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September 14, 2006

Quantum n-Transport

Posted by Urs Schreiber

This is what reading Freed has done to me:

I think we should make manifest the following identifications of familiar concepts in quantum theory with functorial notions:

- system/model -
σmodel: world volume field configuration target space exponentiated action phases charged particle: 1Cob Riem γ P 1(X) tra Vect WZW: 2Cob conf Σ P 2(G) tra 2 2Vect general: parameter space field configuration configuration space exponentiated action phases Chern-Simons: 3Cob V P 3(BG) tra 3 3Vect \array{ \mathbf{\sigma-\text{model:}} && \text{world volume} &\stackrel{\text{field configuration}}{\to}& \text{target space} &\stackrel{\text{exponentiated action}}{\to}& \text{phases} \\ \\ \mathit{\text{charged particle:}} && \href{ } {1\mathrm{Cob}_\mathrm{Riem}} &\stackrel{\gamma}{\to}& P_1(X) & \stackrel{ \href{} { \mathrm{tra}_{\nabla}} }{ \to } & \mathrm{Vect} \\ \\ \href{} {\mathit{\text{WZW:}}} && 2\mathrm{Cob}_\mathrm{conf} &\stackrel{\Sigma}{\to}& P_2( G ) & \stackrel{ \href{ }{ \mathrm{tra}_{\nabla_2} } }{\to} & \href{} { 2\mathrm{Vect} } \\ \\ \mathbf{\text{general:}} && \text{parameter space} &\stackrel{\text{field configuration}}{\to}& \text{configuration space} &\stackrel{\text{exponentiated action}}{\to}& \text{phases} \\ \\ \href{ } { \mathit{\text{Chern-Simons:}} } && \href{ } {3\mathrm{Cob}} &\stackrel{V}{\to}& P_3(B G) &\stackrel{\mathrm{tra}_{\nabla_3}}{\to}& 3\mathrm{Vect} }

- quantization -
general: parameter space integral of exp. action over field conf. amplitudes charged particle: 1Cob γγ *tra Hilb \array{ \mathbf{\text{general:}} && \text{parameter space} &\stackrel{\text{integral of exp. action over field conf.}}{\to}& \text{ amplitudes } \\ \\ \mathit{\text{charged particle:}} && 1\mathrm{Cob} &\stackrel{\bigoplus_\substack{\gamma} \gamma^*\mathrm{tra}}{\to}& \mathrm{Hilb} }

The main issue is to set up the category of morphisms nCobnVectn\mathrm{Cob} \to n\mathrm{Vect} in a way such that the coproduct

(1) γγ *tra \bigoplus_{\substack{\gamma}} \gamma^*\mathrm{tra}

is the nn-functorial version of the path integral. This is supposed to be an operation which swallows an nn-vector nn-transport on a “configuration space” P n(X)P_n(X), together with a collection of field configurations nCobP n(X)n\mathrm{Cob} \to P_n(X) and spits out an nn-QFT nn-functor:

(2)({Φ:nCobP n(X)},P n(X)tranVect)DΦtra(ϕ)= ΦΦ *tra. (\{\Phi : n\mathrm{Cob} \to P_n(X)\},\; P_n(X)\stackrel{\mathrm{tra}}{\to} n\mathrm{Vect}) \mapsto \int D\Phi \; \mathrm{tra}(\phi) = \oplus_{\substack{\Phi}} \Phi^*\mathrm{tra} \,.

I think it works like this:

Path integral over a vector transport 1-functor:

I take the liberty of slightly adapting the definition of the various categories involved, just to avoid some technical issues that are irrelevant for the main idea I want to express.

So consider this:

Instead of using the category of 1-dimensional Riemannian cobordisms, I’ll here use the graph

(3)nn+1n+2n+3. \cdots \bullet \stackrel{n}{\to} \bullet \stackrel{n+1}{\to} \bullet \stackrel{n+2}{\to} \bullet \stackrel{n+3}{\to} \bullet \cdots \,.

Even though it is abuse of notation, I feel like calling this graph

(4)1Cob, 1\mathrm{Cob} \,,

because this graph will play the role of the time line, or the worldline of a non-relativistic particle.

For XX some finite set, let P 1(X)P_1(X) be groupoid freely generated from the free directed graph over XX. So morphisms here are sequences of elements of XX, like x 1x 2x 3x_1 \to x_2 \to x_3, and reversing an edge is inverting the corresponding morphism, (xyx)=Id x(x \to y \to x) = \mathrm{Id}_x.

We want to couple our particle to a gauge field on XX and then find its quantum dynamics.

Let EXE \to X be a vector bundle with connection \nabla on XX. (You should think actually of a complex line bundle, but I won’t need this assumption.) For me, this is a parallel transport functor

(5)tra:P 1(X)Vect. \mathrm{tra} : P_1(X) \to \Vect \,.

This is our “action functional” for a particle coupled to a gauge field.

I take a morphism of graphs

(6)γ:1CobP 1(X) \gamma : 1\mathrm{Cob} \to P_1(X)

to be a map that sends sequences of edges to sequences of edges, like this

(7)γ: n n+1 x y z. \gamma \;\;:\;\; \array{ \bullet \\ n \downarrow\; \\ \bullet \\ n+1\downarrow\;\;\;\;\;\;\; \\ \bullet } \;\,\mapsto\;\; \array{ x \\ \downarrow \\ y \\ \downarrow \\ z } \,.

This is a path traced out in XX, a possible configuration of our “field” (namely the embedding field from the worldline into the target space) on a small piece of worldline.

Composing with our action tra\mathrm{tra}, we find the generalized phase associated with this field configuration

(8)γ *tra:(n)(xγ(n)y)E xtra(γ(n))E y. \gamma^*\mathrm{tra} : (\bullet \stackrel{n}{\to} \bullet) \mapsto (x \stackrel{\gamma(n)}{\to} y) \mapsto E_{x} \stackrel{\mathrm{tra}(\gamma(n))}{\to} E_{y} \,.

Notice how γ *tra\gamma^*\mathrm{tra} naturally extends to a functor from the graph category associated to 1Cob1\mathrm{Cob} to Vect\mathrm{Vect}. (compare Anders Kock’s description of parallel transport #).

Now, the crux of path integral quantization is to “sum” all phases over all possible field configurations.

In our setup everything is categorical and “sum” cannot mean anything but a coproduct-like object. “Field configuration” means “functor γ\gamma” and “phase” means “image of γ *tra\gamma^* \mathrm{tra}”.

Hence “sum of phases over field configurations” must be some morphism, which I’ll denote

(9) γγ *tra. \bigoplus_{\substack{\gamma}} \gamma^* \mathrm{tra} \,.

such that all other such functors have morphisms into it

(10)γ itraf i γγ *tra. \gamma_i\mathrm{tra} \stackrel{f_i}{\to} \bigoplus_{\substack{\gamma}} \gamma^* \mathrm{tra} \,.

and such that it is the universal object with this property.

In order for this to yield the desired result, one has to find a suitable notion of morphism of graph maps. Something that does seem to work, is the following.

Let the morphisms of the category

(11)[1Cob,Vect] gr, [1\mathrm{Cob},\mathrm{Vect}]_\mathrm{gr} \,,

be given on each graph edge by something like a naturality square, but with one arrow reversed, like this:

(12)(fρg)f(x) f(x,y) f(y) ρ(x) ρ˜(y) g(x) g(x,y) g(y). (f \stackrel{\rho}{\to} g) \;\;\Leftrightarrow\;\; \array{ f(x) &\stackrel{f(x,y)}{\to}& f(y) \\ \rho(x)\downarrow\;\; && \;\; \uparrow \tilde \rho(y) \\ g(x) & \stackrel{g(x,y)}{\to} & g(y) } \,.

This does not make sense for functors, but it does make sense for graph maps with codomain a category.


(13) γγ *tra \bigoplus_\gamma \gamma^*\mathrm{tra}

be the object in this category which has morphisms into it from every γ i *tra\gamma_i^*\mathrm{tra}, and such that for any other object QQ with this propert there is a morphism γγ *traQ \bigoplus_\gamma \gamma^*\mathrm{tra} \to Q.

In order to compute this object, we have to consider commuting diagrams of the form

(14)E x tra(γ i(n)) E y f i(x) f˜ i(y) ( γγ *tra)(x) ( γγ *tra)(n) ( γγ *tra)(y) \array{ E_{x} & \stackrel{ \mathrm{tra}(\gamma_i(n)) }{\to} & E_{y} \\ f_i(x)\downarrow \;\; && \;\; \uparrow \tilde f_i(y) \\ (\bigoplus_{\substack{\gamma}} \gamma^*\mathrm{tra})(x) &\stackrel{ (\bigoplus_{\substack{\gamma}}\gamma^*\mathrm{tra})(n) }{\to}& (\bigoplus_{\substack{\gamma}}\gamma^*\mathrm{tra})(y) }

in Vect\mathrm{Vect}.

I think (but you should check it) that this γγ *tra\oplus_{\substack{\gamma}}\gamma^*\mathrm{tra} is precisely the strange kind of sum that we want.


(15)( γγ *tra)(n)=( xXE x)U(δt)( xXE x), \left(\bigoplus_{\substack{\gamma}}\gamma^*\mathrm{tra}\right)\left(n \right) = \left( \bigoplus_{\substack{x \in X}} E_x \right) \stackrel{U(\delta t)}{\to} \left( \bigoplus_{\substack{x \in X}} E_x \right) \,,

where U(δt)U(\delta t) is the |X|×|X||X| \times |X|-matrix whose (x,y)(x,y)-entry is the linear map

(16)U(δt) x,y=E xtra(xy)E y. U(\delta t)_{x,y} = E_x \stackrel{\mathrm{tra}(x\to y)}{\to} E_y \,.

This is indeed the quantum mechnical propagator in the limit where δt\delta t is a tiny period of time.

As we have seen before, we should think of

(17)H:= xXE x H := \bigoplus_{\substack{x \in X}} E_x

as the space of sections of the bundle EXE \to X, hence in fact as the Hilbert space of states of our charged quantum particle.

(In a more detailed derivation we would start with a hermitian vector bundle, which carries a scalar product on each of its fibers. This would then induce the expected scalar product on our Hilbert space of states. Square integrability of sections is one of the technicalities that we are ignoring here, since |X||X| is assumed to be finite.)

Moreover, U(δt)U(\delta t) is evidently that linear endomorphism of this Hilbert space which is obtained by using the classical action along every possible path (of elementary length) between any two points in XX.

Accordingly, the power

(18)U(nδt)=(U(δt)) n U(n \delta t) = (U(\delta t))^n

is the linear operator on HH which is obtained by a path integral over all paths of total parameter length nδtn \delta t.

But that’s precisely the propagator that we are after:

(19) γγ *tra : 1Cob Vect (nn+2) ( xXE x)( γγ *tra)(n)( xXE x)( γγ *tra)(n+1)( xXE x) =HU(2δt)H. \array{ \bigoplus_{\substack{\gamma}} \gamma^*\mathrm{tra} &:& 1\mathrm{Cob} &\to& \mathrm{Vect} \\ && (\bullet \stackrel{n}{\to} \bullet \stackrel{n+2}{\to} \bullet) &\mapsto& \left( \bigoplus_{\substack{x \in X}} E_x \right) \stackrel{\left(\bigoplus_{\substack{\gamma}} \gamma^*\mathrm{tra}\right)(n)}{\to} \left( \bigoplus_{\substack{x \in X}} E_x \right) \stackrel{\left(\bigoplus_{\substack{\gamma}} \gamma^*\mathrm{tra}\right)(n+1)}{\to} \left( \bigoplus_{\substack{x \in X}} E_x \right) \\ &&&& = H \stackrel{U(2\delta t)}{\to} H \,. }

Voilà, the path integral in terms of something like coproducts of transport functors.

Of course the point of this exercise is to go from n=1n=1 to n>1n \gt 1.

Here is a pdf with some related notes.

Posted at September 14, 2006 11:49 AM UTC

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propagation on n-cobordisms from transport on n-paths

I find myself feverishly thinking (more than is good for me, it seems) about the general idea motivating the above, which is this:

There should be a nice canonical construction that reads in an nn-vector nn-transport tra:P n(X)nVect\mathrm{tra} : P_n(X) \to n\mathrm{Vect} from globular nn-paths to nn-vector spaces and spits out a QFT propagator nCobnHilbn\mathrm{Cob} \to n\mathrm{Hilb} describing the quantum mechanics of an nn-particle coupled to tra\mathrm{tra}.

What I wrote before was an attempt to understand the path integral over all field configurations as a coproduct of a bunch of functors, since Freed’s observation # suggested that the ordinary path integral in nn-dimensional QFT is just the top component of an entire list of length nn of various “sums”.

While I already tried to simplify a lot by making everything in sight finite, it seems that the familiar unwieldiness of the path integral still infects the construction.

Usually, the Hamiltonian formulation of the quantum theory is much easier to handle. Is there maybe a nice canonical construction that produces a QFT in Hamiltonian formulation from the input datum of an nn-transport functor?

That’s what I am currently thinking about. One nice trick one can invoke is a generalization of the fact that the Hamiltonian of the charged 1-particle is just one part of the covariant Laplace-Beltrami operator

(1)Δ tra=d tra d tra:Ω 0(E)Ω 0(E), \Delta_\mathrm{tra} = d_\mathrm{tra}^\dagger d_\mathrm{tra} : \Omega^0(E) \to \Omega^0(E) \,,

where d trad_\mathrm{tra} is the obvious covariant exterior derivative associated to tra\mathrm{tra}.

At least in principle this nicely generalizes to n>1n \gt 1 as long as our notion of nn-vector space is such that (n1)(n-1)-morphisms of nn-vector spaces are ordinary vector spaces (with extra structure), such that nn-morphisms are ordinary linear maps.

For instance, let Cat Vect\mathrm{Cat}_\mathrm{Vect} be the 2-category of categories that are equipped with a faithful functor to Vect\mathrm{Vect}, and let a 2-vector space be a Cat Vect\mathrm{Cat}_\mathrm{Vect}-enriched category. This is for instance the case for my favorite example, Bim(Vect)\Bim(\mathrm{Vect}), which is what we need for strings coupled to the Kalb-Ramond field.

So here the basic idea is to pass to the space of loops and find, due to the above assumption on 2Vect2\mathrm{Vect}, a vector bundle over that loop space. By an analogous construction as for n=1n=1, our transport 2-functor tra\mathrm{tra} then provides a Hamiltonian for propagation along paths of loops.

But I want to fomulate this idea in a nice, crisp, canonical way.

Maybe I found one nice solution. I’d be grateful for any comments. Maybe someone has seen something similar somewhere.

Let Disc()\mathrm{Disc}(\bullet) be the discrete nn-category on a single object. Say the category of field configurations is the weak nn-functor category

(2)F=[Disc(),P n(X)] F = [\mathrm{Disc}(\bullet),P_n(X)]

from the single object to “configuration spaceP n(X)P_n(X).

The nice thing about this definition is that automatically

- objects of FF are (closed) (n1)(n-1)-paths in P n(X)P_n(X)

- morphisms of FF are indeed nn-dimensional cobordisms between these objects (due to the nature of pseudonatural transformations of nn-functors, see the pictures here)

- higher order morphisms are various gauge transformations of these, related to the existence of various base points whose choice is a gauge freedom.

Better yet, instead of Disc()\mathrm{Disc}(\bullet) we could take some quiver, like (abc)(a \to b \to c) and find open/closed cobordisms attached to various branes # (at least if the rest of the program goes through as hoped for).

That way of looking at the situation might be helpful. We can now compose any field configuration γ[Disc(),P n(X)]\gamma \in [\mathrm{Disc}(\bullet),P_n(X)] with our “exponentiated actiontra\mathrm{tra} to obtain the generalized “phase”

(3)Disc()γP n(X)tranVect. \mathrm{Disc}(\bullet) \stackrel{\gamma}{\to} P_n(X) \stackrel{\mathrm{tra}}{\to} n\mathrm{Vect} \,.

In particular, by our assumption on the definition of nVectn\mathrm{Vect} and by the nature of objects in [Disc(),P n(X)][\mathrm{Disc}(\bullet),P_n(X)], this does associate

- a vector space to each field configuration

- and a morphism of vector spaces to each (embedded) cobordism between field configurations .

Hence we are, up to some slight nn-categorical details, back in the situation of 1-dimensional quantum mechanics. Instead of propagation on target space, we now have propagation on something like the (n1)(n-1)-torus space of target space (e.g. loop space for n=2n=2).

It should again be possible to canonically obtain from tra\mathrm{tra} a covariant Laplace-Beltrami operator (now on the (n1)(n-1)-torus space of target space) and hence finally a 1-functor

(4)1CobHilb. 1\mathrm{Cob} \to \mathrm{Hilb} \,.

I am trying to spell out more details of this program. But I thought I should check if maybe somebody has a helpful comment on the general idea.

Posted by: urs on September 19, 2006 12:37 PM | Permalink | Reply to this
Read the post Differential n-Geometry
Weblog: The n-Category Café
Excerpt: A quest for arrow-theoretic differential geometry.
Tracked: September 20, 2006 9:17 PM
Read the post The Globular Extended QFT of the Charged n-Particle: Definition
Weblog: The n-Category Café
Excerpt: Turning a classical parallel transport functor on target space into a quantum propagation functor on parameter space.
Tracked: January 24, 2007 8:07 PM

Re: Quantum n-Transport

It is helpful to have the following picture of the propagator in mind:

Posted by: urs on February 8, 2007 11:48 AM | Permalink | Reply to this

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