Wightman Axioms - Consequences of The Axioms

Consequences of The Axioms

From these axioms, certain general theorems follow:

• PCT 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 which 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. See also the ghosts in BRST.

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.