In the early 1950s as a young instructor in the Princeton Department of Physics, Arthur Strong Wightman undertook the study of a major problem. Fundamental physics has been discussed in terms of quantum field theory, in which were united two great revolutions of early twentieth century physics -- special relativity and quantum theory. But what IS a quantized field? Furthermore: can the answer be given in mathematically lucid and consistent terms allowing the presentation of various fundamental physical laws as mathematical implications following from a few basic principles of the theory?
The first major step, achieved by Wightman in collaboration with the Swedish mathematician Lars Gårding, was to produce a good working formulation, in terms of precise mathematical notions, of what a quantum field should be, and to formulate a set of basic axioms which may be required of such an object. Thus was born axiomatic quantum field theory.
Two tests for any such axiomatic theory would be: 1) Consistency - are the axioms satisfied in any non-trivial model that includes interacting matter and fields? 2) Richness - do the axioms carry interesting implications, in particular testable physics laws?
The theory quickly passed the second test. There are three discrete symmetries: P (reflection in space), C (reversing positive and negative charges), and T (reversing future and past). The Swiss mathematical physicist Res Jost, a close friend of Arthur's, proved from the Wightman axioms that PCT, the combination of all three discrete symmetries, must hold for any quantum field. To appreciate this statement it helps to know that it was determined experimentally that individually the three symmetries do not apply to all observable processes, yet the breaking of PCT was never observed. It was known empirically (and theoretically formulated by Fierz and Pauli) that particles with half-integer spin obey Fermi-Dirac statistics, with the Pauli exclusion principle, whereas particles with integer spin obey Bose-Einstein statistics. This too follows from the Wightman axioms. Wightman proved that the vacuum expectation values of products of the field operators, now called the Wightman functions, completely characterize the quantum field.
A classic account of these and other facts is in the book, written together with R. F. Streater, PCT, Spin and Statistics, and All That. The book has also served to focus attention on the need for a mathematical verification of the consistency of the axiomatic quantum field theory. This challenge has spurred the work of more than one generation of mathematical physicists. Other notable publications on quantum field theory include his paper with Bargmann and Hall on analytic continuation of the Wightman functions and his paper with Wick and Wigner introducing superselection rules.
Wightman continued to play an essential role in promoting constructive quantum field theory - the effort to construct models of the QFT axioms. However, his interest and influence in mathematical physics extended far beyond that. He guided the theses of over 34 doctoral students writing on a wide variety of subjects. For more than one of these he undertook to write a substantial introduction or pedagogical preface for a book or compilation. His interests also extended far beyond physics. Being a fund of information and enthusiasm on topics ranging from birds to books, it was only natural for him to undertake writing the history of science at Princeton for the University's 250th anniversary. He also undertook a laborious editorial work on the publication of the collected works and writings of Eugene P. Wigner, his predecessor in the Thomas D. Jones Chair for Mathematical Physics.
Arthur S. Wightman was born March 30, 1922 in Rochester, New York. He received his B.A. from Yale in 1942 and served in the Navy. He received his Ph.D. from Princeton in 1949 with a dissertation written under Professor John Wheeler. The same year he joined the Princeton faculty, where he remained ever since, becoming Thomas D. Jones Professor of Mathematical Physics in 1971 and Professor Emeritus in 1992. He served as chair of the mathematics department from 1987 to 1990. His honors include the Heineman Prize in mathematical physics, which he received in 1969. The following year he was elected a member of the National Academy of Sciences, and in 1997 he received the Henri Poincaré Prize. He died on January 13, 2013, at the age of 90.
Arthur was famous for his habit of answering questions with a detailed account of matters related to the context of the question, for his generosity with his time, and for his kindness in encouraging others. Space does not permit us to convey comments of his students and appreciative colleagues, but it could be of value to repeat the distilled chapter heading in the encomium by Professor Arthur Jaffe of Harvard University. Nine lessons of my Teacher, Arthur Strong Wightman: “I) Modesty, ASW was always speaking of the works of others - rarely of his own. II) Work on Big Problems. III) Distinguish ‘What you Know’ from ‘What you Think you Know’. IV) Do not ignore what physicists think. V) Do not ignore the past. VI) Teach Well. VII) Create a Congenial Working Atmosphere. VIII) Be a Good Citizen. IX) What Next? - Arthur seemed never satisfied with knowledge; always wanting more.”
Along with his students and colleagues we mourn his loss, and are grateful for the legacy he passed on.
Professor Michael Aizenman
Professor Elliott Lieb
Professor Edward Nelson