Wightman Axioms - Rationale

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 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.

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 is unique. Moreover, the axioms assume that the vacuum is "cyclic", i.e., that the set of all vectors which can be obtained 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 over fields smeared over test functions with support in any open subspace of Minkowski space whose causal closure is the whole Minkowski space itself.