Path Integral of Bianchi I models in Loop Quantum Cosmology

TL;DR

This paper develops a path integral formulation for Bianchi I models in loop quantum cosmology, comparing different schemes and confirming the equivalence with the canonical approach.

Contribution

It extends the path integral approach to non-isotropic Bianchi I models and analyzes scheme-dependent invariance properties.

Findings

Path integral formulation for Bianchi I models is constructed.

Invariance under fiducial cell scaling differs between schemes.

Canonical and path integral approaches are shown to be equivalent.

Abstract

A path integral formulation of the Bianchi I models containing a massless scalar field in loop quantum cosmology is constructed. Following the strategy used in the homogenous and isotropic case, the calculation is extended to the simplest non-isotropic models according to the $\bar{\mu}$ and $\bar{\mu}^{\prime}$ scheme. It is proved from the path integral angle that the quantum dynamic lacks the full invariance with respect to fiducial cell scaling in the $\bar{\mu}$ scheme, but it does not in the $\bar{\mu}^{\prime}$ scheme. The investigation affirms the equivalence of the canonical approach and the path integral approach in loop quantum cosmology.