Canonical linearized Regge calculus: Counting lattice gravitons with Pachner moves
Abstract
We afford a systematic and comprehensive account of the canonical dynamics of 4D Regge calculus perturbatively expanded to linear order around a flat background. To this end, we consider the Pachner moves which generate the most basic and general simplicial evolution scheme. The linearized regime features a vertex displacement (diffeomorphism) symmetry for which we derive an Abelian constraint algebra. This permits us to identify gauge invariant lattice "gravitons" as propagating curvature degrees of freedom. The Pachner moves admit a simple method to explicitly count the gauge and graviton degrees of freedom on an evolving triangulated hypersurface, and we clarify the distinct role of each move in the dynamics. It is shown that the 1-4 move generates four "lapse and shift" variables and four conjugate vertex displacement generators; the 2-3 move generates a graviton; the 3-2 move removes one graviton and produces the only non-trivial equation of motion; and the 4-1 move removes four lapse and shift variables and trivializes the four conjugate symmetry generators. It is further shown that the Pachner moves preserve the vertex displacement generators. These results may provide new impetus for exploring `graviton dynamics' in discrete quantum gravity models.
- Publication:
-
Physical Review D
- Pub Date:
- June 2015
- DOI:
- 10.1103/PhysRevD.91.124034
- arXiv:
- arXiv:1411.5672
- Bibcode:
- 2015PhRvD..91l4034H
- Keywords:
-
- 04.60.-m;
- 04.60.Pp;
- 04.20.Fy;
- 11.15.Bt;
- Quantum gravity;
- Loop quantum gravity quantum geometry spin foams;
- Canonical formalism Lagrangians and variational principles;
- General properties of perturbation theory;
- General Relativity and Quantum Cosmology;
- High Energy Physics - Lattice;
- Mathematical Physics
- E-Print:
- 26+13 pages, 2 appendices, many figures. References updated, small clarifications added. This article is fairly self-contained