Since today is Halloween, I really wanted to write a post talking about the spookiest particles in physics, ghosts.
The problem is, in order to explain ghosts I’d have to explain something called gauge symmetry. And gauge symmetry is quite possibly the hardest topic in modern physics to explain to a general audience.
Deep down, gauge symmetry is the idea that irrelevant extra parts of how we represent things in physics should stay irrelevant. While that sounds obvious, it’s far from obvious how you can go from that to predicting new particles like the Higgs boson.
Explaining this is tough! Tough enough that I haven’t thought of a good way to do it yet.
Which is why I was fairly stoked when a fellow postdoc pointed out a recent popular physics article by Juan Maldacena, explaining gauge symmetry.
Juan Maldacena is a Big Deal. He’s the guy who figured out the AdS/CFT correspondence, showing that string theory (in a particular hyperbola-shaped space called AdS) and everybody’s favorite N=4 super Yang-Mills theory are secretly the same, a discovery which led to a Big Blue Dot on Paperscape. So naturally, I was excited to see what he had to say.
The core analogy he makes is with currencies in different countries. Just like gauge symmetry, currencies aren’t measuring anything “real”: they’re arbitrary conventions put in place because we don’t have a good way of just buying things based on pure “value”. However, also like gauge symmetry, then can have real-life consequences, as different currency exchange rates can lead to currency speculation, letting some people make money and others lose money. In Maldacena’s analogy the Higgs field works like a precious metal, making differences in exchange rates manifest as different prices of precious metals in different countries.
It’s a solid analogy, and one that is quite close to the real mathematics of the problem (as the paper’s Appendix goes into detail to show). However, I have some reservations, both about the paper as a whole and about the core analogy.
In general, Maldacena doesn’t do a very good job of writing something publicly accessible. There’s a lot of stilted, academic language, and a lot of use of “we” to do things other than lead the reader through a thought experiment. There’s also a sprinkling of terms that I don’t think the average person will understand; for example, I doubt the average college student knows flux as anything other than a zany card game.
Regarding the analogy itself, I think Maldacena has fallen into the common physicist trap of making an analogy that explains things really well…if you already know the math.
This is a problem I see pretty frequently. I keep picking on this article, and I apologize for doing so, but it’s got a great example of this when it describes supersymmetry as involving “a whole new class of number that can be thought of as the square roots of zero”. That’s a really great analogy…if you’re a student learning about the math behind supersymmetry. If you’re not, it doesn’t tell you anything about what supersymmetry does, or how it works, or why anyone might study it. It relates something unfamiliar to something unfamiliar.
I’m worried that Maldacena is doing that in this paper. His setup is mathematically rigorous, but doesn’t say much about the why of things: why do physicists use something like this economic model to understand these forces? How does this lead to what we observe around us in the real world? What’s actually going on, physically? What do particles have to do with dimensionless constants? (If you’re curious about that last one, I like to think I have a good explanation here.)
It’s not that Maldacena ignores these questions, he definitely puts effort into answering them. The problem is that his analogy itself doesn’t really address them. They’re the trickiest part, the part that people need help picturing and framing, the part that would benefit the most from a good analogy. Instead, the core imagery of the piece is wasted on details that don’t really do much for a non-expert.
Maybe I’m wrong about this, and I welcome comments from non-physicists. Do you feel like Maldacena’s account gives you a satisfying idea of what gauge symmetry is?
I read it a while ago and had to recheck a bit to write something sensible. The monetary gain from doing loops and exploiting exchange rates seems pretty intuitive and easy to understand and I get how that shows how a magnetic field works. The fact that the value of a given currency can be scaled arbitrarily is clear but I don’t really see how that’s anything but completely separate from (orthogonal to?) the moneymaking loops. Is that simply the way it’s supposed to be; U(1) symmetry = rotation plus movement along a line orthogonal to the rotations?
I’m also not really clear on the long wavelengths – massless particles part; I’d have to reread it to make sure I get it. In particular I don’t get why the particle would become massless and why that would imply the existence of another field (the Higgs here). Is there some value in the currency analogy that is the equivalent of mass?
Just after reading it the first time I commented on Motl’s blog something along the lines of “I never really know if I’ve learned something real or just understood an analogy without correctly understanding how it relates to reality”
Well, as Lubos points out, there isn’t a U(1) here, it’s been replaced with an R, the
pirate symmetryreal number line.
I’d have to go through his mathematical appendices in detail to figure out what is supposed to correspond with what, but if I’m remembering right in essence that “orthogonality” you’re noticing is gauge invariance. The physics doesn’t change if you scale one of the currencies uniformly, it’s only if you scale it non-uniformly (more for some exchanges than for others) that you get something physically meaningful. It’s only the “curl” of the potential that has a physical effect.
For masslessness, think about it like this: in general, the longer the wavelength the lower the energy (assuming you’re familiar with that part, ask if you aren’t). If a particle has mass there’s a lower bound, you can’t have energy lower than mc^2. So in order for a particle to be massless it must be able to have arbitrarily long wavelength. This is the case in the default way the example works, you can have patterns of currencies that make the best speculation loop as long as you want it to be. But if gold is in the system then the easiest way to speculate always just involves neighboring countries: you can’t get a higher wavelength so you can have only high energy particles, thus your speculation field has a mass.
(Actually, it looks like Maldacena’s setup gives the field arbitrarily high (Planck-sized anyway) mass, since only the smallest loops survive. I don’t know if he’s got a way to avoid that, but as-is he’s producing a “natural” theory 😉 )
The existence of another particle comes from there being two precious metals, gold and silver. They have a dimensionless (independent of arbitrary stuff like currency) ratio, which can vary from place to place. That (see the logic in my Nature Abhors a Constant post) gives you a new particle, the Higgs boson. This compares in reality to the Higgs field initially having two “isospin” states, one of which gets “eaten” by the W and Z bosons, giving them mass, while the other becomes the Higgs detected at the LHC.
I definitely agree that the big problem with pieces like this is that they give you a nice analogy but if you aren’t already familiar with what it’s talking about it doesn’t really explain anything, you don’t know whether it’s telling you something about reality or not.
It took me a while to respond, sorry about that. In hindsight it’s obvious that the symmetry is the part that changes nothing real, but the movement in a circle does change something and is therefore not a symmetry. I have no clue if this is something a general audience would easily misunderstand or if it’s just me.
For masslessness I can get that the ideal profitable path has a length that goes towards infinity. I’m still not quite sure how the introduction of gold would always make the ideal path as short as possible; surely there’s the possibility of the gold exchange rate mirroring that of the currency? Your parenthesis there says I’m not getting that part correctly. It’s an addition that may but doesn’t have to change things and for a finite mass it should make the optimal path stay at a given distance when you change the scale of the loops?
I’ll still have to read the Nature Abhors a Constant post to see why the two precious metals would imply a new particle; that’s just an “ok, I’ll accept this” part of the analogy for me right now.
Generally, it would be good to tie parts of an analogy (oh lol, I should get off the Internet. I spelled it analorgy at first) to the parts of stuff you’re explaining. I find myself nodding at understanding some parts of the explanation but still not knowing what the real life counterpart is supposed to be.
(got an error message, hope this isn’t a double post)
Yeah, I definitely think that’s the center of the problem with Maldacena’s piece: a good analogy really needs to have links to the science buried throughout so people end up applying it properly.
The thing with gold is, as long as two places have different currencies with respect to gold, it’s always possible to earn money simply by buying gold in one country and selling it in an adjacent one. You can’t do that at all if you just have currency, the shortest loop (not the most profitable one, just the shortest profitable one) is always longer. But since you can always just sell the gold in an adjacent country where it’s more valuable, you can always make money through as short a loop as possible. That, if I’m reading him right, is supposed to be the analogy with mass.
As for the confusion about whether the movement in a circle is the symmetry, I think part of the issue here is that this is a confusing aspect of gauge symmetry in general. Gauge symmetry is both something that doesn’t matter (a symmetry, and one based on our inability to write things in a consistent way) and something that matters a great deal (with the magnetic potential being thought of as a “gauge connection” and thus intimately tied to how gauge theory works). Disentangling those is really the big challenge of doing a good popular physics article on gauge theory, and while Maldacena definitely has some worthwhile success in that direction, I don’t think your confusion is unusual.
I’ll have to read Maldacena’s paper again, but the first read didn’t do much for me. I don’t think the connection between his analogy and gauge symmetry was well-drawn. The introduction of wavelengths and particle masses came suddenly and without much explanation. I set it down after 2/3 intending to have another go at it another time. Can’t say what I read illuminated anything in terms of gauge symmetry.
I did get that, rather than being called the “God boson” the Higgs should be called the “gold” boson…
Ouch, I did not like Maldacena’s article. He doesn’t understand electromagnetism, and instead of sticking with the physics he launched into an inappropriate analogy. Min you, I don’t like AdS/CFT correspondence either.