It is with much sadness and anxiety that I attempt to say a few words in memory of our teacher, colleague and friend Arthur Wightman. What I will have to say is a blend of general remarks of historical nature, comments concerning Arthur’s science and some personal reminiscences. I ask you for your indulgence in case I get something wrong.
I feel that Arthur Jaffe found exactly the right words when, upon learning that Arthur Wightman had died, he said, and I quote him:
“I think of Arthur as the spiritual leader of mathematical physics and his death really marks the end of an era. It's hard to think of who will step into Arthur's shoes with the same wonderful breadth of interests, insights, understanding of people and ability to inspire the best from others."
It may be somewhat unclear to you – just as it is to me – why I have been asked to speak on today’s occasion. Well, it could be argued that, for my own academic career, my stay at Princeton University, from 1974 till the end of 1977, and my interactions with Arthur Wightman were crucial. My experiences with the Princeton of the 1970’s can best be described in the words of the late Ernst Specker of ETH – the Specker of the ‘Kochen-Specker Theorem’: “Princeton was for me what Rome had been for Goethe.” But my personal relationship to Princeton is hardly relevant for today’s event. What may be of some interest is the circumstance that there has been a somewhat surprising and long-standing connection between Zurich and professors at ETH, on one side, and Princeton, the Institute, the University and Arthur Wightman, on the other side. Coming from ETH Zurich I regard it as my task to tell you about this connection.
We may not want to go back as far as the celebrated Princeton geographer and geologist of the 19th Century Arnold Guyot, who was Swiss; although Arthur would surely have had some taste for a story like his. So, let’s start with Albert Einstein, who had studied at ETH Zurich and, during a period when he was working hard and successfully on the general theory of relativity, held a professorship there. He then went on to Berlin. When the Nazis came to power in Germany, he found shelter at the Princeton Institute – just like Hermann Weyl, who had also been a professor at ETH and had made the same mistake to move on to Germany.
As an undergraduate at Yale, Arthur Wightman apparently got the homework task to write an essay about the beginnings of statistical mechanics. He went to the library and found out that he should learn a lot about the works of Ludwig Boltzmann and, in particular, Albert Einstein. But – and he found it later quite embarrassing – he completely missed that famous Yale professor and pioneer of statistical mechanics J. Willard Gibbs. This would never have happened to Arthur in his mature years, when he had become a highly cultured scholar with immensely broad knowledge. His deep interest and wide knowledge in thermodynamics and statistical mecahnics is manifest in his introduction to the book on lattice gases by Robert Bryan Israel, a Princeton PhD student of Barry Simon’s, where Gibbs is featured prominently.
During World War II, after having been refused Swiss citizenship, Wolfgang Pauli took leave from ETH and worked at the Institute for Advanced Study. Although, after the war, he was offered Einstein’s chair, he returned to Zurich. Soon, he not only received a Nobel Prize but also his Swiss citizenship. To his friend and faithful follower Markus Fierz he explained his return to ETH, saying that it was easy to make a lot of money in the United States, but not so easy to spend it in a pleasant way.
Among scientists from Switzerland who had studied at ETH and later went to Princeton, one should also mention the mathematician Armand Borel. One might recall that there was a fairly intense connection between the ETH school of topology founded by Heinz Hopf and the Princeton school of topology. John Milnor and Ernst Specker, among others, profited from this connection as young mathematicians. There are more recent examples of exchanges of mathematical ideas (e.g., Kochen-Specker) and mathematicians between Princeton and Zurich – but I cannot go into details about this.
I believe the fact that Pauli had been a frequent visitor at Princeton, after the war, and that Sonja and Valya Bargmann had strong ties to Zurich may have played some role for Arthur Wightman to develop a close relationship to ETH. But the main reason was undoubtedly that he met Res Jost at Princeton and in Copenhagen, who had been a postdoctoral researcher with Pauli and was sent to the Institute by him. It would turn out that Jost and Wightman were ‘kindred spirits.’ Let me explain their relationship in a little more detail, because this will also shed light on Arthur Wightman’s scientific endeavors and style.
In 1989, shortly before Res Jost died, Arthur Jaffe, then the chief editor of the ‘Communications in Mathematical Physics’, decided to dedicate a special volume to Jost and Wightman. In the introduction, Jaffe writes about them:
“Theirs was a crucial contribution to the challenging -and still not finished- endeavor to bring mathematical rigor and beauty into a branch of theoretical physics [namely particle physics] where it was often not so easy to separate facts from fancies.
Besides being strong researchers both Res and Arthur have left a lasting footprint in the world of mathematical physics through their power of teaching. Each one of them, in his own characteristic and unique way, was the teacher for a whole new generation of young scientists. Their success and their lasting influence cannot be explained just by their being leaders in their field. With all the weight of their strong personalities they impressed upon their disciples a deep sense for what good and honest and beautiful science was. ... But, most important of all, they have offered their personal affection and friendship to all of us.”
In a tribute to Arthur Wightman published in the same volume of Commun. Math. Phys., Res Jost writes about him and the impact of his scientific work as follows:
“Unexpectedly the clouds disperse and free the view on a blue sky and massifs of inaccessible mountains, peaks on which in old times we oriented our work. Such a massif is Arthur Wightman's scientific work and teaching. I had the privilege to roam through some of its modest foothills.
I met Arthur ... shortly after our arrival in the United States in December 1949 in the flat of our unforgettable friends Sonya and Valya Bargmann. I did not, at that time, realize to whom I had just been introduced. I was still ignorant of the wide difference in scientific standards between Zurich, where I came from, and Princeton. As an example, I barely knew the definition of a Hilbert space. ... Arthur, however, moved with ease through the vast domains of functional analysis. During his student days, he had been a member of a congregation, named "The Group Characters", which included the ... mathematician John Tate. Together they studied the difficult modern theory of the unitary representations of non-compact groups, a theory to which Eugene Wigner and Valentine Bargmann had made so decisive initial contributions. No wonder then that our meeting at the Bargmann's did not, for the moment, result in a close connection between Arthur and myself. He was occupied by problems of principle in the domain of theoretical physics, whereas I played with peripheral trifles, paying little attention to the developments in the center of science. So it happened that the fundamental investigation of G.C. Wick, A.S. Wightman, and E.P. Wigner, which introduced the all-important notion of superselection rules, was almost lost on me, as was the analysis of L. Garding and A. S. Wightman on the representations of the canonical anti-commutation and commutation relations. ...
On March 29th 1957, [Arthur] and Anna-Greta visited us in Berne. While the ladies toured the city, Arthur and I worked in front of a blackboard in my old Gymnasium: he explained to me his field theory. That was my initiation.”
Well, Jost was a good student: In the same year, he deduced the general PCT Theorem from the Garding-Wightman axioms of relativistic quantum field theory. Ever after, Jost and Wightman were close friends, who would visit each other regularly. It may be appropriate to mention that, in 1968, Wightman received an honorary degree from ETH Zurich.
Thanks to Res and Arthur, a solid ‘two-way bridge’ of exchange between Princeton and Zurich was built. Zurich sent people like David Ruelle, my PhD advisor Klaus Hepp and the late Walter Hunziker of the ETH physics department to Princeton. Presumably, I was the last envoy from Zurich who originated from the Jost-Wightman tradition, (but certainly not the last mathematical physicist to cross that bridge). Princeton paid back by sending us people like Freeman Dyson (whom Jost had met at the Institute in the late fourties), Ed Nelson, Arthur Jaffe, Barry Simon and Oscar E. Lanford III. Arthur Jaffe, Oscar Lanford and Barry Simon had all been PhD students of Wightman’s. Arthur Jaffe and Barry Simon became frequent visitors and “honorary members” of the “Seminar für theoretische Physik” of ETH Zurich, while Oscar Lanford later joined the ETH math department as a faculty member. None of this would have happened without Res Jost and Arthur Wightman; and, besides other lives, mine would surely have taken a rather different turn.
More recently, exchange between ETH Zurich and Princeton was kept alive by Michael Aizenman, Gian Michele Graf, Elliott Lieb and Thomas Spencer. Tom came to play a particularly important role in my scientific trajectory.
In Jost’s tribute to Arthur Wightman, we have already heard quite a lot about the latter’s scientific interests and accomplishments. But let me attempt to give a somewhat more systematic account of Arthur’s science.
Arthur got his PhD degree from Princeton in 1949, with a thesis entitled “The Moderation and Absorption of Negative Pions in Hydrogen”. His advisor was John Archibald Wheeler. In the same year, Arthur joined the Princeton faculty. His early work concerned more down-to-earth nuclear and particle physics. For example, he collaborated with the late Louis Michel on problems concerning the weak interactions.
[What they missed became clear when the work of Lee and Yang appeared.]
In 1952, Wightman co-authored an important paper with Gian Carlo Wick and Eugene Paul Wigner, entitled “The Intrinsic Parity of Elementary Particles”, which appeared in the ‘Physical Review’ [volume 88]. The abstract of their paper reads as follows:
“The limitations to the concept of parity of quantum-mechanical states and, in particular, of intrinsic parity of elementary particles are discussed. These limitations are shown to follow from "superselection rules," i.e., from restrictions on the nature and scope of possible measurements. The existence of such superselection rules is proved for the case of spinor fields; it is also conjectured that a superselection rule operates between states of different total charge.”
Two themes alluded to in this abstract would remain among Arthur’s favorite ones: The foundations of Quantum Theory, and the “Charge Superselection Rule” in QED. On the latter subject he co-authored an important paper with Franco Strocchi of Pisa. He shared the former interest with Wigner. In 1975 or ‘76, Wigner announced that he would present a physics colloquium at the University about the significance of macroscopic systems in quantum measurements. Unfortunately, shortly before his lecture, he caught a cold and thought he had lost his voice. Perhaps, he actually caught cold feet, realizing that he had not understood the problem completely. So he asked Arthur Wightman to replace him and gave him his notes. Arthur presented the lecture, but, towards the end, weaved in some of his own ideas related to the role of the thermodynamic limit (measurement devices with infinitely many degrees of freedom) in theories of the quantum measurement process and the Coleman-Hepp model of decoherence. At this point, a miracle happened: Wigner’s voice was back, and everybody present heard him speak as follows: “May I make a stupid comment? Arthur Wightman, you are a great man, but you are not infinite!”
Among many other things, Pauli is known for the nasty comments and biting criticism he would utter in other peoples’ talks. Among many other things, Arthur Wightman is known for having succeeded in silencing Pauli in a lecture he gave at the Niels Bohr Institute as a young man. After many interruptions by Pauli, Arthur said that it was notoriously difficult to convince barbarians of the advantages of – in Arthur’s case, mathematical – culture, but he would try it anyway and would go on in his lecture. Pauli remained silent for the remainder of the lecture.
From Copenhagen it is not far to Sweden. Arthur liked Sweden, and his first wife Anna-Greta had Swedish roots. Furthermore, Arthur got to know the Swedish mathematician Lars Garding. With Garding, he studied the representations of the canonical commutation and anti-commutation relations in quantum field theory. This became a point of contact with that other great mathematical physicist Rudolf Haag and the subject matter of two famous PhD theses, by Huzihiro Araki and Robert Powers, both of whom later made major contributions to the theory of operator algebras, including the representation theory of the canonical commutation and anti-commutation relations.
Better known, among Wightman’s scientific achievements, are the axioms for relativistic quantum field theory, known under the name of “(Garding-) Wightman axioms”, which are at the basis of axiomatic quantum field theory. Arthur proved the famous result that a quantum field theory can be reconstructed from the vacuum expectation values of arbitrary products of its field operators. Equally or more important is the so-called “Bargmann-Hall-Wightman Theorem” that describes a very large domain of analyticity [the extended forward tube] of the so-called Wightman functions whose boundary values are those vacuum expectation values of products of field operators. This theorem played an important role in Jost’s proof of the general PCT Theorem (which, in turn, played a crucial role in the work of Bisognano and Wichman that connects Jost’s result to Tomita-Takesaki theory, ...)
Once the general structure of relativistic quantum field theory appeared to be clarified and axiomatic quantum field theory entered a somewhat baroque stage, Wightman decided that it was time to construct models of quantum field theories that satisfy all the Garding-Wightman axioms. He was lucky that, at that time, he had two excellent PhD students, Arthur Jaffe and, shortly after him, Oscar Lanford. Together, they co-authored what was, perhaps, the first paper in a new field called “Constructive Quantum Field Theory”, a field to which other famous Princetonians were to make seminal contributions: Ed Nelson, Francesco Guerra, Lon Rosen and Barry Simon, all of whom would be happy to acknowledge the importance of Arthur’s guidance. Arthur Jaffe remained one of the leading contributors to CQFT, while Oscar became an expert in statistical mechanics and the theory of dynamical systems. The latter, too, belonged to Arthur’s many areas of interest.
Unfortunately, it appeared to be too difficult to attack the construction of field theory models in space-times of dimension more than 2 or 3, except at the level of renormalized perturbation theory. Thus, Arthur decided that Dyson’s program of renormalization theory to arbitrary order in the physical coupling constant ought to be taken up again. At this point the bridge to Zurich comes back into the picture: In 1962, Klaus Hepp had earned his PhD degree under Res Jost with important work on Wightman’s axiomatic field theory, and in 1964 he arrived at the Institute for Advanced Study. He was a very brilliant young man willing to face a real challenge. The challenge turned out to be Arthur Wightman’s program of a mathematical proof that so-called renormalizable quantum field theories in four dimensions, such as QED, could be defined to all orders in a power series expansion in the physical charge if all masses, field strengths and coupling constants are properly renormalized. This led to the BPHZ version of perturbative renormalization theory. Two years later, Arthur had Eugene Speer as his PhD student and guided him to the elegant development of analytic renormalization.
Arthur also nurtured interests in a variety of problems in quantum mechanics. These interests found their concrete incarnation in the work on quantum-mechanical scattering theory of Walter Hunziker, who did postdoctoral research under Arthur Wightman, a PhD thesis of Christian Gruber from Lausanne directed by Arthur, in Barry Simon’s landmark PhD thesis, and in the work of Elliott Lieb and Barry Simon on Thomas-Fermi theory.
Around the mid sixties, Arthur Wightman did not do very much research of his own, anymore. Instead, he very generously shared his time and his many interests, programs and insights with his numerous PhD students, postdocs and the students in his courses. Perhaps, he developed somewhat too many and diverse interests to be able to focus on technical problems. One may, however, argue that Wightman’s mode of operation was more fruitful for the progress of science than if he had attempted to continue to solve technical problems.
Speaking briefly about myself, I can say that I owed the offer to come to Princeton [as the successor of Michael Reed in the math department] to Arthur Wightman, but presumably mainly to Barry Simon, who had noticed me when he visited Zurich and Lausanne as a visiting professor, in 1973, and apparently decided that I had some potential. I arrived at Princeton in the fall of 1974, simultaneously with Elliott Lieb, after having spent a year with Arthur Jaffe at Harvard. During that year, Arthur Wightman had organized a workshop on the infrared problem in QED, which had been the central theme of my PhD thesis, and he invited me to participate in it and present a talk. When I arrived at Princeton, he gave me a wonderfully warm welcome, in spite of the fact that that was a very difficult period in his life, because his first wife was dying of cancer. Although I may not have profited as much from Arthur’s immensely broad knowledge and experience as I should have – he was preoccupied with more existential things – he was always willing to listen to my ideas most of which were presumably crazy, and he would almost invariably find them interesting and encourage me to pursue them, thus creating an optimistic atmosphere. The door to his office was always open, the office was densely packed with books, journals, preprints and boxes of notes and manuscripts, and it was not easy to find an empty spot on the blackboard. The right upper corner of the blackboard was forbidden territory; it carried the inscription “Challifour paper”. (I fear that this paper was never completed.)
Arthur’s life was marked by at least three tragedies: The premature death of his first wife; the premature death of his daughter Robin, whom Arthur was immensely proud of – he visibly admired and loved her – and his own degenerative disease. For all I know, he shouldered these tragedies boldly and courageously.
As Arthur Jaffe said, the disappearance of Arthur Wightman marks the end of an era. I fear it may also mark the gradual disappearance of an attitude and style among scientists that I associate directly with people like Arthur Wightman and Res Jost: Focus on the central problems of your field – even if they may not be doable immediately – generously share your time, insights and ideas with others, especially with young colleagues, generously support the careers of young scientists, maintain unerring intellectual honesty and integrity – in short, try to be a gentleman scientist! More than his scientific oeuvre, I view the latter qualities as Arthur Wightman’s central legacy for which he will be remembered, and which, in a time when they are endangered, we should cherish!
I am glad that, in December of last year, my wife Eva and I decided to visit Captain Arthur Strong Wightman at the New Jersey Veterans Nursing Home in Edison. It was our farewell to him. He seemed to be at peace. But he was not of this world, anymore. One might thus see his passing away as redemption.
I thank you!
P.S. I would like to add with emphasis that I owe my more recent ties to Princeton and the Institute for Advanced Study mainly to my friend and former mentor and collaborator Thomas C. Spencer to whom I am indebted in many ways. He is truly a gentleman scientist, thus continuing the tradition of Jost and Wightman.
Juerg Froehlich, ETH Zürich