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(Broken) Gauge Symmetries and Constraints in Regge Calculus

Benjamin Bahr, Bianca Dittrich

TL;DR

This work analyzes whether discrete Regge calculus preserves diffeomorphism invariance, showing that in 4d with curvature exact discrete gauge symmetries are generically broken. It develops a canonical formulation that mirrors covariant dynamics and reveals pseudo constraints when gauge symmetries are broken, clarifying the relationship between covariant (spin-foam) and canonical (loop quantum gravity) approaches. Using tent moves, the authors derive local evolution and explore how exact constraints arise if the action has exact gauge symmetry, or how pseudo constraints depend on future data otherwise. They also study 3d Regge calculus with a cosmological constant to illustrate limits where constraints can persist at first order and discuss discrete reparametrization-invariant systems to highlight when gauge symmetry can be maintained in discretizations. The results illuminate how symmetry breaking affects discretized gravity and suggest directions (fat triangulations, perfect actions, or summed triangulations) to recover or approximate diffeomorphism invariance in quantum gravity models.

Abstract

We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so--called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.

(Broken) Gauge Symmetries and Constraints in Regge Calculus

TL;DR

This work analyzes whether discrete Regge calculus preserves diffeomorphism invariance, showing that in 4d with curvature exact discrete gauge symmetries are generically broken. It develops a canonical formulation that mirrors covariant dynamics and reveals pseudo constraints when gauge symmetries are broken, clarifying the relationship between covariant (spin-foam) and canonical (loop quantum gravity) approaches. Using tent moves, the authors derive local evolution and explore how exact constraints arise if the action has exact gauge symmetry, or how pseudo constraints depend on future data otherwise. They also study 3d Regge calculus with a cosmological constant to illustrate limits where constraints can persist at first order and discuss discrete reparametrization-invariant systems to highlight when gauge symmetry can be maintained in discretizations. The results illuminate how symmetry breaking affects discretized gravity and suggest directions (fat triangulations, perfect actions, or summed triangulations) to recover or approximate diffeomorphism invariance in quantum gravity models.

Abstract

We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so--called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.

Paper Structure

This paper contains 15 sections, 79 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definition of gauge symmetry
  3. Tent moves
  4. Evolution equations for tent moves
  5. Remarks on discrete ambiguities of solutions to the Regge equations
  6. Construction of a Regge solution via tent moves
  7. Canonical analysis for 3d Regge calculus with cosmological constant
  8. Canonical analysis of a tent move
  9. Relation between symmetries of the action and constraints in the canonical framework
  10. Discrete reparametrization invariant systems
  11. Discussion
  12. An example for physically different solutions with the same boundary conditions
  13. One tent move
  14. Two tent moves
  15. Geometric relations in simplices

Figures (15)

  • Figure 1: A $5$-valent tent move at the vertex $v$.
  • Figure 2: The 3d boundary of the tent move triangulation.
  • Figure 3: The lowest eigenvalue of the Hessian as a function of the deviation parameter $x$.
  • Figure 4: The deficit angle at the triangle $v^0v^1i$ as a function of the deviation parameter $x$.
  • Figure 5: The eigenvalue corresponding to translations in tent-pole direction as a function of $\lambda=0.01\times x$-axis label
  • ...and 10 more figures