Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice

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.

19 pages, Latex

References in corpus (1)

• Lie-algebraic discretization of differential equations

Cited by in corpus (19)

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• On umbral extensions of Stirling numbers and Dobinski-like formulas
• Lattice Gauge Fields and Noncommutative Geometry
• First Observations on Prefab Posets Whitney Numbers
• On Pairs of Difference Operators Satisfying: [P,Q] = Id
• Graded posets zeta matrix formula
• Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
• Integrable maps from Galois differential algebras, Borel transforms and number sequences
• First contact remarks on umbra difference calculus references streams
• Special Functions of Hypercomplex Variable on the Lattice Based on SU(1,1)
• Umbral Deformations on Discrete Spacetime
• Umbral Vade Mecum
• More on the Bernoulli and Taylor Formula for Extended Umbral Calculus
• Almansi Theorems in Umbral Clifford Analysis and the Quantum Harmonic Oscillator
• Ivan Bernoulli Series Universalissima
• Quantum mechanics and umbral calculus