Without much hope of actually understanding anything, Oz decided to listen in on Track 2 one more time. Again he hid under a desk near the back of the classroom during the break. And again the Wiz came in after the break and launched into his explanation without any preliminary chitchat or jokes....
"Okay. Last week we quantized the vacuum Maxwell theory on a cylindrical spacetime. When we did that, we were treating electromagnetism as Yang-Mills theory with gauge group \R. But sometimes people think of electromagnetism as Yang-Mills with some other gauge group! And what's that?"
"U(1)!" said Toby and Miguel simultaneously.
"Right! The group of unit complex numbers, or phases! Now, how does this group change things? It's a bit subtle. But we can work it out. We just need to take everything we did and modify it a tiny bit, replacing the real numbers by U(1) in all the right places.
Now, just as the group R is a line, the group U(1) is a circle. When we did electromagnetism on a cylinder with gauge group R, we saw it was isomorphic to a free point particle on the line. So when we do electromagnetism on a cylinder with gauge group U(1), we'll see... what?"
"It's isomorphic to a free point particle on the circle?" said Toby, dubiously.
"RIGHT! Easy as pie. Or 2pi, anyway. But let's see how it works.
As before, we take spacetime to be R x S1, with the obvious Lorentzian metric for which the circle has circumference equal to L.
Next we put a trivial U(1) bundle on our spacetime, let A be a connection on this bundle, and let F be its curvature. We can still regard A as a 1-form and F as a 2-form. The same formula applies:
F = dA.The vacuum Maxwell equations look just the same, too:
d*F = 0."So what's the difference?!" asked Miguel.
"The difference is that now we treat A as a U(1) connection instead of an R connection. For example, the holonomy along a path is now an element of U(1) instead of R. If we have some curve C, the holonomy along it used to be a real number, namely
\intC A.But now it's a unit complex number, namely
exp(i \intC A).
Another way to put it is that now we only care about \intC A modulo 2pi. By the way, watch out: I'll be constantly jumping back and forth between two ways of thinking about U(1). Sometimes I'll think of it as unit complex numbers with the group operation being multiplication. Other times, when I want to emphasize the analogy to R, I'll think of it as real numbers mod 2pi, with the group operation being addition mod 2pi."
"Sure," said Miguel. "But there's something else that's more confusing. You said a connection on a trivial R-bundle is just a 1-form, and now you're saying a connection on a trivial U(1) bundles is also just a 1-form."
"True," said the Wiz, his eyes twinkling.
"So what's the difference? Now you're saying we think about the holonomies differently. Okay, but so what? The connections are still the same!"
"You're right," laughed the Wiz, "it's possible to get really confused about this if you go about it properly. You're doing a good job. To get unconfused, ponder this. The space of connections is the same -- or isomorphic -- in both theories. But the space of connections mod gauge transformations is different! And that's what really matters."
"Hmm," said Miguel.
"Let's work it out. We need to. The space of connections on S1, modulo gauge transformations, will be the configuration space of our theory. Last time we saw that when the gauge group is R, this configuration space is R. Now that the gauge group is U(1), the configuration space will be...?"
"U(1)?" said Toby.
"Right -- but do you see why, or are you just guessing?"
"Well," said Toby, improvising rapidly, "the holonomy of a connection around a loop is gauge-invariant when the gauge group is abelian, so we can take the holonomy of our connection around the circle, and we an element of U(1). This element can be anything... so at least there's a map from our configuration space onto U(1)."
"Good, so we only need to show it's one-to-one. For that, just take two U(1) connections with the same holonomy around the circle and cook up a gauge transformation sending one to the other. This isn't hard, so I'll leave it as a puzzle.
Okay, so the configuration space is now U(1) instead of R. This changes things only slightly. Our phase space is now the cotangent bundle of the circle... or in other words, the space of pairs (a,e) with a in U(1) and e in R. a is the holonomy of our connection around the circle, while e is the electric field.
As before, Maxwell's equations say e is constant, while a marches along at constant velocity around the circle. More precisely:
a(t) = a + tLe e(t) = ejust like last time. The only difference is that now a(t) lies in U(1), so we have to do the addition mod 2pi. So the theory really is isomorphic to a free point particle marching around the circle!
The classical Hamiltonian looks the same as last time, too: p
H = Le2/2Again, this comes from integrating (E2 + B2)/2 over space. There ain't no B field, E equals e, and space is a circle of circumference L, so we get Le2/2.
So far, so dull. Now let's quantize! The configuration space is U(1), so the Hilbert space is L2(U(1)). We can write out any state using Fourier transforms as before, but now we use a Fourier series instead of an integral:
psi(x) = (1/sqrt(2 pi) sumk exp(ikx) f(k)
where we sum over all integers k. The formula for the electric field operator looks the same as before:
(e psi)(x) = -i psi'(x)
but now its spectrum is discrete instead of continuous! In other words, there's an orthonormal basis of states
psik(x) = (1/sqrt(2 pi)) exp(ikx)
that are eigenvectors of the electric field operator:
e psik = k psik.
Thus the electric field is quantized, just as the momentum of a point particle on the circle is quantized. Here I'm using "quantized" in a very simple-minded sense: I mean it can only take on integer values! This is a big difference between the U(1) theory and the R theory of electromagnetism: in the R theory, k could be any real number, so the electric field could take on any value."
Miguel said "This is a lot like how charge is quantized in the U(1) version of electromagnetism but not the R version."
"Right! Of course, we don't have any charged particles here yet, but we're still getting quantized flux lines of the electric field. This is typical in loop representations of gauge theories. In loop quantum gravity, where the analogue of the electric field determines the metric, these become quantized flux lines of area! When they poke through a surface they give it area, and this area can't be just anything: it takes a discrete spectrum of values. We get this discreteness whenever our gauge group is compact, by the way.
Of course, in our current example space is a circle, so there's only one place for a flux line to go: all the way around the circle. Things will get more interesting in higher dimensions.
Next, let's quantize the Hamiltonian. We take our formula
H = Le2/2
and reinterpret e as an operator, getting
(H psi)(x) = -(L/2) psi''(x)
The states psik form a basis of eigenvectors of this operator, with
H psik = (Lk2/2) psik
So the energy can only take on a discrete set of values in the U(1) theory! Energy is quantized! In the R theory, the quantity k could be any real number, so the energy could be any nonnegative number."
The Wiz paused, and smiled broadly. "Okay, good, that's all for today. We've made some real progress. We're getting a taste of how the loop representation of gauge theories works. We're also starting to see some interesting relationships between local and global, between classical and quantum! For example: as classical theories, the R and U(1) theories of electromagnetism on the cylinder are locally indistinguishable. Locally, a solution just amounts to a constant electric field, which can take on any value whatsoever. But when we quantize the theories, they become distinguishable even locally! The electric field can take on any value in the R theory, but only a discrete set of values in the U(1) theory. Cool, eh?"
The Wiz and his acolytes walked out in a buzz of conversation, and slowly, painfully, Oz extricated himself from the desk. It was really annoying how the Wiz would fly up into the world of mathematical abstractions, and then come down from the clouds for a minute and say something tantalizingly simple.