the value of k. Coefficients of the difference equation, and also asymptotic properties, do however depend on the choice. Detailed investigations are now emerging which can shed light on more precise features of the difference equation and fix some of the freedom in their derivation. In general, one may have to consider non-equidistant difference equations, especially for less symmetric models where a partial difference equation results from anisotropy [10, 23] or even several coupled equations in inhomogeneous situations [22].

3 Quantum Singularity Problem

The first general property of difference equations of loop quantum cosmology which has been studied deals with the singularity problem [8]. In this context, the singularity problem arises as the question of whether or not wave functions solving the functional equations are uniquely extended across classical singularities on the configuration spaces of metrics or related objects [18]. As before, wave functions of loop quantum cosmology are first defined on the whole real axis μ R, which includes the classical singularity μ = 0 as an interior point. Unlike for the Wheeler– DeWitt equation, solutions to the difference equation then extend uniquely from positive μ to negative μ and vice versa, even though coefficients of the difference equation may vanish. There is thus a well-posed initial value problem even across the classical singularity, which makes the quantum evolution non-singular. (The recurrence determined by the difference equation is not guaranteed to extend a wave function in such a way just by the fact that μ = 0 is an interior point. The lowest order coefficient Vμ − Vμ−2 of ψμ−1 vanishes at μ = 1, and the backward evolution will thus not determine the value ψ0 right at the classical singularity. If this value is needed for the further recurrence, additional input would be required and one would be dealing with a boundary value rather than initial value problem. Physically, the behavior of solutions would not be determined by values of the wave function at one side of the classical singularity only.) This extendability, called quantum hyperbolicity for the functional equations, replaces geodesic completeness as the criterion for non-singular behavior. Using symmetry reduction as described above, it is currently verified in many cases based on the difference equations of loop quantum cosmology [13], including inhomogeneous ones [8, 10, 23, 12]. This covers the quantum analogs of the basic classical examples of singular space-times.

In addition to the hyperbolicity issue, which can be rather involved especially in inhomogeneous models with coupled partial difference equations, the analysis of extendability leads to a further class of mathematical issues: To analyze the extendability of solutions across classical singularities in spaces of triads, one needs a characterization of classical singularities by geometrical properties. This is difficult in general as the singularity theorems do not provide much information on the behavior of the singularities they predict. Here, further simplifications occur in symmetric models where classical singularities can often be analyzed completely.

As one of the results, it turned out that the extendability of wave functions hinges on the type of variables used. Densitized triads Eia , related to the spatial metric by Eia Eib = qab det(qcd ), are used as basic variables in loop quantum gravity. They naturally arise in setting up a background independent quantization and have, independently, led to a resolution of singularities in all cases studied so far. On the other hand, the spatial metric or even triads or co-triads, rather than densitized triads, do not provide a general mechanism. While any choice of triad variables offers an extended configuration space due to the sign provided by the triad orientation, the position of classical singularities in superspaces depends sensitively on the type of variables. In isotropic models the difference is not crucial, as demonstrated by the analysis of [36] following [5], but it is essential in anisotropic models as seen by comparing [41] with [2]. This is a consequence of the fact that anisotropic models, which unlike isotropic ones are often expected to show the typical approach to a space-like singularity, have finite densitized triad component but one infinite metric or co-triad component at the singularity. One can see this easily in the Kasner solution with metric components aI (t ) t αI and densitized triad components

general singularities from this perspective can provide important insights for quantum gravity by a combination of mathematical relativity and properties of difference equations, linked by quantum dynamics.

4 Examples for Properties of Solutions

In addition to the singularity issue, questions about solutions to difference equations one is interested in are:

1.Is it possible to find exact special solutions to the difference equations of loop quantum cosmology in some cases? This is not easy in general, although we have a linear difference equation, due to the presence of non-constant coefficients involving an absolute value.

2.If no exact solution is known, what are asymptotic properties (for large |μ|) of general solutions? This is often relevant for a normalization of wave functions in a physical inner product. There are two types of asymptotic behavior which are being investigated:

(a)Oscillations on small scales (such as ψμ (−1)μ), which can often be analyzed by generating functions [29] and

(b)Boundedness of solutions, where continued fractions can advantageously be applied [20].

Let us first look at the difference equation

am+1 − am−1 = 2λm−1am

which arises when the partial difference equation of an anisotropic quantum cosmological model is separated [10, 29]. Here, λ R is the separation parameter. Solutions to this difference equation have generating function

∞

G(x) = am+1xm = c0(1 + x)λ−1(1 − x)−λ−1

m=0

−2λa0 (1 − x)−λ−12F1(1 − λ, −λ; 2 − λ; (1 + x)/2)

λ− 1

which can advantageously be used to determine asymptotic properties. For instance, solutions have (−1)m-oscillations with shrinking amplitude if G(x) is regular at x = −1 because m(−1)mam must then be convergent. This requires special initial values of the solution satisfying

a1/a0 = 1 − λψ (1/2 − λ/2) + λψ (1 − λ/2)

with the digamma function ψ (z) = d log Γ (z)/dz. Moreover, the parameter c0 in the generating function must vanish in this case. The initial values are determined through a1 = G(0) while a0 already appears as a parameter in the generating function. Further applications of generating functions in this context can be found in [27, 30, 28].

Another difference equation is

sn+4 − 2sn + sn−4 = Λnsn

which appears in isotropic models of Euclidean gravity including a cosmological constant Λ (see also [42] for an analysis of this model). Generic solutions to this equation exponentially increase for large n, but it is often important to determine special solutions which are bounded (and thus normalizable as states in an 2 Hilbert space). This also poses conditions on initial values of the solution, such that they equal the continued fraction [20]

Such results are true for more general difference equations than that shown above, but the one used here provides an interesting relation to Bessel functions. In fact, the displayed difference equation can be solved exactly in terms of Bessel functions,

Only the first contribution is bounded which, by comparison with (2) and its analog for s4(m0−1)/s4m0 with integer m0, gives the continued fraction representation

82 Martin Bojowald

These are infinitely many coupled ordinary differential equations for infinitely many variables, equivalent to the partial Schrödinger equation. This system is in general much more complicated to analyze in this form, but in an adiabatic approximation in the quantum variables one can decouple and solve the equations order by order, reproducing the low energy effective action as it is known in particle physics [21, 44]. The reformulation is thus valuable for semiclassical and perturbative aspects.

In general, the high coupling of the equations of motion is barely manageable. The main requirement for the applicability of effective theory in this form is then the availability of an exactly solvable system which one can use as zeroth order of a perturbation expansion. For the solvable model itself, the equations decouple, which can then be exploited in a perturbation analysis even in more complicated systems. For low energy effective actions as mentioned above, the solvable model is the harmonic oscillator or a free quantum field theory. Cosmological models are different, but also here a solvable model is available: a spatially flat isotropic model sourced by a free, massless scalar [16]. According to general relativity it is governed by the Friedmann equation q2√p = 12 p−3/2pφ2 . (The gravitational variables are extrinsic curvature q = a˙ and the spatial volume p3/2 = a3 in terms of the scale factor a.) Solving for the scalar momentum pφ gives the Hamiltonian pφ = H = qp (also known as the Berry–Keating–Connes Hamiltonian), interpreted as generating evolution in φ. Note that there are different sign choices possible in the solution pφ ; see [17] for more details.

difference equation as mentioned before. The analysis of this section, motivated by effective theory, thus provides a further example for studying properties of solutions to a difference equation without actually solving the difference equation.