 
Summary: Time Quantization and qdeformations
Claudio Albanese and Stephan Lawi
Department of Mathematics, 100 St. George Street, University of Toronto, M5S 3G3,
Toronto, Canada
Email: albanese@math.toronto.edu, slawi@math.toronto.edu
Abstract.
We extend to quantum mechanics the technique of stochastic subordination, by
means of which one can express any semimartingale as a timechanged Brownian
motion. As examples, we considered two versions of the qdeformed 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 qdeformations 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
