Concrete statement about QFT not being mathematically rigorous [duplicate]

Question

It is often mentioned that QFT is ill-defined mathematically. I have seen this as stated that QFT can be defined on a lattice, but that it breaks down if the lattice spacing goes to zero. Discussions and other posts (e.g., Rigor in quantum field theory) go into some high-level details about where the trouble comes from, including citing large numbers of in-depth references. But these are not concrete in discussing an actual problem.

I understand the problem is conceptually difficult. Is there a simple, succinct, concrete problem statement that elucidates the issue?

For example, what is the simplest version of a QFT (in terms of type of fields, interactions, # dimensions, etc) in which the problem occurs? And what is the most basic calculation or definition that is problematic?

Answer 1

You'll need someone more experienced than me to check the epsilons and deltas, but to my knowledge we can actually formulate (most) of QFT in a rigorous manner. This is done, for example, through perturbative algebraic quantum field theory (pAQFT). This includes treatments of interactions, renormalization, and etc within perturbation theory.

There is definitely a caveat, though: to my knowledge, the perturbative series in pAQFT is merely formal, in the sense it does not converge (neither do the series in regular QFT, of course). An example is $\lambda \phi^4$ theory: the correlation functions can't be analytic at $\lambda = 0$ because the Hamiltonian is not bounded from below for $\lambda < 0$, implying the vacuum is not well-defined in that case. Hence, the converge radius for the Taylor series is zero.

To my knowledge, there is no rigorous formulation of nonperturbative QFT to this day. I find it one of the most exciting open problems in physics.

If you are interested in mathematical aspects of QFT, you might want to check algebraic QFT in general and also axiomatic QFT.