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
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