Thanksgiving
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.
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.
name that famous physicist.
0)
i
“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.
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.
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?
“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.
“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.