Dirac-like approach for consistent discretizations of classical constrained theories

TL;DR

This paper develops a Dirac-like discretization method for classical constrained systems, ensuring the preservation of constraints and gauge invariance, applicable to lattice field theories like Yang-Mills and gravity.

Contribution

It introduces a general discretization scheme that maintains the structure of constrained theories, clarifying conditions for preserving constraints and gauge invariance in discrete time.

Findings

Canonical transformations preserve the constraint surface in discrete systems.

Some continuum constraints become second class after discretization.

The scheme applies to lattice field theories like Yang-Mills and gravity.

Abstract

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