Fractals and the quantum classical boundary

Early in this century, the arrival of quantum mechanics resulted in a split in the world views of physicists. In classical mechanics the apparent correspondence between mathematical models and their physical counterparts was often so close that little distinction needed to be made between the two. In contrast, quantum mechanics provides a beautifully accurate description of Nature, but itself yields little explanation of microscopic phenomena. An understanding of the mathematics of quantum mechanics leads one to an ability to calculate and predict, but few would argue that it affords a deep understanding of the phenomena being described. In the quantum world, the Heisenberg uncertainty relations seem to put a fundamental limit on what is observable, and until recently, strongly represented the conceptual barrier separating the quantum and classical worlds. However, it may be shown that by extending classical mechanics to allow Fractal trajectories, the uncertainty relations and some of the dynamical equations of quantum mechanics appear in this extended classical domain. This shift of the boundary between the quantum and classical world will be discussed at a general level and illustrated by some exactly solvable statistical mechanical models. (C) 1999 Published by Elsevier Science Ltd. All rights reserved.