Re: Lattice Gauge Theory

"Urs Schreiber" <[EMAIL PROTECTED]> wrote:
> "Tim S" <[EMAIL PROTECTED]> schrieb:
> > on 15/8/03 10:45 pm, Derek Wise at [EMAIL PROTECTED] wrote:
> 
> I'll take the opportunity of Tim's very nice discussion to mention some
> related things.

Urs has been itching to discuss this stuff out in the open for months
now. I guess he just couldn't wait for me to finish the public version
of our notes :)

Since he already released the flood gates, we might as well go all the
way :)

> As it turns out, it is in fact possible to construct the true Hodge star
> operator on (hyper-)cubic lattices with arbitrary metric. Using this it is
> possible to write the EM action on hypercubic lattices in the familiar form.

Ack! :)

By now it is well known (I suppose) that you can construct a nice
graded differential algebra on a directed graph with a map

d: Omega^p -> Omega^{p+1}

satisfying

d^2 = 0

and

d(AB) = (dA)B + (-1)^|A| A(dB).

See for example (one of my all time favorite papers):

http://www.arxiv.org/abs/gr-qc/9808023
Discrete Riemannian Geometry
A. Dimakis, F. Muller-Hoissen

The way I like to think about this is to relate it to elementary
algebraic topology where the graded algebra is the space of co-chains
and the product is cup product (defined on the cochain level) and of
course d being the coboundary map.

However, if you really think about it, cup product defined on the
cochain level is a lot like concatenation of paths (or their duals!).

Therefore, I prefer to call the graded vector space 

P = (+)_r P_r

the space of paths instead of chains. An element of the dual space is
then a "copath", i.e a linear functional on the space of paths in a
directed n-graph. So that if A is a p-"copath" then

<A|i0...ip> 

is the value of A evaluated on the path |i0...ip>.

But what IS the space of paths?!?!

Oh yeah, and what is a directed n-graph?!?!

Ok, let me back up...

A directed n-graph is about the only thing it could be. First of all,
what is an ordinary directed graph?

A directed graph G (for my purposes) consists of countable sets G_0
and G_1 together with maps

s_1,t_1:G_1->G_0.

The elements of G_0 are our "nodes" and the elements of G_1 are our
"directed edges". Given a directed edge |g_1> in G_1, then

s|g_1>

is the "source node" of |g_1> and 

t|g_1> 

is the "target node" of |g_1>.

The space of 0-paths (nodes) is that consisting of formal linear
combinations of nodes and the space of 1-paths (directed edges) is
that consisting of formal linear combinations of directed edges.

Ok. Now a direct n-graph G (for my purposes) consists of n+1 countable
sets

G_p, p in {0,...,n}

together with maps

s_p,t_p: G_p->G_{p-1}

satisfying

s_{p-1} t_p = t_{p-1} s_p,

which essentially says the maps commute and we truncate this by
setting s_0|g_0> = t_0|g_0> = 0, the empty set, for all |g_0> in G_0.

The elements of G_0 and G_1 may again be thought of as nodes and
directed edges, respectively, while the elements of G_p will in
general be referred to as "simple p-paths."

With the intentional flavor of n-catefories, a simple p-path |g_p> may
be thought of as a "p-dimensional arrow" from the source s_p|g_p> to
the target t_p|g_p>. We will eventually interpret the arrows as
providing a sense of time, as in the causal poset approach of Sorkin.

We then define the space of p-paths in the obvious way.

A "boundary map" @_p:P_p(G)->P_{p-1}(G) may be defined on the space of
p-paths on a directed n-graph G via

@_p := t_p + (-1)^p s_p.

The boundary map is not, in general, nilpotent, because

@_{p-1} @_p = t_{p-1} t_p - s_{p-1} s_p

or if we drop the subscripts

@^2 = t^2 - s^2.

Because "the boundary of a boundary is zero" is one of the most
beautiful things in mathematical physics, I can't help but to think it
holds a special place. For this reason, I think of the space of
p-paths as being "pre-geometric." It seems that anything that is
geometrically or topologically realizable should satisfy @^2 = 0.

Therefore, we let the space of a "p-chains" C_p(G) on a directed
n-graph to be that subspace of "p-paths" for which @^2 IS zero, i.e.

|c_p> in C_p(G)  <==>  @^2|c_p> = 0.

For example, consider the 2-paths

|ijk>   -->   s|ijk> := |ij>, t|ijk> := |jk>
|jki>   -->   s|jki> := |jk>, t|jki> := |ki>
|kij>   -->   s|kij> := |ki>, t|kij> := |ij>

and 1-paths

|ij>   -->   s|ij> := |i>, t|ij> := |j>
|jk>   -->   s|jk> := |j>, t|jk> := |k>
|ki>   -->   s|ki> := |k>, t|ki> := |i>.

None of the individual 2-paths are 2-chains because they do not
satisfy @^2 = 0, however, the linear combination

|triangle> := |ijk> + |jki> + |kij>

IS a 2-chain, i.e. it is geometrically realizable. Check it :)

With this definition, clearly the boundary map is closed on the space
of chains, i.e.

@: C_p(G) -> C_{p-1}(G).

Phew!!

Now we have our vector space of paths and a boundary map. The dual
space is the space of copaths.

The product on the dual space can now be defined as

<AB|g_{p+q}> := <A|s^q(g_{p+q})><B|t^p(g_{p+q}>

where A in P^p and B in P^q.

[Note: Urs and I are still arm wrestling over the notation for all
this stuff :)]

The coboundary is defined as usual as

<dA|S> = <A|@S>,

which is supposed to make you think of Stokes theorem :)

=====================================================
Exercise: Prove that

d(AB) = (dA)B + (-1)^|A| A(dB)

for all p-paths A and q-paths B even though d^2 != 0.
=====================================================

As you may guess, the space of p-cochains C^p is the subspace of
p-copaths for which d^2 = 0, i.e.

A in C^p  <==>  d^2A = 0.

Phew^2 !!

If you are interested enough to have read this far, then it might be
worth pointing out that all this is done on an ARBITRARY DIRECTED
N-GRAPH! :)

> Eric Forgy and I have been working on this lately and because it is of some
> interest with respect to this thread I note that a pre-preprint version of
> our notes can be found at http://www-stud.uni-essen.de/~sb0264/p4.pdf . 

Note that the "pre"-preprint was not a typographical error. It really
is a rough sketch even before the preprint is done :)

> On pp. 30 of this text the Hodge star operator on discrete spaces is discussed
> in general and its specific realization on (hyper-)cubic graphs is analyzed
> in detail. 

Ack!! :)

As I said, all of this works for arbitrary graphs. Pictorially, you
might imagine a bunch of paths as "weaving" its way through some
abstract space. However, something especially nice happens if you have
precisely n edges directed both away from each node and toward each
node. Kind of like a conservation of flux lines (which reminds me a
little bit about the U(1) version of loop QG, i.e. loop
electromagnetism :)). THIS is what Urs means by a "hyper cubic graph"!
At each node, it may topological look like a "cubic grid", but it need
not look cubic globally. Very much like a manifold looks locally like
R^n. The particular class of n-graphs for which we worked out the
Hodge star looks locally cubic, at least topologically. The use of the
word "cubic" is also something we arm wrestle over because "cube"
makes me think of a geometrical cube, when what he really means is a
topological cube, which doesn't necessarily have to look geometrically
anything like a cube. Oh well :) The class of directed n-graphs for
which each node has n edges directed both toward and away from each
node serves as a discrete (not "topologically discrete") kind of
manifold.

Hmm... since I brought up "topological discreteness" I might as well
mention that the topologies we put on these directed n-graphs is NOT
the discrete topology. It is more like the topology of a poset. In
fact, the directed n-graphs are very much like (and possible can be
special cases of) Sorkin's finitary topological spaces. Then again,
since we have finitary topologies that are not discrete, this implies
that the topologies we deal with are not Hausdorff!! :) I've mentioned
this a few times over the years and I still find it kind of
interesting. The idea of non-Hausdorff topologies in physics also
comes up in the idea of charges/fermions actually being the
identification of two distinct points in spacetime (making spacetime
non-Hausdorff). I forget who came up with that idea, but Professor
Baez recently brought that up again :) This kind of thing seems
natural on a directed n-graph. Anyway :)

> Volume forms, integration, and everything needed to write down
> the EM action on the lattice with arbitrary (and in particular non-flat)
> background metric as familiar from the continuum is discussed in section 4.

I'll just clarify again that all of these "mimetic" analogues of
continuum objects are defined on quite general directed n-graphs that
can have exotic topologies.

> This approach does not refer to the dual lattice for defining Hodge duals
> but instead proceeds in the spirit of noncommutative geometry by
> representing every object in discrete differential geometry as an operator
> on a suitable Hilbert space. An inner product (in the pseudo-Riemannian
> case) or scalar product (in the Riemannian case) on this Hilbert space then
> induces a notion of metric on the discrete space and the Hodge dual can be
> formulated in terms of operator products and operator adjoints.

Beauty is in the eye of the beholder, but I personally find this to be
very beautiful :)

> Implicitly the discussion has focused on using timelike and spacelike edges.
> Why not use lightlike ones?
> 
> We were kind of surprised to find that the non-commutative-geometry-like
> formulation of differential geometry on discrete spaces singles out a metric
> on (hyper-)cubic lattices with respect to which _all_ edges are _lightlike_
> (pp. 46). A (hyper-)cubic complex with such a metric we call an "n-diamond
> complex" and it turns out that such diamonds enjoy all sorts of nice
> properties. See below.

Again, an n-diamond complex is kind of like a discrete version of
Minkowski space. Not to say that it is Lorentz invariant, but rather
that it has a GLOBAL coordinate patch. Everything in the pre-preprint
can be extended to more exotic spaces that look locally like an
n-diamond complex.

> > Next, I think we should demonstrate conservation of charge.
> 
> When a fully "mimetic" formulation of discrete geometry is available, all
> results such as this charge conservation automatically carry over from the
> continuum. By "mimetic" one means (e.g.
> http://www.math.unm.edu/~stanly/mimetic.html
> http://math.unm.edu/~stanly/mimetic/contmech.html) a discrete framework in
> which all the familiar algebraic relations such as Stokes' theorem and
> various identities involving the Hodge star hold without lattice
> corrections.

Beauty :)

> > Now, lets make waves!
>  [...]
> > HOWEVER, the discretised wave equation tends to have horrible numerical
> > instabilities, so I'm not going to try giving an example.
> 
> I am not an expert on the general case of waves on discrete spaces, but I
> think that on diamond complexes, where all edges are lightlike, the
> discretized wave equation actually gives the exact result (along the
> preferred lattice directions), since the waves can propagate happily along
> the lightlike edges. 

Right, which means that there WILL be numerical "dispersion error"
along directions that are not necessarily aligned with edges. I just
wanted to clarify that it is only exact for edges that are laid out in
a "straight" line, which won't really happen in general. This
corresponds to the fact that there is a "magic" discretization of the
(1+1)d wave equation that is exact, but there is no such "magic"
formulation for (p+1)d wave equations for p > 1.

> The Laplace-Beltrami operator for n-diamond complexes
> is worked out on p. 45 and it clearly has all such waves in its kernel.
> 
> Furthermore hypercubic complexes have the advantage that the (discrete)
> exterior bundle over them does decompose as the product of two (discrete)
> spinor bundles just as in the continuum case (p 47). (This is not true for
> non-hypercubic discrete spaces.) Therefore this formalism also allows to
> write down Dirac-Kaehler actions and equations (pp. 56).  (Some of the
> notation regarding spinors and forms is not explained in the above file but
> in appendix A of http://xxx.lanl.gov/abs/hep-th/0311064 .)

Yep. It has been great fun finally working with someone (Urs) who has
some chance to make my half baked ideas precise :) I'm working hard on
the "pre"print sequel to the "pre pre" print :) Who knows if it will
ever make its way to an actual "print"? :)

Best regards,
Eric