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Time Quantization and q-deformations Claudio Albanese and Stephan Lawi

Summary: Time Quantization and q-deformations
Claudio Albanese and Stephan Lawi
Department of Mathematics, 100 St. George Street, University of Toronto, M5S 3G3,
Toronto, Canada
E-mail: albanese@math.toronto.edu, slawi@math.toronto.edu
We extend to quantum mechanics the technique of stochastic subordination, by
means of which one can express any semi-martingale as a time-changed Brownian
motion. As examples, we considered two versions of the q-deformed Harmonic oscillator
in both ordinary and imaginary time and show how these various cases can be
understood as different patterns of time quantization rules.
Submitted to: J. Phys. A: Math. Gen.
1. Introduction
In a search to unravel the fabric of space at short distances, many authors have
explored variations on ordinary quantum mechanics based on q-deformations of the
canonical commutation relation, q being a parameter in the interval (0, 1) where 1
corresponds to the Bose limit, see for instance [1], [2], [4], [5], [6], [7], [8], [9], [10]. Time
quantization was considered also, see [3], [13]. On an entirely different line of research,
probabilists developed the notion of stochastic time changes (also called stochastic
subordination) as a way of understanding jump processes, see [11], [12], [14]. This work


Source: Albanese, Claudio - Department of Mathematics, King's College London


Collections: Mathematics