Why do we need to make sense of QED in continuous spacetime anyway?

Question

QED is an approximate description of reality. Even if it did give finite predictions in the continuum limit, those predictions would've been incorrect anyway! Newtonian gravity does give finite predictions for high gravity scenarios, but the numbers are incorrect.

QED can be defined in a mathematically rigorously (non-perturbative) way in the discrete spacetime approximation! Then why do we need to make the continuum QED mathematically rigorous, even when its predictions in the tiny length scales will be incorrect anyway?

I also read that a non-perturbative formulation of Yang Mills is a Millenium problem. But Yang Mills is already non-perturbative in discrete spacetime, i.e. in the length scales it works in the first place. In the tinier length scales, it will be incorrect anyway (Gravity effects will kick in).

So it shouldn't be a problem that QFTs at tiny length scales aren't rigorous / contain infinite quantities. Non-perturbative definitions of the theories exist in the discrete spacetime approximation, i.e. in the domain the theories hold in the first place.

Answer 1

1. The premise of the question is flawed - many practitioners of QFT definitely do not really care that people interested in rigor think QFT is lacking. They care that their procedures, whether "rigorous" or not, produce correct models of reality, as QED and the Standard Model demonstrably do. So, in turn, people who are interested in rigor are less interested in producing alternatives (like you seem to think discrete QFT is) and more interested in putting what people in practice actually do on a rigorous foundation.

   In this viewpoint, the Millenium problem is really that: It is a fact that physicists non-rigorously use the language of continuous QFTs to describe the world. The challenge is not "find something else that produces the same predictions within acceptable margins", it's "explain what they're doing". Note that we, historically, didn't start with lattice QFTs and then desparately tried to make them continuous - people started by quantizing a continuous classical field theory and got results, however non-rigorously.

2. The claim that "QED can be defined in a mathematically rigorously (non-perturbative) way in the discrete spacetime approximation!" is certainly true in some senses, but the connection between lattice theories and continuum theories is much more subtle than it might first appear. For instance, $\phi^4$ theory is trivial in the continuum limit, fermion doubling complicates the treatment of fermions and there is the more general unease that a lattice inherently breaks continuous Lorentz symmetry, which is a feature of reality people would like to see reflected in their physical theory.

3. "QED is an approximate description of reality." Perhaps, but this is not the only thing a physical theory is. Yes, physical theories are approximate models of reality. But they're often also supposed to be consistent and perfect models of an idealized world. As far as we can tell, space in reality is not discrete, at least not in the "there's a fixed lattice" way. So a "beautiful" physical theory shouldn't be claiming that it is, even if it produces correct approximative predictions.

Now, none of these points are absolutely convincing arguments. You can still say "I don't care, I like lattice theories and don't think we really need anything else". That's fine - it's probably the exact motivation of at least some people who have devoted their life to lattice theory.