The n-Category Café

The Canonical 1-Particle, Part II

Posted by Urs Schreiber

As I possibly mentioned before, I am interested in understanding how to canonically (in the category-theoretic sense!) quantize a parallel transport $n$-functor to obtain the quantum propagation $n$-functor of the extended QFT describing the $n$-particle charged under the $n$-functor.

I want to understand this in particular in order to systematically understand how Chern-Simons theory arises as a 3-functorial theory, because we have some idea about how that will allow to understand 2-dimensional CFT from first principles.

The first step in this program is to find the right structure $n$-group which makes a parallel transport $n$-functor taking values in it reproduce the desired classical action functional. (This step for Chern-Simons I discussed here.)

The next step is to undertand in detail how a classical parallel transport $n$-functor is quantized to a propagation $n$-functor. That is, how to proceed along the third edge of the cube.

This involves finding the right arrow-theory for “taking sections” and/or (following Freed) “doing the path integral”.

I am testing my ideas on this on the simple case of the 1-particle, i.e. of ordinary quantum mechanics. In The canonical 1-Particle it was pointed out that the path integral, including its measure, might be conceivable entirely in terms of a colimit involving the classical parallel transport functor.

The main observation there was this: suppose, for definiteness, we are looking at the particle propagating on the 2-dimensional plane, which we model by the category of the graph $\mathbb{Z}^2$. Just to get our hands on some concrete and tractable example.

Then, decreeing that the space of “one step histories”, i.e. of paths the particle may trace out in one time interval, is that of paths of at most one edge length, one finds that the path integral over paths of unit temporal length is the colimit of some functor over the category $$\array{ && (x,y+1) \\ &&\uparrow \\ (x-1,y) & \leftarrow & (x,y) & \rightarrow & (x+1,y) \\ && \downarrow \\ && (x,y-1) } \,.$$ One of the important assumptions which I made in part one was that our functor here, which is supposed to be the pullback of some “wave function” to the space of these histories, is actually that: a functor instead of a function, taking values in something like John Baez’s phased sets (a set equipped with a map to a group “of phases”, usually $U(1)$).

My main point here is to not only to justify this assumption, but to actually derive it using arrow-theoretic differential theory. But before getting into that, let me finish saying what the main point of part one had been:

namely, if we assume our functor on the above category to take values in something like sets, and to be “free” in that it sends all morphisms to monomorphisms, then its colimit over the above category is goverened by the Leinster-measure $$\array{ && 1 \\ &&\uparrow \\ 1 & \leftarrow & (-3 = -4 + 1) & \rightarrow & 1 \\ && \downarrow \\ && 1 } \,.$$ This just says, in words, that the colimit will take the disjoint union of all the phased sets but then get rid of the overcounting which is induced by the fact that the set in the center sits inside all the other sets but must only be counted once.

It is clear how the analogous situation looks like in arbitrary dimensions: in $d$ dimensions we find 1s everywhere except in the center, where we get a weight $2d-1$.

So with $\psi(x,y)$ denoting the value of our functor at $(x,y)$, the colimit produces a phased set isomorphic to $$\begin{aligned} U\psi(x,y) := &\;\;\psi(x-1,y) - 2\psi(x,y) + \psi(x+1,y) \\ &+ \psi(x,y-1) - 2\psi(x,y) + \psi(x,y+1) \\ & \;\; + \psi(x,y) \end{aligned} \,.$$

If we take our lattice spacing scale $l$ to be given by Planck’s constant as $$\hbar = 2 l^2 \frac{m}{t} \,,$$ where $m$ is the mass of the 1-particle and $t$ the time unit, then this is precisely the Taylor expansion to first order of the exponential of the lattice Laplace operator $$U\psi = \exp|( \frac{t}{\hbar} \frac{\hbar^2}{2 m} \Delta ) \psi \,.$$ (Here $\exp|( x) := 1 + x$ denotes the first-order expansion of the exponential.)

If we hence take $\mathrm{hist}$ to be the category of “one step histories” and pull-push our wave functions through the correspondence $$\array{ && \mathrm{hist} \\ & \multiscripts{^{\mathrm{out}}}{\swarrow}{} && \searrow^{\mathrm{in}} \\ \mathrm{conf} &&&& \mathrm{conf} }$$ this induces the action of the usual (euclidean) quantum mechanical propagator to first order in our time unit.

To propagate the 1-particle over more than one time step, we continue pull-pushing through the correspondences $$\array{ && \mathrm{hist} && && \mathrm{hist} && \\ & \multiscripts{^{\mathrm{out}}}{\swarrow}{} && \searrow^{\mathrm{in}} & & \multiscripts{^{\mathrm{out}}}{\swarrow}{} && \searrow^{\mathrm{in}} & \\ \mathrm{conf} &&&& \mathrm{conf} &&&& \mathrm{conf} } \,.$$ The map induced by $N$ such steps is $$\psi \mapsto \left( \exp|(\frac{t}{\hbar} \frac{\hbar^2}{2 m} \Delta) \right)^N \psi \,.$$ In the joint limit $$\begin{aligned} & t = 1/N \\ & l^2 \propto t \\ & N \to \infty \end{aligned}$$ this approaches the ordinary continuous (euclidean) propagator.

I had said all this before, if maybe not as coherently. But it deserves to be said again.

Here I will now discuss why indeed wave functions are functors with values in phased sets, using just fundamental (if you wish) arrow-theoretic differential theory.

The main point is this notion of section. Let me give the definition and then say a word about it:

Definition (sections on $n$-functors). Given any $n$-functor $$F : C \to D$$ I say that a section of $F$ is a morphism into its differential (section 3.2) $$\delta F : C \stackrel{F}{\to} D \stackrel{T D}{\to} n\mathrm{Cat}$$ hence a morphism $$E : C \to n\mathrm{Cat}$$ together with a transformation $$e : E \to \delta F \,.$$ Morphisms of two such sections $(E,e) \to (E',e')$ I take to be pairs consisting of a transformation $$f : E \to E'$$ together with a choice of $n$-equivalence $$e' \circ f \simeq e \,.$$ Here by $n$-equivalence I mean equivalence when regarding $n\mathrm{Cat}$ as just an $n$-category.

A couple of remarks

1) We will see that this notion of section actually models the concept of a “section of something like an $n$-vector bundle associated to $F$”. In particular, there always exists a canonical global section, akin to the 0-section of a vector bundle. This is a crucial aspect of what I am trying to get at: usually, in quantization of a charged $n$-particle we’d tend to first consider a principal $n$-bundle, then an $n$-vector bundle associated with that. Not so here: the passage to the associated bundle emerges automatically, albeit in an unfamiliar form: we get something like vector spaces the way John Baez conceived them in the Unfinished Tale of Groupoidification.

2) Notice that passing to the differential of the functor introduces an increase in the categorical dimension (recall that the differential $\delta F$ of the $n$-functor $F$ extends to an $(n+1)$-functor $\mathrm{curv}_F$.) That’s (I am claiming here) the curious shift in dimension which one expects to see.

3) Notice at the same time that I am restricting the morphisms of sections to be insensitive to this shift in dimension in that they form just an $n$-category of sections, not an $(n+1)$-category. That’s important. I should talk about that in more detail, but maybe not right now.

To continue reading, you’d now have to “leave the room” and follow this link:

On the Canonical Quantization of the Charged 1-Particle )
(pdf slides)