“So can we agree that: Sean did a commendable job writing for a lay audience but got all the nits picked out of him by a group with more technical knowledge of the field than the intended audience? ”

Yes, certainly, Sean did a very good job, which we over-scrutinized a bit or two. But it was a nice discussion, anyway.

“http://arxiv.org/abs/1201.5868″

Well, wow! A real tour the force in the SM, impressive! Thanks for the reference, it’s a nice read.

“So then, what is the full picture here ?
Does the symmetry (still vague, symmetry of what exactly) imply that the connection will have special mathematical properties which will result in a force ?”

The key point here is your “symmetry of what exactly?”. Gauge theory involves groups, and when people hear the word, “group”, they automatically think, “symmetry!”. But that isn’t right. Think of linear algebra class: the set of all orthonormal bases of an inner product space is mapped into itself by the orthogonal group. But did your linear algebra instructor talk about the orthogonal group as the group of “symmetries of linear algebra?” Did he or she say that linear algebra exists *because* of this symmetry? Of course not. Likewise, on a manifold (let’s say one with no metric) at each point there exists a set of bases (of the tangent space, or of some more abstract space defined at each point), related to each other by a group G. This G is exactly like the one I have just discussed, ie it is not a symmetry of anything. Now if you want to differentiate things it is true that you need a connection. But this imposes no conditions because G wasn’t a symmetry in the first place. It is true that this connection has certain properties related to the structure of G, but this is not a restriction.

Anyway, I can tell you for sure that the statement, “connections exist *because* of symmetry” is just wrong. In explaining this stuff to the layman, it would be better to say, “connections exist because we need to differentiate things”. Alas, that doesn’t sound as Deep and Meaningful.

BTW, I am myself a physicist, not a mathematician. And this stuff really matters to physics, it’s not just philosophy. People who believe that symmetries are the Key to the Universe are likely to pursue research influenced by that error — for example, you often hear Important People saying that we would understand string theory better if we knew the Symmetry Principle On Which It is Based. Anyone working in that direction is making a big mistake. And there are lots of less grandiose examples.

“AFAIK the beta function for the YM theories has been evaluated only for the UV sector, while the expression in the IR sector is still unknown. I think you’ll have a hard time finding a formula for the IR beta function in a textbook or something. The UV beta function is easy to find, for example here:

http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory#Beta_function

But that does not help you deduce the running of the coupling in the IR regime.”

Well, you can’t trust a perturbative calculation for the beta function when the coupling gets strong. This is true whenever it happens, infrared, ultraviolet, whatever. So yes, the infrared beta function for asymptotically free Yang-Mills is unknown (unless some genius has a non-perturbative calculation I’m unaware of).

By the way, as it turns out the standard model gauge beta functions have been done to three loops (over a million Feynman diagrams!): http://arxiv.org/abs/1201.5868

“asymptotic freedom (or lack thereof) has absolutely nothing to do with the infrared behavior of the theory. Or if it somehow does, please enlighten me.”

It implies that the coupling gets stronger at lower energies, at least to the point where perturbation theory breaks down. So in the far infrared, yes, I agree the behaviour is unknown if you’re at strong coupling.

“Generally, I agree that SU(2)xU(1) YM may have weak-glueballs after all. OTOH, it is so far an open problem, so it would be fair that you and Sean also agree that it might not have them. I also agree that there is ample evidence for QCD confinement, but still…”

I think we’re on the same page now. I’m not sure what Sean thinks on the matter, but I’m okay with saying it’s an open problem. I was basing my earlier statements on the instinct that what works in the strong sector would also work in the electro-weak, though I have no proof. I thinks it’s a healthy conjecture at this point, though I would love to hear that someone could do the calculation to prove me wrong. The more we know about the phases of Yang-Mills the better, even if it doesn’t apply to the real world (yet?).

“the dimensional transmutation through conformal anomaly is a very interesting idea, but I didn’t see any model resembling the SM which implements the idea successfully enough. Still, it is something worth keeping in mind and maybe looking into a bit further.”

I agree.

“In addition, the covariant derivative does not contain an arbitrary connection field, but rather only the gradient of the gauge parameter. That is enough to localize the symmetry, and has nothing whatsoever to do with adding new interactions. To add an interaction you need to promote that gradient into an arbitrary field, and add a kinetic term for it. This has nothing to do with symmetry localization, aside from the _motivation_ for the form of the coupling to matter and self-coupling terms for the new field. So the symmetry localization does not imply interaction, but only suggests its form.”

I understand what you are saying now and I agree completely. But as Andrew @25 says, just go to unitary gauge. You can always add arbitrary arbitrariness for fun and without profit.

“It’s a bit of a nitpicking, yes, but then again people seem to speak of this too loosely to the uninitiated public, which then gets a wrong idea that symmetry is somehow a source of the interaction, and naturally get confused. ”

Nitpicking yes. I’m all for improving science communication to lay audiences, so how would you explain this technical point? Maybe approach the topic from the other direction: we need to introduce interactions into the theory to match nature, and the only way to do that without breaking all the nice things we want (Lorentz invariance, locality, unitarity) is to add some redundancy in the description (using four-vector fields for particles with two helicity states). But to make sure the redundant bits don’t contribute we need to make sure the interactions obey some constraints – different processes have to cancel etc. So we find these constraints and it turns out they have a remarkable geometric interpretation because…? I like this approach, but it gets to a point where you really need the math.

@26:
“Nevertheless, combining the two concepts proves to be useful in physics. While mathematicians are ignorant of this usefulness, physicists are ignorant of properly formulating what exactly is useful. That’s how the whole discussion came about. ”

Fair enough, but I don’t think it’s necessarily a lack of understanding or rigor on the part of working physicists, more a language barrier between physicists and mathematicians. Once we got on the same page I agreed with your point immediately.

So can we agree that: Sean did a commendable job writing for a lay audience but got all the nits picked out of him by a group with more technical knowledge of the field than the intended audience?

0)

i

name that famous physicist.

“However from a physics point of view it really wouldn’t make much sense to have a gauge symmetry if you didn’t give the connection a kinetic term.”

Oh, I completely agree. Physically, gauge symmetries don’t make much sense without connections, and vice versa. However, as Archie said in (11), if you run this by a mathematician, he would take the statement at face value, pick it apart and conclude that you are flat-out wrong. And technically, he’d be right. Gauge symmetries and connections indeed have absolutely nothing to do with each other.

Nevertheless, combining the two concepts proves to be useful in physics. While mathematicians are ignorant of this usefulness, physicists are ignorant of properly formulating what exactly is useful. That’s how the whole discussion came about.

I believe we can put this issue to rest now.

Best,
Marko

Aha I see what you are saying now. A related way to phrase your complaint is that people often say GR is special because it is ‘diffeomorphism invariant.’ Well in some sense that diff invariance is trivial, because obviously you can phrase any theory you like in arbitrary coordinates. The nontrivial aspect (which is usually taken to be implied, but I agree that it is lazy and confusing to do so) is that the gauge fields (connections) have to be dynamical. However from a physics point of view it really wouldn’t make much sense to have a gauge symmetry if you didn’t give the connection a kinetic term.

This is actually explained nicely in an appendix in Sean’s GR book, where he uses path integrals to illustrate the difference. Thinking about GR now, if you have a theory without a dynamical metric, you can say it’s ‘diff invariant’, but when you do the path integral you can simply fix the coordinates on the background manifold to some specific configuration and not worry about the extra redundancy in doing the integral. However with a dynamical metric you can’t do this–you have to integrate over all possible metric configurations, and because of this there is not ‘background manifold’ on which to fix your coordinates and you are forced to deal with the problem of overcounting physical field configurations in the path integral due to the gauge redundancy.

Another way of saying this is that the basic problem that demands you introduce local symmetries is that you want to describe particles with higher spin in a manifestly lorentz covariant way. The fields you use to describe the higher spin particles are the connections (well in GR it’s actually the metric not the connection but they’re closely related). If you don’t give the connections a kinetic term, then the content of the local symmetry is basically trivial–you can just pick a gauge, you will be left with a lorentz covariant theory. For example, if you take QED and say that the connection A_mu is not dynamical, then you just pick a gauge where A_mu=0 and you are left with a theory of massive dirac fermions (up to topological obstructions I guess, but then you are breaking the translation invariance of the vacuum whic is weird). Basically if you don’t give the connection a kinetic term than the local symmetry has no physical content and you might as well throw it out.

@23Pete

That’s a huge can of worms, and at the end of the day I think it’s one of those arguments where people have strong feelings but it’s impossible to settle it conclusively one way or another. I tend to think that given a set of axioms, the theorems and patterns inherent to that set of axioms ‘exists’ in the sense that it’s not a matter of opinion whether a given well formed statement can or can’t be proven from those axioms. But I don’t tend to think that there is a higher plane in which all possible sets of axioms exist. In any case reality only uses a very small subset of all possible mathematical systems.

I’ve hijacked most of the comments for this Sean’s post, so it probably won’t hurt much more if I do it yet again. I happen to have some free time on my hands atm, and the topics are interesting…

“I wondered about the recurring question of the unreasonable effectiveness of mathematics in explaining the fundamental aspects of the world.”

I wouldn’t say the effectiveness of mathematics is unreasonable. IMO, it is quite reasonable, and even to be expected, for the following reason. Math is not something that gets developed by mathematicians for some abstract reasons, and then 50+ years later discovered by physicists to miraculously work when applied to the real world. No, mathematicians and physicists alike work together to formulate various mathematical concepts, for the very purpose of describing the real world. Consequently, math is so effective by design. The motivation for introduction of various math structures came from the need to describe the real world. It is then only to be expected that these structures are well-suited to do their job, and provide effective descriptions of the world. There is nothing unreasonable in this.

There are many examples of various branches of mathematics, that came out of solving problems for physicists — calculus, Riemannian geometry, partial differential equations, functional analysis… The main exception is group theory, which was developed with a completely different motivation, and subsequently found applications in physics, geometry, topology, etc. But in this case, I think that Galois just happened to stumble on a concept which is extremely useful in general, not just in physics.

There are also branches of mathematics, developed by mathematicians without input from physics, and which have motivation independent of physics. Naturally, they do not find so much applications in physics, and cannot be considered to be “unreasonably effective” for describing the real world. For example, formal logics and set theories, number theory, some parts of abstract algebra, etc. None of these are being heavily used for physics, due to a simple reason — they are suited to solving different problems than those present in physics, and consequently serve no central purpose. Maybe some marginal technical purpose can be found on occasion, but they are not central concepts for any physical theory.

So some branches of math are effective for describing the world, while some other branches of math are not. The two correlate very well with the motivation for their introduction having roots in physics or not having roots in physics. I see nothing mysterious in all that.

“I feel like some scientists/mathematicians get uneasy with the religious notions that come up with platonism”

Platonism is a philosophical viewpoint, and we should probably keep religion out of the discussion (religion and God have very little, if anything, to do with all this stuff). If one is not a very hard-core materialist, one is at peace with the idea that abstract concepts (like laws of physics) exist (somewhere). Since their effects in the material world are just instances (i.e. examples) of the laws in action, the place where all such abstract concepts exist is called Plato’s “world of ideas”. Most of mathematical concepts are also elements of that set. AFAIK, Platonism (in one version or another) is typically accepted among scientists and mathematicians.

Of course, there are also die-hard materialists, which mainly deny the existence of any abstract laws and concepts whatsoever. They believe that the observed regularities in nature (from which we try to infer the “existence” of abstract laws) are nothing but a statistical accident. I find that hard to believe, but there are people who would say otherwise (and maybe some of them might speak up on this blog).

HTH,
Marko