Where were we? Ah yes, spontaneous symmetry breaking. When some field takes on a nonzero value even in empty space, and that field is affected by some symmetry transformation, the resulting symmetry is said to be “spontaneously broken,” and becomes hard for us to see directly. The classic example is the electroweak symmetry of the Standard Model, which is purportedly broken by a Higgs field that we have yet to directly detect.

The fields that get expectation values and spontaneously break symmetries are generally taken to be “scalar” fields — that is, they are single functions of spacetime, not something more complicated like a vector field. If a vector field did get a nonzero expectation value, it would have to point somewhere, thereby picking out a preferred direction in spacetime. That means that Lorentz invariance — the physical symmetry corresponding to rotations and changes of velocity — would be broken. Lorentz invariance is a cornerstone of relativity (and thus of all of modern physics), so breaking it is often thought to be bad.

But really, how bad is it? When Einstein put together special relativity on the basis of Lorentz invariance, he was arguing that there was no absolute space nor absolute time in the sense of Sir Isaac Newton. If two physicists traveling freely through empty space passed by each other at a high relative velocity, we couldn’t tell in any universal sense which one was stationary and which was moving — it’s all relative, if you like. If we violated Lorentz invariance by having a vector field get a nonzero value in the vacuum, we could tell who was stationary and who was moving — the vector would define a preferred rest frame.

But that’s not quite the same as going all the way back to Newtonian spacetime. The underlying theory is still Lorentz invariant — if we can’t easily detect this vector field (and we obviously haven’t thus far), Lorentz invariance could be spontaneously violated while remaining in complete accord with all experimental tests.

I was in on the ground floor for this idea — it was the first project I worked on in graduate school (with George Field and Roman Jackiw), and was sufficiently non-mainstream that I worried for my career prospects. Alas, those were more freewheeling times, and you could get a good postdoc without necessarily jumping on a major bandwagon. Subsequently, I was surprised to see Lorentz violation actually become it’s own (relatively tiny) bandwagon! A group of researchers, led by Alan Kostelecky at Indiana, have really pushed the idea of writing down ways to spontaneously violate Lorentz invariance, and have spawned an active experimental program to test these ideas using precision data from astophysics, particle physics, and atomic physics. (Alan has a FAQ on the whole idea of violating Lorentz symmetries.)

So I occasionally return to the idea, as in work with my former graduate student Eugene Lim on the gravitational effects of Lorentz-violating vectors. And now I’ve returned to it again, this time with current student Jing Shu, as we try to understand a fundamental question in physics: why is there more matter than antimatter?

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Lorentz invariance and youRead More »

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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Hidden symmetriesRead More »

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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Escape from the clutches of the dark sector?Read More »

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.