} i.e. the phase is a multiple of π {\displaystyle \pi } . For particles of integer spin (pions, photons, gravitons, ...) one can remove the ± sign by In mathematical physics, the Wightman axioms, also called the Gårding–Wightman axioms, named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 with Lars Gårding after Haag–Ruelle scattering theory by Rudolf Haag and David Ruelle affirmed their s CPT theorem — there is general symmetry under change of parity, particle–antiparticle reversal and time inversion (none of these symmetries alone exists in nature, as it turns out). Connection between spin and statistic — fields that transform according to half integer spin anticommute, while those with integer spin commute (axiom W3). There are actually technical fine details to this theorem. This can be patched up using Klein transformations. See parastatistics and also the ghosts in BRST. The impossibility of superluminal communication – if two observers are spacelike separated, then the actions of one observer (including both measurements and changes to the Hamiltonian) do not affect the measurement statistics of the other observer. Arthur Wightman showed that the vacuum expectation value distributions, satisfying certain set of properties, which follow from the axioms, are sufficient to reconstruct the field theory — Wightman reconstruction theorem, including the existence of a vacuum state; he did not find the condition on the vacuum expectation values guaranteeing the uniqueness of the vacuum; this condition, the cluster property, was found later by Res Jost, Klaus Hepp, David Ruelle and Othmar Steinmann. If the theory has a mass gap, i.e. there are no masses between 0 and some constant greater than zero, then vacuum expectation distributions are asymptotically independent in distant regions. Haag's theorem says that there can be no interaction picture — that we cannot use the Fock space of noninteracting particles as a Hilbert space — in the sense that we would identify Hilbert spaces via field polynomials acting on a vacuum at a certain time.

Z; = Z,, Z; = Z, if the spin s of the two fields is not an integer or a half-integer. This is an aspect … of the spin-statistics theorem. However, in two dimensions it is possible to bypass this theorem. The … g-state Potts model, in which the spin o; takes q different integer values: o; = 1,2,---,q. To each possible

958 (1948) Quantum Field Theory Spin, Statistics, and the TCP Theorem Proc. Natl. Acad. Sci. U.S.A. 44 … photon of finite mass is a spin one particle, not a helicity one particle.’ Feynman!” was more forthright … a proof, for interacting systems, of the spin-statistics theorem, which had been proved by Pauli only