Uncertain Principles

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Posted at 8:04 AM | link | follow-ups | 10 comments

The big burst of long physics-related post earlier this week has given way to general blogging lethargy. It's been a hectic couple of days at work (not terribly productive, but very active), and I haven't felt like posting anything new in the evenings.

To the limited extent that I've done anything blog-like, it's been on other people's sites. Specifically, in the comments at Cosmic Variance, where they're soliciting nominations for The Greatest Physics Paper Ever. Personally, I think Newton's Principia Mathematica ought to win in a walk: it more or less marks the beginning of physics as we know it, it's hugely influential and important, and Newton was British, which matters because they've cribbed the idea from the BBC...

There have been lots of interesting suggestions (I've posted a few of my own), and I'm sure more would be welcome, so go check it out.

Posted at 6:46 PM | link | follow-ups | 1 comment

Boy, I thought that the combination of ASCII-art Feynman diagrams and calling Murray Gell-Mann an enormous dork would get at least one comment... What do you want from me, people?

Here's a slightly different physics topic:

A while back, there was a post on Arcane Gazebo in which the proprietor lamented his over-specialization in physics grad school:

But the more I think about it the less defensible it is that I've not had much contact with GR. I'll spend at least six years in grad school, and for most of that time I won't be taking any classes, just doing research. Surely I could take a little time to audit a GR course? (In fact, a student who recently graduated from our group did exactly this.) It seems more than a little ridiculous to accumulate all the tools necessary to comprehend one of mankind's greatest intellectual achievements, a profound description of nature, and then not make any attempt to learn it. That's a pretty high level of incuriosity. I think there's a certain amount of information overload the first few years in grad school, that made it feel like such a relief to be "done" with classes and just take some data for a while. But I seem to be past that stage.

And while I'm at it, it would be nice to know more about quantum field theory and the Standard Model...

I think there are structural as well as psychological issues contributing to this-- research productivity isn't a perfectly monotonic function of time spent in the lab, but taking time out to audit classes won't get you any closer to graduating, and may not endear you to your advisor, either. Overall, though, I sympathize with the point.

The knowledge gap that I personally regret is not General Relativity-- the "Cosmology for Complete Idiots" class I took as an undergrad included just enough GR that I don't want to learn any more. My personal physics blind spot is symmetry group theory. My PhD is in Chemical Physics, a program at Maryland that I highly recommend to interested students, but that does not require students to take any of the field theory sort of classes where this stuff comes up. There should've been some symmetry group content in the molecular physics class (aka "Quantum Chemistry") that I took for the qualifier, but the guy teaching it did his reasearch on diatomic molecules, so we never got to molecules with even three atoms. I also took a "QED for Idiot Experimentalists" class, but we spent more or less the entire term talking about Casimir forces, and didn't really get to anything else.

At the time, I was perfectly happy with this state of affairs, as the classes that would've covered such topics seemed very dry and formal in a mathematical sense, and mathematical formalism isn't one of my strong suits. Plus, I had data to take in the lab.

Now, though, I'm a little sorry that I never covered that stuff in grad school. Not so much because I feel bad about missing out on the beauty and power of the math (see earlier comment about formalism), but because there are so many questions to which the answer is, apparently, "It's an SU(something) mumble garble blarg." Having never covered symmetry groups in any of my classes, I find those answers singularly unhelpful, but it's really hard to get answers to a lot of field theory/ particle physics questions that are free of group theory jargon. And I would really like to understand more of what's going on in those areas, particularly now that I've embarked on a project that involves regular conversations with nuclear astrophysicists...

Having despaired of getting the entire particle theory field to discuss their work in different terms (my first choice of solution), I've decided to try to correct this on my end, by trying to learn a bit more about the field. To this end, I picked up a copy of Introduction to Elementary Particles by David Griffiths. This is an upper-level undergrad textbook by an author who is famous for writing readable textbooks-- his undergraduate E&M book is regarded by many as the Rosetta Stone for understanding Jackson's infamous graduate text-- so it seemed like a good choice. Also, I was able to borrow it from a colleague.

Attempting to learn this stuff from a textbook has proven deeply frustrating, though. The book was readable enough, until I sat down and started to work through the math in detail (which is the only way to really learn from a physics book), at which point all sorts of problems started to crop up. A lot of these are related to the fundamental problem of textbooks: a great deal of the information you need in order to understand the subject is conveyed through problems and homework assignments, and the answers aren't in the book. Which makes it impossible to check your work.

To give a concrete example (if you're bored or scared by math, skip this bit): Chapter 4 introduces symmetry groups, and explains that, for example, SU(2) describes a group of 2x2 unitary matrices whose determinant is 1, then shows how to arrive at the matrices in question from the Pauli spin matrices (which are unitary matrices whose determinant is -1, not 1, which confused me for a while, but they're not the matrices that really matter). Fine. It also mentions that there is a three-dimensional representation of this group, which would presumably be a 3x3 matrix (which is somehow distinct from an SU(3) group, which would also be a set of unitary 3x3 matrices of determinant 1) without explaining what such a beast would look like. There is, however, a problem in which the student is asked to construct the matrices for a three-dimensional representation of SU(2), which seems straightforward enough, except that the matrices I end up with aren't unitary, have determinant zero, and generally don't seem to behave in a sensible manner. And there's no answer in the book, nor did a Google search turn up anything useful ("useful" is defined for these purposes as "the matrices corresponding to a three-dimensional representation of SU(2) written out in conventional matrix notation"). Ultimately, this is probably a side issue, but if anything, I'm more baffled than I was before I started this whole mess.

Worse yet, I'm still not clear what the point of the whole business is. In the next section, he goes on to introduce isospin, and start talking about symmetries between nucleons, which he helpfully notes make up an SU(2) doublet. But this doesn't seem to be much more than the answer to a trivia question-- that fact is thrown out there, but not really used for anything. All the calculation that goes on in that section is done using isospin numbers and angular momentum addition algebra (this, I'm OK with), with no further mention of group theory. In fact, according to the Index, SU(2) doesn't reappear until Chapter 10.

And then there's an absolutely bizarre paragraph in which it's explained that nucleons are SU(2), but the Eightfold Way baryons are SU(3), except if you include all the different types of quarks, the whole thing is SU(6), except that really doesn't work at all because the masses are so different. Um, huh? You've dragged in all this formalism, and it isn't actually good for anything?

Were I doing this in the context of a class, I would be able to get these questions answered, either in class, or by consulting the professor. As it is, I'm completely at sea-- I've read through the relevant sections a half-dozen times, and nothing I'm seeing there seems to answer my questions. It's really remarkably frustrating.

(I don't want to completely diss the book, by the way. The Standard Model sections were pretty readable, and the introductory explanation of the various forces and interactions was very helpful. It's just not all that helpful on the whole symmetry issue...)

So, in summary, my advice to any grad students out there would be: if you think you might someday be interested in a subject, try to take a class on it while you have the chance, rather than attempting to learn it later from a book. And, to Hannah Shapero: everyone has problems learning from textbooks, so don't be afraid to ask people questions.

Posted at 8:43 PM | link | follow-ups | 22 comments

So, how does looking for a non-zero electric dipole moment for the electron test physics beyond the Standard Model, anyway?

Well, in the previous discussion, I talked about dipole moments in very classical terms-- well-defined particles with well-defined properties interacting with a continuous field that extends through all of space. If you want to really talk about the behavior of electrons in detail, though, you need to move to the world of quantum field theory, where things look completely different. In particular, we need to replace the classical field with a quantized field, and talk about the interactions of electrons and photons.

The most convenient way to talk about these things is through the use of the infamous Feynman Diagrams, which describe the interactions of particles and fields in terms of little pictures showing possible events in the life of an electron. The most basic Feynman diagram showing an electron interacting with an electric field would look like this (again with the ASCII art):

         e                    .
          \                   .
           \                  .
            \                 .
             \                .
              ~~~~~~~~~~~~    .
             /                .
            /                 .
           /                  .
          /                   .
         e

In these diagrams, time flows from bottom to top, so this represents an electron (diagonal lines) in some region of space moving along and either absorbing or emitting a photon ("~"), and changing its motion as a result. This diagram really represents a mathematical expression that describes one part of the interaction between the particle and the field. To fully describe the interaction, you evaluate these expressions for all the possible things that might happen to the electron as it cruises along through space (that is, all the pictures you can draw that satisfy the rules for making such diagrams), and add them all together.

(I make no claim to be presenting a rigorous description of the theory, here-- this is very much the "lies-to-children" version.)

In principle, there are an infinite number of such diagrams, describing various more complicated processes, starting with things like this:

         e
          \ 
           \ 
            \ 
             \ 
              ~~~~~~~~~~~~
              |
              |
              |
              ~~~~~~~~~~~~
             /
            /
           /
          /
         e

where the electron scatters two photons, and getting more intricate as you go. In practice, as the diagrams get more and more complicated, their contribution to the final interaction gets smaller and smaller, so you can stop after some finite number of diagrams (basically, once you've gotten your answer to however many digits you were looking for). When you go through this process for a typical electron, what you find is mathematically just the same as a pure monopole interacting with a classical field. So, no dipole moment.

Buried in these calculations, you also have a bunch of diagrams like this:

         e
          \ 
           \ 
            \ 
             \ 
             |\ 
             | \ 
             |  \ 
             |   ~~~~~~~~~~
             |  /
             | /
             |/
             /
            /
           /
          /
         e

where the vertical line represents the emission and re-absorption of some other sort of particle. Since in quantum field theory everything that isn't forbidden is mandatory, this mystery particle could be just about anything, and the contributions of just about everything need to be included.

These diagrams turn out not to make a difference for any of the particles that we've already observed in nature-- none of them have the right properties to cause a dipole-like contribution to the calculation (or if they do, it's orders of magnitude below the sensitivity of any current or proposed experiment). The minute you start talking about physics beyond the Standard Model, though, you start introducing all sorts of new particles. The most popular class of theories gets the general name "supersymmetry," and these theories actually double the number of new particles, by saying that every particle has a "supersymmetric partner," another as-yet-undiscovered particle that is somehow related to the particle we know.

(Shameful note: these partner particles are often referred to as "sparticles," (or, worse, "selectrons" and "squarks," and "sneutrinos" and so on) because particle physicists are enormous dorks when it comes to naming things. I blame Murray Gell-Mann.)

It turns out that once you start doing introducing new particles, they almost inevitably have the necessary properties to cause a dipole-like contribution in the calculated interaction between the electron and an electromagnetic field. Different versions of these theories predict different sets of particle properties, and you can plug those values in, and see what the theory predicts for an electron (or muon, or neutron) EDM.

When you do that, you find that the simplest version of supersymmetric theory ("Naive Supersymmetry," in physics parlance) predicts a value for the dipole moment of the electron that's around 10^-25 e-cm-- at least an order of magnitude larger than the current experimental limit. If those theories were correct, the experiments should have found a non-zero EDM a long time ago, but there's still no sign of one.

Now, nobody really expects the "naive" version of a theory to work right off, so there are some obvious tricks people play to adjust their models to give smaller values for the electron EDM. And theorists are well known for finding non-obvious tricks, leading to dozens of different models that predict smaller EDM's than "naive" supersymmetry. And the experiments are starting to cut into those, too-- according to a summary slide available in this PowerPoint presentation, the current limit has already excluded the upper end of the next two candidate theories (the charmingly named "Accidental Cancellations" version, and the cryptic "SO(10) GUT"). Experiments currently in production should lower that limit by another 2-4 orders of magnitude, which would either find a non-zero EDM, or rule out pretty much every theory now on the books. I'm told that the theorists who deal with this stuff are starting to sweat a little...

As I said, these are fiendishly difficult experiments to carry out. But the rewards for success are high enough that lots of people are jumping into the game-- new EDM experiments are being set up at Yale, Berkeley, Texas, Penn State, Colorado, Sussex, Harvard, Princeton, and I'm sure I've forgotten a whole bunch of other place. The beautiful thing about this is that they can put tight constraints on some of the most esoteric models in high-energy physics, using apparatus that can fit in a single room-- no giant particle accelerators required.

Of course, the methods used are nowhere near as sexy as billion-dollar atom smashers-- lead oxide might be the least sexy substance in experimental science-- so the big accelerators still get most of the press. But if you want to see people really pushing the limits of current particle theories, EDM experiments are the best place to look right now, and will be for a few more years.

Posted at 9:07 PM | link | follow-ups | 9 comments

Kate pointed out to me this Washington Post column on pickup basketball:

With nine seconds left in the game, I found myself guarding the opposing team's top scorer, a duty I had assigned myself out of some combination of challenge and confidence. Not that this made physiological sense. I'd guess my opponent was 15 years younger than I. Taller, quicker, more athletic.

He knew how to play, too, repeatedly maneuvering to the hoop, either scoring or drawing a foul. And so with the game on the line, our team up by two points in overtime, Mr. Top Scorer planted himself eight feet from the basket and called for the ball.

He might go a bit too far in trying to draw grand conclusions about Human Nature, but it's worth reading if you enjoy the game. Since it will shortly disappear behind a paywall, I figured I should post it sooner rather than later. The author is also doing an online chat about the article.

Posted at 1:20 PM | link | follow-ups | 3 comments

Having bored everybody to tears talking about gravity measurements and their importance, I'd like to spend a post or two talking about the other major group of experiments using atomic physics techniques to test high-energy physics. These are the "EDM search" experiments, looking for Electric Dipole Moments in a variety of subatomic particles-- neutrons, muons, and electrons-- and as I understand it, they're the tightest constraints we have right now on theories beyond the "Standard Model" of particle physics.

Before I try to explain why these experiments work to test new particle theories, it's probably worth spending a few paragraphs on what, exactly, a dipole moment is, and how you look for one.

The idea of dipoles comes up first in dealing with electric and magnetic fields. Most people have seen these in some form or another-- usually they show up as diagrams showing arrows radiating out from little spheres, or lines of iron filings sprinkled on paper over a magnet. These fields represent (loosely speaking) the direction of the force on a particle placed in that field.

The brute-force way of dealing with these fields is to consider each particle involved in creating the field separately, and add up all their contributions. This very quickly becomes really excruciating to do, as anyone who's ever suffered through a basic course in electricity and magnetism quickly discovers. It's the easiest thing to understand, but calculationally, it's a nightmare.

Mathematically, it turns out to be more convenient to describe these fields in terms of a set of standard types of fields-- arrangements of charges or magnets that produce simple and symmetric field patterns that are well known. This process is called a "multipole expansion," and consists of adding together the characteristic fields for the various types, with each given a weight known as the "moment" for that particular multipole. This may seem like a weird way to handle the situation, but in practice, it's relatively easy to determine the moments of a collection of charges or magnets, and once you've got that, you just add together the known fields for each multipole, and you're done. You can also use the same technique to quickly calculate how a collection of charges or magnets will react when placed in some known field.

The first two terms in this expansion are generally the most important (the weights almost always get smaller as you go to more complicated fields, unless you're in an E&M class, and your professor is into really perverse problems). The simplest field is the "monopole," which is just the field for a single charged particle sitting in empty space. It's drawn as a collection of lines radiating out from the particle, indicating that a test particle placed somewhere in the field will either be attracted straight in toward the center, or repelled directly away from the particle. The "monopole moment" is just the magnitude of the charge on the particle. (There are no monopoles in ordinary magnetic systems.)

The next most complicated is the "dipole," the most familiar example of which is probably a simple bar magnet-- if you've seen the demonstration with iron filings sprinkled on a sheet of paper over a magnet, you've seen what a dipole field looks like. You get the same basic pattern with two oppositely charged particles placed near one another: field lines radiate out from one, and into the other, making big closed loops. The magnitude of the "dipole moment" for this simple system is the magnitude of the charge multiplied by the distance between the two particles.

Dipoles show up all over the place, and have a couple of important features that are worth mentioning. First, the field of a dipole drops off much more quickly than the field of a monopole-- that is, the force on a test charge due to a dipole at a given distance is much less than the force due to a monopole of similar magnitude at the same distance. If you think about it a little bit, this makes sense-- if you're very far away (much farther than the separation between the particles), the dipole will look like a neutral particle, as the positive and negative charges cancel each other out.

The other important feature has to do with the way a dipole interacts with an external field. If you put an electric dipole in an electric field, the way that it interacts with the field will depend on its orientation. If you place it near a positive charge, for example, the dipole will feel an attraction toward the particle if its negative end is pointing toward the charge, and it will feel a repulsion if the positive end is toward the positive charge. If the dipole is free to rotate, it will tend to flip itself around to a preferred orientation; if it's not free to move, the interaction with the field will either raise or lower the energy of the dipole, depending on the orientation. The same thing happens with magnetic dipoles, in a magnetic field-- the energy will go up or down depending on the orientation.

Lots of things that we deal with in physics behave like dipoles. Neutral atoms look like fluctuating electric dipoles-- a positive nucleus, with negative electrons orbiting some distance away-- and you can successfully describe a remarkable number of phenomena in optics simply by considering atoms as microscopic electric dipoles. Many atoms and nearly all subatomic particles also behave like magnetic dipoles, with energy states that shift in response to a magnetic field. Dipole moments are incredibly useful mathematical tools, and calculating them is a big business in some areas of theoretical physics.

What does all this have to do with particle physics? Well, consider the case of the electron. The electron is supposed to be a point particle-- to the best of our ability to measure, the radius of an electron is zero. It's also a fundamental particle-- there are no smaller things that get stuck together to make the electron, the way quarks get put together to make a proton or a neutron. As a result, you would expect the electron to have a pure monopole field-- there's no way to look at it as two opposite charges separated by a small distance, so you would expect the electric dipole moment of the electron to be zero.

If you want to look for a failure of our usual understanding of particle physics, then, this might be a good place. If you can find a dipole moment for an electron, that will cause all sorts of problems for the theory-- it means there's got to be something else out there that we haven't already accounted for.

Now, you can't do this just by measuring the electric field of the electron. Because the dipole component of the field drops off much faster than the monopole component, you'd need to get awfully close, and map the field out very precisely, and there's just no way to get an electron to hold still well enough to do that to the necessary precision. What you can do, though, is to measure the orientation dependence that's characteristic of a dipole. If an electron has a dipole moment, its energy should change depending on how that dipole is oriented with respect to an external field.

Of course, you can't do this on free electrons, because they just go zipping away the minute you put a field on, due to the monopole moment. But you can do this with electrons that are bound to atoms-- as long as the field isn't too big, the electrons will continue to be stuck to the atoms, but their energy levels will be shifted very slightly. In particular, the difference between two properly chosen energy levels will change by a very small amount. It's a tiny, tiny shift, but over the last hundred-odd years of atomic physics, we've developed an impressive array of tricks for measuring extremely small shifts in the separations between atomic energy levels. That's the whole basis for atomic clocks, after all, and the state of the art in those energy measurements is equivalent to a clock that'll run for tens of millions of years before dropping a second.

The basic technique for looking for electron EDM's, then, is always the same. You take a bunch of atoms (or molecules-- the same basic idea applies), and you stick them in a big magnetic field, to ensure that all the atoms are aligned in the same way. Then you apply an electric field in some direction, and measure the energy difference between two particular levels. Then you reverse the direction of the field, and measure it again. If the electrons in your sample have dipole moments, the energy difference should increase in one case, and decrease in the other, and you should see a clear difference between them.

These experiments are fiendishly difficult to do. You need to work very hard to make sure that your electric and magnetic fields are very uniform, and that you can switch the direction of the electric field without changing anything else. And for whatever reason, all the systems that are promising candidates for EDM searches seem to be extremely unpleasant to work with-- they're toxic, or expensive, or difficult to handle, or the experiments need to be carried out at high temperatures (900 Celsius, say). The physics potential is high enough, though, that a lot of people have gotten into this field.

No-one has yet detected a non-zero electric dipole moment in an electron, muon, or neutron. The current state of the experimental art is somewhere around the 10^-27 level-- that is, they've measured nothing at all down to that level, so the electric dipole moment is 0.000000000000000000000000001 or smaller, in the units used to measure these things (electron charge times centimeters). That's still roughly ten billion times larger than the Standard Model prediction, but it's a good factor of ten smaller than predicted by a large swath of particle physics theories.

How you get these dipole moments out of particle physics will have to wait for another post, though.

Posted at 4:06 PM | link | follow-ups | 10 comments