Quantum mechanics with difference operators

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 transla...

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Bibliographic Details
Published in:Reports on mathematical physics 2002-12, Vol.50 (3), p.409-431
Main Authors: Dobrev, V.K, Doebner, H.-D, Twarock, R
Format: Article
Language:English
Subjects:
discrete derivative
nonlinear Schrödinger equation
q- deformation
Witt algebra
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Summary: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.
ISSN:0034-4877
1879-0674
DOI:10.1016/S0034-4877(02)80069-6