As the Acolytes settled in after their break, the Wizard could barely contain his joy. "I've got wonderful news for you!" he said "Today, we're going to start on a grand new journey: nonvacuous equations of motion!!!".
Everybody cheered.
"Hic!" added Toby the Acolyte, involuntarily.
"However," added the Wizard "I do want to point out one thing: in the vacuous cases, the action S is independent of the connection A and is therefore a topological invariant of the G-bundle P. This invariant can be interesting in its own right. For example, in nth Chern theory, the action is the nth Chern number.".
"Strictly speaking," said Toby the Acolyte, voluntarily, "we only know S is independent of A on each connected component of the space of all bundles. How do we know that space is connected?".
"That's easy." said the Wizard "The differences between connections form a vector space, the vector space of Ad(P)-valued 1-forms. In other words, the space of connections is an affine space, which is like a vector space that's forgotten its origin. And vector spaces are connected, as I'm sure you know.
The 1st Chern number is particularly nice for principal U(1)-bundles over a connected 2-dimensional manifold. As Toby once worked out on sci.physics.research, \int tr F classifies these bundles up to isomorphism.
Similarly, principal SU(2)-bundles over a 4-dimensional manifold are classified up to isomorphism by the 2nd Chern number. Now, spacetime is a principal SU(2)-bundle, and you might wonder if we can measure it experimentally. Unfortunately, there is no physical process that measures this unless the so-called "theta angle" is nonzero. The theta angle might be nonzero -- but it's not clear yet whether it is or not. In any case, we don't have enough precision to measure the 2nd Chern number of the universe.".
"Aw!" sighed the Acolytes.
"Hic!" added Toby the Acolyte, involuntarily.
"Someone needs to get the radiator fixed." said the Wizard "All right, now let's do electromagnetism!
In electromagnetism, our group G is U(1). Now, the Lie algebra u(1) of U(1) is, strictly speaking, the real vector space of purely imaginary numbers. So, u(1)-valued forms are purely imaginary valued forms. But, if we're sufficiently lowbrow, we can think of these as regular old forms. Then we can just ignore that silly trace. So, the action
S(A) = \int (F ^ *F)describes a universe of pure E&M -- world of pure light.".
"Ooh!" said the Acolytes, a vision of pure light shimmering through their minds.
"Here * is the Hodge star operator, valid because we assume that the spacetime manifold M has a metric.
Now, then, I'm sure you all remember what to do.
DS = D \int (F ^ *F) = \int D(F ^ *F) = \int (DF ^ *F + F ^ D*F).Now what?".
"Well," said Miguel the Acolyte "D and * should commute, because * is a linear operator.".
"Right!" said the Wizard "so let's just pop the * over:
DS = \int (DF ^ *F + F ^ *DF).Now, if only the * were on F both times! Well, there's a little trick we can use. I never remember exactly how the signs go, so I'll just look it up in here.".
With a wave of his hand, the Wizard conjured up a book out of thin air. "Gauge Fields, Knots, and Gravity." said the Wizard "I wrote this book so I'd never have to remember how the signs go -- now I can just look them up in here!". He snapped the book shut. "Got it! The rule is:
\int (a ^ *b) = (-1)(n-|a|)|b| \int (*a ^ b).By "(-1)(n-|a|)|b|", I mean that you take the degrees of a and b, subtract the first from n, multiply these numbers together, and stick in a minus sign if you get an odd number in the end. In our case, F and DF are 2-forms, and 2(n-2) is even no matter what n is, so the sign is even.
So,
DS = 2 \int (DF ^ *F)
= 2 \int (dA DA ^ *F)
= 2 \int (dDA ^ *F).
Now, why could I change dA to just d?
Because U(1) is an Abelian group, the Lie bracket is always 0,
so
dA C = dC + [A,C]
= dC + 0
= dC.
Finally, we move the d off of the DA through integration by parts,
getting
DS = -(-1)(n-2) 2 \int (DA ^ d*F).Actually, I may have gotten that sign wrong too, but it doesn't matter, since we only care when DS is 0.
And when is DS equal to 0? When d*F = 0. And that's the *nonvacuous* vacuum Maxwell equation: d*F = 0!".
Jay the Acolyte looked confused.
"Let me put this in different terms for you physicists out there." said the Wizard "If spacetime is decomposable as R x S, where R is the timeline and S is space, write F as E ^ dt + B. Then
*F = *S E - *S B ^ dt,and
d*F = dS *S E + dt ^ dt *S E - dS *S B ^ dt.By a subscript S, I mean the corresponding operator on the space manifold S, and t, of course, is the time coordinate on R. Splitting this up into time and space components, the equation d*F = 0 becomes
dS *S E = 0and
dt *S E = dS *S B.But dS *S E is just the divergence of E, dt *S E is just the time derivative of E, and dS *S B is just the curl of B. So, we get Maxwell's equations in the vacuum, div E = 0 and E' = curl B. The other equations, div B = 0 and B' + curl E = 0, are automatically satisfied because the form F is closed, being the differential of A.
By the way, does anybody know why the magnetic field is called "B"? Me neither.
Any questions?".
"Hic!" said Toby the Acolyte, involuntarily.
"That's not a question!" roared the Wizard "You've mocked me for the last time with your involuntary hiccups! Take that!!!".
And when the mist had cleared, what used to be Toby the Acolyte was nothing but an ugly green frog.
"Ribbit!" said Toby the Acolyte, involuntarily.
The Wizard grimaced. "I'm not sure that helped.".