# Which of the Wightman axioms are not incorporated by four dimensional quantum Yang-Mills?

I am trying to understand the quantum Yang-Mills existence problem but the best I have seen so far is the statement that there is no known interacting relativist field theory in four dimensions which can be constructed from Wightman axioms.

Which of the Wightman axioms cannot be incorporated in a Quantum Yang-Mills theory such as QCD and how/why they fail? How does traditional QFT deal with these fails?

• It is not possible to construct a theory from the Wightman axioms. The axioms are a set of conditions that a given theory must satisfy. However, they do not tell anything about the theory, apart from being a reasonable and acceptable relativistic quantum theory. Consider, for example, the free scalar field theory in $d+1$ dimensions, with fields of mass $m$. This is a theory satisfying the Wightman axioms. However, we know that there are interacting theories that once renormalized are just free theories with a different mass (this would probably be the case for scalar $\varphi^4$ – yuggib Aug 6 '18 at 7:17
• whenever $d\geq 4$, and it is true for toy models of quadratic interactions in any dimension). Therefore, it is not possible to understand, just looking at the fact that a theory satisfies the Wightman axioms, which theory we are talking about. The problem with Yang-Mills (but also with $\varphi^4$ in $3+1$ dimensions) is that we are not able to formulate rigorously a theory describing such interacting fields, on which we could then check if the Wightman axioms are satisfied. In other words, we do not know what should be the (interacting) vacuum of the theory. – yuggib Aug 6 '18 at 7:25