Author: Sean Carroll

Symmetries, you may have heard, play a crucial role in modern physics. (Leon Lederman and Chris Hill wrote a whole popular book about the subject, if you’re interested.) But one of the things that makes them so interesting is that they can be hidden — the symmetry is secretly there, even though you don’t easily notice. And sometimes you may be interested in the converse situation — it looks like there is an obvious symmetry of nature, but in fact there are tiny violations of it, which we haven’t yet detected.

To physicists, a “symmetry” is a situation where you can rearrange things a bit (values of quantum fields, positions in space, any of the characteristics of some physical state) and get the same answer to any physical question you may want to ask. An obvious example is, in fact, position in space: it doesn’t matter where in the world you set up your experiment to measure the charge of the electron, you should get the same answer. Of course, if your experiment is to measure the Earth’s gravitational field, you might think that you do get a different answer by moving somewhere else in space. But the rules of the game are that everything has to move — you, the experiment, and even the Earth! If you do that, the gravitational field should indeed be the same.

How do such symmetries get hidden? The classic example here is in the weak interactions of particle physics: the interactions by which, for example, a neutron decays into a proton, an electron, and an anti-neutrino. It turns out that a very elegant understanding of the weak interactions emerges if we imagine that there is actually a symmetry (labeled “SU(2)”) between certain particles; examples include the up and down quarks, as well as the electron and the electron neutrino. (This is the insight for which Glashow, Salam and Weinberg won the Nobel Prize in 1979.) If this electroweak symmetry were manifest (or “unbroken” or “linearly realized,” depending on one’s level of fastidiousness), that means that it would be impossible to tell the difference between ups and downs, or between electrons and their neutrinos.

Of course, in reality it’s not so hard to tell. These purportedly-indistinguishable particles have some similar properties, but they have different masses, and even different electric charges. Nobody would ever mistake an electron for an electron neutrino. (They would mistake a red quark for a green quark or a blue quark, as those are related by an unbroken symmetry — the SU(3) of quantum chromodynamics, for which the Nobel came much more recently.)

The reason is that the SU(2) symmetry of the weak interactions is spontaneously broken (or “nonlinearly realized”). The symmetry is firmly embedded in the laws of physics, but is hidden from our view because the particular state in which we find the universe is not invariant under this symmetry. There is something about the vacuum — empty space itself — which knows the difference between an up quark and a down quark, and it’s the influence of the vacuum on these particles that makes them look different to us.

This idea of spontaneous symmetry breaking has a long history in physics — it was elucidated in condensed matter physics by Philip Anderson (Nobel 1977) and others, and in particle physics by my colleague Yochiro Nambu and erstwhile colleague Jeffrey Goldstone (no Nobel yet, which is a shame). And here’s an interesting thing — if the vacuum is not invariant under some symmetry, there must be some field that is making it not invariant, by taking on a “vacuum expectation value.” In other words, this field likes to have a non-zero value even in its lowest-energy state. That’s not what we’re used to; the electromagnetic field, for example, has its minimum energy when the field itself is zero. But “zero” doesn’t break any symmetries; it’s only when a field has a nonzero value in the vacuum that it can affect different particles in different ways.

The way to do that, in turn, is to imagine that the symmetry-breaking field has a “Mexican-Hat Potential,” as illustrated at right. (Image swiped from The Official String Theory Web Site, which also has a nice discussion at a more technical level.) This is a graph of the potential energy of a set of two fields φ₁ and φ₂. Fields like to sit at the minimum of their potentials; notice that in this example, the minimum is not at zero, but along a circle at the brim of the hat. Notice also that there is a symmetry — we can rotate the hat, and everything looks the same. But in reality the field would actually be sitting at some particular point in the brim of the hat. The point is that you should imagine yourself as sitting there along with the field, in the brim of the hat. If you were at the peak in the center of the potential, the symmetry would be manifest — spin around, and everything looks the same. But there in the brim, the symmetry is hidden — spin around, and things look dramatically different in different directions. The symmetry is still there, but it’s nonlinearly realized.

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Dark matter and dark energy make up 95% of the universe — or at least, we think so. Since these components are “dark,” we infer their existence only from their gravitational influences. Some of us have been foolhardy enough to imagine that these observations signal a breakdown of gravity as described by general relativity, rather than new stuff out there in the universe; but so far, the smart money is still on the existence of a dark sector that we have not yet directly detected.

There remains another possibility worth considering — that there is no dark stuff, and that gravity is perfectly well described by general relativity, but that we just aren’t using GR correctly. In other words, that the conventional theory can explain the observations perfectly well without dark matter or dark energy, we just have to be clever enough to figure out how. This would be the most radically conservative approach to the problem, in John Wheeler’s sense: we should push the smallest number of assumptions as far as they can possibly go.

Recently, separate attempts have been made to explain away “dark matter” and “dark energy” by this kind of strategy. In a paper that somehow got mentioned in the CERN Courier and on Slashdot, authors Cooperstock and Tieu have suggested that nonlinear effects in GR could explain flat rotation curves in spiral galaxies (one of the historically important pieces of evidence for dark matter). And in two papers, Kolb, Matarrese, Notari and Riotto and then just Kolb, Matarrese, and Riotto have suggested that nonlinear effects in GR could explain the acceleration of the universe (a key piece of evidence for dark energy). Are these people making sense? Are they crazy? Is this worth thinking about? Have they actually explained away the entire dark sector? (Answers: occasionally, possibly, yes, no.)

In both cases, the relevant technical issue is perturbation theory, specifically in the context of general relativity. Imagine that we have some equation (in particular, Einstein’s equation for the curvature of spacetime), and we’d like to solve it, but it’s just too complicated. But it could be that physically interesting solutions are somehow “close to” certain very special solutions that we can find exactly. That’s when perturbation theory is useful.

Call the solution we are looking for f(x), the special solution we know f₀(x), and the small parameter that tells us how close we are to the special solution ε. For example, gravity is weak, so in GR the small paramter ε is typically something proportional to Newton’s constant G. Then for a wide variety of situations, the sought-after solution can be written as the special solution plus a series of corrections:

f(x) = f₀(x) + ε f₁(x) + ε² f₂(x) + …

So there are a series of functions that come into the answer, each of which is accompanied by a progressively larger power of ε. By only knowing the first one to start, we can often plug that solution into the equation we are trying to solve, and get an equation for the next function f_i(x) that is much simpler than the full equation we are struggling to solve.

The point, of course, is that we don’t really need to get the whole infinite series of contributions. Since ε is by hypothesis small, every time we raise it to a higher power we get smaller and smaller numbers. Often you do more than well enough by just “going to first order” — calculating the εf₁(x) term and forgetting about the rest. But it’s certainly possible to get into trouble — for example, there could be “non-perturbative effects” that this procedure simply can’t capture, or the perturbation series itself could be sick, for example if the function f₂(x) were so huge itself that it overwhelmed the extra factor of ε it comes along with. We would then say that perturbation theory was breaking down.

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NASA’s infrared Spitzer satellite has released these gorgeous new images of the Andromeda galaxy. In infrared, you are directly observing the dust lanes that describe the galactic arms, rather than simply looking at reflected starlight.

Here’s a bigger version. Lyman Spitzer, after whom the telescope is named, was one of the primary movers behind the original Space Telescope idea, which eventually grew into the Hubble Space Telescope. He was also my grand-advisor: George Field was my Ph.D. advisor, and Spitzer was his.

Last year I was fortunate enough to join the folks at Project Exploration on an honest-to-God dinosaur expedition, digging up fossils in Wyoming. PE is a great organization, headed by educator Gabrielle Lyon and paleontologist Paul Sereno, that works to get kids interested in science. I wasn’t able to make it to Wyoming this year (I was enjoying croissants in Paris, as I recall), but I wanted to point to PE’s latest project: a set of field updates on the web about a recent expedition to Niger.

Sereno has led several expeditions to Niger to search for fossils, coming back with such discoveries as an astonishing skeleton of SuperCroc (or Sarcosuchus Imperator, for you sticklers out there). During the 2000 expedition, the team stumbled across a remarkable find: remains of a Neolithic human settlement, perhaps 5,000 years old, with about 200 human skeletons in addition to countless artifacts of various sorts. Not being really equipped to take advantage of the find, the team protected the fossils as well as they could, with the idea of teaming up with archeologists and coming back later to excavate the site.

That return trip was just recently undertaken, and one of the team members was Shureice Kornegay, a graduate of PE’s Junior Paleontologist program who is now attending Norther Illinois University. Shureice and Paul have been writing these field updates that convey some of the excitement and challenge of such a major undertaking as this expedition. It’s great to read along as they cope with tipping water trucks and insect swarms of “biblical proportions.”

Some details about the expedition can be found in this communication to the team (pdf), which will fill you in both on the background of the site, and on what you need to bring with you when you’re about to head out to the Sahara to dig for bones! It’s good to be occasionally reminded that physics isn’t the only exciting science out there.

Yesterday I gave a talk at a Fermilab symposium celebrating the World Year of Physics. It was a great event, aimed mostly at local high-school students and the public more generally, although personally I learned alot from the other talks myself.

My own talk was an overview of special and general relativity; you can see the slides here (warning: large pdf file). Eventually I think all the talks will be in video on the symposium web page. I played an audio file featuring Einstein himself explaining the basics of that equation E = mc² that we were talking about a while back. People were asking me where I stole it from, so here’s the answer: an Einstein exhibit at the American Institute of Physics website. Give it a click; it’s nice to hear the master himself talk about his formula, thick German accent and all.