Fractals and the Quantum Classical Boundary

Abstract

Early in this century, the arrival of quantum mechanics resulted in a split in the worldviews of physicists. In classical mechanics the apparent correspondence between mathematicalmodels and their physical counterparts was often so close that little distinction needed to be madebetween the two. In contrast, quantum mechanics provides a beautifully accurate description ofNature, but itself yields little explanation of microscopic phenomena. An understanding of themathematics of quantum mechanics leads one to an ability to calculate and predict, but few wouldargue that it affords a deep understanding of the phenomena being described. In the quantum world, the Heisenberg uncertainty relations seem to put a fundamentallimit on what is observable, and until recently, strongly represented the conceptual barrierseparating the quantum and classical worlds. However, it may be shown that by extendingclassical mechanics to allow Fractal trajectories, the uncertainty relations and some of thedynamical equations of quantum mechanics appear in this extended classical domain. This shift ofthe boundary between the quantum and classical world will be discussed at a general level andillustrated by some exactly solvable statistical mechanical models.