Unimodular loop quantum gravity and the problems of time

TL;DR

This paper develops a framework for quantizing unimodular gravity within loop quantum gravity, showing that the effective quantum geometry is unaffected by certain energy-momentum tensor terms and exploring the concept of physical time in quantum evolution.

Contribution

It introduces a unimodular quantization approach in loop quantum gravity, extending previous work to incorporate a physical time variable based on four-volume.

Findings

Quantum effective action remains unimodular.

Path integral form aligns with spin foam models.

Discussion on quantum evolution with physical time.

Abstract

We develop the quantization of unimodular gravity in the Plebanski and Ashtekar formulations and show that the quantum effective action defined by a formal path integral is unimodular. This means that the effective quantum geometry does not couple to terms in the expectation value of the energy-momentum tensor proportional to the metric tensor. The path integral takes the same form as is used to define spin foam models, with the additional constraint that the determinant of the four metric is constrained to be a constant by a gauge fixing term. We also review the proposal of Unruh, Wald and Sorkin- that the hamiltonian quantization yields quantum evolution in a physical time variable equal to elapsed four volume-and discuss how this may be carried out in loop quantum gravity. This then extends the results of arXiv:0904.4841 to the context of loop quantum gravity.