Oz hesitated... should he keep secretly attending Track 2? On the one hand, he was having a lot of trouble following the material. On the other hand, how would he ever learn this stuff if he didn't try? He stood pondering the matter until he heard the Wiz and his Acolytes nearing the classroom. At the last minute, more out of panic than anything else, he ducked under a desk.
The Wiz started in: "So far we've been studying 2d electromagnetism in a very rigid way. First we fixed a topology for spacetime: the cylinder. Then we fixed a metric on spacetime. Then we picked a way to foliate spacetime with spacelike slices. We proceeded from there... and found that the theory was isomorphic to the classical mechanics of a free point particle.
But now let's loosen things up! Let's figure out how the theory works with an arbitrary 2-dimensional manifold as spacetime, with an arbitrary metric and an arbitrary slicing.
To do this, we'll make use of a wonderful fact: when the dimension of spacetime is 2, the vacuum Maxwell theory is almost background-free!"
"Almost background-free?" asked Oz, mimicking Toby's voice. "What does that mean?"
"Heh," said the Wiz, turning to Toby and smiling. "Indeed, we often think of background dependence as a yes-or-no business. We say a field theory is "background-dependent" if the Lagrangian involves some fields that are held fixed -- not varied when we work out the equations of motion. These fields are called "background fields". The classic example is the metric, which we fix from the very start in ordinary special-relativistic quantum field theory. We call this a "background metric". So far, we've been using a background metric all over the place in our study of 2d electromagnetism.
When there are background fields around, they usually break diffeomorphism invariance: not all diffeomorphims of spacetime act as symmetries of our theory, but only those that preserve the background fields. For example, in ordinary quantum field theory, the diffeomorphism group gets broken down to the subgroup that preserves the Minkowski metric: the Poincare group.
Now, in our study of the 2d vacuum Maxwell theory, we started out by fixing a metric on the cylinder:
g = dt2 - dx2But in fact, we don't really need the metric in all its full glory! We only need the volume form:
vol = dt ^ dx.This counts as "less" background structure in the following precise sense: the group of diffeomorphisms that preserve the volume form contains the group of diffeomorphisms that preserve the metric. So you see, we can really talk about "more" or "less" background structure!
In fact, the vacuum Maxwell equations in 2d rely on so little background structure that we can easily modify them and get some truly background-free theories. This also works starting from 2d Yang-Mills theory. That's the direction we're heading....
But I'm getting ahead of myself. Let's see why the vacuum Maxwell equations in 2d only depend on the volume form. In any dimension, we can write them as
dA = F d*F = 0The only role of the background metric is to define *F. But in 2 dimensions we have
F = e volfor some function e called the "electric field", and then
*F = -e.This lets us rewrite the vacuum Maxwell equations in a way that uses only the volume form as a background structure:
dA = e vol de = 0We can also write down the Lagrangian in a way that only involves the volume form! Normally we write it as
L = F ^ *Fbut in 2d we can write it as
L = e2 vol.Now, in 2 dimensions the volume form should really be called the "area form", since it's what we use to measure areas. It follows from what we've done that in 2d, vacuum electromagnetism has all *area-preserving* diffeomorphisms as symmetries. These form a really big group. To get a feeling for it, imagine an incompressible fluid in 2 dimensions. As it flows, any particle that starts at any point x will move to some new point x(t) at time t. This map
x |-> x(t)is an area-preserving diffeomorphism. If you visualize a 2d fluid swirling around, you'll see how wiggly and flexible such diffeomorphisms can be!"
"In fact," said Miguel, "just as vector fields form the Lie algebra of the diffeomorphism group, divergence-free vector fields form the Lie algebra of this group of area-preserving diffeomorphisms!"
"Right -- at least if one takes care to deal with these infinite-dimensional groups and Lie algebras correctly," said the Wiz.
"And in V. I. Arnold's book, he shows that the equations for an incompressible fluid with zero viscosity say that --"
The Wiz cut him off. "Wait, we're getting off track here. We're doing electromagnetism, not fluid mechanics!
Let's describe time evolution in 2d electromagnetism when spacetime is a cylinder with an arbitrary Lorentzian metric. To be specific, let's do the version of electromagnetism with gauge group R -- the U(1) version will be similar.
We can slice spacetime any way we want... so we've got a bumpy cylinder with two wiggly slices S and S':
| ........ |
|' ` /
| `.. |
|\ `|
| \____ _/|
/ \____/ \
| S |
\ ... .. |
|' `.....' ` /
| `|
| /|
|\____________/ |
| S' \
| |
The region of spacetime between S and S' is a cobordism M: S -> S', but
with a volume form on it.
We've seen that classically, a state of electromagnetism on the slice S is described by a pair of numbers (a,e). The number a is the integral of the A field around S, and e is the electric field. Similarly, the state on the slice S' is some pair (a',e'). Time evolution is described by the map
(a,e) |-> (a',e').But what is this map like?
Well, Maxwell's equation in 2d says the electric field is constant, so
e' = e.To figure out a', we use Stokes' theorem:
\intM dA = \intdM A.This says that
\intM e vol = a' - a.The left side is just e times the area of M, so
a' = a + Area(M) e.In short, time evolution only depends on the area between our slices. That's just what you'd expect in a 2d theory where the only background field is the volume form... it really couldn't work any other way.
In fact, in 2d electromagnetism, area plays a role very much like "time" in ordinary mechanics. If we use "t" to stand for the area between our slices, our time evolution map is
(a,e) |-> (a + te, e).This is isomorphic to time evolution for a point particle with unit mass:
(q,p) |-> (q + tp, p).Sound familiar? It should! We saw a special case of this in Week 13. And just as we did back then, we can use this isomorphism to quickly figure out the quantum version of our theory. The Hilbert space is L2(R), the Hamiltonian is copied after the Hamiltonian for a particle of unit mass:
(H psi)(x) = -(1/2) psi''(x)and to evolve a state in time from one slice to another, we just hit it with the operator exp(-itH), where t is the area between the slices.
This whole business also works for the U(1) version of electromagnetism. The only difference is that the holonomy a takes values in U(1), so our Hilbert space is L2(U(1)). The formula for the Hamiltonian looks just the same... no big deal.
Okay, great -- we've quantized 2d electromagnetism on a cylinder with arbitrary metric. What's left to do?"
"More general manifolds," said Richard.
"Right! We'd really like to describe time evolution for any cobordism between 1-dimensional manifolds, like this:
___ ____
/ \___/ \
| __ \ S
| / \___ |
|\_/ \___/| |
| | |
| ___ | |
| / \ | |
| | | | | M
| | | | |
| ... \ | .. | |
|' `.| |' ` | |
| | | `| v
| | \ |
\_____/ \__/ S'
If we do this right, we'll get something almost like a 2d TQFT!
We'll get time evolution operators
Z(M): Z(S) -> Z(S')satisfying a lot of the rules that hold in a TQFT. The main difference is that now our cobordism M will have a specified area, and our time evolution operator will depend on this area, not just the topology of M.
Before we can do this, though, there's a little problem we must confront: the problem of "topology change". It's not easy to put nice Lorentzian metrics on cobordisms that go between spaces with different topologies. For example, there's no way to put a Lorentzian metric on this manifold M:
___ ____
/ \___/ \ S
| __ \
| / \___ | |
|\_/ \___/| |
| ___ | |
| / \ | |
| | | | | M
| ... \ | .. | |
|' `.| |' ` | |
| | | `| v
| | \ |
\_____/ \__/ S'
for which S and S' are spacelike. Now, there are different ways to
tackle this problem. One is to allow degenerate Lorentzian metrics.
That's probably the best approach. But another is to pull a dirty
trick: switch to spacetimes equipped with a Riemannian metric instead
of a Lorentzian one! Let me explain this trick, because it's quite
popular.
First consider the cylinder R x S1. Formally, we can turn the Lorentzian metric
-dt2 + dx2into the Riemannian metric
dt2 + dx2by means of this substitution:
t -> -itThis is called a Wick rotation. If we pull this trick, Schroedinger's equation turns into the heat equation --"
Oz was very puzzled, so he interrupted, again mimicking Toby's voice: "What do you mean by formally here?"
"It means: don't ask me what I mean. Physicists say this when they don't really know what they're doing."
"What kind of answer is that???" said Miguel. "Don't tell me you're one of those guys who switch to imaginary time as soon as the going gets tough, and then never come back to the real world."
"Yeah!" said Toby. "I've never been convinced that this is a legitimate way to deal with topology change."
"Yeah!" said Miguel. "Sure, you can relate Schroedinger's equation and the heat equation in situations when there's a fixed time coordinate t around: you can take your formulas, analytically continue them to imaginary values of t, do whatever you want with them, and analytically continue back -- that's fine. But a fixed time coordinate is exactly what you don't have in quantum gravity. Especially in situations with topology change."
"Yeah!" said Toby. "This sucks! I want my money back!" And without exchanging another word, Toby and Miguel stood up and advanced towards the Wiz, brandishing their staffs menacingly. "DOWN WITH WICK ROTATION!" they chanted... and the rest of the class began to join in.
"Wait a minute!" cried the Wizard, backing into a corner.
"DOWN WITH WICK ROTATION!"
"Be patient!" he yelped, as the class began to wave pitchforks in the air.
"DOWN WITH WICK ROTATION!"
"Listen: I AGREE WITH YOU! I'm not one of those evil physicists who permanently flees the Lorentzian world for the imaginary paradise of Riemannian spacetime. I just wanted to show you how it works!"
The class lowered their weapons, grumbling.
"We'll learn some interesting things, and then we'll come back to reality. Honest!"
The class went back to their seats, and the Wiz wiped the sweat off his forehead. "Phew!" he said. "At least you're taking the subject seriously."
"Down with Wick rotation!" yelled Oz.
"Hey!" said the Wiz. "Who said that?" He spotted Oz crouched behind a desk. "Aha, so there's the troublemaker! There's the fellow who's been asking all those questions in your voice, Toby!"
Toby turned and saw Oz quivering with fear. "Why, you little...."
Oz got up and made a break for the door. Furious, Toby chased him out of class and down the hallway.
"I'll deal with them later," said the Wiz. "For now, let me finish today's lecture. If we formally replace t by -it, the standard Lorentzian metric on the cylinder becomes Riemannian, and Schroedinger's equation
d psi / dt = - iH psibecomes the heat equation
d psi / dt = - H psi.Instead of time evolution being described by the one-parameter unitary group exp(-itH), it's described by the one-parameter semigroup exp(-tH). By "semigroup", I mean that exp(-tH) is only a nice bounded operator for nonnegative times t. Basically, what this strange trick amounts to is studying thermal physics rather than quantum physics... but they are related by many useful analogies.
Generalizing this idea to the case of an arbitrary Riemannian metric on the cylinder, we can define a theory where the Hilbert space for a circle is L2(R) and the time evolution operator from one slice S to another slice S':
________
/ \ S
| \__
|\ \ |
| \____ _/| |
/ \____/ \ |
| | | M
\ ... .. | |
|' `.....' ` / |
| `| v
| /
\____________/ S'
is defined to be
Z(M) = exp(-tH)where t is the area of M. Note that we indeed have
Z(MM') = Z(M) Z(M')much as in a TQFT. So one question is: can we extend this idea to more general cobordisms, allowing topology change? Next week, we'll see that the answer is yes. We'll get something a lot like a TQFT, but where any cobordism M must be equipped with an area before we can define Z(M). This sort of theory isn't completely background-free: the area plays the role of a background structure. But as we've seen, it's much closer to being background-free than your average quantum field theory formulated with the help of a background metric. And later we'll see how to modify this idea to get some truly background-free theories... which don't involve this "Wick rotation" idea that you hate so much."
Then the Wizard walked out in search of Oz and Toby, who could be heard scuffling somewhere in the courtyard below.