Arthur Wightman

Arthur Wightman passed away this past January, at age 90. He was one of the great mathematical physicists of the past century.

Two of Arthur’s most renowned students, Arthur Jaffe and Barry Simon, wrote an affectionate obituary. I thought I would add some reminiscences of my own — Wightman was my undergraduate thesis advisor at Princeton.

I loved math in high school, and like many high school students before and since, I became convinced that Gödel’s incompleteness theorem is the coolest insight ever produced by the human mind. I resolved to devote my life to set theory and logic, and somehow I also became convinced that Princeton would be the best place in the world to study the subject. So there I went. I had a plan.

As a freshman, I talked my way into a graduate level course on Advanced Logic taught by Dana Scott. (I cleared the biggest obstacle by writing an essay to pass out of the freshman English requirement.) The course was wonderful, but by the end of it I was starting to accept what I had already sensed while in high school — I lack the talent to be a great mathematician.

A door was closing, but meanwhile another was opening. I was also taking a course on Electricity and Magnetism, based on the extraordinary book by Ed Purcell, taught by the charismatic Val Fitch. Chapter 5 contains an unforgettable argument explaining how electrostatic forces combined with special relativity imply magnetic forces. Meanwhile, while learning advanced calculus from the lovely (but challenging) little book by Michael Spivak, I realized that the Maxwell field is actually a two-form! Physics can be almost as cool as logic, so I would be a physics major! I had a new plan.

By the end of my junior year, I was feeling some dissatisfaction with the paucity of rigor in much of theoretical physics. In those days I liked to browse the physics books in the Princeton U-Store, and occasionally I would treat myself by buying one. One day a book by Streater and Wightman caught my eye, with the charming title PCT, Spin and Statistics, and All That. The content of the book was equally charming, starting with the first sentence of the brief preface: “The idea of this book arose in a conversation with H. A. Bethe, who remarked that a little book about modern field theory which contained only Memorable Results would be a Good Thing.” Ever since, I have been unable to resist capitalizing Good Thing when the situation seemed to call for it.

I didn’t understand everything in the book, but the point was that it was a mathematically rigorous book about Quantum Field Theory! Having learned just that year that an unbounded operator in Hilbert space should be defined on a dense subspace (the operator’s domain), I felt relieved that Streater and Wightman (unlike, say, Bjorken and Drell) were keenly aware of this indispensable mathematical fact.

Every Princeton undergrad writes a Senior Thesis. As I plowed through Streater and Wightman in my spare time during the summer before senior year, I resolved that Arthur Wightman would be my thesis advisor, which led to an acutely embarrassing moment. At the university’s opening exercises that fall, I received an award from President Bowen, and we chatted for a while beforehand. He asked about my thesis, and I told him of my plan to work with Wightman, failing to mention that Wightman and I had never met. Sure enough, when I caught Arthur after his class a week later to introduce myself, he immediately responded that he had “already heard about you from a surprising source,” meaning President Bowen. While Prof. Wightman seemed unfazed by this incident, I was somewhat mortified.

In front of Princeton’s Jadwin Hall during senior year.

My sense of humiliation melted away as it became clear that Prof. Wightman was actually quite receptive to supervising my thesis. Not long earlier, Osterwalder and Schrader had discovered the axioms satisfied by a quantum field theory in imaginary time (most notably, reflection positivity) which correspond to the famous Wightman axioms for a relativistic quantum field theory in real time, thus transforming the task of constructing quantum field theories into a problem in classical statistical mechanics. Wightman was very enthusiastic about these new Euclidean methods, and suggested that I use them to prove there is a phase transition in the pseudoscalar Yukawa theory in two spacetime dimensions. This problem was way too hard for me (it was solved years later by Balaban and Gawedzki), and I didn’t make much headway, but I learned a lot under Arthur’s tutelage.

Though I did not sufficiently appreciate it at the time, Arthur was incredibly generous with his time. He decided I should start by studying Barry Simon’s book The Euclidean (Quantum) Field Theory, so I worked through it, taking full advantage of Arthur’s willingness to instruct me where my background was lacking. A few times, Barry spotted me carrying the book and shot a curious glance my way, but I never asked him any questions about it — he was far too intimidating! (Even now, though we have been colleagues on the Caltech faculty for 30 years, and I realize that Barry is actually a very nice man with an engaging sense of humor, I’m still a little bit scared of him.)

Arthur Wightman was not intimidating; he was avuncular. I made frequent appointments with Arthur, which he would record in the pocket-size little black book that professors used to carry in those days. But sometimes I would find him alone in his office when I passed by, and with an undergraduate’s sense of entitlement (particularly pronounced at Princeton, at least according to the graduate students), I would barge in to ask a question. He almost always received me amiably, and these impromptu visits sometimes turned into long meetings. I sometimes felt uncomfortable, sensing that graduate students and postdocs, who may have had more compelling reasons to claim some of Arthur’s time, were a bit annoyed: That kid is talking to Arthur again! I kept coming back because Arthur always made me feel like he genuinely enjoyed my company. Maybe he made everyone feel that way.

Although our meetings would typically begin with a particular well-honed question I wanted to pose, they would often meander in unexpected directions, with Arthur glad for the opening to tell an ancedote or muse on a related issue in physics or mathematics. After each meeting I would scramble to the library and make copious notes, trying to reconstruct the substance of our discussion. Over the course of the year these notes filled four spiral ring notebooks, which I still have.

There was a memorable moment during my oral thesis defense. I had derived the one-loop effective potential for the Yukawa theory by a laborious method, and Sam Treiman was skeptical of both the method and the result. Arthur made a suggestion: “Maybe we should ask one of the whiz-bang kids?” As if on cue, David Gross went rushing by the open door of the conference room, and Arthur called him in. David, after looking at the formula I had written on the board for just a few seconds, confidently announced that not only was the formula correct, it was obviously correct. It would take another year and a half for me to realize that David was right: the formula was obvious.

The opening paragraph of the Preface to my thesis reads: “I am surely not the first student to discover that seven months is not enough time to write a senior thesis. My ambition to choose a thesis problem that would be both “non-trivial” and “important” forced me to devote more than half of that time to learning the background material that made the statement of the problem intelligible. Even that task might have been impossible if not for the constant guidance of my advisor, Professor Arthur Wightman. Our frequent conversations were the major source of this paper; it would be impossible to acknowledge his influence on every thought expressed here. His lucid explanations, extraordinary patience with my stupidity, and sense of humor made our meetings thoroughly delightful for me. I have never known a better teacher.”

I guess it sounds like I was laying it on pretty thick, yet I was really sincere. Arthur’s example made me want to inspire others as he had inspired me, but I know I will always fall short of that standard.