### 5.2 Quantization of two-Killing vector reductions

#### 5.2.1 Quantization of Einstein–Rosen waves

The first historic attempt to canonically quantize (vacuum) Einstein–Rosen waves goes back to the pioneering paper by Kuchař [141]. There he carefully studied cylindrical metrics for the polarized case and derived the Hamiltonian formulation for the system. The author used the full four-dimensional picture in a very effective way in order to develop a suitable Hamiltonian formalism, in particular, the canonical transformations leading to a convenient coordinatization of the reduced phase space. One of the key achievements of the paper was the identification of a phase-space function that could play the role of a time variable for ER waves. This provides an extrinsic time representation similar to the one used in the spherically-symmetric case [143]. By defining an appropriate canonical transformation it is possible to turn this time into a canonical variable and make it part of a new set of canonical coordinates. In terms of them the action functional takes the particularly simple form of the parameterized formalism for an axially-symmetric scalar field evolving in a (fictitious) Minkowskian background. An interesting comment is that the canonical transformation mentioned above mixes configuration and momentum variables in such a way that the original configuration space is traded for a rather different one, which is not a subset of a space of metrics. The main problem with [141] was that it did not take into account the necessary boundary terms needed to render the variational problem well defined. The results of the derivation given by Kuchař can be obtained in a more systematic and straightforward way by using the principle of symmetric criticality [192], substituting the form of the cylindrically-symmetric metrics corresponding to polarized Einstein–Rosen waves in the Einstein–Hilbert action and getting the Hamiltonian formulation from there.

The Dirac quantization of the Einstein–Rosen waves that Kuchař gives is interesting from a pedagogical and intuitive point of view but arguably quite formal. The main consistency issues related, for example, to the path independence with respect to the foliations interpolating between two given ones, are formally taken into account as well as the definition of the scalar product of Schrödinger picture functionals. However, no attention is payed to the subtle functional analytic and measure theoretic issues that come up. This problem has been addressed in [65, 73]. Cho and Varadarajan [65] have studied the relationship between the Schrödinger and Fock representations and considered several issues related to the unitary implementability of the evolution of the free axisymmetric scalar field in a Minkowskian background. In particular they have discussed the existence of unitary transformations on the Fock space implementing the evolution between two axisymmetric, but otherwise arbitrary, Cauchy slices of the auxiliary flat spacetime in such a way that their infinitesimal version gives the functional Schrödinger equations obtained by Kuchař. The analysis is based on work by Torre and Varadarajan [208, 209] on the evolution of free scalar fields between arbitrary Cauchy surfaces. In this respect it is interesting to remark on the different behaviors in the 1+1 dimensional case and the higher dimensional ones. It is also important to point out that polarized Einstein–Rosen waves are remarkably close to this type of model. The main result of [65] is that in the half-parameterized case, when the radial coordinate is not changed, the dynamics can be unitarily implemented as a consequence of Shale’s theorem [196]. However, if no condition is imposed on the radial coordinate – thus allowing for the possibility of having radial diffeomorphisms – the quantum counterpart of these transformations cannot be unitarily implemented because it is not given by a Hilbert–Schmidt operator. The unitary equivalence of the Schrödinger and Fock quantizations for free scalar fields has been studied in a slightly more general setting by Corichi et al. in [73]. The authors of this paper take into account several functional issues that are relevant for a rigorous treatment of the rather subtle issues that crop up in the quantization of free field theories defined in arbitrary globally-hyperbolic spacetimes.

A different approach to the quantization of Einstein–Rosen waves by standard quantum field theory methods consists in performing a complete gauge fixing and studying the reduced space of the model. This has been done by Ashtekar and Pierri [13]. The quantization of the system is performed after paying special attention to the asymptotic conditions relevant in the 2+1 dimensional case (and, consequently, to the necessary boundary terms in the gravitational action). Specifically, some of the peculiarities of this system come from the fact that, due to the translational part of the isometry group, the (non-trivial) Einstein–Rosen solutions cannot be asymptotically flat in four dimensions (or alternatively, the 2+1 dimensional metric does not approach a Minkowski metric at spatial infinity because a deficit angle is allowed). The numerical value of the Hamiltonian of this system, when the constraints are satisfied, is given by an expression originating in the surface term of the action for Einstein–Rosen waves. As mentioned before, this has the form given by Equation (3) and is a function of H0, the free Hamiltonian corresponding to an axisymmetric massless scalar field in 2+1 dimensions evolving in a Minkowskian background metric. This allows us to use a Fock space to quantize the system. In fact, as the Hamiltonian is just a function of a free one it is possible to obtain the exact quantum evolution operator from the one corresponding to the free auxiliary model and use it to obtain close form expressions for many objects of physical interest such as field commutators and n-point functions. It is important to point out that only after the gauge fixing is performed will the Hilbert space of states take this form. This means that other quantizations or gauge fixings describing the gravitational degrees of freedom in a different way, could lead to a different form for the quantized model. It is important to notice that Ashtekar and Pierri do not perform the canonical transformation used by Kuchař to introduce the extrinsic time representation, but directly fix the gauge in the original canonical formulation. By using this scheme Varadarajan [213] has studied the mathematical properties of the regularized quantum counterpart of the energy of the scalar field in a spherical region of finite ratio of 2+1 flat spacetime. In particular, he has given a proof of the fact that this regularized operator is densely defined and discussed its possible self-adjoint extensions (this is only a symmetric operator).

In the aftermath of [13], several papers have used the quantization presented there to derive physical consequences of the quantization of Einstein–Rosen waves. Among the most influential is [2]. In this paper Ashtekar considers 2+1 gravity coupled to a Maxwell field – with the additional condition of axisymmetry – as a solvable toy model to discuss quantum gravity issues. It is important to point out that in 2+1 dimensions the Maxwell field can be interpreted as a free massless scalar, with the usual coupling to gravity, and dynamics given by the wave equation. This system is equivalent to the symmetry reduction of 3+1 dimensional gravity provided by Einstein–Rosen waves, so the results on the paper are relevant to studying the quantum physics of this midisuperspace model. The main result of [2] is the somewhat surprising appearance of large quantum gravity effects in the system. The presence of these has some importance because, if we trust this quantization, it points to the spurious character of many classical solutions to the model, as they cannot appear in the classical limit. One key element here is the form of the Hamiltonian, given by Equation (3) and the fact that it is a non-linear function of a free Hamiltonian. The large quantum gravity effects manifest themselves in the expectation values of the metric components (that are functions of the scalar field that describes the degrees of freedom). Specifically these do not correspond to the classical values in some limits, in particular for high frequencies. Furthermore the metric fluctuations in this regime are very large even for states that are peaked around a classical configuration of the scalar field and grow with the number of quanta (“photons”) of the scalar field. A reflection of these large quantum gravitational effects is also manifest in the behavior of the field commutators, especially at the symmetry axis, as shown in [25].

An extension of these results to four dimensions has been carried out in the paper by Angulo and Mena [1]. They do this by expressing the four-dimensional metric of the Einstein–Rosen waves in terms of the Maxwell scalar and the 2+1 dimensional metric. It is important to mention at this point that the scalar field enters the four-metric in a highly non-linear way. The most important result of [1] is the actual verification of the possibility to extend the conclusions reached by Ashtekar in the 2+1 dimensional setting to four dimensions (far from the symmetry axis). However, the authors argue that to reach an acceptable classical description in the asymptotic region in four dimensions, it is not mandatory to require – as in three – that the number of quanta of the Maxwell scalar field be small. Also, the behavior on the symmetry axis is interesting because the relative uncertainties of the metric become very large there. This casts some doubts on the appropriateness of the classical regularity conditions usually introduced at the axis.

There are other papers in which the physical consequences of the Fock quantization of Einstein–Rosen waves given in [13] have been considered. In [24, 25, 28, 18, 19, 20] several physical issues have been discussed in some detail, in particular microcausality, n-point functions, 2-point functions, matter couplings and coherent states. Microcausality in Poincaré-invariant models, formulated with the help of a background Minkowskian metric, reflects itself in the vanishing of the commutator of the field operators at spatially separated spacetime points. The fact that Einstein–Rosen waves can be quantized offers the possibility of quantitatively testing some of the expected features of quantum gravity such as the smearing of light cones due to quantum fluctuations of the metric. The presence of this effect was suggested already in the original paper by Ashtekar and Pierri [13]. In [24] the explicit form of the commutator was obtained and a direct numerical analysis showed the expected smearing effect and clear indications about how the classical limit is reached when the effective gravitational constant of the model goes to zero. A quantitative understanding of this phenomenon has been given in [25, 28] by performing an asymptotic analysis of the integrals that define the field propagators. An interesting result coming from these analyses is the different way in which the classical limit is reached on and outside the symmetry axis. Outside the symmetry axis the large scale regime is reached in a rather smooth way, but on the axis large quantum gravitational effects persist even at macroscopic scales. This is probably another manifestation of the kind of effects described by Ashtekar in [2].

A way to incorporate quantum test particles – that could further help explore the quantized geometry of the model – is to couple matter fields to gravity and use their quanta as probes. This is very difficult for generic matter fields but can be remarkably achieved for massless scalars keeping both the classical and quantum resolubility of the system. This was done in [18, 19] – though the classical integrability was understood by several preceding authors, in particular Lapedes, Charach, Malin, Feinstein, Carmeli and Chandrasekhar [145, 62, 63, 60]. It is also fair to mention at this point that the effective decoupling of the gravitational and matter scalar modes (in the flat space picture), that is the key ingredient in the Fock quantization of the Einstein–Rosen waves coupled to massless scalar fields presented in [18], was discussed in essentially the same form by Lapedes in [145], though he treated the quantization in the heuristic way customary at the time.

In [19] Newton–Wigner localized states were used to build actual position space wave functions for the massless quanta in order to study how they evolve in full interaction with the quantized geometry. The resulting picture shows, in a convincing way, that the quantum particles in their motion define approximate trajectories that follow the light cones given by the microcausality analysis. Also, the study of 2-point functions (extended to n-point functions in [20]) gives a consistent picture when they are interpreted as approximate propagation amplitudes; in particular, the persistence of large quantum gravity effects in the symmetry axis is confirmed. Finally, the issue of obtaining coherent states for the Einstein–Rosen waves has been considered in [20].

The results described above refer to polarized ER waves. It is natural then to consider the full non-polarized case. An interesting set of papers by Korotkin, Nicolai and Samtleben [182, 137, 136, 138, 139] explores a family of systems consisting of two-dimensional GR coupled to non-linear sigma models. These generalize the symmetry reductions from 3+1 dimensions that we are considering in this section, in particular the Einstein–Rosen waves, and treat genuine midisuperspace models with non-linear interactions and an infinite number of degrees of freedom. A unified treatment of them appears in [182], where a number of issues related to their classical integrability and quantization are discussed. By assuming the presence of two commuting Killing symmetries it is possible to make a first simplification of the functional form of the metrics for this system by restricting the number of coordinates on which the fields depend to just two. In this way they effectively correspond to 2-dimensional non-linear sigma models coupled to a dilaton and gravity. A key observation is that the resulting models are integrable and their solutions can be obtained from an auxiliary linear system of equations. In fact, the matter dynamics can be derived from the compatibility conditions for this linear system. The Hamiltonian formulation can be written in terms of two constraints generating translations along the light cone that become partial differential equations when quantized. These quantum constraints are precisely the Knizhnik–Zamolodchikov equations that play a fundamental role in conformal field theory. Their solutions, as discussed in [182], provide concrete physical states of the quantized theory. Some of the specific models collectively considered in this paper are individually studied in a series of works by the authors [137, 136, 138]. The limits of this line of work are related to the impossibility of solving the coset constraints for the non-compact spaces [such as ] by using discrete unitary representations [of in the previous example].

We want to mention here that [139] is of particular interest because Korotkin and Samtleben extend the Fock quantization techniques used in [13] to the much harder non-polarized case where the non-linearity of the model shows up in full strength. In this case, the Einstein field equations can be written as the Ernst equation for the unimodular metric on the Killing orbits and an integral expression for the conformal factor of the metric in terms of the solutions to this Ernst equation. The quantization is achieved by finding a complete set of quantum observables and a representation of them in a Fock Hilbert space. This is done by introducing a new set of classical observables. These generalize another set of variables that can be defined in the polarized case in terms of the positive and negative frequency modes that appear in the Fourier decomposition of the axisymmetric scalar field that encodes the gravitational degrees of freedom. The Poisson algebra of the new observables is quadratic (i.e., the Poisson bracket of two basic observables can be written as a linear combination of products of two of them); this introduces some complications in the treatment due to the necessity to deal with operator ordering issues after quantization. However, the solution to this problem is known in the theory of integrable systems [154]. The final step of the process consists in finding a representation of the quantum algebra in a Fock Hilbert space. The availability of this representation opens up the possibility of computing expectation values of important operators, in particular certain components of the metric. In principle this can be used to derive physics from the non-polarized ER waves, although the non-local character of these new observables introduced by the authors and, in particular, their lack of an explicit spacetime dependence, makes it difficult to make contact with other results, especially those related to the study of microcausality in the polarized case [24]. As a final comment on the approach by Korotkin, Nicolai and Samtleben, we want to mention the possibility of mapping the non-polarized cylindrical models to free theories as discussed in [78].

To end this section we want to comment that the quantization of Einstein–Rosen waves has also been considered from other points of view, such as perturbative methods and LQG inspired techniques. The use of perturbative techniques in particular is a very natural path to follow because it offers the possibility of comparing the results with those obtained by non-perturbative methods and ultimately trying to get an additional understanding of the causes leading to the perturbative non-renormalizability of GR. In [183] a thorough study of the quantization of two-Killing vector reductions is carried out. The main goal of this paper is to find out if the perturbative quantization of Einstein–Rosen waves is consistent with the asymptotic safety scenario of Weinberg [184]. In fact, the main result of [183] is to show that two-Killing symmetry reductions of GR are asymptotically safe. However, this result is not straightforward because two-Killing vector reductions are not renormalizable in the standard sense (beyond one loop). Despite this, the model can be declared to be renormalizable if the space of Lagrangians is expanded by allowing conformal factors that are functions of the radion field (the determinant of the pull-back of the metric to the integral manifolds of the Killing vector fields). In fact, when this is done, the renormalization flow has a unique ultraviolet stable fixed point where the trace anomaly vanishes [183]. A similar result has been obtained in the case of non-polarized Einstein–Rosen waves in [27] by using a path integral approach and the algorithm developed by Osborn in [185] to deal with position dependent sigma models. These papers provide an alternative point of view from the esentially non-perturbative methods of Korotkin, Nicolai and Samtleben. A comparison of physical predictions in both approaches would be most interesting. Several other papers can be found in the literature that consider different aspects of some two-Killing vector reductions from the perturbative point of view; in particular [26]

The loop quantization of the Einstein–Rosen midisuperspace is an interesting open problem that deserves some comments. In their seminal paper about Fock quantization of cylindrically symmetric spacetimes [13], Ashtekar and Pierri computed the holonomies around those loops that are integral curves of the rotational Killing vector and showed that their traces are functions of the energy (in a box of finite radius) of the scalar field encoding the reduced degrees of freedom. In particular, in the large radius limit, those traces reduce to a simple function of the total energy of the system. Hence, as they point out in the paper, the question of whether those traces are well-defined operators in quantum theory, reduces to the question of whether the operator corresponding to the energy of a scalar field in a box can be satisfactorily regulated (see [213]). In any case, the dynamic issues of the polymeric quantization of the scalar field (including the classical limit and the relation with standard quantizations) need to be analyzed in detail.

As usual in LQG, one follows a two-step route to quantization by first constructing the kinematical Hilbert space of the theory and then defining the Hamiltonian constraint (and, for ER waves, also the Hamiltonian) of the model. In this last step, the geometric operators (in particular the volume operator) are thought to play a relevant role in the rigorous definition of the Hamiltonian constrain operator. In his Living Review [41], Bojowald discusses the kinematical Hilbert space for ER waves and also certain properties of the volume operator. He pays special attention to the differences of the cylindrically-symmetric sector and the homogeneous cosmologies. There are also other papers, much more qualitative in nature, by Neville [181, 180], in which the construction of a kinematical Hilbert space for the loop quantization of cylindrically-symmetric spacetimes and planar waves is sketched.

#### 5.2.2 Quantization of Gowdy models

Gowdy cosmological models [102, 103] are described by time-oriented, globally-hyperbolic, vacuum spacetimes, which can be constructed from the evolution of U(1) × U(1)-invariant Cauchy data defined on a 3-dimensional closed (compact without boundary) hypersurface . The action of the U(1) × U(1) group of spatial isometries is assumed to be effective and the topology of is restricted to be T3, S2 × S1, S3, or the lens spaces. These spacetimes describe inhomogeneous cosmologies with initial and/or final singularities and represent gravitational waves propagating in a closed universe. For the T3 topology, only one singularity is present, whereas in the case of the S2 × S1, S3 topologies there are both initial and final singularities. The quantization of these models has been considered only in the polarized case (for which both Killing vector fields are hypersurface orthogonal).

The Gowdy T3 model describes an inhomogeneous cosmology with one singularity (that can be thought of as initial or final). Misner was the first researcher to recognize its relevance as a test bed for quantum cosmology [164, 165]. Pioneering work on its quantization was carried out in the 1970s and 1980s by him and Berger [35, 34, 36]. Actually it is fair to say that many of the ideas that have been used to this day in the attempts to achieve a rigorous quantization for this system actually originate in these works. In particular:

• The Hamiltonian analysis for this type of model, including the identification of the extra constraint present for the T3 topology [165, 34].
• The introduction of the deparametrization customarily used to achieve a (partial) gauge fixing [165].
• The identification of a suitable real Fourier expansion that decouples the modes appearing in the Hamiltonian [165].
• The identification of the time-dependent field redefinitions that have been used in recent works to find a unitary implementation of the quantum dynamics [34] (though these were used by Berger in a different context to study the WBK regime).
• The role of the harmonic oscillator with time-dependent frequency in the dynamics of the Gowdy models [165, 34].

The approach to quantization followed in these papers consisted in a rather formal treatment in which the Hilbert space was taken to be the infinite tensor product of the (countably) infinite Hilbert spaces associated with each of the oscillator modes appearing in the Fourier transformation of the fields. The use of an infinite tensor product of Hilbert spaces is problematic, as emphasized by Wald [225], because this Hilbert space is non-separable and the representation of the canonical commutation relations is reducible. Misner and Berger have discussed a number of issues related to graviton pair creation in Gowdy universes and the semiclassical limit. A nice and intuitive picture developed in these papers is the idea that their quantum dynamics can be interpreted as scattering in superspace.

An interesting problem that received significant attention even at this early stage was the issue of the singularity resolution. Berger approached it by working in a semiclassical approximation, where it was possible to describe the gravitational radiation by means of an effective energy momentum tensor depending on some of the metric components. By using this framework it was argued that the classical singularity was replaced by a bounce. This very same question was considered by Husain in [115], where he used the same kind of Hilbert space as Berger but followed a different approach consisting in quantizing the Kretschmann invariant (the square of the Riemann tensor) after finding an appropriate operator ordering. This was done by imposing the sensible requirement that the expectation values of the Kretschmann operator in coherent states equal their classical values sufficiently far from the singularity. The main result of [115] was that, at variance with the findings of Berger, the classical singularity persisted in the quantized model. It is interesting to mention that this quantum version of the Kretschmann invariant for Gowdy T3 was also used by Husain to explore a conjecture by Penrose pointing to a relation between the Weyl curvature tensor and gravitational entropy [116].

In the mid-1990s, the work on the quantization of the Einstein–Rosen waves developed by Ashtekar and Pierri [13] led naturally to the revision of the quantization of other two-Killing midisuperspace models, and in particular the Gowdy T3 cosmologies. This was done, among other reasons, to open up the possibility of using this type of symmetry reductions as toy models for LQG. Since then the system has been considered not only within the traditional geometrodynamical approach but also in the Ashtekar variables framework. The Hamiltonian formalism of the Gowdy T3 model, with a detailed analysis of the gauge fixing procedure, was studied in terms of (complex) Ashtekar variables by Mena in [159] (see also [120]). In that paper, the quantization of the reduced model was also sketched, however, the first attempt to study the Fock quantization of the polarized T3 Gowdy model (in the geometrodynamical setting) appears in [189]. In this paper, Pierri used one of the U(1) subgroups of the isometry group to perform a dimensional reduction and represent the model as 2+1 gravity coupled to a massless scalar field. She showed that the reduced phase space could be identified with the one corresponding to a U(1) symmetric, massless scalar field propagating in a 2+1 background geometry and satisfying a quadratic constraint. By using this description of the reduced phase space, she proposed a Fock quantization that relied on a quantization of the modes of the free field propagating in the 2+1 background geometry. The quadratic constraint was imposed à la Dirac. The main drawback of this approach, as later pointed out by Corichi, Cortez and Quevedo in [72], was that the quantum dynamics of the free scalar field used in Pierri’s quantization does not admit a unitary implementation. It is important to realize that this type of behavior is not a specific pathology of the Gowdy models but actually an expected (and somehow generic) feature of quantum field theories [112]. The results of [72] were confirmed and extended by Torre [204] who was able to show that, even after restricting the quantum dynamics to the physical Hilbert space obtained by imposing the constraint present in the model à la Dirac, the quantum evolution is not given by a unitary operator.

This important problem was tackled and solved in a satisfactory way in a series of papers by Corichi, Cortez, Mena, and Velhinho [69, 68, 70, 71, 74, 76, 77, 75]. These authors have shown that it is actually possible to have unitary dynamics if one redefines the basic scalar field in the description of the Gowdy T3 model [69, 68] by introducing an appropriate time-dependent factor (inspired by a similar field redefinition used by Berger in [34]). An additional important uniqueness result appearing in these papers is that, up to unitary equivalence, this is the only way in which the dynamics can be unitarily implemented in this reduced phase-space quantization of the system.

The quantum description of the S1 × S2 and S3 Gowdy models in terms of a Fock quantization of their reduced phase spaces can be found in [23] (the details of the Hamiltonian formulation for these topologies were studied in [22]). In those cases, the reduced phase spaces can be identified with the ones corresponding to U(1)-symmetric massless scalar fields. The problem of unitarily implementing the quantum dynamics is present also for these topologies but, as in the T3 case, the quantum dynamics can be implemented in a unitary way if the scalar fields are suitably redefined [23, 99, 75].

The description of the reduced phase space of Gowdy models in terms of massless scalar fields has been used to explore the quantum Schrödinger representation of the system in terms of square-integrable functions on a space of distributional fields with a Gaussian probability measure [206, 71, 99, 100]. This representation is, in this case, unitarily equivalent to the Fock one, but in some situations it is actually more convenient due to the availability of a spacetime interpretation.

We want to end this section by commenting that Gowdy models have been used as a test bed for other approaches to quantum gravity. In [91], the T3 model has been studied within the “consistent discretizations” approach (though the paper mostly deals with classical issues aimed at the problem of showing that the discretizations reasonably reproduce the expected classical results, in particular the preservation of constraints). The analysis presented in this paper is relevant because the difficulties to determine the lapse and shift suggest, according to the authors, that the quantization of the non-polarized case will have to rely on numerical methods.

The loop quantization of the non-polarized Gowdy T3 model in terms of complex Ashtekar variables was considered for the first time by Husain and Smolin in [120]. In this work, Husain and Smolin found a loop representation of the unconstrained algebra of observables and gave sense to the (regulated) constrains in this representation. They also constructed a large and non-trivial sector of the physical state space and identified the algebra of physical operators on the state space. The loop quantization of the polarized Gowdy T3 model has been studied recently in terms of real Ashtekar variables by Barnerjee and Date [16, 17], where the authors recast the model in terms of real SU(2) connections as a first step towards quantization and also discuss the gauge fixing procedure. A preliminary description of the kinematical Hilbert space for the polarized Gowdy T3 model and some issues related to the volume operator are given in [17].

The previous papers do not discuss the quantum resolution of the classical singularity, a natural question to consider after the success of LQC in the study of this problem in the simpler setting provided by homogeneous models. This very important issue has been recently considered by using a combination of loop and Fock quantizations in [157, 51]. In [157] the authors use the formulation of the theory as 2+1 gravity coupled to a massless scalar field (with a residual U(1) symmetry). By introducing the usual Fourier mode decomposition of the solutions in terms of the relevant angle variable, the authors define a hybrid quantization consisting in a polymeric quantization for the homogeneous mode (angle-independent) and a Fock quantization for the inhomogeneous (angle-dependent) ones. The main result of the article is that the singularity is resolved. The follow-up study appearing in [51] further considers the quantum dynamics of the polarized Gowdy T3 model, in particular the description of the initial Big Bang singularity that appears to be replaced by a Big Bounce as in the popular LQC models.

#### 5.2.3 Other related models

Two-Killing vector reductions of GR, in the case when the Killing vectors are hypersurface orthogonal, can be classified according to some properties of the gradient of the determinant of the restriction of the metric to the group orbits (the area function). The familiar cases of the Einstein–Rosen waves and the Gowdy cosmologies correspond to spacelike and timelike gradients respectively, whereas plane waves correspond to the null case. The geometrodynamical approach to the quantization of plane wave midisuperspaces was considered in [161], where the authors study both the polarized and non-polarized cases. They show that the reduced phase-space models have vanishing Hamiltonians in the coordinates adopted for their description. In the case of polarized plane waves, the reduced phase space can be described by an infinite set of annihilation- and creation-like variables (that are classical constants of motion) and therefore, it is possible to quantize the system by finding a Fock representation for these variables. In this respect the model is quite similar to the Gowdy cosmologies and Einstein–Rosen waves that can also be quantized by using Fock space techniques. The results of this paper have been used in [162] to study the appearance of large quantum effects in the system (similar to the ones described by Ashtekar for Einstein–Rosen waves in [2]). The plane wave case is rather interesting in this respect because of the focusing of light cones characteristic of this system. Important quantum gravity effects are expected precisely at the places where this focusing takes place. By introducing suitable coherent states, the authors show that the expectation value of a regularized metric operator coincides with a classical plane wave solution whereas the fluctuations of the metric become large precisely in the vicinity of the regions when focusing of light cones occurs (and this happens for every coherent state). These papers nicely complement and expand in a rigorous language the preliminary analysis carried out in [179, 48] in terms of (complex) Ashtekar variables. A complete discussion based in the modern approach to connection dynamics and symmetry reductions in this framework would be interesting indeed.

Finally, we want to mention another cosmological midisuperspace model, due to Schmidt [195], that has some interesting features. The spacetime of this case has the topology R2 × T2, i.e., it is the product of a plane and a torus, and the isometry group is U(1) × U(1) with orbits given by tori. Schmidt cosmologies have initial singularities that are similar to Gowdy T3. Beetle [30] has studied its Hamiltonian formulation (in the polarized case) by choosing appropriate asymptotic conditions for the fields in such a way that the resulting reduced phase space is very similar to the one corresponding to the Gowdy T3 model. In fact, the system can be described as a U(1) symmetric massless scalar field evolving in a fixed time-dependent 2+1 background (topologically R2 × S1) and with no extra constraints (at variance with the Gowdy T3 case). The same unitarity problems that show up in the quantum evolution of the Gowdy models also appear here and can be solved again with a time dependent canonical transformation [21].