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arXiv:1312.1229

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

Gerard ’t Hooft

Published Dec 4, 2013• 2 citations
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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.

DOI
10.48550/arxiv.1312.1229

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The Discretized Hamiltonian Formalism in PQ PQ\mathit{PQ}PQ Theory
In standard quantum mechanics, the construction of the Schrödinger equation requires a Hamiltonian operator, which must obey some important restrictions. Most of our deterministic models are, to some extent, discrete. How do we carry over the standard Hamiltonian formalism to a discretized world? As in the continuum case, we may assume that conjugate pairs ( Pi;QiP_{i}; Q_{i}Pi​;Qi​ ) can be defined, but now PiP_{i}Pi​ and QiQ_{i}Qi​ are integers. How can a Hamiltonian function H(Q⃗;P⃗)H(\vec{Q}; \vec{P})H(Q​;P) be used to define an evolution law?
Discrete quantum mechanics is here defined to be a quantum theory of wave functions defined on integers P_i and Q_i, while canonical quantum mechanics is assumed to be based on wave functions on the real numbers, R^n. We study reversible mappings from the position operators q_i and their quantum canonical operators p_i of a canonical theory, onto the discrete, commuting operators Q_i and P_i. In this paper we are particularly interested in harmonic oscillators. In the discrete system, these turn into deterministic models, which is our motivation for this study. We regard the procedure worked out here as a "canonical formalism" for discrete dynamics, and as a stepping stone to handling discrete deterministic systems in a quantum formalism.
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