Hamiltonian formalism for integer-valued variables and integer time steps and a possible application in quantum physics

TL;DR

This paper develops a Hamiltonian formalism for systems with integer-valued variables and discrete time, providing a framework that converges to classical Hamilton equations in the continuum limit, with potential applications in quantum physics.

Contribution

It introduces a novel Hamiltonian formalism for discrete integer variables and time steps, extending classical mechanics concepts to discrete systems.

Findings

Formalism recovers classical Hamilton equations in the continuum limit

Provides a new perspective on deterministic quantum systems

Enables energy conservation in discrete variable systems

Abstract

Most classical mechanical systems are based on dynamical variables whose values are real numbers. Energy conservation is then guaranteed if the dynamical equations are phrased in terms of a Hamiltonian function, which then leads to differential equations in the time variable. If these real dynamical variables are instead replaced by integers, and also the time variable is restricted to integers, it appears to be hard to enforce energy conservation unless one can also derive a Hamiltonian formalism for that case. We here show how the Hamiltonian formalism works here, and how it may yield the usual Hamilton equations in the continuum limit. The question was motivated by the author's investigations of special quantum systems that allow for a deterministic interpretation. The 'discrete Hamiltonian formalism' appears to shed new light on these approaches.