1 Introduction

with some assumptions it is possible to derive general properties. One of the best known theorems states that for matter satisfying positive energy conditions, spacetime solutions are generically geodesically incomplete: There is always a geodesic which is incomplete and cannot be extended beyond its defined range [34]. Such space-times are called singular due to the presence of boundaries to the motion of freely falling observers. In the best-known examples such as the Friedmann– Robertson–Walker or Schwarzschild solutions, some curvature scalars diverge at such boundaries, although this is not a general property proved in singularity theorems.

Canonical quantum gravity describes space-time by different structures, the metric no longer being the basic object. This requires a different mathematical formulation and poses new problems. A classical state is given by a point in phase space which for gravity is the cotangent bundle of the space of spatial metrics. Momenta, i.e. coordinates along the fibers, are given by the extrinsic curvature tensor of spatial slices Σ , which describes how the slice bends in space-time. A map from spacetime metrics on a 4-dimensional manifold M as solutions to general relativity to points in this phase space is not given canonically but requires a choice of slicing Σt : t = const, given by a global time function t : M → R. Space-time geometries are independent of the time function chosen provided that the cotangent bundle of spatial metrics with its canonical symplectic form is symplectically reduced to the appropriate physical phase space. This is achieved by imposing constraints Ci = 0, i = 1, . . . , 4 on the phase space variables which are equivalent to Einstein’s equation.

In quantum gravity, analogously to quantum mechanics, a state is described by a vector in a (projective) Hilbert space. The Hilbert space is a representation space of the classical Lie algebra of basic variables, such as spatial metrics and extrinsic curvature, under taking Poisson brackets. Alternatively, states can be defined as positive linear functionals (expectation values) on the algebra itself. Such states can usually be represented as wave functionals on the configuration space, such as the space of spatial metrics or related objects for gravity. In analogy to the classical symplectic reduction, the Hilbert space with its representation of basic operators must be reduced by imposing constraints. These are obtained by representing the classical

constraints as operators and asking that they annihilate physical states: ˆ | = 0.

Ci ψ

This implies linear functional differential equations (in a Wheeler–DeWitt quantization [31, 46]) or functional difference equations (in loop quantum gravity [43, 3, 45]) where independent variables of the functional equations are the metric tensors. Issues specific to applications of those equations to quantum cosmology are described in [15].

In general, such operators are difficult to formulate, and the resulting equations difficult to solve. As in the classical theory, one often uses simplifications due to spatial symmetries such as homogeneity and isotropy in cosmological situations. This article describes the derivation of such reduced models as well as aspects of solutions to their quantum constraint equations.

2 Quantum Representation and Dynamical Equations

A classical symmetry reduction simply selects a subspace Psymm of the full phase space P , motivated by symmetry assumptions on metric and extrinsic curvature, for which the pull-back of the full symplectic structure Ω is also symplectic. The model thus inherits a well-defined phase space of its own, which allows one to define its dynamics by pulling back the full constraints Ci , defining the reduced constrained system.

2.1 Quantum Reduction

At the quantum level, one starts with a representation of basic operators on a Hilbert space on which symmetries are to be implemented. Since a classical symmetry condition has to be imposed on both configuration variables and momenta to ensure a symplectic reduced phase space, operators for those conjugate variables do not commute. One thus cannot impose symmetry conditions strongly as operator equations for states. A suitable formulation has been achieved by imposing symmetry conditions based on the consideration of appropriate states as well as operators [19]. Just as the full classical phase space induces a unique reduced phase space once the symmetry has been specified, the full basic quantum algebra, representing the classical configuration and momentum variables as operators, induces a unique reduced representation once the symmetry has been specified. For explicit constructions see [24, 11]. (The procedure is related to Rieffel induction [40, 39]. An alternative procedure, which does not fully remove non-symmetric degrees of freedom but requires that they are unexcited, is described in [32, 33]. At least for free quantum field theories this can be achieved using the usual coherent states.)

Given a Hilbert space representation, a symmetric state is defined as a distributional state in the full Hilbert space which is supported only on invariant configuration variables [19]. Here, one can take advantage of the fact that canonical general relativity can be formulated in terms of connections as configuration variables [1, 7], and of the classification of invariant connections on symmetric principal fiber bundles [38, 26]. Symmetry conditions for momenta are then imposed by the induced algebra: it is generated by all basic operators of the full theory which map the space of symmetric states into itself.

A simple example illustrates the procedure: Given n + 1 degrees of freedom (qi , pi ) from which the (n + 1)st one is to be removed by symmetry reduction, we have to impose qn+1 = 0 = pn+1. While this is straightforward classically, strong operator conditions qˆn+1|ψ = 0 = pˆn+1|ψ in quantum mechanics would be inconsistent due to 0 = [qˆn+1, pˆn+1]|ψ = i |ψ . Instead, one can perform the reduction as follows. Define a symmetric state to be a distribution Ψ (qi ) = δ(qn+1)ψ (qi ) in the dual D of a suitable dense subset D of the Hilbert space. This defines the set

basic operators (qi , pi ) thus obtain a dual action on D , but not all of them fix the

subset Dsymm. The induced algebra of the reduction is now defined as the algebra generated by the basic operators mapping Dsymm into itself, and their induced representation is obtained from the dual action. This leaves us with the correct degrees of freedom: (qˆi , pˆi ), i = 1, . . . , n satisfy the condition and are thus generators of the induced algebra. The derivative operator pˆn+1, however, does not fix Dsymm and is thus not part of the induced algebra, while qˆn+1 becomes the zero operator in the dual action. We thus have successfully derived the reduced algebra and an induced representation. Since the dual D does not carry a natural inner product related to that of the Hilbert space, there is initially no inner product on Dsymm, either. The induced representation nevertheless carries a natural inner product defined by requiring the correct adjointness properties of generators of the induced algebra. In our example, (qˆi , pˆi ) have to be self-adjoint, such that the induced representation fully agrees with the usual quantum representation of the classically reduced system. Note that there is a difference to the classical situation: classical symmetric solutions are exact solutions of the general equations while the induced representation space, in general, is not a subspace of the full representation space. This arises because for the quantum theory it is not the representation space but the algebra of basic operators which is primary.

This procedure is general enough to apply to loop quantum gravity, too. It defines the symmetric sector of the full theory, derived from the full quantum representation. Already the derivation of the induced basic representation is crucial since the Stone– von Neumann theorem does not apply in loop quantum cosmology (holonomies not being weakly continuous in a loop quantization). Thus, there is no unique representation even in finite-dimensional systems and physical properties can be representation dependent. It is thus crucial that the representation of loop quantum cosmology is derived from that of full loop quantum gravity along the lines sketched above. This is the underlying reason for the availability of qualitative physical predictions.

2.2 Dynamics

On the induced representation one then has to formulate the constraints ˆ and solve

Ci

the equations they imply for states. The quantum analog of pulling back the classical constraints is not simple, unless they are linear in basic variables, and is still being developed. Rather than deriving quantum constraints in this way, one currently quantizes the reduced constraints from induced basic operators along the lines followed in a full construction. Since the most important aspects of quantum constraints and their solutions depend on the representation in which the operators are formulated, properties of constraints in the full theory are thus inherited in models through the induction procedure. This suffices to show crucial effects which enter the general quantization scheme, and has by now led to many applications. But, at a detailed level, it leaves several different possibilities for the exact form of

constraints, a degree of non-uniqueness which is exacerbated by the current nonuniqueness of the full constraint in the first place.

In most models introduced so far, the constraints take the form of difference equations for states. Once a model has been specified, loop quantum cosmology thus requires one to solve difference equations for a wave function. This leads to our second mathematical issue, properties of difference equations of a certain type. We will focus here on isotropic models, whose induced representation is given by the space of square integrable functions on the Bohr compactification of the real line [5, 14]. By this compactification, the representation differs from what one would expect naively from a quantum mechanical procedure as it is followed in Wheeler–DeWitt quantizations. This is an example for the importance of the induction procedure to impose symmetry conditions. As a direct implication, the momentum operator of isotropic models, corresponding to a densitized triad component and thus describing spatial geometry, has a discrete spectrum of eigenvalues μ. (Any real value of μ is allowed, but all eigenstates are normalizable: the Hilbert space is non-separable.) This is not only the reason for the occurrence of difference equations but also has further physical implications.

Only one constraint remains to be imposed in an isotropic model due to the symmetry. A corresponding operator has to be constructed in terms of the basic ones, following the steps one would do without assuming symmetries. This is indeed possible, but not in a unique manner. The resulting equation for a wave function ψμ is in general of the form of a difference equation [9]

where coefficients are written in terms of volume eigenvalues Vμ = (|μ|/6)3/2 and

ˆ is a differential operator (quantizing the matter Hamiltonian) acting on the

Hmatter

matter field φ.

This equation, as written here, is the simplest version, based on certain assumptions on how a full constraint operator would reduce to that of the model. The freedom one has in the full construction essentially reduces, in isotropic models, to the form of the step size of the dynamical difference equation. This can be phrased as the question of which function of μ, if any at all, changes equidistantly in the equation, which need not be the eigenvalue μ of the basic triad operator. A more general class can be formulated after replacing according to the canonical transformation (c, p) → (pk c, p1−k /(1 − k)) which can be motivated by lattice refinements occurring in an inhomogeneous state and restricts k to the range −1/2 < k < 0 [14]. Then, μ1−k instead of μ will be equidistant. So far, only the extreme cases k = 0

[6]) have been considered in some detail.

Fully realistic cases are somewhere in between with a μ-dependence which can only be determined from a precise relation to full dynamics. The general statements and techniques described in what follows are insensitive to the precise behavior or