He was one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms.
[1] With his rigorous treatment of quantum field theories, he promoted research on various aspects of modern mathematical physics.
[5] In the years 1951–1952 and 1956–1957 he was a visiting researcher at the University of Copenhagen at the Niels Bohr Institute, where he worked in particular with Gunnar Källén and Lars Gårding.
The Hilbert space carries a unitary representation of the Poincaré group under which the field operators transform covariantly.
[9]: 425 Using this, Res Jost was able to derive the PCT and the spin-statistics theorems, as shown in Wightman's and Streater's book.
[10] Together with Eugene Wigner and Gian-Carlo Wick, he introduced superselection rules and studied the representations of commutator and anti-commutator algebras with the mathematician Lars Gårding.
[11] In 1969 Arthur Wightman was awarded the Dannie Heineman Prize for Mathematical Physics for founding and contributing in developing axiomatic quantum field theory[12] and in 1997 the Henri Poincaré Prize of the International Association of Mathematical Physics[13] for his central role in the foundations of the general theory of quantum fields.