Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice

Details Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice

1995-09-21

arxiv.org/abs/quant-ph/9509014v1

J.Phys.A29:6861-6876,1996

Physics

Quantum Physics

19 pages, Latex

Scientific paper

10.1088/0305-4470/29/21/017

`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.

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