Operator valued distribution

In physics the Wightman axioms (also called Gårding–Wightman axioms, named after Lars Gårding and 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 they 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 isn't 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.

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