Rigor In Quantum Field Theory

Axiomatic QFT is an attempt to make everything absolutely perfectly mathematically rigorous. It is severely handicapped by the fact that it is nearly impossible to get results in QFT that are both interesting and rigorous. Heuristic QFT, on the other hand, is what the vast majority of working field theorists actually do — putting aside delicate questions of whether series converge and integrals are well defined, and instead leaping forward and attempting to match predictions to the data. Philosophers like things to be well-defined, so it’s not surprising that many of them are sympathetic to the axiomatic QFT program, tangible results be damned.

The question of whether or not the interesting parts of QFT can be made rigorous is a good one, but not one that keeps many physicists awake at night. All of the difficulty in making QFT rigorous can be traced to what happens at very short distances and very high energies. And that’s certainly important to understand. But the great insight of Ken Wilson and the effective field theory approach is that, as far as particle physics is concerned, it just doesn’t matter. Many different things can happen at high energies, and we can still get the same low-energy physics at the end of the day. So putting great intellectual effort into “doing things right” at high energies might be misplaced, at least until we actually have some data about what is going on there.

From Sean Carroll at Cosmic Variance (emphasis added).

The rigor issues axiomatic QFT theorists are concerned about is mostly taught in the upper division undergraduate mathematics course called "Real Analysis", which I skipped in favor of applied subjects myself, betraying my biases on this issue.

Nobel prize winning physicist Richard Feynman publicly worried quite a bit about the rigor of quantum field theory and convergent infinite series in particular, but probably didn't lose much sleep over it. If you don't lose sleep over helping to invent the nuclear bomb, infinite series that might not converge are probably not going to do you in either

The emphasized sentence is really the key one from a practical perspective. This is the line between what we can say that we know as a result of the Standard Model with considerable confidence, and what we recognize that we don't know with current quantum mechanical equations.

The very short distance issue, at its root, boils down to assumptions in quantum mechanics that (1) particles are point-like, (2) space-time is continuous rather than discrete, (3) locality in space-time is always well defined, and (4) the effects of general relativity (as distinct from the effects specific to special relativity that is well accounted for in quantum mechanics) on how systems behave is modest. The tools of "analysis" in mathematics (i.e. advance calculus) are pretty much useless in situations where the first three assumptions do not hold. To do interesting mathematics without those three assumptions, you pretty much have to use "numerical methods" which means that you can basically only do calculations by having computers conduct myriad subcalculations that it would be impracticable to do by hand (which helps explain why research programs that abandoned those assumptions never really got very far until powerful computers were available).

The fourth assumption (the absence of extreme general relativity effects) is different in kind and more fundamental. General relativity inherently gives rise to all sorts of singularities, which mathematicians abhor, in the "classical" (meaning non-quantum mechanical) formulation of these equations, and any mathematically rigorous reformulation of general relativity on a quantum mechanical basis needs some sort of asymptotic limit that instrinically prevents the equations from blowing up in the wrong ways as one comes close to what would be the singularities in the normal formulation of general relativity.

Loop quantum gravity is one fairly successful research program (so far) at developing a way of addressing these issues of rigor in assumptions (2), (3) and (4) are dispensed with in lieu of a discrete space-time in which locality is only an imperfect and emergent implication of the theory at super-Planckian scales, and the effects of general relativity are fully accounted for by the theory. LQG is a work in progress, but hasn't yet hit the kind of seemingly insurmountable theoretical roadblocks to progress that have cropped up in String Theory, the other main approach in theoretical physics that is trying to formulate a rigorous treatment of gravity at a quantum scale.

The high energy issue also implicates the effects of general relativity in quantum systems, and also recognizes that there may be beyond the Standard Model physics that are unified or present new particles or forces at high energies rather than displaying the force specific symmetries observed at experimentally accessible energy scales.

The loop quantum gravity program has very little to say about these issues of high energy physics, but these issues are at the heart of String Theory.

The point about the limited utility of theorizing without sufficient data to provide much guidance is an astute one, and the story of String Theory is one of a theory with so many degrees of freedom that no one can find a unique version of it that can be derived from experimental observable and tested with current technology, or even technology conceivable in the next few decades.