Kinematic quantum states for the Teleparallel Equivalent of General Relativity

A space of kinematic quantum states for the Teleparallel Equivalent of General Relativity is constructed by means of projective techniques. The states are kinematic in this sense that their construction bases merely on the structure of the phase space of the theory and does not take into account constraints on it. The space of quantum states is meant to serve as an element of a canonical background independent quantization of the theory.


Introduction
Nowadays there are many approaches [1,2] to quantum gravity but so far no one is fully successful. Therefore it is still worth to take a risk to develop a new approach. It seems that the Teleparallel Equivalent of General Relativity (TEGR) was never used as a point of departure for a construction of a model of quantum gravity and therefore we would like to check whether it is possible to quantize gravity in this formulation (for the latest review of TEGR see [3]). More precisely, we would like to check whether it is possible to quantize TEGR using the method of canonical quantization or, if it is needed, a modification of the method. Since TEGR is a background independent theory we would like to quantize it in a background independent manner.
A. Okołów (B) Institute of Theoretical Physics, Warsaw University, ul. Hoża 69, 00-681 Warsaw, Poland e-mail: oko@fuw.edu.pl TEGR in its canonical formulation is a constrained system (see e.g. [4][5][6][7]). Therefore it is quite natural to attempt to apply the Dirac strategy of canonical quantization of such systems which requires two steps to be carried out: (i) first one neglects constraints and constructs a space of kinematic quantum states, that is, quantum states corresponding to all classical states constituting the whole phase space (ii) then among the kinematic quantum states one distinguishes physical quantum states as those corresponding to classical states satisfying all the constraints. The space of kinematic quantum states is usually a Hilbert space and to carry out the second step one tries to find operators on the Hilbert space corresponding to the constraints and singles out physical quantum states as those annihilated by the operators (this procedure is valid if all the constraints are of the first class).
In this paper we construct a space of kinematic quantum states for TEGR treated as a theory of cotetrad fields on a four-dimensional manifold. More precisely, the construction is valid for any theory of cotetrad fields the phase space of which coincides with that of TEGR-an example of such a theory is the Yang-Mills-type Teleparallel Model (YMTM) considered in [8,9].
The space of quantum states for TEGR, which since now will be denoted by D, will be constructed according to a method presented in [10] combined with some Loop Quantum Gravity (LQG) techniques [11][12][13][14]. This method being a generalization of a construction by Kijowski [15] provides us with a space of quantum states which is not a Hilbert space but rather a convex set of quantum states-these states can be seen as algebraic states (i.e. linear positive normed functionals) on a C * -algebra which can be thought of as an algebra of some quantum observables.
We will also show that spatial diffeomorphisms act naturally on the space D which allows to hope that D can be used as an element of a background independent quantization of TEGR.
The construction of D is similar to a construction of a space of quantum states for the degenerate Plebański gravity (DPG) [10] and the descriptions of both constructions follow the same pattern. It may be helpful to study first the construction in [10] since it is simpler than that of D.
Let us mention that except the space D it is possible to construct other spaces of kinematical quantum states for TEGR-in this paper we will briefly describe the other spaces and comment on their possible application to quantization of TEGR.
To proceed further with quantization of TEGR it is necessary to single out physical quantum states in the space D, that is, to carry out the second step of the Dirac strategy. Since D is not a Hilbert space the standard procedure mentioned above by means of which one distinguishes physical quantum states has to be modified in a way. At this moment we are not able to present a satisfactory and workable modification of the procedure (some remarks on this very important issue can be found in [10]), but we hope that this problem will be solved in the future.
The paper is organized as follows: Sect. 2 contains preliminaries, in Sect. 3 the space of quantum states for TEGR is constructed, in Sect. 4 we define an action of spatial diffeomorphisms on D, Sect. 5 contains a short description of the other spaces of quantum states, in Sect. 6 we discuss the results. Finally, in Appendix 7 we show that the space D is identical to one of the other spaces.
Denote by E the subspace of M spanned by the vectors {v 1 , v 2 , v 3 }. The scalar product η induces on E a positive definite scalar product δ. Its components (δ I J ) in the basis (v 1 , v 2 , v 3 ) form a matrix diag(1, 1, 1). The matrix (δ I J ) and its inverse (δ I J ) will be used to, respectively, lower and raise capital Latin letter indeces I, J, K , L , M ∈ {1, 2, 3}. In some formulae we will use the three-dimensional permutation symbol which will be denoted by ε I J K .

Phase space
The goal of this paper is to construct a space of quantum states for theories of a particular phase space consisting of some fields defined on a three-dimensional oriented manifold -a point in the phase space consists of: 1. a quadruplet of one-forms (θ A ), A = 0, 1, 2, 3, on such that the metric is Riemannian (positive definite); 2. a quadruplet of two-forms ( p B ), B = 0, 1, 2, 3, on .
p A is the momentum conjugate to θ A . The set of all (θ A ) satisfying the assumption above will be called a Hamiltonian configuration space and denoted by , while the set of all ( p A ) will be called a momentum space and denoted by P. Thus the phase space is the Cartesian product P × . The Poisson bracket between two functions f 1 and f 2 on the phase space is given by the following formula -a definition of the variational derivative with respect to a differential form can be found in [9]. As shown in, respectively, [6] and [9] both TEGR and YMTM possess such a phase space.
It turns out [16] that it is possible to construct quantum states via the method presented in [10] starting from the phase space description above (which in a sense is a natural description)-see Sect. 5. However, as it was argued in [16], a space of these quantum states possesses an undesired property. Therefore the space of quantum states D will be constructed starting from another description [17] of the phase space.
Let ι be a function defined on a space of all global coframes on valued in {−1, 1}. Since for every (θ A ) = (θ 0 , θ I ) ∈ the triplet (θ I ) is a global coframe on the manifold [16] ι can be regarded as a function on . Every function ι which is a constant function on every path-connected subset of defines new variables on the phase space [17] which provide new description of the space. According to it a point in the phase space consists of: , is a real function (a zero-form) on and (θ J ), J = 1, 2, 3, are one-forms on constituting a global coframe; 2. a collection (ζ ιI , r J ) ≡ p, where ζ ιI , I = 1, 2, 3, is a three-form on and r J , J = 1, 2, 3, is a two-form on the manifold.
ζ ιI is the momentum conjugate to ξ I ι and r J is the momentum conjugate to θ J . Thus all the (ξ I ι , θ J ) constitute the Hamiltonian configuration space while all the (ζ ιI , r J ) constitute the momentum space P. The Poisson (2.2) reads now as follows Regarding a relation of the latter description to the former let us first express the dependence of ( p A , θ B ) on (ζ ιI , r J , ξ K ι , θ L ) [17]: Here θ I is a vector field on obtained from θ I by raising its index by a metric inverse to the metric q-in a local coordinate frame (x i ) on Since [17] the vector field θ I is a function of both ξ J ι and θ L . The inverse dependence, that is, the dependence of (ζ ιI , r J , ξ K ι , θ L ) on ( p A , θ B ) reads [17] Here * is a Hodge operator defined by the metric q, and (q I J ) are components of q in the basis (θ J ). Let us emphasize that in (2.6) q is treated as a function of (θ A ) (see (2.1)). Moreover, 3 Construction of quantum states for a theory of the phase space P ×

Choice of variables
The construction of a space of quantum states for TEGR we are going to present in this section can be successfully carried out starting from any variables (ζ ιI , r J , ξ K ι , θ L ). However, as proved in [17] unless ι = sgn or ι = − sgn, where sgn is given by (2.7), the constraints of TEGR found in [6] and the constraints of YMTM found in [9] cannot be imposed on the resulting space of quantum states. Therefore it is reasonable to restrict ourselves to variables defined by, respectively, ι = sgn or ι = − sgn. Actually, we will construct the space D using the variables (ζ s I , r J , ξ K s , θ L ), and then we will show that a space D −s built from the variables (ζ −s I , r J , ξ K −s , θ L ) coincides with D. Since now we will use a simplified notation according to which In the next step of the construction we will choose a set such that each element of it is a pair (F, K ), whereF is a finite dimensional linear subspace ofF and K is a finite set of configurational elementary d.o.f.. Then we will define on a relation ≥ equipping it with the structure of a directed set and show that ( , ≥) satisfies some special assumptions. This will finish the construction since at this moment we will refer to [10] where it was shown that from each directed set satisfying the assumptions one can build a space of quantum states.
The construction of the space of quantum states from such a directed set ( , ≥) proceeds as follows. Given (F, K ) ≡ λ ∈ , one uses elements of K to reduce the "infinite-dimensional" space to a space K of finite dimension. Next, one defines (i) a Hilbert space H λ as a space of functions on K square integrable with respect to a natural measure on K and (ii) a space D λ of all density operators on the Hilbert space (i.e. positive operators of trace equal 1). It turns out that assumed properties of the set ( , ≥) unambiguously induce on a set {D λ } λ∈ the structure of a projective family. The space of quantum states is then defined as the projective limit of the family.
Let us emphasize that our choice of elementary d.o.f. as well as application of graphs, cylindrical functions and the operators defined on them by the Poisson bracket is motivated by LQG methods-see [11][12][13][14] and references therein.

Submanifolds of
Each elementary d.o.f. we are going to use will be associated with a submanifold of .
Following the LQG methods since now till the end of this paper we will assume that the manifold is real analytic. 1 An analytic edge is a one-dimensional connected analytic embedded submanifold of with two-point boundary. An oriented one-dimensional connected C 0 submanifold of given by a finite union of analytic edges will be called an edge. The set of all edges in will be denoted by E.
Given an edge e of two-point boundary, its orientation allows to call one of its endpoints a source and the other a target of the edge; if an edge is a loop then we distinguish one of its points and treat it simultaneously as the source and the target of the edge.
An edge e −1 is called an inverse of an edge e if e −1 and e coincide as un-oriented submanifolds of and differ by their orientations. We say that an edge e is a composition of the edges e 1 and e 2 , e = e 2 • e 1 , if (i) e as an oriented manifold is a union of e 1 and e 2 , (ii) the target of e 1 coincides with the source of e 2 and (iii) e 1 ∩ e 2 consists solely of some (or all) endpoints of e 1 and e 2 .
We say that two edges are independent if the set of their common points is either empty or consists solely of some (or all) endpoints of the edges. A graph in is a finite set of pairwise independent edges. Any finite set of edges can be described in terms of edges of a graph [20]: The set of all graphs in is naturally a directed set: γ ≥ γ if each edge of the graph γ is a composition of some edges of the graph γ and the inverses of some edges of γ .
Let S be a two-dimensional embedded submanifold of . Assume that S is (i) analytic, (ii) oriented and (iii) of a compact closure. We moreover require S to be such that every edge e ∈ E can be adapted to S in the following sense [21]: e can be divided into a finite number of edges {e 1 , . . . , e N }, i.e. e = e N • e N −1 • · · · • e 2 • e 1 , each of them either 1. is contained in the closure S; 2. has no common points with S; 3. has exactly one common point with S being one of its two distinct endpoints.
We will call such a submanifold a face. A set of all faces in will be denoted by S.
A three-dimensional submanifold V of of a compact closure and of an orientation inherited from will be called a region. A set of all regions in will be denoted by V.

Elementary degrees of freedom
Note that the variables (ξ I , θ J ) and (ζ K , r L ) parameterizing the phase space P × are, respectively, zero-forms (functions), one-forms, three-forms and two-forms which can be naturally integrated over submanifolds of of appropriate dimensions.
Thus every point y ∈ defines naturally a function on : Similarly, every edge e defines a function on : We choose the set K of configurational elementary d.o.f as follows It is easy to realize that the functions in K separate points in . Note that for every I = 1, 2, 3, every e ∈ E and every pair of edges e 1 , e 2 ∈ E for which the composition e 2 • e 1 makes sense Every region V defines a function on P: Similarly, every face S defines a function on P: We choose the set F of momentum elementary d.o.f as follows It is not difficult to check that the functions in F separate points in P.
Clearly, the relation ∼ K is an equivalence one. Therefore it defines a quotient space Note now that there exist (i) a canonical projection from onto K : and (ii) an injective map 2 from K into R N : 9) where N is the number of elementary d.o.f. constituting K and [θ ] denotes the equivalence class of θ defined by the relation ∼ K .
We will say that elementary d.o.f. in K = {κ 1 , . . . , κ N } are independent if the image ofK is an N -dimensional submanifold of R N . A quotient space K given by a set K of independent d.o.f. will be called a reduced configuration space.  This lemma proven in [16] guarantees that if under the mapK u,γ , i.e.K u,γ is a bijection. It means in particular that the d.o.f. constituting K u,γ are independent and K u,γ is a reduced configuration space. We are also allowed to conclude that if K is a one-element subset of K thenK is a bijection and consequently K is a set of independent d.o.f. and K is a reduced configuration space. . Let u be a set of points defining elements of K of the type (3.2) and let E be a set of edges defining elements of K of the type (3.3). Let γ be a graph related to E as stated in Lemma 3.1.
Since every e ∈ E is a combination of edges of γ and their inverses we can apply equations (3.4)  Note now that if K is a reduced configuration space then the mapK can be used to define a differential structure on the space. It may happen that a set K of independent d.o.f. distinct from K defines the same space: for every θ ∈ , where [θ ] denotes the equivalence class of θ defined by the relation ∼ K . Assume that then the differential structures on K = K given byK and K coincide (we will prove soon that this is the case). Then following [20] we can introduce the notion of cylindrical functions: Definition 3. 4 We say that a function : → C is a cylindrical function compatible for some smooth function ψ : K → C.
being the number of elements of K and the mapK defines a linear structure on K being the pull-back of the linear structure on R N ; if K = K for some other set K ∈ K then the linear structures defined on the space byK andK coincide. 2. if a cylindrical function compatible with a set K ∈ K can be expressed as where K ∈ K and ψ is a complex function on K then ψ is smooth and consequently is compatible with K ; 3. for every element ∈ Cyl there exists a set K ∈ K such that is compatible with K .
It follows from Lemmas 3.2 and Corollary 3.3 that a subset of K consisting of all sets K u,γ , where u runs through all finite subsets of and γ runs through all graphs in satisfies the requirement imposed on the set K by the proposition. Thus, according to Assertion 1 of the proposition, on every reduced configuration space K there exists a natural linear structure and, consequently, a natural differential structure. This means that the space Cyl introduced just above the proposition is well defined, Assertions 2 holds and by virtue of Assertion 3 for every element ∈ Cyl there exist a finite set u of points in and a graph γ such that is compatible with K u,γ .
A simple but useful consequence of the results above is that on every reduced configuration space K , where K = {κ 1 , . . . , κ N }, one can define a linear coordinate frame (x 1 , . . . , x N ): in other words, (3.14) The frame (3.13) will be called natural coordinate frame on K .
Consider a finite collection u = {y 1 , . . . , y M } of points in and a graph γ = {e 1 , . . . , e N } such that either u or γ is not an empty set (N , M ≥ 0 but N + M > 0). Let us introduce a special notation for natural coordinates (3.13) defined on a reduced configuration space K u,γ : we will denote the coordinates by . The coordinates define vector fields We know already (see the discussion just below Proposition 3.5) that is compatible with a set K u,γ , i.e. = pr * K u,γ ψ for a function ψ defined on K u,γ . Assume that u = {y 1 , . . . , y M }. Then where {∂ z L l } are vector fields on K u,γ defined by the natural coordinates (3.15). To find an explicite expression for {ϕ V I , κ L y l } note that where I V is the characteristic function of the region V and δ y is the Dirac distribution supported at y ∈ . Hence Let us emphasize thatφ V I κ L y is a constant real cylindrical functions which since now will be treated as a real number. Thus finallŷ which means thatφ V I preserves the space Cyl. Thusφ V I is a linear operator on the space.

Operators corresponding to d.o.f. (3.6)
With every elementary d.o.f. ϕ S J ∈ F we associate a flux operatorφ S J [11]-it is a linear operator on Cyl defined via a suitable regularization of the Poisson bracket {ϕ S J , }, where ∈ Cyl. Again, we express the cylindrical function as = pr * K u,γ ψ for some set K u,γ and a function ψ on K u,γ . Assume that γ = {e 1 , . . . , e N }. Then the operatorφ J S acts on as follows: where {∂ x J j } are vector fields on K u,γ given by the coordinate frame (3.15) and each (S, e j ) is a certain real number.
To define the number (S, e j ) we adapt the edge e j to S obtaining thereby a set of edges {e j1 , . . . , e jn } and define a function on this set: (e ja ) = 0 if e ja is contained inS or has no common points with S; in remaining cases 1. (e ja ) := 1 2 if e ja is either 'outgoing' from S and placed 'below' the face or is 'incoming' to S and placed 'above' the face; 2. (e ja ) := − 1 2 if e ja is either 'outgoing' from S and placed 'above' the face or is 'incoming' to S and placed 'below' the face.
Here the terms 'outgoing' and 'ingoing' refer to the orientation of the edges (which is inherited from the orientation of e j ) while the terms 'below' and 'above' refer to the orientation of the normal bundle of S defined naturally by the orientations of S and . Then we define It is not difficult to realize that for every edge e ∈ Ê which means thatφ S J κ L e is a constant real cylindrical function which since now will be regarded as a real number.

Linear space of the operators
Let us introduce a spaceF as a real linear space spanned by all operators (3.16) and (3.21):F Thus an elementφ ofF is of the following form where A I i , B J j are real numbers and both sums are finite. Let = pr * K u,γ ψ be a cylindrical function compatible with K u,γ . Then where in the first step we used (3.20) and (3.21) and in the second one we applied (3.19) and (3.22).

A directed set ( , ≥)
All considerations above were preparatory steps to the crucial one which is a choice of a directed set ( , ≥)-once such a set is chosen properly the prescription described in [10] can be used to build from it a unique space of quantum states.
∈ then the image of the mapK given by (3.9) is R N (where N is the number of elements of K )-in other words,K is a bijection and consequently 3. if (F, K ) ∈ , then (a) for everyφ ∈F and for every cylindrical function where {∂ x α } are vector fields on K given by the natural coordinate frame (3.13); (b) for everyφ ∈F and for every κ ∈ K the cylindrical functionφκ is a real constant function on ; is a basis ofF then an N × N matrix G = (G βα ) of components

Speckled graphs
In the considerations above an important role was played by sets {K u,γ }. Therefore one may try to use these sets to define a set as one consisting of pairs (F, K u,γ ) which satisfy all Assumptions listed in the previous section. However, we will not use all sets K u,γ to define but will restrict ourselves to some of them.
To justify our decision let us refer to the general construction presented in [10] (see its outline in Sect. 3.2). According to it for every (F, K u,γ ) ≡ λ ∈ we will have to associate a Hilbert space H λ of some square integrable functions on K u,γ (then density operators on all such Hilbert spaces will be used to build the space D). It seems to us that it would be highly desirable if one could define on each Hilbert space H λ a sort of quantum geometry related to the Riemannian geometry of . But to achieve this we have to guarantee that the d.o.f. in K u,γ can be used to extract some consistent information about the geometry. Since the geometry is given by the metric q we have to require that the d.o.f. provide some consistent information about the metric. Let us now analyze this issue more carefully.
The metric q is defined by (2.1) in terms of the variables (θ 0 , θ J ), can be approximated modulo the factor sgn(θ I ) by If, however, y ∈ e then in general we cannot obtain from the d.o.f. {κ I y , κ J e } a good approximation of the integral (3.25). Thus we conclude that to define the set we should use sets {K u,γ } such that each point y ∈ u belongs to an edge of γ and for each edge e of γ the intersection e ∩ u consists of exactly one point.
However, this conclusion may seem to be a bit premature because while drawing it we neglected the lack of the factor sgn(θ I ) in the formula (3.26). It turns out that, given K u,γ , the d.o.f. in it do not contain any information about the factor-this fact is a consequence of the following lemma [16]: The lemma means that for every θ ≡ (ξ I , θ J ) ∈ the equivalence class [θ ] defined by K u,γ contains points of given by global coframes compatible as well as global coframes incompatible with the orientation of . Hence no function on K u,γ can be an approximation of sgn(θ I ).
Is the impossibility to approximate sgn(θ I ) by functions on K u,γ a problem? It could be if relevant quantities like ones describing the geometry of as well as constraints and Hamiltonians depended on sgn(θ I ).
Note that the metric q is a quadratic function of θ 0 . Therefore the sign sgn(θ I ) is irrelevant for the metric-see the formula (2.5) expressing the metric in terms of (ξ I , θ J ). Consequently, geometric quantities on including the Hodge operator * given by q and the orientation of do not depend on sgn(θ I ).
Regarding the constraints (and the Hamiltonians 3 ) of TEGR and YMTM, an important observation is that they are quite specific functions of (θ A , p B ) and a variable ξ A defined as a solution of the following equation system 4 [22]: Namely, these three variables appear in the constraints exclusively in the form of either (i) a contraction with respect to the index A e.g. ξ A dp A or (ii) scalar products defined by η (or its inverse) e.g. η AB dθ A ∧ * dθ B (or η AB p A ∧ * p B ). Since the matrix (η AB ) (and its inverse) is diagonal two time-like components (that is, components with A = 0) of the variables always multiply each other e.g.
On the other hand every time-like component of the variables under consideration is proportional to sgn(θ I ) [17]: and the space-like components (that is, components with A ∈ {1, 2, 3}) of the variables are independent of the factor. Thus the factor appears in the constraints exclusively as [sgn(θ I )] 2 ≡ 1 and, consequently, the constraints expressed in terms of (ζ I , r J , ξ K , θ L ) are independent of sgn(θ I )-see [17] for explicite expressions of them. Thus (at least at this stage) the impossibility to express sgn(θ I ) by functions on K u,γ does not seem to cause any problem.
Let us turn back to the conclusion placed just below the formula (3.26). It motivates us to introduce a special kind of graphs: is a pair (u, γ ), where u is a finite set of points in and γ is a graph such that there exists a surjective map χ : γ → u such that χ(e) ∈ e for every e ∈ γ . Letγ = (u, γ ) be a speckled graph. We will denote K u,γ ≡ Kγ . Now the conclusion mentioned above can be reformulated: to define a set for TEGR we should use sets {Kγ } given by all speckled graphs in . Let us now take a closer look at these graphs.

Properties of speckled graphs
Consider now a pair (u, γ ), where u is a finite set of points and γ is a graph. Of course, (u, γ ) may not be a speckled graph because it may happen that (i) there exist elements of u which do not belong to any edge of γ ; (ii) there are edges of γ such that no point in u belongs to the edges or (iii) there are edges of γ such that two or more distinct points of u belong to each of the edges. Note however that (u, γ ) can be easily transformed to a speckled graph as follows: in a case of a point y in u of the sort (i) one can choose an edge e such that y is the only point of u which belongs to e and γ ∪ {e} is a graph; in a case of an edge of the sort (ii) one can single out a point in e and add it to u; in a case of an edge of the sort (iii) one can divide it into smaller edges such that each smaller one contains exactly one point belonging to u. (u, γ ), where u is a finite set of points and γ is a graph, there exists a speckled graphγ = (u , γ ) such that u ⊃ u and γ ≥ γ .

Corollary 3.8 For every pair
We will say that a speckled graphγ is greater or equal to a speckled graphγ , if u ⊃ u and γ ≥ γ .

Lemma 3.9
The set of all speckled graphs in equipped with the relation (3.27) is a directed set.
Proof The relation (3.27) is obviously transitive. Let us then show that for every two speckled graphsγ = (u , γ ),γ = (u, γ ) there exists a speckled graphγ such thaṫ γ ≥γ andγ ≥γ . Let u 0 := u ∪ u and let γ 0 be a graph such that γ 0 ≥ γ , γ . Due to Corollary 3.8 there exists a speckled graphγ = (u , γ ) such that u ⊃ u 0 and γ ≥ γ 0 . Thusγ is the desired speckled graph. Due to Lemmas 3.2 and 3.10 a set of all sets Kγ , whereγ runs through all speckled graphs in , meets the requirement satisfied by the set K in Proposition 3.5. This means that for every ∈ Cyl there exists a speckled graphγ such that is a cylindrical function compatible with Kγ . Thus γ ≥ γ and, taking into account the previous result u ⊃ u, we see thaṫ γ ≥γ .

Definition 3.13
The set is a set of all non-degenerate pairs (F, Kγ ) ∈F×K, wherė γ runs through all speckled graphs in . Using the multi-labels α = (i, I ) and β = ( j, J ) we can write  Regarding a proof of the lemma it would be perhaps enough to refer to the proof of an analogous lemma in [10] concerning a set constructed for DPG saying that a proof of Lemma 3.16 is a modification of the proof of that lemma in [10]. But yet taking into account the importance of Lemma 3.16 to avoid any doubt we decided to present the proof explicitly.
Before we will prove the lemma let us state some facts which will be used in the proof. Let be a subset of Cyl. Then operators inF restricted to are maps from into Cyl. Since both Cyl andF are linear spaces the restricted operators are maps valued in a linear space and the space of all the restricted operators is a linear space. Consequently, the notion of linear independence of the restricted operators is well defined-below this notion will be called linear independence of the operators on . Both the lemma and the proposition are proven in [10].

Proof of Theorem 3.16
The transitivity of the relation ≥ is obvious. Thus we have to prove only that for any two elements λ , λ ∈ there exists λ ∈ such that λ ≥ λ and λ ≥ λ. To achieve this we will refer to Lemma 3.17 and Proposition 3.18. Therefore first we have to show that we are allowed to use them.
By virtue of Lemmas 3.10 and 3.14 the set satisfies Assumption 1a. On the other hand, Eq.  is a speckled graph. As in the proof of Lemma 3.14 one can find a face S i such thatφ

Checking assumptions
for every J, L = 1, 2, 3. Let  (3.29) The measure provides a Hilbert space together with a set D λ of all density operators (i.e. positive operators of trace equal 1) on H λ . It was shown in [10] that given two elements λ , λ of such that λ ≥ λ there exists a distinguished projection π λλ from D λ onto D λ . The projection is defined as follows.
An important observation is that the projection π λλ is fully determined by the projection pr K K and the subspace [F] . It turns out that for every triplet λ , λ , λ ∈ such that λ ≥ λ ≥ λ the corresponding projections satisfy the following consistency condition π λλ = π λλ • π λ λ , (3.34) which means that {D λ , π λλ } λ∈ is a projective family. The space D of quantum states for a theory of the phase space P × is the projective limit of the family: 3.9 C * -algebra of quantum observables Let us recall briefly a construction of a C * -algebra of quantum observables [15] associated with the space D. Denote by B λ the C * -algebra of bounded linear operators on the Hilbert space H λ given by (3.30). Each density operator ρ ∈ D λ defines an algebraic state (that is, linear C-valued positive normed functional) on the algebra B λ via a trace: This fact guarantees that for every pair λ ≥ λ of elements of there exists a unique injective * -homomorphism π * λ λ : B λ → B λ dual to the projection π λλ : D λ → D λ in the following sense: for every a ∈ B λ and every ρ ∈ D λ tr(π * λ λ (a)ρ ) = tr(a π λλ (ρ )), By virtue of (3.34) for every triplet λ ≥ λ ≥ λ which means that {B, π * λ λ } is an inductive family of C * -algebras associated with the projective family {D λ , π λλ }. Its inductive limit is naturally a unital C * -algebra which can be interpreted as an algebra of quantum observables. It can be shown that each element ρ of the space D defines an algebraic state on B.

Action of spatial diffeomorphisms on D
Since we would like to quantize TEGR in a background independent manner it is natural to follow LQG methods (see e.g. [12][13][14]) to define an action of diffeomorphisms of the manifold on the space D. Since represents a spatial slice of the original space-time the diffeomorphisms of can be regarded as spatial diffeomorphisms.

Action of diffeomorphisms on elementary d.o.f
Let Diff be a group of all analytic diffeomorphisms of which preserve the orientation of the manifold. Consider an element τ of Diff. Since the fields (ζ I , r J , ξ K , θ L ) are differential forms on the diffeomorphism acts on them naturally as the pull-back τ * . Thus the pull-back define the action of the diffeomorphism on the phase space P × . Since elementary d.o.f. in K and F are functions on the phase space it is natural to define an action of τ on K and F as follows. Given κ ∈ K, the result τ κ of the action of τ on κ is a function on such that Similarly, given ϕ ∈ F, the result τ ϕ of the action of τ on ϕ is a function on P such that Obviously, which mean that both sets K and F are preserved by the action of τ .

Action of diffeomorphisms on reduced configuration spaces
In the next step let us define an action of diffeomorphisms on reduced configuration spaces. Let us fix a finite set K = {κ 1 , . . . , κ N } of independent d.o.f. and a diffeomorphism τ . Denote by τ K the set {τ κ 1 , . . . , τ κ N }. Moreover, let ∼ and ∼ τ be the equivalence relations on defined by, respectively, K and τ K (see Sect. 3.5) and let [θ ] and [θ ] τ denote equivalence classes of θ ∈ given by the corresponding relations.
This means that θ ∼ θ if and only if (τ −1 * θ) ∼ τ (τ −1 * θ ). Consequently, the following map is well defined and is a bijection. Consider now the projections pr K and pr τ K defined by (3.8) and the mapsK and τ K defined by (3.9). We have On the other hand, It follows from the this result that τ K is a bijection onto R N which means that τ K is a set of independent d.o.f. and τ K is a reduced configuration space.
Let (x α ) be the natural coordinate frame (3.13) on K and let (x α ) be the natural coordinate frame on τ K . Denote by {∂ x α } and {∂x α } vector fields defined by the coordinate frames on, respectively, K and τ K . Equations (3.14) and (4.4) imply that the map T τ when expressed in the coordinate frames is an identity map. Hence T τ * ∂ x α = ∂x α and consequently where ψ is a function on K .

Action of diffeomorphisms on cylindrical functions and momentum operators
Now we will extend the action τ from K onto Cyl: given ∈ Cyl we define -this definition guarantees that τ acts linearly on Cyl. Assume that ∈ Cyl is compatible with a set K of independent d.o.f., that is, = pr K ψ. Then by virtue of (4.3) The corollary means that the action of τ preserves the space Cyl-recall that every element of Cyl is a finite linear combination of functions such that each function is compatible with a set of independent d.o.f.. Thus → τ is a linear automorphism on Cyl.
In the next step we define an action of diffeomorphisms on the linear spaceF of the momentum operators: givenφ ∈F, It follows immediately from the definition thatφ → τφ is a linear map.
Let us now calculateφ V I (τ ). We know already that for every ∈ Cyl there exists a finite set K ≡ K u,γ and a complex function ψ on K such that = pr K ψ. Obviously, Let u = {y 1 , . . . , y N } and let (z I i ,x J j ) be the natural coordinates (3.15) on τ K . Then by virtue of (3.20) and Corollary 4.1 Using in turn (4.5) and (4.3) we obtain where (z I i ) are coordinates being a part of the natural coordinate frame on K . Note now that by virtue of (3.18) We conclude that 5 -the latter equation can be proven similarly. The result means that the action of τ preserves the spaceF. Thusφ → τφ is a linear automorphism onF.
Consider now a pair λ = (F , Kγ ) and λ = (F, Kγ ) such that λ ≥ λ. Using Definition 3.15 and properties of the action of τ onF we obtain which means that Consequently, the relation ≥ is preserved by the action τ on . We conclude that the directed set ( , ≥) is preserved by the action of diffeomorphisms.

Action of diffeomorphisms on D
Consider λ = (F, Kγ ) ∈ . Recall that the map T τ : Kγ → τ Kγ when expressed in the natural coordinate frames (x α ) on Kγ and (x α ) on τ Kγ is an identity map. This means that T τ maps the measure dμ λ on Kγ defined by (3.29) to the measure dμ τ λ on τ Kγ defined analogously. Therefore the map is a map onto D τ λ . Consider now an element ρ of D-by virtue of the definition of a projective limit ρ is a family {ρ λ } λ∈ such that ρ λ ∈ D λ for every λ and π λλ ρ λ = ρ λ for every pair λ ≥ λ. It is natural to define an action of diffeomorphisms on ρ as follows τρ := {u τ ρ λ } λ∈ , (4.8) but is τρ an element of D? Clearly, τρ ∈ D if for every λ ≥ λ where we denotedλ ≡ τ λ andλ ≡ τ λ to keep the notation compact. Thus to prove that the action of diffeomorphisms on D preserves the space we should show that Recall the projection π λλ is determined by the projection pr K K and the subspace [F] of K (see Sect. 3.8). Similarly, the projection πλλ is constructed from prKK and the subspace [τF] of τ K , wherē K ≡ τ K ,K ≡ τ K andφ → [φ] is the linear map fromF onto τ K defined in Sect. 3.8. Taking into account that the map u τ appearing (4.9) is defined by T τ (see (4.7) and (4.6)) we conclude that to prove (4.9) it is enough to show that (4.10) Let us denote elements of K and K as it was done in Sect. 3.8. It follows from (3.31) that τ κ α = B β α τ κ β , hence by virtue of (3.32) Using (4.4) we obtain the first equation in (4.10) An operatorφ ∈F defines on K the constant vector field (3.33) which means that in the natural coordinate frame (x β ) the point [φ] ∈ K is represented by (φκ β ). On the other hand, the operator τφ defines on τ K a constant vector field where (x β ) are the natural coordinates on τ K . Thus the point [τφ] ∈ τ K is represented by in the frame (x β ). Since the map T τ expressed in the coordinates (x β ) and (x β ) is an identity we conclude that which means that the second equation in (4.10) is true.
In this way we showed that τρ ∈ D, that is, that the action (4.8) of diffeomorphisms preserves the space D.

Spaces built from other variables on the phase space
A spaceD of quantum states similar to D can be constructed by applying the natural description of the phase space [16], that is, the description in terms of fields (θ A , p B ) (see Sect. 2.2). In this case elementary d.o.f. are given by integrals of one-forms (θ A ) over edges and by integrals of two-forms ( p B ) over faces. To define a directed set (¯ , ≥) which underlies the construction ofD it is enough to use the directed set of all usual (non-speckled) graphs. In other words, this construction is fully analogous to the construction of quantum states for DPG presented in [10]-the only difference between these two constructions is that in the case ofD the canonical variables are four one-forms (θ A ) and four two-forms ( p B ) while in [10] the canonical variables are one one-form and one two-form.
Thus the construction ofD is simpler than that of D. Unfortunately, the spaceD possesses an undesirable property: as shown in [16] quantum states inD correspond not only to elements of by also to all quadruplets (θ A ) which define via (2.1) non-Riemannian metrics on . Since we do not see any workable method which could distinguish inD states corresponding only to elements of we prefer to base the quantization of TEGR on the space D.
Let us emphasize that constructing the space D we never applied the fact that the variables (ζ I , r J , ξ K , θ L ) are defined by ι = sgn-this fact was used in the discussion in Sect. 3.7.2, but the only goal of this discussion was to show that the impossibility to approximate sgn(θ I ) by means of functions on Kγ is not an obstacle for defining quantum geometry operators and quantum constraints as counterparts of classical constraints of TEGR and YMTM. In other words, the discussion concerned not the very construction of D but rather further applications of D. This means that a space D ι of quantum states for TEGR can be built in the same way starting from any variables (ζ ιI , r J , ξ K ι , θ L ), however, as shown in [17] the constraints of TEGR and YMTM derived in [6] and [9] cannot be imposed on D ι unless ι = sgn or ι = − sgn.
In particular, the variables (ζ −s I , r J , ξ K −s , θ L ) given by ι = − sgn can be used to construct a space D −s . By virtue of (2.6) where we used the original notation for the variables (ζ I , r J , ξ K , θ L ) (see (3.1)). These simple relations imply that the space D −s and D are the same: D ≡ D −s -a proof of this statement can be found in Appendix 7.

Hilbert spaces built from some almost periodic functions
It was shown in [10] that for every theory for which it is possible to apply the general method presented in that paper to obtain a convex set of quantum states there exists another space of quantum states. This space is a Hilbert space built from almost periodic functions defined on those reduced configuration spaces which are isomorphic to R N . Thus in the case of TEGR there exist Hilbert spaces {H ι } andH: the former ones associated with the spaces {D ι } and the latter one withD. However, in order to proceed with the second step of the Dirac strategy we would have to define on such a Hilbert space operators corresponding to the constraints and we expect this to be quite difficult. The source of the difficulty is the fact that on a Hilbert space of almost periodic functions on R N the standard quantum operator of position is ill defined because an almost periodic function multiplied by a Cartesian coordinate on R N is no longer an element of this Hilbert space.  [6,17] it is hard to expect that such "exponentiated position operators" can be used to represent the constraints. Thus the spaces {H ι } andH do not seem to be very promising for canonical quantization of TEGR.

General remarks
The main results of this paper is the space D of quantum states and the related C *algebra B of quantum observables. The space D is not a Hilbert space but a convex set and each element of it naturally defines an algebraic state on the algebra, hence a Hilbert space can be obtained [15] from any state in D and the algebra via the GNS construction. Although for every λ ∈ the space D λ is a set of all density operators on the Hilbert space H λ we do not expect that there exists a Hilbert space such that D is a set of density operators on it.  (3.5) and (3.6) used in the construction are defined as natural integrals of the canonical variables (ζ I , r J , ξ K , θ L ) being differential forms on the manifold . Recall that the natural variables (θ A , p B ) on the phase space are functions (2.4) of (ζ I , r J , ξ K , θ L ) involving the factor sgn(θ I ) defined by (2.7). Since the factor cannot be expressed or even approximated by the elementary d.o.f. (see Lemma 3.6) the spaces D and B may be useful only for a class of theories: the Hamiltonian (and possible constraints) of a theory belonging to this class when expressed in terms of the variables (ζ I , r J , ξ K , θ L ) may not depend on the factor. As shown in [17], both TEGR and YMTM belong to this class.

Diffeomorphism invariant states
Since D is a space of kinematic quantum states to proceed further with the canonical quantization of TEGR we have to find a procedure by means of which we could single out physical quantum states for TEGR-an outline of such procedure was presented in [10]. Because TEGR is a diffeomorphism invariant theory it is reasonable to require that each physical state is invariant with respect to the natural action of the spatial diffeomorphisms on the space D defined in Sect. 4 as it is required in the case of LQG [12][13][14]. Existence of such states in D and possible uniqueness are open questions-at this moment it is difficult to predict whether a theorem of existence and uniqueness of such a state analogous to those presented in [18,19] can be proven; let us only note that in the case of a space of quantum states for DPG constructed in [24] there are plenty of diffeomorphism invariant states, however that construction does not follow the general pattern described in [10] and differs significantly from both the present construction of D and the construction of the space of quantum states for DPG described in [10].

The space D versus the kinematic Hilbert space of LQG
The space D is a space of kinematic quantum states meant to serve as an element of a background independent canonical quantization of general relativity (GR) in the teleparallel formulation. Let us compare the space with its counterpart in LQG since LQG is a result of a background independent canonical quantization of an other formulation of GR.
The counterpart is the Hilbert space H LQG defined as a space of some wave functions. These wave functions are defined on a space A of so called generalized SU (2)connections [23] over a three-dimensional manifold and the scalar product on H LQG is defined by an integral with respect to the Ashtekar-Lewandowski (AL) measure dμ AL [20] on A: Alternatively, the space H LQG can be seen as the inductive limit of an inductive family of Hilbert spaces {H γ , p γ γ } labeled by the directed set of (usual) graphs in (for some details of this alternative description see [24]). Each Hilbert space H γ is defined as follows: given graph γ , one reduces the Hamiltonian configuration space A of LQG being the space of all SU (2)-connections over obtaining a reduced configuration space A γ isomorphic to SU (2) N , where N is the number of edges of γ . Next, one defines where dμ γ is a measure on A γ given uniquely by the normed Haar measure on SU (2) N .
It is easy to find some close similarities between elements of the construction of H LQG and those of the construction of D: A corresponds to the Hamiltonian configuration space , the spaces {A γ } are counterparts of the reduced configuration spaces { Kγ }, likewise the Hilbert spaces {H γ } are counterparts of the spaces {H λ }. Note also that the measure dμ λ given by (3.29) which defines H λ via (3.30) is in fact a Haar measure on Kγ (the latter space being a real linear space is naturally a Lie group). Moreover, as shown in [10] for the space there exists a space¯ related to in the same way as A is related to A.
One may ask now why we did not define a Hilbert space for TEGR in the same way as the space H LQG is defined? The answer is very simple: each space A γ is compact and this fact enables to define the AL measure on A and, alternatively, it enables to define the embeddings { p γ γ : H γ → H γ } which allow to "glue" the Hilbert spaces {H γ } into H LQG via the inductive limit. On the other hand, every space Kγ is noncompact and this fact turns out to be an obstacle for defining a measure on¯ as a counterpart of the AL measure and, alternatively, it turns out to be an obstacle for defining embeddings p λ λ : H λ → H λ which would allows us to "glue" the spaces {H λ } into a larger one by means of an inductive limit. In other words, non-compactness of the spaces { Kγ } precludes the use of the inductive techniques but on the other hand linearity of the spaces allows us to apply the projective techniques according to the original idea by Kijowski [15].
Note however that the compactness of the spaces {A γ } is in fact obtained by means of a reduction of the natural Lorentz symmetry of GR done at the level of the classical theory-this symmetry is reduced to its "sub-symmetry" described by the group of three-dimensional rotations. Technically it is achieved by a passage from the complex Ashtekar-Sen connections [25,26] of the non-compact structure group SL(2, C) to the real Ashtekar-Barbero connections [27] of the compact structure group SU (2). Let us emphasize that the construction of D does not require any reduction of the Lorentz symmetry of the classical theory, however it is still to early to claim that there are no obstacles for defining local Lorentz transformations on D-this issue needs to be analyzed carefully.
Let us finally mention an important difference between the spaces D and H LGQ (for a similar discussion see [24]). Both spaces D and H LGQ are built from some spaces associated with (speckled or usual) graphs in : in the former case these spaces are {D λ } (λ = (F, Kγ ), but D λ does not depend actually on the space F), in the latter one these spaces are {H γ }. Since D is the projective limit of {D λ } each state ρ ∈ D is a collection {ρ λ } of states such that ρ λ ∈ D λ . This means that, given λ, the state ρ λ contains only a partial information about ρ and therefore it can be treated merely as an approximation of ρ [15]. On the other hand, in the case of H LQG defined as the inductive limit of {H γ } for every graph γ there exists a canonical embedding p γ : H γ → H LQG and consequently each element of H γ can be treated as a rightful element of H LQG . Acknowledgments This work was partially supported by the Grant N N202 104838 of Polish Ministerstwo Nauki i Szkolnictwa Wyższego.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. The space D −s is built exactly in the same way as the space D is, the only difference is that the starting point of the construction of D −s are the variables (ζ −s I , r J , ξ K −s , θ L ). Let us then trace all steps of both constructions noting differences and similarities between them. In what follows the variable (ζ I , r J , ξ K , θ L ) will be called first variables while (ζ −s I , r J , ξ K −s , θ L ) will be called second variables. LetK u,γ be a set of d.o.f. {κ I y ,κ J e } distinguished by the finite set u ⊂ and the graph γ . Using the formulae (7.1) it is easy to realize that although K u,γ =K u,γ the equivalence relations ∼ K u,γ and ∼K u,γ coincide hence K u,γ = K u,γ , pr K u,γ = prK u,γ Consider now λ andλ as above andλ = r (λ ), where λ = (F , Kγ ) and assume that λ ≥ λ; thenλ ≥λ. Our goal now is to show that π λλ = πλλ . To prove the second equation in (7.9) we assume that elements of Kγ andKγ are ordered as above and note that due to (7.1) and (7.5) eachφ ∈F defines the same constant vector field on Kγ = Kγ regardless we use the first or the second variables: The fact that and¯ are isomorphic, the second equation (7.7) and equation (7.8) just proven mean that D = D −s .