"That's good," said the Wiz, "because today I'm talking about some far-out stuff, and I wouldn't want it to go to his head.
Today our theme is... When Worlds Collide! We've studied the classical and quantum versions of the vacuum Maxwell equations when spacetime is a 2-dimensional cylinder, either Lorentzian or Riemannian. Now let's see what happens for more general 2d spacetimes. We'll start with the collision of two circular universes:
____ ___
/ \ / \
| | | |
| | | |
|\____/| |\____/|
| | | |
\ \ / /
\ \ / /
\ \_/ /
\ /
\ /
\ /
| ..... |
|' `|
| |
| |
\_____/
This is a nice example of "topology change": space starts out as two
circles, but winds up as one! Topology change is a fascinating but
controversial topic in quantum gravity. Was there really nothing before
the Big Bang? Would there be nothing after a Big Crunch if the universe
recollapsed? Where, if anywhere, does stuff go when it falls into a
black hole? Do black holes give rise to new "baby universes", as some
have suggested? Do virtual black holes cause the entropy of the universe
to rise, as Hawking's calculations seem to indicate?
Nobody knows the answers to any of these questions. But in general relativity spacetime becomes dynamical, so it's natural to wonder whether not only the geometry of space, but also its topology, can change with the passage of time -- and what would happen if it did!
Serious physicists have written on these subjects. Some have studied the possibility of creating wormholes. Some have even pondered the possibility of creating baby universes in the lab!"
Jay's eyes gleamed with fascination. The Wizard, reading his mind, raised a warning finger.
"Don't even think about it! I would not take these speculations too seriously. Indeed, I wouldn't even mention them, except that now we can study them in a simple toy model: 2d vacuum Maxwell theory. It's not physically realistic, but it will give us some idea of how physics gets modified in the presence of topology change. For example, we'll need to say goodbye to the idea that time evolution is described by unitary operators. We've already seen this in our work on TQFTs, but it's nice to see how it works in a familiar theory like electromagnetism.
Before we study two colliding universes, remember what happens if spacetime is a cylinder of a given area:
CLASSICAL QUANTUM QUANTUM
TIME EVOLUTION TIME EVOLUTION TIME EVOLUTION
(real time) (imaginary time)
________
/ \ (a,e) psi psi
| | _____ ___ ___
| | | | |
|\________/| | | |
| | | | |
| area = t | | | |
| ........ | | | |
|' `| | | |
| | v v v
| |
\________/ (a + te, e) exp(-itH) psi exp(-tH) psi
We call the area "t" because it plays a role much like time here. A
classical state is a pair (a,e), where a is the holonomy around the
circle and e is the electric field. A quantum state is a wavefunction
in L2(R) if we're doing the version of
electromagnetism with gauge group R, or L2(U(1)) if
we're doing the U(1) version. We'll do the R version at first,
but the two are very similar. Either way, in the quantized theory the
electric field is an operator like this:
(e psi)(x) = -i psi'(x)and the Hamiltonian looks like this:
(H psi)(x) = -(1/2) psi''(x).Both these operators are diagonalized by this basis:
psik(x) = exp(ikx)where k is any real number in the R version of electromagnetism, but an integer in the U(1) version.
Now suppose spacetime is this cobordism M:
____ ____
/ \ / \
| | | |
| ^ a | ^ a'
|\____/| |\____/|
| | | |
\ \ / /
\ \ / /
\ \_/ /
\ /
\ area = t /
\ /
| ..... |
|' `|
| ^ a''
| |
\_____/
The holonomies around the two incoming circles are a and a', and
the holonomy around the outgoing one is a''. How are these related?
Well, we have
dA = e volwhere the electric field e is constant over the whole spacetime. Stokes' theorem says
\intM dA = \intdM Aso
\intM e vol = \intdM A
= a'' - a - a'.
The left side is just e times the area of M, so
a'' = a + a' + et.This is simple enough, but note some peculiarities. When space is a circle, the classical phase space consists of pairs (a,e), and when space is a pair of circles, it consists of 4-tuples (a,e,a',e'). Thus, when spacetime goes from a pair of circles to a single one, naively you might think that the corresponding classical time evolution would be described by a function from R4 to R2. But it's not!
First, not any state (a,e,a',e') in R4 has the ability to evolve to a state in R2. Since the electric field must be constant throughout each component of spacetime, we're stuck unless e = e'. And even if e = e', there's another peculiarity: lots of different states in R4 evolve to the same state in R2.
The moral is this: when spacetime is a cobordism where the topology of space changes, a single classical state can evolve to many or no states. Mathematically speaking, this means that time evolution is not one-to-one, and not even a map! It's just a relation.
Now let's turn to the quantum theory. To do this systematically we'd need to know how to quantize relations between classical phase spaces. People have actually studied this, but it would be a big digression to talk about it now, so let's take a more ad hoc approach.
Let's start by assuming the area of M is zero: t = 0. This is a somewhat degenerate case, but it's nice because then the holonomy a'' depends on just the holonomies a and a', not the electric field:
a'' = a + a'.This means we get a map between classical configuration spaces:
____ ____
/ \ / \
| | | | CLASSICAL CONFIGURATION
| | | | SPACE = R2 (a,a')
|\____/| |\____/| ______
| | | | | |
\ \ / / | |
\ \ / / | |
\ \_/ / | |
\ / | f | f
\ area = 0 / | |
\ / | |
| ..... | | v
|' `| v
| | a + a'
| | CLASSICAL CONFIGURATION
\_____/ SPACE = R
Since the Hilbert space of the quantum system is L2 of the classical
configuration space, this suggests that there's a nice time evolution
operator between Hilbert spaces... but going the other way, namely:
f*: L2(R) -> L2(R2)where f* is the pullback:
(f* psi)(a,a') = psi(f(a,a')) = psi(a + a').Since pulling back is contravariant, this operator tells us how to evolve quantum states backwards in time."
"That's weird," said Jay.
"Well, you're probably more used to quantizing systems with time-reversible dynamics, where you don't notice these subtleties. When two universes collide, that's about as irreversible as it gets."
"Hey, wait a minute -- you were lying!" said Miguel, looking up from a calculation he'd just scribbled in his notebook. "If psi is a square-integrable function, f* psi won't usually be! In fact,
\int |psi(a + a')|2 da da'will be infinite unless psi is zero. Just do a change of variables and it's obvious."
"Excellent point," said the Wiz. "But I wasn't lying: I just said our ponderings suggest there's an operator
f*: L2(R) -> L2(R2)I didn't said there actually was! The problem is that the measure of the real line is infinite. Quantum field theory is plagued with infinities, and this is one. Can anyone see how to get around it?"
"Use the U(1) version of electromagnetism!" said Toby.
"Right! All the formulas we've derived for the R version also hold for the U(1) version as long as we treat the holonomies as real numbers modulo 2pi. And in the U(1) version, we really do get
f*: L2(U(1)) -> L2(U(1)2),since if psi is a square-integrable function on U(1) then
\int |psi(a + a')|2 da da'is finite. Physicists can make sense of nonnormalizable states, so we could probably learn to live with the R version of electromagnetism if we had to. But we don't... so let's switch to the U(1) version!
We then get this picture:
____ ____
/ \ / \
| | | | CLASSICAL CONFIGURATION
| | | | SPACE = U(1)2 (a,a')
|\____/| |\____/| ______
| | | | | |
\ \ / / | |
\ \ / / | |
\ \_/ / | |
\ / | f | f
\ area = 0 / | |
\ / | |
| ..... | | v
|' `| v
| | a + a'
| | CLASSICAL CONFIGURATION
\_____/ SPACE = U(1)
_____
/ \
| | QUANTUM HILBERT SPACE =
| | L2(U(1)) psik
|\_____/| _______
| | | |
| | | |
| | | |
/ \ | |
/ area = 0 \ | f* | f*
/ _ \ | |
/ / \ \ | |
/ / \ \ | |
/ / \ \ | |
| .... | | .... | v v
|' `| |' `|
| | | | QUANTUM HILBERT SPACE = psik (x) psik
\____/ \____/ L2(U(1)) (x) L2(U(1))
Here I've worked out what f* does to our basis states psik:
(f* psik)(a,a') = psik(a + a')
= exp(ik(a + a'))
= exp(ika) exp(ika')
= psik(a) psik(a')
so
f* psik = psik (x) psik.So now we know what happens in 2d vacuum quantum electrodynamics when the universe splits in two."
"And just as you warned us," Miguel pointed out, "this time evolution operator isn't unitary. It's norm-preserving, at least if you normalize things properly... but it's not onto!"
"Yes. Of course, we got this result in an ad hoc way. You might worry that we screwed up somewhere. But we can check it! For example, we know that psik is the state where the electric field has value k. Our calculation says that as the universe splits, this state evolves to the state psik (x) psik, where the electric field has value k on each of the two circles. So the electric field is constant on all of spacetime! This is consistent with what we've already seen in the classical theory.
Also, we can consider a cobordism like this:
QUANTUM
TIME EVOLUTION
_________ (imaginary time)
/ \
| | psik
| | _____
|\_________/| |
| | v
| area = t |
| | exp(-tk2/2) psik
/ \_________/ \ _____
/ \ |
/ area = 0 \ v
/ _ \
/\________/ \________/\ exp(-tk2/2) psik (x) psik
/ / \ \ _____
/ / \ \ |
| ....... | | ....... | v
|' `| |' `|
|area = t'| |area = t"| exp(-tk2/2) exp(-t'k2/2) psik (x)
| | | | exp(-t"k2/2) psik
\_______/ \_______/
Our cobordism has a fixed area, but we can calculate its time evolution
operator in various ways, depending on how we chop it up. Let's chop
it into a cylinder of area t, then a piece of area 0, and finally two
cylinders with areas t' and t". These areas can be whatever we like, as
long as they add up to right total area.
Over on the right I've worked out what time evolution does to the state psik, working in imaginary time. If you stare at the result, you'll see it's just
exp(-(t + t' + t")k2/2) ( psik (x) psik ).This is great. See what's so great about it? It just depends on the total area of our cobordism, not on how we chopped it up! This makes an excellent consistency check on our calculation of what happens when the universe splits.
By the way, while we did the calculation in imaginary time, we can analytically continue the result back to real time, and we get this:
exp(-i(t + t' + t")k2/2) ( psik (x) psik ).In fact, we can also get this result by working directly in real time. Any Lorentzian metric on our cobordism must be degenerate if the boundary is spacelike, but for the vacuum Maxwell's equations in 2d that's okay. The volume form vanishes, but it's not a problem: all the formulas still work fine. We can go ahead and quantize the theory even if the metric is degenerate, and we get the same answer as above. So this calculation also makes a nice consistency check on the Wick rotation idea."
"What?" said Toby. "You mean you can actually skip the Wick rotation business and study topology change directly in the Lorentzian version of the theory, even though the metric is degenerate?"
"Yes. I hadn't realized that last week, but now I see it's true."
"So Wick rotation is useless!" crowed Toby.
"Umm, well..." said the Wiz, "In this situation we don't need it, but at least it gives the right answer."
Miguel stood up, shook his first in the air and yelled: "DOWN WITH WICK ROTATION!"
"DOWN WITH WICK ROTATION!" responded Toby. Soon the whole class took up the chant, marching in circles and waving protest signs, banging pots and pans.
The Wiz sighed. "Youth," he muttered. Then he flipped a small glowing object into the air with his thumb. It slowly sailed up, tracing out a parabolic arc. Knowing the Wizard's love for pyrotechnics, the acolytes stopped their chant and gazed at it in terror, expecting it to explode with devastating force. It gradually fell back down.... and fizzled out with an almost inaudible hiss.
"Now listen," said the Wizard. "Wick rotation may or may not be a good idea, but you should see what it does in various contexts before getting all worked up over it! We'll see later that it really does help, sometimes."
Grumbling, the Acolytes returned to their seats.
"Okay," continued the Wiz. "We wanted to see what happens when worlds collide, but due to the contravariance of pullback, we wound up studying what happens when the world splits. This is a toy model of the "baby universe" idea.
But what really happens in the quantum theory when worlds collide? For that, let's assume that 2d vacuum electromagnetism shares this property with unitary TQFTs: if we time-reverse a cobordism, we take the adjoint of the corresponding time evolution operator. A little calculation then gives this:
QUANTUM
TIME EVOLUTION
____ ____ psij (x) psik
/ \ / \ ___________
| | | | |
| | | | |
|\____/| |\____/| |
| | | | |
\ \ / / |
\ \ / / |
\ \_/ / |
\ / |
\ area = 0 / |
\ / |
| ..... | |
|' `| v
| |
| | deltajk psik
\_____/
"So the transition amplitude for the state psij (x)
psik to become the state psii is zero unless i = j
= k," said Toby. "That's good: since these states psik are
eigenstates of the electric field, it goes along with our idea that the
electric field is constant throughout each connected component of
spacetime."
The Wiz nodded. "Now let's do the other two basic cobordisms. First, the birth of a circle:
______
/ \
/ \
| |
| area = t |
| ........ |
|' `|
| |
| ^ a
\________/
or in cosmological terms, a Big Bang. I've drawn in the area of this
spacetime and the holonomy around the outgoing circle. Let's work out
everything, using the same strategy as before.
Stokes' theorem says that a = et, where e is the electric field on this spacetime. The classical space for the circle consists of pairs (a,e), while the classical phase space for the empty set is...?"
"The one-element set," said Toby, "because there's only way for nothing to be!"
"Right. There's one state for the empty set, and it can evolve to any state (a,e) with a = et. So as before, classical time evolution is just a relation between these classical phase spaces, not a function.
And as before, we quantize by considering the t = 0 case, where there's a well-defined function between the classical configuration spaces. The classical configuration space for the empty set consists of one point. If we call that point x, we have:
______ CLASSICAL CONFIGURATION
/ \ SPACE = {x} x
/ \ _______
| area = 0 | | |
| | | |
| ........ | | |
|' `| | f | f
| | | |
| | | |
\________/ v v
CLASSICAL CONFIGURATION 0
SPACE = U(1)
Here 0 stands for the identity element of U(1), which we're treating as the
additive group of real numbers modulo 2pi.
Next, our strategy says to quantize using pullbacks. This gives us the upside-down picture:
________
/ \
| |
| |
| |
|\________/|
| |
| |
| area = 0 |
| |
\ /
\______/
namely the death of a circle."
"The death of a circle," Toby mused. "Wasn't that a play by Arthur Miller?"
The Wizard threw a fireball at Toby. "No, that was the death of a salesman. I'm talking about something far more dramatic: in terms of cosmology, nothing less than the Big Crunch -- the end of the universe! Anyway, here's what it does to quantum states:
________ QUANTUM HILBERT SPACE =
/ \ L2(U(1)) psik
| | _____
| | | |
| | | |
|\________/| | |
| | | f* | f*
| | | |
| area = 0 | | |
| | v v
\ /
\______/ QUANTUM HILBERT SPACE = 1
L2({x}) = C
The quantum state psik gets mapped to the state 1, as we can easily
check:
(f* psik)(x) = psik(f(x))
= psik (0)
= exp(ik0)
= 1.
Finally, to figure out what the birth of a circle does to quantum
states, we compute the adjoint operator:
QUANTUM
______ TIME EVOLUTION
/ \
/ \ 1
| | _____
| ........ | |
|' `| |
| | |
| area = 0 | |
| | v
\________/
sumk psik
In short, in this model the Big Bang produces a superposition of all
possible energy eigenstates, all with coefficient 1."
Miguel's eyes bugged out. "What?? That's not a normalizable state!"
"Yes, you're right. If we think of it as a wavefunction on U(1), it's a delta function sitting at a = 0. This makes sense: in the classical theory we also get a = 0 in this situation."
"But -- but --" Miguel spluttered, "this means our time evolution operator from C to L2(U(1)) doesn't really map into L2(U(1))! And that means our other time evolution operator, the one for the death of a circle, must also have been bad somehow!" He did a furious little calculation and said, "Hey! It's not calculation and said, "Hey! It's an unbounded operator!"
The Wiz smiled. "Yes, I'm afraid so. I was wondering whether someone would notice that. So you see, working with the U(1) version of electromagnetism eliminates some but not all the infinities in this theory."
"But wait a minute!" continued Miguel. "It's all okay if we consider the death of a circle as a cobordism with positive area! We can chop it into two parts and work it out like this:
QUANTUM
TIME EVOLUTION
________ (imaginary time)
/ \
| | psik
| | _____
| | |
|\________/| |
| | v
| area = t |
| | exp(-tk2/2) psik
|\________/| _____
| | |
| area = 0 | |
| | v
\ /
\______/ exp(-tk2/2)
This a nice bounded operator from L2(U(1)) to C,
thanks to the exponential decay. We can also work out the birth of the
circle the same way:
QUANTUM
TIME EVOLUTION
______ (imaginary time)
/ \
/ \ 1
| | _____
| | |
| area = t | |
| | v
|\________/|
| | sumk psik
| | _____
| ........ | |
|' `| |
| area = 0 | v
| |
\________/ sumk exp(-tk2/2) psik
Now we get a normalizable state."
"Excellent!" said the Wiz. "But note: this does not work in real time, since then we do not get the exponential decay."
"True," said Toby, "but I hope you're not touting this fact as some virtue of Wick rotation! The imaginary time theory is better-behaved than the real time theory here, but so what? It's still not helping us do actual real-time physics!"
"DOWN WITH WICK ROTATION!" came a cry from outside the classroom.
"Hmm," said the Wiz. "Who was that? Anyway, you're right so far, but consider the partition function of the 2-sphere. We can compute it in imaginary time by chopping the sphere into 3 parts:
QUANTUM
TIME EVOLUTION
______ (imaginary time)
/ \
/ \ 1
| ........ | _____
|' `| |
| area = 0 | |
| | v
|\________/|
| | sumk psik
| | _____
| ........ | |
|' `| |
| area = t | v
| |
|\________/| sumk exp(-tk2/2) psik
| | _____
| | |
| area = 0 | |
\ / v
\______/
sumk exp(-tk2/2)
and we get a nice convergent sum when t > 0:
Z(t) = sumk exp(-tk2/2)But in fact, this sum converges whenever the real part of t is positive, and we can analytically continue it to the whole imaginary axis except for the point t = 0. Now since t here stands for imaginary time, this means the partition function is well-defined for the real time theory, too -- by means of analytic continuation!
If we tried to do the calculation directly in real time, we'd get
Z(t) = sumk exp(-itk2/2)for the partition function of the Lorentzian theory. But this diverges! The advantage of Wick rotation is that it picks out a sensible meaning for this divergent sum."
The Acolytes began to protest, but the Wiz shut them off. "That's all I want to say today. We've quantized the 2d vacuum Maxwell equations on arbitrary topologies, and we've seen some of the effects of topology change. Next time we'll see how all this is related to topological quantum field theory. Class dismissed."
The Wiz and the Acolytes left the classroom, but Jay lagged behind a bit, deep in thought. As he walked around a bend, a hooded figure motioned to him from a dark recess in the torchlit hallway.
"Who's that?" asked Jay.
"It's me! Oz!" The figure lifted its hood to reveal a familiar face.
"So it was you out there. What were you up to?"
"Listening in on the class. Say, you know that part about creating baby universes in the lab?"
"Yeah, wasn't that cool?"
"Yeah! And I thought: Jay is an Acolyte of Physics, he has access to the lab; maybe we could try it sometime! We could sneak in some night when the Wiz isn't there...."
"Hmm!" said Jay.
Whispering plans, the two went downstairs....