I just heard today that mathematical physicist Arthur Wightman passed away earlier this month, at the age of 90. Wightman was one of the leading figures in the field of rigorous quantum field theory, the effort to try and make precise sense of the often heuristic methods used by physicists when they deal with quantum fields. He was a well-liked and very respected professor at Princeton during the years 1979-84 that I was a graduate student there, but unfortunately I don’t think I ever made an effort to talk to him, to my loss. The university has something about him here, the department here.
Wightman is most well-known for the “Wightman Axioms”, which are an attempt to formalize the fundamental assumptions of locality and transformation under space-time symmetries that any sensible quantum field theory should satisfy. His 1964 book with Raymond Streater, PCT, Spin, Statistics and All That, explains these axioms and shows how they lead to some well-known properties of quantum field theories such as PCT invariance and the Spin-Statistics relation. When this work was being done during the 1950s and early 60s, quantum field theory was considered something that couldn’t possibly be fundamental. All sorts of discoveries about strong interaction physics were being made, and it seemed clear that these did not fit into the quantum field theory framework (this only changed in 1973 with asymptotic freedom and QCD). In any case, problems with infinities of various sorts plagued any attempt to come up with a completely consistent way of discussing interacting quantum fields, providing yet another reason for skepticism.
Wightman was one of a small group of mathematical physicists who reacted to this situation by trying to come to grips with the question of exactly what a quantum field theory was, in an attempt to find both the implications of the concept and its limitations. After the early 60s, attention moved from the axioms and their implications to the question of “constructive quantum field theory”: could one explicitly construct something that satisfied the axioms? Examples were found in 2 and 3 space-time dimensions, but unless I’m missing something, to this day there is no rigorous construction of an interacting QFT in 4 space-time dimensions. There is every reason to believe that Yang-Mills theory, constructed with a lattice cut-off, has a sensible continuum limit that would provide such an example, but this remains to be shown (and there’s a one million dollar prize if you can do this).
Thinking back to the early 1980s and my days as a graduate student, it’s clear what some of the reasons were why I didn’t spend time going to talk to Wightman. With the triumph of the Standard Model, attention had turned to questions about quantum gravity, as well as questions about the non-perturbative behavior of QCD. I certainly spent some time trying to read and understand the Streater-Wightman volume, but its emphasis on the role of the Poincaré group meant it had little to say about QFT in curved space-time, much less how to think about quantized general relativity. Gauge theories in general did not seem to fit into the Streater-Wightman framework, with the tricky issue of how to handle gauge symmetry something their methods could not address. For non-perturbative QCD, we had new semi-classical computation methods, and I was happily programming computers do numerical simulations of Yang-Mills theory. Why pay attention to the difficult analysis needed to say anything rigorous about quantum fields, when the path integral method seemed to indicate one could just put them on a computer and have the computer tell you the answer?
In later years I became much more sensitive to the fact that quantum fields can’t just be understood by a Monte-Carlo calculation, as well as the importance of some of the questions that Streater and Wightman were addressing. As particle theory continues to suffer deeply from the fact that the SM QFT is just too good, anything that can be done to better understand the subtleties of QFT may be worthwhile. It remains true that gauge theories require new methods way beyond what is in Streater-Wightman, but looking back at the book I see it as largely devoted to understanding the role of space-time symmetries in the structure of the theory. The importance of such understanding of how symmetries govern QFT may be a lesson still not completely absorbed, with gauge symmetries and diffeomorphism symmetries part of a story extending Streater and Wightman to the Standard Model, in a way that we have yet to understand.