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October 5, 2006

Quantization and Cohomology (Week 1)

Posted by John Baez

This fall I’m also teaching a seminar on quantization and cohomology. Derek Wise is taking notes, which you can get as PDF files here:

I’ll create a blog entry for each week’s lecture, and anyone with something to say or ask can post comments. So, it’s a bit like an online course. Here are this Tuesday’s notes:

  • Week 1 (Oct. 3) - How the dynamics of p-branes resembles the statics of (p+1)-branes.

Next week’s notes are here. For a syllabus, read on…

In these lectures I hope to talk about:

  • the Lagrangian approach to classical mechanics,
  • the path-integral approach to quantum mechanics,
  • symplectic geometry,
  • geometric quantization,
  • how going from point particles to strings makes us “categorify” all the above.

Here’s a little refresher course on classical mechanics for mathematicians, if you want it:

Posted at October 5, 2006 5:56 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/961

37 Comments & 8 Trackbacks

Re: Quantization and Cohomology (Week 1)

Week 1 (Oct. 3) - How the dynamics of p-branes resembles the statics of (p+1)-branes.

Cool. That touches upon many things that I have been thinking about lately.

Let me propose this general way of looking at the situation:

Let P n(X)P_n(X) be our geometric nn-category of nn-paths in target space XX.

Let d p1d^{p-1} be a pp-particle. This means: let d p1d^{p-1} be the (p1)(p-1)-category which encodes the internal structure of a pp-particle.

For p=1p=1 we usually choose

(1)d 0={} d^0 = \{\bullet\}

to be the one-object 0-category.

For p=2p=2 we usually choose

(2)d 1={0π} d^1 = \{0 \to \pi\}

to be the poset of the oriented interval [0,π][0,\pi] for the “open string”, or, similarly for the “closed string”.

And so on.

The configuration space of the pp-particle with target space P n(X)P_n(X) is the functor nn-category

(3)[d p1,P n(X)]. [d^{p-1},P_n(X)] \,.

Objects in here are images of the pp-particle in target space.

Morphisms in here are worldvolumes traced out by the pp-particle.

By the nature of pseudonatural transformations, this automatically ensures that the endpoints of the pp-particle need not be nailed down. In fact, these worldvolumes are automatically (by the logic of nn-functor categories) cobordisms cobounding the source and target pp-particle image.

There are also higher morphisms in [d p1,P n(X)][d^{p-1},P_n(X)], which encode various gauge invariances (reparameterizations and generalizations thereof) of the worldvolume of a pp-particle.

Next, we want to associate phases to morphisms in [d p1,P n(X)][d^{p-1},P_n(X)].

To get that, let

(4)tra:P n(X)T \mathrm{tra} : P_n(X) \to T

be an nn-bundle with connection on XX, under which our pp-particle shall be charged.

(Take the trivial bundle with trivial connection if in your application the pp-particle is not charged.)

Composing with tra\mathrm{tra} provides us with an nn-functor

(5)tra *:[d p1,P n(X)][d p1,T]. \mathrm{tra}_* : [d^{p-1},P_n(X)] \to [d^{p-1},T] \,.

This functor reads in a pp-particle in target space and spits out a “fiber” over it.

It reads in a worldvolume of a pp-particle and spits out the “phase” associated to this due to the charge of the pp-particle.

Now let triv\mathrm{triv} be a transport functor which factors through the category {}\{\bullet\} with a single object and no nontrivial morphisms.

(6)triv:P n(X){}T. \mathrm{triv} : P_n(X) \to \{\bullet\} \to T \,.

The space of states of our pp-particle is the nn-category

(7)[triv *,tra *]. [\mathrm{triv}_*, \mathrm{tra}_*] \,.

(Recall that tra *\mathrm{tra}_* was the functor from configuration space [d p1,P n(X)][d^{p-1},P_n(X)] to phases [d p1,T][d^{p-1},T].)

An object in

(8)[triv *,tra *] [\mathrm{triv}_*, \mathrm{tra}_*]

is a section of the bundle of “fibers” over the configuration space of the pp-particle.

In order to be able to study the quantum mechanics of our pp-particle charged under tra\mathrm{tra}, we need to assume that the category of generalized phases

(9)[d p1,T] [d^{p-1},T]

is a category with duals.

If that is the case, we can form

(10)(tra,tra):[d p1,P n(X)]tra * ×tra *[d p1,T] op×[d p1,T]HomT˜. (\mathrm{tra},\mathrm{tra}) : [d^{p-1},P_n(X)] \stackrel{ \mathrm{tra}_*^\dagger \times \mathrm{tra}_* }{\to} [d^{p-1},T]^\mathrm{op} \times [d^{p-1},T] \stackrel{\mathrm{Hom}}{\to} \tilde T \,.

(Here T˜\tilde T is whatever the Hom\mathrm{Hom} takes values in, depending on what [d n1,T][d^{n-1},T] is enriched over.)

Similarly for triv\mathrm{triv}.

The point of this is that given any two sections

(11)e 1 :triv *tra * e 2 :triv *tra * \begin{aligned} e_1 & : \mathrm{triv}_* \to \mathrm{tra}_* \\ e_2 & : \mathrm{triv}_* \to \mathrm{tra}_* \end{aligned}

we get a morphism

(12)(triv *,triv *)(e 1,e 2)(tra *,tra *). (\mathrm{triv}_*,\mathrm{triv}_*) \stackrel{(e_1,e_2)}{\to} (\mathrm{tra}_*,\mathrm{tra}_*) \,.

On objects, this encodes the scalar product on the space of sections.

On morphisms, this encodes a scalar product on covariant derivatives

(13)(d trae 1,d trae 2). (d_\mathrm{tra} e_1, d_\mathrm{tra} e_2) \,.

Notice that this is the e 1e_1, e 2e_2 matrix element of the Hamiltonian

(14)Δ tra=d tra * d tra * \Delta_\mathrm{tra} = d_{\mathrm{tra}_*}^\dagger d_{\mathrm{tra}_*}

of the charged pp-particle.


I have more details of this discussion scattered on notes flying around on a couple of tables of the nn-Café, for instance here and here.

I think I have checked that for ordinary charged 1-particles, the above prescription indeed reproduces the ordinary quantization of the particle.

I am in the process of working out what the above says for strings charged under an abelian gerbe. I think everything works as expected, but this is work in progress.

One further aspect one should be able to discuss along these lines is interaction of pp-particles for p>1p \gt 1, by passing from d p1d^{p-1} to suitable interaction diagrams. For instance for the triangle one gets the multiplicative structure on the space of sections over the configuration space of the open string, as described above.

Posted by: urs on October 6, 2006 11:29 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

let tra:P n(X)T \mathrm{tra} : P_n(X) \to T be an nn-bundle…

So TT is an nn-category?

Now let triv\mathrm{triv} be a transport functor which factors through the category {}\{\bullet\} with a single object and no nontrivial morphisms. triv:P n(X){}T. \mathrm{triv} : P_n(X) \to \{\bullet\} \to T \,.

Isn’t {}\{\bullet\} the trivial nn-category?

Posted by: David Corfield on October 6, 2006 12:32 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

So TT is an nn-category?

In principle it could be anything - if we agree on implicitly thinking throughout in terms of ω\omega-categories with only identity pp -morphisms for p>mp \gt m, for some mm.

The main point of the formulation I proposed is to get a completely diagrammatic way to conceive the quantization of a charged pp-particle. So it’s an intended feature - not a bug - that the concrete implementation is left undefined.

But in concrete applications that I have come across, it will usually be an (n+1)(n+1)-category (possibly obtained by throwing in unique (n+1)(n+1)-morphisms in between any ordered pair of parallel nn-morphisms of a given nn-category, as discussed here).

Isn’t {•} the trivial n-category?

Yes, exactly. The point is that triv\mathrm{triv} is comletely trivial. It assigns the identity morphism to everything.

This is supposed to ensure that morphisms triv *tra *\mathrm{triv}_* \to \mathrm{tra}_* can indeed be addressed as appropriate sections. This is illustrated by the diagram shown here.

Possibly there is a better way to encode the necessary requirement on triv *\mathrm{triv}_*. Suggestions are welcome.

Posted by: urs on October 6, 2006 12:53 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

So TT is an nn-category?

In principle it could be anything - if we agree on implicitly thinking throughout in terms of ω\omega-categories with only identity pp -morphisms for p>mp \gt m, for some mm.

Might anything be gained by considering some nn-category of suitable TT? What might be learned from the existence of a morphism between TT and TT' in terms of the respective dynamics? I’m fishing for something along the lines of the morphism of rigs which corresponds to mapping from the cost of different routes from A to B to the existence of a path from A to B.

Do people use homotopy theory to explore possible trajectories in phase space? Richer rigs than truth values would presumably provide more interesting comparisons.

Posted by: David Corfield on October 7, 2006 9:11 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Might anything be gained by considering some nn-category of suitable TT?

Assuming a satisfactory arrow-theoretic picture like the one I am after here has been obtained, one should certainly want to understand which choice of TT provides which kind of concrete implementation.

For (n=1)(n=1) ordinary quantum mechanics is obtained for something like T=ModT = \mathbb{C}-\mathrm{Mod}.

For (n=2)(n=2) it seems that the standard physics applications (“quantum strings”) are obtained for something like T=Bim(C)T = \mathrm{Bim}(C) (bimodules internal to a monoidal category CC), which may be regarded as sitting inside CModC-\mathrm{Mod}.

Both of these vector cases can be regarded as associated to corresponding principal cases.

Clearly there seems to be a pattern here. But there should be much more structure than has been identified so far.

What might be learned from the existence of a morphism between TT and TT' in terms of the respective dynamics?

There might be several good answers to that. Right now I can only point out that the behaviour of the above setup under pullback along morphisms

(1)i:TT i : T' \to T

is at the very heart of the general framework of local trivialization of transport which I consider here.

I’m fishing for something along the lines of the morphism of rigs which corresponds to mapping from the cost of different routes from AA to BB to the existence of a path from AA to BB.

Yes, I recall we talked about this before, in the context of Lagrangian quantization. What I described above is a slightly different approach, which essentially amounts to Hamiltonian quantization.

As a consequence, the “generalized phases” I mentioned above contain only the contribution from what would in the ordinary case be the charge of the pp-particle - but not the contribution from the kinetic term. (Because instead of describing the full action functional, the above rests on describing the covariant derivative from which the Hamiltonian is built.)

Still, it should be intersting to see what happens in the above setup when tra(γ)\mathrm{tra}(\gamma) is, for instance, no longer an invertible morphism.

I haven’t thought about this before.

Let’s see. The covariant derivative of a section ee along a (infinitesimal) path xγyx \stackrel{\gamma}{\to} y is

(2)e ytra(γ)(e x). e_y - \mathrm{tra}(\gamma)(e_x) \,.

For instance, if e xe_x and e ye_y can be assumed to live in the same space and tra(γ)\mathrm{tra}(\gamma) is the identity, then this leads to the ordinary derivative.

Now, if we allow tra(γ)\mathrm{tra}(\gamma) to be the zero-map, the above expression reduces to just e ye_y.

This can be interpreted as the limit of a covariant derivative of the form

(3)e y(1+A)e x, e_y - (1 + A) e_x \,,

where AA is allowed to be a number that tends to 1-1 (instead of one being imaginary, as for an ordinary U(1)U(1)-covariant derivative).

I am not sure if anything like that can be used to model what you have in mind. But maybe it can.

Posted by: urs on October 8, 2006 6:32 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

What I described above is a slightly different approach, which essentially amounts to Hamiltonian quantization.

Is there some higher quantized reduction?

Posted by: David Corfield on October 9, 2006 12:49 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Is there some higher quantized reduction?

Interesting. Lots of intriguing questions arise once one starts seriously thinking about higher quantization. :-) There should be no need to worry about a lack of thesis topics #.

Certainly I don’t know the answer. On the other hand, Landsman’s very point in the paper you linked to is to emphasize that (Marsden-Weinstein) symplectic reduction is a special case of some general nonsense involving just tensor products of bimodules.

Hence as far as our higher quantum theory still involves gadgets that look like Hilbert spaces acted on by algebras (and I think it does), it seems quite likely that after sorting out a bunch of technicalities one does indeed obtain a higher version of quantized symplectic redution.

But that’s just my guess.

Posted by: urs on October 9, 2006 1:32 PM | Permalink | Reply to this

Technical notes.

John wrote in part:

Here’s a little refresher course on classical mechanics for mathematicians, if you want it:

You’ll need to fix the permissions so people can read the errata.

Also, I notice that you have some homeworks from the 2002 course, including 2 solutions written by me. I have all 5 solutions, if you want them.

Posted by: Toby Bartels on October 7, 2006 7:24 PM | Permalink | Reply to this

Re: Technical notes.

Toby wrote:

You’ll need to fix the permissions so people can read the errata.

Done. Thanks! There’s just one erratum so far… if more people read these classical mechanics notes, I’m sure more mistakes will turn up.

Also, I notice that you have some homeworks from the 2002 course, including 2 solutions written by me. I have all 5 solutions, if you want them.

Yes, definitely! Please email them to me in TeX - and also PDF if you want to save me a little work.

What I really need is for Derek to scan in the rest of the notes, like the stuff on symplectic geometry. He said he would. This symplectic stuff will become important soon in our quantization and cohomology course, when I talk about geometric quantization and the cohomological classification of U(1)\mathrm{U}(1) bundles.

I guess you learned that stuff before you came to UCR! Then you came here and started categorifying it…

Posted by: John Baez on October 7, 2006 10:19 PM | Permalink | Reply to this

2-Geometric Quantization

This symplectic stuff will become important soon in our quantization and cohomology course, when I talk about geometric quantization and the cohomological classification of U(1)U(1) bundles.

For those who were not around then: we had some interesting discussion related to the categorification of geometric quantization in Gerbes, Strings and Nambu Brackets as well as in Classical Canonical, Stringy.

It would be really interesting if one could find a general categorical framework in which Rovelli’s geometric string quantization using a 3-form can be understood as a categorification of the ordinary geometric quantization using a 2-form.

Posted by: urs on October 8, 2006 6:46 PM | Permalink | Reply to this

Re: 2-Geometric Quantization

Maybe it pays to look at the geometric quantization of the ordinary particle coupled to the electromagnetic field, in order to make contact with the kind of approach I was promoting above - where we obtain our Hilbert space as that of square integrable sections of the bundle the particle is charged under.

For the particle on XX, charged under the line bundle EXE \to X with connection \nabla we may use the symplectic form

(1)ω=dqdp \omega = d q\wedge d p

on phase space and the Hamiltonian

(2)H=(pA) 2, H = (p-A)^2 \,,

but, unless I am hallucinating, we can also use

(3)ω=dqdp+F \omega = d q \wedge d p + F

and

(4)H=p 2, H = p^2 \,,

which amounts to passing from canonical to kinetic momentum.

Now, since

(5)ω=F+dsomething \omega = F + d \text{something}

the line bundle classified by ω\omega on phase space is the pullback of EE to phase space.

We should be able to choose a polarization of phase space such that our Hilbert space spit out by the geometric quantization procedure is the space of square integrable sections of EE.

Right?

Posted by: urs on October 8, 2006 9:23 PM | Permalink | Reply to this

Re: 2-Geometric Quantization

Urs wrote:

For the particle on XX, charged under the line bundle EXE \to X with connection \nabla we may use the symplectic form

(1)ω=dqdp \omega = d q\wedge d p

on phase space and the Hamiltonian

(2)H=(pA) 2, H = (p-A)^2 \,,

but, unless I am hallucinating, we can also use but, unless I am hallucinating, we can also use

(3)ω=dqdp+F \omega = d q \wedge d p + F

and

(4)H=p 2, H = p^2 \,,

which amounts to passing from canonical to kinetic momentum.

Yes, something sort of like this is right - Guillemin and Sternberg talk about this in their book Symplectic Techniques in Physics, but in the context of the “extended phase space”, where XX is spacetime.

I can’t tell if you’re intending XX to be space or spacetime here. If the former, I guess you’re studying a particle in a static field. If the latter, I guess the Hilbert space of square-integrable sections of EXE \to X is at best a “kinematical Hilbert space”, bigger than the physical Hilbert space: we need to impose a constraint to get the physical Hilbert space.

I’m going to talk about this later in the course, so I have to learn more about it!

And, of course I need to talk about the “extended phase space” formalism as a warmup for discussing Rovelli’s ideas.

Posted by: John Baez on October 9, 2006 1:10 AM | Permalink | Reply to this

Re: 2-Geometric Quantization

Yes, something sort of like this is right

In fact, for a nontrivial bundle, the second prescription

(1)ω =dqdp+F H =p 2 \begin{aligned} \omega &= dq \wedge dp + F \\ H &= p^2 \end{aligned}

is the only one that works, since (pA) 2(p-A)^2 isn’t even defineable for a nontrivial bundle.

Also, I should add that the polarization one wants to take here is simply that coming from embedding the original tangent space of XX into that of phase space.

Guillemin and Sternberg talk about this in their book Symplectic Techniques in Physics,

Thanks! I should take a look at that.

I can’t tell if you’re intending X to be space or spacetime here.

I can’t either. :-) I was in a real hurry when posting this, and only cared about getting the central symbols across.

But as you say, it’s the same general mechanism in both cases, modulo some differences in the physical interpretation.

I’m going to talk about this later in the course, so I have to learn more about it!

What made me get into the above observation is that it might shed a certain light on geometric quantization, which helps get the categorification right:

To some degree, we can turn the above argument around and argue that any geometric quantization is the quantization of an (abelian) charged point. (That point might be the abstract point in configuration space of a complicated system consisting of many partcicles - a point in “superspace” - but still.)

Maybe I am exaggerating here. But in as far as we are interested in quantizing and categorifying the charged point, it seems to me that the central message of geometric quantization useful for categorification is:

  • the quantum Hilbert space is that of (square integrable) sections of the bundle that the point is charged under
  • the Hamiltonian is the covariant (with respect to the connection on the bundle) Laplace operator acting on these sections

These two statement lend themselves to categorification very nicely, as I tried to indicate above #.

This also allows to generalize a lot: for instance from abelian to nonabelian charges, away from smooth target spaces, etc.

Posted by: urs on October 9, 2006 10:09 AM | Permalink | Reply to this

Re: 2-Geometric Quantization

Urs wrote:

It would be really interesting if one could find a general categorical framework in which Rovelli’s geometric string quantization using a 3-form can be understood as a categorification of the ordinary geometric quantization using a 2-form.

That’s where this course is heading - at least ideally. I don’t know how to categorify the whole framework of geometric quantization, but there’s lots of stuff I need to explain, starting from scratch, to show the students why that’s a sensible thing to want to do. First I need to explain symplectic mechanics, line bundles and geometric quantization; then I need to explain Rovelli’s 3-form mechanics and gerbes… and maybe by then I’ll understand something about categorified geometric quantization!

I figure that even if I don’t succeed, I will bump into some nice thesis topics for my students.

Posted by: John Baez on October 8, 2006 10:21 PM | Permalink | Reply to this

Re: Technical notes.

I wrote long ago:

You’ll need to fix the permissions so people can read the errata.

And now you’ll need to fix the permissions so people can download the TeX sources.

Posted by: Toby Bartels on August 27, 2008 5:44 AM | Permalink | Reply to this

Re: Technical notes.

Hi! I think I’ve got them all fixed now. Thanks.

nn-Café customers worldwide will be pleased to know that Toby, famous for his starring role as the Acolyte of Mathematics, has returned to Southern California — and may attend my seminar at UCR this fall, and talk with Jim and me about math!

Posted by: John Baez on August 28, 2008 8:00 AM | Permalink | Reply to this

Re: Technical notes.

John wrote in part:

I think I’ve got them all fixed now.

Yes, they are.

Posted by: Toby Bartels on August 30, 2008 12:36 AM | Permalink | Reply to this
Read the post Philosophy of Physics
Weblog: The n-Category Café
Excerpt: Publication of 'Philosophy of Physics'
Tracked: October 9, 2006 9:59 AM

Re: Quantization and Cohomology (Week 1)

Urs wrote:

Maybe I am exaggerating here. But in as far as we are interested in quantizing and categorifying the charged point, it seems to me that the central message of geometric quantization useful for categorification is:

  • the quantum Hilbert space is that of (square integrable) sections of the bundle that the point is charged under;
  • the Hamiltonian is the covariant (with respect to the connection on the bundle) Laplace operator acting on these sections.

These two statement lend themselves to categorification very nicely, as I tried to indicate above #.

I could be wrong, but it seems you’re not talking about full-fledged geometric quantization - just a special case that’s a bit more general than Schrödinger quantization.

For readers not up on their quantization techniques, I should say a bit more. Take a little nap, Urs…

In Schrödinger quantization we take a “configuration space” - a space of possible positions for our particle, say a Riemannian manifold QQ - and build a Hilbert space L 2(Q)L^2(Q) which serves to describe states of the quantum version of the particle.

In geometric quantization we take a “phase space” - a space of possible positions and momenta for our particle, say a symplectic manifold XX - and build a Hilbert space which serves to describe states of the quantum version of the particle.

What is this Hilbert space? It’s not so easy to describe. Our first guess, L 2(X)L^2(X), is wrong in several ways.

The tricky thing to realize is that when the symplectic structure on XX is “integral”, it determines a line bundle

EXE \to X

with a connection on it (at least up to isomorphism). Instead of thinking about L 2(X)L^2(X), we need to think about L 2(X,E)L^2(X,E) - the Hilbert space of square-integrable sections of this line bundle. I guess the easiest way to see this is to think very hard about the old quantum mechanics - Bohr and Sommerfeld’s approach to quantization before Schrödinger came along.

But, L 2(X,E)L^2(X,E) is still not right; it doesn’t match the Schrödinger prescription in the case when both strategies apply, namely when

X=T *Q.X = T^* Q.

So, we call L 2(X,E)L^2(X,E) the prequantum Hilbert space. It’s too big: we need to chop it down to something smaller to get the right answer. To do this, we need to choose an extra structure on XX, called a “polarization”, which picks out certain allowed sections of EE, giving a subspace of L 2(X,E)L^2(X,E), the quantum Hilbert space H(X,E)H(X,E).

This matches Schrödinger’s prescription when X=T *QX = T^* Q: we have a nice isomorphism

H(T *Q,E)L 2(Q).H(T^*Q, E) \cong L^2(Q).

Anyway, that’s a short version of a long story - for more, try this.

Okay, Urs, you can wake up!

In your discussion of the point particle, you seem to be going a bit beyond Schrödinger quantization, but not all the way to general geometric quantization, since your Hilbert space is L 2(Q,E)L^2(Q,E) - you’re still using configuration space, not phase space - but you’ve got a nontrivial line bundle over it.

This seems like something I could categorify. And, I guess you already have, to some extent. Do you follow Dan Freed’s strategy, and try to build a 2-Hilbert space of sections of a 2-line bundle associated to a gerbe?

(I’m sorry, I should read your notes, but I’m a lazy guy; I like to just sit around and chat.)

This sort of thing is already fascinating, but I can’t help wanting to categorify full-fledged geometric quantization, something like this:

We start with a phase space XX with a closed integral 3-form ω\omega on it. This determines a gerbe EXE \to X connection on it, at least up to equivalence.

Following Freed, EE this should have some 2-Hilbert space of sections L 2(X,E)L^2(X,E). I don’t actually know how to do this rigorously, and I don’t think Freed did either, but I’ve learned a lot about infinite-dimensional 2-Hilbert spaces recently, in my work with Laurent Freidel, so I feel optimistic.

But then, I want some concept of “2-polarization” - some geometrical structure I can put on XX, which determines a sub-2-space of L 2(X,E)L^2(X,E), the “quantum 2-Hilbert space”.

To figure out this concept, we need to look at examples, starting with the Schrödinger quantization of the string - as before, geometric quantization should be backwards-compatible with Schrödinger quantization!

But, one wants to go quite a bit further… I guess the trick is finding more examples: that’s how they invented geometric quantization in the first place.

PS - the above link about the old quantum mechanics doesn’t say anything about the Bohr-Sommerfeld quantization condition and its relation to the crucial “integrality” condition on symplectic structures - it’s all implicit in the charmingly quaint pictures of electron orbitals. But, it leads to a nice story involving Sommerfeld and a young student named Heisenberg, who was defending his dissertation on turbulent fluid flow:

Trouble began when acceptance of the dissertation brought admission of the candidate to the final orals. The examining committee consisted of Sommerfeld and Wien in physics, along with representatives in Heisenberg’s two minor subjects, mathematics and astronomy. Much was at stake, for the only grades a candidate received for his studies were those based on the dissertation and final oral: one grade for each subject and one for overall performance. The grades ranged from I (equivalent to an A) to V (an F).

As the 21-year-old Heisenberg appeared before the four professors on July 23, 1923, he easily handled Sommerfeld’s questions and those in mathematics, but he began to stumble on astronomy and fell flat on his face on experimental physics. In his laboratory work Heisenberg had to use a Fabry-Perot interferometer, a device for observing the interference of light waves, which the class had studied extensively. But Heisenberg had no idea how to derive the resolving power of the interferometer nor, to Wien’s surprise, could he derive the resolution (ability to distinguish objects) of such common instruments as the telescope or the microscope. When an angry Wien asked him how a storage battery works, the candidate was still lost. Wien saw no reason to pass the young man, no matter how brilliant he might be in other branches of physics. An argument arose between Sommerfeld and Wien over the relative importance of theoretical physics in relation to experimental physics. The result was that Heisenberg received a III, equivalent to a C, in physics and for the overall grade for his doctorate. Both of these grades were probably averages between Sommerfeld’s grade (an A) and Wien’s grade (an F).

Sommerfeld was shocked. Heisenberg was mortified. Accustomed to being always at the top of his class, Heisenberg found it hard to accept a mediocre grade for his doctorate. Sommerfeld held a small party at his home later that evening for the new Dr. Heisenberg, but Heisenberg excused himself early, packed his bag, and took the midnight train to Göttingen, showing up in Max Born’s office the next morning. Born had already hired Heisenberg as his teaching assistant for the coming school year. After informing Born of the debacle of his orals, Heisenberg asked sheepishly, “I wonder if you still want to have me.”

Born did not answer until he had gone over the questions Heisenberg had missed. Convincing himself that the questions were “rather tricky,” Born let his employment offer stand. But that fall Heisenberg’s worried father wrote to the famed Göttingen experimentalist James Franck, asking Franck to teach his boy some experimental physics. Franck did his best, but he could not overcome Heisenberg’s complete lack of interest and gave up the effort. If Heisenberg was going to survive at all in physics it would be only as a theorist.

Posted by: John Baez on October 9, 2006 2:04 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

it seems you’re not talking about full-fledged geometric quantization

Let’s see. I first tried to talk about full-fledged geometric quantization - but applied to the special case of a single charged particle on SS (space(time)), coupled to the line bundle (ES,)(E \to S,\nabla).

I observed that the line bundle on phase space that we get from the geometric quantization procedure is - in this case - nothing but the pullback of the bundle on space(time).

Moreover, we may pick the polarization on T *ST^* S induced from the tangent space of SS, thereby restricting our sections on phase space to be constant in the momentum direction - so that they are just sections of EE on SS.

This way the line bundle and the space of sections on phase space T *ST^* S descend to SS, and the result of our geometric quantization is that the Hilbert space of the charged particle is the space of square integrable sections of ESE \to S.

I think. Is that not right? I should look at the book by Guillemin and Sternberg which you mentioned.

If it’s right, we have found that for the special case of the charged particle space(time) SS is the “space of leaves” (or whatever it is called) of the polarization chosen on phase space X=T *SX = T^* S, and that the bundle on XX descends to SS.

Next I wanted to argue (but that’s the part where I am essentially guessing) that we can regard the geometric quantization of an arbitrary symplectic space XX as the quantization of an abstract charged point on the space of leaves of the chosen polarization.

Probably this does not work in general.

But the reason why I am interested in such a point of view is that I would like to replace the description of bundles with connection in terms of their classifying 2-form by something more arrow-theoretic. Otherwise I don’t know how to categorify.

Do you follow Dan Freed’s strategy, and try to build a 2-Hilbert space of sections of a 2-line bundle associated to a gerbe?

As you know, I first tried to find an arrow-theoretic way to derive what Freed is talking about #, by unifying the space of sections with the path integral of the theory into a single sort of coproduct.

I think in principle this works, but I wasn’t really satisfied with the result. It’s not all that elegant.

So after thinking quite a while about this, I came up with the framework I sketched in my first comment here. This is quite elegant:

The space of sections of an nn-vector bundle with connection tra\mathrm{tra} is the functor nn-category

(1)[triv *,tra *]. [\mathrm{triv}_*,\mathrm{tra}_*] \,.

I think there is a canonical way to put a scalar product on the space of objects of this category and one can see in simple special cases that the result is the expected Hilbert space.

For the 2-particle, this Hilbert space of objects of the category of sections is the 1-Hilbert space (with the structure of an algebra on it) on the bundle over path space that the gerbe gives rise to.

(So it’s the Hilbert space you would want to associate to the boundaries of cobordisms in the quantum theory.)

Posted by: urs on October 9, 2006 2:47 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Urs wrote:

John wrote:

it seems you’re not talking about full-fledged geometric quantization

Let’s see. I first tried to talk about full-fledged geometric quantization - but applied to the special case of a single charged particle on SS (space(time)), coupled to the line bundle (ES,)(E \to S,\nabla).

Right; maybe I didn’t express myself clearly. My point was this:

You chose a polarization such that the quantum Hilbert space was the Hilbert space of all square-integrable sections of a line bundle. This special case of geometric quantization seems like the easiest one to categorify, since I can imagine understanding the 2-Hilbert space of all square-integrable sections of a 2-line bundle. And, it sounds like you’ve made a lot of progress on this!

This may be all we really need for physics.

But, the case of geometric quantization that’s mathematically most fruitful involves a Kähler polarization, giving a quantum Hilbert space that’s the Hilbert space of all holomorphic sections of a line bundle. This seems much harder to categorify - but tremendously exciting if we could do it.

In short, you’re doing something sensible, while I’m dreaming a crazy dream.

Posted by: John Baez on October 9, 2006 3:01 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Okay, I get it. We want to understand holomorphic 2-bundles.

But before I abandon it, it would be great if I could draw a line underneath the intuition I was talking about:

Phrased as a question:

Given a symplectic space (X,ω)(X,\omega) with ω\omega nondegenerate and integral, as well as a polarization on (X,ω)(X,\omega); and given the corresponding Hilbert space HH of sections, compatible with the polarization, of the line bundle EXE \to X classified by ω\omega - under which conditions is there a space SS such that XT *S X \simeq T^* S and such that HH and EE descend to SS?

Is anything known about this?

Posted by: urs on October 9, 2006 3:39 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Urs wrote:

Is anything known about this?

I’m sure there is, but if I really wanted to know, I’d ask the big boys: Victor Guillemin and Alan Weinstein.

I can ponder the question in an amateur way, though:

For starters you need XX to be equipped with a Lagrangian polarization. A polarization is a purely local structure, and being Lagrangian is a purely local condition on this. But I guess the Frobenius theorem implies any Lagrangian polarization gives a foliation of XX by Lagrangian submanifolds.

This lets you define the space SS of leaves of this foliation. In nasty cases SS could be non-Hausdorff; in nice cases it’s a manifold, and you want the nice cases. I could be completely wrong, but this may be sufficient to get some sort of symplectic map T *SXT^* S \to X.

In any event, there are still plenty of cases where we have such a map but it’s not a diffeomorphism. For example, XX could be a torus, SS a circle, and T *ST^* S a cylinder. Then there’s a covering T *SXT^* S \to X.

You want conditions that force T *SXT^* S \cong X. I bet the big boys know some theorems along these lines.

But, in most of the really interesting examples of geometric quantization - interesting to mathematicians, anyway - XX is compact and thus not diffeomorphic to a cotangent bundle. That’s because all the really beautiful theorems of geometric quantization apply to the case where XX is compact and equipped with a Kähler polarization. We want compactness to be sure that every holomorphic section of the line bundle over XX is square-integrable. Otherwise that usually fails!

In Bargmann-Segal quantization, where X= nX = \mathbb{C}^n, there are no holomorphic square-integrable sections - not if we use the obvious measure. But, we can pull a clever trick by using a different measure: not the Liouville measure, but the Liouville measure times a Gaussian! Then we get a nice Hilbert space of square-integrable sections.

People have tried to generalize the Bargmann-Segal trick to other noncompact phase spaces; so far the best results seem to show up when XX is very symmetrical, like T *GT^* G for a compact Lie group GG. That’s the case Brian Hall is famous for tackling.

Hmm - it looks like he’s gone further now!

Anyway, I’m rambling now.

Posted by: John Baez on October 9, 2006 4:51 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

I wrote:

The space of sections of an nn-vector bundle with connection tra\mathrm{tra} is the functor nn-category [triv *,tra *][\mathrm{triv}_*,\mathrm{tra}_*].

Since I think this is closely related to some points John makes in his notes, I should maybe unwrap what this says for the 2-particle.

(For the 1-particle I have described this here.)

Take the 2-particle to be something like an open string, modeled by the category

(1)d 1={0π} d^1 = \{0 \to \pi\}

whose objects are the points of the interval [0,π][0,\pi] with one morphism per every pair (a,b)(a,b) with 0a<bπ0 \leq a \lt b \leq \pi.

Let target space be XX, or rather P 2(X)P_2(X), the category of 2-paths in XX.

The configuration space is the path space of XX, or rather the functor category

(2)[d 1,P 2(X)]. [d^1 , P_2(X)] \,.

Objects here are paths in XX, morphisms are (open) cobordisms between these paths.

Let this 2-particle be coupled to a gerbe with connection. This is encoded in a 2-functor #

(3)tra:P 2(X)Bim(Vect ), \mathrm{tra} : P_2(X) \to \mathrm{Bim}(\mathrm{Vect}_\mathbb{C}) \,,

with values in bimodules of the algebra of compact operators (or, in fact, a 3-category extending this).

In order not to get distracted by details not relevant for the main point here, let’s assume the gerbe is trivializable and tra\mathrm{tra} takes only values in \mathbb{C}-bimodules, i.e. in Σ(Vect )\Sigma(\mathrm{Vect}_\mathbb{C}).

So tra\mathrm{tra} assigns a (1-dimensional) vector space to each 1-morphism in P 2(X)P_2(X), and a linear map between these to each 2-morphism. It’s nothing but a line bundle with connection on path space!

Let triv:P 2(X)Σ(Vect)\mathrm{triv} : P_2(X) \to \Sigma(\mathrm{Vect}) be the trivial such 2-functor which sends every path to \mathbb{C} itself and every surface to the identity map.

I claimed that the space of sections, which is to become the Hilbert space for our 2-particle, is the functor category

(4)triv *tra *, \mathrm{triv}_* \to \mathrm{tra}_* \,,

where

(5)tra *:[d 1,P 2(X)][d 1,Σ(Vect)]. \mathrm{tra}_* : [d^1,P_2(X)] \to [d^1,\Sigma(\mathrm{Vect})] \,.

To see that this makes good sense, notice that an object in this category is nothing but an assignment of an element in V γV_\gamma for every vector space V γV_\gamma living over a path γ\gamma (i.e. V γ=tra(γ)V_\gamma = \mathrm{tra}(\gamma)). So it’s just an ordinary section of the line bundle on path space.

The general situation is more involved, with gerbe modules (D-branes) playing a role and a couple of other things a didn’t mention here.

The main point I wanted to illustrate here is that and how the concept [triv *,tra *] [\mathrm{triv}_*,\mathrm{tra}_*] correctly captures the notion of sections of a 2-vector bundle that we want.

Posted by: urs on October 10, 2006 9:33 AM | Permalink | Reply to this

nor […] could he derive the resolution […] of […] the microscope.

This is a fun example for the value of making mistakes (preferably in front of an audience) - it hurts, but you learn something:

I assume Heisenberg had a real close look at the textbook description of the microscope some time after his examination - only to realize what nobody else had fully appreciated before: the uncertainty principle

(Heisenberg’s microscope).

Posted by: urs on October 9, 2006 2:57 PM | Permalink | Reply to this

Heisenberg’s Microscope

My god, Urs, that’s fascinating! I wonder if any historians of science have thought about this? I can really imagine Heisenberg brooding over the humiliation of his doctoral exam, where he failed to answer Wien’s question about the resolving power of a telescope… and eventually realizing that this question is related to the uncertainty principle!

I wonder if there’s any evidence for this train of thought.

The perennial conflict between experimentalists and theorists is very much a part of this story. Maybe Heisenberg was lousy at experiments… but he proved himself good at thought experiments.

Posted by: John Baez on October 9, 2006 3:19 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Great Week 1 notes on quantization and cohomology. I had never fully appreciated before the great unity of the subject: that the dynamics of one dimension is the statics of the next, and so on, and that you can think of the action as a one form on “path space” just like the force is a one form on configuration space.

This latter idea ties in perfectly with another great set of notes I’ve recently seen:

Of course, his talks on A mathematician looks at quantum field theory are just as inspiring, but slightly off the topic for now. But in lecture 1 he does make some fascinating comments on qft from a calculus of variations point of view - relating this to the “renormalizability” of the theory.

Posted by: Bruce Bartlett on October 10, 2006 12:06 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

you can think of the action as a one form on “path space”

Hi Bruce, did you mean to say this?

I’d think the action (of the 1-particle) should be 0-form on path space.

I know you all know this, but I like to say it anyway:

The action (of the 1-particle) is in fact not just a 0-form on path space, but (when exponentiated) even a functor from Moore paths # to U(1)U(1).

Part of this 0-form on path space comes from the pullback of a 1-form AA on configuration space. That’s the contribution from the charge of the particle.

But the other part does not come from integrating a 1-form along the path. That’s the kinetic part of the action.

However, when we pass from configuration space to phase space, then the entire action comes from integrating a 1-form (pdq+Hdt\sim p d q + H d t)

Posted by: urs on October 10, 2006 9:08 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Woops, yes you’re right - I should have said that the action is a 0-form!

What was going through my mind at the time was that Segal gives a slightly different angle on this in his notes. I refer here to the section “2. Variational Problems” of the notes I mentioned above.

I’d actually appreciate some help comparing the two approaches.

In John’s lecture, one speaks about the space of *arbitrary* paths with fixed boundary points into X, and considers the action as a 0-form on this space. The differential of the action is then a closed 1-form; setting it equal to zero gives the Euler-Lagrange equations.

In Segal’s notes, one speaks about the *solution manifold* X to the Euler Lagrange equations, and you consider variations with *no* boundary conditions. This allows one to set up a genuine (non-exact) family of one forms on X, whose differential (it doesn’t matter which one you choose) is then a closed 2-form - which turns X into a symplectic manifold.

At the end of the day, if one wishes, one can make the “Legendre transform” which identifies X symplectically with T*X, by sending
phi -> (phi(0), p(0))
where p is the dual space partner of the tangent vector d/dt phi (0), obtained by using the metric.

It seems there are conceptual advantages to both approaches : the former ties in nicely with the statics/dynamics analogy John was making, while symplectic geometry seems to emerge more naturally in the latter.

Posted by: Bruce Bartlett on October 10, 2006 12:52 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

In Segal’s notes, one speaks about the solution manifold XX to the Euler Lagrange equations,

I haven’t yet had the time to look at Segal’s text, but I’d believe this is what I said at the end of my last comment #, namely that the action does become a 1-form on phase space - which is nothing but the space of solutions to the equations of motion.

For simple cases this 1-form is just the canonical 1-form pdqp d q (or pdq+Hdtp d q + H d t if we are on exteded phase space) on a cotangent bundle, and it’s derivative is the symplectic form

(1)ω=dpdq. \omega = d p \wedge d q \,.

(But if that’s not what Segal is talking about, please correct me.)

Posted by: urs on October 10, 2006 1:36 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Well, you’re certainly right Urs - these are just different ways of slicing the same cake, I suppose.

But I think one should appreciate how strange this can seem to a novice who encounters it for the first time… the aforementioned person may or not be me.

For, at least naively, in John’s approach the role of the action is ultimately to produce a closed 1-form (the differential of the action), which we set to zero to obtain the equations of motion. In this picture, one considers variations with fixed boundary conditions.

While, in the second approach, the role of the action is ultimately to produce a closed 2-form - the symplectic form - together with the Hamiltonian function, of course. Moreover, this arises from considering variations with no boundary conditions.

These are ultimately equivalent, but perhaps this second way would allow John’s lecture to treat the statics of a particle and its dynamics in an even-handed way.

For while in the statics part, one does not initally assume that F = -dV for some potential, and subsequently set dV = 0; yet in the dynamics part, we skip directly to the case where the 1-form on path space is given by dS, and then we set dS = 0.


Posted by: Bruce Bartlett on October 10, 2006 3:41 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

But I think one should appreciate how strange this can seem to a novice who encounters it for the first time…

Sure, yes, I certainly didn’t mean to imply the contrary. Sorry if it came across differently.

treat the statics of a particle and its dynamics in an even-handed way.

This analogy between dynamics of the pp-particle and statics of the (p+1)(p+1)-particle is quite curious. I wonder what it really means in the end. I haven’t thought about it before.

One reason why I haven’t thought about it is closely related to my above desire to restrict the categorification of geometric quantization to the charged particle #:

Namely, in common applications, the 2-particle is not subject to a scalar potential on path space (or loop space).

Rather, the forces the 2-particle couples to are, in familiar applications, of the form of gauge field forces.

Part of the reason for that, again, is that in most applications the 2-particle is treated relativistically, and even for the ordinary particle the scalar potential of the electric field becomes part of a 1-form coupling when treated relativistically.

Hm, let’s see. On the other hand, for instance a gauge trivial Kalb-Ramond field for the string is induced by a “potential”

(1)LXU(1) L X \to U(1)

which sends every loop to the holonomy of a U(1)U(1)-connection on XX around that loop #.

Hm…

Posted by: urs on October 10, 2006 4:15 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Urs wrote:

Sure, yes, I certainly didn’t mean to imply the contrary. Sorry if it came across differently.

No problem - I didn’t mean it that way. Your comment about a gauge trivial Kalb-Ramond field is interesting… perhaps John will touch on such structures during his lectures.

John Armstrong wrote:

I’m not sure about Segal’s notes, but it sounds similar to Zuckerman’s approach which is completely local and doesn’t talk about boundary conditions at all. I’ll describe it so you can say if it’s more or less Segal’s or not.

Zuckerman’s approach seems rather intricate, although I have thought along similar lines before - especially for a reformulation of Noether’s theorem. From what I can understand of it though, it does seem like much the same stuff as Segal’s. The crucial point is that one considers the variation of the action at the end-points - as you point out, Segal obtains this from integration by parts, while Zuckerman gets it from the magic of his setup.

Posted by: Bruce Bartlett on October 10, 2006 9:32 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

The crucial point is that one considers the variation of the action at the end-points - as you point out, Segal obtains this from integration by parts, while Zuckerman gets it from the magic of his setup.

I’ll admit I’m not an expert at the Zuckerman approach, but I’m pretty sure that you can understand variations of boundaries by sticking them into γ\gamma.

Anyhow, Noether’s theorem comes out very nicely in this approach. Pick a local vector field ξ\xi on \mathcal{F} and a local α ξ\alpha_\xi in Ω 0,|1|\Omega^{0,|1|} with Lie(ξ)L=dα ξ\mathrm{Lie}(\xi)L = d\alpha_\xi on ×M\mathcal{F}\times M. Then there is a β ξ\beta_\xi in Ω 1,|2|\Omega^{1,|2|} so that Lie(ξ)γ=δα ξ+dβ ξ\mathrm{Lie}(\xi)\gamma = \delta\alpha_\xi + d\beta_\xi on ×M\mathcal{M}\times M. Then the Noether current j ξ=ι(ξ)γα ξ\j_\xi = \iota(\xi)\gamma-\alpha_\xi is conserved (dj ξ=0dj_\xi = 0) on ×M\mathcal{M}\times M and δj ξ=ι(ξ)ω+dβ ξ\delta j_\xi = -\iota(\xi)\omega + d\beta_\xi.

It’s all just juggling the cohomology operations. Even better, it’s all local so you never have to bother about all that messy analysis.

Posted by: John Armstrong on October 10, 2006 11:03 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

in the second approach, the role of the action is ultimately to produce a closed 2-form - the symplectic form - together with the Hamiltonian function, of course. Moreover, this arises from considering variations with no boundary conditions.

I’m not sure about Segal’s notes, but it sounds similar to Zuckerman’s approach which is completely local and doesn’t talk about boundary conditions at all. I’ll describe it so you can say if it’s more or less Segal’s or not.

Zuckerman considers the deRham bicomplex of ×M\mathcal {F}\times M where MM is the manifold and \mathcal{F} is the space of fields. Variations are tangent vectors to \mathcal{F}. Actually, he twists by the orientation bundle of MM so that he gets densities rather than volume forms.

Anyhow, a Lagrangian LL lives in Ω 0,|0|\Omega^{0,|0|}, which assigns an nn-density to each field. Its exterior derivative in the MM direction dies and the one in the \mathcal{F} direction is the old δL\delta L which assigns an nn-density to every variation. In general this is not “linear on functions”. That is, multiplying the variation by a function doesn’t multiply δL\delta L by the same function. However, it is linear on functions up to an MM-exact term dγd\gamma. This essentially manages to handle “integration by parts” without any integration or boundary terms.

So, we also have a γ\gamma in our theory living in Ω 1,|1|\Omega^{1,|1|}, which assigns an n1n-1-density to every variation. The inhomogenous element L+γL+\gamma has δL+dγ\delta L+d\gamma linear on functions, and this gives the old Euler-Lagrange equations. The space \mathcal{M} where δL+dγ\delta L+d\gamma vanishes is the space of classical solutions, the “solution manifold” referred to before.

So we pass down to ×M\mathcal{M}\times M. On this space, (d+δ)(L+γ)=dL+(δL+dγ)+δγ=δγ (d+\delta)(L+\gamma) = dL + (\delta L + d\gamma) + \delta\gamma = \delta\gamma which we now call ω\omega. This lives in Ω 2,|1|\Omega^{2,|1|}, and assigns an n1n-1-density to every bivariation. That is, it’s an n1n-1-density-valued 2-form on \mathcal{M}. Even better, it’s symplectic.

Now, if MM is split into time and space, we can integrate ω\omega over any space slice to get a symplectic form Ω\Omega on \mathcal{M}, which is the symplectic manifold discussed above for the Hamiltonian setting. We can also get the Hamiltonian function on \mathcal{M} out of this setup.

In short, the Zuckerman approach seems to get both the Euler-Lagrange equations of motion and the symplectic geometry picture. If this is more or less what Segal does, then so does his approach.

Posted by: John Armstrong on October 10, 2006 4:16 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Bruce wrote:

It seems there are conceptual advantages to both approaches: the former ties in nicely with the statics/dynamics analogy John was making, while symplectic geometry seems to emerge more naturally in the latter.

I’ll try to fill you in on the “big picture” as time goes by. For now you should note that potential energy is a 0-form on XX, its differential is a 1-form called force, the action is a 0-form on PXP X, and its differential is a 1-form called… well, maybe the Euler-Lagrange 1-form, since its vanishing is called “the Euler-Lagrange equations”.

Today - I’m about to go to class - I’ll cover the Euler-Lagrange equations, and with any luck I’ll get to the Legendre transform mapping TXT X to T *XT^* X. There’s a god-given 1-form on T *XT^* X called the canonical 1-form, and its differential is a 2-form called the symplectic structure.

So, we have a bunch of closely related differential forms and our job is to understand deeply how they’re related.

The fun part will be categorifying this whole story, and seeing how the degrees of all the forms go up one notch!

Posted by: John Baez on October 10, 2006 4:47 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 1)

Thanks for the link to Segal’s talks! I’ll add that to my course webpage.

One of the cool things about this blog is that you can easily insert links. If you type something

<a href = “http://likethis.com”>like this<a>

we’ll see something

like this

In this busy age, people are more likely to look something up if it takes a mere click. So, I’ve taken the liberty of enhancing your post with links.

LaTeX isn’t much harder. If you get stuck, check the TeXnical FAQ or ask questions there - I’ll see them.

Posted by: John Baez on October 10, 2006 5:05 PM | Permalink | Reply to this
Read the post Quantization and Cohomology (Week 2)
Weblog: The n-Category Café
Excerpt: The Lagrangian approach to classical mechanics
Tracked: October 11, 2006 2:01 AM
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Excerpt: A remark on describing string composition and string field products in terms of adjunctions in categories of paths.
Tracked: October 16, 2006 2:16 PM
Read the post D-Branes from Tin Cans: Arrow Theory of Disks
Weblog: The n-Category Café
Excerpt: On disk holonomy and boundary conditions.
Tracked: October 18, 2006 3:41 PM
Read the post Flat Sections and Twisted Groupoid Reps
Weblog: The n-Category Café
Excerpt: A comment on Willerton's explanation of twisted groupoid reps in terms of flat sections of n-bundles.
Tracked: November 8, 2006 11:47 PM
Read the post QFT of Charged n-particle: Chan-Paton Bundles
Weblog: The n-Category Café
Excerpt: Chan-Paton bundles from the pull-push quantization of the open 2-particle.
Tracked: February 7, 2007 9:56 PM
Read the post Arrow-Theoretic Differential Theory
Weblog: The n-Category Café
Excerpt: We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
Tracked: July 27, 2007 5:04 PM

Re: Quantization and Cohomology (Week 1)

Dear John Baez,

I am going through all your QC courses again, now in more detail. I hope to be posting my comments and questions here in your blog as time allows. I have very simple questions and suggestions for now, here they are:

Questions for qc.pdf - week 1
====================

- First of all, I think it would be nice to state what the pre-requisites for the course are. It would also be interesting to include references to some terms which are introduced without definition. For instance, on page 5, references to cohomology and homotopy classes. I know these are quite basic terms in topology, but it would be a plus to have a pointer to them (an appendix with some basic definitions or a list of references). Or something like, for instance, Srednicki has done in his QFT book - see his “Preface for Students”, in which he lists a few equations that the students should recognise and be comfortable with them before proceeding. I thought that was very nice!

- Second paragraph of page 5:

Thus, we have a “configuration space” X (…)

I think this is a bit confusing. The term “configuration space X^n” fits nicely with what is usually defined in classical mechanics, for instance, applied to a N-body system: the whole system is represented in the *configuration space* - this space is 6N-dimensional - as a single point, and in the *phase space* - which is 6-dimensional - as N points, each point representing a single particle of the system, and so on. But here you say that the space X (and the path space PX, and so on) is a configuration space as well right? This is just a terminology issue of course, but I found it confusing at first, because for me it is like that the configuration space only makes sense when you include all degrees of freedom of the system. The other spaces are then just projections…

- page 6, first paragraph: please define what \Delta_X is.

- page 7, last line: include a point - “function. In short…”

- page 8: “not exactly isomorphic, but close to it at any rate.”

This sounds very interesting but mysterious to me. Close in what sense?

Thanks,
Christine


Posted by: Christine Dantas on July 28, 2007 1:35 PM | Permalink | Reply to this
Read the post BV for Dummies (Part V)
Weblog: The n-Category Café
Excerpt: Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
Tracked: October 30, 2007 10:09 PM

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