Quantum Mechanics with Difference Operators
Abstract
A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle and its N-point discretisation; the connection to q-deformations of the Witt algebra is discussed. For a "natural" choice of the q-kinematics a corresponding q-difference evolution equation is obtained. This study shows general difficulties for a generalisation of a physical theory from a known one to a "new" framework.
Cite
@article{arxiv.quant-ph/0207077,
title = {Quantum Mechanics with Difference Operators},
author = {V. K. Dobrev and H. -D. Doebner and R. Twarock},
journal= {arXiv preprint arXiv:quant-ph/0207077},
year = {2009}
}
Comments
19 pages, LaTeX, using packages: pictex, amssymb; v.2: small corrections, to appear in Rep. Math. Phys