In Quantum Field Theory quantum fields are operator valued distributions. Namely, given the Schwartz space $\mathcal{S}(M)$ defined on Minkowski spacetime $M$, fields are continuous linear maps $\phi : \mathcal{S}(M)\to \mathcal{L}(\mathcal{H})$ which gives back for each function one operator in some Hilbert space.
When just free fields are considered this is just fine. Now, when one considers interactions, the issue that appears is that we have to deal with products of quantum fields.
For instance, the $\phi^4$ theory has the interaction given by $\lambda \phi^4$. The scalar Yukawa theory also has products of fields.
The field equations become non-linear in the fields. The issue now is that if quantum fields are distributions, then necessarily we can't deal with these terms, because products of distributions is not defined. Physicists just ignore this most of the time.
My question here is: is there a rigorous known way to deal with this? Can we make this rigorous someway, even if we need to go into approximate theories? Or there is no way currently known to make this rigorous?
Edit: For more information on this matter, I've asked on Physics.SE about the issues with interacting QFT and found out the main problems. This product problem seemed the worst, and since the point here is about whether or not a mathematical construct can be made rigorous I thought that Math.SE could be a good place to discuss the matter further.
Best Answer
I'm definitely not a QFT expert, but I believe this is what Martin Hairer has been working on. In particular, his Regularity Structures allow for a rigorous treatment of products of distributions that arise in QFT and other nonlinear stochastic PDE scenarios. He gives a fairly easy to follow talk on the subject here, where I seem to recall he discusses $\Phi^4$ renormalization theory. I don't recall if he addresses operator valued distributions, but it doesn't seem too difficult to believe that if you can work out the theory for scalar valued distributions, operators are not too far off via spectral theory.