This year we give thanks for an idea that is central to our modern understanding of the forces of nature: gauge symmetry. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, the Spin-Statistics Theorem, conservation of momentum, effective field theory, and the error bar.)

When you write a popular book, some of the biggest decisions you are faced with involve choosing which interesting but difficult concepts to tackle, and which to simply put aside. In *The Particle at the End of the Universe*, I faced this question when it came to the concept of gauge symmetries, and in particular their relationship to the forces of nature. It’s a simple relationship to summarize: the standard four “forces of nature” all arise directly from gauge symmetries. And the Higgs field is interesting because it serves to hide some of those symmetries from us. So in the end, recognizing that it’s a subtle topic and the discussion might prove unsatisfying, I bit the bullet and tried my best to explain why this kind of symmetry leads directly to what we think of as a force. Part of that involved explaining what a “connection” is in this context, which I’m not sure anyone has ever tried before in a popular book. And likely nobody ever will try again! (Corrections welcome in comments.)

Physicists and mathematicians define a “symmetry” as “a transformation we can do to a system that leaves its essential features unchanged.” A circle has a lot of symmetry, as we can rotate it around the middle by any angle, and after the rotation it remains the same circle. We can also reflect it around an axis down the middle. A square, by contrast, has some symmetry, but less — we can reflect it around the middle, or rotate by some number of 90-degree angles, but if we rotated it by an angle that wasn’t a multiple of 90 degrees we wouldn’t get the same square back. A random scribble doesn’t have any symmetry at all; anything we do to it will change its appearance.

That’s not too hard to swallow. One layer of abstraction is to leap from symmetries of a tangible physical object like a circle to something a bit more conceptual, like “the laws of physics.” But it’s a leap well worth making! The laws of physics as we experience them here on Earth are, like the circle, invariant under rotations. We can do an experiment — say, the Cavendish experiment to measure the strength of gravity between two test bodies — in some given laboratory configuration. Then we can take the entire laboratory, rotate it by a fixed angle, and do the experiment again. If you do it right, you will get the same result, up to experimental errors. (Note that the Cavendish experiment is wickedly hard, so don’t try this at home unless you’re really up to it.) Likewise for other kinds of experiments, like measuring the charge of the electron. The laws of physics are invariant under rotations: you can rotate your experiment and get the same result, just like rotating the circle leaves you with the same geometrical figure.

Now to kick it up an additional notch, imagine you have a friend located in the lab down the hall, doing the same experiment. They will get the same results that you do for the strength of gravity or charge of the electron. That’s due to another symmetry — the laws of physics are invariant under translations (changes of position). And, of course, the invariance under rotations still holds; if anyone were crazy enough to pick up both labs at once, rotate the whole building by some fixed amount, put them back down, and do the experiments again, we would once again expect the same answer.

Your intuition tells you that there’s more to it than that, and your intuition is right. We don’t have to pick up the whole building with both labs inside; we should be able to rotate the apparatus in just one of the buildings, leaving the other one unchanged, and *still* get the same experimental results. But notice that this isn’t a single rotation of the whole world, as in our previous examples; now we’re rotating the two experiments separately, so their orientation changes with respect to each other.

That’s a gauge symmetry: when a symmetry transformation can be separately carried out at different points in space. Gauge symmetries are sometimes called *local* symmetries, since we can do them independently (locally) at every point; they are to be contrasted with *global* symmetries, which need to be done in a uniform way all over the place. It can be confusing, because “local” sounds like it’s less than “global,” whereas really a local/gauge symmetry represents enormously more symmetry than a mere global symmetry — infinitely more, since the transformations can happen completely independently at every point.

Fair enough, and hopefully it all makes sense. Here’s the subtle point: how do you *know* if one laboratory has been rotated with respect to another one? How are you able to compare the orientations of laboratories at different locations?

Doesn’t sound like it’s too difficult a question; you can use some surveying equipment, or for that matter just look at the other experiment if they’re close enough together. But while doing that you are taking advantage of the structure of space itself, something so fundamental that we typically don’t even notice it’s there. In particular, we have the means for comparing locations and orientations of distant circumstances, by traveling back and forth between them or sending signals of some sort. As we travel (or signals propagate), we are able to keep track of the location and orientation of the circumstances we left behind. Pretty amazing, when you think about it.

In order to compare things that are set up at different locations, what we are implicitly relying on is a *field* that stretches between the locations. The mathematical name for the kind of field we need is a *connection*, because it helps connect what’s going on at different points. In physics it’s called a *gauge field*, because Hermann Weyl introduced an (unhelpful) analogy with the “gauge” measuring the distance between rails on railroad tracks.

You might think of a gauge field as a latticework of invisible lines running through the universe, keeping track of what counts as “staying parallel” and “moving on a straight line” as we travel through space. But it’s a venerable principle of quantum field theory that, once you have a field, that field can have its own dynamics — it can bend and twist through space, typically in response to other fields that it interacts with. And when your gauge field starts twisting, you feel it as a force of nature.

Think of you and your friend doing separate experiments. If you were just in different rooms in the same building, you can travel between them on a flat floor, and you aren’t feeling any forces. But if you’re doing your experiments outdoors on a rolling hillside, the ground beneath your feet pushes you back and forth as you walk over the hills. In this case, the structure of the ground itself defines a connection field, and its curvature gives rise to a force.

That’s literally a down-to-Earth example. More fundamentally, there is a connection field on spacetime itself, which tells us how to walk on straight lines (geodesics) and compare orientations at different points. And this connection can be curved, and that curvature gives rise to a force of nature, one we call “gravity.” We’ve just invented the theory of general relativity.

General relativity is based on a rather straightforward set of symmetries: the rotations and translations we’ve already mentioned, plus “boosts” relating frames of reference moving with respect to each other. (All told, the Poincaré group.) What about the other forces — electromagnetism and the strong and weak nuclear forces? Nothing nearly so tangible, I’m afraid. These are all based on “internal” symmetries — they don’t transform things within space, but rather rotate different fields into each other. For example, you may have heard that quarks come in three different colors: red, green, and blue. It doesn’t matter what color you call a particular quark; therefore, there is a symmetry in which you rotate different colors into each other. Mathematically it takes the structure of the group SU(3), and the gauge field associated with it gives rise to the strong interactions. Electromagnetism and the weak interactions follow a simple pattern. Gluons, photons, and W/Z bosons all arise from different kinds of connection fields relating the symmetry transformations at different points in space.

Electromagnetism, indeed, was the first force for which we were able to understand that it was based on a gauge symmetry. General relativity was next, but interestingly the fact that GR is based directly on spacetime symmetries rather than internal symmetries actually makes it something of a special case, so the connection (pardon the pun) wasn’t as obvious. (Although it’s right there in my GR book.) It was Yang and Mills in the 1950’s who took the bold step of suggesting that gauge theories might be at the heart of the nuclear forces as well, although similar notions had been contemplated before.

The reason why the Yang-Mills idea wasn’t tried earlier, and didn’t catch on right away, is that forces based on gauge symmetries seem at first blush to have a universal and immediately-noticeable feature: they stretch over infinitely long ranges. That is the case for both general relativity and electromagnetism, and the mathematical structure of connection fields seems to imply that is should always be true. (This is a statement I could not for the life of me think of how to justify at a hand-waving level — anyone have any ideas?) In particle-physics language, the boson particle you get by quantizing the gauge field should be massless, like the photon and the graviton. But the nuclear forces are manifestly short-range, so the idea wasn’t immediately successful.

The answer to this dilemma is a little something called … the Higgs mechanism! By introducing yet another field (the Higgs field) that has a nonzero value everywhere in space, you can give the gauge bosons a mass in a way that is completely compatible with the mathematics. It’s the triumph of that idea has been seemingly vindicated by the discovery of the Higgs boson.

Interestingly, it turns out that Yang-Mills theories don’t have to give rise to long-range forces even if the bosons do stay massless. Imagine there were no Higgs field (and also no other effect that led to spontaneous symmetry breaking), so that the W and Z bosons of the weak interactions (or their pre-symmetry-breaking precursors) remained exactly massless. Unlike the photon, these bosons interact directly with each other, and at low energies those interactions would become very strong. Sufficiently strong that weakly-interacting particles would be confined, and the weak force wouldn’t be able to stretch over long distances. This is, of course, exactly what does happen with the strong nuclear force; gluons are massless, but the strong force is confined and therefore short-range. Perhaps we’re lucky that the physics of confinement wasn’t discovered until after the Higgs mechanism, or the latter might have taken a long time to figure out.

Well, now, the strong nuclear force is not directly from the confined chromodynamic interactions, its the residual force from pion exchange. Which would be long-range if the pion were massless, would it not? So the short range of the strong nuclear force comes about because quarks are not exactly massless, so pions are pseudo-Nambu-goldstone bosons and have a small mass.

Sean, the clarity with which you explain physics never seizes to amaze me. I am thankful that science has communicators like you amongst us. I finally understand what a gauge symmetry is.

Sean, you are the best writer who can explain so well about physics. I am still reading your new book. Happy Thanksgiving to you. By the way, what is the latest about “moving naturalism forward”? I have not seen the latest post on the last workshop.

Last paragraph:

“Unlike the photon, these bosons interact directly with each other, and at low energies those interactions would become very strong. Sufficiently strong that weakly-interacting particles would be confined, and the weak force wouldn’t be able to stretch over long distances.”

This is not quite correct. The strength of the interactions is dictated by the values of the coupling constants. The coupling constant of the strong force is of order one or greater, and implies color confinement (btw, even this is still under a question mark, an open problem). However, the coupling constant of the weak force is of the same order as electromagnetic force (which is 1/137), so it is too weak to imply any confinement at all. The W and Z bosons do interact among themselves, but this interaction is too weak to keep them confined into a massive bounded state. There certainly exist solutions in which W and Z are free (i.e. not bounded), and if there is no Higgs field to provide them with mass, they are massless and consequently long-range, like electromagnetism.

So the interactions between W and Z are *not* sufficiently strong to imply confinement. Btw, that’s one of the reasons why the weak force is called “weak”, and the strong force is called “strong”. The weak interaction is just not strong enough to require a confined bound-state for W and Z.

HTH,

Marko

Marko,

The coupling constants change with distance due to renormalisation. The strong and weak

interactions are both “asymptotically free,” meaning that as you go to very short distances the couplings go to zero and quarks/weak doublets become essentially free particles. Conversely, as you go to large distances the couplings increase and you end up in a strong coupling phase, the exact nature of which depends on the number of flavours of quarks/leptons and whether or not the Higgs mechanism is operating. In the case where there is no Higgs and there aren’t too many flavours you end up in a confining phase. That’s what happens for quarks and (I believe) it would happen for the weak doublets as well without a Higgs, though I haven’t checked the math.

I’m curious if Sean or anyone else has a reference? I would be interested in the low energy phenomenology of an unbroken standard model.

Cheers,

Michael

Michael is right; coupling constants are not constant in QFT, they depend on the scale at which they are measured. Whether or not they actually blow up at zero energy, leading to confinement, depends on the size of the gauge group, as well as the number fermion fields and scalars. To be honest I had in my mind that the weak interactions would be confining in the absence of a Higgs expectation value (but with the Higgs field), but I don’t have the formula for the beta function (including scalars) in front of me, so that might not be general — there might be some hidden assumptions. If I remember I’ll try to look it up after the holidays.

I vaguely remember at some point seeing that the weak force is confining with a 0 vev Higgs or without a Higgs entirely. I feel like it’s just SU(2) general.

Bravo! Thank you very much for this excellent explanation on gauge symmetry! Another unfortunately complicated name that makes it intimidating to learn about, but Sean makes it easy to understand. The only bad thing about the post is I have to wait a whole year to hear what we are giving thanks to next – but it is more than worth the wait

I got lost at one point:

So, I am thinking of a local transformation as a bunch of arrows distributed over space, each pointing in different directions.

A connection allows me to pick up an arrow at one point, and place it “parallel to itself” at another point.

If this analogy holds, what does a local symmetry imply ? Some kind of restriction on the connection ? And how does that give you a force ?

@5. Michael:

I am aware of the renormalization flow of coupling constants. But I am not so sure that the weak coupling would grow so large in the infrared regime. For one, in QED it doesn’t happen — the fine structure constant stays small at zero energies (that’s essentially where we measure it to be 1/137). In addition, even if the weak coupling would be large in the infrared, we still don’t know that it requires confinement. We don’t know it even for QCD, where we observe confinement experimentally. Confinement in Yang-Mills theories is an open problem, one of the unsolved Millenium problems in fact.

So Sean’s statement that I complained about is not an established result, but rather just a guess.

As for the unbroken standard model, the phenomenology can be worked out, but that is not neccesary since the whole theory is then conformally invariant, i.e. scale-free, i.e. wrong. Unbroken conformal invariance is enough to rule the model out — we do not observe it in experiments, but rather only Poincare invariance. Conformal symmetry would lead to additional conservation laws, scale-free nature, etc. It doesn’t work.

In order to introduce scale (and break the conformal group down to Poincare group), you need either a Higgs field with nonzero mass, or a SM coupled to gravity. If we ignore the gravity story (which is IMO a bad idea), a massive Higgs is the only way to reduce the spacetime symmetry down to Poincare, and this mass will also break the rest of SM gauge group in the usual way.

HTH,

Marko

OK, you explain two different things and it is very unclear what they have to do with each other.

(a) Gauge “symmetries” (actually redundancies, not symmetries, but let’s leave that aside.)

(b) Connections.

The standard lore is that somehow (b) exist *because* of (a), and that is implicit in the idea that gauge fields exist “because” of gauge symmetries. But that of course is false, as you know: it is perfectly possible to define a linear connection on a manifold that has nothing like Poincare symmetry, in fact it need not even have a metric.

I know that people just love this idea that connections exist “because” of gauge symmetry. As therapy for that I suggest that people run this one past a mathematician. “Hey, do you know why connections *have* to be defined on principal bundles? Well, it’s because principal bundles have this symmetry, you know, the group of all vertical automorphisms preserving a flat connection….”

Hint: they won’t be sympathetic.

This idea is just one of those strange fictions (another one is that the principle of equivalence is the reason for the existence of gravitational fields, and all the poppycock that goes with that…) which linger out a spectral life in the minds of physicists, because of historical accidents. Repeat after me: History is Bunk!

Anyway, you did a good job of explaining (a) and (b) separately. Which is how they should be kept.

Archie @11:

Ok then, so I’m not the only one who found Sean’s explanation very confusing.

I thought I saw two different concepts and felt the connection (ahem) was missing.

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 ?

Sean, gauge symmetries are NOT physical symmetries, they are required only to eliminate unphysical degrees of freedom. One can instead start with a gauge fixed action and derive everything. So they can’t possibly be symmetries if they don’t exist in the lagrangian. Only global symmetries are physical and are always present in the lagrangian, even after gauge fixing.

Cosmonut@12:

symmetry of the dynamical properties of a system of space while it is changing at different rates (symmetry of the eigen-stuffs?). the properties of distortions in fields; or some crap like that.

I think bosons are the result of field connections interacting with each other, bosons causing the resulting forces of nature experienced by matter interacting with those bosons. So yeah, after a few derivative concepts, they do have special properties that result in a force.

I may be incorrect; but I think the idea of connections being a product of local symmetries is necessary to explain Physics as we know it. But I guess this could quickly deteriorate into an argument about the structure of spacetime. As well as an argument over the validity of physics “as we know it”.

I thought there were redundant degrees of freedom in the lagrangian ?

@Archie11

I am a physicist, and I don’t know much about what a pure mathematician would say about physics (other than I imagine they would probably get angry and rant about things like ‘rigor’), so I have a different perspective from you. But to me it seems relatively clear that if you want a local symmetry then you will end up needing a connection. I believe this what people mean when they say ‘connections exist because of gauge symmetry’, maybe you are upset that sometimes the statement is said somewhat ambiguously and could be taken to mistakenly claim the converse?

The intuitive reason that local symmetries require the use of connections (which can be backed up with equations) is simply that you will want to take derivatives of your fields if you want to have a theory that does anything interesting. To take a derivative of something means comparing its value at two different spacetime points, but we know that we can independently ‘rotate’ the fields at the two different points. So our derivative must correct for our freedom to rotate the fields, and the way to make this correction is to use a covariant derivative instead of a regular derivative. In order to correct for the freedom to rotate, the covariant derivative must have a piece that itself rotates, and this is of course the connection. So the need to differentiate fields with a local symmetry means that we need to introduce a connection to make covariant derivatives. All of this is what Sean said in his post just with more technical sounding words.

Having said all that, it is true that local symmetries are somewhat fake and really reflect the fact that describing particles with spin >1/2 in a manifestly lorentz covariant way is awkward. I’m uncomfortable with saying that the photon (in this context the U(1) gauge connection) exists * because * of the need to have a U(1) covariant derivative acting on fermions, to me that seems a little bit like mixing up the formalism for the physics, but I’m sure others disagree with me.

wasn’t it Feynman who used to joke about how often the term ‘trivial’ is used in mathematics.

@12. Cosmonut:

“Ok then, so I’m not the only one who found Sean’s explanation very confusing.

I thought I saw two different concepts and felt the connection (ahem) was missing.

So then, what is the full picture here ?”

The local gauge symmetries give you rules how each quantity in your theory should transform under the action of the local symmetry transformation. Based on those rules, one can *generalize* the theory by introducing connections, chosen approprietly, which describe new fields which are called “forces”.

In other words, the structure of local gauge symmetry provides a “motivation” to introduce new forces in a particular way, with some particular properties. Those forces are described by connection fields. However, as Archie (11) has remarked, the connection fields themselves have nothing else in common with the gauge symmetry group. Only motivation about how the equations of the interacting theory should be written down.

HTH,

Marko

@15. Andrew:

“But to me it seems relatively clear that if you want a local symmetry then you will end up needing a connection.”

This is incorrect. You can have a theory with local symmetry without a connection (i.e. with a trivial one). See below.

“In order to correct for the freedom to rotate, the covariant derivative must have a piece that itself rotates, and this is of course the connection.”

No, it isn’t. One certainly does need an additional piece in the derivative, but a necessary and sufficient condition for local symmetry is that this piece is a pure gauge term. Putting an

arbitraryconnection term in its place is an extra step, one that introduces interaction. This is much stronger — it is sufficient for local symmetry, but not necessary, and it represents a generalization of the original covariant derivative.If you are familiar with general relativity, the introduction of gravity is precisely like that — start from some theory for matter fields in flat Minkowski space. Then localize the translational symmetry by rewriting your theory in curvilinear coordinates. You end up with a theory that has local symmetry (it becomes diffeomorphism-invariant), but you have not introduced any new interaction (the spacetime is still flat). However, when you promote the “pure gauge” term in the covariant derivative to a completely arbitrary quantity (a connection), the theory becomes nontrivial, the spacetime curvature becomes nonzero, and the connection gives rise to a new interaction (called “gravity”).

So there is an extra step from a covariant derivative which obeys local gauge symmetry to a covariant derivative which has a gravitational connection. Ditto for the strong and electroweak force.

HTH,

Marko

@11 Archie

I didn’t get from Sean’s explanation of the expediency of connections when making use of local gauge symmetries to characterize forces the assertion that local gauge symmetries are indispensable machinery if one is to define connections on a manifold. I can’t speak for Sean, but I think you missed the point.

Are you just flaunting the generality of your mathematical knowledge to be pedantic? If so, that’s fine of course. It’s one of the perks of being a mathematician, I know. Don’t waste it, though. It makes us seem petty. ;-p

Last time I checked, humans do not convey “thanks” to terrestrial concepts, least of all allocate a special holiday for it. But Sean uses it figuratively, not literally, as an excuse to write a tutorial on gauge theory, about which my subject is a noted expert.

However, today, there is a Great theoretical physicist in our community who has No reason to thank anyone, & is going thru A Hell most of us might have nightmares about. I’m of course referring to Prof. Paul Frampton, U.N.Carolina, who’s just been sentenced to 5 yrs imprisonment in a Argentinian prison. He is ill, & looking at a hopeless situation, despite small efforts by physicists in the US to assist him. UNC has done nothing to help, & has abandoned him:

http://physicsworld.com/cws/article/news/2012/nov/22/paul-frampton-hit-by-56-month-drugs-sentence

Whether or not you believe him guilty, it is unconscionable that the US govnt. has let such a distinguished American rot in jail in a foreign country for almost 300 days, & exerted zero effort to help him. If you want to give “Thanks”, just imagine yourself in his place, & you’ll have plenty of reasons to appreciate your life & freedom, & realize that finally, the US-led international drug war has taken one of our own as its latest victim.

When the Soviet Union put Andrei Sakharov under house arrest, American physicists exerted considerable influence on the US govnt. to do likewise on the USSR govnt. Can we not do the same for our fallen comrade ?

@6. Sean:

I don’t have the beta function in front of me either. Thanks for a good thanksgiving discussion.

@10. Marko:

Ok. To find the evolution of the coupling constants you need to know their beta functions, which I don’t have in front of me. But QED is _not_ asymptotically free. In fact it’s infrared free. If the electron didn’t have a mass to give an infrared cutoff to the RG flow the coupling would go to zero at large distances (the reference I have for this is Shifman, Advanced Topics in Quantum Field Theory), and it goes to infinity at the Landau pole. This is the opposite behaviour of nonabelian theories that don’t have too many fermions. This may or may not include the weak sector of the standard model with a zero Higgs vev – I haven’t checked the beta function.

You’re correct that strong coupling does not imply confinement, but the one example we have where we can look (QCD) is confined. Also, lattice gauge calculations and phenomenological (i.e., hand-wavy) arguments (e.g., likening the QCD vacuum to a chromo-magnetic superconductor and talking about flux tubes etc.) support the picture. Also, I read a good discussion, I think by t’Hooft, about how confinement does not necessarily imply that you never see free quarks. It has to do with the relation between the quark masses and the QCD scale. I can chase up the reference. So no, there is no mathematically rigorous proof of confinement in Yang-Mills theory, but the circumstantial evidence is stacked in its favour.

I’m aware that the unbroken standard model is not the correct theory of nature – I’m interested in it as a mental exercise. I’m not sure it’s conformal though. It needs to be checked. The Lagrangian is conformal – definitely. But there is an anomaly and a possibly non-trivial beta function. This has the effect of introducing a scale (dimensional transmutation). The simple fact that there are no parameters with non-zero mass dimension in the Lagrangian is not enough to call a theory conformal, at least at a quantum level. There have been attempts to develop unified field theories that only get a scale through the conformal anomaly – no Higgs and no gravity in the bare Lagrangian. I’m sure you’re aware of this.

@11 Archie and @17-18 Marko:

I’m not a mathematician, so forgive me for trying to rephrase what you are saying. Are you simply saying that you can write down a theory with covariant derivatives but no kinetic term for the connection? That’s true of course, but also kind of boring. It violates the spirit of relativity: everything is dynamical. Fixed backgrounds are to be avoided. Whether it’s necessary or not is a matter for experiment, so it’s good to raise the possibility from time to time. Also, I’m not sure this is consistent with quantum mechanics… wouldn’t matter fluctuations induce a kinetic term for the connection anyways…?

@21. Michael:

“To find the evolution of the coupling constants you need to know their beta functions, which I don’t have in front of me.”

I agree, but 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.

“But QED is _not_ asymptotically free.”

Asymptotic freedom is again a UV effect, deduced from the UV sector of the beta function. While I agree that QED, being Abelian, may be a poor example for what I wanted to say, 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.

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…

“There have been attempts to develop unified field theories that only get a scale through the conformal anomaly – no Higgs and no gravity in the bare Lagrangian.”

Yes, 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.

“Are you simply saying that you can write down a theory with covariant derivatives but no kinetic term for the connection?”

Pretty much, yes. 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 implyinteraction, but only suggests its form.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.

Best,

Marko

While reading this article I wondered about the recurring question of the unreasonable effectiveness of mathematics in explaining the fundamental aspects of the world. I know a vast number of mathematicians, physicists, and philosophers subscribe to the idea of mathematical realism or platonism (I myself do as well), but I wanted to get some thoughts from Sean and others who have commented. I know some people think mathematics is simply a “useful tool for explanation,” but I feel like that sentiment amongst nominalists gets crushed with examples of physicists utilizing abstract mathematical objects decades after they’re discovered and finding that they predict certain things found in nature. I feel like some scientists/mathematicians get uneasy with the religious notions that come up with platonism, but I think its possible to think about mathematical platonism as the universe (or multiverse) being inherently mathematical in some deep and abstract way (God is mathematics if thats how you want to label it). I know this comment doesn’t pertain much to gauge symmetries directly, but this article just got me thinking more about mathematics and its relation to reality. Anybody have any ideas?

@23. Pete:

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

@22vmarko

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.