Quantization and Cohomology (Week 2)

October 11, 2006

Quantization and Cohomology (Week 2)

Posted by John Baez

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Here are the notes for this week’s class on Quantization and Cohomology:

Week 2 (Oct. 10) - The Lagrangian approach to classical mechanics. Action as the integral of a 1-form (prelude).
- Homework: A spring in imaginary time.

Last week’s notes are here; next week’s notes are here.

A spring in imaginary time, you ask? Read on…

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In last week’s lecture we learned - rather abstractly - how the dynamics of particles was analogous to the statics of strings. This time we introduce the basics of the Lagrangian formalism: enough to make the analogy more concrete, by considering an example.

What’s a simple problem involving the dynamics of a particle? How about the motion of a thrown rock in a constant gravitational field? We all know it traces out a parabola.

What’s the analogous problem involving the statics of a string? It’s just the problem of determining the equilibrium state of a string hung between fixed endpoints in a constant gravitational field. Instead of calling it a “string”, let’s call it a “spring”. Imagine a spring stretched out with its ends nailed to two posts… what curve does it trace out?

As you’ll see when you do this homework problem, the analogy is very cute. But there’s a funny wrinkle - obvious if you think about it. The thrown rock arcs up and then back down. The hung spring curves down and then back up! There’s a minus sign somewhere…

And, we can understand this minus sign by treating the spring as a thrown rock in imaginary time. The sign comes from

$i^2 = -1.$

The idea of relating dynamics and statics using imaginary time is well known in quantum field theory - it’s called “Wick rotation”. But, it works in classical field theory too, and here we see it in an even simpler context: classical mechanics!

Posted at October 11, 2006 1:19 AM UTC

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15 Comments & 4 Trackbacks

Re: Quantization and Cohomology (Week 2)

What’s a simple problem involving the dynamics of a particle? How about the motion of a thrown rock in a constant gravitational field? We all know it traces out a parabola.

What’s the analogous problem involving the statics of a string? It’s just the problem of determining the equilibrium state of a string hung between fixed endpoints in a constant gravitational field.

I assume we’re neglecting the weight of the string itself, because otherwise “we all know” the hanging string actually traces out a catenary. Well, that’s true by definition. Specifically it’s not the graph if a quadratic function, but of a hyperbolic cosine.

So if we neglect the weight of the string in the statics problem, what are we neglecting in the dynamics problem? Not the weight, or the rock would move in a straight line. Not the air resistance, because that wouldn’t make the path of the rock a catenary, and we’ve neglected air for the string so that would mean something else we’re neglecting for the rock.

Posted by: John Armstrong on October 11, 2006 3:04 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

I leave it as an extra puzzle to figure out the answer to John Armstrong’s dilemma here.

If you get really stuck, reread the sentence of mine just after the last one he quoted above.

Or, read the actual statement of the homework problem, instead of the deliberately tantalizing statement in this blog. But only do that if you give up on this puzzle - it makes it too easy.

Posted by: John Baez on October 11, 2006 3:30 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

I’ve looked through the homework and tried to figure something out to handle my question. I’ll try to be vague so as not to spoil the homework for the rest of the class.

Basically, there are two different differential equations that would lead to the hyperbolic cosine. One is pretty simple to find a Lagrangian for, but it doesn’t look much like gravity. In fact, it looks like if you hung a spring between two points and then drew it towards the ground (sans gravity) by a bunch of other springs.

The other equation is much weirder from an Euler-Lagrange point of view. This is the one you get if you try to derive the equation for the catenary from a Newtonian (rather than Lagrangian) viewpoint. I tried for a while but I don’t think it’s possible to get a Lagrangian depending only on first derivatives, and I don’t think it’s possible with any finite number. On further consideration, the problem is that the derivation of the force on an element of the chain is not local, and depends heavily on how the rest of the chain is hanging. Wick rotating to time, it seems that that setup would describe a “nonlocal” action principle depending on how the rock flew in the past and future.

So, at first I thought I’d just missed something, but evidently there’s something odd going on here in the analogy. I find it interesting that the space of solutions of a nonlocal energy functional can be reproduced by a different local one. Is it possible that seemingly nonlocal interactions in quantum physics are also simply “misinterpretations” of truly local phenomena?

As a side note: is this sort of thing the reason we use the term “local” to describe a functional that only depends on the k-jet of a field for some finite k?

Posted by: John Armstrong on October 11, 2006 6:21 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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I guess by now it’s okay to give away the answer to John Armstrong’s dilemma.

We’re not considering a flexible but inelastic rope hanging between two posts. That would trace out a catenary.

Instead, we’re considering a flexible and elastic spring hanging between posts, idealized so its energy of stretching is proportional to

$\int |\frac{d q}{d s}|^2 d s$

where $q(s)$ is the position of the spring as a function of the parameter $s$ that “counts atoms” as we move along the spring. Such a spring would have the least energy when it shrinks down to zero length.

And, such a spring traces out a parabola when you hang it between two posts!

Posted by: John Baez on October 18, 2006 4:11 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

You can now see a bunch of answers to the homework problem about A Spring in Imaginary Time:

Answers by Garett Leskowitz.
Answers by Jeffrey Morton.
Answers by Erin Pearse.
Answers by Alex Hoffnung.

Posted by: John Baez on October 18, 2006 4:15 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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At least heuristically, there is a deep and important statement very closely related to the correspondence

$n$ -particle dynamics $\leftrightarrow$ $(n+1)$ -particle statics.

Just translate this to the quantum regime, proper, where it reads

correlators of the $n$ -particle $\leftrightarrow$ states of the $(n+1)$ -particle

For $n=2$ , that’s the deep relationship between 3D TFT and 2D CFT, of which the

WZW $\leftrightarrow$ Chern-Simons

originally found by Witten is just a special case.

Here, states of the 3D TFT associated to a 2-dimensional space $\Sigma$ are “the same as” the “conformal blocks” (this are essentially the same as correlators on $\Sigma$ ) of the 2D CFT.

That’s a pretty deep statement. (By the way, the lecture by Hopkins that I mentioned is about this.)

On heuristic grounds, it seems clear that this related to $n$ -dynamics $\leftrightarrow$ $(n+1)$ -statics.

Can we understand it conceptually, but in a more precise manner?

In my latest comment on the 2-particle, I propose what I think is a good arrow-theoretic notion of

- the space of sections of the 2-particle (def. 3)

- the disk correlator of the 2-particle (def. 5)

(actually, I don’t yet talk there explicitly about the quantum 2-particle, but just about it’s classical coupling to a gerbe, but the claim in the background is that the formalism is almost the same in both cases).

Now, I guess that def. 5 will look pretty obscure, unless you have spent some time staring at all these notes.

On the other hand, I claim that if you stare at it long enough, a rather interesting fact becomes obvious:

my def. 5 of the disk holonomy is essentially what would be def 3b, namely the definition of a section of a 3-particle!.

See? That’s the

$n$ -correlator $\leftrightarrow$ $(n+1)$ -section

correspondence.

Or so I think. I’ll have to sleep over this.

Posted by: urs on October 18, 2006 9:04 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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Urs wrote:

At least heuristically, there is a deep and important statement very closely related to the correspondence

$n$ -particle dynamics $\leftrightarrow$ $(n+1)$ -particle statics.

Just translate this to the quantum regime, proper, where it reads

correlators of the $n$ -particle $\leftrightarrow$ states of the $(n+1)$ -particle

That’s excellent. Truly excellent!

It’s especially nice if it “explains” the relation between the WZW model in 2 dimensions and Chern-Simons theory in 3 dimensions, which has fascinated me ever since I read Witten’s paper. In fact, that paper was how Louis Crane got excited about the “ladder of dimensions” which relates field theories in neighboring dimensions through a process of categorification. And he, in turn, got me interested in n-categories. So, maybe it’s finally all fitting together.

It will take me quite a while before I reach these more advanced topics in my class, but there are some quantum topologists attending, so I will let them know now that this is one of the hoped-for payoffs of our work!

You’ll have everything figured out by the time I’m ready to teach it.

Posted by: John Baez on October 18, 2006 11:18 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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In fact, that paper was how Louis Crane got excited about the “ladder of dimensions” which relates field theories in neighboring dimensions through a process of categorification. And he, in turn, got me interested in n-categories. So, maybe it’s finally all fitting together.

Interesting. I had not known this part of the story.

I am wondering how many other people in the world seriously think about the step WZW to CS in terms of categorification. Willerton and Bartlett, inspired by Freed, do #. According to the first slide of Gukov’s talk at Strings 06, possibly also Witten does. But it’s not clear. There is still a difference between using a category here and there and taking things seriously.

In Vienna both Anton Kapustin as well as Tony Pantev, after their brilliant talks on Langlands (Pan., Kap. ) found little appreciation for the fact that the eigenbrane condition is the condition for an eigen-2-vector in categorified linear algebra - a very simple and obvious statement that makes the entire field appear in a different light. To me, it’s the difference between happening to use a category and taking things seriously.

Anyway. Above # I wrote:

actually, I don’t yet talk there explicitly about the quantum 2-particle

I was hoping to find the time to work more today. But the blissful period between semesters is coming to an end, causing lots of distractions (and I guess I was also being distracted by thinking about Euler characteristics).

But I should maybe indicate how using the puzzle pieces that I talked about a coherent picture seems to emerge.

Let the 3-particle be the 2-category

(1)

p_3 = \left\lbrace \array{ & \nearrow \searrow \\ \bullet & \Downarrow& \circ \\ & \searrow \nearrow } \right\rbrace

with two objects, two nontrivial 1-morphisms and 1 nontrivial 2-morphism.

Its configuration space is the space of objects of the functor category

(2)

[p_3,P_3(X)] \,.

(To be compared with the first couple of definitions here.)

For $C_2$ a braided monoidal category, let

(3)

\mathrm{End}(C_2)

be the 3-category of its endomorphisms.

According to this we want to look at lax 2-functors

(4)

\mathrm{tra} : P_3(X) \to \mathrm{End}(C_2)

into that, which assign algebras internal to $C_2$ to points, left-right induced bimodules to edges.

By the general logic # a state of the 3-particle coupled to the 2-bundle with connection represented by $\mathrm{tra}$ is a section $e$ of that 3-bundle, which again is a morphism

(5)

\left\lbrace \array{ & \nearrow \searrow^ {\mathrm{tra}^0_*} \\ [p_3,P_3(X)] & \;\Downarrow e& [p_3, \mathrm{Bim}(\mathrm{End}(C_2))] \\ & \searrow \nearrow_{\mathrm{tra}_*} } \right\rbrace \,,

where $\mathrm{tra}^0$ sends everything to the idendity on the tensor unit.

Meditating over that for a while reveals that such a morphism is more or less determined by the same form of disk diagrams that are featured at the end of that text on disk holonomy #.

In fact, I believe the resulting disk diagram should be essentially that which appears in section 4.3 of my FRS notes #, except that the $U$ - and $V$ - ribbons which were inserted by hand there now appear automatically due to # the nature of $\mathrm{End}(C_2)$ . Very nice.

So this story starts as a description of a state for the 3-particle and ends as a description of the disk correlator of a 2-particle.

Now I need the time to work this out in detail.

Posted by: urs on October 19, 2006 7:49 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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I wrote about

[…] little appreciation […]

It might be noteworthy that in spite of that, everybody is aware of Kapranov’s idea in

M. Kapranov
Analogies between the Langlands correspondence and topological quantum field theory
in: “Functional analysis on the eve of 21st century” (S. Gindikin, J. Lepowsky, R. Wilson Eds.), vol. 1 (Progress in Math. v. 131), p. 119-151, Birkhauser, 1995.

Therein, M. Kapranov discusses how passing from strings to membranes in geometric Langlands would make us consider moduli spaces not of vector bundles, but of 2-bundles.

So I was told: not what you are talking about (that Hecke eigensheaves are 2-eigenvectors), but what Kapranov says in the above text makes the relation between Langlands and 2-vector spaces.

But - no. Kapranov’s observation goes even a step further. Instead of realizing that geometric Langlands is already the categorification of something (which is my point), he categorifies geometric Langlands itself!

That, in turn, would lead me to speak about 3-eigenvectors (whose entries, yes, would be the 2-vector spaces that Kapranov talks about).

Posted by: urs on October 19, 2006 8:15 PM | Permalink | Reply to this

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 19, 2006 3:14 PM

Re: Quantization and Cohomology (Week 2)

Could you gives us a sense of the larger category theoretic picture? Is there a category of Lagrangian systems? What would its morphisms be? Might they include reduction, imposition of constraints? Is Wick rotation an endofunctor? Is there also a category of Hamiltonian systems? What then is the Legendre transform doing?

Posted by: David Corfield on October 30, 2006 12:10 PM | Permalink | Reply to this

Read the post Quantization and Cohomology (Week 3)
Weblog: The n-Category Café
Excerpt: Sorry for the long pause! Here are the notes for the October 17th class on Quantization and Cohomology: Week 3 (Oct. 17) - From Lagrangian to Hamiltonian dynamics. Momentum as a cotangent vector. The Legendre transform. The Hamiltonian. Hamilton's equa...
Tracked: November 8, 2006 5:30 AM

Read the post D-Branes from Tin Cans, III: Homs of Homs
Weblog: The n-Category Café
Excerpt: Sections of sections, their pairing and n-disk correlators.
Tracked: January 19, 2007 9:10 AM

Read the post Quantization and Cohomology (Week 1)
Weblog: The n-Category Café
Excerpt: How the dynamics of p-branes resembles the statics of (p+1)-branes.
Tracked: January 31, 2007 3:00 AM

Re: Quantization and Cohomology (Week 2)

Dear John Baez,

Here are my questions/comments for week 02.

Page 12, section 3.5, second paragraph:

I think it would be interesting to elaborate a little bit more on the motivations behind writing the action as an integral of a 1-form. Is it to make S independent of parametrization? Perhaps an example would clarify this? Also, why is this interesting from the point of de Rham cohomology? Does this restrict the types of manifolds for the configuration space?

Thanks,
Christine

Posted by: Christine Dantas on July 29, 2007 7:44 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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Thanks for your comments and questions.

One reason for wanting to make the action into the integral of a 1-form is just as you say: to make the integral reparametrization-invariant.

More philosophically, a 1-form is precisely what people invented to formalize the idea of integration along a path in a nice coordinate-free way. The idea of the ‘principle of least action’ is that a physical system should undergo something called ‘action’ as it traces out a path. So, one can’t help wanting this to be the integral of a 1-form!

The other great thing about a 1-form is that its exterior derivative is a 2-form. If you apply this to the 1-form that defines the action, you’re led straight to the concept of symplectic structure.

But, all these matters only become clear if we think of action as something defined for paths in $T^* M$ — the cotangent bundle of the ‘extended configuation space’ $M$ . For a single particle moving around in spacetime, this $M$ is just spacetime itself!

So, the fall session eventually moved in the direction of explaining symplectic geometry and the extended configuration space, starting from a more low-brow approach to classical mechanics.

Posted by: John Baez on July 31, 2007 3:02 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

Thanks for your answer. These are good motivations and I’ll keep them in mind. I’m soon going back in more detail to the part on symplectic structures.

Best regards,
Christine

Posted by: Christine Dantas on July 31, 2007 3:54 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

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You could say that a lot of the first quarter of this course is an attempt to understand the relation between Hamiltonian and Lagrangian mechanics. Of course people understand the relation already. But, some aspects of this relation still feel like ‘tricks’, and they shouldn’t — everything should be very clear.

In particular, it seems a bit ‘tricky’ how Hamiltonian mechanics is usually formalized in terms of a 2-form on phase space, the symplectic structure $\omega$ . What’s the physical meaning of this 2-form?

A 2-form is something that you integrate over a surface… so what does it mean when you integrate $\omega$ over a surface in phase space?

The quantity you get has units of action, since the symplectic structure looks like $\omega = d p_i \wedge d q^i,$ and momentum times position has units of action.

But, why should a surface in phase space have an action associated to it?

In the Lagrangian approach, what gets an action is a path, not a surface.

So, it’s all a bit mysterious.

However, we can imagine a system tracing out a periodic orbit. We get a path in phase space that’s a loop. This loop might be the boundary of a surface. If the loop is the boundary of some surface, we could try to define the action of that surface to be the action of the loop!

This is definitely a good idea, and Bohr and Sommerfeld thought about it a lot when quantizing the hydrogen atom. They realized that the hydrogen atom seemed to enjoy orbits that go around surfaces where the integral of $\omega$ is an integer times Planck’s constant. This is the Bohr-Sommerfeld quantization condition.

So, there’s something important about surfaces in phase space where the integral of $\omega$ is an integer. Maybe that’s why integrals are called integrals: Nature likes them best when they’re integral!

But, there are a bunch of things that don’t seem to gell at first. For example, action is really the integral of $H d t - p_i d q^i$ along a path. What does this have to do with the integral of $\omega = d p_i \wedge d q^i$ over a surface? How do time and energy get unified with position and momentum, exactly? Clearly this has something do with relativity… but what?

In the notes I try to answer these questions.

However, I should improve all my explanations, so people can see my motivations better! Your questions are already helping me do this.

By the way, the notes from my course on classical mechanics could be helpful. I stick to more standard subjects, so everything is better organized… less of a personal quest into unknown territory.

Posted by: John Baez on August 1, 2007 3:28 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 2)

This “personal quest into unknown territory” is what makes the course very interesting by itself and I’m enjoying it. And exposing your motivations is part of making this a unique course.

Concerning Lagrangian and Hamiltonian mechanics, I can tell you from an astrophysicist point of view that of course we go into all that material, but in a very standard way (e.g., Goldstein). Then, during my graduate years, I found Arnold’s Mathematical Methods of Classical Mechanics. Only then I realized there was a more elegant approach to mechanics.

Best,
Christine

Posted by: Christine Dantas on August 1, 2007 4:51 PM | Permalink | Reply to this

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October 11, 2006