Approximating the physical inner product of Loop Quantum Cosmology

TL;DR

This paper explores methods to approximate the physical inner product in Loop Quantum Cosmology, demonstrating that coherent states can effectively replicate analytical solutions and potentially aid in more complex quantum gravity theories.

Contribution

It introduces a novel approach using complexifier coherent states to approximate the physical inner product in a simple cosmological model, validating the method against analytical results.

Findings

Coherent states successfully approximate the analytic physical inner product.

The approximation method is promising for more complex quantum gravity models.

Analytical and numerical approaches are consistent in this context.

Abstract

In this article, we investigate the possibility of approximating the physical inner product of constrained quantum theories. In particular, we calculate the physical inner product of a simple cosmological model in two ways: Firstly, we compute it analytically via a trick, secondly, we use the complexifier coherent states to approximate the physical inner product defined by the master constraint of the system. We will find that the approximation is able to recover the analytic solution of the problem, which solidifies hopes that coherent states will help to approximate solutions of more complicated theories, like loop quantum gravity.