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Axiomatization of quantum field theory

In mathematical physics, the Wightman axioms (also called 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 after Haag–Ruelle scattering theory 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.

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.

For each test function f, i. e. for a function with a compact support and continuous derivatives of any order, there exists a set of operators which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum. The fields A are operator valued tempered distributions. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).

If the supports of two fields are space like separated, then the fields either commute or anticommute. Cyclicity of a vacuum and uniqueness of a vacuum are sometimes considered separately. Also, there is property of asymptotic completeness that Hilbert state space is spanned by the asymptotic spaces and , appearing in the collision S matrix. The other important property of field theory is mass gap, which is not required by the axioms that energy–momentum spectrum has a gap between zero and some positive number.

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 does not 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, this is not a great loss of generality. However, this assumption does leave out finite energy states like solitons, which cannot 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" with an indefinite norm (hence not truly a Hilbert space, which requires a positive definite norm, but physicists call it a Hilbert space nonetheless), 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.

One can generalize the Wightman axioms to dimensions other than 4. In dimension 2 and 3, interacting (i. e. non free) theories that satisfy the axioms have been constructed. Currently, there is no proof that the Wightman axioms can be satisfied for interacting theories in dimension 4. In particular, the Standard Model of particle physics has no mathematically rigorous foundations. 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.

Under certain technical assumptions, it has been shown that a Euclidean QFT can be Wick rotated into a Wightman QFT, see Osterwalder–Schrader theorem. This theorem is the key tool for the constructions of interacting theories in dimension 2 and 3 that satisfy the Wightman axioms.