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Quantum Mechanics with Difference Operators

V. K. Dobrev·H. -D. Doebner·R. Twarock
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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.

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V. K. Dobrev, H. -D. Doebner, R. Twarock. 2002-08-06. Quantum Mechanics with Difference Operators. https://doi.org/10.1016/s0034-4877(02)80069-6

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