A No-Go Theorem

by I. T. Grodsky and R. F. Streater; Phys. Rev. Lett., 20, 695-698, 1968.

ABSTRACT

We show that in the covariant approach there exist no systems of single-particle functions satisfying a completeness property which generate any acceptable single-particle representations of current algebras and superconvergence relations. The non-existence of any acceptable local infinite-component field theory or wave equation is also demonstrated.

Note. The work shows that local field theories with infinitely many components must have infinite multiplicity in the mass spectrum or else will violate some other required property. It also showed that a technique advocated at the time by Gell-Mann, Horn and Weyers for constructing representations of current algebra (ref.62 as listed on the link) will not work. Gell-Mann, who was lecturing on his method at Coral Gables, remarked that he had just been shown a paper, "written in Pidgin English", which, if correct, could be important.

My coauthor, known as Jack, was a colourful character; for example, he danced on our grand piano during a party we gave for visitors to the research group. He was sure that our work was important enough to be published in Phys. Rev. Lett., and he oversaw the typing, as he knew how fussy the office at PRL was that the form of the typescript conform to their requirements. Nevertheless, the typescript was rejected by the office. After many hours of scrutiny, we found the reason: the acknowledgements to the US European Office of Aerospace Research had not been typed in double spacing. It was accepted when this was altered.

Jack and I were at the opposite poles also in our style of doing physics, and our paper shows the good that can arise from such people's joining forces. He was a new post-doc at Imperial, and had become interested in the problem of Gell-Mann, Horn and Weyers, on representing current algebras at infinity. I saw him around, and out of politeness asked what he was working on. He formulated the question, for my benefit, as a concrete mathematical problem: we need to find some analytic functions with a given set of properties. I immediately said, that I knew that there were no such functions. This follows from the "finite covariance theorem" conjectured in my 1962 paper and (then) recently proved by Bros, Epstein and Glaser, (Commun. in Math. Phys., 6, 77-, 1967). Jack was very startled by my claimed result, because I clearly knew nothing about current algebra at infinity. When I had convinced him, he wrote up the paper.

John W. Clark writes

"In 1988 Jack died tragically in northern California from a heart-attack combined with a one-car accident (the order is not clear). He was in his late forties and making ends meet by consulting in the Bay area, after resigning his tenured professorship at Cleveland State. He is still missed by many whose lives he touched".

John was the Chairman of Jack's Ph. D. Exam Committee.