Dirac-type approach for consistent discretizations of classical constrained theories

Di Bartolo C., Gambini R., Porto R.A.

We analyze the canonical treatment of classical constrained mechanical systemsformulated with a discrete time. We prove that under very general conditions, it ispossible to introduce nonsingular canonical transformations that preserve the constraintsurface and the Poisson or Dirac bracket structure. The conditions for thepreservation of the constraints are more stringent than in the continuous case and asa consequence some of the continuum constraints become second class upon discretizationand need to be solved by fixing their associated Lagrange multipliers.The gauge invariance of the discrete theory is encoded in a set of arbitrary functionsthat appear in the generating function of the evolution equations. The resultingscheme is general enough to accommodate the treatment of field theories on thelattice. This paper attempts to clarify and put on sounder footing a discretizationtechnique that has already been used to treat a variety of systems, including Yang-Mills theories, BF theory, and general relativity on the lattice