Again Oz hid under a desk during the break, hoping to glean a little wisdom from Track 2. This time he resolved not to ask any questions, though.
The Wiz was excited. His grand scheme for explaining quantum gravity was gradually starting to pick up speed. It was taking a long time, but the stuff was finally starting to sound like physics. He plunged in:
"Last time we quantized the free point particle on the line. Now let's turn to electromagnetism. If we look at the vacuum Maxwell equations on a 2-dimensional spacetime shaped like a cylinder, we'll get a classical system that's isomorphic to the free point particle on the line! Let's see why, and use this to quantize the electromagnetic field in 2d.
Suppose spacetime is
M = R x S1and think of S1 as the interval [0,L] with its endpoints identified. We use the obvious coordinates (t,x) where the time t is a real number and x lies in the circle, and we give spacetime the metric
g = -dt2 + dx2.In short, we're assuming spacetime is a cylinder with circumference L.
The basic field in electromagnetism is a 1-form A. We can think of this as a connection on a trivial R-bundle over spacetime if we like. The electromagnetic field strength F is the curvature of this connection:
F = dA,and Maxwell equations say that
d*F = 0.Let's see what this looks like in coordinates! If
A = Ax dt + Ax dxthen
F = (dt Ax - dx Ax) dt ^ dxYou're probably used to Maxwell's equations in 4d spacetime, where F has six components: 3 electric and 3 magnetic. Here it has just one! So: what is this one component? Is it the electric field, or magnetic field?"
"It's the electric field," said Miguel.
"Right. In general, the time-space components of F correspond to the electric field, while space-space components correspond to the magnetic field. In n-dimensional spacetime the electric field thus has n-1 components, while the magnetic field has (n-1)(n-2)/2. This means that for n > 4, the magnetic field has more components than the electric field, while for n < 4, the electric field has more. And in 2d spacetime, there's no magnetic field at all!
So let's write
F = e dt ^ dxwhere
e = dt Ax - dx Axis the electric field. From this we get
*F = -eso the vacuum Maxwell equation says that
de = 0.What does this mean?"
"The electric field is constant!" said Jay.
"Right: constant throughout spacetime. Weird, huh? No electromagnetic waves in this dimension. But it makes sense if you think about it."
"Yeah," said Jay. "There's no charged matter in our theory, so electric field lines can never end... and space is a circle, so there's just a single electric field line wrapping around this circle. There's nothing it can do, nowhere it can go!"
"Pretty dull, eh?" asked the Wiz. "But there's a bit more to it than that. Remember, the basic field in electromagnetism is not F, but A. Two A fields count as physically "the same" if they differ by a gauge transformation:
A |-> A + dfGauge theory is about connections mod gauge transformations!
If you think about this carefully, you'll see that in addition to the electric field, there's another degree of freedom: the integral of A A all the way around the circle. If we think of A as a connection, this is called the holonomy of A around the circle. It's a global, rather than local, degree of freedom. In other words, you can't detect it with experiments in a small patch of spacetime -- but you can detect it by carrying a quantum charged particle all the way around the universe! The phase of its wavefunction will change.
You may have heard of the Bohm-Aharonov effect, where we carry a charged particle around a solenoid and its wavefunction changes phase thanks to the nonzero A field, even though it never gets close to a nonzero magnetic field. This is just like that."
Jay blurted out, "But there are no charged particles in this theory!"
"True," conceded the Wiz. "Our theory a bit like the vacuum Einstein equation, where the curvature of spacetime says what would happen to a test particle in free fall, *were there to be such a particle*. The difference is that here we need a *quantum* test particle to detect the integral of A around the circle."
Jay looked skeptical. "Test particles aren't really real, though."
The Wizard nodded. "Let's face it: these are just toy models; we need matter before we get realistic physics. But they're still interesting. And we have to learn to crawl before we can walk!
So, let's carefully work out what Maxwell's equations say about the connection A modulo gauge transformations.
First, we can always do a gauge transformation that makes the time component of A vanish -- then we say A is in "temporal gauge". We just replace A by A + df where f is chosen so its time derivative cancels the time component of A.
This gives
e = dt Axand since e is constant throughout spacetime,
Ax(t,x) = Ax(0,x) + te.This says that in temporal gauge, A is determined by its value at time zero together with the constant e.
Next, we can do a gauge transformation that makes the A field constant on the circle at time zero:
Ax(0,x) = c dxThis is obvious with a little deRham cohomology: by adding an exact 1-form df where f depends only on x, we can get A to be whatever we want at t = 0 as long as we don't change its integral around the circle.
At this point, we've used up all our gauge freedom: we can't do any gauge transformation
A |-> A + dfwhere f depends on t without screwing up the temporal gauge condition, and we can't do one where f depends on x without making Ax depend on x!
It follows that any solution of the vacuum Maxwell theory on the cylinder is gauge-equivalent to a unique one of this form:
A = (c + te) dxHere e is the electric field on spacetime and c is related to the integral of A around the circle at t = 0. So it takes only two numbers to describe a solution of this theory! In other words, the phase space is R2."
"That's just like the particle on the line," noted Miguel.
"RIGHT! But let's make the analogy more precise. Let a(t) be the integral of A around the circle at time t, and let a = a(0). Our formula for A implies
a(t) = a + teLwhere L is the length of the circle. Let e(t) be the electric field at time t. This is just constant:
e(t) = e.These equations are exactly like those saying how the classical free point particle evolves with time:
q(t) = q + pt/m p(t) = pThe holonomy a(t) marches along at constant velocity just like the particle's position; the electric field e(t) is constant just like the particle's momentum! And as you can see, the analogue of the mass m is 1/L. So here's our translation dictionary:
FREE POINT PARTICLE VACUUM MAXWELL EQUATIONS
ON THE LINE ON THE CYLINDER
q (position) a (holonomy around circle)
p (momentum) e (electric field)
m (mass) 1/L (inverse length)
Using this dictionary we can guess all sorts of things... and they're
all true. For example, since the energy of a free point particle is
H = p2/2mwe might guess the energy of the electromagnetic field is
H = Le2/2.And it is!
To see this, remember that in general, the energy of the electromagnetic field is the integral over all space of (E2 + B2)/2, where E is the electric field and B is the magnetic field. Here there's no magnetic field and the electric field is e. Integrate e2/2 around our circle of circumference L and -- presto! -- you get H = Le2/2.
In short, the analogy is perfect. And what do we call a perfect analogy?"
Toby said, "An isomorphism!"
"Right! The point particle on the line and 2d vacuum Maxwell theory on a cylinder are isomorphic classical systems. Since we've already quantized the first system, we can just transfer our results over to the second one. So let's do it. We just copy everything we said last week, making the substitutions
q -> a p -> e m -> 1/LThe configuration space of the classical vacuum Maxwell equations on the cylinder is R: the space of all values of a. So when we quantize, we get the Hilbert space L2(R).
Besides our Hilbert space, we get various observables, which are self-adjoint operators on this Hilbert space. The most important observables are a and e. They work just like the position and momentum operators for the quantum particle on the line:
(a psi)(x) = x psi(x) (e psi)(x) = -i psi'(x)From these we can build fancier operators, like the Hamiltonian:
H = Le2/2Here's how it acts:
(H psi)(x) = -(L/2) psi''(x)Using the Hamiltonian, we can see how states evolve in time. The formulas are just like last week, so I won't bother writing them down. Here's the point, though: if we start with a wavefunction psi(x) that's peaked near a particular value of x, this represents a state where the holonomy a has close to a definite value. As time passes, this wavefunction will typically spread out, so the holonomy will become more uncertain. However, the expected value of the holonomy will keep marching along at a constant rate. And what's this rate?"
"Umm," said Miguel, "for the free particle on the line we had q' = p/m, so now we have a' = Le."
"Right! The expected value of the holonomy keeps marching along at a rate equal to the expected value of the e field times the circumference of the universe.
This is easy to see if we work in the Heisenberg picture. If we let a(t) be the holonomy around the circle at time t, and let e(t) be the electric field at time t, our formulas from last time translate into:
a(t) = a + tLe e(t) = eso the expected value of the holonomy at time t is
<psi, a(t) psi> = <psi, a psi> + <psi, e psi> tL,confirming what Miguel just said.
Okay... that's all there is to it! But there are some morals to this story. First of all, note that to get going, we chopped spacetime into space and time. The connection on space, the A field, is the analogous to position, while the E field is analogous to momentum. We'll see this happening over and over again in lots of gauge theories: Yang-Mills theory, BF theory, various formulations of general relativity....
Since the A field is analogous to position, you might think the configuration space in these gauge theories is the space of A fields -- i.e., connections on space. People often use that; if what we just did looks weird to you, it's because we're taking a different approach. The idea here is that we're really interested in connections mod gauge transformations, so the configuration space is really a space of equivalence classes: connections modulo gauge transformations.
When we quantize these gauge theories, quantum states will be wavefunctions on the configuration space. Observables involving the A field get described as multiplication operators, while observables involving the E field get described as differentiation operators.
To do get anywhere in this game, we've got to understand the configuration space: the space of connections mod gauge transformations. How did we do that just now? In the theory we just considered, space was a circle. That makes things very simple! To describe a connection mod gauge transformations on the circle, all we need is its holonomy around the circle -- i.e., an element of the gauge group. If the gauge group is abelian, this is exactly right. That's why our configuration space was R just now: space was a circle, and our gauge group was R! If the gauge group G is nonabelian, the holonomy around a loop isn't gauge invariant, so things get a bit more complicated. But we'll worry about that later....
Even when space is not a circle, it's good to describe connections mod gauge transformations using their holonomies around loops. This is the idea behind the "loop representation" of quantum gravity. We'll be talking about this a lot.
But enough moralizing. Next time I'll see if you really understood what I just said."
The class ended. Some of the Acolytes went out to dinner with the Wiz; others went to the library to study. When they had all left, Oz slipped out from under the desk and stared sadly at the blackboard. He didn't follow all this talk of "bundles", "connections modulo gauge transformations" and so on. It was frustrating. Gloomily he went back to his cell....