On kinematical constraints in Regge calculus

TL;DR

This paper investigates the kinematical constraints in (3+1)D Hamiltonian Regge calculus, showing that certain derivative-containing constraints are consequences of equations of motion and can be omitted, simplifying the formulation.

Contribution

It demonstrates that derivative-containing kinematical constraints are derivable from equations of motion and do not introduce new dynamical variables in Regge calculus.

Findings

Derivative constraints are consequences of equations of motion.

Omitting these constraints does not affect the set of dynamical variables.

Simplifies the Hamiltonian formulation of Regge calculus.

Abstract

In the (3+1)D Hamiltonian Regge calculus (one of the coordinates, $ t$, is continuous) conjugate variables are (defined on triangles of discrete 3D section $ t=const$) finite connections and antisymmetric area bivectors. The latter, however, are not independent, since triangles may have common edges. This circumstance can be taken into account with the help of the set of kinematical (that is, required to hold by definition of Regge manifold) bilinear constraints on bivectors. Some of these contain derivatives over $ t$, and taking them into account with the help of Lagrange multipliers would result in the new dynamical variables not having analogs in the continuum theory. It is shown that kinematical constraints with derivatives are consequences of eqs. of motion for Regge action supplemented with the rest of these constraints without derivatives and can be omitted; so the new dynamical variables do not appear.