Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discretisations, Constraints and Diffeomorphisms in Quantum Gravity

Benjamin Bahr, Rodolfo Gambini, Jorge Pullin

TL;DR

The article addresses how discretization affects diffeomorphism symmetry in quantum gravity by contrasting canonical (LQG) and covariant (Spin Foam/Regge) formalisms. It analyzes two main strategies: Consistent/Uniform Discretizations and Master Constraint methods to control constraints and approach the correct physical Hilbert space, and Perfect Actions with coarse-graining/renormalization to restore continuum diffeomorphism invariance in the path integral. Through mechanical toy models and gravity-inspired cases, it shows when zero-eigenvalue (physical) sectors arise or when fundamental discreteness can persist, and it discusses the feasibility and implications of RG fixed points for diffeomorphism symmetry in 4d gravity. The work emphasizes that restoring continuum gauge symmetry in discrete quantum gravity likely hinges on renormalization group insights and selective coarse-graining, with significant open questions in 4d GR and spin-foam renormalization. These approaches collectively illuminate how discretization artifacts can be tamed to yield a viable quantum gravity theory with the correct symmetry structure.

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.

Discretisations, Constraints and Diffeomorphisms in Quantum Gravity

TL;DR

The article addresses how discretization affects diffeomorphism symmetry in quantum gravity by contrasting canonical (LQG) and covariant (Spin Foam/Regge) formalisms. It analyzes two main strategies: Consistent/Uniform Discretizations and Master Constraint methods to control constraints and approach the correct physical Hilbert space, and Perfect Actions with coarse-graining/renormalization to restore continuum diffeomorphism invariance in the path integral. Through mechanical toy models and gravity-inspired cases, it shows when zero-eigenvalue (physical) sectors arise or when fundamental discreteness can persist, and it discusses the feasibility and implications of RG fixed points for diffeomorphism symmetry in 4d gravity. The work emphasizes that restoring continuum gauge symmetry in discrete quantum gravity likely hinges on renormalization group insights and selective coarse-graining, with significant open questions in 4d GR and spin-foam renormalization. These approaches collectively illuminate how discretization artifacts can be tamed to yield a viable quantum gravity theory with the correct symmetry structure.

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.

Paper Structure

This paper contains 17 sections, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Canonical approach
  3. Covariant approach
  4. Consistent discretisation
  5. Systems with discrete time
  6. Uniform discretizations
  7. Examples
  8. Model with continuum limit
  9. A model with non-compact gauge group
  10. Other examples
  11. Improved and perfect actions
  12. Diffeo breaking in Regge
  13. The coarse graining idea: classical
  14. Feasibility
  15. Coarse graining: quantum
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Deficit angle $\epsilon_h$ situated at a $(d-2)$-dimensional simplex (hinge) $h$.
  • Figure 2: The lowest eigenvalue of the Hessian as a function of deviation from flatness (proportional to $\epsilon$), in a particular $4d$-triangulation with one inner vertex, discussed in bahrdittrich1.
  • Figure 3: A tetrahedron $\Sigma$ subdivided into smaller tetrahedra $\sigma$. The length $L_E$ of a coarse edge $E$ is the sum of all the lengths $l_e$ of edges $e\subset E$ that constitute $E$.