Loop quantization of the Gowdy model with local rotational symmetry
Abstract
We provide a full quantization of the vacuum Gowdy model with local rotational symmetry. We consider a redefinition of the constraints where the Hamiltonian Poissoncommutes with itself. We then apply the canonical quantization program of loop quantum gravity within an improved dynamics scheme. We identify the exact solutions of the constraints and the physical observables, and we construct the physical Hilbert space. It is remarkable that quantum spacetimes are free of singularities. New quantum observables naturally arising in the treatment partially codify the discretization of the geometry. The preliminary analysis of the asymptotic future/past of the evolution indicates that the existing Abelianization technique needs further refinement.
I Introduction
Making realistic predictions on effects of quantum gravity in the cosmological context—an element needed in particular to solve the singularity problem in cosmology—requires investigating models admitting inhomogeneous spacetimes, preferably at the nonperturbative level. Among such settings, Gowdy spacetimes Gowdy (1971) in vacuum are particularly interesting since they are a natural extension of Bianchi I cosmologies Bianchi (1897) and admit nonperturbative inhomogeneities in the form of gravitational waves.
As they capture essential properties of full general relativity (GR) and at the same time are relatively simple, these models have brought over the years a lot of attention of researchers delving upon various aspects of gravity quantization. For instance, a quantum (geometrodynamics) description was already considered in the 1970s Misner (1973); Berger (1974). Further they were explored in the specific context of gravitational waves quantization Pierri (2002) which description was later shown to admit a unitary dynamics Corichi et al. (2002); *unit2; *unit3; *unit4; *unit5. Besides, it was possible to prove that this representation where the dynamics is unitarily implementable is indeed unique Corichi et al. (2006c); *uniq2 if in addition it is compatible with the symmetries of the equation of motion, i.e. the symmetries of the dynamics. The canonical quantization of these models employing the (complex) Ashtekar variables was carried out in Ref. Mena Marugan (1997). These studies however, as treating the homogeneous background either classically or via geometrodynamics could not ’cure’ the singularity problem.
The Gowdy model with linear polarization and spatial slices has been subsequently studied in terms of real Ashtekar–Barbero variables Banerjee and Date (2008a, b) via the midisuperspaces techniques Bojowald and Swiderski (2004). There, however, the difficulties in applying the conventional loop quantum gravity (LQG) techniques Thiemann (2008) did not allow to complete the quantization program and probe the dynamics. Fortunately, the problems hampering prior approaches were successfully addressed in Ref. MartinBenito et al. (2008a); MartínBenito et al. (2010); *hybrid2 via the socalled hybrid quantization program. This approach, also suitable for perturbative cosmological scenarios FernandezMendez et al. (2012); FernándezMéndez et al. (2012); *hybinf2; *hybinf3; *hybinf4; *hybinf5; *hybinf6, combines the standard Fock quantization for the gravitational waves with a polymeric quantization of the homogeneous degrees of freedom. It is furthermore quite convenient for the study of quantum gravity in the presence of matter MartinBenito et al. (2011), and provides an arena for unveiling novel quantum phenomena on some sectors of the theory MartínBenito et al. (2014); *hybrid5; *hybrid6. All these models allow for a convenient partial gauge choice which reduces the set of local constraints to a global Hamiltonian and diffeomorphism constraints. The latter are sufficiently simple to allow finding their solutions (at least formally) and to construct the physical Hilbert space. This approach however, while successful, by the very nature of the hybrid quantization cannot be easily related with the standard LQG.
In this work we follow a more orthodox approach, expanding upon the original midisuperspace program of Ref. de Blas et al. (2015). In order to test new techniques we study a slight simplification of the full polarized, threetorus Gowdy model, namely its locally rotational symmetric (LRS) version, where one identifies the two directions orthogonal to the inhomogeneous one. While in vacuo the model, being diffeomorphic to a homogeneous Bianchi I spacetime (Kasner solutions), features just one free global degree of freedom, in presence of matter (for example a massless scalar field) it admits genuine inhomogeneous solutions, containing however a homogeneous and even isotropic sector. This feature makes the model viable for cosmology applications and particularly useful in testing the results of the perturbative approaches against nonperturbative effects as well as in the studies of the relation of loop quantum cosmology (LQC) Banerjee et al. (2012) with LQG. For instance, Bianchi I spacetimes in loop quantum cosmology Ashtekar and WilsonEwing (2009); MartinBenito et al. (2009a), after imposing local rotational symmetry, or the hybrid quantization of the polarized Gowdy model in the threetorus MartinBenito et al. (2008a); MartínBenito et al. (2010); Garay et al. (2010); MartinBenito et al. (2011) are clearly interesting for this purpose. Gowdy LRS has also been consider in the context of the (loop) consistent algebra approach Bojowald and Brahma (2015a). Our quantization program, unlike the previous approaches, will not involve gaugefixing. Instead, we will be working with the constraint algebra, featuring (as in full GR) local structure functions, and will be forced to employ the Dirac program in a manner featuring the same level of complication as in full LQG. To deal with the known difficulties in its implementation we will follow a strategy already adopted in studies of spherically symmetric spacetimes Gambini and Pullin (2013); Gambini et al. (2014) (see Ref. Bojowald and Brahma (2015b) for a discussion of the full polarized Gowdy model). That strategy is based on a specific redefinition of the constraints and consequently of their algebra structure, which makes the Hamiltonian constraint Abelian. Furthermore, in the construction of the quantum counterpart of this constraint we implement, for the first time in a loop quantized inhomogeneous model, an improved dynamics scheme. The solutions to the Hamiltonian constraint can be explicitly determined and can be equipped with a well defined Hilbert space structure, which in turn, together with the application of the standard LQC treatment of the spatial diffeomorphisms, allows us to unambiguously probe the dynamical sector of the model. It is remarkable that the resulting spacetimes are free of singularities. Furthermore, the area of the Killing orbits is quantized due to the discreteness of the spectrum of a new observable emerging in this quantization. This treatment and its results open a new window for the quantization of cosmological scenarios in LQC featuring a natural connection with the full theory by means of more realistic models admitting nonperturbative inhomogeneities.
The paper is organized as follows. In Sec. II we introduce the classical polarized Gowdy model on the threetorus. Then we consider the model with local rotational symmetry in Sec. III, where we also provide the classical Dirac observables of the model. In Sec. IV we define the Abelian constraint. In Sec. V we describe the kinematical quantum framework and the basic kinematical observables, whereas the Hamiltonian constraint is discussed in Sec. VI. The physical Hilbert space as well as the observables (together with a discussion about the semiclassical sectors of the theory) are provided in Sec. VII. We conclude with Sec. VIII. Furthermore, in App. A we discuss an alternative construction for the Abelianized Hamiltonian constraint, and in App. B a partial spectral analysis of some operators relevant for our treatment.
Ii Classical Polarized Gowdy model in AshtekarBarbero variables
Let us start by summarizing the description of the Gowdy model with spatial sections isomorphic to the threetorus and linear polarization for the gravitational wave content. We will provide this description in terms of real AshtekarBarbero variables with the notation introduced in Ref. Banerjee and Date (2008a). The polarized Gowdy model in this case consists of three local degrees of freedom ^{1}^{1}1The connection in our work is decreased by a factor of with respect to the one introduced in Banerjee and Date (2008a)., and , all of them with support on the circle of angular coordinate . They are pairs of connections and densitized triads, respectively, such that the spatial metric components are , and , and where the function corresponds to the area of the (here 2dimensional) orbits of the spatial Killing vectors.
The symplectic structure is given by
(1) 
with^{2}^{2}2In order to restore the Immirzi parameter Immirzi (1997) in our studies we drop the rescaling by introduced in Banerjee and Date (2008a).
(2) 
Besides, the Hamiltonian of the model is a linear combination of two first class constraints: the diffeomorphism one
(3) 
and the Hamiltonian constraint
(4) 
where . This set of constraints has a nontrivial algebra
(5a)  
(5b)  
(5c) 
As we see, the constraint algebra involves structure functions, as it happens in the full theory. This is one of the main handicaps preventing us from attaining a complete quantization of the full polarized Gowdy model so far. One of the strategies that can be adopted in this type of field theories is to replace the original form of the Hamiltonian constraint with its Abelian version (see Bojowald and Brahma (2015b) for the discussion of this method and its limitations). This procedure was originally suggested and successfully applied in spherically symmetric spacetimes Gambini and Pullin (2013); Gambini et al. (2014). However, at the quantum level, the anomalies that a loop quantization introduces are not yet well understood in the context of the full polarized Gowdy model Bojowald and Brahma (2015b), thus preventing the completion of the traditional loop quantization program there. Fortunately, as we will show further in the manuscript, the Abelianized version of the Hamiltonian constraint can be easily represented at the quantum level as an operator with a constraint algebra free of anomalies provided that we introduce an additional symmetry: local rotational symmetry.
Iii Classical model with local rotational symmetry
Let us consider a restriction of the standard polarized Gowdy model via imposing the requirement that the rotations on the plane are isometries. An easy way to represent the restricted geometries is to identify the degrees of freedom represented by the pairs and . The original symplectic structure (1) reduces then to
(6) 
The phase space is now coordinatized by two pairs of canonical variables and . The spatial metric components become then
(7) 
In the specified notation^{3}^{3}3The reader must be concerned about the fact that we keep here the same notation for the Hamiltonian and the constraints than in the full polarized Gowdy model. In order to avoid any possible confusion, in what follows we will refer only to the scenario with local rotational symmetry. the Hamiltonian is again a linear combination of the constraints , where
(8a)  
(8b) 
The constraint algebra remains unchanged with respect to Eq. (5)
(9a)  
(9b)  
(9c) 
In this restricted setting we are still dealing with a 1+1 field theory with a nontrivial constraint algebra as in the full polarized Gowdy model, sharing with the latter the level of complication and set of difficulties in implementing the Dirac quantization procedure. On the other hand, the structure of the degrees of freedom changes drastically with respect to the original Gowdy model, as the local rotational symmetry fixes an infinite number of physical degrees of freedom. On shell, instead of two global and one local degree of freedom the model we study features only one global and no local ones—the degrees of freedom of Bianchi I Kasner solutions. While in itself it does not admit genuinely inhomogeneous spacetimes^{4}^{4}4Although the solutions represent Kasner spacetimes, these spacetimes are foliated by Cauchy surfaces which are not invariant with respect to finite symmetry transformations., it describes the geometry in a diffeomorphism invariant manner resembling the one of full LQG and not tied to the homogeneity of the physical solutions. Because of that, the treatment can be extended in a straightforward way to LRS Gowdy models including matter content, in particular a massless scalar field. The latter, while it is in general genuinely inhomogeneous, it admits homogeneous (Bianchi I) and isotropic (Friedmann–Robertson–Walker solutions) spacetimes. Hence, it is strongly relevant for the study of our Universe.
Since we are dealing with a constrained system, in order to provide the classical description, one needs to perform one more step: the identification of Dirac observables. For as long as we consider just the vacuum solutions (which is the case of this article), they just admit one global degree of freedom, thus one needs to identify just a single observable. One of the straightforward choices follows from the treatment of spherically symmetric black holes Gambini et al. (2014) (of which techniques we implement in this work), where one could choose the phase space function encoding the ADM mass. Using its form in AshtekarBarbero variables we can propose its “analog” (just in form but not in the physical meaning) in the LRS Gowdy context. It is defined as
(10) 
By a quite lengthy but straightforward calculation one can show that its Poisson bracket with the total Hamiltonian
(11) 
thus it vanishes on shell.^{5}^{5}5We must notice that, as one can deduce from the analysis in Sec. IV, the spatial derivative of the integrand in the righthand side of Eq. (10) is already a combination of constraints. Therefore, only its homogeneous mode, given by , is nonvanishing on shell. As a consequence, encodes the diffeomorphisminvariant constant of motion.
The classical description constructed above could be used as the basis for the (loop) quantization. In its present form, however, due to the nontrivial Poisson bracket (9c), the resulting quantum Hamiltonian constraint would not be commuting with itself, which poses certain challenges in the completion of the quantization program. Therefore, following Gambini and Pullin (2013); Gambini et al. (2014) we will perform the modification of the classical description known as Abelianization procedure.
Iv Abelianization of the scalar constraint in the reduced model
In this section we will consider a redefinition of the scalar constraint so that the analog of the Poisson bracket (9c) (with the redefined constraint) vanishes strongly.
It is well known Hajicek and Kuchar (1990); *teitel that for diffeomorphism invariant systems, transformations like

, where is a function of the configuration variables only (here denoted as for simplicity) and ,

with ,

with representing momentum variables (conjugate to ) and ,
provide a new set of constraints with the same constraint surface. Of course, these transformations are not canonical transformations since the constraint algebra is changing (i.e. the Poisson brackets between constraints change). They can be understood as redefinitions of the lapse and shift functions. In our case we will consider the last type of transformations in order to achieve a new scalar constraint such that it commutes with itself, while the rest of the constraint algebra structure remains that of a proper Lie algebra (with structure constants instead of structure functions).
This method was developed (and successfully applied) in the context of spherically symmetric black hole spacetimes in Gambini and Pullin (2013); Gambini et al. (2014). Following the construction presented there we consider the following transformation
(12) 
which is the unique available transformation removing the dependence on from and not involving rescaling . This transformation is unfortunately singular—not defined on any phase space point on which vanishes on any point in space. Due to the compactness of the spatial slices this is the case for all the considered geometries ( for at least two isolated values of ). By invoking certain completion constructions, one can still work with it at the cost of cutting off from the phase space all the geometries where on an open interval. We will discuss that treatment and its limitations in Appendix A.
To avoid cuttingoff a physically relevant portion of the phase space, we propose another transformation supplementing Eq. (12) with a rescaling by , that is
(13) 
Such transformation has been considered in the context of full polarized Gowdy model in Ref. Bojowald and Brahma (2015b). While the term is regular outside of the strong classical singularity, still makes the transformation singular, this time enlarging the constraint surfaces. This is caused by the fact that the new sets of constraints are satisfied automatically on homogeneous configurations (they vanish identically) while the old set of constraints selects a proper subsurface (within the surface of the homogeneous geometries) of the phase space. However, the spatial continuity of the classical fields (here playing the role of the phase space variables) implies the equivalence between the two sets of constraints on a subset of geometries (points in the phase space) admitting an open region where , ^{6}^{6}6By continuity the old Hamiltonian constraint has to be satisfied at the boundary of the set , then by the diffeomorphism constraint the interior of the complement of this set has to correspond to the fully homogeneous spatial regions and all the variables must remain constant (in particular the same as on the boundary) at each connected subset of that complement. Thus the old constraints have to be satisfied also in the complement. , thus the expansion is nontrivial on the surface corresponding to the homogeneous geometries only. Therefore, at this point, it is necessary to stress that we are modifying our classical theory, which thus will not fully coincide with GR in its globally homogeneous sector. In the classical theory such geometries are highly nongeneric. Unfortunately the discreteness of the variables introduced by the polymeric quantization will make the set of states corresponding to these geometries a nonzero measure one and a certain level of care will be needed when studying the properties of the physical states.
Under the new transformation the total Hamiltonian takes the form
(14) 
The above redefinition of the constraint algebra does not affect the properties of the Dirac observable defined in Eq. (10), which still remains a weak Dirac observable. Since the new Hamiltonian constraint is a linear combination of the original Hamiltonian and diffeomorphism constraints, one can check explicitly that commutes (under Poisson brackets) with the new set of constraints on shell. Actually, one can see that the integrand in the righthand side of Eq. (10) is related with the new scalar constraint—see the phase space function inside the square brackets in Eq. (14).
At this point a few remarks on the range of the classical evolution generated by the new Hamiltonian are necessary. For that, let us compare the Hamiltonian flow generators or the vectors in both approaches. The original lapse function is related to the new one as follows
(15) 
Since to be well defined the infinitesimal time translation vector does need to be finite, so does . On the other hand the topology of the reduced manifold enforces the existence of at least two points on where . At each of these points the new time translation generators will necessarily vanish—will not generate the evolution in the original formulation of the theory as the reduction of GR. Furthermore, for the canonical formalism to work, the constant time slices have to remain spatial. As a consequence, the range of the classical evolution after Abelianization is severely limited, as the constant time slices must stay causally disconnected from any point at which vanishes. The time evolution determines the geometry only within spacetime regions containing the initial data slice and bounded by both the future and past light cones of the points (on the initial data surface) where . In Fig. 1 we provide a schematic diagram of the time evolution of the spacetime.
As the attempt to increase the range of the evolution one can consider alternative Abelianization procedure, skipping for example the step (13). We discuss such approach in Appendix A. Unfortunately, due to the singular nature of the transformation, also inherited in the quantum theory, we have decided to keep away of this alternative in the principal treatment of this manuscript.
In summary, in this section we obtained a complete classical description with a strongly selfcommuting Hamiltonian constraint. This setting will be used in the remaining part of the article to build a (loop) quantum description of this LRS Gowdy model. The first step in this building process is the socalled kinematical level quantization, where one ignores the constraints.
V Quantum kinematical theory
We will proceed now with the quantization of the above system within loop quantum gravity Thiemann (2008). It has been already specified in Refs. Banerjee and Date (2008a, b), following the ideas of Ref. Bojowald (2004); Bojowald and Swiderski (2004, 2006).
Here the basic objects of the description are:

dimensional closed oriented graphs embedded on the reduced manifold (Where are dimensional surfaces generated by isometries). Each graph contains a collection of disjoint edges terminating in vertices .

An algebra of holonomies along disjoint (oriented) edges of these graphs and the intertwiners on their vertices.

A basis of states which in the connection representation is given by
(16) where and are valences of edges^{7}^{7}7The signs of the (otherwise natural) valences encode the orientation of a particular edge. and vertices , respectively. These valences enumerate the representations of holonomy subalgebras (correspondingly, holonomies along and point holonomies encoding parallel transports along curves on ).
In this manuscript, we will adhere to the socalled improved dynamics scheme. It was originally proposed in the context of isotropic LQC Ashtekar et al. (2006) in order to equip the description with a proper infrared regulator removal limit and thus give robustness to the physical predictions of these symmetry reduced models. On the level of midisuperspace models it has been already considered in spherically symmetric gravity in Ref. Chiou et al. (2012), although there the studies have been restricted to the heuristic level of the socalled semiclassical effective dynamics Singh and Vandersloot (2005) only and no attempt to build a quantum description was made.
The key idea of the scheme is based on a set of properties of reduced loop quantization for symmetric spacetimes. These properties can be summarized in: the existence of distinguished geometrical (fiducial) structures provided by the symmetries that allow a partial diffeomorphism gauge fixing (in consequence providing a physical meaning to the embedding data) ^{8}^{8}8In the standard diffeomorphism invariant formulation of LQG the embedding data is averaged out and has no physical meaning., nonexistence of well defined curvature operators which thus have to be built via Thiemann’s regularization process Thiemann (1998). The technical implementation of in symmetry reduced treatment involves approximating the classical curvature by holonomies of finite closed loops along the edges generated by the Killing vectors, whereas fixes the embedding data as determined by the physical area of the chosen regularization loop. Following Ashtekar et al. (2006) that area is heuristically fixed as the st nonzero eigenvalue of the full LQG area operator^{9}^{9}9Outside of isotropic models this procedure is substantially more involved. See the discussion in Ashtekar and WilsonEwing (2009)., known as the area gap and denoted by , with the (square of the) Planck length.
Here, the regularization loops, built at each vertex of the graph, are squares. The length (with respect to certain fiducial metric) of each side is fixed such that the physical area of the loop equals exactly . Following LQG, these areas are determined by the action of the area operator —with eigenvalues —built in turn out of the flux operators (see Ref. Thiemann (2008) for details). The area of the surface enclosed by the loop (at vertex ) with fiducial sides is given by .^{10}^{10}10Here we follow the convention where and are dimensionless. In this situation, the action of the point holonomies of length will produce a shift in a state by a “length” which depends on the phase space variables. Therefore, it will be convenient to adopt a more appropriate state labeling .
After this relabeling, the kinematical Hilbert space is constructed as the closure of the space spanned by Eq. (16) with respect to the inner product , further generalized by the rule that basis states belonging to different graphs are mutually orthogonal^{11}^{11}11This is actually guaranteed if we allow vanishing labels for the quantum numbers and ..
Once the kinematical Hilbert space is constructed we promote a set of classical variables to operators in which we follow (modulo minor refinements) the proposals in Bojowald (2004); Bojowald and Swiderski (2004). These are:

The “triad component” operator, in the precise definition following the ideas of LQC Ashtekar et al. (2003) as the (appropriately rescaled) flux of the triad component across the Killing orbit surface, which can be done due to the symmetries of the model. Its action takes a very simple form
(17) where is the index corresponding to the edge going through . We extend the definition of to the vertices of the graph by considering the contributions of both edges connecting the vertex (with weight ), i.e. . It is also convenient to define an operator
(18) corresponding to the flux over a surface intersected by the edge , and
(19) if the intersection is at a vertex.
At this moment it is necessary to recall that associating the operator to a particular edge/vertex, although it is standard in midisuperspace models and spin foam approaches, it does not follow from the standard quantization procedure as one cannot construct a classical observable distinguishing it. Instead, it constitutes an additional nontrivial component of the implemented treatment.

The operator corresponding to the area of the Killing orbit surface, defined as
(20) For lying in the interior of an edge its action reads
(21) whereas on the vertex it has (as in full LQG) contributions from both edges
(22) 
The operator corresponding to the “volume of a region” , of which action reads
(23) Again, it is convenient to introduce the volume operator associated to a vertex by choosing the interval so that is the only vertex it contains. Its action reads
(24) 
The point holonomy operator defined on a vertex has an action on the corresponding subspace given by
(25) being the identity on the remaining subspaces.
For mathematical convenience, adopting the construction commonly used when dealing with the volume operator in LQG Ashtekar and Lewandowski (1998) and in the context of midisuperspace models postulated originally in Eq. (29) of Bojowald and Swiderski (2006) we can think of the operator (23) as the integral of the distribution valued “volume form operator density”
(26) 
Note however that, unlike Eq. (23), this “volume density” is not in itself a well defined operator. In the remaining part of the paper we will omit its dependence unless otherwise specified.
Due to the absence (thanks to the abelianization procedure of Sec. IV) of the variable in the Hamiltonian constraint, it will be not necessary to construct the operator corresponding to the holonomy along edges .
The Hilbert space and basic operators defined in this section will serve as an arena for the second step in the Dirac program, promoting the (relevant) classical constraint to an operator and finding its kernel. Following the treatment of full LQG this procedure will be applied to the Hamiltonian constraint.
Vi The scalar constraint
As in all applications of the loop quantization to either full GR or symmetry reduced models, the components of the Hamiltonian constraint cannot be directly promoted to operators as most of them do not exist on the kinematical level of quantization. It is necessary to first express or approximate them via variables of which quantum counterparts we have at our disposal (which here means the operators given in the Eqs. (17) and (23), and their powers) or their Poisson brackets. This method is known in the literature as the Thiemann’s regularization Thiemann (2008, 1998). Subsequently, the quantum constraint operator will be defined by directly promoting the components of the regularized constraint to operators.
As in other loop quantized models, the procedure mentioned above permits a series of ambiguities related with either the details of the regularization (which is not uniquely defined) or the factor ordering. These ambiguities are fixed in the following way:

To express the connection in terms of holonomies we employ the simplest possible approximation: by the difference of a holonomy and its inverse. Mathematically this procedure amounts to the substitution .

The chosen factor ordering is a straightforward generalization of the socalled MMO scheme MartinBenito et al. (2008b, 2009b); Mena Marugan et al. (2011), which ensures decoupling distinct orientations of and the “classical singularity states”—basis states for which any component of or equals zero—from the dynamics.

Following the invariance of the Hamiltonian constraint operator with respect to the orientationreflection symmetry, the operator is represented as , thus becoming a power of an area operator.

The spatial derivative has been promoted to an operator after the following simple observation: the operator is diagonal
(27) with each nontrivial coefficient constant on each edge of the graph and discontinuous jumps on the graph vertices. Thus, calculating directly the derivative of this coefficient on the embedded graph would yield the following result
(28) where , is the position of the vertex and we follow the numbering convention where is the lefthand side boundary of the edge . Since it is a distribution, taking it as a coefficient of a diagonal operator would not give a well defined result. It does however provide a correct definition of a smeared operator (similarly to )
(29) This, in turn, by selecting , such that it contains only the vertex , allows us to define the “vertex difference operator”
(30) A quite surprising (and counterintuitive for the quantum counterpart of the spatial derivative operator) property of such definition is its independence of the action on the coordinate length of the involved edges of the graph.

Since the derivatives of operators defined on the open intervals on cannot be defined in a straightforward way, thus the “global” spatial derivative present in the Abelian Hamiltonian constraint (see Eq. (14)) has to be regularized. One of the ways (not necessarily optimal) to achieve that is to apply the mathematical shortcut discussed in the context of the volume operator when the “distributional operator” has been defined as in Eq. (26). Such procedure will replace the derivative in question with the difference of the “interior” operators between two vertices of a given edge.
The application of the listed choices results in the operator form of the Hamiltonian constraint taking the form
(31) 
where
(32a)  
(32b)  
(32c)  
(32d) 
An analogous Hamiltonian constraint operator, taking very similar form, has been already studied in spherically symmetric spacetimes Gambini et al. (2014). It commutes with itself and is free of anomalies. Besides, due to the factor ordering choice, the states with either or , or both, are trivially annihilated by the constraint. In consequence, they will be irrelevant for the dynamics, and can be safely removed from the space of solutions to the constraint. As a consequence, the quantum theory is then able to cure those coordinate and curvature singularities ^{12}^{12}12It is well established that the considered Gowdy model possesses a curvature singularity when the area of the Killing orbits vanishes Berger (2002), i.e., when vanishes. arising either at and/or .
The quantum Hamiltonian constraint constructed here will be next used (after suitably dealing with the diffeomorphism group) to construct the physical Hilbert space, further allowing to study the dynamical sector of the theory.
Vii Physical Hilbert space, dynamical sector
At this point we have at our disposal the kinematical Hilbert space, i.e., welldefined operator(s) acting on that space and the operator form of the Hamiltonian constraint. The remaining tasks in completing the quantization program are: finding the physical Hilbert space as the (dual of the) kernel of the constraint operators and constructing a set of physically meaningful observables acting on this space. Its first step: solving the constraints requires an additional effort as at present we do not have (nor we intend to build) a quantum diffeomorphism constraint operator. While recently a construction of such quantum diffeomorphism generator has been proposed in the full theory Laddha and Varadarajan (2011), the standard treatment of diffeomorphisms in loop quantization just uses the finite diffeomorphisms. Here we employ the same philosophy, applying the constraintsolving program of the full LQG. There, the constraints are implemented in hierarchy: first, the diffeomorphisminvariant Hilbert space is constructed out of the kinematical one by the socalled group averaging procedure Ashtekar et al. (1995), next the actual physical Hilbert space is defined as a space annihilated by the Hamiltonian constraint operator acting on the diffeomorphisminvariant space ^{13}^{13}13A condition necessary for this step is that the Hamiltonian operator is well defined on this space, which is exactly the case in the full LQG.. We repeat this exact procedure in the context of our model.
First, let us identify the diffeomorphisminvariant sector of the theory via group averaging.
vii.1 Averaging over the spatial diffeomorphisms
Consider a general situation of a compact group of transformations (classically generated by a set of constraints) represented in the quantum theory by unitary operators acting on certain Hilbert space. In such situation one is usually interested in finding a sector of the quantum theory under study, which is invariant with respect to these transformations. It usually lies in the algebraic dual of a dense subspace of the original Hilbert space and roughly speaking consists of “averages” of states transformed over the whole transformation group. Because of this principal idea the procedure of building such states is known as the group averaging technique Ashtekar et al. (1995); Marolf (1995a); *mgave2; *mgave3; *mgave4. In a precise sense the averaging is achieved by constructing for the group of unitary transformations parametrized by the index (such that is the biinvariant measure on ) the antilinear rigging map such that
(33) 
The states (when nontrivial) span a transformationinvariant Hilbert space with induced inner product
(34) 
Our goal in this section is to define such rigging map averaging over the group of spatial diffeomorphisms of the Cauchy slice of the LRS Gowdy spacetime. This group is relatively large, however, as we will see below, for the purpose of averaging it can be reduced to a small (in fact finite dimensional) subgroup. To perform this reduction we first note that, due to a natural requirement, that the transformations preserve the symmetries of the selected class of spacetimes, the group of diffeomorphisms reduces to a subgroup of diffeomorphisms on the reduced manifold . We will denote this subgroup by .
Each diffeomorphism acts on a given graph yielding a new state such that it drags the vertices of the graphs (together with all the points of ) in such a way that the sets and . Then it is the position of any point in that is dragged to the new position , while preserving the order of the points, that is: . Thus induces a unitary operator on the kinematical Hilbert space such that . We will parametrize the group of the operators with the index , such that provides the biinvariant measure on the group.
Next, we note that due to the orthogonality (with respect to the kinematical inner product) of the disjoint graphs, for the evaluation of the integrals in Eqs. (33) and (34) it is sufficient to consider an action of the diffeomorphism transformation group on each single closed graph separately. Consider now a subgroup of diffeomorphisms preserving the positions of all the edges and vertices of a particular graph. The action of the unitary transformation operator corresponding to each element of this subgroup is the identity operator on . Furthermore, we can group all the elements of the basis of into classes of equivalence with respect to transformations from . These classes of equivalence will span a Hilbert space with an inner product induced in a straightforward way from since the latter is invariant with respect to parametrizations of graph edges by . The only difference of the new space with respect to is that now the information about the parametrization of the graph edges (which in embedded graph are curves on ) is removed. As a consequence we can reformulate the integrals in Eqs. (33) and (34) as integrals over the quotient group of the transformations acting on ,
(35a)  
(35b) 
The considered quotient group is already finitedimensional. The transformations are just shifts of positions of the edges of the graph preserving their order. To characterize the group we note that, once the information about the parametrization of edges is removed (which happened in going to the auxiliary Hilbert space) each embedded graph can be characterized (up to a global phase rotation) by a sequence of fiducial lengths of the graph edges, thus be represented as a point on the surface in the space . Then (after further dividing by the subgroup of the abovementioned rigid rotations) each transformation from the considered quotient group is just a shift on this surface.
To build a meaningful rigging map we select a discrete measure . Then the integrals (35) become uncountable sums and can be calculated quite easily. Indeed, the inner product in the diffeomorphism invariant Hilbert space becomes now
(36) 
Since each transformation preserves the labels , order of vertices and edges (and orientation of the latter) of the auxiliary state, the basis of can be constructed out of states now understood as living on the abstract graph (with only ordering of the vertices instead of the embedding). The diffeomorphism invariant inner product will mathematically take the same form (modulo vertex positions) as the kinematical one
(37) 
Once has been constructed, the next step is to build analogs of the elementary operators (originally acting in ) which will be well defined on a dense domain in . For that we again employ the group averaging, defining for a given an operator (where are dense domains in their respective Hilbert spaces) in the following way
(38) 
An application of this procedure to Eqs. (25) and (30) yields operators which are mathematically identical to their kinematical predecessors, although now they act on the labels of the abstract graphs. However, the families of operators defined by Eqs. (17), (23) and (28) will not provide useful definitions as they are parametrized by either points or regions on . Instead, one needs to take their versions in Eqs. (18), (24) and (30), parametrized by the vertex label which plays the role of the ordering index of the graph elements. Thus, we end up with the following set of diffeomorphism invariant elementary operators
(39a)  
(39b)  
(39c)  
(39d) 
and
(40) 
Given these operators, one can immediately write a version of the quantum Hamiltonian constraint defined on the dense domain of by averaging [via Eq. (38)] the operators in Eqs. (31) and (32). This procedure will yield a constraint in which all the elementary operators are replaced with their diffeomorphism invariant counterparts defined in Eqs. (39) and (40). This result is critical in performing the next step in building the physical Hilbert space—finding the kernel of this constraint (as the only one remaining).
vii.2 Solutions to the Hamiltonian constraint
Unlike the diffeomorphisms, in loop approaches the Hamiltonian constraint is implemented directly as a generator, thus (as stated earlier) solving it corresponds to finding the states annihilated by its quantum counterpart. Technically, it amounts to solving the equation for the (generalized) wave function in the algebraic dual of a suitable dense subset of
(41) 
This can be again achieved by the group averaging technique. Here, however, unlike in the case of differomphisms we have at our disposal the generator of the transformations (in this case time reparametrizations). Thus, the rigging map can be written as
(42) 
In order to construct this map explicitly we will perform the spectral decomposition of . First, we note that this operator [see (31)] is a linear combination of mutually commuting component operators [defined in (32b)], which thus can be simultaneously diagonalized. It is thus enough to perform the spectral decomposition of each . To do so we introduce a set of auxiliary operators (where ) of the mathematical form analogous to (32b)
(43a)  
(43b)  
(43c) 
where . Since in the eigenvalue problem the dependence on is algebraic only one can split it onto a set of independent eigenvalue problems on involving the auxiliary operators and parametrized by values of . Precisely, we realize this split via introducing the auxiliary Hilbert space (corresponding to the vertex degrees of freedom) and a set of the projection/embedding operators
(44a)  
(44b)  
(44c) 
The component operators can now be written as
(45) 
where and the product in the parenthesis must be understood as a Cartesian product. The eigenvalue problem reduces to the set of equations for the eigenfunctions
(46) 
The properties of the operators have been analyzed in detail in Appendix B. They are essentially selfadjoint (barring an extreme finetuning of the Barbero–Immirzi parameter), the spectrum of each of them is nondegenerate and its continuous part is .
Consider now a set of Dirac delta normalized solutions to (46), denoted further by . Then, the functions of the form
(47) 
are the (normalized) eigenfunctions of , where is any normalized function on the dimensional space of vectors such that corresponds to any vector with the th coordinate removed. As a consequence, the spectrum of also has a continuous part , although now it has a continuous degeneracy (originating from the freedom in the choice of function in (47)). We note however that this degeneracy becomes spurious when we consider the complete Hamiltonian constraint operator.
The mutual eigenfunctions of corresponding to the vector of eigenvalues (parametrized by ) are the linear combinations of products of the (Dirac delta normalized) solutions to (46)
(48) 
By construction they are also eigenfunctions of , thus they diagonalize all the component operators of .
Consider now the diffeomorphism invariant state decomposed in the above basis
(49) 
When acting on it, the rigging map (42) (well defined as is essentially selfadjoint, which follows from essential selfadjointness of the component operators ) produces the following physical state
(50) 
As a consequence the group averaging procedure selects out the states of the products taking the same value on all the vertices of the graph. It is convenient to represent this constraint by introducing the dependent variable such that
(51) 
which allows to determine the frequencies as functions of in the following way
(52) 
Since the variables are nonnegative and bounded from above by the function of