## January 13, 2007

### The First Part of the Story of Quantizing by Pushing to a Point…

#### Posted by Urs Schreiber …in which the author entertains himself by computing the space of states of a charged particle by pushing its parallel transport forward to a point. Just for fun.

Let

(1)$X$

be a space and let

(2)$\array{ V \\ \downarrow \\ X }$

be vector bundle over $X$ with connection

(3)$\nabla \,.$

Equivalently this means # that we have a locally smoothly trivializable functor

(4)$\mathrm{tra}_{(V,\nabla)} : P_1(X) \to \mathrm{Vect}$

that sends paths in $X$ to the parallel transport along them obtained from the connection $\nabla$.

Quantizing the single particle charged under this bundle with connection consists of a kinematical and of a dynamical aspect: In this first part of the story we amuse ourselves by just doing the trivial kinematics – but in a nice way.

So, let’s forget the connection immediatey by just looking at constant paths in $X$. I’ll write

(5)$\mathrm{tra}_V : \mathrm{Disc}(X) \to \mathrm{Vect}$

for the above functor restricted to constant paths, ie. to the discrete category over $X$. It does nothing but sending each point in $X$ to the vector space sitting over it – but smoothly so.

We can play the same game on the base space

(6)$\{\mathrm{pt}\}$

that consists of nothing but a single point.

A trivial rank one vector bundle over a point is a functor

(7)$I_{\mathrm{pt}} : \mathrm{Disc}(\{\mathrm{pt}\}) \to \mathrm{Vect}$

that does nothing but sending the single point to the complex numbers:

(8)$I_{\mathrm{pt}} : x \mapsto \mathbb{C} \,.$

I admit that I am presupposing a certain tolerance for fancy-looking trivialities here. But enduring these will pay off eventually.

Using the uniqe functor from $X$ to the point

(9)$p : \mathrm{Disc}(X) \to \mathrm{Disc}(\{\mathrm{pt}\})$

we can pull back the trivial vector bundle over the point to $X$. The result

(10)$I_X := p^* I_{\mathrm{pt}} : \mathrm{Disc}(X) \stackrel{p}{\to} \mathrm{Disc}(\{\mathrm{pt}\}) \stackrel{I_{\mathrm{pt}}}{\to} \mathrm{Vect}$

is the trivial rank one bundle on $X$. This functor simply sends each point of $x$ the typical fiber $\mathbb{C}$:

(11)$I_X : x \mapsto \mathbb{C} \,.$

In as far as any of this is interesting at all, it is for the following simple fact:

a morphism of functors:

(12)$e : I_X \to \mathrm{tra}_V$

is precisely a section of the vector bundle $V$: $e$ is nothing but an assignment

(13)$e : x \mapsto (e_x : \mathbb{C} \tp V_x)$

of a linear map from $\mathbb{C}$ to the fiber $V_x$ for each point $x$. That’s nothing but a choice of vector in each fiber.

So, the space of all such functor morphisms

(14)$\Gamma(V) = \mathrm{Hom}(I_x, \mathrm{tra}_V)$

from the trivial one into the one defining our vector bundle is nothing but the space of sections of $V$.

Since $\Gamma(V)$ is a vector space, and since vector bundles over the point are nothing but vector spaces, I want to think of $\Gamma(V)$ as a vector bundle over the point. So I regard it as a functor

(15)$q(\mathrm{tra}_V) := \mathrm{pt} \mapsto \Gamma(V) \,.$

On top of all these trivialities, I’ll finally allow mysef to think of $\Gamma(V)$ as morphisms from the trivial line bundle on the point into this guy:

(16)$\Gamma(V) \simeq \mathrm{Hom}(I_{\mathrm{pt}}, q(\mathrm{tra}_V)) \,.$

The upshot is that, taken together, we get the isomorphism

(17)$\mathrm{Hom}(p^* I_{\mathrm{pt}}, \mathrm{tra}_V) \simeq \mathrm{Hom}(I_{\mathrm{pt}}, q(\mathrm{tra}_V)) \,.$

If you like, you can convince yourself that this isomorphism of Hom-spaces in indeed natural in both arguments. But this means that pulling back functors from points to $X$

(18)$[\mathrm{Disc}(\{\mathrm{pt}\}),\mathrm{Vect}] \stackrel{p^*}{\to} [\mathrm{Disc}(X),\mathrm{Vect}]$

is the adjoint of taking sections

(19)$[\mathrm{Disc}(\{\mathrm{pt}\}),\mathrm{Vect}] \stackrel{q(\cdot)}{\leftarrow} [\mathrm{Disc}(X),\mathrm{Vect}] \,.$

This, in turn, says that forming the space of sections of $\mathrm{tra}_V$ is the result of pushing $\mathrm{tra}_V$ forward to a point.

Of course that’s neither new nor very deep. But part of a nice story that still needs to be told.

Posted at January 13, 2007 7:16 PM UTC

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