A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.

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.

Loop quantum cosmology, the symmetry reduction of quantum geometry for the study of various cosmological situations, leads to a difference equation for its quantum evolution equation. To ensure that solutions of this equation act in the expected classical manner far from singularities, additional restrictions are imposed on the solution. In this paper, we consider the Bianchi I model, both the vacuum case and the addition of a cosmological constant, and show using generating function techniques that only the zero solution satisfies these constraints. This implies either that there are technical difficulties with the current method of quantizing the evolution equation, or else loop quantum gravity imposes strong restrictions on the physically allowed solutions.PACS numbers: 04.60. Pp, 04.60.Kz, 98.80.Qc When a new method of looking at physical situations arises, it is frequently too complicated to look at the general case, and thus it is necessary to look at simpler versions as test models. For example, in quantum mechanics one considers the hydrogen atom before examining more involved systems. These reduced models not only are easier to solve, but frequently provide information about what to expect from the more generic situation. This is the case with loop quantum cosmology [1], a symmetry reduced version of the full theory of quantum geometry [2] (also known as loop quantum gravity), which is intended to provide a framework joining together general relativity and quantum mechanics.Much like the Bohr quantization condition for the electrons of the atom, space-time itself is no longer continuous in quantum geometry, but instead is discrete. Since this means the states representing geometry are also discrete, the quantum evolution equation becomes a difference equation. In general, however, solutions to a recursion relation will oscillate in sign as one moves in a particular direction in the parameter space. Ordinarily, unphysical states like these would be eliminated by the imposition of an inner product, but this is unavailable at the moment in loop quantum cosmology. To get around this, a notion of pre-classicality [3] is used to pick out those quantum solutions to match a semi-classical state far away from the singularity. This condition will eliminate all states which vary greatly at Planck-length scales. We will see here how the pre-classicality of a solution can be imposed as a global property.Not only does loop quantum cosmology allow us to study simple situations of physical interest -for example, the quantized version of the isotropic FriedmanRobertson-Walker (FRW) space-times that serves as the basis of modern cosmology -so that physical predictions can be made, but it can give information about the full theory. In particular, it is closer in spirit to full quantum geometry than the minisuperspace models considered in standard quantum cosmology would be to a quantized general relativity. For the latter, the metric was first reduced to a specific model, then the equations of motion were quantized. Lo...

Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for pre-classical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter γ to the quantum ambiguity δ appearing in the model. This bound is violated when we use a natural value for δ based on loop quantum gravity together with a recent proposal for γ. We also present numerical solutions of the constraint.

In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the pre-classicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology.Comment: 5 pages, 3 figures, published versio