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Umbral calculus, discretization, and quantum mechanics on a lattice

A Dimakis-+, F Müller-Hoissen++ and T Striker++

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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.


PACS

03.65.Fd Algebraic methods

05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

02.10.Ud Linear algebra

MSC

05A40 Umbral calculus

81S05 Commutation relations and statistics

Subjects

Mathematical physics

Statistical physics and nonlinear systems

Quantum information and quantum mechanics

Dates

Issue 21 (7 November 1996)

Received 24 June 1996



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