1 Introduction
2 The Viewpoint of Loop Quantum Cosmology
Loop quantum cosmology is based on quantum Riemannian geometry, or loop quantum gravity [172, 22, 195, 174] , which is an attempt at a non-perturbative and background independent quantization of general relativity. This means that no assumptions of small fields or the presence of a classical background metric are made, both of which is expected to be essential close to classical singularities where the gravitational field would diverge and space degenerates. In contrast to other approaches to quantum cosmology there is a direct link between cosmological models and the full theory [38, 66] , as we will describe later in Section 6 . With cosmological applications we are thus able to test several possible constructions and draw conclusions for open issues in the full theory. At the same time, of course, we can learn about physical effects which have to be expected from properties of the quantization and can potentially lead to observable predictions. Since the full theory is not completed yet, however, an important issue in this context is the robustness of those applications to choices in the full theory and quantization ambiguities. The full theory itself is, understandably, extremely complex and thus requires approximation schemes for direct applications. Loop quantum cosmology is based on symmetry reduction, in the simplest case to isotropic geometries [46] . This poses the mathematical problem as to how the quantum representation of a model and its composite operators can be derived from that of the full theory, and in which sense this can be regarded as an approximation with suitable correction terms. Research in this direction currently proceeds by studying symmetric models with less symmetries and the relations between them. This allows to see what role anisotropies and inhomogeneities play in the full theory. While this work is still in progress, one can obtain full quantizations of models by using basic features as they can already be derived from the full theory together with constructions of more complicated operators in a way analogous to what one does in the full theory (see Section 5 ). For those complicated operators, the prime example being the Hamiltonian constraint which dictates the dynamics of the theory, the link between model and the full theory is not always clear-cut. Nevertheless, one can try different versions in the model in explicit ways and see what implications this has, so again the robustness issue arises. This has already been applied to issues such as the semiclassical limit and general properties of quantum dynamics. Thus, general ideas which are required for this new, background independent quantization scheme, can be tried in a rather simple context in explicit ways to see how those constructions work in practice. At the same time, there are possible phenomenological consequences in the physical systems being studied, which is the subject of Section 4 . In fact it turned out, rather surprisingly, that already very basic effects such as the discreteness of quantum geometry and other features briefly reviewed in Section 3 , for which a reliable derivation from the full theory is available, have very specific implications in early universe cosmology. While quantitative aspects depend on quantization ambiguities, there is a rich source of qualitative effects which work together in a well-defined and viable picture of the early universe. In such a way, as illustrated later, a partial view of the full theory and its properties emerges also from a physical, not just mathematical perspective. With this wide range of problems being investigated we can keep our eyes open to input from all sides. There are mathematical consistency conditions in the full theory, some of which are identically satisfied in the simplest models (such as the isotropic model which has only one Hamiltonian constraint and thus a trivial constraint algebra). They are being studied in different, more complicated models and also in the full theory directly. Since the conditions are not easy to satisfy, they put stringent bounds on possible ambiguities. From physical applications, on the other hand, we obtain conceptual and phenomenological constraints which can be complementary to those obtained from consistency checks. All this contributes to a test and better understanding of the background independent framework and its implications. Other reviews of loop quantum cosmology at different levels can be found in [56, 55, 199, 50, 69, 51, 96] . For complementary applications of loop quantum gravity to cosmology see [140, 141, 2, 114, 152, 1] .3 Loop Quantum Gravity
Since many reviews of full loop quantum gravity [172, 195, 22, 174, 161] as well as shorter accounts [9, 10, 173, 190, 167, 198] are already available, we describe here only those properties which will be essential later on. Nevertheless, this review is mostly self-contained; our notation is closest to that in [22] . A recent bibliography can be found in [93] .3.1 Geometry
General relativity in its canonical formulation [6] describes the geometry of space-time in terms of fields on spatial slices. Geometry on such a spatial slice is encoded in the spatial metric , which presents the configuration variables. Canonical momenta are given in terms of extrinsic curvature which is the derivative of the spatial metric under changing the spatial slice. Those fields are not arbitrary since they are obtained from a solution of Einstein's equations by choosing a time coordinate defining the spatial slices, and space-time geometry is generally covariant. In the canonical formalism this is expressed by the presence of constraints on the fields, the diffeomorphism constraint and the Hamiltonian constraint. The diffeomorphism constraint generates deformations of a spatial slice or coordinate changes, and when it is satisfied spatial geometry does not depend on which coordinates we choose on space. General covariance of space-time geometry also for the time coordinate is then completed by imposing the Hamiltonian constraint. This constraint, furthermore, is important for the dynamics of the theory: Since there is no absolute time, there is no Hamiltonian generating evolution, but only the Hamiltonian constraint. When it is satisfied, it encodes correlations between the physical fields of gravity and matter such that evolution in this framework is relational. The reproduction of a space-time metric in a coordinate dependent way then requires to choose a gauge and to compute the transformation in gauge parameters (including the coordinates) generated by the constraints. It is often useful to describe spatial geometry not by the spatial metric but by a triad which defines three vector fields which are orthogonal to each other and normalized in each point. This yields all information about spatial geometry, and indeed the inverse metric is obtained from the triad by where we sum over the index counting the triad vector fields. There are differences, however, between metric and triad formulations. First, the set of triad vectors can be rotated without changing the metric, which implies an additional gauge freedom with group SO(3) acting on the index . Invariance of the theory under those rotations is then guaranteed by a Gauss constraint in addition to the diffeomorphism and Hamiltonian constraints. The second difference will turn out to be more important later on: We can not only rotate the triad vectors but also reflect them, i.e., change the orientation of the triad given by . This does not change the metric either, and so could be included in the gauge group as O(3). However, reflections are not connected to the unit element of O(3) and thus are not generated by a constraint. It then has to be seen whether or not the theory allows to impose invariance under reflections, i.e., if its solutions are reflection symmetric. This is not usually an issue in the classical theory since positive and negative orientations on the space of triads are separated by degenerate configurations where the determinant of the metric vanishes. Points on the boundary are usually singularities where the classical evolution breaks down such that we will never connect between both sides. However, since there are expectations that quantum gravity may resolve classical singularities, which indeed are confirmed in loop quantum cosmology, we will have to keep this issue in mind and not restrict to only one orientation from the outset.3.2 Ashtekar variables
To quantize a constrained canonical theory one can use Dirac's prescription [105] and first represent the classical Poisson algebra of a suitable complete set of basic variables on phase space as an operator algebra on a Hilbert space, called kinematical. This ignores the constraints, which can be written as operators on the same Hilbert space. At the quantum level the constraints are then solved by determining their kernel, to be equipped with an inner product so as to define the physical Hilbert space. If zero is in the discrete part of the spectrum of a constraint, as e.g., for the Gauss constraint when the structure group is compact, the kernel is a subspace of the kinematical Hilbert space to which the kinematical inner product can be restricted. If, on the other hand, zero lies in the continuous part of the spectrum, there are no normalizable eigenstates and one has to construct a new physical Hilbert space from distributions. This is the case for the diffeomorphism and Hamiltonian constraints. To perform the first step we need a Hilbert space of functionals of spatial metrics. Unfortunately, the space of metrics, or alternatively extrinsic curvature tensors, is mathematically poorly understood and not much is known about suitable inner products. At this point, a new set of variables introduced by Ashtekar [7, 8, 30] becomes essential. This is a triad formulation, but uses the triad in a densitized form (i.e., it is multiplied with an additional factor of a Jacobian under coordinate transformations). The densitized triad is then related to the triad by but has the same properties concerning gauge rotations and its orientation (note the absolute value which is often omitted). The densitized triad is conjugate to extrinsic curvature coefficients :| (1) |
| (2) |
| (3) |
| (4) |
| (5) |
| (6) |
| (7) |
3.3 Representation
The key new aspect is now that we can choose the space of Ashtekar connections as our configuration space whose structure is much better understood than that of a space of metrics. Moreover, the formulation lends itself easily to a background independent quantization. To see this we need to remember that quantizing field theories requires one to smear fields, i.e., to integrate them over regions in order to obtain a well-defined algebra without -functions as in Equation ( 3 ). Usually this is done by integrating both configuration and momentum variables over three-dimensional regions, which requires an integration measure. This is no problem in ordinary field theories, which are formulated on a background such as Minkowski or a curved space. However, doing this here for gravity in terms of Ashtekar variables would immediately spoil any possible background independence since a background would already occur at this very basic step. There is now a different smearing available that does not require a background metric. Instead of using three-dimensional regions we integrate the connection along one-dimensional curves and exponentiate in a path-ordered manner, resulting in holonomies| (8) |
| (9) |
3.4 Function spaces
A connection 1-form can be reconstructed uniquely if all its holonomies are known [118] . It is thus sufficient to parameterize the configuration space by matrix elements of for all edges in space. This defines an algebra of functions on the infinite dimensional space of connections , which are multiplied as -valued functions. Moreover, there is a duality operation by complex conjugation, and if the structure group is compact a supremum norm exists since matrix elements of holonomies are then bounded. Thus, matrix elements form an Abelian -algebra with unit as a subalgebra of all continuous functions on . Any Abelian -algebra with unit can be represented as the algebra of all continuous functions on a compact space . The intuitive idea is that the original space , which has many more continuous functions, is enlarged by adding new points to it. This increases the number of continuity conditions and thus shrinks the set of continuous functions. This is done until only matrix elements of holonomies survive when continuity is imposed, and it follows from general results that the enlarged space must be compact for an Abelian unital -algebra. We thus obtain a compactification , the space of generalized connections [23] , which densely contains the space . There is a natural diffeomorphism invariant measure on , the Ashtekar–Lewandowski measure [19] , which defines the Hilbert space of square integrable functions on the space of generalized connections. A dense subset of functions is given by cylindrical functions , which depend on the connection through a finite but arbitrary number of holonomies. They are associated with graphs formed by the edges , . . . , . For functions cylindrical with respect to two identical graphs the inner product can be written as| (10) |
| (11) |
3.5 Composite operators
From the basic operators we can construct more complicated ones which, with growing degree of complexity, will be more and more ambiguous for instance from factor ordering choices. Quite simple expressions exist for the area and volume operator [177, 20, 21] , which are constructed solely from fluxes. Thus, they are less ambiguous since no factor ordering issues with holonomies arise. This is true because the area of a surface and volume of a region can be written classically as functionals of the densitized triad alone, and . At the quantum level, this implies that, just as fluxes, also area and volume have discrete spectra showing that spatial quantum geometry is discrete. (For discrete approaches to quantum gravity in general see [150] .) All area eigenvalues are known explicitly, but this is not possible even in principle for the volume operator. Nevertheless, some closed formulas and numerical techniques exist [149, 103, 102, 83] . The length of a curve, on the other hand, requires the co-triad which is an inverse of the densitized triad and is more problematic. Since fluxes have discrete spectra containing zero, they do not have densely defined inverse operators. As we will describe below, it is possible to quantize those expressions but requires one to use holonomies. Thus, here we encounter more ambiguities from factor ordering. Still, one can show that also length operators have discrete spectra [192] . Inverse densitized triad components also arise when we try to quantize matter Hamiltonians such as| (12) |
| (13) |
3.6 Hamiltonian constraint
Similarly to matter Hamiltonians one can also quantize the Hamiltonian constraint in a well-defined manner [194] . Again, this requires to rewrite triad components and to make other regularization choices. Thus, there is not just one quantization but a class of different possibilities. It is more direct to quantize the first part of the constraint containing only the Ashtekar curvature. (This part agrees with the constraint in Euclidean signature and Barbero–Immirzi parameter , and so is sometimes called Euclidean part of the constraint.) Triad components and their inverse determinant are again expressed as a Poisson bracket using the identity ( 13 ), and curvature components are obtained through a holonomy around a small loop of coordinate size and with tangent vectors and at its base point [176] :| (14) |
| (15) |
3.7 Open issues
For an anomaly-free quantization the constraint operators have to satisfy an algebra mimicking the classical one. There are arguments that this is the case for the quantization as described above when each loop contains exactly one vertex of a given graph [191] , but the issue is still open. Moreover, the operators are quite complicated and it is not easy to see if they have the correct expectation values in appropriately defined semiclassical states. Even if one regards the quantization and semiclassical issues as satisfactory, one has to face several hurdles in evaluating the theory. There are interpretational issues of the wave function obtained as a solution to the constraints, and also the problem of time or observables emerges [143] . There is a wild mixture of conceptual and technical problems at different levels, not at least because the operators are quite complicated. For instance, as seen in the rewriting procedure above, the volume operator plays an important role even if one is not necessarily interested in the volume of regions. Since this operator is complicated, without an explicitly known spectrum, it translates to complicated matrix elements of the constraints and matter Hamiltonians. Loop quantum gravity should thus be considered as a framework rather than a uniquely defined theory, which however has important rigid aspects. This includes the basic representation of the holonomy-flux algebra and its general consequences. All this should not come as a surprise since even classical gravity, at this level of generality, is complicated enough. Most solutions and results in general relativity are obtained with approximations or assumptions, one of the most widely used being symmetry reduction. In fact, this allows access to the most interesting gravitational phenomena such as cosmological expansion, black holes and gravitational waves. Similarly, symmetry reduction is expected to simplify many problems of full quantum gravity by resulting in simpler operators and by isolating conceptual problems such that not all of them need to be considered at once.4 Loop Cosmology
4.1 Isotropy
Isotropy reduces the phase space of general relativity to be 2-dimensional since, up to SU(2)-gauge freedom, there is only one independent component in an isotropic connection and triad, respectively, which is not already determined by the symmetry. This is analogous to metric variables, where the scale factor is the only free component in the spatial part of an isotropic metric| (16) |
| (17) |
4.2 Isotropy: Connection variables
Isotropic connections and triads, as discussed in Appendix 10.2 , are analogously described by single components and , respectively, related to the scale factor by| (18) |
| (19) |
| (20) |
| (21) |
| (22) |
| (23) |
The author is grateful to Ghanashyam Date and Golam Hossain for discussions and correspondence on this issue.
4.3 Isotropy: Implications of a loop quantization
We are now dealing with a simple system with finitely many degrees of freedom, subject to a constraint. It is well known how to quantize such a system from quantum mechanics, which has been applied to cosmology starting with DeWitt [104] . Here, one chooses a metric representation for wave functions, i.e., , on which the scale factor acts as multiplication operator and its conjugate , related to , as a derivative operator. These basic operators are then used to form the Wheeler–DeWitt operator quantizing the constraint ( 17 ) once a factor ordering is chosen. This prescription is rooted in quantum mechanics which, despite its formal similarity, is physically very different from cosmology. The procedure looks innocent, but one should realize that there are already basic choices involved. Choosing the factor ordering is harmless, even though results can depend on it [142] . More importantly, one has chosen the Schrödinger representation of the classical Poisson algebra, which immediately implies the familiar properties of operators such as the scale factor with a continuous spectrum. There are inequivalent representations with different properties, and it is not clear that this representation, which works well in quantum mechanics, is also correct for quantum cosmology. In fact, quantum mechanics is not very sensitive to the representation chosen [18] and one can use the most convenient one. This is the case because energies and thus oscillation lengths of wave functions described usually by quantum mechanics span only a limited range. Results can then be reproduced to arbitrary accuracy in any representation. Quantum cosmology, in contrast, has to deal with potentially infinitely high matter energies, leading to small oscillation lengths of wave functions, such that the issue of quantum representations becomes essential. That the Wheeler–DeWitt representation may not be the right choice is also indicated by the fact that its scale factor operator has a continuous spectrum, while quantum geometry which is a well-defined quantization of the full theory, implies discrete volume spectra. Indeed, the Wheeler–DeWitt quantization of full gravity exists only formally, and its application to quantum cosmology simply quantizes the classically reduced isotropic system. This is much easier, and also more ambiguous, and leaves open many consistency considerations. It would be more reliable to start with the full quantization and introduce the symmetries there, or at least follow the same constructions of the full theory in a reduced model. If this is done, it turns out that indeed we obtain a quantum representation inequivalent to the Wheeler–DeWitt representation, with strong implications in high energy regimes. In particular, just as the full theory such a quantization has a volume or operator with a discrete spectrum, as derived in Section 5.2 .4.4 Isotropy: Effective densities and equations
The isotropic model is thus quantized in such a way that the operator has a discrete spectrum containing zero. This immediately leads to a problem since we need a quantization of in order to quantize a matter Hamiltonian such as ( 23 ) where not only the matter fields but also geometry are quantized. However, an operator with zero in the discrete part of its spectrum does not have a densely defined inverse and does not allow a direct quantization of . This leads us to the first main effect of the loop quantization: It turns out that despite the non-existence of an inverse operator of one can quantize the classical to a well-defined operator. This is not just possible in the model but also in the full theory where it even has been defined first [193] . Classically, one can always write expressions in many equivalent ways, which usually result in different quantizations. In the case of , as discussed in Section 5.3 , there is a general class of ways to rewrite it in a quantizable manner [41] which differ in details but have all the same important properties. This can be parameterized by a function [47, 50] which replaces the classical and strongly deviates from it for small while being very close at large . The parameters and specify quantization ambiguities resulting from different ways of rewriting. With the function| (24) |
| (25) |
| (26) |
| (27) |
| (28) |
| (29) |
| (30) |
| (31) |
| (32) |
4.5 Isotropy: Properties and intuitive meaning
As a consequence of the function , the effective equations have different qualitative behavior at small versus large scales . In the effective Friedmann equation ( 29 ), this is most easily seen by comparing it with a mechanics problem with a standard Hamiltonian, or energy, of the form restricted to be zero. If we assume a constant scalar potential , there is no -dependence and the scalar equations of motion show that is constant. Thus, the potential for the motion of is essentially determined by the function . In the classical case, and the potential is negative and increasing, with a divergence at . The scale factor is thus driven toward , which it will always reach in finite time where the system breaks down. With the effective density , however, the potential is bounded from below, and is decreasing from zero for to the minimum around . Thus, the scale factor is now slowed down before it reaches , which depending on the matter content could avoid the classical singularity altogether. The behavior of matter is also different as shown by the effective Klein–Gordon equation ( 32 ). Most importantly, the derivative in the -term changes sign at small since the effective density is increasing there. Thus, the qualitative behavior of all the equations changes at small scales, which as we will see gives rise to many characteristic effects. Nevertheless, for the analysis of the equations as well as conceptual considerations it is interesting that solutions at small and large scales are connected by a duality transformation [147] , which even exists between effective solutions for loop cosmology and braneworld cosmology [90] . We have seen that the equations of motion following from an effective Hamiltonian are expected to display qualitatively different behavior at small scales. Before discussing specific models in detail, it is helpful to observe what physical meaning the resulting modifications have. Classical gravity is always attractive, which implies that there is nothing to prevent collapse in black holes or the whole universe. In the Friedmann equation this is expressed by the fact that the potential as used before is always decreasing toward where it diverges. With the effective density, on the other hand, we have seen that the decrease stops and instead the potential starts to increase at a certain scale before it reaches zero at . This means that at small scales, where quantum gravity becomes important, the gravitational attraction turns into repulsion. In contrast to classical gravity, thus, quantum gravity has a repulsive component which can potentially prevent collapse. So far, this has only been demonstrated in homogeneous models, but it relies on a general mechanism which is also present in the full theory. Not only the attractive nature of gravity changes at small scales, but also the behavior of matter in a gravitational background. Classically, matter fields in an expanding universe are slowed down by a friction term in the Klein–Gordon equation ( 32 ) where is negative. Conversely, in a contracting universe matter fields are excited and even diverge when the classical singularity is reached. This behavior turns around at small scales where the derivative becomes positive. Friction in an expanding universe then turns into antifriction such that matter fields are driven away from their potential minima before classical behavior sets in. In a contracting universe, on the other hand, matter fields are not excited by antifriction but freeze once the universe becomes small enough. These effects do not only have implications for the avoidance of singularities at but also for the behavior at small but non-zero scales. Gravitational repulsion can not only prevent collapse of a contracting universe [187] but also, in an expanding universe, enhance its expansion. The universe then accelerates in an inflationary manner from quantum gravity effects alone [45] . Similarly, the modified behavior of matter fields has implications for inflationary models [77] .4.6 Isotropy: Applications
There is now one characteristic modification in the matter Hamiltonian, coming directly from a loop quantization. Its implications can be interpreted as repulsive behavior on small scales and the exchange of friction and antifriction for matter, and it leads to many further consequences.4.6.1 Collapsing phase
When the universe has collapsed to a sufficiently small size, repulsion becomes noticeable and bouncing solutions become possible as illustrated in Figure 1 . Requirements for a bounce are that the conditions and can be fulfilled at the same time, where the first one can be evaluated with the Friedmann equation, and the second one with the Raychaudhuri equation. The first condition can only be fulfilled if there is a negative contribution to the matter energy, which can come from a positive curvature term or a negative matter potential . In those cases, there are classical solutions with , but they generically have corresponding to a recollapse. This can easily be seen in the flat case with a negative potential where ( 30 ) is strictly negative with at large scales. The repulsive nature at small scales now implies a second point where from ( 29 ) at smaller since the matter energy now decreases also for . Moreover, the modified Raychaudhuri equation ( 30 ) has an additional positive term at small scales such that becomes possible. Matter also behaves differently through the modified Klein–Gordon equation ( 32 ). Classically, with the scalar experiences antifriction and diverges close to the classical singularity. With the modification, antifriction turns into friction at small scales, damping the motion of such that it remains finite. In the case of a negative potential [68] this allows the kinetic term to cancel the potential term in the Friedmann equation. With a positive potential and positive curvature, on the other hand, the scalar is frozen and the potential is canceled by the curvature term. Since the scalar is almost constant, the behavior around the turning point is similar to a de Sitter bounce [187, 203] . Further, more generic possibilities for bounces arise from other correction terms [100, 97] .4.6.2 Expansion
Repulsion can not only prevent collapse but also accelerates an expanding phase. Indeed, using the behavior ( 26 ) at small scales in the effective Raychaudhuri equation ( 30 ) shows that is generically positive since the inner bracket is smaller than for the allowed values . Thus, as illustrated by the numerical solution in the upper left panel of Figure 2 , inflation is realized by quantum gravity effects for any matter field irrespective of its form, potential or initial values [45] . The kind of expansion at early stages is generically super-inflationary, i.e., with equation of state parameter . For free massless matter fields, usually starts very small, depending on the value of , but with a non-zero potential such as a mass term for matter inflation is generically close to exponential: . This can be shown by a simple and elegant argument independently of the precise matter dynamics [101] : The equation of state parameter is defined as where is the pressure, i.e., the negative change of energy with respect to volume, and energy density. Using the matter Hamiltonian for and , we obtain and thus in the classical case as usually. In the modified case, however, we have4.6.3 Model building
It is already clear that there are different inflationary scenarios using effects from loop cosmology. A scenario without inflaton is more attractive since it requires less choices and provides a fundamental explanation of inflation directly from quantum gravity. However, it is also more difficult to analyze structure formation in this context while there are already well-developed techniques in slow role scenarios. In these cases where one couples loop cosmology to an inflaton model one still requires the same conditions for the potential, but generically gets the required large initial values for the scalar by antifriction. On the other hand, finer details of the results now depend on the ambiguity parameters, which describe aspects of the quantization that also arise in the full theory. It is also possible to combine collapsing and expanding phases in cyclic or oscillatory models [148] . One then has a history of many cycles separated by bounces, whose duration depends on details of the model such as the potential. There can then be many brief cycles until eventually, if the potential is right, one obtains an inflationary phase if the scalar has grown high enough. In this way, one can develop ideas for the pre-history of our universe before the Big Bang. There are also possibilities to use a bounce to describe the structure in the universe. So far, this has only been described in effective models [137] using brane scenarios [151] where the classical singularity has been assumed to be absent by yet to be determined quantum effects. As it turns out, the explicit mechanism removing singularities in loop cosmology is not compatible with the assumptions made in those effective pictures. In particular, the scalar was supposed to turn around during the bounce, which is impossible in loop scenarios unless it encounters a range of positive potential during its evolution [68] . Then, however, generically an inflationary phase commences as in [148] , which is then the relevant regime for structure formation. This shows how model building in loop cosmology can distinguish scenarios that are more likely to occur from quantum gravity effects. Cyclic models can be argued to shift the initial moment of a universe in the infinite past, but they do not explain how the universe started. An attempt to explain this is the emergent universe model [110, 112] where one starts close to a static solution. This is difficult to achieve classically, however, since the available fixed points of the equations of motion are not stable and thus a universe departs too rapidly. Loop cosmology, on the other hand, implies an additional fixed point of the effective equations which is stable and allows to start the universe in an initial phase of oscillations before an inflationary phase is entered [160, 53] . This presents a natural realization of the scenario where the initial scale factor at the fixed point is automatically small so as to start the universe close to the Planck phase.4.6.4 Stability
Cosmological equations displaying super-inflation or antifriction are often unstable in the sense that matter can propagate faster than light. This has been voiced as a potential danger for loop cosmology, too [94, 95] . An analysis requires inhomogeneous techniques at least at an effective level, such as those described in Section 4.12 . It has been shown that loop cosmology is free of this problem, because the modified behavior for the homogeneous mode of the metric and matter is not relevant for matter propagation [129] . The whole cosmological picture that follows from the effective equations is thus consistent.4.7 Anisotropies
Anisotropic models provide a first generalization of isotropic ones to more realistic situations. They thus can be used to study the robustness of effects analyzed in isotropic situations and, at the same time, provide a large class of interesting applications. An analysis in particular of the singularity issue is important since the classical approach to a singularity can be very different from the isotropic one. On the other hand, the anisotropic approach is deemed to be characteristic even for general inhomogeneous singularities if the BKL scenario [31] is correct. A general homogeneous but anisotropic metric is of the form with left-invariant 1-forms on space , which, thanks to homogeneity, can be identified with the simply transitive symmetry group as a manifold. The left-invariant 1-forms satisfy the Maurer–Cartan relations with the structure constants of the symmetry group. In a matrix parameterization of the symmetry group, one can derive explicit expressions for from the Maurer–Cartan form with generators of . The simplest case of a symmetry group is an Abelian one with , corresponding to the Bianchi I model. In this case, is given by or a torus, and left-invariant 1-forms are simply in Cartesian coordinates. Other groups must be restricted to class A models in this context, satisfying since otherwise there is no Hamiltonian formulation. The structure constants can then be parameterized as . A common simplification is to assume the metric to be diagonal at all times, which corresponds to a reduction technically similar to a symmetry reduction. This amounts to as well as for the extrinsic curvature with . Depending on the structure constants, there is also non-zero intrinsic curvature quantified by the spin connection components| (33) |
| (34) |
4.8 Anisotropy: Connection variables
A densitized triad corresponding to a diagonal homogeneous metric has real components with if [48] . Connection components are and are conjugate to the , . In terms of triad variables we now have spin connection components| (35) |
| (36) |
4.9 Anisotropy: Applications
4.9.1 Isotropization
Matter fields are not the only contributions to the Hamiltonian in cosmology, but also the effect of anisotropies can be included in this way to an isotropic model. The late time behavior of this contribution can be shown to behave as in the shear energy density [156] , which falls off faster than any other matter component. Thus, toward later times the universe becomes more and more isotropic. In the backward direction, on the other hand, this means that the shear term diverges most strongly, which suggests that this term should be most relevant for the singularity issue. Even if matter densities are cut off as discussed before, the presence of bounces would depend on the fate of the anisotropy term. This simple reasoning is not true, however, since the behavior of shear is only effective and uses assumptions about the behavior of matter. It can thus not simply be extrapolated to early times. Anisotropies are independent degrees of freedom which affect the evolution of the scale factor. But only in certain regimes can this contribution be modeled simply by a function of the scale factor alone; in general one has to use the coupled system of equations for the scale factor, anisotropies and possible matter fields.4.9.2 Bianchi IX
Modifications to classical behavior are most drastic in the Bianchi IX model with symmetry group such that . The classical evolution can be described by a 3-dimensional mechanics system with a potential obtained from ( 34 ) such that the kinetic term is quadratic in derivatives of with respect to a time coordinate defined by . This potential| (37) |
4.9.3 Isotropic curvature suppression
If we use the potential for time coordinate rather than , it is replaced by , which in the isotropic reduction gives the curvature term . Although the anisotropic effective curvature potential is not bounded it is, unlike the classical curvature, bounded from above at any fixed volume. Moreover, it is bounded along the isotropy line and decays when approaches zero. Thus, there is a suppression of the divergence in when the closed isotropic model is viewed as embedded in a Bianchi IX model. Similarly to matter Hamiltonians, intrinsic curvature then approaches zero at zero scale factor. This is a further illustration for the special nature of isotropic models compared to anisotropic ones. In the classical reduction, the in the anisotropic spin connection cancel such that the spin connection is a constant and no special steps are needed for its quantization. By viewing isotropic models within anisotropic ones, one can consistently realize the model and see a suppression of intrinsic curvature terms. Anisotropic models, on the other hand, do not have, and do not need, complete suppression since curvature functions can still be unbounded.4.10 Anisotropy: Implications for inhomogeneities
Even without implementing inhomogeneous models the previous discussion allows some tentative conclusions as to the structure of general singularities. This is based on the BKL picture [31] whose basic idea is to study Einstein's field equations close to a singularity. One can then argue that spatial derivatives become subdominant compared to time-like derivatives such that the approach should locally be described by homogeneous models, in particular the Bianchi IX model since it has the most freedom in its general solution. Since spatial derivatives are present, though, they lead to small corrections and couple the geometries in different spatial points. One can visualize this by starting with an initial slice which is approximated by a collection of homogeneous patches. For some time, each patch evolves independently of the others, but this is not precisely true since coupling effects have been ignored. Moreover, each patch geometry evolves in a chaotic manner, which means that two initially nearby geometries depart rapidly from each other. The approximation can thus be maintained only if the patches are subdivided during the evolution, which goes on without limits in the approach to the singularity. There is, thus, more and more inhomogeneous structure being generated on arbitrarily small scales, which leads to a complicated picture of a general singularity. This picture can be taken over to the effective behavior of the Bianchi IX model. Here, the patches do not evolve chaotically even though at larger volume they follow the classical behavior. The subdivision thus has to be done also for the initial effective evolution. At some point, however, when reflections on the potential walls stop, the evolution simplifies and subdivisions are no longer necessary. There is thus a lower bound to the scale of structure whose precise value depends on the initial geometries. Nevertheless, from the scale at which the potential walls break down one can show that structure formation stops at the latest when the discreteness scale of quantum geometry is reached [60] . This can be seen as a consistency test of the theory since structure below the discreteness could not be supported by quantum geometry. We have thus a glimpse on the inhomogeneous situation with a complicated but consistent approach to a general classical singularity. The methods involved, however, are not very robust since the BKL scenario, which even classically is still at the level of a conjecture for the general case [32, 168] , would need to be available as an approximation to quantum geometry. For more reliable results the methods need to be refined to take into account inhomogeneities properly.4.11 Inhomogeneities
Allowing for inhomogeneities inevitably means to take a big step from finitely many degrees of freedom to infinitely many ones. There is no straightforward way to cut down the number of degrees of freedom to finitely many ones while being more general than in the homogeneous context. One possibility would be to introduce a small-scale cut-off such that only finitely many wave modes arise (e.g., through a lattice as is indeed done in some coherent state constructions [184] ). This is in fact expected to happen in a discrete framework such as quantum geometry, but would at this stage of defining a model simply be introduced by hand. For the analysis of inhomogeneous situations there are several different approximation schemes:4.12 Inhomogeneous matter with isotropic quantum geometry
Inhomogeneous matter fields cannot be introduced directly to isotropic quantum geometry since after the symmetry reduction there is no space manifold left for the fields to live on. There are then two different routes to proceed: One can simply take the classical field Hamiltonian and introduce effective modifications modeled on what happens to the isotropic Hamiltonian, or perform a mode decomposition of the matter fields and just work with the space-independent amplitudes. The latter is possible since the homogeneous geometry provides a background for the mode decomposition. The basic question, for the example of a scalar field, then is how the metric coefficient in the gradient term of Equation ( 12 ), , would be replaced effectively. For the other terms, one can simply use the isotropic modification, which is taken directly from the quantization. For the gradient term, however, one does not have a quantum expression in this context and a modification can only be guessed. The problem arises since the inhomogeneous term involves inverse powers of , while in the isotropic context the coefficient just reduces to , which would not be modified at all. There is thus no obvious and unique way to find a suitable replacement. A possible route would be to read off the modification from the full quantum Hamiltonian, or at least from an inhomogeneous model, which requires a better knowledge of the reduction procedure. Alternatively, one can take a more phenomenological point of view and study the effects of possible replacements. If the robustness of these effects to changes in the replacements is known, one can get a good picture of possible implications. So far, only initial steps have been taken and there is no complete programme in this direction. Another approximation of the inhomogeneous situation has been developed in [70] by patching isotropic quantum geometries together to support an inhomogeneous matter field. This can be used to study modified dispersion relations to the extent that the result agrees with preliminary calculations performed in the full theory [115, 3, 4, 181, 182] even at a quantitative level. There is thus further evidence that symmetric models and their approximations can provide reliable insights into the full theory.4.13 Inhomogeneity: Perturbations
With a symmetric background, a mode decomposition is not only possible for matter fields but also for geometry. The homogeneous modes can then be quantized as before, while higher modes are coupled as perturbations implementing inhomogeneities [120] . As with matter Hamiltonians before, one can then also deal with the gravitational part of the Hamiltonian constraint. In particular, there are terms with inverse powers of the homogeneous fields which receive modifications upon quantization. As with gradient terms in matter Hamiltonians, there are several options for those modifications which can only be restricted by relating them to the full Hamiltonian. This would require introducing the mode decomposition, analogously to symmetry conditions, at the quantum level and writing the full constraint operator as the homogeneous one plus correction terms. An additional complication compared to matter fields is that one is now dealing with infinitely many coupled constraint equations since the lapse function is inhomogeneous, too. This function can itself be decomposed into modes , with harmonics according to the symmetry, and each amplitude is varied independently giving rise to a separate constraint. The main constraint arises from the homogeneous mode, which describes how inhomogeneities affect the evolution of the homogeneous scale factors.4.14 Inhomogeneous models
The full theory is complicated at several different levels of both conceptual and technical nature. For instance, one has to deal with infinitely many degrees of freedom, most operators have complicated actions, and interpreting solutions to all constraints in a geometrical manner can be difficult. Most of these complications are avoided in homogeneous models, in particular when effective classical equations are employed. These equations use approximations of expectation values of quantum geometrical operators which need to be known rather explicitly. The question then arises whether one can still work at this level while relaxing the symmetry conditions and bringing in more complications of the full theory. Explicit calculations at a level similar to homogeneous models, at least for matrix elements of individual operators, are possible in inhomogeneous models, too. In particular the spherically symmetric model and cylindrically symmetric Einstein–Rosen waves are of this class, where the symmetry or other conditions are strong enough to result in a simple volume operator. In the spherically symmetric model, this simplification comes from the remaining isotropy subgroup isomorphic to U(1) in generic points, while the Einstein–Rosen model is simplified by polarization conditions that play a role analogous to the diagonalization of homogeneous models. With these models one obtains access to applications for black holes and gravitational waves, but also to inhomogeneities in cosmology. In spherical coordinates , , a spherically symmetric spatial metric takes the form with . This is related to densitized triad components by [196, 136] which are conjugate to the other basic variables given by the Ashtekar connection component and the extrinsic curvature component : Note that we use the Ashtekar connection for the inhomogeneous direction but extrinsic curvature for the homogeneous direction along symmetry orbits [75] . Connection and extrinsic curvature components for the -direction are related by with the spin connection component| (38) |
| (39) |
| (40) |
4.15 Inhomogeneity: Results
There are some results obtained for inhomogeneous systems. We have already discussed glimpses from the BKL picture, which used loop results only for anisotropic models. Methods described in this section have led to some preliminary insights into possible cosmological scenarios.4.15.1 Matter gradient terms and small-a effects
When an inhomogeneous matter Hamiltonian is available it is possible to study its implications on the cosmic microwave background with standard techniques. With modifications of densities there are then different regimes since the part of the inflationary era responsible for the formation of currently visible structure can be in the small- or large- region of the effective density. The small- regime below the peak of effective densities has more dramatic effects since inflation can here be provided by quantum geometry effects alone and the matter behavior changes to be anti-frictional [45, 77] . Mode evolution in this regime has been investigated for a particular choice of gradient term and using a power-law approximation for the effective density at small , with the result that there are characteristic signatures [130] . As in standard inflation models the spectrum is nearly scale invariant, but its spectral index is slightly larger than one (blue tilt) as compared to slightly smaller than one (red tilt) for single-field inflaton models. Since small scale factors at early stages of inflation generate structure which today appears on the largest scales, this implies that low multipoles of the power spectrum should have a blue tilt. The running of the spectral index in this regime can also be computed but depends only weakly on ambiguity parameters. The main parameter then is the duration of loop inflation. In the simplest scenario, one can assume only one inflationary phase, which would require huge values for the ambiguity parameter . This is unnatural and would imply that the spectrum is blue on almost all scales, which is in conflict with present observations. Thus, not only conceptual arguments but also cosmological observations point to smaller values for , which is quite remarkable. In order to have sufficient inflation to make the universe big enough one then needs additional stages provided by the behavior of matter fields. One still does not need an inflaton since now the details of the expansion after the structure generating phase are less important. Any matter field being driven away from its potential minimum during loop inflation and rolling down its potential thereafter suffices. Depending on the complexity of the model there can be several such phases.4.15.2 Matter gradient terms and large-a effects
At larger scale factors above the peak of effective densities there are only perturbative corrections from loop effects. This has been investigated with the aim of finding trans-Planckian corrections to the microwave background, also here with a particular gradient term. In this model, cancellations have been observed that imply that corrections appear only at higher orders of the perturbation series and are too weak to be observable [126] . A common problem of both analyses is that the robustness of the observed effects has not yet been studied. This is in particular a pressing problem since one modification of the gradient term has been chosen without further motivation. Moreover, the modifications in both examples were different. Without a more direct derivation of the modifications from inhomogeneous models or the full theory one can only rely on a robustness analysis to show that the effects can be trusted. In particular the cancellation in the second example must be shown to be realized for a larger class of modifications.4.15.3 Non-inflationary structure formation
Given a modification of the gradient term, one obtains effective equations for the matter field, which for a scalar results in a modified Klein–Gordon equation. After a mode decomposition, one can then easily see that all the modes behave differently at small scales with the classical friction replaced by anti-friction as in Section 4.5 . Thus, not only the average value of the field is driven away from its potential minimum but also higher modes are being excited. The coupled dynamics of all the modes thus provides a scenario for structure formation, which does not rely on inflation but on the anti-friction effect of loop cosmology. Even though all modes experience this effect, they do not all see it in the same way. The gradient term implies an additive contribution to the potential proportional to for a mode of wave number , which also depends on the metric in a way determined by the gradient term modification. For larger scales, the additional term is not essential and their amplitudes will be pushed to similar magnitudes, suggesting scale invariance for them. The potential relevant for higher modes, however, becomes steeper and steeper such that they are less excited by anti-friction and retain a small initial amplitude. In this way, the structure formation scenario provides a dynamical mechanism for a small-scale cut-off, possibly realizing older expectations [165, 166] .4.15.4 Stability
As already noted, inhomogeneous matter Hamiltonians can be used to study the stability of cosmological equations in the sense that matter does not propagate faster than light. The modified behavior of homogeneous modes has led to the suspicion that loop cosmology is not stable [94, 95] since other cosmological models displaying super-inflation have this problem. A detailed analysis of the loop equations, however, shows that the equations as they arise from modifications are automatically stable. While the homogeneous modes display super-inflationary and anti-frictional behavior, they are not relevant for matter propagation. Modes relevant for propagation, on the other hand, are modified differently in such a manner that the total behavior is stable [129] . Most importantly, this is an example where an inhomogeneous matter Hamiltonian with its modifications must be used and the qualitative result of stability can be shown to be robust under possible changes of the effective modification. This shows that reliable conclusions can be drawn for important issues without a precise definition of the effective inhomogeneous behavior.4.16 Summary
Loop cosmology is an effective description of quantum effects in cosmology, obtained in a framework of a background independent and non-perturbative quantization. There is mainly one change compared to classical equations coming from modified densities in matter Hamiltonians or also anisotropy potentials. These modifications are non-perturbative as they contain inverse powers of the Planck length and thus the gravitational constant, but also perturbative corrections arise from curvature terms, which are now being studied. The non-perturbative modification alone is responsible for a surprising variety of phenomena, which all improve the behavior in classical cosmology. Nevertheless, the modification had not been motivated by phenomenology but derived through the background independent quantization. Details of its derivation in cosmological models and its technical origin will now be reviewed in Section 5 , before we come to a discussion of the link to the full theory in Section 6 .5 Loop quantization of symmetric models
5.1 Symmetries and backgrounds
It is impossible to introduce symmetries in a completely background independent manner. The mathematical action of a symmetry group is defined by a mapping between abstract points, which do not exist in a diffeomorphism invariant setting (if one, for instance, considers only equivalence classes up to arbitrary diffeomorphisms). More precisely, while the full theory has as background only a differentiable or analytic manifold , a symmetric model has as background a symmetric manifold consisting of a differentiable or analytic manifold together with an action of a symmetry group . How strong the additional structure is depends on the symmetry used. The strongest symmetry in gravitational models is realized with spatial isotropy, which implies a unique spatial metric up to a scale factor. The background is thus equivalent to a conformal space. All constructions in a given model must take its symmetry into account since otherwise its particular dynamics, for instance, could not be captured. The structure of models thus depends on the different types of background realized for different symmetry groups. This can not only lead to simplifications but also to conceptual differences, and it is always instructive to keep the complete view on different models as well as the full theory. Since the loop formalism is general enough to encompass all relevant models, there are many ways to compare and relate different systems. It is thus possible to observe characteristic features of (metric) background independence even in cases where more structure is available.5.2 Isotropy
Isotropic models are described purely in terms of the scale factor such that there is only a single kinematical degree of freedom. In connection variables, this is parameterized by the triad component conjugate to the connection component . If we restrict ourselves to invariant connections of a given form, it suffices to probe them with only special holonomies. For an isotropic connection (see Appendix 10.2 ) we can choose holonomies along one integral curve of a symmetry generator . They are of the form| (41) |
| (42) |
| (43) |
| (44) |
| (45) |
| (46) |
5.3 Isotropy: Matter Hamiltonian
We now know how the basic quantities and are quantized, and can use the operators to construct more complicated ones. Of particular importance, also for cosmology, are matter Hamiltonians where now not only the matter field but also geometry is quantized. For an isotropic geometry and a scalar, this requires us to quantize for the kinetic term and for the potential term. The latter can be defined readily as , but for the former we need an inverse power of . Since has a discrete spectrum containing zero, a densely defined inverse does not exist. At this point, one has to find an alternative route to the quantization of , or else one could only conclude that there is no well-defined quantization of matter Hamiltonians as a manifestation of the classical divergence. In the case of loop quantum cosmology it turns out, following a general scheme of the full theory [193] , that one can reformulate the classical expression in an equivalent way such that quantization becomes possible. One possibility is to write, similarly to ( 13 ) where we use holonomies of isotropic connections and the volume . In this expression we can insert holonomies as multiplication operators and the volume operator, and turn the Poisson bracket into a commutator. The result| (47) |
| (48) |
5.4 Isotropy: Hamiltonian constraint
Dynamics is controlled by the Hamiltonian constraint, which classically gives the Friedmann equation. Since the classical expression ( 28 ) contains the connection component , we have to use holonomy operators. In the quantum algebra we only have almost periodic functions at our disposal, which does not include polynomials such as . Quantum expressions can therefore only coincide with the classical one in appropriate limits, which in isotropic cosmology is realized for small extrinsic curvature, i.e., small in the flat case. We thus need an almost periodic function of , which for small approaches . This can easily be found, e.g., the function . Again, the procedure is not unique since there are many such possibilities, e.g., , and more quantization ambiguities ensue. In contrast to the density , where we also used holonomies in the reformulation, the expressions are not equivalent to each other classically but only in the small curvature regime. As we will discuss shortly, the resulting new terms have the interpretation of higher order corrections to the classical Hamiltonian. One can restrict the ambiguities to some degree by modeling the expression on that of the full theory. This means that one does not simply replace by an almost periodic function, but uses holonomies tracing out closed loops formed by symmetry generators [42, 46] . Moreover, the procedure can be embedded in a general scheme that encompasses different models and the full theory [194, 42, 58] , further reducing ambiguities. In particular models with non-zero intrinsic curvature on their symmetry orbits, such as the closed isotropic model, can then be included in the construction. One issue to keep in mind is the fact that “holonomies” are treated differently in models and the full theory. In the latter case, they are ordinary holonomies along edges, which can be shrunk and then approximate connection components. In models, on the other hand, one sometimes uses direct exponentials of connection components without integration. In such a case, connection components are approximated only when they are small; if they are not, the corresponding objects such as the Hamiltonian constraint receive infinitely many correction terms of higher powers in curvature (similarly to effective actions). The difference between both ways of dealing with holonomies can be understood in inhomogeneous models, where they are both realized for different connection components. In the flat case the construction is easiest, related to the Abelian nature of the symmetry group. One can directly use the exponentials in ( 41 ), viewed as 3-dimensional holonomies along integral curves, and mimic the full constraint where one follows a loop to get curvature components of the connection . Respecting the symmetry, this can be done in the model with a square loop in two independent directions and . This yields the product , which appears in a trace, as in ( 15 ), together with a commutator using the remaining direction . The latter, following the general scheme of the full theory reviewed in Section 3.6 , quantizes the contribution to the constraint, instead of directly using the simpler . Taking the trace one obtains a diagonal operator in terms of the volume operator, as well as the multiplication operator In the triad representation where instead of working with functions one works with the coefficients in an expansion , this operator is the square of a difference operator. The constraint equation thus takes the form of a difference equation [46, 77, 15]| (49) |
| (50) |
5.5 Semiclassical limit and correction terms
When replacing by holonomies we have modified the constraint as a function on the classical phase space. This is necessary since otherwise the function cannot be quantized, but is different from the quantization of densities because now the replacements are not equivalent to the original constraint classically. Also the limit , which would give the classical result, does not exist at the operator level. This situation is different from the full theory, again related to the presence of a partial background [15] . There, the parameter length of edges used to construct appropriate loops is irrelevant and thus can shrink to zero. In the model, however, changing the edge length with respect to the background does change the operator and the limit does not exist. Intuitively, this can be understood as follows: The full constraint operator ( 15 ) is a vertex sum obtained after introducing a discretization of space used to choose loops . This classical regularization sums over all tetrahedra in the discretization, whose number diverges in the limit where the discretization size shrinks to zero. In the quantization, however, almost all these contributions vanish since a tetrahedron must contain a vertex of a state in order to contribute non-trivially. The result is independent of the discretization size once it is fine enough, and the limit can thus be taken trivially. In a homogeneous model, on the other hand, contributions from different tetrahedra of the triangulation must be identical owing to homogeneity. The coordinate size of tetrahedra drops out of the construction in the full background independent quantization, as emphasized in Section 3.6 , which is part of the reason for the discretization independence. In a homogeneous configuration the number of contributions thus increases in the limit, but their size does not change. This results in an ill-defined limit as we have already seen within the model itself. The difference between models and the full theory is thus only a consequence of the symmetry and not of different approaches. This will also become clear later in inhomogeneous models where one obtains a mixture between the two situations. Moreover, in the full theory one has a situation similar to symmetric models if one does not only look at the operator limit when the regularization is removed but also checks the classical limit on semiclassical states. In homogeneous models, the expression in terms of holonomies implies corrections to the classical constraint when curvature becomes larger. This is in analogy to other quantum field theories where effective actions generally have higher curvature terms. In the full theory, those correction terms can be seen when one computes expectation values of the Hamiltonian constraint in semiclassical states peaked at classical configurations for the connection and triad. When this classical configuration becomes small in volume or large in curvature, correction terms to the classical constraint arise. In this case, the semiclassical state provides the background with respect to which these corrections appear. In a homogeneous model, the symmetry already provides a partial background such that correction terms can be noticed already for the constraint operator itself.5.5.1 WKB approximation
There are different procedures to make contact between the difference equation and classical constraints. The most straightforward way is to expand the difference operators in a Taylor series, assuming that the wave function is sufficiently smooth. On large scales, this indeed results in the Wheeler–DeWitt equation as a continuum limit in a particular ordering [44] . From then on, one can use the WKB approximation or Wigner functions as usually. That this is possible may be surprising because as just discussed the continuum limit does not exist for the constraint operator. And indeed, the limit of the constraint equation, i.e., the operator applied to a wave function, does not exist in general. Even for a wave function the limit does not exist in general since some solutions are sensitive to the discreteness and do not have a continuum limit at all. When performing the Taylor expansion we already assumed certain properties of the wave function such that the continuum limit does exist. This then reduces the number of independent wave functions to that present in the Wheeler–DeWitt framework, subject to the Wheeler–DeWitt equation. That this is possible demonstrates that the constraint in terms of holonomies does not have problems with the classical limit. The Wheeler–DeWitt equation results at leading order, and in addition higher order terms arise in an expansion of difference operators in terms of or . Similarly, after the WKB or other semiclassical approximation there are correction terms to the classical constraint in terms of as well as [99] . This procedure is intuitive, but it is not suitable for inhomogeneous models where the Wheeler–DeWitt representation becomes ill-defined. One can evade this by performing the continuum and semiclassical limit together. This again leads to corrections in terms of as well as , which are mainly of the following form [29] : Matter Hamiltonians receive corrections through the modified density , and there are similar terms in the gravitational part containing . These are purely from triad coefficients, and similarly connection components lead to higher order corrections as well as additional contributions summarized in a quantum geometry potential. A possible interpretation of this potential in analogy to the Casimir effect has been put forward in [128] . A related procedure to extract semiclassical properties from the difference operator, based on the Bohmian interpretation of quantum mechanics, has been discussed in [185] .5.5.2 Effective formulation
In general, one does not only expect higher order corrections for a gravitational action but also higher derivative terms. The situation is then qualitatively different since not only correction terms to a given equation arise, but also new degrees of freedom coming from higher derivatives being independent of lower ones. In a WKB approximation, this could be introduced by parameterizing the amplitude of the wave function in a suitable way, but it has not been worked out yet. An alternative approach makes use of a geometrical formulation of quantum mechanics [26] , which not only provides a geometrical picture of the classical limit but also a clear-cut procedure for computing effective Hamiltonians in analogy to effective actions [73] . Instead of using linear operators on a Hilbert space, one can formulate quantum mechanics on an infinite-dimensional phase space. This space is directly obtained from the Hilbert space where the inner product defines a metric as well as a symplectic form on its linear vector space (which in this way even becomes Kähler). This formulation brings quantum mechanics conceptually much closer to classical physics, which also facilitates a comparison in a semiclassical analysis. We thus obtain a quantum phase space with infinitely many degrees of freedom, together with a flow defined by the Schrödinger equation. Operators become functions on this phase space through expectation values. Coordinates can be chosen by suitable parameterizations of a general wave function, in particular using the expectation values and together with uncertainties and higher moments. The projection defines the quantum phase space as a fiber bundle over the classical phase space with infinite-dimensional fibers. Sections of this bundle can be defined by embedding the classical phase space into the quantum phase space by means of suitable semiclassical states. For a harmonic oscillator this embedding can be done by coherent states which are preserved by the quantum evolution. This means that the quantum flow is tangential to the embedding of the classical phase space such that it agrees with the classical flow. The harmonic oscillator thus does not receive quantum corrections as is well known from effective actions for free field theories. Other systems, however, behave in a more complicated manner where in general states spread. This means that additional coordinates of the quantum phase space are dynamical and may become excited. If this is the case, the quantum flow differs from the classical flow and an effective Hamiltonian arises with correction terms that can be computed systematically. This effective Hamiltonian is given by the expectation value in approximate coherent states [16, 206, 188] . In these calculations, one can include higher degrees of freedom along the fibers, which, through the effective equations of motion, can be related to higher derivatives or higher curvature in the case of gravity. For a constrained system, such as gravity, one has to compute the expectation value of the Hamiltonian constraint, i.e., first go to the classical picture and then solve equations of motion. Otherwise, there would simply be no effective equations left after the constraints would already have been solved. This is the same procedure as in standard effective actions, which one can also formulate in a constrained manner if one chooses to parameterize time. Indeed, also for non-constrained systems agreement between the geometrical way to derive effective equations and standard path integral methods has been shown for perturbations around a harmonic oscillator [73] .5.6 Homogeneity
A Hamiltonian formulation is available for all homogeneous models of Bianchi class A [111] , which have structure constants fulfilling . The structure constants also determine left-invariant 1-forms in terms of which one can write a homogeneous connection as (see Appendix 10.1 ) where all freedom is contained in the -independent . A homogeneous densitized triad can be written in a dual form with coefficients conjugate to . As in isotropic models, one absorbs powers of the coordinate volume to obtain variables and . The kinematics is the same for all class A models, except possibly for slight differences in the diffeomorphism constraint [25, 36] . Connection components define a distinguished triple of su(2) elements , one for each independent direction of space. Holonomies in those directions are then obtained as with parameters for the edge lengths. Cylindrical functions depend on those holonomies, i.e., are countable superpositions of terms . A basis can be written down as spin network states where the matrix specifies how the representation matrices are contracted to a gauge invariant function of . There are uncountably many such states for different and thus the Hilbert space is non-separable. In contrast to isotropic models, the general homogeneous theory is genuinely SU(2) and therefore not much simpler than the full theory for individual calculations. As a consequence of homogeneity we observe the same degeneracy as in isotropic models where both spin and edge length appear similarly as parameters. Spins are important to specify the contraction and thus appear, e.g., in the volume spectrum. For this one needs to know the spins, and it is not sufficient to consider only products . On the other hand, there is still a degeneracy of spin and edge length and keeping both and independent leaves too many parameters. It is therefore more difficult to determine what the analog of the Bohr compactification is in this case.5.7 Diagonalization
The situation simplifies if one considers diagonal models, which is usually also done in classical considerations since it does not lead to much loss of information. In a metric formulation, one requires the metric and its time derivative to be diagonal, which is equivalent to a homogeneous densitized triad and connection with real numbers and (where coordinate volume has been absorbed as described in Appendix 10.1 ) which are conjugate to each other, , and internal directions as in isotropic models [48] . In fact, the kinematics becomes similar to isotropic models, except that there are now three independent copies. The reason for the simplification is that we are able to separate off the gauge degrees of freedom in from gauge invariant variables and (except for remaining discrete gauge transformations changing the signs of two of the and together). In a general homogenous connection, gauge-dependent and gauge-invariant parameters are mixed together in , which both react differently to a change in . This makes it more difficult to discuss the structure of relevant function spaces without assuming diagonalization. As mentioned, the variables and are not completely gauge invariant since a gauge transformation can flip the sign of two components and while keeping the third fixed. There is thus a discrete gauge group left, and only the total sign is gauge invariant in addition to the absolute values. Quantization can now proceed simply by using as Hilbert space the triple product of the isotropic Hilbert space, given by square integrable functions on the Bohr compactification of the real line. This results in states expanded in an orthonormal basis Gauge invariance under discrete gauge transformations requires to be symmetric under a flip of two signs in . Without loss of generality one can thus assume that is defined for all real but only non-negative and . Densitized triad components are quantized by which directly give the volume operator with spectrum Moreover, after dividing out the remaining discrete gauge freedom the only independent sign in triad components is given by the orientation , which again leads to a doubling of the metric minisuperspace with a degenerate subset in the interior, where one of the vanishes.5.8 Homogeneity: Dynamics
The Hamiltonian constraint can be constructed in the standard manner and its matrix elements can be computed explicitly thanks to the simple volume spectrum. There are holonomy operators for all three directions, and so in the triad representation the constraint equation becomes a partial difference equation for in three independent variables. Its (lengthy) form can be found in [48] for the Bianchi I model and in [62] for all other class A models. Simpler cases arise in so-called locally rotationally symmetric (LRS) models, where a non-trivial isotropy subgroup is assumed. Here, only two independent parameters and remain, where only one, e.g., can take both signs if discrete gauge freedom is fixed, and the vacuum difference equation is, e.g., for Bianchi I,| (51) |
| (52) |
5.9 Inhomogeneous models
Homogeneous models provide a rich generalization of isotropic ones, but inhomogeneities lead to stronger qualitative differences. To start with, at least at the kinematical level one has infinitely many degrees of freedom and is thus always dealing with field theories. Studying field theoretical implications does not require going immediately to the full theory since there are many inhomogeneous models of physical interest. We will describe some 1-dimensional models with one inhomogeneous coordinate and two others parameterizing symmetry orbits. A general connection is then of the form (with coordinate differentials and depending on the symmetry)| (53) |
5.10 Einstein–Rosen waves
One class of 1-dimensional models is given by cylindrically symmetric gravitational waves, with connections and triads| (54) |
| (55) |
5.10.1 Canonical variables
A difference to homogeneous models, however, is that the internal directions of a connection and a triad do not need to be identical, which in homogeneous models with internal directions is the case as a consequence of the Gauss constraint . With inhomogeneous fields, now, the Gauss constraint reads| (56) |
| (57) |
| (58) |
| (59) |
5.10.2 Representation
With the polarization condition the kinematics of the quantum theory simplifies. Relevant holonomies are given by along edges in the 1-dimensional manifold and in vertices with real . Cylindrical functions depend on finitely many of those holonomies, whose edges and vertices form a graph in the 1-dimensional manifold. Flux operators, i.e., quantized triad components, act simply by| (60) |
| (61) |
| (62) |
| (63) |
| (64) |
5.11 Spherical symmetry
For spherically symmetric models, a connection has the form (Appendix 10.3 )| (65) |
5.12 Loop inspired quantum cosmology
The constructions described so far in this section follow all the steps in the full theory as closely as possible. Most importantly, one obtains quantum representations inequivalent to those used in a Wheeler–DeWitt quantization, which results in many further implications. This has inspired investigations where not all the steps of loop quantum gravity are followed, but only the same type of representation, i.e., the Bohr Hilbert space in an isotropic model, is used. Other constructions, based on ADM rather than Ashtekar variables, are then done in the most straightforward way rather than a way suggested by the full theory [131] . In isotropic models the results are similar, but already here one can see conceptual differences. Since the model is based on ADM variables, in particular using the metric and not triads, it is not clear what the additional sign factor , which is then introduced by hand, means geometrically. In loop quantum cosmology it arose naturally as orientation of triads, even before its role in removing the classical singularity, to be discussed in Section 5.15 , had been noticed. (The necessity of having both signs available is also reinforced independently by kinematical consistency considerations in the full theory [117] .) In homogeneous models the situation is even more complicated since sign factors are still introduced by hand, but not all of them are removed by discrete gauge transformations as in Section 5.7 (see [158] as opposed to [14] ). Those models are useful to illuminate possible effects, but they also demonstrate how new ambiguities, even with conceptual implications, arise if guidance from a full theory is lost. In particular the internal time dynamics is more ambiguous in those models and thus not usually considered. There are then only arguments that the singularity could be avoided through boundedness of relevant operators, but those statements are not generic in anisotropic models [62] or even the full theory [85] . Moreover, even if all curvature quantities could be shown to be bounded, the evolution could still stop (as happens classically where not any singularity is also a curvature singularity).5.13 Dynamics
5.14 Dynamics: General construction
Not all steps in the construction of the full constraint can be taken over immediately to a model since symmetry requirements have to be respected. It is thus important to have a more general construction scheme that shows how generic different steps are, and whether or not crucial input in a given symmetric situation is needed. We have already observed one such issue, which is the appearance of holonomies but also simple exponentials of connection components without integration. This is a consequence of different transformation properties of different connection components in a reduced context. Components along remaining inhomogeneous directions, such as for Einstein–Rosen waves, play the role of connection components in the model, giving rise to ordinary holonomies. Other components, such as and in Einstein–Rosen waves or all components in homogeneous models, transform as scalars and thus only appear in exponentials without integration. In the overall picture, we have the full theory with only holonomies, homogeneous models with only exponentials, and inhomogeneous models in between where both holonomies and exponentials appear. Another crucial issue is that of intrinsic curvature encoded in the spin connection. In the full theory, the spin connection does not have any covariant meaning and in fact can locally be made to vanish. In symmetric models, however, some spin connection components can become covariantly well-defined since not all coordinate transformations are allowed within a model. In isotropic models, for instance, the spin connection is simply given by a constant proportional to the curvature parameter. Of particular importance is the spin connection when one considers semiclassical regimes because intrinsic curvature does not need to become small there in contrast to extrinsic curvature. Since the Ashtekar connection mixes the spin connection and extrinsic curvature, its semiclassical properties can be rather complicated in symmetric models. The full constraint is based on holonomies around closed loops in order to approximate Ashtekar curvature components when the loop becomes small in a continuum limit. For homogeneous directions, however, one cannot shrink the loop and instead works with exponentials of the components. One thus approximates the classical components only when arguments of the exponential are small. If these arguments were always connection components, one would not obtain the right semiclassical properties because those components can remain large. In models one thus has to base the construction for homogeneous directions on extrinsic curvature components, i.e., subtract off the spin connection from the Ashtekar connection. For inhomogeneous directions, on the other hand, this is not possible since one needs a connection in order to define a holonomy. At first sight this procedure seems rather ad hoc and even goes half a step back to ADM variables since extrinsic curvature components are used. However, there are several places where this procedure turns out to be necessary for a variety of independent reasons. We have already seen in Section 5.10 that inhomogeneous models can lead to a complicated volume operator when one insists on using all Ashtekar connection components. When one allows for extrinsic curvature components in the way just described, on the other hand, the volume operator becomes straightforward. This appeared after performing a canonical transformation, which rests non-trivially on the form of inhomogeneous spin connections and extrinsic curvature tensors. Moreover, in addition to the semiclassical limit used above as justification one also has to discuss local stability of the resulting evolution equation [59] : Since higher order difference equations have additional solutions, one must ensure that they do not become dominant in order not to spoil the continuum limit. This is satisfied with the above construction, while it is generically violated if one were to use only connection components. There is thus a common construction scheme available based on holonomies and exponentials. As already discussed, this is responsible for correction terms in a continuum limit, but also gives rise to the constraint equation being a difference equation in a triad representation, whenever it exists. In homogeneous models the structure of the resulting difference equation is clear, but there are different open possibilities in inhomogeneous models. This is intimately related to the issue of anomalies, which also appears only in inhomogeneous models. With a fixed choice, one has to solve a set of coupled difference equations for a wave function on superspace. The basic question then always is what kind of initial or boundary value problem has to be used in order to ensure the existence of solutions with suitable properties, e.g., in a semiclassical regime. Once this is specified one can already discuss the singularity problem since one needs to find out if initial conditions in one semiclassical regime together with boundary conditions away from classical singularities suffice for a unique solution on all of superspace. A secondary question is how this equation can be interpreted as evolution equation for the wave function in an internal time. This is not strictly necessary and can be complicated owing to the problem of time in general. Nevertheless, when available, an evolution interpretation can be helpful for interpretations.5.15 Singularities
5.16 Initial/boundary value problems
In isotropic models the gravitational part of the constraint corresponds to an ordinary difference operator which can be interpreted as generating evolution in internal time. One thus needs to specify only initial conditions to solve the equation. The number of conditions is large since, first, the procedure to construct the constraint operator usually results in higher order equations and, second, this equation relates values of a wave function defined on an uncountable set. In general, one thus has to choose a function on a real interval unless further conditions are used. This can be achieved, for instance, by using observables that can reduce the kinematical framework back to wave functions defined on a countable discrete lattice [202] . Similar restrictions can come from semiclassical properties or the physical inner product [162] , all of which has not yet been studied in generality. The situation in homogeneous models is similar, but now one has several gravitational degrees of freedom only one of which is interpreted as internal time. One has a partial difference equation for a wave function on a minisuperspace with boundary, and initial as well as boundary conditions are required [48] . Boundary conditions are imposed only at non-singular parts of minisuperspace such as in LRS models ( 51 ). They must not be imposed at places of classical singularities, of course, where instead the evolution must continue just as at any regular part. In inhomogeneous models, then, there are not only many independent kinematical variables but also many difference equations for only one wave function on midisuperspace. These difference equations are of a similar type as in homogeneous models, but they are coupled in complicated ways. Since one has several choices in the general construction of the constraint, there are different possibilities for the way how difference equations arise and are coupled. Not all of them are expected to be consistent, i.e., in many cases some of the difference equations will not be compatible such that there would be no non-zero solution at all. This is related to the anomaly issue since the commutation behavior of difference operators is important for properties and the existence of common solutions. So far, the evolution operator in inhomogeneous models has not been studied in detail, and solutions in this case remain poorly understood. The difficulty of this issue can be illustrated by the expectations in spherical symmetry where there is only one classical physical degree of freedom. If this is to be reproduced for semiclassical solutions of the quantum constraint, there must be a subtle elimination of infinitely many kinematical degrees of freedom such that in the end only one physical degree of freedom remains. Thus, from the many parameters needed in general to specify a solution to a set of difference equations, only one can remain when compatibility relations between the coupled difference equations and semiclassicality conditions are taken into account. How much this cancellation depends on semiclassicality and asymptotic infinity conditions remains to be seen. Some influence is to be expected since classical behavior should have a bearing on the correct reproduction of classical degrees of freedom. However, it may also turn out that the number of solutions to the quantum constraint is more sensitive to quantum effects. It is already known from isotropic models that the constraint equation can imply additional conditions for solutions beyond the higher order difference equation, as we will discuss in Section 5.18 . This usually arises at the place of classical singularities where the order of the difference equation can change. Since the quantum behavior at classical singularities is important here, the number of solutions can be different from the classically expected freedom, even when combined with possible semiclassical requirements far away from the singularity. We will now first discuss these requirements in semiclassical regimes, followed by more information on possibly arising additional conditions for solutions.5.17 Pre-classicality and boundedness
The high order of difference equations implies that there are in general many independent solutions, most of which are oscillating on small scales, i.e., when the labels change only slightly. One possibility to restrict the number of solutions then is to require suppressed or even absent oscillations on small scales [40] . Intuitively, this seems to be a pre-requisite for semiclassical behavior and has thus been called pre-classicality. It can be motivated by the fact that a semiclassical solution should not be sensitive to small changes of, e.g., the volume by amounts of Planck size. However, even though the criterion sounds intuitively reasonable, there is so far no justification through more physical arguments involving observables or measurement processes to extract information from wave functions. The status of pre-classicality as a selection criterion is thus not final. Moreover, pre-classicality is not always consistent in all disjoint classical regimes or with other conditions. For instance, as discussed in the following section, there can be additional conditions on wave functions arising from the constraint equation at the classical singularity. Such conditions do not arise in classical regimes, but they nevertheless have implications for the behavior of wave functions there through the evolution equation [87, 86] . Pre-classicality also may not be possible to impose in all disconnected classical regimes. If the evolution equation is locally stable – which is a basic criterion for constructing the constraint – choosing initial values in classical regimes, which do not have small-scale oscillations, guarantees that oscillations do not build up through evolution in a classical regime [59] . However, when the solution is extended through the quantum regime around a classical singularity, oscillations do arise and do not in general decay after a new supposedly classical regime beyond the singularity is entered. It is thus not obvious that indeed a new semiclassical region forms even if the quantum evolution for the wave function is non-singular. On the other hand, evolution does continue to large volume and macroscopic regions, which is different from other scenarios such as [124] where inhomogeneities have been quantized on a background. A similar issue is the boundedness of solutions, which also is motivated intuitively by referring to the common probability interpretation of quantum mechanics [119] but must be supported by an analysis of physical inner products. The issue arises in particular in classically forbidden regions where one expects exponentially growing and decaying solutions. If a classically forbidden region extends to infinite volume, as happens for models of recollapsing universes, the probability interpretation would require that only the exponentially decaying solution is realized. As before, such a condition at large volume is in general not consistent in all asymptotic regions or with other conditions arising in quantum regimes. Both issues, pre-classicality and boundedness, seem to be reasonable, but their physical significance has to be founded on properties of the physical inner product. They are rather straightforward to analyze in isotropic models without matter fields, where one is dealing with ordinary difference equations. However, other cases can be much more complicated such that conclusions drawn from isotropic models alone can be misleading. Moreover, numerical investigations have to be taken with care since in particular for boundedness an exponentially increasing contribution can easily arise from numerical errors and dominate the exact, potentially bounded solution. One thus needs analytical or at least semi-analytical techniques to deal with these issues. For pre-classicality one can advantageously use generating function techniques [87] if the difference equation is of a suitable form, e.g., has only coefficients with integer powers of the discrete parameter. The generating function for a solution on an equidistant lattice then solves a differential equation equivalent to the difference equation for . If is known, one can use its pole structure to get hints for the degree of oscillations in . In particular the behavior around is of interest to rule out alternating behavior where is of the form with for all (or at least all larger than a certain value). At we then have , which is less convergent than the value for a non-alternating solution resulting in . One can similarly find conditions for the pole structure to guarantee boundedness of , but the power of the method depends on the form of the difference equation. More general techniques are available for the boundedness issue, and also for alternating behavior, by mapping the difference equation to a continued fraction which can be evaluated analytically or numerically [71] . One can then systematically find initial values for solutions that are guaranteed to be bounded.5.18 Dynamical initial conditions
5.19 Summary
There is a general construction of a loop representation in the full theory and its models, which is characterized by compactified connection spaces and discrete triad operators. Strong simplifications of some technical and conceptual steps occur in diverse models. Such a general construction allows a view not only on the simplest case, isotropy, but on essentially all representative systems for gravity. Most important is the dynamics, which in the models discussed here can be formulated by a difference equation on superspace. A general scheme for a unique extension of wave functions through classical singularities is realized, such that the quantum theory is non-singular. This general argument, which has been verified in many models, is quite powerful since it does not require detailed knowledge of or assumptions about matter. It is independent of the availability of a global internal time, and so the problem of time does not present an obstacle. Moreover, a complicated discussion of quantum observables can be avoided since once it is known that a wave function can be continued uniquely one can extract relational information at both sides of the classical singularity. (If observables would distinguish both sides with their opposite orientations, they would strongly break parity even on large scales in contradiction with classical gravity.) Similarly, information on the physical inner product is not required since there is a general statement for all solutions of the constraint equation. The uniqueness of an extension through the classical singularity thus remains even if some solutions have to be excluded for the physical Hilbert space or factored out if they have zero norm. This is far from saying that observables or the physical inner product are irrelevant for an understanding of dynamical processes. Such constructions can, fortunately, be avoided for a general statement of non-singular evolution in a wide class of models. For details of the transition and to get information of the precise form of space-time at the other side of classical singularities, however, all those objects are necessary and conceptual problems in their context have to be understood. So far, the transition has often been visualized by intuitive pictures such as a collapsing universe turning its inside out when orientation is reversed. An hourglass presents a picture for the importance of discrete quantum geometry close to the classical singularity and the emergence of continuous geometry on large scales: Away from the bottleneck of the hourglass, its sand seems to be sinking down almost continuously. Directly at the bottleneck with its small circumference, however, one can see that time measured by the hourglass proceeds in discrete steps – one grain at a time. The main remaining issue for the mechanism to remove singularities then is the question how the models, where it has been demonstrated, are related to the full theory and to what extent they are characteristic for full quantum geometry.6 Models within the Full Theory
6.1 Symmetric states
One can imagine to construct states that are invariant under a given action of a symmetry group on space by starting with a general state and naively summing over all its possible translates by elements of the symmetry group. For instance on spin network states, the symmetry group acts by moving the graph underlying the spin network, keeping the labels fixed. Since states with different graphs are orthogonal to each other, the sum over uncountably many different translates cannot be normalizable. In simple cases, such as for graphs with a single edge along a symmetry generator, one can easily make sense of the sum as a distribution. But this is not clear for arbitrary states, in particular for states whose graphs have vertices, which on the other hand would be needed for sufficient generality. A further problem is that any such action of a symmetry group is a subgroup of the diffeomorphism group. At least on compact space manifolds where there are no asymptotic conditions for diffeomorphisms in the gauge group, it then seems that any group averaged diffeomorphism invariant state would already be symmetric with respect to arbitrary symmetries, which is obviously not sensible. In fact, symmetries and (gauge) diffeomorphisms are conceptually very different, even though mathematically they are both expressed by group actions on a space manifold. Gauge diffeomorphisms are generated by first class constraints of the theory, which in canonical quantum gravity are imposed in the Dirac manner [105] or following refined algebraic quantization [28] , conveniently done by group averaging [153] . Symmetries, however, are additional conditions imposed on a given theory to extract a particular sector of special interest. They can also be formulated as constraints added to the theory, but these constraints must be second class for a well defined framework: One obtains a consistent reduced theory, e.g., with a non-degenerate symplectic structure, only if configuration and momentum variables are required to be symmetric in the same (or dual) way. In the case of gravity in Ashtekar variables, the symmetry type determines, along the lines of Appendix 9 the form of invariant connections and densitized triads defining the phase space of the reduced model. At the quantum level, however, one cannot keep connections and triads on the same footing since a polarization is required. One usually uses the connection representation in loop quantum gravity such that states are functionals on the space of connections. In a minisuperspace quantization of the classically reduced model states would then be functionals only of invariant connections for the given symmetry type. This suggests to define symmetric states in the full theory to be those states whose support contains invariant connections as a dense subset [66, 38] (one requires only a dense subset because possible generalized connections must be allowed for). As such, they must necessarily be distributional, as already expected from the naive attempt at construction. Symmetric states thus form a subset of the distributional space . In this manner, only the reduced degrees of freedom are relevant, i.e., the reduction is complete, and all of them are indeed realized, i.e., the reduction is not too strong. Moreover, an “averaging” map from a non-symmetric state to a symmetric one can easily be defined by restricting the non-symmetric state to the space of invariant connections and requiring it to vanish everywhere else. This procedure defines states as functionals, but since there is no inner product on the full this does not automatically result in a Hilbert space. Appropriately defined subspaces of , nevertheless, often carry natural inner products, which is also the case here. In fact, since the reduced space of invariant connections can be treated by the same mathematical techniques as the full space, it carries an analog of the full Ashtekar–Lewandowski measure and this is indeed induced from the unique representation of the full theory. The only difference is that in general an invariant connection is not only determined by a reduced connection but also by scalar fields (see Appendix 9 ). As in the full theory, this space of reduced connections and scalars is compactified to the space of generalized invariant connections on which the reduced Hilbert space is defined. One thus arrives at the same Hilbert space for the subset of symmetric states in as used before for reduced models, e.g., using the Bohr compactification in isotropic models. The new ingredient now is that these states have meaning in the full theory as distributions, whose evaluation on normalizable states depends on the symmetry type and partial background structure used. That the symmetric Hilbert space obtained in this manner is identical to the reduced loop quantization of Section 5 does not happen by definition but is a result of the procedure. The support of a distribution is by definition a closed subset of the configuration space, and would thus be larger than just the set of generalized invariant connections if would not be a closed subset in . In such a case, the reduction at the quantum level would give rise to more degrees of freedom than a loop quantization of the classically reduced model. As shown in [66] , however, the set of invariant connections is a closed subset of the full space of connections such that loop quantum cosmology can be interpreted as a minisuperspace quantization.6.2 Basic operators
In the classical reduction, symmetry conditions are imposed on both connections and triads, but so far at the level of states only connections have been taken into account. Configuration and momentum variables play different roles in any quantum theory since a polarization is necessary. As we based the construction on the connection representation, symmetric triads have to be implemented at the operator level. (There cannot be additional reduction steps at the state level since, as we already observed, states just implement the right number of reduced degrees of freedom.) Classically, the reduction of phase space functions is simply done by pull back to the reduced phase space. The flow generated by the reduced functions then necessarily stays in the reduced phase space and defines canonical transformations for the model. An analog statement in the corresponding quantum theory would mean that the reduced state space would be fixed by full operators such that their action (or dual action on distributions) could directly be used in the model without further work. This, however, is not the case with the reduction performed so far. We have considered only connections in the reduction of states, and also classically a reduction to a subspace , where connections are invariant but not triads, would be incomplete. First, this would not define a phase space of its own with a non-degenerate symplectic structure. More important in this context is the fact that this subspace would not be preserved by the flow of reduced functions. As an example (see also [52] for a different discussion in the spherically symmetric model) we consider a diagonal homogeneous model, such as Bianchi I for simplicity, with connections of the form and look at the flow generated by the full volume . It is straightforward to evaluate the Poisson bracket already used in ( 13 ). A point on characterized by and an arbitrary triad thus changes infinitesimally by which does not preserve the invariant form: First, on the right hand side we have arbitrary fields such that is not homogeneous. Second, even if we would restrict ourselves to homogeneous , would not be of the original diagonal form. This is the case only if since only the are canonical variables. The latter condition is satisfied only if vanishes, which is not the case in general. This condition is true only if , i.e., if we restrict the triads to be of diagonal homogeneous form just as the connections. A reduction of only one part of the canonical variables is thus incomplete and leads to a situation where most phase space functions generate a flow that does not stay in the reduced space. Analogously, the dual action of full operators on symmetric distributional states does not in general map this space to itself. Thus, an arbitrary full operator maps a symmetric state to a non-symmetric one and cannot be used to define the reduced operator. In general, one needs a second reduction step that implements invariant triads at the level of operators by an appropriate projection of its action back to the symmetric space. This can be quite complicated, and fortunately there are special full operators adapted to the symmetry for which this step is not necessary. From the above example, it is clear that those operators must be linear in the momenta , for otherwise one would have a triad remaining after evaluating the Poisson bracket, which on would not be symmetric everywhere. Fluxes are linear in the momenta, so we can try where is a surface in the -plane at position in the -direction. By choosing a surface along symmetry generators and this expression is adapted to the symmetry, even though it is not fully symmetric yet since the position has to be chosen. Again, we compute the Poisson bracket resulting in
