Wightman axioms
Arthur Wightman formulated the axioms in the early 1950s,[4] but they were first published only in 1964[5] after Haag–Ruelle scattering theory[6][7] affirmed their significance.One basic idea of the Wightman axioms is that there is a Hilbert space, upon which the Poincaré group acts unitarily.There is also a stability assumption, which restricts the spectrum of the four-momentum to the positive light cone (and its boundary).For that, the Wightman axioms have position-dependent operators called quantum fields, which form covariant representations of the Poincaré group.To get around this, the Wightman axioms introduce the idea of smearing over a test function to tame the UV divergences, which arise even in a free field theory.The Wightman axioms restrict the causal structure of the theory by imposing either commutativity or anticommutativity between spacelike separated fields.Moreover, the axioms assume that the vacuum is "cyclic", i.e., that the set of all vectors obtainable by evaluating at the vacuum-state elements of the polynomial algebra generated by the smeared field operators is a dense subset of the whole Hilbert space.Lastly, there is the primitive causality restriction, which states that any polynomial in the smeared fields can be arbitrarily accurately approximated (i.e. is the limit of operators in the weak topology) by polynomials in smeared fields over test functions with support in an open set in Minkowski space whose causal closure is the whole Minkowski space.Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space.The second part of the zeroth axiom of Wightman is that the representation U(a, A) fulfills the spectral condition—that the simultaneous spectrum of energy–momentum is contained in the forward cone: The third part of the axiom is that there is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincaré group.The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).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.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.(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.)There is a million-dollar prize for a proof that the Wightman axioms can be satisfied for gauge theories, with the additional requirement of a mass gap.This theorem is the key tool for the constructions of interacting theories in dimension 2 and 3 that satisfy the Wightman axioms.