# Variables suitable for constructing quantum states for the teleparallel equivalent of general relativity I

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## Abstract

We present the first part of an analysis aimed at introducing variables which are suitable for constructing a space of quantum states for the Teleparallel Equivalent of General Relativity via projective techniques—the space is meant to be applied in a canonical quantization of the theory. We show that natural configuration variables on the phase space of the theory can be used to construct a space of quantum states which however possesses an undesired property. We introduce then a family of new variables such that some elements of the family can be applied to build a space of quantum states free of that property.

## Keywords

Teleparallel equivalent of general relativity Canonical quantization Space of kinematic quantum states## 1 Introduction

A formulation of general relativity called Teleparallel Equivalent of General Relativity (TEGR)^{1} has not been yet used as a starting point for a quantization of gravity [2, 3]. Since nowadays no existing approach to quantum gravity seems to be fully successful it is worth to check whether it is possible to construct a model of quantum gravity based on TEGR. In this paper we will address an issue of constructing a space of quantum states for TEGR which could be applied in the procedure of canonical (or a canonical-like) quantization of the theory.

A Hamiltonian analysis of TEGR [4, 5, 6, 7, 8] shows that it is a constrained system. Since we do not expect that constraints on the phase space of TEGR can be solved classically we would like to apply the Dirac’s approach to canonical quantization of constrained systems. According to this approach one first constructs a space of kinematic quantum states, that is, quantum states which correspond to classical states constituting the unconstrained phase space, next among kinematic quantum states one distinguishes physical quantum states as those corresponding to classical states which satisfy all constraints. Thus our goal is to construct a space of kinematic quantum states for TEGR.

Since TEGR is a background independent theory it is desirable to construct a space of quantum states for it in a background independent manner. Methods which provide a construction of this sort are known from Loop Quantum Gravity (LQG)—see e.g. [9, 10] and references therein—but because of a reason explained below they are rather not applicable to TEGR. Therefore we are going to construct the desired space for TEGR by means of a general method [11] deliberately developed for this purpose. This method works as follows.

The starting point for the method is a phase space of a theory of the form \(P\times \Theta \), where \(P\) is a space of momenta, and \(\Theta \) is a (Hamiltonian) configuration space (that is, a space of “positions”). One starts the construction by choosing a set \(\mathcal{{K}}\) of real functions on \(\Theta \) called *configurational elementary degrees of freedom*. Analogously, one chooses a set of *momentum elementary degrees of freedom* consisting of some real functions on \(P\). Next, one defines a special directed set \((\Lambda ,\ge )\)—each element of this set corresponds to a finite collection of both configurational and momentum elementary d.o.f.—and with every element \(\lambda \) of \(\Lambda \) one associates a set of quantum states denoted by \(\mathcal{D}_\lambda \).

Given \(\lambda \in \Lambda \), the set \(\mathcal{D}_\lambda \) of quantum stated is constructed as follows. The element \(\lambda \) corresponds to a finite set \(K\) of configurational d.o.f.. One uses the d.o.f. in \(K\) to reduce “infinite-dimensional” space \(\Theta \) to a finite dimensional space \(\Theta _K\)—this reduction consists in identifying all points of \(\Theta \) for which each d.o.f. in \(K\) gives the same value. Then one defines a Hilbert space of functions on \(\Theta _K\) square integrable with respect to a measure on \(\Theta _K\). The set \(\mathcal{D}_\lambda \) is a set of all density operators (i.e. positive operators of trace equal 1) on this Hilbert space—because density operators represent some (mixed, in general,) quantum states one can treat \(\mathcal{D}_\lambda \) as a set of such states.

In this way one obtains a family \(\{\mathcal{D}_\lambda \}_{\lambda \in \Lambda }\) of sets of quantum states. If the set \((\Lambda ,\ge )\) is chosen properly then it naturally generates on \(\{\mathcal{D}_\lambda \}_{\lambda \in \Lambda }\) the structure of a projective family. Finally, the desired space of kinematic quantum states related to the original phase space \(P\times \Theta \) is defined as the projective limit of the family.

As shown in [11], the task of constructing such a space of quantum states reduces to a construction of a directed set \((\Lambda ,\ge )\) satisfying some assumptions—these assumptions are imposed both on elementary d.o.f. constituting elements of \(\Lambda \) and the relation \(\ge \). Since now a directed set \((\Lambda ,\ge )\) satisfying all these assumption will be called *proper directed set* \((\Lambda ,\ge )\).

- 1.
d.o.f. in \(\mathcal{{K}}\) separate points of \(\Theta \);

- 2.
d.o.f. in \(\mathcal{{K}}\) are defined via integrals of functions of components of the variables; the functions are polynomials of the components of degree 1;

- 3.
there exists a directed set elements of which are finite subsets of \(\mathcal{{K}}\) such that for every element \(K\) of the directed set there exists a natural bijection from \(\Theta _K\) onto \(\mathbb {R}^N\), where \(N\) is the number of d.o.f. in \(K\);

- 4.
d.o.f. in \(\mathcal{{K}}\) are defined in a background independent way i.e. without application of any background field.

Results of our inquiries can be summarized as follows: we will find two kinds of variables on the configurations space \(\Theta \) of TEGR which not only satisfy the four assumptions above but can be actually used in a background independent manner to construct two distinct spaces of quantum states for TEGR. One of these variables are natural configurational variables on the phase space of TEGR, that is, one-forms \((\theta ^A),\,A=0,1,2,3\), defined on a three-dimensional manifold being a space-like slice of a spacetime. We will show, however, that the space of quantum states derived from these variables possesses an undesired property. Therefore we will transform the natural variables obtaining a family of new variables such that some elements of the family can be used to build a space of quantum states for TEGR free of that property—a construction of this space will be presented in [12].

Let us emphasize that the analysis of variables suitable for constructing a space of quantum states for TEGR will be continued in an accompanying paper [13] where we will analyze more closely the family of new variables.

Some constructions presented in the present paper are similar to (elements of) a construction of a space of kinematic quantum states for a simple background independent theory called Degenerate Plebański Gravity (DPG)—the latter construction is described in [11]. It seems to us that it may be quite helpful for the reader to study first the construction in [11] since it is simpler that ones described here.

Let us finally explain why the LQG methods of constructing quantum states do not seem to be applicable to TEGR. The reason is quite simple: the methods require finite dimensional spaces \(\{\Theta _K\}\) to be *compact* ^{2} and it is rather difficult to obtain naturally such spaces in the case of TEGR.

The paper is organized as follows: Sect. 2 contains preliminaries, in Sect. 3 we consider the natural variables \((\theta ^A)\) and explain why the space of quantum state constructed from them does not seem to be very promising for canonical quantization of TEGR. In Sect. 5 we present the family of new variables. Section 6 contains a short summary and an outline of the analysis to be presented in the accompanying paper [13]. In Appendix we prove two very important lemmas which guarantee that both kinds of variables considered in this paper provide d.o.f. satisfying Assumption 3 above.

## 2 Preliminaries

### 2.1 Vector spaces with scalar products

Let such \({\mathbb {M}}\) be a four-dimensional oriented vector space equipped with a scalar product \(\eta \) of signature \((-,+,+,+)\). We fix an orthonormal basis \((v_A)~(A=0,1,2,3)\) of \({\mathbb {M}}\) such that the components \((\eta _{AB})\) of \(\eta \) given by the basis form the matrix \(\mathrm{diag}(-1,1,1,1)\). The matrix \((\eta _{AB})\) and its inverse \((\eta ^{AB})\) will be used to, respectively, lower and raise capital Latin letter indices \(A,B,C,D\in \{0,1,2,3\}\).

Denote by \({\mathbb {E}}\) the subspace of \({\mathbb {M}}\) spanned by the vectors \(\{v_1,v_2,v_3\}\). The scalar product \(\eta \) induces on \({\mathbb {E}}\) a positive definite scalar product \(\delta \)—its components \((\delta _{IJ})\) in the basis \((v_1,v_2,v_3)\) form a matrix \(\mathrm{diag}(1,1,1)\). The matrix \((\delta _{IJ})\) and its inverse \((\delta ^{IJ})\) will be used to, respectively, lower and raise capital Latin letter indices \(I,J,K,L,M\in \{1,2,3\}\).

### 2.2 Phase space

^{3}manifold \(\Sigma \). A point in the phase space consists of:

- 1.a quadruplet of smooth one-forms \((\theta ^{A})\equiv \theta \) on \(\Sigma \) such that
^{4}- (a)
at each point \(y\in \Sigma \) three of four one-forms \((\theta ^A(y))\) are linearly independent,

- (b)the metricon \(\Sigma \) is Riemannian (positive definite).$$\begin{aligned} q=\eta _{AB}\theta ^{A}\otimes \theta ^{B} \end{aligned}$$(2.1)

- (a)
- 2.
a quadruplet of smooth two-forms \((p_A)\) on \(\Sigma \); \(p_A\) is the momentum conjugate to \(\theta ^A\).

*Conditions*above and \(P\) will denote the space of all momenta \((p_A)\). We will call the space \(\Theta \)

*(Hamiltonian) configuration space*.

The phase space under consideration is then a Cartesian product \(P\times \Theta \). As shown in [8] and [15] this is a phase space of both TEGR and a simple theory of the teleparallel geometry called Yang-Mills-type Teleparallel Model^{5} (YMTM) [16].

### 2.3 Reduced configuration spaces

As mentioned above we are going to construct quantum states for TEGR by means of the method described in [11]. Let us recall some notions used in that paper.

^{6}injective map from \(\Theta _K\) into \(\mathbb {R}^N\):

*independent*if the image of \(\tilde{K}\) is an \(N\)-dimensional submanifold of \(\mathbb {R}^N\). The set \(\Theta _K\) given by a set \(K\) of independent d.o.f. will be called a

*reduced configuration space*.

- 3.there exists a directed set elements of which are finite subsets of \(\mathcal{{K}}\) such that for every element \(K\) of the directed set the map \(\tilde{K}\) given by (2.2) is a bijection or, equivalently,under \(\tilde{K}\), where \(N\) is the number of elements of \(K\).$$\begin{aligned} \Theta _{K}\cong {\mathbb {R}}^N \end{aligned}$$(2.3)

## 3 Natural variables on \(\Theta \)

### 3.1 Configurational elementary d.o.f.

^{7}variables \((\theta ^A)\) on \(\Theta \) to define configurational elementary d.o.f.. Since the variables are one-forms we follow the LQG methods (see [9, 10]) and define the following real function on \(\Theta \):

*edge*

^{8}in \(\Sigma \). Let

Now we have to check whether the set \(\bar{\mathcal{{K}}}\) satisfies Assumptions listed in Sect. 1. It is clear that functions in \(\bar{\mathcal{{K}}}\) separate points of \(\Theta \), thus \(\bar{\mathcal{{K}}}\) meets Assumption 1. The function \(\kappa ^A_e(\theta )\) can be easily expressed in terms of components of the one-form \(\theta ^A\) given by local coordinate frames on \(\Sigma \). It follows immediately from such expressions that \(\bar{\mathcal{{K}}}\) satisfies Assumption 2.

*graphs*

^{9}in \(\Sigma \)—it is known from LQG that under a technical requirement

^{10}all graphs in \(\Sigma \) form a directed set. Consider then a graph \(\gamma \) being a collection \(\{e_1,\ldots ,e_N\}\) of edges in \(\Sigma \). The graph defines a finite set

There holds the following lemma proven in Appendix 7:

### **Lemma 3.1**

Let us now comment on the lemma. Recall now that Condition 1b of the phase space description presented in Sect. 2.2 means that for every \(\theta \in \Theta \) and for every nonzero vector \(X\) tangent to \(\Sigma \) the values \((\theta ^A(X))\) form a space-like vector in \({\mathbb {M}}\cong \mathbb {R}^4\). On the other hand, given edge \(e\) and \(\theta \in \Theta \), we can interpret a quadruplet \((\kappa ^A_e(\theta ))\) as a vector in \({\mathbb {M}}\). Naively thinking, one could expect that \((\kappa ^A_e(\theta ))\) should be space-like also. However, a sum—and then an integral—of space-like vectors in \({\mathbb {M}}\) may be any other vector in \({\mathbb {M}}\) and this is exactly why the lemma is true.

We conclude that the set \(\bar{\mathcal{{K}}}\) of configurational d.o.f. defined by the natural variables \((\theta ^A)\) satisfies all Assumptions presented in Sect. 1. Thus the set \(\bar{\mathcal{{K}}}\) seems to be suitable for constructing in a background independent way a set of quantum states for TEGR. In fact, the directed set \((K_\gamma ,\ge )\) can be extended to a proper directed set \(({\Lambda },\ge )\) for TEGR—the construction of the latter set is fully analogous to the construction of a set \((\Lambda ,\ge )\) for DPG [11]. Since the resulting set \(({\Lambda },\ge )\) for TEGR is proper it generates *a space of kinematic quantum states* for TEGR which will be denoted by \(\bar{\mathcal{D}}\).

### 3.2 An undesired property \(\bar{\mathcal{D}}\)

Unfortunately, the space \(\bar{\mathcal{D}}\) of kinematic quantum states for TEGR seems to be too large to be used in a canonical quantization of TEGR. The space is “too large” in the following sense.

Note that the space \(\Theta '\) contains quadruplets \((\theta ^A)\) which via the formula (2.1) define on \(\Sigma \) not only Riemannian metrics but also metrics which (locally or globally) are Lorentzian (i.e. of signature \((-,+,+)\)). Thus the kinematic quantum states in \(\bar{\mathcal{D}}\) correspond also to a large set of quadruplets \((\theta ^A)\) which have nothing to do with elements of \(\Theta \)—note that it is rather not possible for a quadruplet defining a Lorentzian metric to be a limit of a sequence of elements of \(\Theta \).

Is it possible to isolate quantum states in \(\bar{\mathcal{D}}\) which do not correspond to Lorentzian metrics on \(\Sigma \)? Perhaps it is, but we expect this to be rather difficult because of the following reason. By means of d.o.f. belonging to a finite subset \(K_\gamma \) of \(\bar{\mathcal{{K}}}\) we are not able to distinguish between elements of \(\Theta \) and those of \(\Theta '\!\setminus \!\Theta \)—see the first Eq. (3.2). On the other hand, all d.o.f. in \(\bar{\mathcal{{K}}}\) separate points not only in \(\Theta \) but also in \(\Theta '\). Thus the all d.o.f. in \(\bar{\mathcal{{K}}}\) distinguish between elements of \(\Theta \) and \(\Theta '\!\setminus \!\Theta \). Consequently, we are not able to isolate quantum states which do not correspond to Lorentzian metrics by means of a family \(\{R_\lambda \}_{\lambda \in \Lambda }\) of restrictions such that each restriction \(R_\lambda \) is imposed on elements of \(\mathcal{D}_{{\lambda }}\) but would have to isolate desired states at the level of the whole \(\bar{\mathcal{D}}\). Taking into account the complexity of \(\bar{\mathcal{D}}\), this task seems to be very difficult. Therefore we prefer to find other variables which could give us a space of quantum states free of the undesired property of \(\bar{\mathcal{D}}\).

## 4 New variables on \(\Theta \)

### 4.1 New variables—preliminary considerations

The undesired property of \(\bar{\mathcal{D}}\) just described follows from the fact that the variables \((\theta ^A)\) can be used to parameterize not only the configuration space \(\Theta \) but also the larger space \(\Theta '\) (provided Condition 1b has been omitted). Thus to obtain a space of kinematic quantum states for TEGR free of the property of \(\bar{\mathcal{D}}\) we can try to find new variables which parameterize the space \(\Theta \) and cannot be used to describe those elements of \(\Theta '\!\setminus \!\Theta \) which correspond to Lorentzian metrics on \(\Sigma \). Below we present some preliminary considerations results of which will be used in the next subsection to define such new variables.

Condition 1a of the phase space description together with continuity of the fields mean that three of four one-forms \((\theta ^A)\) define a local coframe on \(\Sigma \) and consequently the remaining one-form can be expressed as a linear combination of the three ones. It turns out that Condition 1b allows to formulate a stronger statement:

### **Lemma 4.1**

- 1.
the forms \((\theta ^1(y),\theta ^2(y),\theta ^3(y))\) are linearly independent,

- 2.where \(\alpha _I(y)\) are real numbers satisfying$$\begin{aligned} \theta ^0(y)=\alpha _I(y)\theta ^I(y) \end{aligned}$$(4.1)$$\begin{aligned} \alpha _I(y)\alpha ^I(y)<1. \end{aligned}$$(4.2)

### *Proof*

Let us fix a point \(y\in \Sigma \). For the sake of simplicity till the end of this proof we will omit the symbol “\(y\)” in the notation i.e. we will denote \(\theta ^A(y)\) by \(\theta ^A,\,\alpha _I(y)\) by \(\alpha _I\) and \(q(y)\) by \(q\). As before we will refer to the two conditions imposed on the elements of \(\Theta \) in Sect. 2.2 as to, respectively, Condition 1a and Condition 1b and to the two assertions of the lemma as, respectively, Assertion 1 and Assertion 2.

**Step 1: Conditions 1a and 1b imply Assertion 1**Condition 1a means that either \((i)~(\theta ^1,\theta ^2,\theta ^3)\) or \((ii)~(\theta ^0,\theta ^{I},\theta ^J),\,I\ne J\), are linearly independent. Let us show that \((i)\) is true even if \((ii)\) holds. Without loss of generality we assume that \((\theta ^0,\theta ^1,\theta ^2)\) are linearly independent. Then for some real numbers \(a,b,c\)

**Step 2: Condition 1b and Assertion 1 are equivalent to Assertions 1 and 2**If Assertion 1 is true then there exists real numbers \((\alpha _I),\,I=1,2,3\), such that

**Step 3: final conclusion** Clearly, Assertion 1 implies Condition 1a. This fact together with the result of Step 1 ensure that Conditions 1a and 1b are equivalent to Condition 1b and Assertion 1. Now to finish the proof it is enough to take into account the result of Step 2. \(\square \)

### **Corollary 4.2**

If \((\theta ^A)\in \Theta \) then the triplet \((\theta ^1,\theta ^2,\theta ^3)\) is a global coframe on \(\Sigma \).

### *Proof*

The corollary follows immediately from Assertion 1 of Lemma 4.1. \(\square \)

Let us finally reformulate Lemma 4.1 in the following way:

### **Lemma 4.3**

- 1.real functions \(\alpha _I,\,I=1,2,3\), on \(\Sigma \) such that$$\begin{aligned} \alpha _I\alpha ^I<1, \end{aligned}$$(4.8)
- 2.
one-forms \(\theta ^{J},\,J=1,2,3\), on \(\Sigma \) constituting a global coframe on the manifold.

Let us check whether the set \(\bar{\mathcal{{K}}}'\) satisfies all Assumptions presented in Sect. 1. It obviously meets Assumption 1. Note that the r.h.s. of (4.10) can be treated as an integral of the function \(\alpha ^I\) over the set \(\{y\}\subset \Sigma \) and, consequently, \(\bar{\mathcal{{K}}}'\) satisfies Assumption 2.

Now we have to find the image of the map \(\tilde{K'}_{u,\gamma }\) (see (2.2)). It is obvious that there holds the following lemma

### **Lemma 4.4**

The next lemma is proven in Appendix 6:

### **Lemma 4.5**

The following conclusion is a simple consequence of Lemmas 4.3, 4.4, and 4.5:

### **Corollary 4.6**

*does not*satisfy Assumption 3.

The conclusion is that the set \(\bar{\mathcal{{K}}}'\) does not meet Assumptions 3 but satisfies all remaining ones. Moreover, the variables \((\alpha _{I},\theta ^{J})\) can be used to define Lorentzian metrics on \(\Sigma \) provided we give up the condition (4.8)—if \(\alpha _{I}\alpha ^{I} > 1\) then the eigenvalue (4.7) of the matrix (4.5) is negative and the resulting metric (4.4) is Lorentzian. However, there is a progress with respect to the previously considered variables \((\theta ^{A})\) and the corresponding d.o.f. in \(\bar{\mathcal{{K}}}\), because now if a sextuplet \((\alpha _{I},\theta ^{J})\) defines a metric which is Lorentzian on a subset of \(\Sigma \) then any triplet \(\{\kappa ^{\prime I}_{y} | I=1,2,3 \}\) of d.o.f. with \(y\) belonging to the subset can be used to distinguish between this sextuplet \((\alpha _{I},\theta ^{J})\) and ones belonging to \(\Theta \).

However, there are many diffeomorphisms of this sort and the question is which one should we use? Or, is there a distinguished diffeomorphism? As we will show below a pair of such diffeomorphisms is distinguished by an ADM-like Hamiltonian framework of TEGR.

### 4.2 New variables and new d.o.f.

Note that at every point \(y\in \Sigma \) the values \((\xi ^A(y))\) of a solution of (4.14) form a time-like vector in \({\mathbb {M}}\) which means that the value \(\xi ^0(y)\) cannot be 0. Taking into account the assumed smoothness of \(\xi ^A\) we can expect that there exist exactly two distinct solutions of (4.14) which can be distinguished by the sign of \(\xi ^0\). As shown in [15] by presenting explicite solutions of (4.14) the expectation is correct.

In this way we obtain new variables on the Hamiltonian configuration space \(\Theta \):

### **Lemma 4.7**

- 1.
functions \(\xi ^I,\,I=1,2,3\), on \(\Sigma \),

- 2.
one-forms \(\theta ^J,\,J=1,2,3\), on \(\Sigma \) constituting a global coframe on the manifold.

### *Proof*

Corollary 4.6 and the relation between \((\xi ^I)\) and \((\alpha _J)\) (see (4.21) and (4.22)) allows us to formulate the following lemma:

### **Lemma 4.8**

^{11}set \((K_{u,\gamma },\ge )\) given by all finite subsets of \(\Sigma \) and all graphs in the manifold meets Assumption 3.

It is also worth to note that if \(\xi ^I=0\) then \((\theta ^J)\) is an orthonormal coframe with respect to \(q\)—this fact can be easily deduced from (4.25). Thus we can regard \((\xi ^I)\) as variables indicating how much the coframe \((\theta ^J)\) deviates from being orthonormal with respect to \(q\) (of course, the same can be said about \((\alpha _I)\)).

We conclude that for every function \(\iota \) the set \(\mathcal{{K}}\) of d.o.f. defined by corresponding new variables \((\xi ^{I},\theta ^J)\) satisfy all Assumptions listed in Sect. 1. Moreover, the variables cannot define Lorentzian metrics on \(\Sigma \).

## 5 Summary

In this paper we showed that the natural variables \((\theta ^A)\) on the Hamiltonian configuration space \(\Theta \) of TEGR (and YMTM) can be used to build via the general method described in [11] the space \(\bar{\mathcal{D}}\) of kinematic quantum states. The space \(\bar{\mathcal{D}}\) is constructed in a background independent manner. It turned out that states constituting this space correspond not only to elements of \(\Theta \), but also to quadruplets \((\theta ^A)\) which define Lorentzian metrics on the manifold \(\Sigma \) being a space-like slice of a spacetime. Since the task of isolating quantum states in \(\bar{\mathcal{D}}\) which do not correspond to Lorentzian metrics seems to be very difficult we decided to look for other more suitable variables.

The results of our inquiry is the family \(\{(\xi ^{I},\theta ^J)\}\) of variables parameterized by functions \(\{\iota \}\) defined on the set of all global coframes on \(\Sigma \) and valued in the set \(\{-1,1\}\). Each element of the family satisfies all Assumptions presented in Sect. 1 and cannot define any Lorentzian metric on \(\Sigma \). Therefore we expect that at least some of the variables can be used to define in a background independent way a space of kinematic quantum states for TEGR free of the undesired property of the space \(\bar{\mathcal{D}}\). We will show in [12] by an explicite construction that the expectation is correct.

However, at this moment we are not completely ready for a construction of a space of quantum states from variables \((\xi ^{I},\theta ^J)\) because of the following reason. Recall that we would like to apply the Dirac’s approach to a canonical quantization of TEGR which means that once a space of kinematic quantum states is constructed we will have to impose on the states “quantum constraints” as counterparts of constraints on the phase space of TEGR—this is the second step of the Dirac’s quantization procedure. The problem is that it is not obvious whether every element of the family \(\{(\xi ^{I},\theta ^J)\}\) generates a space of quantum states suitable for defining “quantum constraints” on it.

Although at this stage we are not able to solve this problem completely, we will address the issue in the accompanying paper [13]—we will show there that indeed some variables \((\xi ^{I},\theta ^J)\) are quite problematic. Namely, the constraints of TEGR (and YMTM) when expressed in terms of these variables depend on a special function defined on \(\Theta \). It turns out that this function cannot be even approximated by functions on any \(\Theta _{K_{u,\gamma }}\). This means that in the case of a space of kinematic quantum states built from such variables we will not be able to define “quantum constraints” by means of a family of restrictions such that each restriction is imposed on elements of a single space \(\mathcal{D}_\lambda \). It is clear that if we are not able to define “quantum constraints” in such a way then this task becomes much more difficult.

Fortunately, as it will be proven in [13], there exist exactly two closely related elements of the family \(\{(\xi ^{I},\theta ^J)\}\) for which the problem just described does not appear—the elements are closely related in this sense that functions \(\{\iota \}\) distinguishing them differ from each other by a factor \(-1\). Using one of these two elements we will construct in [12] a space \(\mathcal{D}\) of kinematic quantum states for TEGR. The space \(\mathcal{D}\) will be obviously free of the undesired property of \(\bar{\mathcal{D}}\) and we hope that \(\mathcal{D}\) will be also suitable for carrying out the second step of the Dirac’s procedure.

## Footnotes

- 1.
See [1] for the newest review on TEGR.

- 2.
See [14] for a discussion of obstacles which appear if one tries to apply the LQG methods for non-compact spaces \(\{\Theta _K\}\).

- 3.
Throughout the paper “smooth” means “of \(C^\infty \) class”.

- 4.
Conditions 1a and 1b are not independent—in fact, the former is implied by the latter [13], but for further considerations it will be convenient to formulate them separately.

- 5.
In [15] while describing the phase space of YMTM we imposed only the weaker and insufficient Condition 1a and overlooked Condition 1b.

- 6.
The set \(K\) is unordered, thus to define the map \(\tilde{K}\) one has to order elements of \(K\). Thus the map \(\tilde{K}\) is natural modulo the ordering. However, every choice of the ordering is equally well suited for our purposes and nothing essential depends on the choice. Therefore we will neglect this subtlety throughout the paper.

- 7.
- 8.
A

*simple edge*is a one-dimensional connected \(C^\infty \) submanifold of \(\Sigma \) with two-point boundary. An edge is an*oriented*one-dimensional connected \(C^0\) submanifold of \(\Sigma \) given by a finite union of simple edges. - 9.
We say that two edges are

*independent*if the set of their common points is either empty or consist of one or two endpoints of the edges. A*graph*in \(\Sigma \) is a finite set of pairwise independent edges. - 10.
One assumes \(\Sigma \) to be a real-analytic manifold and restrict oneself to edges built from

*analytic*simple edges. - 11.
The relation \(\ge \) is defined as described just below the formula (4.12).

- 12.
Of course, we have to choose the neighborhood \(U'\) “small” enough to ensure that every integral curve in the bunch intersects the set \(e\cap U'\) exactly once.

- 13.
To satisfy this requirement \(W_\beta \) may be defined as an open coordinate ball of non-zero radius.

- 14.
To satisfy this requirement \(W^{{j}a}_\beta \) may be defined as an open coordinate ball of non-zero radius.

- 15.
Note that for \(n>\sigma \) the \(n\)-th derivative of \(\varphi ^{{j}a}_{\sigma }\) contains a factor \((\varvec{1}-\phi ^{{j}a})^{\sigma -n}\) which could be a source of non-differentiability of \(\varphi ^{{j}a}_{\sigma }\) if a value of a function \((\varvec{1}-\phi ^{{j}a})\) was zero. This is, however, not the case—the function is positive everywhere.

## Notes

### Acknowledgments

This work was partially supported by the Grant N N202 104838 of Polish Ministerstwo Nauki i Szkolnictwa Wyższego.

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