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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 X

be a space and let

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

be vector bundle over XX with connection

(3). \nabla \,.

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

(4)tra (V,):P 1(X)Vect \mathrm{tra}_{(V,\nabla)} : P_1(X) \to \mathrm{Vect}

that sends paths in XX 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 XX. I’ll write

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

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

We can play the same game on the base space

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

that consists of nothing but a single point.

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

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

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

(8)I pt:x. 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 XX to the point

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

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

(10)I X:=p *I pt:Disc(X)pDisc({pt})I ptVect 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 XX. This functor simply sends each point of xx the typical fiber \mathbb{C}:

(11)I X:x. 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 Xtra V e : I_X \to \mathrm{tra}_V

is precisely a section of the vector bundle VV: ee is nothing but an assignment

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

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

So, the space of all such functor morphisms

(14)Γ(V)=Hom(I x,tra V) \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 VV.

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

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

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

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

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

(17)Hom(p *I pt,tra V)Hom(I pt,q(tra V)). \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 XX

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

is the adjoint of taking sections

(19)[Disc({pt}),Vect]q()[Disc(X),Vect]. [\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 tra V\mathrm{tra}_V is the result of pushing tra V\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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