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January 31, 2007

Quantization and Cohomology (Week 12)

Posted by John Baez

This week’s class on Quantization and Cohomology introduced the theme of ‘rigs’ (rings without negatives), foreshadowed last week:

  • Week 12 (Jan. 30) - Classical, quantum and statistical mechanics as ‘matrix mechanics’. In quantum mechanics we use linear algebra over the ring ℂ\mathbb{C}; in classical mechanics everything is formally the same, but we instead use the rig ℝ min=ℝ∪{+∞}\mathbb{R}^{min} = \mathbb{R} \cup \{+\infty\}, where the addition is min and the multiplication is +. As a warmup for bringing statistical mechanics into the picture - and linear algebra over yet another rig - we recall how the dynamics of particles becomes the statics of strings after Wick rotation.

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

Posted at January 31, 2007 1:56 AM UTC

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13 Comments & 2 Trackbacks

Read the post Quantization and Cohomology (Week 11)
Weblog: The n-Category Café
Excerpt: What's really going on with quantization?
Tracked: January 31, 2007 2:28 AM

Hamilton-Jacobi Equation

So, where do I go to learn about the Hamilton-Jacobi equation? Assume that I am mathematically sophisticated, and I understand on a mathematical level what people mean when they describe mechanics through symplectic forms and Hamiltonians, but my formal training in mechanics was mostly directly from Newton’s laws with a quick reference to Lagrangians at the very end of the course.

I ask because I’ve been doing tropical math for five years now, so I understand very well the degeneration of (C,+,*) to (R,min,+). I’ve always know that people describe the reverse process as quantization, but I had the impression that this was just in the vague way that mathematicians like to refer to any situation where you perturb an algebraic object into something more complicated and useful as “quantization”. This is the first time that anyone laid out for me a real physical analogy, so now I want to learn more.

Thanks!

Posted by: David Speyer on January 31, 2007 4:36 PM | Permalink | Reply to this

Re: Hamilton-Jacobi Equation

So, where do I go to learn about the Hamilton-Jacobi equation?

The Hamilton-Jacobi equation is equation (3) in the review by Litvinov that John mentioned.

There, Litvinov discusses how we can understand it as coming from the Schrödinger equation by a change of rig.

Posted by: urs on January 31, 2007 4:45 PM | Permalink | Reply to this

Re: Hamilton-Jacobi Equation

You could also try my introduction to the Hamilton-Jacobi equation, which is lecture 7 in my course on classical mechanics for math grad students.

Posted by: John Baez on February 1, 2007 2:44 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

Isn’t there a problem with restricting from extremal to minimal paths/actions only?

Posted by: xyz on February 5, 2007 11:16 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

One can minimize the action, extremize it, or ‘criticize’ it (find its critical points). They’re all different, and all worth pondering! I discuss this a bit back in week 11 of the course, but there’s much more one could say.

Posted by: John Baez on February 7, 2007 3:00 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

One can minimize the action, extremize it, or ‘criticize’ it (find its critical points).

What are the good examples to show that sometimes one must do that? I seem to recall that some exist, but I can’t recall what they are.

Posted by: Toby Bartels on February 7, 2007 9:59 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

Consider a free particle on a round sphere. At time t=0t = 0 it’s at the north pole. At time t=1t = 1 it’s again at the north pole. What did it do in the meantime?

Even more confusing: a free particle is bouncing around in a cubical box with perfectly elastic walls. At t=0t = 0 it’s in the center of the box. At t=1t = 1 it’s also in the center of the box. What did it do in the meantime?

(The second problem is more confusing because it seems we need to study subtle issues like configuration spaces that are manifolds with boundary, boundary conditions, and non-smooth paths. But, we need to understand this problem before we can study the statistical mechanics of the famous ‘classical ideal gas in a box’. And in this particular case, there’s a cheap trick to avoid a full-fledged study of all those subtle issues.)

Posted by: John Baez on February 8, 2007 6:12 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

Consider a free particle on a round sphere. At time t=0 it’s at the north pole. At time t=1 it’s again at the north pole. What did it do in the meantime?

Hum, OK. Of course, these paths are still all local minima. Are there examples with local (or global!) maxima, or with saddle points? It would be good for me to memorise these.

Posted by: Toby Bartels on February 9, 2007 4:41 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

Toby wrote:

Of course, these paths are still all local minima.

No they’re not! If your particle circles the sphere 43 times between t=0t = 0 and t=1t = 1, tracing out the same great circle 43 times, you could have slightly shortened its route by changing it slightly.

Such paths are merely critical points of the action. They’re sort of like ‘saddle points’.

More precisely: think of the action as a function on the loop space of the sphere. Find a critical point. Look at the matrix of second derivatives of the action — the ‘Hessian’. Diagonalize it. Then you’ll see finitely many of the diagonal entries are negative, while infinitely many are positive.

I believe in the example I gave there will be 43 negative entries on the diagonal.

If you want an action functional with a nice local maximum, take your favorite action with a local minimum and multiply it by -1!

Posted by: John Baez on February 10, 2007 3:35 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

If your particle circles the sphere 43 times […], you could have slightly shortened its route by changing it slightly.

Wait, really?

I believe in the example I gave there will be 43 negative entries on the diagonal [of the Hessian when diagonalised].

Ah, so there are infinitely many ways to lengthen the path slightly, but just 43 ways to shorten it. I think that I see them now, thanks!

If you want an action functional with a nice local maximum, take your favorite action with a local minimum and multiply it by -1!

Any natural examples, that is where an ordinary physicist would be likely to write down an action that is maximised, rather than one that is minimised? (But perhaps this isn’t so important, since it’s saddle points that really prove the point.)

Posted by: Toby Bartels on February 11, 2007 12:32 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

Well, in relativity the signature of the metric is somewhat of a convention. If you take the arc-length (or “proper time” with the opposite signature) of a path as your action, one is locally maximized on physical paths and the other is locally minimized. Which one you get is sort of arbitrary.

Posted by: John Armstrong on February 11, 2007 12:42 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

In general relativity, if we count proper time the usual way clocks do, a freely falling particle takes the path that maximizes the proper time to get from one point in spacetime to another.

This may seem counterintuitive, but note that the particle could always lessen its proper time by wiggling rapidly back and forth, thanks to relativistic time dilation.

Or, consider the twin paradox. When they reunite, the freely falling twin is older than the one who zipped back and forth in his rocket.

So, if you want to stay young — travel!

Posted by: John Baez on February 11, 2007 1:04 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 12)

OK, thanks, John and John! Maybe I will remember these now …

Posted by: Toby Bartels on February 12, 2007 3:54 AM | Permalink | Reply to this
Read the post Quantization and Cohomology (Week 13)
Weblog: The n-Category Café
Excerpt: Statistical mechanics and temperature-dependent mathematics.
Tracked: February 6, 2007 9:04 PM

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