Diffusive motion--displacement due to the cumulative effect of irregular
fluctuations--has been a fundamental concept in mathematics and physics
since Einstein's work on Brownian motion. It is also relevant to
understanding various aspects of quantum theory. This book explains
diffusive motion and its relation to both nonrelativistic quantum theory
and quantum field theory. It shows how diffusive motion concepts lead to
a radical reexamination of the structure of mathematical analysis. The
book's inspiration is Princeton University mathematics professor Edward
Nelson's influential work in probability, functional analysis,
nonstandard analysis, stochastic mechanics, and logic. The book can be
used as a tutorial or reference, or read for pleasure by anyone
interested in the role of mathematics in science. Because of the
application of diffusive motion to quantum theory, it will interest
physicists as well as mathematicians.
The introductory chapter describes the interrelationships between the
various themes, many of which were first brought to light by Edward
Nelson. In his writing and conversation, Nelson has always emphasized
and relished the human aspect of mathematical endeavor. In his
intellectual world, there is no sharp boundary between the mathematical,
the cultural, and the spiritual. It is fitting that the final chapter
provides a mathematical perspective on musical theory, one that reveals
an unexpected connection with some of the book's main themes.
