Monday, December 10, 2018

In Physics, Rigor Is Often Late To The Party

Physicists often move forward using equations and mathematical models before knowing with certainty that they are consistent and have all of the necessary properties.

For example, Feynman himself suspected that renormalization, the process used to make the mathematics of quantum mechanics tractable, might not be mathematically valid in a rigorous way. He was wrong on that score. But, the proof of that came only a few years ago, about fifty years after he came up with the technique.

Another property of the Standard Model that has long been assumed, but has still only been partially established in a rigorous manner, is that it gauge invariant. Now a new paper has established that for an substantial and important subset of Standard Model processes.
For gauge theory, the matrix element for any physical process is independent of the gauge used. Since this is a formal statement and examples are known where gauge invariance is violated, for any specific process this gauge invariance needs to be checked by explicit calculation. In this paper, gauge invariance is found to hold for a large non-trivial class of processes described by tree diagrams in the standard model -- tree diagrams with two external W bosons and any number of external Higgs bosons. This verification of gauge invariance is quite complicated, and is based on a direct study of the difference between different gauges through induction on the number of external Higgs bosons.
Tai Tsun Wu, Sau Lan Wu, Gauge invariance for a class of tree diagrams in the standard model (December 6, 2018).

The fact that the Standard Model can be rigorously proven to have the mathematical properties that it should may seem like thankless work for a model that has worked well in applied applications for half a century. But, this is still important work because there are some kinds of mathematical defects that would be capable of going undetected for a long time period in practical applications that could point the way towards new physics that would otherwise have been assumed to involve possibilities that are believed to be impossible.

For example, the possibility that singularities could exist in General Relativity was determined to exist mathematically not long after the theory was proposed and was assumed to be a mere mathematical defect in the theory, but eventually, it has turned out that singularities like black holes and the Big Bang in General Relativity have physical meaning and are critically important phenomena necessary to understand the universe. 

A quote from Professor Susskind about string theory is relevant:
My guess is, the theory of the real world may have things to do with string theory but it’s not string theory in it’s formal, rigorous, mathematical sense. We know that the formal, by formal I mean mathematically, rigorous structure that string theory became. It became a mathematical structure of great rigor and consistency that it, in itself, as it is, cannot describe the real world of particles. It has to be modified, it has to be generalized, it has to be put in a slightly bigger context. The exact thing, which I call string theory, which is this mathematical structure, is not going to be able to, by itself, describe particles… 
We made great progress in understanding elementary particles for a long time, and it was always progressed, though, in hand-in-hand with experimental developments, big accelerators and so forth. We seem to have run out of new experimental data, even though there was a big experimental project, the LHC at CERN, whatever that is? A great big machine that produces particles and collides them. I don’t want to use the word disappointingly, well, I will anyway, disappointingly, it simply didn’t give any new information. Particle physics has run into, what I suspect is a temporary brick wall, it’s been, basically since the early 1980s, that it hasn’t changed. I don’t see at the present time, for me, much profit in pursuing it.

1 comment:

Mitchell said...

The earlier work by these two authors - ref 1 in this paper - has been challenged as incorrect.

I would have to think about it a little, but I think that the gauge invariance of the standard model would widely be regarded as trivially proven, on the grounds that once you understand how anomalies work, that's it. Either the theory has local anomalies and it breaks down, or it doesn't and it's OK.

On the other hand, I can find papers from the early 1990s saying that proving gauge invariance in QED is quite difficult, so I may be missing something.

One detail may be, whether these proofs are in the context of a particular calculation scheme. Then you have to worry about ghost fields, renormalization methods, etc. The attitude might be that proving that a QFT is gauge-invariant "in principle" is (relatively) simple, but then you also have to check that a particular calculational framework also gives gauge-invariant results. And if it doesn't, you don't say the theory is not gauge-invariant, you say that the method of calculation is incorrect.

Arnold Neumaier might be someone within our communities who knows the formal side of this topic.