Chapter Three. Ed Nelson's Work in Quantum Theory

Barry Simon A

It is a pleasure to contribute to this celebration of Ed Nelson’s scientific work, not only because of the importance of that work but because it allows me an opportunity to express my gratitude and acknowledge my enormous debt to Ed. He and Arthur Wightman were the key formulative influences on my education, not only as a graduate student but during my early postdoctoral years. Thanks, Ed!

OpenStax

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.

Steve Adams

Jochen Rau

The quantum realm exhibits phenomena that run counter to conventional logic. This chapter discusses the evidence from a particularly simple class of experiments: measurements or combinations of measurements which each have only two possible outcomes. It reviews the classical notions of logic and pins down where exactly they fail in the quantum realm. In particular, it highlights the importance of the order in which measurements are performed. The experimental evidence also calls for a critical review of the rules of probability theory. For these, too, this chapter identifies those which continue to hold in the quantum realm and those which need to be modi ed. Moreover, it introduces the pivotal concept of a ‘state’.