize that the coordinate representation of ( ˜Ψ | ˜Ψ) in the coordinate x , diverges to infinity as ( x ) − , as x goes to zero from the right. This means that the density ( ˜Ψ | ˜Ψ) is not continuous at the point x = 0 and is also non-integrable over M . ˜Ψ ∈ H c , the scalardensity ( ˜Ψ | ˜Ψ) is continuous and non-negative i.e., for every x ∈ M , ( ˜Ψ | ˜Ψ)( x ) ≥ (seethe formula (2.16)). Therefore h ˜Ψ | ˜Ψ i ≥ . Suppose that ˜Ψ( x ) = 0 for a point x ∈ M .Then, by continuity of ( ˜Ψ | ˜Ψ) its support contains a non-empty open set. Consequently, h ˜Ψ | ˜Ψ i > and (3.18) is positive definite.The completion H of H c in the norm induced by the inner product (3.18), is a Hilbertspace built over the set Q ( M ) .Let us end this section by a remark concerning the standard Hilbert space H QM ofthe ordinary quantum mechanics. This space is usually defined as L ( R , dµ L ) , where dµ L is the Lebesgue measure. Note, however, that wave functions in H QM can be viewed asHilbert half-densities of special sort defined on R . Indeed, the set C of complex numbersequipped with the map C ∋ ( z, z ′ ) ¯ zz ′ ∈ C is a one-dimensional complex Hilbert space. If ˜ C / x , ( x ∈ R ), is the set of all half-densities over T x R valued in the Hilbert space C , then a section of S x ∈ R ˜ C / x is a sortof a Hilbert half-density. It is clear that each wave function in H QM , if expressed in aCartesian coordinate system on R , can be understood as a coordinate representation ofsuch a half-density. It is also not difficult to convince oneself that the inner product of twowave functions in H QM , can be expressed in terms of ( i ) pairing of corresponding Hilberthalf-densities to a scalar density and ( ii ) integrating of the resulting scalar density over R . Thus, counterintuitively, the Hilbert spaces H and H QM are fairly similar. H Given manifold M and signature ( p, p ′ ) , the only choice we have to make in order to obtainthe Hilbert space H , is the choice of a diffeomorphism invariant field (2.11) of invariantmeasures. 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 d ˇ µ x are two such measure fields on M , and H and ˇ H the resulting Hilbert spaces, then it follows from (2.13) that H ∋ ˜Ψ ˜Ψ √ c ∈ ˇ H (3.19)is a unitary map.Thus all Hilbert spaces constructed according to the prescription presented in Section3 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 is unique up to natural isomorphisms. H Let ˜ F be a scalar density on M , and θ : M → M a diffeomorphism. The diffeomorphism θ acts on the density ˜ F by means of the following pull-back: ( θ ∗ ˜ F )( x, e x ) := ˜ F (cid:0) θ ( x ) , θ ′ e x (cid:1) , (4.1)16here e x ≡ ( e xi ) i =1 ,..., dim M is a basis of T x M , θ ′ : T x M → T θ ( x ) M is the tangent mapgenerated by θ and θ ′ e x ≡ (cid:0) θ ′ ( e xi ) (cid:1) 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 ( θ ∗ ˜ F )( x, Λ e x ) = ˜ F (cid:0) θ ( x ) , θ ′ Λ e x (cid:1) , where Λ ≡ (Λ j i ) is any non-singular matrix. If e x = ( e xi ) , then Λ e x = (Λ j i e xj ) . Hence byvirtue of linearity of θ ′ θ ′ Λ e x = (cid:0) θ ′ (Λ j i e xj ) (cid:1) = (cid:0) Λ j i θ ′ ( e xj ) (cid:1) = Λ (cid:0) θ ′ ( e xj ) (cid:1) = Λ θ ′ e x . (4.2)Consequently, ( θ ∗ ˜ F )( x, Λ e x ) = ˜ F ( θ ( x ) , Λ θ ′ e x ) = | det Λ | ˜ F ( θ ( x ) , θ ′ e x ) = | det Λ | ( θ ∗ ˜ F )( x, e x ) . Let ˜ F be a scalar density integrable over M . Then for every diffeomorphism θ of M [10] Z M θ ∗ ˜ F = Z M ˜ F . (4.3) H c Let us consider a Hilbert half-density ˜Ψ being an element of H c . If θ is a diffeomorphismof M , θ ′ : T x M → T θ ( x ) M the corresponding tangent map, and e x a basis of T x M , then ˜Ψ( θ ( x ) , θ ′ e x ) (4.4)is an element of the Hilbert space H θ ( x ) , that is, an equivalence class of functions on Γ θ ( x ) . Since ˜Ψ ∈ H c , the equivalence class (4.4) can be represented by a unique continuouscompactly supported function on Γ θ ( x ) (see Lemma 2.2).If θ − ′ : T θ ( x ) M → T x M is the inverse of θ ′ , then the pull-back θ − ′∗ : Γ x → Γ θ ( x ) is adiffeomorphism. Thus θ − ′∗ can be used to pull-back functions on Γ θ ( x ) to ones on Γ x .To pull-back the equivalence class (4.4) of functions on Γ θ ( x ) to an equivalence classof functions on Γ x being an element of H x , we will proceed as follows. First we will pull-back by means of θ − ′∗ the unique continuous representative of (4.4), obtaining therebya continuous compactly supported function on Γ x . Then we will find an element of H x defined by this resulting function, denote it by θ − ′∗ ⋆ ˜Ψ( θ ( x ) , θ ′ e x ) (4.5) In [10] scalar densities are defined in a different way than in the present paper, but it is not difficultto realize that both definitions are equivalent. The standard symbol for the pull-back of a metric q under a diffeomorphism θ is θ ∗ q . To be consistentwith the standard notation we should use in the formula (4.5) the simpler symbol θ − ∗ ⋆ instead of θ − ′∗ ⋆ .However, here we work with a multi-level construction: the baseline level is the manifold M , and thenext levels are in turn: the tangent space T x M , the space Γ x of scalar products on the tangent space andfinally the functions on Γ x constituting H x . In the symbol θ − ′∗ ⋆ and the like, each superscript ′ , ∗ and ⋆ corresponds to each level above the baseline one and makes it easier to keep track where we are. of (4.4).Using (4.2) and linearity of the pull-back θ − ′∗ ⋆ one easily shows that θ − ′∗ ⋆ ˜Ψ( θ ( x ) , θ ′ Λ e x ) = | det Λ | / θ − ′∗ ⋆ ˜Ψ( θ ( x ) , θ ′ e x ) , which means that the following map e x → θ − ′∗ ⋆ ˜Ψ( θ ( x ) , θ ′ e x ) 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 thatfor every x ∈ M and for every basis e x of T x M , ( θ ∗ ˜Ψ)( x, e x ) = θ − ′∗ ⋆ ˜Ψ( θ ( x ) , θ ′ e x ) . (4.6)We will say that θ ∗ ˜Ψ is pull-back of ˜Ψ under the diffeomorphisms θ . Lemma 4.1. The Hilbert half-density θ ∗ ˜Ψ is an element of H c .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 assumedthat ˜Ψ ∈ H c .Regarding ( i ): if supp ˜Ψ is the M -support of ˜Ψ , then θ − (supp ˜Ψ) is the M -supportof θ ∗ ˜Ψ . Since supp ˜Ψ is compact and θ − a continuous map, θ − (supp ˜Ψ) is also compact.Regarding ( ii ): suppose that U is an open subset of M and that a map ϕ : U R dim M defines a coordinate system ( x i ) on U . Then the diffeomorphism θ can be used to pull-back ( x i ) to a coordinate system (¯ x i ) on θ − ( U ) defined by the map ϕ ◦ θ : θ − ( U ) → R dim M .Now let us find a relation between the (continuous) coordinate representation ψ of ˜Ψ in the system ( x i ) (see (3.7)), and a coordinate representation θ ∗ ψ of θ ∗ ˜Ψ defined on theset ( ϕ ◦ θ )( θ − ( U )) × Γ R by the system (¯ x i ) .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 ( θ ∗ ˜Ψ)( x, e x , γ x ) = ˜Ψ( θ ( x ) , θ ′ e x , θ − ′∗ γ x ) , (4.7)where the number at the r.h.s. is a value of the continuous representative of ˜Ψ( θ ( x ) , θ ′ e x ) .Assume now that x ∈ θ − ( U ) . Then x = ( θ − ◦ ϕ − )(¯ x i ) for some value (¯ x i ) ∈ ϕ ( U ) and if the basis e x = ( ∂ ¯ x i ) , then θ ′ e x = ( ∂ x i ) . Moreover, ( θ − ′∗ γ x )( ∂ x i , ∂ x j ) = γ x ( θ − ′ ∂ x i , θ − ′ ∂ x i ) = γ x ( ∂ ¯ x i , ∂ ¯ x j ) and, consequently, if γ x = ¯ γ ij d ¯ x i ⊗ d ¯ x j , then θ − ′∗ γ x = ¯ γ ij dx i ⊗ dx j . Using all these results we can transform (4.7) obtaining ( θ ∗ ˜Ψ) (cid:0) ( θ − ◦ ϕ − )(¯ x i ) , ( ∂ ¯ x i ) , ¯ γ ij d ¯ x i ⊗ d ¯ x j (cid:1) = ˜Ψ (cid:0) ϕ − (¯ x i ) , ( ∂ x i ) , ¯ γ ij dx i ⊗ dx j (cid:1) . Applying this procedure we avoid to prove that if R Γ θ ( x ) ¯ΨΨ dµ θ ( x ) = 0 then R Γ x θ − ′∗ ⋆ ( ¯ΨΨ) dµ x = 0 . ϕ ( U ) × Γ R = ( ϕ ◦ θ )( θ − ( U )) × Γ R ( θ ∗ ψ )(¯ x i , ¯ γ ij ) = ψ (¯ x i , ¯ γ ij ) . (4.8)Note that ψ at the r.h.s. of the equation above, is the continuous coordinate representationof ˜Ψ in the system ( x i ) —to be continuous, for every value (¯ x i ) ∈ ϕ ( U ) the representationmust come from the unique continuous representative of ˜Ψ( ϕ − (¯ 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 continuityof ψ . This is sufficient for θ ∗ ˜Ψ to be continuous, since (¯ x i ) is an arbitrary local coordinatesystem on M .Regarding ( iii ): let us consider now a point θ ( x ) ∈ U . Let U ⊂ U be an openneighborhood of θ ( x ) and K a compact subset of Γ R such that for every value (¯ x i ) ∈ ϕ ( U ) ,the support of the function ψ (¯ x i ) related to ψ via the formula (3.10), is contained in K —theexistence of U and K follows from the assumption, that ˜Ψ is an element of H c .By virtue of (4.8), for every value (¯ x i ) ∈ ( ϕ ◦ θ )( θ − ( U )) , 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 R -support of the pull-back θ ∗ ˜Ψ around x is compact and slowly changing in the coordinatesystem (¯ x i ) . But x is an arbitrary point of M and (¯ x i ) an arbitrary local coordinatesystem on the manifold. Therefore θ ∗ ˜Ψ is a Hilbert half-density on M of compact andslowly changing Γ R -support. H Lemma 4.2. Suppose that ˜Ψ , ˜Ψ ′ ∈ H c and θ is a diffeomorphism of M . Then the pull-back θ ∗ preserves the inner product (3.18) on H c : h θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ i = h ˜Ψ ′ | ˜Ψ i . Proof. Consider first the scalar density ( θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ) (see (3.13) for the definition). If e x isa basis of T x M , then ( θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ)( x, e x ) = h ( θ ∗ ˜Ψ ′ )( x, e x ) | ( θ ∗ ˜Ψ)( x, e x ) i x == h θ − ′∗ ⋆ ˜Ψ ′ ( θ ( x ) , θ ′ e x ) | θ − ′∗ ⋆ ˜Ψ( θ ( x ) , θ ′ e x ) i x == Z Γ x θ − ′∗ ⋆ (cid:0) ˜Ψ ′ ( θ ( x ) , θ ′ e x ) ˜Ψ( θ ( x ) , θ ′ e x ) (cid:1) dµ x == Z Γ θ ( x ) (cid:0) ˜Ψ ′ ( θ ( x ) , θ ′ e x ) ˜Ψ( θ ( x ) , θ ′ e x ) (cid:1) ( θ − ′∗ ) ⋆ dµ x == Z Γ θ ( x ) (cid:0) ˜Ψ ′ ( θ ( x ) , θ ′ e x ) ˜Ψ( θ ( x ) , θ ′ e x ) (cid:1) dµ θ ( x ) —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 ′ ( e x )) and ˜Ψ( θ ( x ) , θ ′ ( e x )) , in the forth step we applied (2.5), finally in the last19tep we used the diffeomorphism invariance of the measure field x dµ x (see Equation(2.12)). Thus ( θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ)( x, e x ) = h ˜Ψ ′ ( θ ( x ) , θ ′ e x ) | ˜Ψ( θ ( x ) , θ ′ e x ) i θ ( x ) = ( ˜Ψ ′ | ˜Ψ)( θ ( x ) , θ ′ e x ) . Comparing this with (4.1) we see that ( θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ) = θ ∗ ( ˜Ψ ′ | ˜Ψ) . By virtue of this result and Equation (4.3) h θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ i = Z M ( θ ∗ ˜Ψ ′ | θ ∗ ˜Ψ) = Z M θ ∗ ( ˜Ψ ′ | ˜Ψ) = Z M ( ˜Ψ ′ | ˜Ψ) = h ˜Ψ ′ | ˜Ψ i . Let us define an operator on H c : H c ∋ ˜Ψ u θ ˜Ψ := θ − ∗ ˜Ψ ∈ H c . It follows from (4.7) that u θ is linear. Manifestly, for every ˜Ψ ∈ H c Ψ = u θ ( θ ∗ Ψ) , which means that u θ is surjective. Lemma 4.2 guarantees that u θ preserves the innerproduct on H c being a dense linear subspace of H . Taking into account all these facts,we see that the operator u θ can be uniquely extended to a unitary operator U ( θ ) on H .It is not difficult to check that for two diffeomorphisms θ and θ , U ( θ ) ◦ U ( θ ) = U ( θ ◦ θ ) . Thus θ U ( θ ) (4.9)is a unitary representation of the group Diff( M ) of all diffeomorphisms of M on the Hilbertspace H . H Suppose that the set Q ( M ) is non-empty. Then the following surjective [1] map Q ( M ) ∋ q κ x ( q ) := q x ∈ Γ x , 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 aquantum 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 ) shouldcontain tensor products of the Hilbert spaces { H x } .20t is easy to realize that the structure of H does not meet this expectation. Indeed,the inner product (3.18) is an integral, that is, an “uncountable sum” of values of innerproducts on the Hilbert spaces { H x } . Therefore H is more like a direct integral of Hilbertspaces [11]: Z ⊕M H x . This fact suggests that H (in the case of signature (3 , ) is not well suited for quantizationof the ADM formalism. Fortunately, it is relatively easy to find a way around this problem.Namely, consider the set N N of all N -element subsets of M : N N := (cid:8) { x , . . . , x N } ≡ { x K } ⊂ M | x I = x J for I = J (cid:9) . (5.1)It is possible to define on N N a differential structure in such a way that the set becomesa smooth paracompact manifold locally diffeomorphic to M N —for details see AppendixA.1. We will associate with each point y = { x K } of N N a Hilbert space H ⊗ y ∼ = H x ⊗ . . . ⊗ H x N (5.2)and then construct a Hilbert space H N using half-densities on N N valued in the spaces { H ⊗ y } .Next, we will merge all the spaces {H N } into the desired Hilbert space H . To dothis we will follow some feature of states in the kinematical Hilbert space H LQG of LoopQuantum Gravity (see e.g. [12]): if a spin-network state Ψ ∈ H LQG depends non-triviallyon a classical d.o.f., a spin-network state Ψ ′ ∈ H LQG is independent of, then Ψ and Ψ ′ are orthogonal. Note now, that if N < N ′ and H ⊗ y and H ⊗ y ′ are the Hilbert spaces (5.2)associated with, respectively, y ∈ N N and y ′ ∈ N N ′ , then wave functions in H ⊗ y ′ depend oncertain d.o.f. κ x , wave functions in H ⊗ y are independent of. Therefore we would like H ⊗ y and H ⊗ y ′ to be orthogonal as building blocks of H . To achieve this goal, we will define H asthe orthogonal sum H := ∞ M N =1 H N . (5.3)Note also that the structure of this H will resemble to a certain degree the structure of theFock space.Obviously, the Hilbert space H will contain all finite tensor products of the Hilbertspaces { H x } . Therefore the space H constructed in the case of signature (3 , , seems tobe better suited for quantization of the ADM formalism than H alone. (5.3) of H A rigorous construction of H N , we are going to present below, is fairly long and technicallyinvolved (this concerns in particular the construction of a smooth atlas on the set N N ).Let us note that an other method to take into account the tensor products of { H x } , is touse in (5.3) the tensor product N N H of N copies of H , instead of H N . This may seemto be the simplest way to achieve the goal, which does not require any effort, but a closerlook at this construction makes clear that it is not the case.Namely, it is easy to realize that, given { x K } ∈ N N , the space N N H contains N ! tensor products of Hilbert spaces { H x , . . . , H x N } —these tensor products differ from each21ther by the ordering of the factors and are given by all possible orderings. From a physicalpoint of view the ordering of the factors in H x ⊗ . . . ⊗ H x N is irrelevant—each such tensorproduct describes a space of quantum states of the same quantum system, obtained bya quantization of a classical system whose configuration space is given by the collection { κ x , . . . , κ x N } of classical d.o.f..Consequently, a generic state in N N H consists of N ! distinct states in H N . It isthen clear that working with such a state would be rather cumbersome. To avoid this, onewould have to impose on elements of N N H some restrictions in order to isolate thosestates, which fit exactly single states in H N .Moreover, for many purposes it would be convenient or even necessary to expresselements of N N H as half-densities on M N valued in the tensor products of { H x } and toimpose some regularity conditions on the half-densities. But in order to do this, one wouldhave to repeat some steps of the construction of H N .Thus the application of N N H does require some additional effort, which makes thisspace not as attractive as it seemed to be at the very beginning.In our opinion the advantage of H N over N N H is conceptual simplicity of H N : theidea of its construction is simple and natural, and complies with the fact that the orderingof factors in H x ⊗ . . . ⊗ H x N is physically irrelevant. Consequently, each state in H N fitswell our need to take into account the tensor products of { H x } , without the necessity toimpose on it any extra conditions.Moreover, once the space H N is rigorously built, one can work with it without theneed to refer to many technical details of its construction like e.g. the construction of thesmooth atlas on N N . For convenience of the reader not interested in such details, we willplace a considerable part of these technicalities in the appendix to this paper. H N The construction of the Hilbert space H N will follow as closely as possible the constructionof the Hilbert space H described in Section 3.Let ( p, p ′ ) be the signature of metrics on M fixed at the very beginning of Section 3for the sake of the construction of the Hilbert space H . Let us fix additionally a naturalnumber N ≥ . Suppose that V ⊕ is a real vector space of dimension N ( p + p ′ ) and that a decomposition V ⊕ = N M I =1 V I (5.4)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 ∈ { , . . . , N } let us choose γ I ∈ Γ I and define V ⊕ × V ⊕ ∋ ( v, ˇ v ) γ ⊕ ( v, ˇ v ) := N X I =1 γ I ( v I , ˇ v I ) ∈ R , (5.5)22here v = N X I =1 v I , ˇ v = N X I =1 ˇ v I , v I , ˇ v I ∈ V I . 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 (5.5) (with the fixeddecomposition (5.4)). Let us emphasize that Γ ⊕ is a proper subset of the set of all scalarproducts 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(5.5). Obviously, this assignment, Γ ⊕ ∋ γ ⊕ b ( γ ⊕ ) := ( γ , . . . , γ N ) ∈ Γ × . . . × Γ N ≡ ą Γ K , (5.6)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 constructa map Γ ⊕ ∋ γ ⊕ ( χ × . . . × χ N )( b ( γ ⊕ )) ∈ Γ N R ⊂ R N dim Γ I (5.7)which define a global coordinate system on Γ ⊕ . This map can be used to “pull-back” thetopology from R N dim Γ I onto Γ ⊕ . Obviously, coordinate systems given by all maps (5.7)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 dµ × . . . × dµ N ≡ ą dµ K is well-defined. This product is also a regular Borel measure on the Cartesian product Γ × . . . × Γ N —this is because [9] each dµ I is a regular Borel measure on a second countablel.c.H. space Γ I . We can use the bijection (5.6) to push-forward this measure obtainingthereby a (regular Borel) measure on Γ ⊕ , which will be denoted by dµ × : dµ × := ( b − ) ⋆ (cid:0) ą dµ K (cid:1) . Let H ⊗ := L (Γ ⊕ , dµ × ) ∼ = L (Γ × . . . × Γ N , dµ × . . . × dµ N ) . Recall that each L (Γ I , dµ I ) is separable [1] and each measure dµ I is σ -finite. Theseproperties of the space and the measure guarantee [13] that H ⊗ ∼ = L (Γ , dµ ) ⊗ . . . ⊗ L (Γ N , dµ N ) . (5.8)Let us emphasize that the Cartesian product Γ × . . . × Γ N , unlike the constructionof Γ ⊕ , does require the spaces { Γ I } to be ordered. However, neither the topology nor thedifferential structure nor the measure dµ × induced on Γ ⊕ depends on the ordering. Thusthe 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 ( α ⋆ dµ ) × ( β ⋆ dν ) = ( α × β ) ⋆ ( dµ × dν ) . (5.9)23his 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 toAppendix B.The lemma above can be easily generalized to a product of any finite number of suitablemeasures since [9] ( i ) a product X × Y of second countable l.c.H. spaces X and Y is sucha 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 × interms of an integral over Γ N R . Recall first that each Γ I in (5.6) and Γ R are second countablel.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 > and dµ Q I is the invariant measure on Γ I givenby the natural metric on this space (see (2.9)). On the other hand, we can treat ther.h.s. of (2.7) as the definition of a positive functional on C c (Γ R ) , which (by virtue of theRiesz 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 tomake the following transformations: Z Γ ⊕ ( b ⋆ Ψ) dµ × = Z Ś Γ K Ψ (cid:0) ą dµ K (cid:1) = c Z Ś Γ K Ψ (cid:0) ą dµ Q K (cid:1) == c Z Ś Γ K Ψ (cid:0) ą χ − K⋆ (∆ dµ L ) (cid:1) = c Z Ś Γ K Ψ (cid:16)(cid:0) ą χ − K (cid:1) ⋆ (cid:0) ą ∆ dµ L (cid:1)(cid:17) == c Z Γ N R (cid:16) ( ą χ − K ) ⋆ Ψ (cid:17) (cid:0) ą ∆ dµ L (cid:1) , (5.10)where b ⋆ Ψ ∈ C c (Γ ⊕ ) , c = c · . . . · c N , the product ą χ − K ≡ χ − × . . . × χ − N is given by the maps appearing in (5.7), and Ś ∆ dµ L is the product of N copies of ∆ dµ L .Equation (5.10) can be now used to prove the following generalization of Lemma 2.2: Lemma 5.2. Let Ψ : Γ ⊕ → C be continuous. If Z Γ ⊕ ¯ΨΨ dµ × = 0 , then Ψ = 0 . N N Here we will apply the construction of the Hilbert space H ⊗ just presented to associatethe Hilbert space H ⊗ y with every point y ≡ { x K } of the manifold N N (let us recall thatthe set N N is defined by the formula (5.1), and a smooth atlas on N N is introduced inAppendix A.1).As shown in Appendix A.3, for every y ∈ N N , there exists a distinguished decomposi-tion of the tangent space T y N N into a direct sum of (linear subspaces naturally isomorphicto) the tangent spaces { T x I M} x I ∈ y : T y N N = M x K ∈ y T x K M . (5.11)24uppose 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 decomposi-tion (5.11) define a scalar product γ ⊕ y on T y N N of signature ( N p, N p ′ ) according to theprescription (5.5). We will denote by Γ ⊕ y the set of all such scalar products on T y N 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 . Denote by dµ × y the measure on Γ ⊕ y defined as the push-forward of the measure dµ x × . . . × dµ x N given by the inverse of the natural bijection (see (5.6)) b y : Γ ⊕ y → Γ x × . . . × Γ x N . (5.12)This allows us to associate with the point y the following Hilbert space (see (5.8)): H ⊗ y := L (Γ ⊕ y , dµ × y ) ∼ = L (Γ x , dµ x ) ⊗ . . . ⊗ L (Γ x N , dµ x N ) = H x ⊗ . . . ⊗ H x N . (5.13)Let us emphasize that the measure dµ × y does not depend on the choice of the ordering ofthe spaces { Γ x K } in (5.12). Consequently, the Hilbert space H ⊗ y does not distinguish anyordering of the Hilbert spaces { H x K } in (5.13). Definition Let ˜ H ⊗ y denote the pseudo-Hilbert space of all half-densities over T y N N val-ued in H ⊗ y . We will use the symbol ( ·|· ) y to represent the density product on ˜ H ⊗ y . A map ˜Ψ from N N to ˜ H ⊗ := [ y ∈N N ˜ H ⊗ y (5.14)such that ˜Ψ( y ) ∈ ˜ H ⊗ y , will be called Hilbert half-density on N N . Equivalently, one canthink of ˜Ψ as of a section of the bundle-like set ˜ H ⊗ . Regularity conditions Again, we would like to impose on the Hilbert half-densitiesjust defined, some regularity conditions, which ( i ) would be helpful while defining physicaloperators on H N and ( ii ) will ensure that the half-densities paired by means of the densityproducts { ( ·|· ) y } y ∈N N , give integrable scalar densities on N N .The regularity conditions, we are going to introduce here, will be analogous to thosepresented in Section 3.1.2, however some differences will be unavoidable. The reason isthat each space Γ ⊕ y does not consist of all scalar products on T y N N of signature ( N p, N p ′ ) ,but contains only some special ones. Therefore, introducing and working with these newregularity conditions, we will restrict ourselves to some coordinate systems on N N , which,in a sense, are compatible with decompositions (5.11) and, thereby, with the special formof the elements of Γ ⊕ y .We showed in Appendix A.1 that for every y ∈ N N , there exist local charts { ( U K , ϕ K ) } K =1 ,...,N on M such that ( i ) the sets { U K } are pairwise disjoint and ( ii ) there exists adistinguished diffeomorphism from an open neighborhood Z y of y onto U × . . . × U N . Thecomposition Φ of this diffeomorphism with the map ϕ × . . . × ϕ N and the set Z y form achart ( Z y , Φ) on N N (see (A.11)). All charts of this sort constitute a smooth atlas on themanifold denoted in the appendix by A . Here we will extend this atlas by admitting all25harts obtained by restricting domains of charts in A . The extended atlas will be denotedby A ′ .Let Σ be the set of all permutation of the sequence (1 , . . . , N ) . Every chart in A ′ defines a local coordinate system ( x i , . . . , x i N N ) ≡ ( x a ) (5.15)on N N , which is compatible with the decomposition (5.11) in the following sense: thereexists σ ∈ Σ such that the tangent vectors ( ∂ x iI ) (with fixed I ) form a basis of T x σ ( I ) M in (5.11) (see Appendix A.3 for a justification of this claim). Then each element of Γ ⊕ y , y = { x K } , reads γ ⊕ y = ( γ ⊕ y ) ab dx a ⊗ dx b = N X K =1 ( γ x σ ( K ) ) i K j K dx i K K ⊗ dx j K K . (5.16)Thus the map Γ ⊕ y ∋ γ ⊕ y (cid:0) ( γ x σ (1) ) i j , . . . , ( γ x σ ( N ) ) i N j N (cid:1) ∈ Γ N R is of the sort of the map (5.7).The description (5.16) of γ ⊕ y in terms of the components (cid:0) ( γ x σ ( K ) ) i K j K (cid:1) is more ex-plicit than that in terms of (cid:0) ( γ ⊕ y ) ab (cid:1) . However, the symbols (cid:0) ( γ x σ ( K ) ) i K j K (cid:1) are fairlycomplex and thereby somewhat unreadable. Therefore we would like to use the compo-nents (cid:0) ( γ ⊕ y ) ab (cid:1) 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 willallow us to treat the set (cid:0) ( γ ⊕ y ) ab (cid:1) (being in fact an element of R ( N dim M ) ) as an elementof Γ N R and identify the two sets of components under consideration: (cid:0) ( γ ⊕ y ) ab (cid:1) ≡ (cid:0) ( γ x σ (1) ) i j , . . . , ( γ x σ ( N ) ) i N j N (cid:1) . Let us now introduce a coordinate representation of a Hilbert half-density ˜Ψ on N N .To this end consider a chart ( Z, Φ) ∈ A ′ and the corresponding coordinate system (5.15).Given y ∈ Z , ˜Ψ( y ) is a half-density over T y N N valued in H ⊗ y , which means that the valueof ˜Ψ( y ) on the basis ( ∂ x i , . . . , ∂ x iNN ) ≡ ( ∂ x a ) of the tangent space, is an element of theHilbert space: ˜Ψ (cid:0) y, ( ∂ x a ) (cid:1) ∈ H ⊗ y . Thus ˜Ψ (cid:0) y, ( ∂ x a ) (cid:1) is an equivalence class of a function Γ ⊕ y ∋ γ ⊕ ˜Ψ (cid:0) y, ( ∂ x a ) , γ ⊕ (cid:1) ∈ C . Expressing y by means of the coordinates and γ ⊕ as in (5.16) we obtain a coordinaterepresentation of ˜Ψ being the map Φ( Z ) × Γ N R ∋ ( x a , γ ⊕ ab ) ψ ( x a , γ ⊕ ab ) := ˜Ψ (cid:0) Φ − ( x a ) , ( ∂ x a ) , γ ⊕ ab dx a ⊗ dx b (cid:1) ∈ C . (5.17)Now we can introduce the notion of continuous Hilbert half-densities on N N of compactand slowly changing Γ N R -support exactly as we did in Section 3.1.2 in the case of Hilberthalf-densities on M with only three exceptions:26. the scalar product components ( γ ⊕ ab ) in (5.17) do not describe arbitrary scalar prod-ucts on T y N N , y = Φ − ( 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 N N , but onlythose given by charts in A ′ .3. 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 N N in the system is unique (provided it exists) and ( ii ) appropriate counterparts of Lemmas 3.2, 3.4 and 3.5 can beproven in the same way without any essential changes. Suppose that ˜Ψ and ˜Ψ ′ are Hilbert half-densities on N N . Clearly, the map N N ∋ y ( ˜Ψ ′ | ˜Ψ)( y ) := ( ˜Ψ ′ ( y ) | ˜Ψ( y )) y ∈ ˜ C (5.18)is a scalar density on N N ( ˜ C here is defined analogously to ˜ C in Section 3.1.1).As before, if we assume that both half-densities ˜Ψ and ˜Ψ ′ are continuous and that the Γ N R -support of one of them is compact and slowly changing, then the scalar density (5.18)is continuous. This fact can be proven analogously to the proof of Lemma 3.6. The onlyessential difference is a bit more complicated passage from the counterpart of Equation(3.14) to the counterpart of Equation (3.16), where now Equation (5.10) should be used. H N Let H cN be the vector space of all continuous Hilbert half-densities on N N of compact N N -support and of compact and slowly changing Γ N R -support. For any two elements ˜Ψ and ˜Ψ ′ of H cN , the scalar density ( ˜Ψ ′ | ˜Ψ) on N N is continuous and of compact support andtherefore the density can be naturally integrated over this paracompact manifold. Thefollowing map H cN × H cN ∋ ( ˜Ψ ′ , ˜Ψ) 7→ h ˜Ψ ′ | ˜Ψ i := Z N N ( ˜Ψ ′ | ˜Ψ) ∈ C , (5.19)where the integral at the r.h.s. is the integral of the scalar density ( ˜Ψ ′ | ˜Ψ) , is an innerproduct on H cN .By definition, the Hilbert space H N is the completion of H cN in the norm induced bythe inner product (5.19). The regularity conditions imposed on elements of H cN are expressed in terms of coordinate systemsdefined by charts in A ′ . Therefore the result of this analogous proof will be a conclusion that a coordinaterepresentation of the density (5.18) in every coordinate system of this special sort, is continuous. But thisis sufficient to claim that all coordinate representations of the density are continuous (see Equation (3.5))and thereby the density is continuous. .3.5 Uniqueness of H N Let us recall that to construct the Hilbert spaces H and H N we used the same diffeomor-phism invariant field x dµ x of invariant measures. If for this purpose we used an othersuch field x d ˇ µ x instead, then we would obtain an other Hilbert space ˇ H N . It is notdifficult to realize (see Equation (2.13)) that there exists a positive number c such that H N ∋ ˜Ψ ˜Ψ √ c N ∈ ˇ H N (5.20)is a unitary map.Thus we conclude that the Hilbert space H N is unique up to natural isomorphisms(5.20). M on the Hilbert space H N Let θ be a diffeomorphism of M . It induces the following map: N N ∋ { x , . . . , x N } 7→ Θ( { x , . . . , x N } ) := { θ ( x ) , . . . , θ ( x N ) } ∈ N N being a diffeomorphism on N N —see Appendix A.4. Diffeomorphisms of this sort form asubgroup of the diffeomorphism group on N N . We will denote this subgroup by Diff M ( N N ) .We showed in Appendix A.4 that each Θ ∈ Diff M ( N N ) preserves1. the atlas A on N N ,2. the decompositions (5.11),3. the spaces { Γ ⊕ y } y ∈N N .Let ˜Ψ be a Hilbert half-density belonging to H cN , and let Θ ∈ Diff M ( N N ) . Followingthe definition (4.6) of the pull-back of a Hilbert half-density on M , we define the pull-backof ˜Ψ (Θ ∗ ˜Ψ)( y, e y ) := Θ − ′∗ ⋆ ˜Ψ(Θ( y ) , Θ ′ e y ) , (5.21)where e y is a basis of T y N N , and Θ ′ is the tangent map given by the diffeomorphism Θ .It can be shown that the pull-back Θ ∗ ˜Ψ is again an element of H cN for every Θ ∈ Diff M ( N N ) . A proof of this fact is similar to the proof of Lemma 4.1, one has only takeinto account that each element of Diff M ( N N ) preserves the atlas A ′ (since it preserves theatlas A ).Moreover, every diffeomorphism Θ ∈ Diff M ( N N ) preserves the inner product (5.19).This can be proven analogously to Lemma 4.2. The only extra work, which has to be done,is 1. to note that if ˜Ψ ∈ H cN , then for every y ∈ N N and for every basis e y of T y N N , theequivalence class ˜Ψ( y, e y ) ∈ H ⊗ y possesses a unique continuous compactly supportedrepresentative;2. to prove that the measure field on N N y → dµ × y (5.22)is invariant with respect to the action of all diffeomorphisms in Diff M ( N N ) .28o this latter end recall that if y = { x K } ∈ N N , then the measure dµ × y is given by thepush-forward of the measure Ś dµ x K under the bijection b − y (see (5.12)). Let y ′ = { x ′ K } be a point of N N such that x K = θ ( x ′ K ) , which means that y = Θ( y ′ ) . As shown inAppendix A.4, the bijections b y and b y ′ intertwine the pull-back Θ ′∗ : Γ ⊕ y → Γ ⊕ y ′ andthe pull-back Ś θ ′∗ : Ś Γ x K → Ś Γ x ′ K (see Equation (A.29) for definition of Ś θ ′∗ andEquation (A.30) for the relation between the bijections and the pull-backs). Consequently, (Θ ′∗ ) ⋆ dµ × y = (Θ ′∗ ) ⋆ ( b − y ) ⋆ ą dµ x K = ( b − y ′ ) ⋆ (cid:0) ą θ ′∗ (cid:1) ⋆ ą dµ x K == ( b − y ′ ) ⋆ ą ( θ ′∗ ⋆ dµ x K ) = ( b − y ′ ) ⋆ ą dµ x ′ K = dµ × y ′ —here in the third step we used Lemma 5.1, and in the forth step the diffeomorphisminvariance of the measure field x dµ x on M (see Equation (2.12)). We thus concludethat, indeed, the measure field (5.22) is invariant with respect to the action of elements of Diff M ( N N ) .As in the case of the Hilbert space H , the pull-back (5.21) given by a diffeomorphism Θ ∈ Diff M ( N N ) corresponding to θ ∈ Diff( M ) , can be unambiguously extended to aunitary operator on the Hilbert space H N . It is convenient to denote this operator by U N ( θ − ) —then θ U N ( θ ) (5.23)is a unitary representation of Diff( M ) on H N . H We assumed that each space H N for N ≥ is built using the same diffeomorphism invariantfield (2.11) of invariant measures on M . Consequently, the resulting Hilbert space H definedby the orthogonal sum (5.3) stems from this field.Suppose now, that a Hilbert space ˇ H is constructed in the same way from an othersuch a measure field—the other field is related to the former one by the formula (2.13).Then taking into account the distinguished unitary maps (3.19) and (5.20) we see thatthere exists a distinguished unitary map (isomorphism) between H and ˇ H . If ˜Ψ N ∈ H N then this unitary map is given by the following formula: H ∋ ( ˜Ψ N ) (cid:16) ˜Ψ N √ c N (cid:17) ∈ ˇ H . We are then allowed to state that the Hilbert space H is unique up to natural (ordistinguished) isomorphisms. Diff( M ) on H The unitary representations (4.9) and (5.23) of Diff( M ) on, respectively, H and H N , N ≥ , can be used to define the following unitary representation of the diffeomorphismgroup on the Hilbert space H : θ ∞ M N =1 U N ( θ ) , where θ ∈ Diff( M ) . 29 .7 Hilbert spaces { H } built over M = R In this section we will consider the Hilbert space H built in the case M = R for signatureeither (1 , or (0 , and will show that this space is separable (regardless of signature).To this end we will show first that for every N ≥ the Hilbert space H N constructed over M = R is separable.In Appendix A.2 we considered the manifold N N , N ≥ , constituted of points of M = R and constructed a bijection ι : N N → R N> , where R N> = { ( x , . . . , x N ) ∈ R N | x > x > . . . > x N − > x N } is an open subset of R N . We have further demonstrated that ι defines a global coordinatesystem on N N . Setting M ≡ N , R > ≡ R and ι ≡ id on N ≡ R > will allow us to treatthe case N = 1 together with all the cases N ≥ in the considerations below.Let us then fix N ≥ and a Hilbert half-density ˜Ψ ∈ H cN . As shown in AppendixA.2, for every y ∈ N N there exists its open neighborhood Z such that the chart ( Z, ι | Z ) belongs to the atlas A on N N and thereby to the atlas A ′ . This together with continuityof ˜Ψ mean that if a coordinate system is given by ( Z, ι | Z ) , then there exists a continuouscoordinate representation of ˜Ψ in this system. Moreover, this continuous representation isunique (see the last sentence of Section 5.3.2). This uniqueness allows us to merge all suchcontinuous coordinate representations of ˜Ψ into one continuous representation R N> × Γ N R ∋ ( x a , γ ⊕ ab ) ψ ( x a , γ ⊕ ab ) ∈ C (5.24)of ˜Ψ in the global coordinate system ( x a ) defined by the map ι (see (3.7) and (5.17)) .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 ∈ N N there existsits 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 (5.24) by an obviousgeneralization of (3.10).It is easy to convince oneself that, the other way round, if a Hilbert half-density ˜Ψ ofcompact N N -support is ( i ) continuous in the coordinate system ( x a ) and ( ii ) of compactand slowly changing Γ N R -support in the same system, then ˜Ψ ∈ H cN . Lemma 5.3. The map H cN ∋ ˜Ψ ψ, (5.25) where ψ is the function (5.24) , is a linear bijection from H cN 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 themap (5.25) is linear.Let us fix ˜Ψ and ψ related by the map (5.25). We know already that ψ is continuous.Let us then show that ψ is compactly supported.If supp ˜Ψ denotes the N N -support of ˜Ψ , then obviously supp ˜Ψ ⊂ [ y ∈ supp ˜Ψ Z y , If N = 1 , then the existence of the continuous representation (5.24) follows directly from the definitionof continuous Hilbert half-densities on M . Moreover, for N = 1 the superscript ⊕ in γ ⊕ in (5.24), issuperfluous. { Z y } are defined in the paragraph just above the lemma. In other words, { Z y } y ∈ supp ˜Ψ is an open cover of supp ˜Ψ . By definition of H cN , the support of every itselement is compact. Therefore the cover { Z y } y ∈ supp ˜Ψ contains a finite open subcover { Z y n } n =1 ,...,m : supp ˜Ψ ⊂ m [ n =1 Z y n . (5.26)Suppose now that ψ ( x a , γ ⊕ ab ) = 0 for some ( x a , γ ⊕ ab ) ∈ R N> × Γ N R . By continuity of ψ , ˜Ψ( y, ∂ x a ) is a non-zero element of H ⊗ y , where y = ι − ( x a ) . Consequently, ( x a ) ∈ ι (supp ˜Ψ) .On the other hand, if ψ ( x a , γ ⊕ ab ) = 0 , then ( γ ⊕ ab ) ∈ supp ψ ( x a ) . But since ( x a ) ∈ ι (supp ˜Ψ) , by virtue of (5.26) there exists n ∈ { , . . . , m } such that ( x a ) ∈ ι ( Z y n ) . Then supp ψ ( x a ) ⊂ K y n (the sets { K y } are introduced just above the lemma). Consequently, ( γ ⊕ ab ) ∈ K y n .We are then allowed to state that if ψ ( x a , γ ⊕ ab ) = 0 , then ( x a ) ∈ ι (supp ˜Ψ) , ( γ ⊕ ab ) ∈ m [ n =1 K y n . (5.27)Note now that ι (supp ˜Ψ) is compact, because it is the image of a compact set undera continuous map. The set S mn =1 K y n is compact being a union of a finite number ofcompact sets. The Cartesian product of ι (supp ˜Ψ) and S mn =1 K y n is then a compact subsetof R N> × Γ N R and therefore it is closed. Because it is closed and ψ ( x a , γ ⊕ ab ) = 0 implies(5.27), then supp ψ ⊂ (cid:16) ι (supp ˜Ψ) × m [ n =1 K y n (cid:17) . (5.28)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 (5.25) is valued in C c ( R N> × Γ N R , C ) . To show that themap is injective note that if ψ is the value of the map at ˜Ψ , then ˜Ψ can be unambiguouslyreconstructed from ψ by means of the obvious generalization of the formula (3.8).To finish the proof it remains to show that the map (5.25) is surjective. To this endassume that ψ ∈ C c ( R N> × Γ N R , C ) . If π : R N> × Γ N R → R N> , π : R N> × Γ N R → Γ N R , ( x a , γ ⊕ ab ) ( x a ) , ( x a , γ ⊕ ab ) ( γ ⊕ ab ) , are canonical projections, then both sets π (supp ψ ) and π (supp ψ ) are compact beingimages 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 (supp ψ ) —see the reasoning concerning the supportof a function h x in Appendix B. This means that the Hilbert half-density ˜Ψ defined on N N by ψ with the help of the generalization of (3.8), is of compact and slowly changing Γ N R -support.On the other hand, if ( x a ) π (supp ψ ) , then ψ ( x a ) = 0 . Therefore, if ˜Ψ is defined by ψ as above, then ˜Ψ( y ) = 0 implies y ∈ ι − (cid:0) π (supp ψ ) (cid:1) . supp ˜Ψ is then contained in thecompact (and closed) set ι − (cid:0) π (supp ψ ) (cid:1) and ˜Ψ is a half-density of compact N N -support.Thus every ψ ∈ C c ( R N> × Γ N R , C ) defines by means of the generalization of (3.8) anelement ˜Ψ ∈ H cN . The map (5.25) is then surjective.31y definition of H N , the space H cN is dense in H N . We will show now that H N isisomorphic to a Hilbert space, which contains the space C c ( R N> × Γ N R , C ) as its densesubset.Let us begin by expressing the inner product (5.19) in terms of an iterated integral over R N> × Γ N R . To this end consider an integrable scalar density ˜ F on N N . If f is its coordinaterepresentation in the global coordinate system ( x a ) defined by ι (see (3.4)), then Z N N ˜ F = Z R N> f dµ NL , where dµ NL is the Lebesgue measure on R N . If ˜ F = ( ˜Ψ ′ | ˜Ψ) , where ˜Ψ ′ , ˜Ψ ∈ H cN , thencombining Equations (3.15), (3.16) and (5.10) we get f ( x a ) = ˜ F ( ι − ( x a ) , ∂ x a ) = Z Γ N R ψ ′ ( x a ) ψ ( x a ) c N (cid:0) ą ∆ dµ L (cid:1) . Here ψ ′ ( x a ) is the function on Γ N R related by the obvious generalization of (3.10) to thecoordinate representation ψ ′ of ˜Ψ ′ 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 (5.19),we see that the inner product h ˜Ψ ′ | ˜Ψ i = Z R N> (cid:16) Z Γ N R ψ ′ ( x a ) ψ ( x a ) c N (cid:0) ą ∆ dµ L (cid:1)(cid:17) dµ NL . (5.29) R N> × Γ N R is an open subset of R N , and thereby a l.c.H. space. Therefore by virtueof the Riesz representation theorem, there exists a regular Borel measure dν on R N> × Γ N R such that integrals defined by both dν and dµ NL × c N ( Ś ∆ dµ L ) coincide on C c ( R N> × Γ N R ) .The measure dν defines the Hilbert space L ( R N> × Γ N R , dν ) , and C c ( R N> × Γ N R , C ) turnsout to be a dense subset [9] of this Hilbert space.Let us denote by h·|·i dν the inner product on L ( R N> × Γ N R , dν ) . If ψ ′ , ψ are related to,respectively, ˜Ψ ′ , ˜Ψ ∈ H cN by the map (5.25), then h ψ ′ | ψ i dν = Z R N> × Γ N R ψ ′ ψ dν = Z R N> × Γ N R ψ ′ ψ (cid:0) dµ NL × c N ( ą ∆ dµ L ) (cid:1) == Z R N> (cid:16) Z Γ N R ψ ′ ( x a ) ψ ( x a ) c N (cid:0) ą ∆ dµ L (cid:1)(cid:17) dµ NL , where in the last step we used the Fubini-Tonelli theorem. Comparing the result abovewith (5.29) we see that for every ˜Ψ ′ , ˜Ψ ∈ H cN h ˜Ψ ′ | ˜Ψ i = h ψ ′ | ψ i dν . We conclude that the map (5.25) ( i ) is a linear bijection between linear dense subspacesof H N and L ( R N> × Γ N R , dν ) and ( ii ) preserves the inner products. Therefore the map canbe unambiguously extended to a unitary map from H N onto L ( 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 N . As eachregular measure on such a space is σ -finite [9], so is dν . On the other hand, each σ -finite Proposition 7.4.3 in [9] concerns real functions. But by separating a complex function into its real andimaginary parts one can show that the proposition holds true in the complex case as well. L space [13]. This meansthat L ( R N> × Γ N R , dν ) is separable. H N being isomorphic to the former Hilbert space, isseparable as well. Consequently, H built over M = R is separable since it is defined as thecountable orthogonal sum (5.3) of separable Hilbert spaces.There is also another important conclusion, which can be drawn from the results justobtained. Let us recall that on elements of H cN there are imposed seemingly strong condi-tions of compact N N -support and of compact and slowly changing Γ N R -support. This factmay raise concerns about whether the resulting Hilbert space H N is “large enough” froma physical point of view. But since H N built over M = R turned out to be isomorphic to L ( R N> × Γ N R , dν ) , then at least in this case we can regard these concerns to be unfounded. K Let us fix a manifold M , a metric signature ( p, p ′ ) such that p + p ′ = dim M and adiffeomorphism invariant field (2.11) of invariant measures on M . This field defines via(5.13) the Hilbert space H ⊗ y for every y ∈ N N , where N ≥ . Note however that thedefinition of N N applied to the case N = 1 , gives the original manifold M (provided weidentify y = { x } with x ∈ M ). Then H ⊗ y coincides with H y defined by (3.1) (underthe same identification). This observation allows us to simplify the presentation below byconsidering spaces {N N } and corresponding Hilbert spaces for all N ≥ .Let us then fix an integer N ≥ and consider a bundle-like set H ⊗ := [ y ∈N N H ⊗ y . (6.1)Let K N be a set, which consists of some special sections of H ⊗ : a section Ψ of H ⊗ belongsto K N if1. the set { y ∈ N N | Ψ( y ) = 0 } (6.2)is countable;2. the sum X y ∈N N || Ψ( y ) || y ≡ || Ψ || , (6.3)where || · || y is the norm on H ⊗ y , is finite (note that by virtue of the previous assump-tion, the uncountable sum above reduces to a sum of countable number of positiveterms). Lemma 6.1. The map K N × K N ∋ (Ψ ′ , Ψ) 7→ h Ψ ′ | Ψ i := X y ∈N N h Ψ ′ ( y ) | Ψ( y ) i y ∈ C (6.4) is well-defined. K N equipped with this map is a Hilbert space. Let us recall that h·|·i y in (6.4) is the inner product on H ⊗ y . Note also that the map (6.4)can be expressed alternatively as h Ψ ′ | Ψ i = Z N N h Ψ ′ ( y ) | Ψ( y ) i y dµ , dµ is the counting measure on N N , being a diffeomorphism invariant measure onthe manifold.A proof of Lemma 6.1, as following the well-known case of the Hilbert sequence space l , is relegated to Appendix C.For many practical purposes it would be convenient to have a dense linear subspaceof K N , which would contain sufficiently regular elements of the Hilbert space. Denote by K cN a set, which consists of all elements { Ψ } of K N of the following property: for every y ∈ N N , the value Ψ( y ) ∈ H ⊗ y is (an equivalence class of) an element of C c (Γ ⊕ y , C ) , i.e., acomplex continuous function on Γ ⊕ y of compact support. Let K cfN be the set consisting ofall elements of K cN , for which the set (6.2) is finite. Lemma 6.2. K cfN is a dense linear subspace of K N .Proof. Since every set C c (Γ ⊕ y , C ) is a linear space, then both K cN and K cfN are linear sub-spaces of K N . It remains then to prove that K cfN is dense in K N . We will first show that K cN is dense in K N .Let us fix Ψ ∈ K N . Then the set (6.2) is countable and all its elements can be orderedto 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 (Γ ⊕ y , dµ × y ) [9] .This means that if ψ is a non-zero element of H ⊗ y then there exists a sequence ( ψ ′ m ) ofnon-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 ψ m = || ψ || y || ψ ′ m || y ψ ′ m ∈ C c (Γ ⊕ y , C ) 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 > . The conclusions above allows us to choose forevery y n a function ψ nm ∈ C c (Γ ⊕ y n , C ) such that || ψ nm − Ψ( y n ) || y n < √ n m , || ψ nm || y n = || Ψ( y n ) || y n . Define now a section Ψ m of H ⊗ : Ψ m ( y ) = ( ψ nm if y = y n otherwise . Obviously, for every Ψ m the set (6.2) is countable and || Ψ m || = X y ∈N N || Ψ m ( y ) || y = X y ∈N N || Ψ( y ) || y = || Ψ || < ∞ . Thus Ψ m is an element of K cN . See Footnote 11. 34n the other hand, the norm of Ψ m − Ψ in the Hilbert space K N , can be bounded fromabove as follows: || Ψ m − Ψ || = X y ∈N N || Ψ m ( y ) − Ψ( y ) || y = ∞ X n =1 || ψ nm − Ψ( y n ) || y n < ∞ X n =1 n m = 1 m . This result means that K cN is dense in K N . Note now that for every Ψ ∈ K cN and forevery ǫ > , there exists Ψ f ∈ K cfN such that || Ψ − Ψ f || < ǫ —if the set (6.2) for Ψ isinfinite, then to obtain the desired Ψ f it is enough to zero out values of Ψ at appropriatelychosen points in N N . Consequently, K cfN is also dense in K N .Taking into account experiences gained from the study of the spaces {H N } , it is easyto realize that1. if the Hilbert spaces K N and ˇ K N are constructed as above, starting from two distinctdiffeomorphism invariant measure fields, then there exists a number c > such that K N ∋ Ψ Ψ √ c N ∈ ˇ K N (6.5)is a unitary map.2. if θ ∈ Diff( M ) and Θ ∈ Diff M ( N N ) are related diffeomorphisms, then the pull-back Ψ Θ ∗ Ψ , (cid:0) Θ ∗ Ψ (cid:1) ( y ) := Θ − ′∗ ⋆ Ψ (cid:0) Θ( y ) (cid:1) defined on K cfN is a linear bijection onto K cfN and preserves the inner product (6.4).Consequently, with help of Lemma 6.2 the pull-back can be uniquely extended to aunitary map U N ( θ − ) on K N . Moreover, θ U N ( θ ) is a unitary representation of Diff( M ) on the Hilbert space.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 Ψ yn ( y ′ ) = ( ψ yn if y ′ = y, otherwise . It is not difficult to demonstrate that { Ψ yn } y ∈N N , n ∈ N is an orthonormal basis 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: K := ∞ M N =1 K N . Let us recall that all the Hilbert spaces {K N } above stem from the same diffeomorphisminvariant field (2.11) of invariant measures.It is now a simple exercise To prove that the linear span of { Ψ yn } y ∈N N , n ∈ N is dense in K N , one can use a reasoning similar tothat used in the proof of Lemma 6.2. 35. 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 Hilbertspaces {K N } . In this paper we constructed two Hilbert spaces H and K over the set Q ( M ) of all metricsof arbitrary signature ( p, p ′ ) , defined on a (smooth connected paracompact) manifold M . Each space was obtained by merging the tensor products { H x ⊗ . . . ⊗ H x N } N =1 , ,... —every state in H was built of an uncountable number of elements of these products, whileevery state in K from a countable number of them.The Hilbert spaces { H x } were defined by means of a diffeomorphism invariant field x dµ x of invariant measures. The diffeomorphism invariance of this measure fieldresulted in existence of a unitary representation of the diffeomorphism group Diff( M ) oneach Hilbert space H and K . On the other hand, the measure field is unique up to amultiplicative constant, which resulted in uniqueness of each of the Hilbert spaces H and K up to distinguished isomorphisms. The two Hilbert spaces { H } built over M = R turnedout to be separable, while all the Hilbert spaces { K } to be non-separable.Let us now present an outlook to future research.The most important question is whether either H or K in the case of signature (3 , ,can be used for quantization of the ADM formalism. As emphasized in the introduction tothis paper, there is no guarantee that the answer to this question is in affirmative. The firststep to be done to clarify this issue, is an attempt to define on the Hilbert spaces operators[8] 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 ofoperators 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 N N , ( ii ) the question whether each space H N generated by the space H cN of special Hilberthalf-densities, is “large enough” from a physical point of view and ( iii ) the questionwhether all the Hilbert spaces { H } are separable.In this paper we considered the bundle-like sets ˜ H , ˜ H ⊗ and H ⊗ defined, respectively,by the formulas (3.2), (5.14) and (6.1). It is interesting, at least from a mathematical pointof view, whether these spaces can be endowed with local trivializations, which would makethem genuine bundles. In particular, it is interesting, whether the set H ⊗ is a Hilbertbundle (see e.g. [14]) over N N .Let us emphasize that the Hilbert spaces H and K are distinctly different from thespace S of quantum states built in [1] by means of the Kijowski’s projective methodover the same set Q ( M ) of metrics. To construct S , we extended each Hilbert spacein { H x ⊗ . . . ⊗ H x N } N =1 , ,... to a larger space S λ , where λ ≡ { x , . . . , x N } . Namely, thespace S λ was defined as the set of all algebraic states on the C ∗ -algebra B λ of all bounded Note, however, that both spaces H and K do exist, even if the set Q ( M ) is empty. Recall that in the case M = R the answer to this question is in affirmative. H λ ≡ H x ⊗ . . . ⊗ H x N . Since all the sets {S λ } form naturally a projectivefamily, the space S were obtained as the projective limit of the family. As a result, thespace 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 ofthe same building blocks (being the Hilbert spaces { H x ⊗ . . . ⊗ H x N } N =1 , ,... ) and in ouropinion it is worthwhile to explore more closely the relations between these three spaces. Acknowledgments I am very grateful for Jerzy Kijowski and Piotr Sołtan for valuablediscussions and help. This work was partially supported by the Polish National ScienceCentre grant No. 2018/30/Q/ST2/00811. A The set N N as a manifold Let us fix an integer N ≥ and a smooth connected paracompact manifold M and define N N := (cid:8) { x , . . . , x N } ≡ { x K } ⊂ M | x I = x J for I = J (cid:9) . A.1 Smooth atlas on N N Our goal in this section is to define a smooth atlas on N N , which will allow us to treat thisset a smooth manifold. A submanifold of M N To this end let us consider the following set: M N := { ( x , . . . , x N ) ≡ ( x K ) ∈ M N | x I = x J for I = J } . M N is an open subset of M N . Indeed, if ( x K ) is an arbitrary point in M N , then for each x I ∈ ( x K ) there exists an open neighborhood U I ⊂ M of x I such that ∀ I = J U I ∩ U J = ∅ . (A.1)Then U × . . . × U N ≡ ą U K is an open subset of M N , contains the point ( x K ) and is contained in M N .Consequently, M N is a smooth manifold being an open subset of M N . Action of permutations on M N Let Σ be the group of all permutations of the finitesequence (1 , , , . . . , N ) . Given σ ∈ Σ , the following map M N ∋ ( x K ) ¯ σ ( x K ) := ( x σ (1) , . . . , x σ ( N ) ) ≡ ( x σ ( K ) ) ∈ M N (A.2)is a diffeomorphism on M N .To see this let us fix a point ( x K ) ∈ M N and for every x I ∈ ( x K ) choose its openneighborhood 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 ( x iI ) i =1 ,..., dim M on U I . Then the map ą ϕ K ≡ ϕ × . . . × ϕ N : U × . . . × U N → R N dim M (A.3)37efines a local coordinate system ( x i , . . . , x i N N ) ≡ ( x i K K ) on Ś U K .Clearly, the following map ( ϕ σ (1) × . . . × ϕ σ ( N ) ) ◦ ¯ σ ◦ ( ϕ − × . . . × ϕ − N ) : ( x i , . . . , x i N N ) ( x i σ (1) , . . . , x i N σ ( N ) ) (A.4)between appropriate open subsets of R N dim M is smooth. Since ϕ σ (1) × . . . × ϕ σ ( N ) is a mapon U σ (1) × . . . × U σ ( N ) of the same sort as (A.3), smoothness of (A.4) means that the map(A.2) is smooth as well. Consequently, the inverse map ¯ σ − is also smooth, since it is givenby the inverse permutation σ − . Thus we see that, indeed, (A.2) is a diffeomorphism. Natural projection from M N onto N N The map M N ∋ ( x K ) π ( x K ) := { x K } ∈ N N , (A.5)is a natural surjection (projection) from M N onto N N , which “forgets” about the orderingof points in ( x K ) . This map will be used to define a smooth atlas on N N .It follows immediately from (A.5) that for every { x K } ∈ N N and for every subset U of M N , π − ( { x K } ) = { ¯ σ ( x K ) | σ ∈ Σ } , π − (cid:0) π ( U ) (cid:1) = [ σ ∈ Σ ¯ σ ( U ) . (A.6) Topology on N N In order to define a smooth atlas on N N , let us first equip the setwith a suitable topology: we will say that a set Z ⊂ N N is open, if its preimage under π , π − ( Z ) , is an open subset of M N . This immediately means that π becomes a continuousmap. Moreover, the image π ( U ) of every open set U ⊂ M N is open—this is because byvirtue of the second of Equations (A.6), the preimage of π ( U ) is a union of open sets in M N .To show that the topology just introduced is Hausdorff, consider two distinct elements { x K } and { x ′ K } of N N . Since M is Hausdorff, for every x ∈ { x K } ∪ { x ′ K } there existsits open neighborhood U x ⊂ M such that U x ∩ U ˇ x = ∅ if x = ˇ x . Because { x K } 6 = { x ′ K } ,there exists x ′ ∈ { x ′ K } , which does not belong to { x K } . Consequently, U x ′ is disjoint withevery U x I , x I ∈ { x K } , and therefore for each two permutation σ, σ ′ ∈ Σ , ¯ σ (cid:0) ą U x K (cid:1) ∩ ¯ σ ′ (cid:0) ą U x ′ K (cid:1) = ∅ . (A.7)It is clear that π (cid:0) Ś U x K (cid:1) and π (cid:0) Ś U x ′ K (cid:1) are open neighborhoods of, respectively, { x K } and { x ′ K } . Suppose that the neighborhoods are not disjoint. Then their preimagesunder the surjection π are also not disjoint: ∅ = π − (cid:0) π (cid:0) ą U x K (cid:1)(cid:1) ∩ π − (cid:0) π (cid:0) ą U x ′ K (cid:1)(cid:1) == (cid:16) [ σ ∈ Σ ¯ σ (cid:0) ą U x K (cid:1)(cid:17) ∩ (cid:16) [ σ ′ ∈ Σ ¯ σ ′ (cid:0) ą U x ′ K (cid:1)(cid:17) (here we used the second of Equations (A.6)). But this contradicts Equation (A.7), whichmeans that the neighborhoods under consideration are disjoint. The topology on N N isthus Hausdorff. If ( x K ) ∈ M N , then the set { ¯ σ ( x K ) | σ ∈ Σ } ⊂ M N represents the unordered set { x K } of pairwise distinct points of M . Under this identification, N N is theset of all orbits of the action (A.2) of the group Σ on M N . n atlas on N N Consider again the domain of the map (A.3), keeping in mind thatfor I = J the sets U I and U J are disjoint. It turns out that the map π | Ś U K , that is, themap π restricted to Ś U K , is a bijection onto its image. Indeed, it follows from the firstof Equations (A.6) that π ( x K ) = π ( x ′ K ) (A.8)if and only if ( x ′ K ) = ¯ σ ( x K ) (A.9)for some σ ∈ Σ . But since the sets { U I } are pairwise disjoint, the intersection (cid:0) ą U K (cid:1) ∩ ¯ σ (cid:0) ą U K (cid:1) = ∅ (A.10)for every σ ∈ Σ except the identity permutation. This means that if (A.8) holds for twoelements of Ś U K , then the elements coincide.Denote by π − Ś U K the map from π ( Ś U K ) onto Ś U K inverse to the map π | Ś U K . Let U be an open subset of the (open) set Ś U K . Then the preimage of U under π − Ś U K ( π − Ś U K ) − ( U ) = π ( U ) and therefore is an open subset of N N (see the second of Equations (A.6)). This means that π − Ś U K is a continuous map. Since the map π is also continuous, π − Ś U K is a homeomorphism.This property of π − Ś U K allows us to define a local coordinate system on N N : given amap (A.3), the composition Φ ≡ (cid:0) ą ϕ K (cid:1) ◦ π − Ś U K : π (cid:0) ą U K (cid:1) → R N dim M (A.11)is a homeomorphism onto its image and defines thereby a local coordinate system on N N .Domains of all maps of the form (A.11) constitute an open cover of N N . Therefore chartsgiven by all maps (A.11) and their domains, form a continuous atlas A on N N . 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: Φ ′ ≡ (cid:0) ą ϕ ′ K (cid:1) ◦ π − Ś U ′ K : π (cid:0) ą U ′ K (cid:1) → R N dim M , (A.12)and suppose that the domains of Φ and Φ ′ are not disjoint: Z ≡ π (cid:0) ą U K (cid:1) ∩ π (cid:0) ą U ′ K (cid:1) = ∅ . (A.13)Our goal now is to show that the transition function Φ ′ ◦ Φ − related to Z is smooth.Fix ( x ′ K ) ∈ Ś U ′ K . Then π ( x ′ K ) ∈ Z if and only if there exists ( x K ) ∈ Ś U K satisfyingEquation (A.8). We know already that (A.8) is equivalent to Equation (A.9) holding forsome σ ∈ Σ . We are then allowed to state, that π ( x ′ K ) ∈ Z if and only if there exists σ ∈ Σ such that ( x ′ K ) ∈ (cid:0) ą U ′ K (cid:1) ∩ ¯ σ (cid:0) ą U K (cid:1) ≡ U σ , or, equivalently, if and only if ( x ′ K ) ∈ [ σ ∈ Σ U σ ⊂ ą U ′ K . Z ⊂ π ( M N ) (see (A.13)) mean that Z = π (cid:16) [ σ ∈ Σ U σ (cid:17) = [ σ ∈ Σ π ( U σ ) . (A.14)Applying (A.10) we see that if σ = σ ′ , then U σ ∩ U σ ′ = ∅ . (A.15)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 − on each non-empty set π ( U σ ) .Consider then ( x ′ K ) = ¯ σ ( x K ) ∈ U σ , (A.16)where ( x ′ K ) ∈ Ś U ′ K and ( x K ) ∈ Ś U K . Then { x ′ K } = π ( x ′ K ) = π ( x K ) = { x K } and Φ ′ ( { x ′ K } ) = (cid:0) ą ϕ ′ K (cid:1) ( x ′ K ) = ( x ′ i K K ) ∈ R N dim M , Φ( { x K } ) = ( ą ϕ K (cid:1) ( x K ) = ( x i K K ) ∈ R N dim M . Thus ( x ′ i K K ) and ( x i K K ) are values of coordinates defined by, respectively, Φ ′ and Φ , of thesame point { x ′ K } = { x K } . Using (A.16) we obtain the following relation between thevalues: ( x ′ i K K ) = (cid:0) ą ϕ ′ K (cid:1) ( x ′ K ) = (cid:16)(cid:0) ą ϕ ′ K (cid:1) ◦ ¯ σ (cid:17) ( x K ) = (cid:16)(cid:0) ą ϕ ′ K (cid:1) ◦ ¯ σ ◦ (cid:0) ą ϕ K (cid:1) − (cid:17) ( x i K K ) (A.17)Obviously, this relation is nothing else but the value of the transition function Φ ′ ◦ Φ − at ( x i K K ) . We thus conclude that Φ ′ ◦ Φ − on the set Φ (cid:0) π ( U σ ) (cid:1) is smooth, since it coincideswith the coordinate expression (A.17) of the diffeomorphism ¯ σ —given I , ( x ′ i I I ) = ϕ ′ I ◦ ϕ − σ ( I ) ( x j I σ ( I ) ) . (A.18)Thus the transition map is smooth on its whole domain Φ( Z ) = S σ ∈ Σ Φ (cid:0) π ( U σ ) (cid:1) . Conclusions Consequently, the atlas A is smooth and the set N N is a smooth manifold.But since we are going to integrate densities over this manifold we extend A to the maximalsmooth atlas on N N , since restricting ourselves to the atlas A would be inconvenient.For every collection { U ′ K } K =1 ,...,N of pairwise disjoint open subsets of M , the map π − Ś U ′ K : π (cid:0) ą U ′ K (cid:1) → ą U ′ K 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 coordinateexpression for π − Ś U ′ K : (cid:0) ą ϕ K (cid:1) ◦ π − Ś U ′ K ◦ Φ − is the identity on ( Ś ϕ K ( U K )) ⊂ R N dim M .This fact allows us to state that ( i ) N N is locally diffeomorphic to M N and ( ii ) theprojection π is smooth. 40 N is paracompact A manifold is paracompact if and only if each connected compo-nent of the manifold is second countable [15]. We assumed that M is paracompact andconnected, which means that M is second countable. Thus M N and M N are secondcountable as well.Let B be a countable base for the topology of M N . Suppose that Z is an open subsetof N N . Then, by definition of the topology on N N , the preimage π − ( Z ) is open subsetof M N . Consequently, the preimage is a union of open sets { U α } ⊂ B . We know already,that π maps open subsets of M N onto open ones in N N . Thus Z = π (cid:0) π − ( Z ) (cid:1) = π (cid:16) [ α U α (cid:17) = [ α π ( U α ) . The conclusion is that every open subset of N N is a union of open sets being images ofelements of B under π —these images form a countable base for the topology of N N . N N is thus second countable and thereby paracompact. A.2 An example of N N Here we will find an explicit description of N N for M = R by means of a global coordinatesystem on N N .Let us fix an integer N ≥ . If M = R , then for every y ≡ { x K } ∈ N N it is possibleto form a decreasing sequence from all elements of y i.e. there exists a permutation σ ∈ Σ such that x σ (1) > x σ (2) > . . . > x σ ( N − > x σ ( N ) . This observation allows us to define the following map N N ∋ { x K } 7→ ι ( { x K } ) := the decreasing sequence of elements of { x K } ∈ M N ⊂ R N . It is obvious that the map is a bijection onto its image such that π ◦ ι = id , ι ◦ π = id , (A.19)where the last equation holds on ι ( N N ) ⊂ M N . Let R N> := { ( x , . . . , x N ) ∈ R N | x > x > . . . > x N − > x N } ⊂ M N . It is clear that ι ( N N ) = R N> .Fix an arbitrary z ≡ ( x , . . . , x N ) ∈ R N> and define ǫ := min { x K +1 − x K | x K +1 , x K ∈ z } . If U K := ] x K − ǫ, x K + ǫ [ , then Ś U K ∋ z is an open subset of R N . Since Ś U K ⊂ R N> , the latter set is an opensubset of R N .Moreover, since the sets { U K } just defined satisfy (A.1) it is not difficult to realizewith help of Equations (A.19) that ι restricted to π (cid:0) Ś U K (cid:1) is a map of the sort of themap (A.11) (with ϕ K being the identity on U K ). This means that every y ∈ N N possessesan open neighborhood Z such that ( Z, ι | Z ) is a chart belonging to the atlas A on themanifold. This is sufficient to conclude that the map ι defines a global coordinate systemon the manifold N N .Thus if M = R , then N N can be identified with R N> being an open subset of R N .41 .3 Natural decomposition of tangent spaces to N N Let us fix a point y ≡ { x K } ∈ N N , the numbering of elements of y and open subsets { U K } of M satisfying (A.1) such that x K ∈ U K . Let y I be a subset of y obtained by removingfrom y the point x I : y I = { x , . . . , x I − , x I +1 , . . . , x N } . Define a map U I ∋ x ξ y I ( x ) := { x } ∪ y I ∈ N N . (A.20)To demonstrate that this map is smooth consider the following smooth map U I ∋ x ξ y I ( x ) := ( x , . . . , x I − , x, x I +1 , . . . , x N ) ∈ M N . Clearly, ξ y I = π ◦ ξ y I , (A.21)which means that ξ y I is a composition of two smooth maps.If ξ ′ y I is the tangent map defined by ξ y I , then ξ ′ y I ( T x I M ) is a linear subspace of T y N N .This subspace is generated by all curves in N N of the following form: t ξ y I (cid:0) x ( t ) (cid:1) = { x ( t ) } ∪ y I ∈ N N , (A.22)where t x ( t ) , x (0) = x I , is a differentiable curve in U I ⊂ M .It is evident that T ( x K ) M N = N M K =1 ξ ′ y K ( T x K M ) , where ξ ′ y K denotes the tangent map given by ξ y K . Let us now act on both sides of thisequation by the tangent map π ′ defined by π . Since π restricted to Ś U K is a (local)diffeomorphism to N N , we thus obtain T y N N = π ′ (cid:0) T ( x K ) M N (cid:1) = N M K =1 π ′ ξ ′ y K ( T x K M ) = N M K =1 ξ ′ y K ( T x K M ) , (A.23)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 todefine the map Φ via (A.11). Denote by ( x i K K ) the value at y = { x K } of the coordinatesdefined by Φ . Then the following curve t (cid:0) π ◦ ( ą ϕ − K ) (cid:1) ( x , . . . , x i − I , x iI + t, x i +1 I , . . . , x dim M N ) == (cid:8) ϕ − I ( x I , . . . , x i − I , x iI + t, x i +1 I , . . . , x dim M I ) (cid:9) ∪ { y I } ∈ N N defines the tangent vector ∂ x iI ∈ T y N N . Taking into account the formula (A.22) we concludethat ∂ x iI ∈ ξ ′ y I ( T x I M ) . Hence the vectors ( ∂ x kI ) k =1 ,..., dim M (with the fixed index I ) form abasis of ξ ′ y I ( T x I M ) .To simplify the notation, in the main body of the paper we will identify ξ ′ y I ( T x I M ) ≡ T x I M (A.24)and write T y N N = N M K =1 T x K M . This is because π − Ś U K is a local diffeomorphism as proven at the end of Appendix A.1. .4 Diffeomorphisms of N N induced by those of M Definition Let θ be a diffeomorphism on M . It induces a map on N N as follows: N N ∋ { x K } 7→ Θ( { x K } ) := { θ ( x K ) } ∈ N N (A.25)(if { x K } consists of pairwise distinct points of M , then the points in (cid:8) θ ( x K ) (cid:9) are pairwisedistinct too). Let us show now that Θ is a diffeomorphism on N N .To this end consider the following map M N ∋ ( x K ) θ ( x K ) := (cid:0) θ ( x K ) (cid:1) ∈ M N Clearly, Θ ◦ π = π ◦ θ . (A.26)Consider now maps (A.11) and (A.12) assuming that U ′ K = θ ( U K ) and ϕ ′ K = ϕ K ◦ θ − .Then using (A.26) we obtain Φ ′ ◦ Θ ◦ Φ − = Φ ′ ◦ Θ ◦ ( π − Ś U K ) − ◦ (cid:0) ą ϕ K (cid:1) − = Φ ′ ◦ Θ ◦ π ◦ ą ϕ − K == (cid:0) ą ϕ ′ L (cid:1) ◦ π − Ś U ′ K ◦ π ◦ θ ◦ ą ϕ − K = (cid:0) ą ( ϕ L ◦ θ − ) (cid:1) ◦ ą ( θ ◦ ϕ − K ) = id (A.27)on (cid:0) Ś ϕ L (cid:1) Ś U K . This means that for every diffeomorphism θ of M , the map Θ issmooth. But Θ − exists and is smooth, since it is given via (A.25) by θ − . Θ is thus adiffeomorphism of N N .If Θ is induced by θ via (A.25), then Diff( M ) ∋ θ Θ ∈ Diff( N N ) is a homomorphism. Its image is a subgroup of the diffeomorphism group of N N . Thissubgroup will be denoted by Diff M ( N N ) .Note also that it follows from (A.27) that Φ ◦ Θ − coincides with Φ ′ . This means that Θ − maps a chart in A to an other one in A . In other words, the atlas A is preserved byall diffeomorphisms in Diff M ( N N ) . Diffeomorphisms in Diff M ( N N ) preserve the decompositions (A.23) Let θ ∈ Diff( M ) generates Θ ∈ Diff M ( N N ) . Using the notation introduced in Appendix A.3,we have for every x ∈ U I : (Θ ◦ ξ y I )( x ) = { θ ( x ) } ∪ Θ( y ) I = ( ξ Θ( y ) I ◦ θ )( x ) , where Θ( y ) I is the set obtained by removing the point θ ( x I ) from Θ( y ) . If Θ ′ denotes thetangent map given by Θ , then by virtue of the equation above, Θ ′ ξ ′ y I ( T x I M ) = ξ ′ Θ( y ) I θ ′ ( T x I M ) = ξ ′ Θ( y ) I ( T θ ( x I ) M ) . (A.28)This means that Θ maps the decomposition (A.23) at y into the decomposition (A.23) at Θ( y ) . 43 iffeomorphisms in Diff M ( N N ) preserve the spaces { Γ ⊕ y } Let θ ∈ Diff( M ) , and Θ be the corresponding element of Diff M ( N N ) . To simplify the notation, in line withthe identification (A.24), we will denote elements of both ξ y I ( T x I M ) and T x I M by thesame symbols v I , ˇ v I . Similarly, taking into account Equation (A.28) we will identify Θ ′ v I being an element of Θ ′ ξ ′ y I ( T x I M ) with θ ′ v I being an element of T θ ( x I ) M . Then for every v ∈ T y N N Θ ′ v = Θ ′ (cid:16) N X I =1 v I (cid:17) = N X I =1 θ ′ v I . Let y = { x K } ∈ N N and let x ′ K = θ − ( x K ) . Then y ′ = { x ′ K } = Θ − ( 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 (5.5). Consider now thepull-back Θ ′∗ γ ⊕ y : (cid:0) Θ ′∗ γ ⊕ y (cid:1) ( v, ˇ v ) = γ ⊕ y (cid:0) Θ ′ v, Θ ′ ˇ v (cid:1) = N X I =1 γ x I ( θ ′ v I , θ ′ ˇ v I ) = N X I =1 ( θ ′∗ γ x I )( v I , ˇ v I ) , where v, ˇ v ∈ T y ′ N N . We see that the pull-back Θ ′∗ γ ⊕ y is a scalar product on T y ′ N N constructed of scalar products { θ ′∗ γ x I } via (5.5). Therefore Θ ′∗ γ ⊕ y ∈ Γ ⊕ y ′ .We are then allowed to conclude that diffeomorphisms in Diff M ( N N ) preserve thespaces { Γ ⊕ y } y ∈N N . Moreover, regarding the correspondence given by the bijection (5.12),we see that the pull-back Θ ′∗ : Γ ⊕ y → Γ ⊕ y ′ corresponds to the pull-back ą θ ′∗ ≡ θ ′∗ × . . . × θ ′∗ : Γ x × . . . × Γ x N → Γ x ′ × . . . × Γ x ′ N . (A.29)More precisely, b y ′ ◦ Θ ′∗ = (cid:0) ą θ ′∗ (cid:1) ◦ b y (A.30)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 ′ [9]. 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 [9]. Thereforeboth product measures, which appear in (5.9) are regular and Borel. That being thecase, by virtue of the Riesz representation theorem, it is enough to show that for every h ∈ C c ( X × Y ) , Z X × Y h (cid:0) ( α ⋆ dµ ) × ( β ⋆ dν ) (cid:1) = Z X × Y h (cid:0) ( α × β ) ⋆ ( dµ × dν ) (cid:1) . (B.1)Each second countable l.c.H. space is metrizable [9]. Let then δ X and δ Y be corre-sponding metrics on X and Y . Then δ (cid:0) ( x, y ) , (ˇ x, ˇ y ) (cid:1) := q δ X ( x, ˇ x ) + δ Y ( y, ˇ y ) 44s 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 andonly if for every ǫ > there exists ˇ y ∈ Y such that h x (ˇ y ) = 0 and δ Y ( y, ˇ y ) < ǫ , or,equivalently, if and only if for every ǫ > there exists a pair ( x, ˇ y ) ∈ X × Y such that h ( x, ˇ y ) = 0 and δ (cid:0) ( x, y ) , ( x, ˇ y ) (cid:1) < 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 underthe continuous map π Y . We see that supp h x is a closed subset of a compact set andtherefore is compact as well .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) Z Y h x ( β ⋆ dν ) = Z Y ′ ( β ⋆ h x ) dν. (B.2)Consider now the following function X ∋ x ˇ h ( x ) := Z Y h x ( β ⋆ dν ) ∈ R . (B.3)Suppose that x π X (supp h ) . This means that for every y ∈ Y , h x ( y ) = h ( x, y ) = 0 and consequently ˇ h ( x ) = 0 . Thus if ˇ h ( x ) = 0 then x ∈ π X (supp h ) . Therefore supp ˇ h ⊂ π X (supp h ) , since π X (supp h ) is compact and thereby closed. We conclude then that supp ˇ h 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 ) , otherwise . Since π Y (supp h ) is compact and the measure β ⋆ dν is regular, h is integrable with respectto the measure. Moreover, for every x ∈ X , | h x | ≤ h . These two facts allow us to use theLebesgue’s dominated convergence theorem to conclude that the function ˇ h is continuous .We thus showed that ˇ h ∈ C c ( X ) and consequently by virtue of (2.5) Z X ˇ h ( α ⋆ dµ ) = Z X ′ ( α ⋆ ˇ h ) dµ. (B.4) More precisely, supp h x ⊂ π Y (supp h ) is a closed subset of Y and therefore is a closed subset of π Y (supp h ) (i.e. is closed with respect to the subspace topology on π Y (supp h ) ). This means that supp h x is a compact subset of π Y (supp h ) and therefore it is a compact subset of Y . Note that since X is metrizable we can apply here the criterion of continuity formulated in terms ofsequences of arguments and values of a function. Z X × Y h (cid:0) ( α ⋆ dµ ) × ( β ⋆ dν ) (cid:1) = Z X h Z Y h x ( β ⋆ dν ) i ( α ⋆ dµ ) = Z X ˇ h ( α ⋆ dµ ) == Z X ′ ( α ⋆ ˇ h ) dµ = Z X ′ h Z Y h α ( x ′ ) ( β ⋆ dν ) i dµ == Z X ′ h Z Y ′ ( β ⋆ h α ( x ′ ) ) dν i dµ = Z X ′ × Y ′ (cid:0) ( α × β ) ⋆ h (cid:1) ( dµ × dν ) == Z X × Y h (cid:0) ( α × β ) ⋆ ( dµ × dν ) (cid:1) (note also that α × β is a homeomorphism and therefore ( α × β ) ⋆ h is continuous andcompactly 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 aninner 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 analogouslemma concerning the Hilbert sequence space l (see e.g. [16, 17]). K N is a linear space If Ψ , Ψ ′ ∈ K N and z ∈ C , then ( z Ψ)( y ) := z Ψ( y ) , (Ψ + Ψ ′ )( y ) := Ψ( y ) + Ψ ′ ( y ) . It is now obvious, that z Ψ ∈ K N for every z ∈ C . Regarding the sum Ψ + Ψ ′ note firstthat the set { y ∈ N N | Ψ( y ) = 0 or Ψ ′ ( y ) = 0 } (C.1)is countable. Therefore all its elements can be ordered into a sequence ( y n ) . The value || Ψ + Ψ ′ || can be now bounded from above as follows: || Ψ + Ψ ′ || = ∞ X n =1 || Ψ( y n ) + Ψ ′ ( y n ) || y n ≤ ∞ X n =1 (cid:0) || Ψ( y n ) || y n + || Ψ ′ ( y ) || y n (cid:1) = ∞ X n =1 (cid:0) || Ψ( y n ) || y n + 2 || Ψ( y n ) || y n || Ψ ′ ( y n ) || y n + || Ψ( y n ) || y n (cid:1) , where || · || y is the norm on H ⊗ y . Note now that ab ≤ a + b for every real numbers a and b . Therefore || Ψ + Ψ ′ || ≤ ∞ X n =1 (cid:0) || Ψ( y n ) || y n + 2 || Ψ ′ ( y n ) || y n (cid:1) = 2 (cid:0) || Ψ || + || Ψ ′ || (cid:1) < ∞ . Thus Ψ + Ψ ′ ∈ K N and K N is a linear space.46 he 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 h Ψ ′ | Ψ i = X y ∈N N h Ψ ′ ( y ) | Ψ( y ) i y := ∞ X n =1 h Ψ ′ ( y n ) | Ψ( y n ) i y n , where h·|·i y is the inner product on H ⊗ y . Let us show now that the series above is absolutelyconvergent—then its sum does not depend on the ordering of points in (C.1) into a sequenceand, consequently, h Ψ ′ | Ψ i is well defined.To this end we will apply the Schwarz inequality to every term in the following series: ∞ X n =1 (cid:12)(cid:12) h Ψ ′ ( y n ) | Ψ( y n ) i y n (cid:12)(cid:12) ≤ ∞ X n =1 || Ψ ′ ( y n ) || y n || Ψ( y n ) || y n ≤≤ ∞ X n =1 (cid:0) || Ψ ′ ( y n ) || y n + || Ψ( y n ) || y n (cid:1) = 12 (cid:0) || Ψ ′ || + || Ψ || (cid:1) < ∞ (here in the second step we again used the inequality ab ≤ a + b ).Thus the map (6.4) is well-defined. It is now an easy exercise to show that it is aninner product on K N and that the norm defined on the set by the inner product coincideswith (6.3). K N is complete It remains to show that K N equipped with the norm is a completespace. Let us then suppose that (Ψ m ) m =1 , ,... is a Cauchy sequence of elements of K N .Then the set { y ∈ N 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 > there exists m such that for every m, m ′ > m || Ψ m − Ψ m ′ || = ∞ X n =1 || Ψ m ( y n ) − Ψ m ′ ( y n ) || y n < ǫ . (C.2)Consequently, for every n and every ǫ > , there exists m such that for each m, m ′ > m , || Ψ m ( y n ) − Ψ m ′ ( y n ) || y n < ǫ . This implies that for every (fixed) n the sequence (cid:0) Ψ m ( y n ) (cid:1) has a limit in (the completespace) H ⊗ y n —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 otherwise . We will show now that ( i ) the sequence (Ψ m ) converges to Ψ in the norm (6.3) and ( ii ) Ψ ∈ K N .It follows from (C.2) that for every l and for every m, m ′ > m l X n =1 || Ψ m ( y n ) − Ψ m ′ ( y n ) || y n < ǫ . l and for every m > m lim m ′ →∞ l X n =1 || Ψ m ( y n ) − Ψ m ′ ( y n ) || y n = l X n =1 || Ψ m ( y n ) − Ψ( y n ) || y n ≤ ǫ . Consequently, for m > m , passing to the limit as l tends to the infinity, we obtain ∞ X n =1 || Ψ m ( y n ) − Ψ( y n ) || y n = X y ∈N N || Ψ m ( y ) − Ψ( y ) || y = || Ψ m − Ψ || ≤ ǫ . (C.3)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 isof 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 elementof this space. The space is then complete. 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