Edward Nelson: 1995 Steele Prize for Seminal Contribution to Research

The 1995 Leroy P Steele award for research of seminal importance was awarded to Professor Edward Nelson of Princeton University for the following two papers in mathematical physics characterised by leaders of the field as extremely innovative:

"A quartic interaction in two dimensions" in Mathematical Theory of Elementary Particles, MIT Press, 1966, pages 69-73;
"Construction of quantum fields from Markoff fields" in Journal of Functional Analysis 12 (1973), 97-112.

In these papers he showed for the first time how to use the powerful tools of probability theory to attack the hard analytic questions of constructive quantum field theory, controlling renormalisations with

L^{p}

estimates in the first paper and, in the second, turning Euclidean quantum field theory into a subset of the theory of stochastic processes.

Citation

The interaction of mathematics with relativistic quantum field theory is, in many respects, one of the signal mathematical developments of the second half of this century. Edward Nelson was one of the pioneers in this development. From the earliest attempts to turn quantum field theory into rigorous mathematics, it had been clear that operator algebras and distribution theory would play prominent roles. What Nelson realised, and implemented in these two fundamental papers, is that probabilistic techniques could provide critical additional tools. In the first of the two papers recognised by the award, Nelson overcame the infinities associated with Wick-ordering renormalisation in two-dimensional field theories by a combination of measure theory and Lp estimates of semigroups. The techniques that he introduced for establishing in two dimensions the stability of the quartic interaction were fundamental, strongly influencing the further development by Glimm and Jaffe of rigorous quantum field theory in dimension three. They continue to be pertinent today in, for example, the theory of the nonlinear Schrödinger equation. Renormalisation modifies a formal, nominally positive fourth power by the subtraction of an infinite constant, so that the positivity of the result was not at all clear, which was contested at the time the paper appeared. Nelson resolved the controversy, and in so doing devised the mathematical tools later generalised to a much larger class of Hamiltonians.

In the second paper recognised by the award, Nelson fired one of the first shots in what became known as the Euclidean revolution. The analytic continuation of relativistic field theory to imaginary time transforms formally the Minkowskian field theory into a Euclidean theory. Nelson realised that this was not only a formal trick, but provided a mathematical interpretation of certain stochastic processes. The concepts he introduced thus furnished a mathematically rigorous approach that combined the operator formalism in Minkowski space with the use of a Markov property symmetric with respect to space and time.

Response

I was introduced to probability theory in a graduate course taught by Irving Segal from galley proofs of Doob's "Stochastic Processes". Irving presented his own viewpoint in addition to Doob's, and it was an exciting course. Once he drove me down to Urbana so we could talk with Doob. It was a memorable trip. Maintaining that the probability of an accident is directly proportional to the time spent on the road, Irving drove in such a way as to minimise that time.

Despite having Irving Segal as thesis adviser, I did not learn physics at the University of Chicago. I took one course in the physics department but was defeated by the lab; I didn't really know how to explain the 457 percent error in my result for the mechanical equivalent of heat. But when I got to Princeton University, I attended several of Arthur Wightman's courses and pored over the papers of Richard Feynman and Kurt Symanzik, and after a while I began to learn the difference between a Lagrangian and a Hamiltonian.

In the first cited paper I put the field in a spatial box and proved that certain operators were bounded. But it was James Glimm who then proved that the bound is in fact 1, a result essential to removing the box. In the sequel to the second paper, when I studied the free Markov field, I omitted to refer to the work of Loren Pitt, who first introduced this field and proved the Markov property for it. He had sent me a preprint, but when I wrote the paper, I did not consciously remember it - these things can happen. No one who knows Loren will be surprised to hear that when I apologised to him, he was very gracious indeed.

One pleasant feature of receiving this prize is that it reminds me of how much fun I had working on those problems, almost as much fun as I am having now in my work. My advice to any young mathematician approaching the age of fifty who wants to continue having fun doing mathematics is this: change field.

Now I come to the main point, which is to express my thanks to the AMS and the Selection Committee for this Steele Prize. It was a great surprise to me. (Notice the shade of difference between that statement and "The committee made a very surprising choice.") It is a great honour, it is great fun, and I am grateful. Thank you. Also, thanks to the AMS for reserving a room for us with a jacuzzi.

Biographical Sketch

Edward Nelson was born 4 May 1932, in Decatur, Georgia. After first grade in Rome, Italy, he returned to Georgia in September 1939 and moved to New York in 1942, where children from Georgia were put back a half grade and required to undergo speech therapy. After secondary schooling at the Bronx High School of Science and the Liceo Scientifico Giovanni Verga in Rome, Nelson enrolled at the University of Chicago, where he obtained a Ph.D. in 1955 with a thesis on Markov processes written under Irving Segal.

Nelson worked two years as a conscientious objector in the Methodist Hospital of Gary, Indiana, and then spent three years at the Institute for Advanced Study. Since 1959 he has been at Princeton University, where he served the mathematics department for six years as director of graduate studies and now as webmaster.

His wife of thirty-five years, Nancy Wong Nelson, died in 1988. Since 1990 he has been married to Sarah Jones Nelson. He has two children and three grandchildren.

Nelson is a member of the American Academy of Arts and Sciences and doctor honoris causa of the Université Louis Pasteur in Strasbourg. His current research interests are logic and foundations.

Last Updated September 2025