Diffusion, Quantum Theory, and Radically Elementary Mathematics

by Nelson, Edward; Faris, William G.

• ISBN13: 9780691125459
• ISBN10: 0691125457
• Format: Paperback
• Copyright: 2006-08-21
• Publisher: Princeton Univ Pr

Summary

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.

Table of Contents

Preface	ix
Chapter 1. Introduction: Diffusive Motion and Where It Leads
William G. Faris	1	(44)
Chapter 2. Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys
Leonard Gross	45	(30)
Chapter 3. Ed Nelson's Work in Quantum Theory
Barry Simon	75	(20)
Chapter 4. Symanzik, Nelson, and Self-Avoiding Walk
David C. Brydges	95	(22)
Chapter 5. Stochastic Mechanics: A Look Back and a Look Ahead
Eric Carlen	117	(24)
Chapter 6. Current Trends in Optimal Transportation: A Tribute to Ed Nelson
Cédric Villani	141	(16)
Chapter 7. Internal Set Theory and Infinitesimal Random Walks
Gregory F. Lawler	157	(26)
Chapter 8. Nelson's Work on Logic and Foundations and Other Reflections on the Foundations of Mathematics
Samuel R. Buss	183	(26)
Chapter 9. Some Musical Groups: Selected Applications of Group Theory in Music
Julian Hook	209	(20)
Chapter 10. Afterword
Edward Nelson	229	(4)
Appendix A. Publications
Edward Nelson	233	(8)
Index	241