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August 14, 2007

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 nn-functor to obtain the quantum propagation nn-functor of the extended QFT describing the nn-particle charged under the nn-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 nn-group which makes a parallel transport nn-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 nn-functor is quantized to a propagation nn-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 2\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 (x,y+1) (x1,y) (x,y) (x+1,y) (x,y1). \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)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 1 1 (3=4+1) 1 1. \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 dd dimensions we find 1s everywhere except in the center, where we get a weight 2d12d-1.

So with ψ(x,y)\psi(x,y) denoting the value of our functor at (x,y)(x,y), the colimit produces a phased set isomorphic to Uψ(x,y):= ψ(x1,y)2ψ(x,y)+ψ(x+1,y) +ψ(x,y1)2ψ(x,y)+ψ(x,y+1) +ψ(x,y). \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 ll to be given by Planck’s constant as =2l 2mt, \hbar = 2 l^2 \frac{m}{t} \,, where mm is the mass of the 1-particle and tt the time unit, then this is precisely the Taylor expansion to first order of the exponential of the lattice Laplace operator Uψ=exp|(t 22mΔ)ψ. U\psi = \exp|( \frac{t}{\hbar} \frac{\hbar^2}{2 m} \Delta ) \psi \,. (Here exp|(x):=1+x\exp|( x) := 1 + x denotes the first-order expansion of the exponential.)

If we hence take hist\mathrm{hist} to be the category of “one step histories” and pull-push our wave functions through the correspondence hist out in conf conf \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 hist hist out in out in conf conf conf. \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 NN such steps is ψ(exp|(t 22mΔ)) Nψ. \psi \mapsto \left( \exp|(\frac{t}{\hbar} \frac{\hbar^2}{2 m} \Delta) \right)^N \psi \,. In the joint limit t=1/N l 2t N \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 nn-functors). Given any nn-functor F:CD F : C \to D I say that a section of FF is a morphism into its differential (section 3.2) δF:CFDTDnCat \delta F : C \stackrel{F}{\to} D \stackrel{T D}{\to} n\mathrm{Cat} hence a morphism E:CnCat E : C \to n\mathrm{Cat} together with a transformation e:EδF. e : E \to \delta F \,. Morphisms of two such sections (E,e)(E,e)(E,e) \to (E',e') I take to be pairs consisting of a transformation f:EE f : E \to E' together with a choice of nn-equivalence efe. e' \circ f \simeq e \,. Here by nn-equivalence I mean equivalence when regarding nCatn\mathrm{Cat} as just an nn-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 nn-vector bundle associated to FF”. 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 nn-particle we’d tend to first consider a principal nn-bundle, then an nn-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 δF\delta F of the nn-functor FF extends to an (n+1)(n+1)-functor curv F\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 nn-category of sections, not an (n+1)(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)

Posted at August 14, 2007 5:38 PM UTC

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2 Comments & 3 Trackbacks

Re: The Canonical 1-Particle, Part II

Any idea how do QFT infinities, i.e. anomalies and the need for renormalization, arise in your scheme?

Posted by: Thomas Larsson on August 15, 2007 1:04 PM | Permalink | Reply to this

Re: The Canonical 1-Particle, Part II

Any idea how do QFT infinities, i.e. anomalies and the need for renormalization, arise in your scheme?

An idea, yes.

If we accept the idea that the path integral may come from a categorical colimit, with something like the Leinster-measure on categories playing the role of the path integral measure…

…then it appears to be striking that evaluating the “integral over the Leinster measure” which is called the “Euler characteristic of a category” involves, when that category is not finite, precisely the kind of divergent sums and resummation techniques which one runs into in quantum field theory.

This was discussed in Return of the Euler Characteristic of a Category, where Tom Leinster’s latest paper The Euler characteristic of a category as the sum of a divergent series was the topic.

Remarkably, Tim Silverman here on the Café had pioneered the converse approach: applying QFT techniques to get a handle on the a priori ill defined Euler characteristic of certain categories. See this, this and this entry by Tim Silverman (all three of them quite amazing), and maybe also Bruce Bartlett’s remark concerning this.

So, yes, that makes me think that if we do for true QFT what I am trying to do here for the toy example of the particle on the lattice, then we will need to do certain colimits over “infinite” categories (of histories) which become meaningful (only) if certain resummation techniques are used, of the kind which are familiar from QFT.

Posted by: Urs Schreiber on August 15, 2007 2:05 PM | Permalink | Reply to this
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