I recently stumbled onto something that could potentially open up
mountains of research/reading and before starting the journey I was hoping
someone might say a few words summarizing what this *-product is that pops
up in noncommutative geometry. It looks suspiciously like something I have
cooked up on my own. If that is the case, then I have reinvented another
wheel :)

Does the *-product have anything at all to do with composition of strings
or paths. A quick lanl search in addition to a quick spr search turned up
tons of hits regarding NCG and string theory. I saw Aaron mention
something about string algebra and Prof Baez mention it in regard to
products on phase space.

What exactly is a string algebra? Is there some operation in string theory
that composes strings/paths? A path algebra is a very obvious
construction and I reinvented it on my own and I even called it "path
algebra". Is it at all related? Could path algebra be a kind of discrete
string algebra? A path algebra is simply a directed graph with product of
paths in the graph defined by concatenation when such a concatenation
makes sense. When it doesn't, i.e. the end of the first path doesn't
correspond to the beginning of the next path, then the product is zero.

It would be really fascinating to me if the work I have been doing on path
algebras is remotely related to string theory. I already know that my
stuff is related to NCG, and I have heard rumors that NCG and string
theory were related, so it would be quite cool if string theory came into
the picture. I'm fairly certain I would be the first electrical engineer
to write his thesis on string theory :)

Anyway, I'd really appreciate some words of wisdom from people who know
about this stuff.

(f * g)(z) = e^(i theta_{uv} d^x_u d^y_v) f(x) g(y) | x=y=z

In bigger words, this can be thought of as the deformation quantization
of R^{2n} wrt the symplectic form theta.

[...]

[...]

Note that this only works nicely for open strings. Doing something
similar for closed strings in a nice way would be very nice. (There
exist bosonic closed string field theories, but they're not tremendously
aesthetically appealing and it's questionabl whether they actually
contain more information than just the perturbative expansion. One of
the really interesting discoveries in the past few years is that this
open string field theory defined by Witten really does contain more than
just the perturbation expansion, and it can be used to actually
calculate things like the potential for the open string tachyon in
bosonic string theory. See the archives for longer posts on this
subject.)

Thanks for your reply.

Now, in LQG you start out with a manifold and then define a spin network
on it, do a bunch of fancy mathematics and when all is said and done, you
discover that you never really needed the manifold at all to begin with.
You could have started with this purely combinatorial spin network.

Is it possible something similar to this could happen in string theory?
That is the impression I get when I read about Witten product of strings.
Like you said, you first split this into two halves and then concatenate
the middle pieces like

(aL,aR) * (bL,bR) = (aL,bR)

Now if you distill out what the pure algebra of this is, you can almost
look at (aL,aR) as an abstract 1-simplex {{aL},{aR},{aR,aL}}. Notice how
the abstract 1-simplex is already in a sense composed of two "halves".

I'm wondering if there is some relation between string theory and
abstract graph theory where instead of continuously parameterized strings,
you have abstract sets like these abstract 1-simplices. Then the string
algebra seems closely related to the path algebra defined on a quiver.

I spent a couple of days in a swirl reading up on representation theory of
quivers and there is just tons of material on the subject. In fact, quite
a bit of the recent material has been in relation to string theory. I
won't pretend to understand the implications of it all, but at least I
know that the connections are there.

I might just be spewing nonsense, but from what I have seen so far is that
working with these abstract spaces automatically gets you this "background
independence" that is held so dear in this place :) Putting coordinates on
these abstract spaces is highly artificial but can be done. When you do
so, it is pretty straight forward to see that what 'labels' you choose to
lay on the space really is of no consequence to any physics.

Even general relatvity is not completely background independent because
you do have to specify a manifold before you can do anything. Then the
metric and everything is else free to wiggle as much as it likes but you
are still stuck with the manifold you originally chose. This is similar to
these abstract spaces. Ultimately, the physics will depend on the
underlying abstract space you began with, but all other quantities defined
ON the abstract space are free to vary as they wish. Even if I am not 100%
correct about my claims regarding GR, since I am no expert (not even a
novice) on the subject, whatever loopholes there are in GR for getting
away from the original manifold you started with, there is probably a
corresponding loophole for the abstract space approach. If topology can
change in GR, there is nothing saying topology of a graph cannot change
either.

Could graph theory come to the rescue here?

[snip]

[...]

exp(i m.x) exp(i n.x) = exp(i (m+n).x)

can be illustrated by vector addition of the Fourier modes. If
exp(i m.x) is the vector m and exp(i n.x) is the vector n, then
exp(i (m+n).x) is of course the vector sum m+n.

     m
   ----->
    \   |      
 m+n \  |n      
      \ |      
       \|
        V

The star-product is

exp(i m.x) * exp(i n.x) = exp(ih A(m,n)) exp(i (m+n).x)

where A(m,n) is the area enclosed by the vectors m, n and m+n and
h is Planck's constant/2 pi. When you add three vectors m, n and r,
the enclosed area can be split in two ways. This is somewhat difficult
to draw in ascii, but it is easy to convince oneself that

A(m,n) + A(m+n,r) = A(m,n+r) + A(n,r).

This gives associativity of the star product. I am pretty sure that
the Moyal product is the unique associative deformation of the
ordinary commutative product.

That is pretty amazing :)

How on earth does it work out like that? Any ideas?

It is really fascinating that the area shows up that like. Do you know
what that looks like to me?

It reminds me of holonomies!!

You are parallel transporting a vector once along n, then along m. This
is related to parallel transporting the vector along m+n by the curvature,
which is proportional to the area enclosed! :) (at least for a small
enough triangle and/or constant curvature)

It's beautiful! :)

It also helps to confirm the fact that what I've been doing is somehow
related. If only I were smarter! :)

It might even hint of a relation to spin networks and LQG.

It would be even more suggestive if:

exp[i m.x] * exp[i n.x] * exp[-i(m+n).x] = exp[ih A(m,n)]

which I imagine is true. Then you are truly parallel transporting around a
closed loop.

Well, instead of imagining, let's go ahead and see :) I've already sunk
enough time into this post :)

First, note that

exp[i m.x] * exp[-i m.x]
= exp[ih A(m,-m)] exp[i (m-m).x]
= 1

since A(m,-m) denotes a degenerate triangle of zero area. (Just like the
loop group.)

Then,

[exp(i m.x) * exp(i n.x)] * exp[-i (m+n).x]
= exp[ih A(m,n)] exp[i (m+n).x] exp[-i (m+n).x]
= exp[ih A(m,n)]

Voila!

Ok. That wasn't so bad :)

If you look at it as parallel transporting vectors around triangles, the
associativity becomes pretty obvious :)

exp(i m.x) * [exp(i n.x) * exp(i r.x)]
= exp[ih A(n,r)] exp(i m.x) * exp[i (n+r).x]
= exp[ih (A(n,r) + A(m,n+r))] exp[i (m+n+r).x]

[exp(i m.x) * exp(i n.x)] * exp(i r.x)
= exp[ih A(m,n)] exp[i (m+n).x] * exp(i r.x)
= exp[ih (A(m,n) + A(m+n,r))] exp[i (m+n+r).x]

Therefore, like you said, * is associative if

A(m,n) + A(m+n,r) = A(m,n+r) + A(n,r)

which it obviously is. However, I suspect that these A's are not simply
numbers but that they have an orientation associated with them such that
the overlapping areas with opposite orientation cancel leaving only the
hourglass figure (see below).

This can be seen (hopefully) by considering the figure below.

         m
    +--------->
    |\       /|
    | \     / |
    |  \   /  |
    |   \ /   |
m+n |    /    | n+r
    |   / \   |
    | n/   \m+n+r
    | /     \ |
    |/       \V
    V--------->
         r

The term A(m,n) corresponds to transporting around the loop:

         m
    +--------->
    |        /
    |A(m,n) /
    | (o)  /     Note: (o) designates clockwise orientation
    |     /            (+) designates counter clockwise orientation
m+n |    /
    |   /
    |  /n
    | /
    |/
    V

The term A(m+n,r) corresponds to transporting around the loop:

    +
    |\
    | \
    |  \
    |   \
m+n |    \ m+n+r
    |     \
    | (+)  \
    |       \
    |A(m+n,r)\
    V--------->
         r

When you compose the two, i.e. A(m,n) + A(m+n,r), you get cancellations
and the result is equivalent to transporting around the hourglass loop:

         m
    +--------->
     \       /
      \ (o) /
       \   /
        \ /
         /
        / \
      n/   \m+n+r
      / (+) \
     /       \
    V--------->
         r

Note, this should be pretty obvious once you have the parallel transport
interpretation at your disposal because the hourglass loop may be written
as

exp(i m.x) * exp(i n.x) * exp(i r.x) * exp[-i (m+n+r).x].

This adds to my conviction that A(m,n) are oriented areas.

Similarly, the term A(m,n+r) corresponds to transporting around the loop:

         m
    +--------->
     \A(m,n+r)|
      \       |
       \  (o) |
        \     |
   m+n+r \    | n+r
          \   |
           \  |
            \ |
             \|
              V

Finally, the term A(n,r) corresponds to transporting around the loop:

              +
             /|
            / |
           /  |
          /   |
       n /    | n+r
        /     |
       /  (+) |
      / A(n,r)|
     /        V
    V--------->
         r

Again, when you compose the two, i.e. A(m,n+r) + A(n,r) you get
cancellations and the result is equivalent to transporting around the
hourglass loop:

         m
    +--------->
     \       /
      \ (o) /
       \   /
        \ /
         /
        / \
      n/   \m+n+r
      / (+) \
     /       \
    V--------->
         r

PHEW!! ASCII art is tough :)

This is another way to view the associativity of *

Nifty, eh?! :)

Thanks for a beautiful post! Wouldn't it be really fascinating if the
*-product were related to curvature for the more general cases as well?

When I first saw the *-product it reminded me of holonomies, which is why
I thought it was related to my stuff. Now that you explicitly show the
area popping up, which obviously (to me anyway) signifies curvature and
holonomies, then I am nearly convinced :)

I had a lot of fun writing this post! :)

Eric

PS: Hmm... with a bit of afterthought, this parallel transport idea is
extremely awesome. Quantization can be thought of phase space with
(constant?) curvature! :) Add that to the list of reinvented wheels :)

PPS: This algebra works equally well for open paths (open string?) and
closed loops (closed strings?).

I believe it is the right way to look at the product; just an
averaging of areas.

When you have a charged particle in a thin film in a constant transverse
magnetic field, it picks up a phase when you parallel translate it.
When you parallel translate it around the triangle enclosed by vectors
m, n, and m+n, it picks up a phase given by your formula - but with some
extra factors of the magnetic field and the charge floating around in
the exp(ih A(m,n)) term.

For a few more details and also some references, try:

http://math.ucr.edu/home/baez/braids/node8.html