Classification of constraints and degrees of freedom for quadratic discrete actions

TL;DR

This paper classifies constraints and degrees of freedom in quadratic discrete systems, analyzing how discretization changes affect the system's dynamical content in classical and quantum contexts.

Contribution

It provides a comprehensive classification based on null vectors of the Lagrangian two-form, applicable to systems with varying or constant discretization, including evolving backgrounds.

Findings

Constraints and observables depend on discretization changes.

Discretization evolution affects phase space and Hilbert space structures.

Results are relevant for discrete gravity and field theories on evolving lattices.

Abstract

We provide a comprehensive classification of constraints and degrees of freedom for variational discrete systems governed by quadratic actions. This classification is based on the different types of null vectors of the Lagrangian two-form and employs the canonical formalism developed in arXiv:1303.4294 [math-ph] (J. Math. Phys. 54, 093505 (2013)) and arXiv:1401.6062 [gr-qc] (J. Math. Phys. 55, 083508 (2014)). The analysis is carried out in both the classical and quantum theory and applies to systems with both temporally varying or constant discretization. In particular, it is shown explicitly how changes in the discretization, e.g. resulting from canonical coarse graining or refining operations or an evolving background geometry, change the dynamical content of the system. It is demonstrated how, on a temporally varying discretization, constraints, Dirac observables, symmetries, reduced phase spaces and physical Hilbert spaces become spacetime region dependent. These results are relevant for free field theory on an evolving lattice and linearized discrete gravity models.