Models of Discrete Linear Evolution for Quantum Systems

TL;DR

This paper explores discrete linear evolution models for quantum systems, focusing on irregular evolution, constraints, and non-unitary quantum dynamics, with applications to a scalar field on a spacetime lattice.

Contribution

It introduces two distinct approaches to quantizing irregular discrete linear systems, including detailed evolution maps and path integral formulations.

Findings

Irregular discrete systems may be best described by non-unitary quantum evolution.

The formalism is applied to a massless scalar field on a 2D lattice.

The approaches handle constraints and non-uniqueness in solutions.

Abstract

Discrete canonical evolution is a key tool for understanding the dynamics in discrete models of spacetime, in particular those represented by a triangular Regge lattice. We consider a finite-dimensional system whose evolution is realized by a series of discrete-time evolution steps governed by Hamiltonian equations of motion that are linear in the canonical coordinates. The evolution is allowed to be irregular, which produces constraints as well as non-uniqueness of solutions. We provide two independent and fundamentally different approaches to canonical quantization of this system, including detailed description of the evolution maps, measurement and path integrals. It is argued that some irregular discrete systems may be most naturally described by a non-unitary quantum evolution. The formalism is then applied to a simple yet physically relevant model of massless scalar field on a two-dimensional spacetime lattice.