On the first day of class, the Wizard walked in and began absent-mindedly erasing the blackboard by waving his wand over it as he mentally rehearsed his lecture. Then turned towards the class....
"Hey! Where's Amanda? And what are you doing here?"
Indeed, Amanda was gone - and in her place sat a fellow no one had seen before!
"Amanda has decided to concentrate on topology for her qualifier exam this quarter," said Alissa, the Acolyte Who Had Never Been Mentioned Until Now, Due to Her Wise Policy of Avoiding Unnecessary Attention.
"And I," said the fellow, somewhat nonplussed, "I just thought I'd take this course."
"Without consulting me first?"
"Umm, no, it didn't say anything about prerequisites or instructor's consent in the course catalog..."
"Have you studied general relativity? Quantum theory?"
"Then what made you decide to take a seminar on quantum gravity?"
"Umm, I don't know. I think I'll leave now, if you don't mind..."
"Begone!" bellowed the Wizard, pounding his staff against the floor. Sparks shot in all directions as its iron tip hit the stone. The poor fellow skulked out the door. Frowning, the Wizard strode over and slammed it shut.
"There!" he said. "We wouldn't want anyone unprepared listening in on this course, now, would we? After all, it's the educator's duty to protect students from knowledge for which they aren't yet ready!"
Turning towards him (and smiling invisibly beneath his long gray beard), the Wizard said "No, I'm serious! We couldn't get students to learn physics if they knew they'd have to keep unlearning everything. In classical mechanics they learn the world is a bunch of point particles interacting instantaneously at a distance. In electromagnetism they discover: no, the forces are carried by fields, which move at finite speed! Then in quantum mechanics they learn that the particles aren't really points after all... and in quantum field theory, they learn that fields aren't what they thought they were, either: they're made of particles! Or is the other way round? -- I always forget.
In special relativity, they find to their shock that space and time aren't absolute: everything is Lorentz-invariant. But in general relativity they learn that no, this Lorentz-invariance business is only a crude approximation. And finally, in this quantum gravity seminar you'll see that..."
He paused to open the door slightly, peer out, and look both ways to make sure nobody was eavesdropping. Then he slammed it shut again.
Putting his finger to his lips, raising his bushy eyebrows and whispering loudly, he continued:
"... WE'RE COMPLETELY CONFUSED AND HAVEN'T THE FOGGIEST CLUE!"
He let out a cackle of wild laughter. Some of the class laughed along; others looked depressed.
Feeling a sense of responsibility, Toby interjected: "None of the theories you mentioned are really wrong, they are just approximations good in certain circumstances - so they're as "right" as most anything ever is!"
The Wizard grew sober and replied, "Yes, that's right, I was just kidding. Now where was I...? Hmm. Oh yes:
Last quarter we finished by proving a marvelous identity called the Biedenharn-Elliot identity. Remember how this went. First, we developed a recipe for getting operators from spin networks - and in particular, numbers from closed spin networks. Then we considered this tetrahedral spin network:
/|\ / | \ j1/ | \j3 / | \ / j2| \ / | \ / | j4 \ -------|------- \ | / \ | / \ | / \ | / j5\ | /j6 \ | / \|/We saw that it was related to the 6j symbols, which are the coefficients need to express this operator:
j1\ j2/ /j3 \ / / \ / / \/ / \ / j5\ / \ / | |j4 |as a linear combination of these:
j1\ j2\ /j3 \ \ / \ \ / \ \/ \ / \ /j6 \ / | |j4 |By playing around with these pictures, we saw a wonderful fact: the number we get from 2 tetrahedra stuck together along a triangle is equal to the number by taking this shape and chopping it into 3 tetrahedra instead. Even better, this "2-3 move" is one of the two so-called "Pachner moves" that take us between any two triangulations of a given compact 3-dimensional manifold.
Now, such harmony between algebra and topology cannot be a mere coincidence. There must be an explanation for it. And there must be something good we can *do* with it!
And indeed, to the wise, this clue is a signal that we should use the 6j symbols to define a topological quantum field theory, or "TQFT"."
He looked around the room to see if anyone displayed signs of familiarity with this concept. Alas, nervous silence prevailed.
"No? Well then, let's get wise! It's time to talk about TQFTs.
I'll start at the beginning. Last quarter I mentioned an analogy between general relativity and quantum theory which is (I believe) our main hope for figuring out how to reconcile these two theories. In general relativity the basic concepts are SPACE, which is an (n-1)-dimensional manifold, and also SPACETIME, which is an n-dimensional manifold going from one (n-1)-manifold to another, like this:
___ ____ / \___/ \ | __ \ space S | / \___ | |\_/ \___/| | | | | | ___ | | | / \ | | | | | | spacetime M | | | | | | | ... \ | .. | | |' `.| |' ` | | | | | `| v | | \ | \_____/ \__/ space S'Technically we call M a "cobordism": an n-manifold with boundary whose boundary is the disjoint union of manifolds S and S'. We write M: S -> S' to indicate that M is a spacetime going from the space S to the space S'. I've drawn a picture where n = 2, and just for fun, I have shown a situation where the universe splits in two: S is topologically a circle, but S' is a disjoint union of two circles.
Over in quantum theory, we have analogous concepts of HILBERT SPACE and OPERATOR. We describe states of a system, i.e. the different ways it can be, by unit vectors in a Hilbert space, which we call "state vectors". And we describe processes, i.e. the ways the system can change, by linear operators from one Hilbert space to another. In ordinary quantum mechanics we fix a single Hilbert space and stick with that, but in fancier situations it can change, so we get a picture like this:
Hilbert space H | | operator T | | v Hilbert space H'I hope the analogy is clear! "Space" and "Hilbert space" are used to describe ways the universe can be, while "spacetimes" and "operators" describe how the universe can become.
We can make this more precise using a little category theory. A category C is a gadget consisting of:
1. a collection of "objects".
2. for any pair of objects x and y, a set hom(x,y) of "morphisms" from x to y. If f is a morphism from x to y, we write f: x -> y. We call x the "source" of f, and call y the "target" of f.
3. for any object x, a morphism 1x: x -> x called the "identity of x".
4. for any pair of objects x and y, a function called "composition", taking a morphism f: x -> y and a morphism g: y -> z to a morphism gf: x -> y.
such that these laws hold:
5. the "left and right unit laws": for any morphism f: x -> y we have 1y f = f = f 1x.
6. the "associative law": for any morphisms f: w -> x, g: x -> y and h: y -> z we have (hg)f = h(gf).
Toby interrupted: "Yuck! You're using that backwards way of writing composite morphisms."
The Wizard sighed. "Yes. In a feeble attempt to win over the hidebound traditionalists among you, I am writing the composite of f: x -> y and g: y -> z as "gf" rather than the more logical "fg". Personally I prefer "fg", but it freaks some people out, so in this class I'll use "gf"... until I change my mind.
Now: the notation is suppose to remind you of sets and functions, but that is just an example of a category; in general f: x -> y does NOT mean that f is a function.
What are some examples of categories? There are millions: whenever a mathematician is talking about some kind of gadget, there is at least one category lurking in the background: the category of those gadgets and suitable morphisms between them.
For starters, there's the category "Set", where the objects are sets and the morphisms are functions, with composition and identity functions defined in the usual way. This is the category most mathematicians take as their starting-point. There's also the category "Vect", where the objects are vector spaces and the morphisms are linear maps, again with the usual composition and identity morphisms. This is the world of linear algebra.
In quantum theory we use a category related to Vect, but slightly different: namely "Hilb", where the objects are Hilbert spaces and the morphisms are linear maps. Remember that in this course, we often restrict attention to finite-dimensional complex vector spaces and Hilbert spaces. We do this in our definition of "Vect" and "Hilb", unless I state otherwise.
In general relativity we use "nCob", where the objects are (n-1)-dimensional manifolds and the morphisms are n-dimensional cobordisms between these. This is a nice example where the morphisms aren't functions!
I hope you already see that in general, the objects of a category represent ways things can "be", while the morphisms represent ways they can "become". BEING and BECOMING: we put them on an equal footing in category theory, unlike traditional set theory, which emphasizes being and tries to make everything into sets - even functions!
Now, I won't define a TQFT yet, but I'll tell you this: a TQFT is, among other things, a functor from nCob to Hilb."
"What's a functor?" asked Toby. The Wiz laughed. "You know that perfectly well! What's come over you?"
"Don't you remember? As a fictional character, part of my job is to push the exposition along by asking dumb questions which you were just about to answer anyway."
The Wizard slapped his forehead. "I forgot! I fictionalized you last quarter as punishment for your sins! Hmm, it's getting a bit annoying. Oh well. So -- let me define a "functor". It's just the obvious thing going from one category to another, preserving all the structure in sight, but I should be precise, so....
Given categories C and D, a functor F: C -> D consists of:
1. a map sending any object x in C to an object F(x) in D.
2. for any pair of objects x and y, a map sending morphisms f: x -> y to morphisms F(f): F(x) -> F(y).
such that these laws hold:
In short: F sends objects to objects, morphisms to morphisms, and preserves sources, targets, identities and composition."
3. for any object x in C, F(1x) = 1F(x).
4. for any pair of morphisms f: x -> y and g: y -> z, F(gf) = F(g)F(f).
Toby said "Hey! That reminds me of a group homomorphism!"
The Wizard growled. "Come off it, Toby -- you know perfectly well why! A category with one object, all of whose morphisms are invertible, is nothing but a group! And a functor between categories of this sort is nothing but a homomorphism! You've known this since kindergarten!"
"Yeah," said Toby. "But I just thought it would help the exposition --"
"Give it a break!" said the Wizard. "When I need help, I'll make you ask a question." He collected himself and continued.
"Now, the cool thing is that the categories we use in general relativity and quantum theory, namely nCob and Hilb, have a lot of features in common. In fact, they share a bunch of features which Set lacks!
For example, we can take any cobordism M: S -> S' and decree the input to be the output and vice versa, getting a cobordism that goes the other way, which we call M*: S' -> S. Pictorially this corresponds to reflecting M in the time direction, which is the vertical direction in our pictures. Physicists call this "time reversal".
We can do the same thing in Hilb! This is one place where the inner product comes in handy. Given a linear operator between Hilbert spaces, say T: H -> H', we get one going the other way, called T*: H' -> H. It works as follows: if x is any vector in H and x' is any vector in H', we have
<x', T x> = <T* x', x>
This guy T* is called the "Hilbert space adjoint" of the operator T. Don't mix it up with the "vector space adjoint"! If T: V -> V' is a linear operator between mere vector spaces, we can define a kind of adjoint T*: V'* -> V*, but this brings in the dual vector spaces."
Toby pointed out, "Another way to think of it is that the inner product allows us to identify a Hilbert space with its dual, so the dual is still there, but we just don't see it."
The Wiz glared angrily. "There's nothing worse than a fictional character with initiative! Anyway, you're right, but I like to think of the two kinds of adjoint as quite different things, one corresponding to reflection in the time direction, the other to rotation -- as we've already discussed.
Anyway, what's cool is that nCob and Hilb share this ability to switch the source and target of any morphism f: x -> y and get a morphism f*: y -> x. We don't have this in Set! This suggests that general relativity and quantum theory have things in common that traditional set-based mathematics underemphasizes. For more, try this paper of mine, which was written so that even philosophers could read it. It's called Higher-Dimensional Algebra and Planck-Scale Physics."
In an extravagant use of magic he tossed a sheaf of papers out to the class, each one landing neatly in front of one of the students. "Technically, we say that nCob and Hilb are not just categories, but "*-categories", and the TQFTs most relevant to physics, the "unitary" ones, will be, not just functors, but "*-functors" -- meaning that they preserve this extra structure. The definitions go like this:
A *-category is a category C equipped with
such that these laws hold:
1) a map * sending each morphism f: x -> y to a morphism f*: y -> x.
2) for any pair of morphisms f: x -> y and g: y -> z we have (gf)* = f*g* .
3) for any object x we have (1x)* = 1x.
4) for any morphism f: x -> y we have f** = f.
Given *-categories C and D, a *-functor F: C -> D is a functor such that for every morphism f: x -> y in C we have F(f*) = F(f)*."
A strange light shone in Toby's eyes, and he raised his hand and asked: "Isn't law 3) up there redundant?"
The Wiz smiled and replied, "Yes - I was hoping you'd ask that! It's a cute little exercise to deduce it from the other stuff.
Okay. Since there are some of you still learning quantum theory, I should say a bit about what else the inner product in a Hilbert space is good for. In elementary quantum mechanics, we assign a single Hilbert space H to any physical system. States of this system are then described using unit vectors in H: we call these "state vectors".
The amazing thing about quantum theory is that if you put a system in some state x, and immediately check to see if it's in the state y, sometimes the answer is "yes" - even if x and y are different! It's a random sort of thing: the probability the answer is "yes" is given by
We call this quantity the "transition probability" to go from x to y. The inner product itself
is called the "transition amplitude".
Now you can see why we use unit vectors to describe states. We want the probabilities |<y,x>|2 to lie between 0 and 1; the Cauchy-Schwarz inequality guarantees this. In particular, if you put a system in some state and immediately check to see if it's in the *same* state, you get the answer "yes" with probability 1.
Also, note that the transition probabilities are unchanged if we multiply one of our state vectors by a "phase" - a unit complex number. In fact, multiplying a state vector by a phase doesn't change the state it describes, so states are equivalence classes of unit vectors.
Now, in physics we have "observables" as well as states. Roughly speaking, an observable is something you can measure about the system and get a real number as an answer. In quantum theory we describe observables using self-adjoint operators. Why is that?
Well, at least in the finite-dimensional case, a linear operator A: H -> H is self-adjoint if and only if it has an orthonormal basis of eigenvectors with real eigenvalues: say eigenvectors ei with
A ei = ai ei.
This has the following physical interpretation. The numbers a_i are the possible outcomes of an experiment in which you measure the observable corresponding to A. The state vector ei has the property that when you measure the observable A when the system is in this state, you always get the outcome ai.
So we see why self-adjoint operators are nice. The eigenvalues being real corresponds to the results of our measurement being real. The eigenvectors ei being orthonormal corresponds to... what?"
The Acolyte Richard struggled to come up with an answer, but the Wizard quickly cut him off and answered the question himself. After all that time alone up the castle, the Wiz had become rather impatient and grumpy when it came to letting other people talk!
"Here's the deal. The fact that the vectors ei are normalized means they represent states. The fact that they're orthogonal means that if we put the system in the state ei, and check to see if it's in the state ej, we get the answer "yes" with probability
which is zero unless i = j. So it all hangs together nicely!
Okay, so much for our lightning review of categories and quantum theory. Next time we'll discuss other things nCob and Hilb have in common, and use these to finish defining the concept of "TQFT"."