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arXiv:quant-ph/9509014 · Journal of Physics A Mathematical and General

Umbral calculus, discretization, and quantum mechanics on a lattice

Aristophanes Dimakis, Folkert Müller-Hoissen, T Striker

Published Nov 7, 1996• 31 citations
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Abstract

`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail.

DOI
10.1088/0305-4470/29/21/017

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`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of Lie algebras on a lattice. Related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus. This observation then suggests various generalizations of these examples. A special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail.
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schrödinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schrödinger equation with a potential depending only in the space variable is discussed.
The umbral calculus is an old mathematical tool that began its first steps in the XVII century [1]. Since the second half of XIX [2, 3] it was applied in a systematical way although until the second half of XX [4, 5] there was not a formal theory. Recently it has been used to provide discrete representations of canonical commutation relations [6]–[12]. In this approach it can be used to map equations and solutions from a continuous frame to a discrete one. The theory works very well for polynomials, however this is not the case for its extension to non-polynomial functions. A complicated, and then interesting, problem in this formalism is the harmonic oscillator. Due to the non preservation of the complete set of symmetries after discretization, some divergences appear. To second physical solutions of the discrete harmonic oscillator we use a non standard form of the umbral calculus discretization. Some surprising results are obtained.
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.
The set of all pairs of polynomials X(x,p) and P(x,p) in the position x and momentum p are found such that X and P have the same commutator as x and p. Several other types of representations of the commutation relations are studied. The reducibility, time, and space inversion properties of these are discussed. The unitary transformations connecting different representations are exhibited. The use of different representations to solve scattering and bound state problems is indicated. The application of one of the transformations to a partial diagonalization of a perturbed harmonic oscillator Hamiltonian is made.
"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.