Regge calculus in the canonical form

TL;DR

This paper reformulates (3+1) Regge calculus into Hamiltonian form, classifies constraints, and discusses classical and quantum implications, revealing that in empty space the dynamics are trivial with fixed spacelike links.

Contribution

It introduces a Hamiltonian formulation of Regge calculus using connection matrices and area tensors, with a quasipolynomial action, and analyzes the constraint classification changes.

Findings

Number of degrees of freedom equals the number of links in the 3D foliation

In empty space, the scale of timelike links becomes zero

Classical dynamics are trivial in empty space with fixed spacelike links

Abstract

(3+1) (continuous time) Regge calculus is reduced to Hamiltonian form. The constraints are classified, classical and quantum consequences are discussed. As basic variables connection matrices and antisymmetric area tensors are used supplemented with appropriate bilinear constraints. In these variables the action can be made quasipolinomial with $\arcsin$ as the only deviation from polinomiality. In comparison with analogous formalism in the continuum theory classification of constraints changes: some of them disappear, the part of I class constraints including Hamiltonian one become II class (and vice versa, some new constraints arise and some II class constraints become I class). As a result, the number of the degrees of freedom coincides with the number of links in 3-dimensional leaf of foliation. Moreover, in empty space classical dynamics is trivial: the scale of timelike links become zero and spacelike links are constant.