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November 15, 2006

Quantization and Cohomology (Week 7)

Posted by John Baez

Here are yesterday’s notes on Quantization and Cohomology:

  • Week 7 (Nov. 14) - From particles to strings and membranes. Generalizing everything we’ve done so far from particles (p=1p = 1) to strings (p=2p = 2) and membranes that trace out pp-dimensional surfaces in spacetime (p0p \ge 0). The concept of "pp-velocity". The canonical pp-form on the extended phase space Λ pT *M\Lambda^p T^*M, where MM is spacetime.

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

Now we’re getting to the cool stuff: after reviewing classical mechanics, now we can generalize it from particles to strings and higher-dimensional membranes following Rovelli’s ideas on extended phase space. We’ll even go beyond his ideas in various ways, like introducing the concept of “pp-velocity” (I don’t think he mentions that - I could be wrong) - and, much more importantly, relating all this stuff to categorification.

In this lecture, the fun begins.

Posted at November 15, 2006 8:50 PM UTC

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6 Comments & 3 Trackbacks

Re: Quantization and Cohomology (Week 7)

Just for the record:

Schuller and Wohlfarth did reformulate the Nambu-Goto action of the string in a way close in spirit to the last remarks about multivelocities in the above notes in their paper Canonical differential structure of string backgrounds.

Posted by: urs on November 16, 2006 12:40 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 7)

Thanks! I sort of knew that; I will have to reread this paper now that I’m finally digging into the details.

Posted by: John Baez on November 16, 2006 4:59 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 7)

Is page 4 supposed to be blank?

Posted by: Toby Bartels on November 16, 2006 9:45 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 7)

There shouldn’t have been a page 4 at all. I’ll get rid of that blank page someday… but at least you’re not missing anything.

Posted by: John Baez on November 19, 2006 12:22 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 7)

I just came across another paper that at least mentions the idea of generalizing geometric quantization on a symplectic manifold to an operation using gerbes and 3-forms:

Marius Crainic, Prequantization and Lie brackets, Journal of Symplctic Geometry, Volume 2, Number 4, 579–602, 2005

There on p. 584 it says:

let us point out here that the construction of the central extension of the loop group (section 6.4 in [3]) is completely analogous to our construction of the prequantizing bundle, but one level higher (2-homotopies instead of 1-homotopies). Such similarities cannot be accidental, and we believe they are just some of the small pieces of a whole (more geometric) picture based on prequantization of groupoids with a 3-form background, gerbes over such groupoids etc.

Posted by: urs on November 24, 2006 10:38 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 7)

Thanks for the reference! Of course this example is the only reason I’ve been interested in what would otherwise seem like a completely insane project - extending the analogy

2-form : 3-form ::

line bundle : gerbe ::

symplectic structure: ???? ::

Kähler structure : ???? ::

holomorphic section : ???? ::

Kähler quantization : ????

Posted by: John Baez on November 26, 2006 12:02 AM | Permalink | Reply to this
Read the post Quantization and Cohomology (Week 7)
Weblog: The n-Category Café
Excerpt: Generalizing classical mechanics from particles to strings and higher-dimensional membranes.
Tracked: December 1, 2006 1:12 AM
Read the post Quantization and Cohomology (Week 8)
Weblog: The n-Category Café
Excerpt: From particles to membranes, continued.
Tracked: December 1, 2006 1:16 AM
Read the post Quantization and Cohomology (Week 6)
Weblog: The n-Category Café
Excerpt: The canonical 1-form, extended phase space and Rovelli's covariant formulation of classical mechanics.
Tracked: December 1, 2006 1:27 AM

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