Quantization and Cohomology (Week 7)
Posted by John Baez
Here are yesterday’s notes on Quantization and Cohomology:
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Week 7 (Nov. 14) - From particles to strings and membranes. Generalizing everything we’ve done so far from particles () to strings () and membranes that trace out -dimensional surfaces in spacetime (). The concept of "-velocity". The canonical -form on the extended phase space , where 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 “-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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Re: Quantization and Cohomology (Week 7)
Re: Quantization and Cohomology (Week 7)
Is page 4 supposed to be blank?
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.
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 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
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.