Quantum Mechanics with Difference Operators

Details Quantum Mechanics with Difference Operators Quantum Mechanics with Difference Operators

2002-07-13

arxiv.org/abs/quant-ph/0207077v2

Rep. Math. Phys. 50 (2002) 409-431

Physics

Quantum Physics

19 pages, LaTeX, using packages: pictex, amssymb; v.2: small corrections, to appear in Rep. Math. Phys

Scientific paper

10.1016/S0034-4877(02)80069-6

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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