Constraint analysis for variational discrete systems

TL;DR

This paper develops a canonical formalism and constraint analysis for variational discrete systems, clarifying the roles of constraints, observables, and symplectic structure, applicable to various fields including quantum gravity and numerical analysis.

Contribution

It introduces a comprehensive formalism for discrete systems with evolving phase spaces, unifying covariant and canonical approaches, and clarifies the nature of constraints and observables in such systems.

Findings

Constraints depend on initial and final steps in evolving phase spaces.

The formalism applies to discrete mechanics, lattice field theory, and quantum gravity.

Observable definitions require initial and final step specification.

Abstract

A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves and naturally incorporates both constant and evolving phase spaces, the latter of which is necessary for a time varying discretization. The different roles of constraints in the discrete and the conditions under which they are first or second class and/or symmetry generators are clarified. The (non-) preservation of constraints and the symplectic structure is discussed; on evolving phase spaces the number of constraints at a fixed time step depends on the initial and final time step of evolution. Moreover, the definition of observables and a reduced phase space is provided; again, on evolving phase spaces the notion of an observable as a propagating degree of freedom requires specification of an initial and final step and crucially depends on this choice, in contrast to the continuum. However, upon restriction to translation invariant systems, one regains the usual time step independence of canonical concepts. This analysis applies, e.g., to discrete mechanics, lattice field theory, quantum gravity models and numerical analysis.