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.
Read more about this topic: Wightman Axioms
Famous quotes containing the words consequences of the, consequences of, consequences and/or axioms:
“We are still barely conscious of how harmful it is to treat children in a degrading manner. Treating them with respect and recognizing the consequences of their being humiliated are by no means intellectual matters; otherwise, their importance would long since have been generally recognized.”
—Alice Miller (20th century)
“Cultivate the habit of thinking ahead, and of anticipating the necessary and immediate consequences of all your actions.... Likewise in your pleasures, ask yourself what such and such an amusement leads to, as it is essential to have an objective in everything you do. Any pastime that contributes nothing to bodily strength or to mental alertness is a totally ridiculous, not to say, idiotic, pleasure.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)
“We are still barely conscious of how harmful it is to treat children in a degrading manner. Treating them with respect and recognizing the consequences of their being humiliated are by no means intellectual matters; otherwise, their importance would long since have been generally recognized.”
—Alice Miller (20th century)
“The axioms of physics translate the laws of ethics. Thus, the whole is greater than its part; reaction is equal to action; the smallest weight may be made to lift the greatest, the difference of weight being compensated by time; and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (18031882)