"What's wrong with a metric?" asked Jay the Acolyte "This is a course on quantum gravity; we should expect metrics.".
The Wizard glowered. "I said a *fixed* metric. The lesson of general relativity is that physics should be background-free: there should be no fields that just sit around affecting things while remaining aloof and unaffected themselves. Maxwell's equations are great, but they don't incorporate this lesson.
Of course, we could take the Lagrangian for the vacuum Maxwell equations and let the metric be *variable*. If you stuck in an extra term, you'd get the Einstein-Maxwell theory: electromagnetism coupled to gravity. This is background-free, since all fields involved are treated as variables. But when we quantize it, nasty things happen! One reason is that, indeed, a metric is a funny thing. The definition of "metric" secretly involves an inequality, lurking in the condition that it be nondegenerate. It's tough to quantize inequalities! So, when we learn to do general relativity without mentioning metrics, it'll be easier to quantize gravity. And the first step is ... EF theory!".
The Acolytes had more questions, but they could see the Wizard wanted to talk about EF theory.
The Wizard looked around. "No more questions? Good. We've already seen the Lagrangian for EF theory; now we'll work out the equations of motion. But, first, a quick review.
As usual, one of our fields is a connection A on the principal G-bundle P over our n-dimensional spacetime manifold M. Its curvature F is an Ad(P)-valued 2-form.".
Miguel the Acolyte stifled a yawn. Glaring at him, the Wizard continued
"But now we also have another field: E! This is an Ad(P)-valued (n-2)-form. Why n - 2, you ask? So that tr(E ^ F) is an n-form. This is the Lagrangian for EF theory. You can think of E as a substitute for the *F in the Lagrangian for the vacuum Maxwell equations: tr(*F ^ F). The * secretly involves a metric, but the E field doesn't!
Okay, now figure out the equations of motion. You know the routine. What do you do first?".
"Well, you write down the variation of the action ... ." said Jay the Acolyte.
"Right:
S = \int tr (E ^ F),so
DS = D \int tr (E ^ F).Great -- now what?".
John the Acolyte said "First you push the D past the integral and the trace ... .".
"Right:
DS = D \int tr (E ^ F) = \int D tr (E ^ F) = \int tr (D(E ^ F)).So far so good. What do we use next?".
The entire class shouted: "The Leibniz law!".
"Good! In other words, the product rule. The first day Leibniz thought about differentiating products, he wrote this down in his notebook: (fg)' = f' g'. The classic freshman mistake! But the next day, he wrote down this: (fg)' = f' g + f g'. Newton got it right the first time. But persistence deserves honor too, so we call this the Leibniz law.
So: using the Leibniz law, we get
DS = \int tr (D(E ^ F)) = \int tr (DE ^ F + E ^ DF).The DE here is good; it's a variation of one of our basic fields. But the DF still needs work. Here we use the formula DF = dA DA which we derived in Week 5. It's not named after anyone yet, so I just call it the "magic formula". We get
DS = \int tr (DE ^ F + E ^ dA DA).Next?".
Miguel the Acolyte said "When in doubt, integrate by parts! Push the dA over to the other side!".
"Right. Neglecting boundary terms, this gives
DS = \int tr (DE ^ F + (-1)^{n-2} dA E ^ DA).
Don't worry about the sign there; it won't matter.
This only vanishes for *all* variations DA and DE if these equations hold:
F = 0
dA E = 0
These are the basic equations of EF theory!
The first one says our connection is flat.
The second says that the E field has vanishing exterior covariant derivative.
They're pretty dull -- but at least they're nontrivial,
and background-free to boot!
These equations have a lot of symmetry. First, they're invariant under gauge transformations:
A |-> gAg^{-1} + gdg^{-1}
E |-> gEg^{-1}
Second, they're invariant under transformations like this:
A |-> A E |-> E + dA Xwhere X is an Ad(P)-valued (n-3)-form. I won't prove this; you can do it yourself.
Both these symmetries are "gauge symmetries". This means they map any solution to a new solution that's "the same" for the purposes of physics: no experiment could tell the difference. Someday I'll tell you how we stare at a symmetry and tell whether it's a "gauge symmetry" or not. For now, just take my word for it. Also, remember this: "gauge symmetries" means something different from "gauge transformations".
Here's another thing I won't prove: Inside any contractible open subset of spacetime, you can map any solution of the equations of EF theory to any other using the gauge symmetries I just listed. This means that *locally*, all solutions look alike. Thus we say that EF theory has "no local degrees of freedom". In this theory, all the interesting physics is global.".
Jay the Acolyte looked frustrated. "What does *that* mean?".
"It means that you can only tell the difference between different solutions of the field equations by doing experiments where you go around a noncontractible loop in spacetime, or something like that. A good example of a theory without local degrees of freedom is general relativity in 3D spacetime. In their usual formulation involving a metric, the vacuum Einstein equations in 3 dimensions say simply that this metric is flat. But if space were a torus and you parallel translated a vector all the way around, it could come back rotated!
When you quantize theories like this, you tend to get "topological quantum field theories" or "TQFTs". Right now, everything in Track 1 is leading up to theories like these. They're no fun unless the topology of spacetime is interesting. They can't describe the real world, since they have no local degrees of freedom, like the classical field theories they came from. Nonetheless, they are a step towards full-fledged quantum gravity, as you shall eventually see.
In fact, 3D general relativity is a special case of EF theory, where the gauge group is SO(3) or its double cover SU(2) if we're doing Riemannian GR, or SO(2,1) or its double cover SL(2,C) if we're doing the more realistic Lorentzian GR.
Instead of a metric, our basic fields are now A and E. But the metric is still hiding there - it's determined by the E field as follows:
g(v,w) = tr((Ev) (Ew)).See? The E field eats the tangent vector v and turns it into an element of the Lie algebra g, which is just a matrix. Ditto for w. Then we multiply these matrices and take the trace. Out pops a number. So we really do get a metric."
Miguel interrupted, asking "Couldn't this metric be degenerate? What if E were zero, for example?"
The Wizard smiled. "YES! The metric can be degenerate. But that's actually GOOD. As I said, the nondegeneracy condition is tough to quantize. But EF theory gets around that: it allows degenerate metrics. The usual formulation of general relativity can't handle them!
So... where was I? Right: in 3d gravity the E field determines the metric, and the A field turns out to determine a metric-compatible connection. The equation dA E = 0 says this connection is torsion-free, so it's the Levi-Civita connection. The equation F = 0 says this connection is flat, so taken together, the equations of EF theory say the metric is flat. This is just what the vacuum Einstein equations say in 3d spacetime!
I'll explain all this stuff better some other day. The main point now is this: EF theory is a generalization of the vacuum Einstein equations in 3d spacetime; it describes a world with no local degrees of freedom. Okay -- class dismissed.".
A bit puzzled and rather tired, the class filed out.