Anisotropic loop quantum cosmology with self-dual variables

TL;DR

This paper develops a loop quantum cosmology model for Bianchi universes using self-dual variables, demonstrating singularity resolution and providing explicit Hamiltonian constraint operators.

Contribution

It introduces a novel loop quantization approach for diagonal Bianchi models with self-dual variables, including explicit Hamiltonian operators and singularity resolution.

Findings

Singularities are resolved with decoupling of singular states.

Explicit form of the Hamiltonian constraint operator is provided.

The model extends loop quantum cosmology to complex self-dual variables.

Abstract

A loop quantization of the diagonal class A Bianchi models starting from the complex-valued self-dual connection variables is presented in this paper. The basic operators in the quantum theory correspond to areas and generalized holonomies of the Ashtekar connection and the reality conditions are implemented via the choice of the inner product on the kinematical Hilbert space. The action of the Hamiltonian constraint operator is given explicitly for the case when the matter content is a massless scalar field (in which case the scalar field can be used as a relational clock), and it is shown that the big-bang and big-crunch singularities are resolved in the sense that singular and non-singular states decouple under the action of the Hamiltonian constraint operator.