Path Integrals and the WKB approximation in Loop Quantum Cosmology

TL;DR

This paper develops a path integral formulation for loop quantum cosmology, revealing how quantum geometry effects alter the measure and affect the WKB approximation, offering new insights into differences from Wheeler-DeWitt theory.

Contribution

It introduces a path integral approach to loop quantum cosmology starting from the Hilbert space, highlighting how quantum geometry modifies the measure and impacts the WKB approximation.

Findings

Quantum geometry effects modify the path integral measure.

The WKB approximation remains well-defined in loop quantum cosmology.

Differences between loop quantum cosmology and Wheeler-DeWitt theory are clarified.

Abstract

We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.