## Arthur Wightman

*March 30, 1922 - January 13, 2013*

**Arthur Strong Wightman (1922–2013)**

*C**-algebras, classification of von Neumann algebras, and higher spin equations.

*Communications in Mathematical Physics*, as well as editing book series for Benjamin and for Princeton Press. Arthur had a compelling interest in the history of science as seen, for example, in the long introduction to Robert Israel’s book on the history of convexity in physics. As a member of the board of Princeton Press, he was a key early supporter in getting the Einstein Papers project under way. Arthur was the editor of Wigner’s complete works published by Springer.

*The Moderation and Absorption of Negative Pions in Hydrogen*under John Wheeler, also with input from his old mentor, Robert Marshak in Rochester. Arthur spent the rest of his academic career at Princeton, eventually as Thomas D. Jones Professor of Mathematical Physics jointly in Mathematics and Physics. During 1951–52 and 1956–57 he visited Copenhagen, where he interacted strongly with scientists in nearby Lund, especially with Gunnar Källén physics and with Lars Gårding in mathematics. With the latter, he framed the mathematical foundations for the quantum field theory axioms that bear their names.

Silvan Samuel Schweber 1952 |
Jerrold Marsden 1968 |

Richard Ferrell 1952 |
Gerald Goldin 1969 |

Douglas Hall 1957 |
Eugene Speer 1969 |

Oscar W. Greenberg 1957 |
George Svetlichny 1969 |

Huzihiro Araki 1960 |
Barry Simon 1970 |

John S.Lew 1960 |
Charles Newman 1971 |

William Stanley Brown 1961 |
Stephen Fulling 1972 |

James McKenna 1961 |
Robert Baumel 1979 |

Peter Nicholas Burgoyne 1961 |
Alan Sokal 1981 |

John Dollard 1963 |
Vincent Rivasseau 1982 |

Eduard Prugovecki 1964 |
Rafael de la Llave Canosa 1983 |

Arthur Jaffe1966 |
Steven Bottone 1984 |

Oscar Lanford, III 1966 |
Thea Pignataro 1984 |

Anton Capri 1967 |
Jan Segert 1987 |

Robert Powers 1967 |
Jay Epperson 1988 |

Lawrence Schulman 1967 |
Marek Radzikowski 1992 |

Christian Gruber 1968 |
Jan Westerholm 1996 |

By Prof. Arthur Jaffe and Prof. Barry Simon for the International Association of Mathematical Physics News Bulletin (January 2013).

### Rembrances by:

- Michael Aizenman, Professor, Princeton University
- Juerg Froehlich, ETH Zürich
- Wally Greenberg - University of Maryland
- Farshid Hajir - University of Massachusetts, Amherst
- Chiara Nappi, Professor, Princeton University
- Edward Nelson - Professor, Princeton University
- John Preskill - Professor, Caltech

During my four years as a math major at Princeton, I didn't get to know Professor Wightman well; indeed I spoke with him only once, on my way out, so to speak, at a small reception the Department held for graduating seniors and their parents after the graduation ceremony. Though we had never spoken before, he introduced himself to me and my parents and surprised me by not just knowing who I was but by speaking quite knowledgeably about work I had done with Professor Dwork for my senior thesis. He spoke to me as if I were already a fellow mathematician: he was enormously kind and encouraging. The whole conversation lasted maybe ten minutes, but it was my last interaction in the mathematics department as an undergraduate and made a lasting impression. In many ways, it was emblematic of the Princeton experience, where even a fledgeling student on his way to learning his craft was not only noticed but treated like an equal by a giant of the field. Historians of Mathematics and Physics will remember Professor Wightman as a founder of mathematical physics -- I'll remember him as a Professor who was as classy and elegant in his human interactions as he was in his mathematics. - * Farshid Hajir, Associate Professor, University of Massachusetts, Amherst *

I grieve for the loss of my friend and mentor, Arthur Wightman.

*Edward Nelson, Professor, Princeton University*

Here are a few words about Art as a thesis advisor: Art Wightman was always available. We often had lunch together, sometimes with other faculty, for example Eugene Wigner. These lunches were both enjoyable and informative. Art would often climb the spiral staircase to my work area under one of the banked lecture halls in Palmer Lab, to discuss his work as well as mine. He provided a solid foundation in quantum field theory. *Wally Greenberg. Professor, University of Maryland*

Arthur was a very impressive guy, both physically and intellectually. He was tall and strong (Strong was his middle name). He went to Yale on an academic scholarship, but he ended up playing basketball for Yale (probably it was his size the decisive element, rather than his athletic skills). He enjoyed the large steaks that came with it, and the travels with the team.

I used to visit him in Acorn Glen, an assisted living facility on Mount Lucas. One day, while we were sitting in a visitor’s area, he picked up a framed picture from the coffee table. It was a picture of physicist Donald Hamilton, he said, and he proceeded to tell me everything about Hamilton. Arthur was a local history buff and an expert on Princeton history. He wrote the history of science in Princeton for the 250th anniversary of the university and gave a series of public lectures about it . At Acorn Glen, when he finished with Donald Hamilton, he started with Eugene Wigner. He did not miss the opportunity to say that his job at Princeton as a graduate student had been to explain ``Wignerism” to his fellow graduate students. Only at the end he explained to me that the picture was there because Mrs (Pat) Wigner (who previously had been Mrs Hamilton) was also living in the same facility. I had not known until then that Mrs Wigner had previously been married to Professor Hamilton.

*Chiara Nappi, Professor, Princeton University*

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.

*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."*

*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.

^{th}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.*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.

*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.

*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.

*‘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.”*

*“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.”*

*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.

*“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.

*Lee*and

*Yang*appeared.]

*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.”*

*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!”*

*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.

*“(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*, ...)

*“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.

**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.

*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.

*“Challifour paper”*. (I fear that this paper was never completed.)

*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.

*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!

*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*

Please click here for remembrances by Michael Aizenman, Professor Princeton University