In mathematical physics, the Wightman axioms, also called the Gårding–Wightman axioms, named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the early 1950s, but they were first published only in 1964 with Lars Gårding after Haag–Ruelle scattering theory by Rudolf Haag and David Ruelle affirmed their significance. The axioms exist in the context of constructive quantum field theory and are meant to provide a basis for rigorous treatment of quantum fields and strict foundation for the perturbative methods used. One of the Millennium Problems is to realize the Wightman axioms in the case of Yang–Mills fields.
Rationale One basic idea of the Wightman axioms is that there is a Hilbert space, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) are implemented. There is also a stability assumption, which restricts the spectrum of the four-momentum to the positive light cone (and its boundary). However, this is not enough to implement locality. For that, the Wightman axioms have position-dependent operators called quantum fields, which form covariant representations of the Poincaré group. Since quantum field theory suffers from ultraviolet problems, the value of a field at a point is not well-defined. 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. Because the axioms are dealing with unbounded operators, the domains of the operators have to be specified. The Wightman axioms restrict the causal structure of the theory by imposing either commutativity or anticommutativity between spacelike separated fields. They also postulate the existence of a Poincaré-invariant state called the vacuum and demand it to be unique. 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.
Axioms
W0 (assumptions of relativistic quantum mechanics) 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. In the following, the scalar product of Hilbert space vectors Ψ and Φ is denoted by ⟨ Ψ , Φ ⟩ {\displaystyle \langle \Psi ,\Phi \rangle } , and the norm of Ψ is denoted by ‖ Ψ ‖ {\displaystyle \lVert \Psi \rVert } . The transition probability between two pure states [Ψ] and [Φ] can be defined in terms of non-zero vector representatives Ψ and Φ to be
P ( [ Ψ ] , [ Φ ] ) = | ⟨ Ψ , Φ ⟩ | 2 ‖ Ψ ‖ 2 ‖ Φ ‖ 2 {\displaystyle P{\big (}[\Psi ],[\Phi ]{\big )}={\frac {|\langle \Psi ,\Phi \rangle |^{2}}{\lVert \Psi \rVert ^{2}\lVert \Phi \rVert ^{2}}}}
and is independent of which representative vectors Ψ and Φ are chosen. The theory of symmetry is described according to Wigner. This is to take advantage of the successful description of relativistic particles by E. P. Wigner in his famous paper of 1939; see Wigner's classification. Wigner postulated the transition probability between states to be the same to all observers related by a transformation of special relativity. More generally, he considered the statement that a theory be invariant under a group G to be expressed in terms of the invariance of the transition probability between any two rays. The statement postulates that the group acts on the set of rays, that is, on projective space. Let (a, L) be an element of the Poincaré group (the inhomogeneous Lorentz group). Thus, a is a real Lorentz four-vector representing the change of spacetime origin x ↦ x − a, where x is in the Minkowski space M4, and L is a Lorentz transformation, which can be defined as a linear transformation of four-dimensional spacetime preserving the Lorentz distance c2t2 − x⋅x of every vector (ct, x). Then the theory is invariant under the Poincaré group if for every ray Ψ of the Hilbert space and every group element (a, L) is given a transformed ray Ψ(a, L) and the transition probability is unchanged by the transformation:
⟨ Ψ ( a , L ) , Φ ( a , L ) ⟩ = ⟨ Ψ , Φ ⟩ . {\displaystyle \langle \Psi (a,L),\Phi (a,L)\rangle =\langle \Psi ,\Phi \rangle .}
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