Quantum Gravity Seminar

Week 17, Track 2

John Baez

March 5, 2001

As soon as the break between classes started, Jay slipped out and ran down to the laboratory. He knocked thrice on the door, and after a short while he heard it being unlocked from the inside. It opened on a crack, and then a face cautiously emerged - the face of Oz!

"How's it going?" whispered Jay.

"Come in," Oz said, opening the door further and motioning to a large contraption in the middle of the room. "Look: I've got it all set up! One flick of the switch and we've got our very own baby universe. I just need some exotic matter to prop it open long enough to look inside... otherwise we'll never really know if this worked."

"There must be exotic matter somewhere around here," said Jay, scanning the room. The laboratory was full of gadgets, from alembics and retorts to scanning tunnelling microscopes and tabletop cyclotrons. The floor was littered with math and physics papers, and the walls were lined with glass-doored cabinets full of bottles, jars, and boxes. "Let's look in some of the cabinets!"

"I did, but I guess it's worth another try. They're not organized in any way, as far as I can tell." Oz opened one of the cabinets and started rummaging around through bottles, all labelled in the Wizard's inimitable and nearly unreadable scrawl. Lifting them up one at a time, he read out the labels: "Aqua Regia. Arsenic. Arsine. Astatine. Guano... damn, just when it looked like alphabetical order!"

"Maybe he just misplaced that one," muttered Jay, who was looking through another cabinet. "Hey! Quarks, up. Quarks, down. This is more like particle physics stuff! Gluons. Muons...."

Ignoring him, Oz continued: "Guanine. Guar Gum. Guava Paste. Christ, he seems to have skipped straight to the letter G! Gulose. Hafnium. Hahnium. Halazone. Harmine. I've never even heard of half this stuff!"

Jay started looking through the next shelf. "Axions. Photinos. Squarks. Quintessence. Caloric. Phlogiston. Nonbaryonic Dark Matter. Exotic - hey! - exotic matter!!!"

"Let me see that!" cried Oz, rushing over. When he tried to grab the bottle from Jay's hands, it got loose and slowly started float towards the ceiling. "Whoa!" Just in the nick of time, they both caught it. "The real stuff, obviously."

"Yeah," said Jay. "Be careful when you open that." He glanced at the clock. "Whoops, I gotta go. Remember, you have only an hour to this experiment. And make sure not to stand too close. You might get sucked in!"

Oz snickered. "You think I'm that dumb? Come on, get out of here. If the Wiz catches on, we're in serious trouble."

Jay left and rushed back up the stairs. By the time he got to the classroom, the Wizard was already lecturing, so he tiptoed in quietly....

"But now," the Wiz was saying, "let's see how all this stuff about Maxwell's equations relates to what we've been doing in Track 1 with topological quantum field theories.

As we've seen, if we work in imaginary time, the quantized 2d vacuum Maxwell equations almost give us a TQFT - but not quite, because we only get a time evolution operator for a 2d cobordism

M: S -> S'
after we equip it with a little extra structure. We don't need a metric on M; we just need some positive numbers representing the area of each connected component. So from now on, at least today, we'll let M stand for a cobordism equipped with this extra structure. As we'll see, this gives us a time evolutions operator
Z(M): Z(S) -> Z(S')
where Z(S) is the Hilbert space for the 1-manifold S and Z(S') is the Hilbert space for the 1-manifold S'.

"Hold on," said Jay. "What do these Hilbert spaces look like, exactly?"

The Wizard glowered at him. "I already explained that - you were late for class! We saw last week that if S is a circle, Z(S) is the Hilbert space L2(U(1)). In general, if S is a disjoint union of n copies of the circle, Z(S) will be the tensor product of n copies of L2(U(1)). Here we're using the version of electromagnetism with gauge group U(1), which is better behaved than the R version.

Last week, we described the operators Z(M) when M was one of these basic "building blocks": the birth of a circle, the death of a circle, the upside-down pair of pants, and the pair of pants. Birth, death, marriage and divorce! That's what life is made of - at least in the category 2Cob. Every morphism in 2Cob can be built from these building blocks using the structure present in any symmetric monoidal category: composition, tensoring, and the braiding and identity morphisms. Technically, we say these 4 basic building blocks "generate" 2Cob as a symmetric monoidal category. You may have seen groups described via generators and relations; well, we can do the same thing for symmetric monoidal categories.

So, once we know the operators Z(M) for the 4 basic building blocks, it's uniquely defined for any cobordism M: S -> S' whose components are all equipped with areas, if we require these laws:

Z(M) Z(M') = Z(MM')

Z(M) (x) Z(M') = Z(M (x) M')

Z(BS,S') = BZ(S),Z(S')

These say that Z gets along with composition, tensoring, and the braiding."

"Umm, why is this true?" asked Miguel.

"I'll let you prove it - it's not hard," replied the Wiz. "You just - "

"What's really going on here," interrupted Toby, "is that we've got a symmetric monoidal category 2CobA whose objects are 1-manifolds and whose morphisms are cobordisms between these, each whose components is equipped with an area. The Wiz is claiming that our quantum field theory gives a symmetric monoidal functor

Z: 2CobA -> Hilb

It's like a TQFT, but not quite, since we're using 2CobA instead of 2Cob. Anyway, this symmetric monoidal functor Z is unique once we know it on the 4 basic building blocks, since these generate 2Cob as a symmetric monoidal category. To show it exists we'd need to check that some relations hold - but the Wiz is apparently going to wimp out and skip this part."

The Wiz hurled a fireball at Toby, which missed only because he had the good sense to duck in time. "Wimp out?! No, it's for your *own good* that I'm not proving everything in detail here - you need *something* to occupy yourself with during those long dull evenings."

"Anyway, Toby is right except for one nuance: we need to work a bit to make 2CobA into a category... we need to give it identity morphisms! Given an object S, the cylinder [0,1] x S with any positive area is not an identity morphism for S, since when we compose this with some other cobordism M, we get a cobordism that's topologically the same as M, but with more area."

The Wiz pondered a moment. "Whoops! The braidings BS,S' suffer from the same problem, actually - they don't really satisfy the laws they should unless they have zero area! I'm sorry for not pointing this out earlier.

So at first glance, it looks like we're in trouble: 2CobA doesn't have identity or braiding morphisms. But it's not really a problem: we can just formally throw in identity and braiding morphisms, which you can think of as cylinders with zero area, and then everything is okay! We get a symmetric monoidal category 2CobA, and the quantized 2d vacuum Maxwell equations give us a symmetric monoidal functor

Z: 2CobA -> Hilb.

Another nuance: here Hilb is the category of possibly infinite-dimensional Hilbert spaces and bounded linear operators. This is different from how we'd been defining Hilb when working with TQFTs, where all the Hilbert spaces are finite-dimensional. It's the infinite-dimensionality of L2(U(1)) that forces us to do this. This is also why the operators Z(M) usually blow up if we try to let the area on M go to zero."

"Blow up?" asked Richard.

"Yes - or in other words, diverge. Remember, last week we computed Z(M) when M was the 2-sphere with area equal to t. We got

Z(M) = sumk exp(-tk2/2) 
where we sum over all integers k. When t is positive this is a nice convergent sum. But if we set t equal to zero 0, this sum BLOWS UP!"

At these words, an explosion rocked the room. Miguel jumped up in shock, and Toby, who was leaning back in his chair, fell right over. Jay looked horrified. John looked around suspiciously and asked "What was that?" He wondered if the Wiz had done this to dramatize his point about divergent sums.

"I don't know!" said the Wiz. "But it felt like it came from the lab, and I have a guess who is to blame - Oz! I've caught him before, sneaking around where he had no right to be. And his mysterious absence... come on, let's see what happened!"

Grabbing his staff and pulling a burning torch from the wall, he strode out the door with a swirl of his robe, and the Acolytes hurried to follow him down the dimly-lit hallway and then downstairs to the lab. He tried to turn the doorknob. "Locked!" he exclaimed. He muttered a spell and the door burst open.

Inside the lab there was a strong smell of ozone. All the machines and gadgets were knocked over, and most of the cabinets' glass doors were shattered. In the center of the room was a large crater. There was no sign of Oz anywhere.

Suddenly the Wizard turned around and gave each of the acolytes a penetrating glance. "This is a serious matter. Do any of you know anything about this? Miguel?"

"No!" said Miguel, shocked that the Wiz would accuse him of such a thing. "I'm your coauthor, remember?"

The Wiz turned his glare to Richard. "How about you?"

"No, sir!" he replied with his usual honesty and candor.

"How about you, John?"

"I hardly ever do anything in this story," said John. "Why would I be responsible for a massive explosion in the lab? It doesn't make narrative sense!"

"Good point," conceded the Wiz. "Toby? Do you know what this is all about?"

"No!" said Toby. "And I can prove it. First of all, I've shunned Oz ever since he got me in trouble in Week 15. Remember that episode? Secondly, I'm just a fictional character - I have been ever since you turned me into one in Week 8."

"Really?" said the absent-minded Wiz. "I forgot."

"Yes," said Toby. "As a fictional character, I have no free will. I just do whatever you want. So how could I be responsible for this - unless, of course, you wanted me to?"

"Hmm," said the Wiz, nodding at this persuasive logic. "So that leaves... YOU!" Eyebrows bristling with fury, he swiveled and stabbed his finger towards Jay. "As I've often said: when you have eliminated the improbable, whatever remains, however impossible, must be the truth!"

"Me?" cried Jay, pointing at his chest and looking as innocent as possible.

"Yes, you!" said the Wizard. "In fact," he said, sitting down on an overstuffed armchair, "I've suspected you all along. I've heard you whispering furtively to Oz all week, and I noticed you coming in late today. Furthermore, of all the acolytes, only you, being a physicist, had the keys to the lab. So - confess! It's the only way out for you now, I'm afraid." He pulled out a pipe and lit it up while waiting for Jay's reply.

Jay thought a few moments, then glanced back and forth. All of a sudden, he jumped up and made a run for the door. However, Miguel sprang up and caught him before he escaped.

"Good work, Miguel!" said the Wiz, puffing vigorously. "By the way, could you pour me a brandy while you're up? - it's in that cabinet near the door, under B."

Turning his stern gaze back to Jay, he continued: "So, you're obviously guilty. The only question is - of what?"

Feeling pangs of conscience, Jay broke down and told the Wiz of his crazy plan with Oz to create a baby universe in the lab, following the instructions given in a paper they had downloaded from the hep-th preprint archive. "I told him not to stand too close, but he must have forgot and got pulled in!"

The Wizard frowned. "This is looking very bad. It's extremely difficult to extricate people from baby universes after they've fallen in. I'll try, but I haven't much hope. That means we may never see Oz again. Worse yet, we've got this huge mess on our hands!" He gestured at the crater and heaved a sigh. Eying Jay fiercely, he said, "I'll deal with you later. For now, let's just finish the class. Handcuff him, Miguel, so he doesn't bolt."

So. Where were we? Ah yes: we've quantized the vacuum Maxwell equations in 2d and gotten something like a TQFT, but applicable to spacetimes each of whose connected components is equipped with an area:

Z: 2CobA -> Hilb.

Of course, this is all for the imaginary-time theory. It's good to see what happens when we try to analytically continue this thing and get the more physical real-time theory... but I'll leave that to you. We did one case last week: the partition function of the 2-sphere. Figure out what happens for other manifolds!

But now I want to change course, and sketch how everything we've done generalizes to the 2d Yang-Mills equations! This will lead us into interesting new mathematics - which happens to be exactly the math we need to understand loop quantum gravity.

Maxwell's equations are just the Yang-Mills equations with gauge group U(1) - or R, but compact gauge groups are nicer, so we'll stick with them for now. We started this quarter by thinking about Maxwell's equations on the cylinder R x S1. We saw that in this case, the classical configuration space was just U(1). So: what do you think the classical configuration space will be for Yang-Mills theory with some compact gauge group G?"

Very dubiously, Toby guessed "G?"

"Right!" said the Wiz. "Or at least, *almost* right. Remember why we got U(1) for the configuration space of Maxwell's equations: the only gauge-invariant function of a U(1) connection on the circle was its holonomy all around the circle - an element of U(1). More precisely, all other gauge-invariant functions can be written as functions of this holonomy. The same logic works for any other gauge group, but now the holonomy around the circle is an element of G."

"But wait!" said Miguel. "If our gauge group is nonabelian, the holonomy of a connection around a loop isn't gauge-invariant. When we do a gauge transformation, the holonomy gets conjugated by a group element."

The Wiz smiled. "That's why I said Toby was almost right. The configuration space really consists of equivalence classes of elements of G, where two elements g and g' are deemed equivalent if they're conjugate:

g' = hgh-1
for some h in G. Fancy mathematicians call this action of G on itself by conjugation the "adjoint action", so they call this space of equivalence classes G/AdG. The configuration space of Yang-Mills theory on the cylinder R x S1 is not really G, it's G/AdG."

"Is that thing a manifold?" asked Miguel.

"No, it can have singularities," said the Wiz. "For example, when G = SU(2), G/AdG is the closed unit interval. That's a manifold with boundary, but G/AdG can be worse than that, too. This makes life a bit tough. However, we can avoid this issue if we're clever. Here's how. Suppose we quantize 2d Yang-Mills theory on the cylinder. What Hilbert space do we get?"

Jay tried to raise his hand but was unable to, so he raised his eyebrows instead. "When we quantize, our Hilbert space is L2 of the classical configuration space, so we should get L2(G/AdG)."

"Right," said the Wiz. "Of course this begs the question of what measure to put on G/AdG, so we can define the inner product in L2. But we can sidestep this question via the following trick. Functions on G/AdG are the same as functions on G that are constant on conjugacy classes:

psi(g) = psi(ghg-1).

We know how to define L2(G), because there's a god-given best measure on any compact group, called "Haar measure" - take my word for it. This lets us define L2(G/AdG) to be the subspace of L2(G) consisting of all functions that are constant on conjugacy classes."

"You could also take Haar measure on G and push it forwards using the quotient map

G -> G/AdG

to get a measure on G/AdG," noted Toby. "Then you could define L2(G/AdG) directly."

"Right," said the Wiz. "In fact, that would be equivalent to the trick I just described! But the nice thing about this trick is that we can work with functions on G and never need to think about G/AdG. Okay, so we've got our Hilbert space. What's next?"

"The Hamiltonian!" said Miguel.

"Right! In the quantized Maxwell theory, the Hamiltonian was just the Laplacian on U(1)... well, times -1/2. So what do you think the Hamiltonian will be in the quantized version of 2d Yang-Mills theory?"

"The Laplacian on G/AdG?" asked Toby dubiously. "Can we define the Laplacian on some nasty space with singularities?"

"Probably, but using our trick we can avoid this issue. We just use the Laplacian on G. This is a self-adjoint operator on L2(G), and we can restrict it to the subspace of functions that are constant on conjugacy classes, and get a self-adjoint operator on this subspace... which I'm calling L2(G/AdG). That's our Hamiltonian!"

"Are you just making this stuff up as you go along," asked Toby, "or is there some way to systematically derive it?"

The Wiz glared. "Of course you can derive it all using geometric quantization! But right now, I just want you to guess the answers. It's always easier to derive things systematically when you already know what answer you're trying to get.

So! Everything about 2d Yang-Mills theory with compact gauge group is very much like 2d electromagnetism. If we work hard enough we can even make 2d Yang-Mills theory into a symmetric monoidal functor

Z: 2CobA -> Hilb

But to calculate anything in this theory, we need a nice basis of L2(G/AdG): a basis that diagonalizes the Hamiltonian. We'll get our hands on this next quarter. And we'll see the answer is very pretty: there's one basis vector for each irreducible representation of G!

Okay, that's enough for now. Next week, for the final class of the quarter, I'll say a bit about how Maxwell's equations and the Yang-Mills equations are related to 2d topological lattice field theories."

Then the Wiz left to read up on saving people trapped in baby universes. Miguel took Jay to the dungeon, and the rest of the Acolytes set to work filling in the crater and cleaning up the lab.

baez@math.ucr.edu © 2001 John Baez