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November 13, 2006

Quantization and Cohomology (Week 6)

Posted by John Baez

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

  • Week 6 (Nov. 7) - The canonical 1-form. The symplectic structure and the action of a loop in phase space. Extended phase space: the cotangent bundle of (configuration space) × time. The action as an integral of the canonical 1-form over a path in the extended phase space. Rovelli’s covariant formulation of classical mechanics, as a warmup for generalizing classical mechanics from particles to strings.

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

Now we are almost ready to leave well-trodden ground behind, and approach the point of the class.

The classical mechanics of a point particle can nicely be described by a 2-form (the “symplectic structure”), which in quantum mechanics gets reincarnated as the curvature 2-form of a line bundle (the “prequantum line bundle”). In his paper, Rovelli shows that all this can be boosted up a dimension - or as we’ll eventually see, categorified. Namely, the classical mechanics of a string can nicely be described by a 3-form, which in quantum mechanics gets reincarnated as the curvature 3-form of a gerbe!

But to see how this works, it seems good to follow Rovelli and switch from the ordinary phase space - the space where position and momentum live - to the “extended phase space”, where position, time, momentum and energy live. So, this week we started by doing that.

Posted at November 13, 2006 8:00 AM UTC

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Re: Quantization and Cohomology (Week 6)

One noteworthy aspect of the “higher” quantization in terms not of symplectic 2-forms but of higher forms (as in Rovelli’s work) is - that it reduces to ordinary quantization after suitable “transgression”.

For instance, quantizing the 2-particle (the string) can lead one to something described by a 3-form. But as we pass from target space to its loop space (for closed strings) the 3-form “transgresses” to a 2-form, and we are left with ordinary (albeit infinite dimensional) mechanics on loop space.

For closed strings this is all very easy to see. For open strings it becomes more subtle. For open/closed membranes it becomes even harder.

People thinking about what is called “extended” QFT are trying to figure out what a dd-dimensional QFT would assign to a (n<d)(n \lt d)-dimensional piece of parameter space, depending on whether that has a boundary or not. The result are tables of the sort # that Hopkins, Freed et al. write down for Chern-Simons theory, or Simon Willerton for Dijkgraaf-Witten theory #.

The situation might be described as follows:

Let d nd^n be a category addressed as parameter space

(1)par=d n. \mathrm{par} = d^n \,.

For the point particle this would be just the discrete category on a single element. For the closed string it might be Σ()\Sigma(\mathbb{Z}). Or {ab}\{a \to b\} for the open string. For the membranbe it might be some 2-category.

Let PP be a (nn-)category addressed as target space. For the point particle this might be some category of paths in a space XX. For GG-Dijkgraaf-Witten theory this would be Σ(G)\Sigma(G). For Chern-Simons theory it might be something like INN(String G)\mathrm{INN}(\mathrm{String}_G).

Now the functor category

(2)[d n,P] [d^n,P]

is our configuration space

(3)conf:=[d n,P]. \mathrm{conf} := [d^n,P] \,.

On target space there is usually some nn-bundle with connection under which our d nd^n is “charged”. Denote the corresponding parallel transport by

(4)tra:PnVect. \mathrm{tra} : P \to n\mathrm{Vect} \,.

For the point particle this might be an ordinary bundle with connection. For the string it might be a gerbe with connection. For Dijkgraaf-Witten it would be a pseudo-rep of GG, for Chern-Simons a rep of INN(String G)\mathrm{INN}(\mathrm{String}_G).

We should call nVectn\mathrm{Vect} the space of phases (or the space of amplitudes)

(5)phas:=nVect. \mathrm{phas} := n\mathrm{Vect} \,.

By what might be called “general transgression”, the parallel transport on target space gives rise to one on configuration space, simply by sending any configuration

(6)d nP d^n \to P

to

(7)d nPtranVect, d^n \to P \stackrel{\mathrm{tra}}{\to} n\mathrm{Vect} \,,

so that we get an assignment

(8)[d n,P]tra *[d n,nVect]. [d^n,P] \stackrel{\mathrm{tra}_*}{\to} [d^n,n\mathrm{Vect}] \,.

A state of the system is a section of the nn-bundle on configuration space, which again is # a morphism

(9) 1 [d n,P] e [d n,nVect] tra *. \array{ & \nearrow \searrow^{1} \\ [d^n,P] &e \Downarrow & [d^n,n\mathrm{Vect}] \\ & \searrow \nearrow_{\mathrm{tra}_*} } \,.

The space of sections is hence the category

(10)sect=1Hom(1,tra *)tra * \mathrm{sect} = 1 \stackrel{\mathrm{Hom}(1,\mathrm{tra}_*)}{\to} \mathrm{tra}_*

Fine. Now the question is: how to we get an extended QFT from this data? In other words, how do we assign (transgressed) spaces of sections to parts of parameter space

(11)QFT:d n(?) QFT : d^n \to (?)

in a natural way?

There is one canonical way to do so, which uses just the given data and some abstract nonsense.

Notice that our space of sections sect\mathrm{sect} is a subcategory of the space of all morphism from configuration space to [d n,phas][d^n,\mathrm{phas}]:

(12)sectHom(conf,[d n,phas]), \mathrm{sect} \stackrel{\subset}{\to} \mathrm{Hom}(conf,[d^n,\mathrm{phas}]) \,,

hence it is an element of

(13)Hom(sect,Hom(conf,Hom(par,phas))). \mathrm{Hom} (\mathrm{sect},\mathrm{Hom}( \mathrm{conf}, \mathrm{Hom}(\mathrm{par}, \mathrm{phas}) )) \,.

This is the abstract nonsense for a very familiar statement:

A state is something

that assigns to a configuration something

that assigns to a parameter

an amplitude.

You might want to think of the point particle, where there is no parameter. Then compare this to the string, for instance, where a state is something that assigns an amplitude to every mode of a given configuration of the string. The mode number here would be the parameter.

Technically slightly trickier but conceptually much more transparent is the case where we take the parameter on the string to be literally the parameter σ\sigma which parameterizes the string as a map

(14)σγ(σ). \sigma \mapsto \gamma(\sigma) \,.

In this example the element in the Hom-space

(15)Hom(sect,Hom(conf,Hom(par,phas))) \mathrm{Hom} (\mathrm{sect},\mathrm{Hom}( \mathrm{conf}, \mathrm{Hom}(\mathrm{par}, \mathrm{phas}) ))

that we are talking about would schematically read

(16)|ψ(γ|(σγ(σ)|ψ)). |\psi\rangle \mapsto ( \langle \gamma| \mapsto ( \sigma \mapsto \langle \gamma(\sigma)| \psi\rangle ) ) \,.

A state is something that sends a configuration to something that sends a parameter to an amplitude.

The nice thing about the above setup is that it interprets this concept nn-functorially, with all data assigned to the parameter “space” category taken care of.

In particular, realizing our space of states as an element in

(17)Hom(sect,Hom(conf,Hom(par,phas))) \mathrm{Hom} (\mathrm{sect},\mathrm{Hom}( \mathrm{conf}, \mathrm{Hom}(\mathrm{par}, \mathrm{phas}) ))

allows us to canonically extract a (nn-)functor which should represent the associated extended QFT.

(18)QFT:par(?). QFT : \mathrm{par} \to (?) \,.

We simply regard the image of our space of states under the canonical equivalence

(19) Hom(sect,Hom(conf,Hom(par,phas))) Hom(par,Hom(sect,Hom(conf,phas))) \begin{aligned} & \mathrm{Hom} (\mathrm{sect},\mathrm{Hom}( \mathrm{conf}, \mathrm{Hom}(\mathrm{par}, \mathrm{phas}) )) \\ \simeq& \mathrm{Hom} (\mathrm{par},\mathrm{Hom}( \mathrm{sect}, \mathrm{Hom}( \mathrm{conf}, \mathrm{phas}) )) \end{aligned}

and thus obtain an (nn-)functor

(20)par[sect,[conf,phas]]. \mathrm{par} \to [\mathrm{sect}, [\mathrm{conf},\mathrm{phas}]] \,.

It assigns to objects of parameter space an image of the space of states in [conf,phas][\mathrm{conf},\mathrm{phas}]. It assigns to edges in parameter space natural transformations between these images, and so on.

To convince oneself that this functor is really what we are looking for, one should look at a couple of examples.

In particular, applying this to Dijkgraaf-Witten and Chern-Simons theory, we could check if that functor on the parameter space of the 3-particle (membrane) yields state spaces of an extended 3-d QFT which coincide with correlators of a 2-d QFT.

I have one consistency check for Dijkgraaf-Witten, which however might be too top secret to talk about here.

I have also some ideas about the CS-case. But I need to work them out.

Posted by: urs on November 13, 2006 5:18 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

Great stuff, Urs! I hope my class will eventually reach this height of abstraction. I’ll start by describing Rovelli’s setup in the case of a string, and showing how his “symplectic 3-form” gives, by transgression, the more familiar “symplectic 2-form” on the familiar phase space of classical string theory. Of course the students in my class are anything but familiar with these “familiar” things, so it will take a while.

I have a new student, Alex Hoffnung, who seems interested in working on this stuff.

Posted by: John Baez on November 13, 2006 5:35 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

I wrote:

We simply regard the image of our space of states under the canonical equivalence

(1) Hom(sect,Hom(conf,Hom(par,phas))) Hom(par,Hom(conf,Hom(conf,phas))) \begin{aligned} & \mathrm{Hom}(sect,\mathrm{Hom}( conf,\mathrm{Hom}(par,phas))) \\ \simeq & \mathrm{Hom}(par,\mathrm{Hom}( conf,\mathrm{Hom}(conf,phas))) \end{aligned}

I am trying to work out an example - and it almost seems to work the way I expected. But not quite. I am hoping that I am just overlooking something.

But maybe - before I waste my time trying to solve the unsolvable - I should find out if the above is really true for 2-categories.

I am working in the 3-category Gr\mathrm{Gr} of strict 2-categories, strict 2-functors between these, pseudonatural transformations and modifications between these.

For AA, BB and CC strict 2-categories, is it even true what I assumed above, that we have a canonical equivalence of 2-categories

(2)Hom Gr(A×B,C)Hom Gr(A,Hom Gr(B,C)) \mathrm{Hom}_\mathrm{Gr}(A\times B, C) \simeq \mathrm{Hom}_\mathrm{Gr}(A, \mathrm{Hom}_\mathrm{Gr}(B,C))

?

Hm, I guess one problem is that Hom Gr(B,C)\mathrm{Hom}_\mathrm{Gr}(B,C) is not strict itself, in general.

But it is biequivalent to its strictification Hom Gr(B,C) str\mathrm{Hom}_\mathrm{Gr}(B,C)_{\mathrm{str}}. So maybe

(3)Hom Gr(A×B,C)Hom Gr(A,Hom Gr(B,C) str) \mathrm{Hom}_\mathrm{Gr}(A\times B, C) \simeq \mathrm{Hom}_\mathrm{Gr}(A, \mathrm{Hom}_\mathrm{Gr}(B,C)_\mathrm{str})

?

Or what if I step outside Gr\mathrm{Gr} and work in, er, the tricategory of bicategories?

Oh dear, this is harder than I thought it would be. And what I really need is the same for 3-categories…

Posted by: urs on November 15, 2006 4:00 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

Urs wrote:

This is the abstract nonsense for a very familiar statement:

A state is something

that assigns to a configuration something

that assigns to a parameter

an amplitude.

Indeed… Reading this poetic sentence is a real treat. It envokes mystical sensations akin to what Chris Isham must have been referring to when he described his first encounter with the Gelfand-Naimark theorem:

He said that as a student, when he’d learned about this theory, he was really excited, because it completely depends on the fact that if we have a space X, we can think of any point x in X as a functional on the space of functions on X, basically defining by defining

x(f)

to be

f(x).

He said this with a laugh, but I knew what he meant, because I too had found this idea tremendously exciting when I first learned the Gelfand-Naimark theory. I guess it’s something about how what seems at first like some sort of bizarre joke can turn out to be very useful….

On the subject of Prof. Isham, I have had the occasion recently to try to understand his work on topos theory and quantum mechanics. I think there are all sorts of interesting things in there! The latest version of his ideas appears to be quite elegant.

Posted by: Bruce Bartlett on November 13, 2006 9:24 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

Bruce said he liked my Haiku

A state is something

that assigns to a configuration something

that assigns to a parameter

an amplitude.

because it reminded him of how the Haiku

A function is something

that assigns to a point

a number.

is related to

A point is something

that assigns to a function

a number.

I should maybe emphasize that this sort of transformation is precisely why I think this is interesting in the context of extended QFT.

We want to get extended QFTs that assign (nd)(n-d) categorical data to dd-dimensional parts of parameter space. Hence we want a functor from parameter space to something.

The point is that the transformation of my above Haiku achieves precisely that

An extended QFT is something

that assigns to a parameter

something that…

What sounds like a triviality in terms of this crude language turns out to be rather subtle when you plug in the nn-categorical details along the lines I sketched above #. We need to repeatedly use the canonical equivalence

(1)Hom(A×B,C)Hom(A,Hom(B,C)) \mathrm{Hom}(A \times B , C) \simeq \mathrm{Hom}(A , \mathrm{Hom}(B,C))

with everything living in homs of nn-categories for some nn. In particular, a state itself is defined to be a morphism of nn-functors. Hence we need this equivalence at the level of higher morphisms, where the standard intuition for “Currying” becomes a little shaky.

Fortunately, the abstract nonsense is there to guide us.

Posted by: urs on November 14, 2006 10:52 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

As Rovelli and Lagrange pointed out, phase space, as a vector space, can be defined covariantly as the space of histories. I prefer the antifield formulation of the covariant phase space, described in ch. 16 of Henneaux and Teitelboim (although it is slightly flawed, since the higher cohomology groups do not vanish).

However, phase space is not just a vector space; it also comes with a Poisson bracket. To define this we must identify momenta and velocities, which AFAIU requires that we introduce some gadget which picks out a time direction. Doing so breaks manifest covariance. Otherwise you get four momenta p_abcd in field theory, as Rovelli does in the beginning of section 3. Unless he somewhere singles out one of them as the physical momentum, he will probably end up with something like de Donder-Weyl theory, which is known to be incorrect.

There is no time-selecting gadget in the antifield approach, and hence no honest Poisson bracket, only an antibracket. But that is OK if you use path integrals.

We also need a preferred time to quantize canonically. In field theory, the Heisenberg algebra admits inequivalent Fock representations, and the correct choice is the Bargmann representation, where energy is bounded from below. To define energy we again need to single out time.

What IMO should be done is to elevate this time-selecting gadget, which I call “the observer’s 4-velocity”, to the status of a physical object to be quantized together with the fields. This leads to a modification of QFT but not of QM.

Posted by: Thomas Larsson on November 14, 2006 9:59 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

Thomas wrote:

—————-
What IMO should be done is to elevate this time-selecting gadget, which I call the observer’s 4-velocity; to the status of a physical object to be quantized together with the fields. This leads to a modification of QFT but not of QM.
——————

I remember having an idea about this kind of thing. Isn’t it clear that the input for both classical and quantum physics is not just Minkowski space, but a worldline gamma (the observer) inside Minkowski space?

It seems that one can do everything so much more `elegantly’ and `covariantly’ this way.

For instance : take the `time evolution’ of the metric - the problem of classical and quantum gravity. In the normal setup, someone has to, at some point, fix a `time’ direction, and this is a sad moment for everyone involved.

But if we take as input worldine + manifold, the worldline gives us a canonical way to split up the manifold into space+time.

The same thing works for quantum field theory.

You can take the whole thing much further, and talk about eg. the Dirac equation as giving a flat connection on the tangent bundle of this space, etc. `Flatness’ here means that it is relativistically invariant :

If two observers begin at the same point (with same four velocity), and then one runs around the block while the other stays put, and then they both return to the same point (with same four velocity), then they will still agree on the wave function of their particle.

This would not be true of the Schrodinger equation, etc.

When I was a physicist, I remember asking Renate Loll about this stuff. She seemed interested, but also gave me that sad look suggesting I’d missed the point somewhere, which I undoubtedly did.

Posted by: Bruce Bartlett on November 14, 2006 11:48 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

On the first page (page 33) of the notes, I don’t understand the ‘last step’ of the derivation of S(γ), involving −E(t1) + E(t0). The explanation for this is that γ maps into the subset of T*X where H(qp) = E, and this phrasing suggests that E is a constant (as is common in physics). But the ‘last step’ treats E as if it were a function of t. Shouldn’t that step involve −t1E + t0E = −E(t1 − t0) instead?

It can seem strange that the conjugate momentum for time is minus the energy. There are many ways to see that this is true, some of which appear in my youthful post on imaginary time to sci.physics. While I’m on the subject, you can remove this minus sign by making energy the coordinate and time the momentum, as in another youthful paper.

Posted by: Toby Bartels on November 14, 2006 7:03 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

That last step, the one that’s puzzling you, is a typo. I’d already noted it in the errata. For some reason Derek took what I wrote on the board:

(1)E(t 1t 0) -E(t_1 - t_0)

and wrote it as

(2)E(t 1)+E(t 0). -E(t_1) + E(t_0) .

I knew this would fool people into thinking E was a function of tt, instead of a number which happens to be multiplied by (t 1t 0)(t_1 - t_0).

Thanks for reminding us of your paper on imaginary time! The minus sign this explains has been bugging me throughout the course - and your explanation will not stop it from bugging me, since it’s pesky even if it’s morally correct. Folks should probably read your paper together with my homework problems on a spring in imaginary time and a pendulum in imaginary time - although what I’m calling “imaginary time” in those problems, you’d call “real”.

Posted by: John Baez on November 15, 2006 5:22 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

I’d already noted it in the errata.

But it’s not in the errata! Perhaps you have yet to upload a new version of the errata?

Posted by: Toby Bartels on November 16, 2006 4:28 PM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 6)

Whoops. I could mention this problem in the errata to the errata, but what the heck - I’ll just fix the errata.

You’ll now see two other irksome tiny little errors listed there, as well.

Posted by: John Baez on November 16, 2006 5:30 PM | 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: November 15, 2006 9:10 PM
Read the post Categorical Trace and Sections of 2-Transport
Weblog: The n-Category Café
Excerpt: A general concept of extended QFT and its relation to the Kapranov-Ganter 2-character.
Tracked: November 17, 2006 5:17 PM
Read the post Quantization and Cohomology (Week 5)
Weblog: The n-Category Café
Excerpt: The canonical 1-form on the cotangent bundle of a manifold, and what it does for classical mechanics.
Tracked: January 17, 2007 2:28 PM
Read the post That Shift in Dimension
Weblog: The n-Category Café
Excerpt: What makes the Kontsevich-Cattaneo-Felder theorem tick? How can it be that an n-dimensional quantum field theory is encoded in an (n+1)-dimensional one?
Tracked: August 25, 2007 2:32 AM

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