Discrete Linear Canonical Evolution

TL;DR

This paper extends discrete canonical evolution models to linear dynamical systems with irregular equations, analyzing constraints and symplectic structure, and applies the formalism to a scalar field on a lattice.

Contribution

It introduces a framework for analyzing irregular discrete linear systems, revealing their internal structure and simplifying their evolution map.

Findings

Constraints and symplectic form analyzed

Adjusted coordinates uncover phase space structure

Application to scalar field on lattice demonstrates formalism

Abstract

This work builds on an existing model of discrete canonical evolution and applies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $ and linear equations of motion. The system is assumed to evolve in discrete time steps. The most distinctive feature of the model is that the equations of motion can be irregular. After an analysis of the arising constraints and the symplectic form, we introduce adjusted coordinates on the phase space which uncover its internal structure and result in a trivial form of the Hamiltonian evolution map. For illustration, the formalism is applied to the example of massless scalar field on a two-dimensional spacetime lattice.