Generating function techniques for loop quantum cosmology

Abstract

Loop quantum cosmology leads to a difference equation for the wave function of a universe, which in general has solutions changing rapidly even when the volume changes only slightly. For a semiclassical regime such small-scale oscillations must be suppressed, by choosing the parameters of the solution appropriately. For anisotropic models this is not possible to do numerically by trial and error; instead, it is shown here for the Bianchi I LRS model how this can be done analytically, using generating function techniques. Those techniques can also be applied to more complicated models, and the results gained allow conclusions about initial value problems for other systems.

Homogeneous loop quantum cosmology

“…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 [166,167]. Pre-classicality also may not be possible to impose in all disconnected classical regimes.…”

“…For pre-classicality one can advantageously use generating function techniques [166] if the difference equation is of a suitable form, e.g. has only coefficients with integer powers of the discrete parameter.…”

“…Special techniques, such as those in [188,166,170,189,196] have to be developed so as to apply to more general systems. In particular for including inhomogeneities, both for solving equations and interpreting solutions, a new area of numerical quantum gravity has to be developed.…”

Wave functions for the Schwarzschild black hole interior

“…arising as a special case in the Bianchi I model [10]. Note that there is always a scaling freedom s m → cs m , so choosing the ratio s 1 /s 0 is the only free information we have in the initial values of the sequence.…”

“…far away from any singular points), we must have a way of restricting the wave functions to those that act semi-classically in the appropriate regime. With this in mind, the technique of using generating functions to solve difference equations has been developed in the context of loop quantum cosmology [10]; for a review of these methods in a generic context, see Wilf [13]. Our goal will be to find a generating function B(y), whose Taylor series 2 gives the sequence β n , i.e.…”