2+1 dimensional loop quantum cosmology of Bianchi I models

TL;DR

This paper explores 2+1 dimensional loop quantum cosmology of Bianchi I models, demonstrating how scalar fields serve as internal clocks, deriving Hamiltonian constraints, and constructing exact solutions, thus extending LQC to lower dimensions.

Contribution

It provides a detailed analysis of anisotropic Bianchi I models in 2+1 dimensions within loop quantum cosmology, including new solutions and the role of scalar fields as internal time.

Findings

Scalar field acts as an internal clock in 2+1 D LQC.

Hamiltonian constraint leads to a difference equation for evolution.

Exact solutions to Friedmann equations are constructed.

Abstract

We study the anisotropic Bianchi I loop quantum cosmology in 2+1 dimensions. Both the $\mubar$ and $\mubar'$ schemes are considered in the present paper and the following expected results are established: (i) the massless scalar field again play the role of emergent time variables and serves as an internal clock; (ii) By imposing the fundamental discreteness of length operator, the total Hamiltonian constraint is obtained and gives rise the evolution as a difference equation; and (iii) the exact solutions of Friedmann equation are constructed rigorously for both classical and effective level. The investigation extends the domain of validity of loop quantum cosmology to beyond the four dimensions.