Relation To Other Frameworks and Concepts in Quantum Field Theory
The Wightman framework does not cover infinite energy states like finite temperature states.
Unlike local quantum field theory, the Wightman axioms restrict the causal structure of the theory explicitly by imposing either commutativity or anticommutativity between spacelike separated fields, instead of deriving the causal structure as a theorem. If one considers a generalization of the Wightman axioms to dimensions other than 4, this (anti)commutativity postulate rules out anyons and braid statistics in lower dimensions.
The Wightman postulate of a unique vacuum state doesn't necessarily make the Wightman axioms inappropriate for the case of spontaneous symmetry breaking because we can always restrict ourselves to a superselection sector.
The cyclicity of the vacuum demanded by the Wightman axioms means that they describe only the superselection sector of the vacuum; again, that is not a great loss of generality. However, this assumption does leave out finite energy states like solitons which can't be generated by a polynomial of fields smeared by test functions because a soliton, at least from a field theoretic perspective, is a global structure involving topological boundary conditions at infinity.
The Wightman framework does not cover effective field theories because there is no limit as to how small the support of a test function can be. I.e., there is no cutoff scale.
The Wightman framework also does not cover gauge theories. Even in Abelian gauge theories conventional approaches start off with a "Hilbert space" (it's not a Hilbert space, but physicists call it a Hilbert space) with an indefinite norm and the physical states and physical operators belong to a cohomology. This obviously is not covered anywhere in the Wightman framework. (However as shown by Schwinger, Christ and Lee, Gribov, Zwanziger, Van Baal, etc., canonical quantization of gauge theories in Coulomb gauge is possible with an ordinary Hilbert space, and this might be the way to make them fall under the applicability of the axiom systematics.)
The Wightman axioms can be rephrased in terms of a state called a Wightman functional on a Borchers algebra equal to the tensor algebra of a space of test functions.
Read more about this topic: Wightman Axioms
Famous quotes containing the words relation to, relation, concepts, quantum, field and/or theory:
“Among the most valuable but least appreciated experiences parenthood can provide are the opportunities it offers for exploring, reliving, and resolving one’s own childhood problems in the context of one’s relation to one’s child.”
—Bruno Bettelheim (20th century)
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)
“Germany collapsed as a result of having engaged in a struggle for empire with the concepts of provincial politics.”
—Albert Camus (1913–1960)
“A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
—Hubert C. Heffner (1901–1985)
“When white men were willing to put their own offspring in the kitchen and corn field and allowed them to be sold into bondage as slaves and degraded them as another man’s slave, the retribution of wrath was hanging over this country and the South paid penance in four years of bloody war.”
—Rebecca Latimer Felton (1835–1930)
“Don’t confuse hypothesis and theory. The former is a possible explanation; the latter, the correct one. The establishment of theory is the very purpose of science.”
—Martin H. Fischer (1879–1962)