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Fundamental theories of physics

The Discretized Hamiltonian Formalism in PQ $\mathit{PQ}$ Theory

Gerard ’t Hooft

Published Jan 1, 2016• 0 citations
Publisher

Abstract

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 ( $P_{i}; Q_{i}$ ) can be defined, but now $P_{i}$ and $Q_{i}$ are integers. How can a Hamiltonian function $H(\vec{Q}; \vec{P})$ be used to define an evolution law?

DOI
10.1007/978-3-319-41285-6_19

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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?
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.
What is badly needed is an elegant mathematical scheme that replaces position and momentum operators qqq and ppp by discrete, commuting counterparts, QQQ and PPP , in such a way that mappings can be defined between conventional, quantum mechanical systems and deterministic, discrete systems. Here the foundations for such a scheme is suggested. An essential role is played by the Jacobi theta functions.
A procedure allowing to construct rigorously discrete as well as continuum deterministic evolution equations from stochastic evolution equations is developed using a Dirac's bra and ket notation. This procedure is an extension of an approach previously used by the authors coined Discrete Stochastic Evolution Equations. Definitions and examples of discretes as well as continuum one-dimensional lattices are developed in detail in order to show the basic tools that allows to construct Schrödinger-like equations. Extension to d-dimensional lattices are studied in order to provide a wider exposition and the one-dimensional cases are derived as special cases, as expected. Some variants of the procedure allows to construct other evolution equations. Also, using a limiting procedure, it is possible to derive the Schrödinger equations from the Schrödinger like equations.
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.
Without wasting time and effort on philosophical justifications and implications, we write down the conditions for the Hamiltonian of a quantum system for rendering it mathematically equivalent to a deterministic system. These are the equations to be considered. Special attention is given to the notion of `locality'. Various examples are worked out, followed by a systematic procedure to generate classical evolution laws and quantum Hamiltonians that are exactly equivalent. What is new here is that we consider interactions, keeping them as general as we can. The quantum systems found, form a dense set if we limit ourselves to sufficiently low energy states. The class is discrete, just because the set of deterministic models containing a finite number of classical states, is discrete. In contrast with earlier suspicions, the gravitational force turns out not to be needed for this; it suffices that the classical system act at a time scale much smaller than the inverse of the maximum scattering energies considered.
Without wasting time and effort on philosophical justifications and implications, we write down the conditions for the Hamiltonian of a quantum system for rendering it mathematically equivalent to a deterministic system. These are the equations to be considered. Special attention is given to the notion of 'locality'. Various examples are worked out, followed by a systematic procedure to generate classical evolution laws and quantum Hamiltonians that are exactly equivalent. What is new here is that we consider interactions, keeping them as general as we can. The quantum systems found, form a dense set if we limit ourselves to sufficiently low energy states. The class is discrete, just because the set of deterministic models containing a finite number of classical states, is discrete. In contrast with earlier suspicions, the gravitational force turns out not to be needed for this; it suffices that the classical system act at a time scale much smaller than the inverse of the maximum scattering energies considered.