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.
Together with the mathematician John Tate, Wightman was engaged in the work on the Lorentz and Poincaré groups representations.
The Hilbert space carries a unitary representation of the Poincaré group under which the field operators transform covariantly.
[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.