I am interested in a rigorous understanding of field theory. To be clear, I, personally, have very little such understanding. There are some reasonable axiomatic frameworks, such as the Wightman axioms, and certain theories can be shown to satisfy them, but Yang-Mills (aka. “gauge”) theories in 4 space-time dimensions are not among them. In other words, nobody so far has been able to show that Y-M theories in 4d space-time satisfy the Wightman axioms (or that they don’t). The Wightman axioms assume the existence of a Hilbert space that satisfies certain conditions. So a starting point in having an axiomatic formulation for a theory via Wightman is the understanding of what the Hilbert space is (and if it exists at all).

On the other hand, we have lattice gauge theory. This was originally formulated by Kenneth Wilson on a discrete 4d euclidean (i.e. with imaginary time) lattice of finite size. Kogut and Susskind derived a Hamiltonian version from the Wilson theory. A Hamiltonian theory is, by definition, something that includes a Hilbert space. However, in the case of gauge theories, this Hilbert space is unphysical, or larger than it needs to be. To obtain a physical space, one needs to impose gauge constraints. Doing so, however, does not necessarily make it clear what the resulting physical space is. Even if one has a nicely formulated physical Hilbert space on the lattice, in order to obtain the space of a continuum Lorentzian theory (aka. a proper field theory), one needs two more steps: taking the limit of infinite lattice with infinitesimal spacing between lattice points and moving from Euclidean space to Lorentzian space. It is known that if the Euclidean theory satisfies a certain set of axioms, called the Osterwalder–Schrader axioms, then there exists a mechanism to obtain a Lorentzian theory that satisfies the Wightman axioms.

To summarize, there are three main difficulties in getting from the unphysical Hilbert space of a Kogut-Susskind lattice theory (which we know well) to a well defined Yang-Mills theory:

1. Obtain the physical Hilbert space of the Kogut-Susskind lattice theory
2. Take the continuum limit
3. Obtain the Lorentzian theory from the Euclidean theory, possibly via Osterwalder–Schrader

I have worked quite a bit on the first part and it appears that much meaningful work has been done on the third. While this avenue seems promising, it remains to be seen if much will come of it.