The n-Category Café

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 $d$-dimensional QFT would assign to a $(n \lt d)$-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 $d^n$ be a category addressed as parameter space

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

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

Now the functor category

is our configuration space

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

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 $G$, for Chern-Simons a rep of $\mathrm{INN}(\mathrm{String}_G)$.

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

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

to

so that we get an assignment

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

The space of sections is hence the category

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

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

hence it is an element of

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 $\sigma$ which parameterizes the string as a map

In this example the element in the Hom-space

that we are talking about would schematically read

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 $n$-functorially, with all data assigned to the parameter “space” category taken care of.

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

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

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

and thus obtain an ($n$-)functor

It assigns to objects of parameter space an image of the space of states in $[\mathrm{conf},\mathrm{phas}]$. 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.

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.

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.

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.

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(q, p) = 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.