Re: Lattice Gauge Theory

Tim S <[EMAIL PROTECTED]> wrote
> on 15/8/03 10:45 pm, Derek Wise at [EMAIL PROTECTED] wrote:
> 
> > [...about path integrals on a lattice]
> 
> OK, I want to learn more about this stuff, so I'm going to try to revive
> this thread, which has gotta be more productive than the Superstring Wars
> being fought over in some other threads.

Hi Tim! :)

Good luck! I also tried without much luck to revive this thread.
Perhaps we can keep it alive just amongst ourselves. Urs let the cat
out of the bag that we have been secretly working together to conquer
the world... err I mean develop a algebraic/combinatorial version of
differential geometry on what may be called a "directed n-graph",
which smells a lot like an n-category. An especially nice case of
which might be called an "n-diamond complex."

With any luck Derek and Professor Baez may make an appearance here and
there if we can make things interesting enough :)

> Now off onto our first side road. I want to understand the classical theory
> on this lattice a bit better before we start thinking about quantisation
> (or, with Gerald Westendorp, thermodynamics).
> 
> I've taken an irrational dislike to the way we've written our action:
> 
> Sum_p F(p)^2
> 
> where p are our plaquettes and F is our curvature dA.
> 
> Aesthetically, I'd like one of those factors of F(p) to be something sort
> of dual to F rather than F itself. Let's call it *F, because it vaguely
> reminds us of something by that name that we saw somewhere else a while
> ago. 

Of course this is the same objection I had when the thread first began
:) However, Professor Baez managed to convince me that to worry about
this Hodge podge business would only add unnecessary complications to
the formulations. The proof of the complications is in that Herculean
effort it must have taken you to draw those ascii diagrams. Try doing
that with a simplicial complex! :)
There is a lot of material you can learn with this unphysical metric
and then we can always go back and insert a more physically motivated
metric later on.

> These dual quantities live on the Poincare dual lattice. We get this
> by replacing each 2-cell by a 0-cell placed at its centre, each 1-cell by a
> 1-cell crossing it, and each 0-cell by a 2-cell surrounding it -- or, on an
> arbtrary n-complex, replacing each p-cell by an n-p cell. Then we can also
> dualise our p-forms to get n-p forms on the dual lattice. On a square
> lattice, it looks a bit like this:

This approach to Hodge duality is well known at least in my field
(applied computational EM). When you first start playing around with
it, results come cheaply enough that you feel you are really onto
something. I gaurantee that it is just an illusion. When you dig
deeper, this Poincare dual <-> Hodge dual is going to get you into
trouble. My vote is to follow the principle of "KISS" :)
 
> (Our original lattice has edges delineated with | and - and vertices marked
> +, while the dual lattice has edges marked x and vertices marked O.)
> 
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
> -----+-------------------------+-----------------------------+--------
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x This is     |              x              |
>      |           x Plaquette n |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
> xxxxx|xxxxxxxxxxxOxxxxxxxxxxxxx|xxxxxxxxxxxxxxOxxxxxxxxxxxxxx|xxxxxxxx
>      |           x \           |              x              |
>      |           x  \          |              x              |
>      |           x  Vertex dual|              x  Edge dual   |
>      |           x  to         |              x  to E        |
>      |           x  Plaquette n|   Edge E     x <-           |
>      |           x             |      |       x              |
>      |           x             |     \ /      x              |
> -----+-------------------------+-----------------------------+--------
>      |           x             |\             x              |
>      |           x             | \            x              |
>      |           x             |  This vertex x              |
>      |           x             |  has as dual x              |
>      |           x             |  the x-and-O x              |
>      |           x             |  plaquette   x              |
>      |           x             |  around it   x              |
> xxxxx|xxxxxxxxxxxOxxxxxxxxxxxxx|xxxxxxxxxxxxxxOxxxxxxxxxxxxxx|xxxxxxxx
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
> -----+-------------------------+-----------------------------+--------
>      |           x             |              x              |
>      |           x             |              x              |
>      |           x             |              x              |
> 

I decided not to delete this because it hurts to think of the effort
to draw it :)

> We can now form the product (*F(p) /\ F(p)). 

Uh huh. Good luck making this product well defined :) Let's not build
a house of cards :)

> It lives on some sort of local
> product lattice between the 2-cells of the original lattice and the
> corresponding 0-cells of its dual. However, we won't be needing to think
> about this for a bit, because we're going to be taking a shortcut across a
> muddy field,

Danger :) Is your product going to end up being associative? Graded
commutative? Does it satisfy the graded Leibniz rule?

> by skipping over the bit where we integrate the Lagrangian and
> calculate the variation in the action, and going straight to the
> (classical) field equations, so we can try to hook up to something
> familiar.

If you REALLY want some sense of duality that is not going to cause
too much of a headache, I suggest taking a look at what I outlined in
a previous post in this thread

http://groups.google.com/groups?q=g:thl3725983303d&dq=&hl=en&lr=&ie=UTF-8&selm=3fa8470f.0307311003.12f45e69%40posting.google.com&rnum=11

Then again, we might just be better off forgetting it like the master
says :) At least for the time being.

[snip]
 
> Next time, I think I'll ramble a little about scalar matter fields, U(1)
> reprentations and the action principle. Then, depending on how I feel and
> what I've been working on in the meantime, I might talk about Brownian
> motion and heat baths, or I might talk about quantisation.

Like I said in another prior post in this thread, I am always very
happy to hear what anyone has to say about this subject that is so
close to my heart. I'm looking forward to learning more about whatever
you have time to say :)

Best regards,
Eric