Dirac-like approach for consistent discretizations of classical constrained theories

TL;DR

This work develops a Dirac-like canonical framework for classical constrained systems discretized in time, proving that under broad conditions one can implement nonsingular canonical transformations that preserve the constraint surface and the Poisson/Dirac bracket structure. Discretization introduces stricter constraint preservation, potentially turning some continuum first-class constraints into second-class that must be fixed via Lagrange multipliers, while gauge invariance is encoded by arbitrary functions in the evolution-generating function. The authors classify constraints into first- and second-class in the discrete setting, illustrate the method with a detailed second-class constraint example, and extend the formalism to Type II generating functions, including pseudo-constraints. The framework is general enough to accommodate lattice gauge theories and field theories, providing a robust, quantization-ready approach to discrete constrained dynamics and clarifying discretization techniques used in Yang–Mills, BF theory, and lattice 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.