Where do actually the Wightman axioms fit in quantum field theory?

Question

The Wightman axioms are a set of postulates for quantum field theory that make it a little bit more rigorous than usual, being usually presented as:

1. The states of the system are unit rays in a Hilbert space that carries a unitary representation of the Poincaré group.
2. The $4$-momentum observable derived from the action of the Poincare group has its spectrum contained within the closed future light cone.
3. There exists a unique Poincaré-invariant state called the vacuum.
4. The quantum fields are operator-valued distributions defined on a dense domain $\mathcal{D}\subset \mathcal{H}$ that is invariant under the actions of both the Poincaré group and of the fields and their adjoints.
5. The fields transform in a covariant manner under Poincaré transformations
6. At spacelike separations fields either commute or anticommute.

These axioms seem to be capable of solving a lot of problems arising from the lack of rigor in QFT. However, they seem to not be able to deal with gauge theories as far I have heard.

Honestly, I'm confused on where this formalism fits in. I mean, if this can't deal with gauge theories and hence the standard model, this is certainly worse than the textbook approach. After all, the textbook approach - although not rigorous - is able to deal with gauge theories which are extremely important these days.

Do, where do the Wightman axioms really fit in QFT? How do they relate to the traditional non-rigorous textbook approach seen in most textbooks like Peskin's, Matthew Schwartz's, etc.?

Answer 1

The point of the Wightman axioms isn't to make a quantum field theory, it's to make a bare minimum of what a quantum field theory should obey, on some rather broad notion of what is probably reasonable. You might have noticed that the Wightman axioms don't include a fair bit of rules that are fairly common in most QFTs, like canonical commutation relations, energy positivity, etc.

What it does not tell you, on the other hand, is how to construct a quantum field theory that obeys those axioms. While it's reasonably simple to construct a theory for free fields that obeys the Wightman axioms, doing it for an interacting theory, such as a gauge QFT, is another matter. It's possible that one exists, or that it needs to be based on different principles, but that remains a rather tough problem to solve.

Answer 2

This is just a complement to Slereah's answer.

I think you are confused by the word "axioms" in Wightman axioms. These are not axioms like ZFC that mathematicians see as basic accepted truths and then go on to build on them proofs of various theorems. Here you are dealing with analogues of "axioms" of a vector space like associativity of $+$, distributive property of scalar multiplication with respect to $+$, etc. Wightman axioms are just a definition of what a QFT is, in a sense that is mathematically precise. Making physics textbook QFT rigorous is an entirely different story. You have to rigorously construct, e.g., by controlling limits of removing cutoffs, objects which satisfy the Wightman axioms. This is a whole area of mathematics called constructive QFT or nowadays rigorous renormalization group theory, since rigorous implementations of the RG are the main technique for producing such examples of nontrivial QFTs in the sense of Wightman.