It doesn’t make much sense to be concentrating on deriving something from first principles when we have no clue as to what those principles are or what they should be deriving. That sounds seriously under-constrained.

This isn’t true at all. If it were, physics would only matter to physicists. And there would be no reason for the physicists to expect us to pay anything for their hobby. Axiomatization is a procedure for understanding, which is what physics (indeed, all science) really is about. As such, axiomatization of a true theory must be desirable. The issue is whether axiomatization is feasible, and if it is, whether QFT is the true theory that should be axiomatized in order to understand. It is doubtful that axiomatization is the only way to understand a theory, so it is doubtful that QFT must be axiomatized as part of validating and interpreting the theory.

Sometimes you argue as if Oldershaw were predicting a “solid-body” electron with a radius of 4 x 10^-17 cm. Of course, we know he is doing no such thing.

You say: “Either it is a naked singularity, or it has a substructure.”

But that is a “dumbell argument” with all the weight on two opposite extremes, i.e., it is an excluded middle argument. Imagine, if you can, a singular electron that is shrouded in a low-density envelope of charged and relatively infinitessimal particles, with the envelope having a radius of 4 x 10^-17 cm.

So there you have a viable model for a virtually singular electron that does have substructure, which would be very hard to detect but not impossible.

If you have a vendetta against Oldershaw, fine. But I think it is a mistake to summarily rule out his discrete fractal paradigm without a more open-minded and balanced evaluation. It has many positive features, in addition to a small number of unresolved issues [and what theory does not?].

Why focus on a couple of moles and ignore the global sweep, potential for unification and elegance of the paradigm?

Albert Z

A quick web search reveals otherwise. By chance, just today I was reading an article on the substructure of the proton as measured at HERA (an electron-proton accelerator in Hamburg) and the comparison with QCD predictions. The agreement was great, and certainly wouldn’t be possible if the electron had a structure a couple of orders of magnitude bigger than that probed in the proton.

(2) You conveniently and unscientifically ignore the fact that Oldershaw mentioned in the same paragraph of the 1987 paper that it was also possible that the electron was a naked singularity.

Either it is a naked singularity, or it has a substructure. A ‘prediction’ which predicts both predicts neither.

(3) Since 1987, which was quite a while ago, Oldershaw has settled on an electron model wherein the electron is a virtually naked singularity that has a low-density envelope of charged subquatum particles, with an envelope radius of 4 x 10^-17 cm.

There have been a few blog posts on the Superbowl recently. I think this is termed ‘moving the goalposts’ in football jargon.

Please reread comment 17. One certainly *can* formulate a set of working rules and have a well-defined theory, as long as one of the rules says “don’t apply these rules beyond a certain point”. In the QED example, the “certain point” would be the 136th order of perturbation, or so. This extra rule limits the applicability of the theory in a very ad hoc way, without properly understanding why.

If you want to have a theory which doesn’t have such axioms (that limit the applicability of other axioms), then in QFT you may easily run into a problem with the theory being (a) self-contradictory, and/or (b) non-predictive, and/or (c) experimentally incorrect.

If you are a mathematician, you might not care about (c), but you indeed must care about (a) and (b), which is the very problem of axiomatic QFT. The solution to this problem might or might not exist.

If you are a physicist, you must care about (c) first, and about (a) and (b) only if (c) is true. In the case of QFT, (c) is known to have failed, ie. QFT does have predictions that contradict observations (just calculate anything in QED beyond the 137th order of perturbation).

Therefore, given that (c) fails, physicists consider QFT as an “effective theory”, ie. an approximation which is valid only up to a certain point. In that setting, caring about axiomatic formulation of QFT is just an academic exercise, and an irrelevant question.

So as a bottomline, the relevance of axiomatic formulation of QFT depends on your standpoint and perspective (and taste).

HTH

“You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won’t believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!” — Six Easy Pieces

The winner of the toss gets to DECIDE whether to kick or receive.

Despite what people say about gifts, NFL coaches usually find it more blessed to receive.

I know Sean is no stranger to these results and is probably restricting the discussion to role of QFT in particle physics.

As for some simple underlying mathematical structure… now that sounds hard. But just producing a mathematically well-defined set of rules for predictions should be simple. And that makes QFT “rigorous”, if not beautiful.

The Sigma for Higgs taken in context of the Bayesian analysis of what the pre-test expectations for the experiment are, and accounting for 2 separate experiences coinciding makes the chance of the phenomenon being real much higher than simply saying that they are likely to be wrong 1/1000 times, and that kind of error is pretty common.

By the way, same goes for the superluminal neutrinos. There are more than one set of experiments revealing the same result, which raises the Bayesian chance of it being a true finding. That does not mean that they beat the speed of light, only that they traverse known space faster than photons can. There may be other reasons besides speed that they do that (like unknown space shortcuts only they have access to). Either way, the street would say the odds of the findings being real are higher than a single result would indicate.

I think my point is that this actually may *be* a physics problem, and not strictly speaking a mathematics issue. I say that for a number of reasons!

First there is the observation that strictly speaking we must take infinite volume limits when defining these theories. Many of the big problems occur there. Well, that may or may not be physically justified in principle if you believe in holographic complementarity.

Two, take QED as an example. I could very easily say that we need 136 terms in the asymptotic expansion, throw away the rest and simply define the theory that way. Its almost well defined (modulo some subtleties) and we are done! The mathematicians will cry foul, b/c in some sense what they want is a systematic method that spits out fundamental theories with well defined objects that work to arbitrarily high energies. They really believe that there is a nonperturbative answer that has ontological value (naively it seems that most exact answers to the path integral are simply infinity). Except that we now know that such an object is meaningless physically. It would need to be completed into the electroweak theory!

Three, it may simply be the case that physical QFT’s simply dont exist as well defined mathematical objects b/c for instance the world is really made out of strings (or some other fundamental theory) and they only spit out QFTs as low energy approximations. So perhaps the mathematics question should be about something else!

I don’t know! It just seems to me that the old attempts at defining QFTs were so unlovely, contrived and ugly (mathematically) that it is likely the case that it was humans putting some structure onto nature that she didn’t want. At least that was my read when I studied them.

“Quoting from this abstract: “Two definitive predictions are also pointed out: (1) the model predicts that the electron will be found to have structure with radius of about 4 x 10 to the -17th cm, at just below the current [1987] empirical resolution capability”. Since current (2012) upper limits on the substructure of the electron are a few orders of magnitude lower than this, and there is no circumstantial evidence for substructure, it appears that the discrete self-similar paradigm has been ruled out since a definitive prediction was falsified. Maybe that’s why no-one takes it seriously.”

Albert Z says:

(1) No one has direct and/or reliable evidence for or against structure, especially regarding low-density envelopes or clouds of virtual particles, below a resolution of 10^-16 cm.

(2) You conveniently and unscientifically ignore the fact that Oldershaw mentioned in the same paragraph of the 1987 paper that it was also possible that the electron was a naked singularity.

(3) Since 1987, which was quite a while ago, Oldershaw has settled on an electron model wherein the electron is a virtually naked singularity that has a low-density envelope of charged subquatum particles, with an envelope radius of 4 x 10^-17 cm.

Less emotion, more accurate information, please.

Best,
Albert Z