Absence of Preclassical Solutions in Bianchi I Loop Quantum Cosmology

Abstract

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...

Wave functions for the Schwarzschild black hole interior

“…In particular, V µ+δ,τ − V µ−δ,τ = 0 when µ = 0, and V µ,τ +δ − V µ,τ −δ = 0 when τ = 0. This situation arises in previous work in loop quantum cosmology, such as the isotropic [4] and the Bianchi I and IX models [8,9,10]. If we write the wave function Ψ solving the Hamiltonian constraint as a sum of eigenstates…”

Observational issues in loop quantum cosmology

“…Many studies have already been devoted to Bianchi-I LQC [97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117]. In particular, it was shown that the bounce prediction is robust.…”

Lattice refining loop quantum cosmology, anisotropic models, and stability

“…In such models, limitations similar to that of a cosmological constant have been observed as possible instabilities of solutions in classical regions or the lack of a sufficient number of semiclassical states [27,28,29]. For the partial difference equations of anisotropic models in loop quantum cosmology, stability issues can be much more severe than in isotropic models and thus lead to further consistency tests which might help to restrict possible quantization freedom (see, e.g., [30]).…”