Discretisations, Constraints and Diffeomorphisms in Quantum Gravity

TL;DR

This paper reviews how discretization methods, constraint implementation, and diffeomorphism symmetry are interconnected in Loop Quantum Gravity and Spin Foam models, focusing on approaches that preserve symmetry and construct the physical Hilbert space.

Contribution

It compares the Consistent Discretizations and Perfect Actions approaches, highlighting their roles in maintaining diffeomorphism symmetry in quantum gravity models.

Findings

Consistent Discretizations effectively implement constraints in canonical quantum gravity.

Perfect Actions aim to find path integral measures that respect diffeomorphism symmetry.

The review clarifies the relationship between discretization schemes and symmetry preservation.

Abstract

In this review we discuss the interplay between discretization, constraint implementation, and diffeomorphism symmetry in Loop Quantum Gravity and Spin Foam models. To this end we review the Consistent Discretizations approach, which is an application of the master constraint program to construct the physical Hilbert space of the canonical theory, as well as the Perfect Actions approach, which aims at finding a path integral measure with the correct symmetry behavior under diffeomorphisms.