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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