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December 1, 2006

Quantization and Cohomology (Week 8)

Posted by John Baez

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

  • Week 8 (Nov. 28) - From particles to membranes, continued. A coordinate-free definition of pp-velocity. The action for a charged point particle in general relativity, versus the action for a charged membrane. The electromagnetic field versus its pp-form generalization.

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

I’m only giving one more class in this course this quarter. I’ll spend that class sketching what we’re aiming at and what we’ll do next quarter.

Posted at December 1, 2006 1:06 AM UTC

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5 Comments & 2 Trackbacks

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 1:20 AM

Re: Quantization and Cohomology (Week 8)

For the charged particle in flat space, the naive path integral

(1)[Dγ]exp(|γ| 2+i γA) \int [D\gamma] \exp(- \int |\gamma'|^2 + i\int_\gamma A )

is made rigorous by realizing that the combination

(2)[Dγ]exp(|γ| 2) [D\gamma]\exp(-\int |\gamma'|^2)

is to be interpreted as the Wiener measure dμd\mu on the space of paths, such that we should read the above as

(3) W(X)dμ(γ)exp(i γA) \int_{W(X)} d\mu(\gamma)\, \exp(i\int_\gamma A)

or rather

(4) W(X)dμ(γ)hol (γ). \int_{W(X)} d\mu(\gamma)\, \mathrm{hol}_\nabla(\gamma) \,.

A similar formula is still true for particles on curved Riemannian spaces. Only that here the Wiener measure receives a certain deformation for nonvanishing Ricci curvature.

I expect that an analogous statement should hold for the string, but I don’t recall having seen it stated anywhere.

Is there a rigourous measure on the space of continuous maps [0,1] 2X[0,1]^2 \to X, analogous to the Wiener measure for paths [0,1]X[0,1] \to X, such that it would analogously yield the kinetic part of the string action, reducing the path integral for the charged string to

(5) W(Σ)dμ(Σ)hol (Σ), \int_{W(\Sigma)} d\mu(\Sigma) \;\;\mathrm{hol}_\nabla(\Sigma) \,,

where now hol \mathrm{hol}_\nabla would in genereal be the surface holonomy of a gerbe (possibly with boundary contribution) but which for the purpose of this question might be just the integral of a global 2-form.

Posted by: urs on December 1, 2006 1:34 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 8)

Hi Urs,

This topic is actually getting more and more interesting and, in fact, this question of yours just added a whole new twist to it. Let me explain myself (by parts):

(1) Regarding your question: I’m not sure whether what i’ll say properly answers it, but… take a look at The wiener-askey polynomial chaos for stochastic differential equations. Or, more precisely, at: The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations (PDF). As i understand it, the authors generalize Wiener chaos to d dimensions (using d-dimensional generalized sets of orthogonal polynomials in order to span/define the measure). I believe a construction along these lines can be used in order to construct “higher dimensional Wiener measures” along the lines you described in your post above.

(2) Personally, this is a “twist” because we’ve been using this technique (“polynomial chaos”) in order to numerically compute QFTs in different phases (what’s not usually possible with Monte Carlo simulations) – in a method that we call “Source-Galerkin” (which, now, is also gaining some ‘stochastic quantization’ flavor). In a clear sense, properly defining this measure implies picking out the phase of a QFT.

Cheers.

Posted by: Daniel Doro Ferrante on December 1, 2006 4:16 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 8)

Hi Urs,

Just replying to myself with a reference that may be more directly related/relevant to your question: J. Yeh. Orthogonal developments of functionals and related theorems in the Wiener space of functions of two variables. Pacific Journal of Mathematics, 13:1427–1436, 1963.

HTH.

Cheers.

Posted by: Daniel Doro Ferrante on December 1, 2006 7:12 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 8)

Hi Daniel,

nice to hear from you!

Thanks for all the links. Good to know that people do have thought about this.

Currently, I am oscillating with a period of roughly one week between different opinions on what the most natural way to think about these things might be.

From one point of view, I’d be fond of the idea that the action of the charged nn-particle should just be

(1) W n(X)dμ(V n)hol(V n), \int_{W^n(X)} d\mu(V_n)\;\; \mathrm{hol}(V_n) \,,

where W n(X)W^n(X) is something like the nn-Wiener space of continuous nn-paths in XX, dμd\mu the corresponding nn-Wiener measure and hol\mathrm{hol} an nn-bundle (n1n-1-gerbe) holonomy of the hypervolume V nV_n. The kinetic part would be entirely encoded in the nn-Wiener measure.

Conceptually, that looks rather nice. But one thing disturbs me:

For n=2n=2, there are plenty of indication that we want to treat the kinetic and the gauge coupling part of the action on the same footing. This nice symmetry is of course completely non-manifest in the 2-Wiener description.

I am not sure yet if that is telling us something or not.

Posted by: urs on December 3, 2006 4:45 PM | Permalink | Reply to this

Coordinate Independence vs. Implementation Independence

In this week’s lecture note John describes

A coordinate-free definition of pp-velocity.

I am trying to find an “implementation independent” definition of pp-vector fields.

By this I mean a purely arrow-theoretic description, such that it reproduces the ordinary description when these arrows are interpreted internal to a suitable ambient category.

For p=1p=1 I think I found a nice solution.

I am claiming that an “implementation independet” formulation of the concept of an ordinary vector field is the following #

Let P 1(X)P_1(X) be a 1-category, supposed to model the idea of a category of paths in some space XX.

Let F(P 1(X))F(P_1(X)) be the sub-category of Σ(Aut(P 1(X)))\Sigma(\mathrm{Aut}(P_1(X))) which has the single object P(X)P(X) , whose morphisms are natural transformations of the form

(1) Id P 1(X) P 1(X) h \array{ & \nearrow\searrow^{\mathrm{Id}} \\ P_1(X) &\Downarrow& P_1(X) \\ &\searrow \nearrow_{h} }

and where composition of morphisms is horizonatal composition of these natural transformations.

For RR any group, I now say that an RR-flow in P 1(X)P_1(X) is a morphisms

(2)Σ(R)F(P 1(X)). \Sigma(R) \to F(P_1(X)) \,.

I claim that if we internalize this arrow theory as follows, we recover the ordinary notion of (flow of a) vector field on a smooth space XX:

Let XX be a smooth space and let P 1(X)P_1(X) be the groupoid of thin-homotopy classes of paths in XX. Let R=R = \mathbb{R} and require all functors to be smooth.

Now the question is: how would we categorify this to describe not a flow of points, but a flow of strings.

I have one way how to do this, which is fine for most applications that I am currently dealing with. But I am hoping there is another way, one which would more directly be connected to the concept of 2-velocity that John describes in his lecture.

The one way I have is simply obtained by looking at 1-flows on something like the path space of XX. This can be easily formulated in a “implementation independent”, i.e., in an arrow-theoretic way.

But what I would also like to have is something like this:

Replace P 1(X)P_1(X) by a 2-category P 2(X)P_2(X). Then consider the category F(P 2(X))F(P_2(X)) essentially as above, but now also including higher morphisms in the obvious way.

What next? How do we have to proceed such that internalizing in smooth path categories as before produces for us the notion of something dual to a 2-form field?

Posted by: urs on December 3, 2006 5:04 PM | Permalink | Reply to this
Read the post Quantization and Cohomology (Week 9)
Weblog: The n-Category Café
Excerpt: A glimpse of geometric quantization.
Tracked: December 21, 2006 3:29 AM

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