TIME QUANTIZATION AND q-DEFORMATIONS

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] 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 gave rise to a representation of Levy processes, a family of translation invariant jump processes, as subordinated Brownian motions whereby the time change is uncorrelated to the underlying process. More generally, Monroe proved that all semimartingales can be represented as time-changed Brownian motions, as long as one allows for the subordinator to be correlated. In this paper we bring together ideas from all these lines of research and show that one can interpret qdeformations in terms of stochastic time changes, albeit of a new type which is designed in such a way to preserve quantum probability. We find that these representations provide new insights in the notion of qdeformation and indicate an alternative, physically intuitive path to understand short-scale deformations of quantum field theory. Stochastic subordination is a procedure to construct a stochastic process from another by means of a stochastic time change. If Xt is a stochastic process, the subordinated process X̃t is defined as follows: