Re: local flow of Euler characteristic

David Hillman wrote:


Suppose just for fun that space is a simplicial complex. (By space I mean some spacelike hypersurface.) Its Euler characteristic is zero, 

I think it is zero for a hypertorus. I remember deriving this some
time ago. I should redo it here, so I' ll not loose my notes:

Suppose you have an n-dimensional hypercubical lattice. Vertices
are 0-cells, edges are 1-cells, faces are 2-cells, going up to
n-cells.

Suppose the hypercube has m vertices per dimension. So the number
of vertices is m^n.
To determine the number of edges, we have to decide on what to
do with the boundaries. The easiest is to build a hypertorus, ie.
connect index 1 to index m for each dimension. In that case the number of edges is 

 V_1 = m^n B(n,2).

B(n,2) is the binomial coefficient.
For the number of q-cells (q<=n):

 V_q = m^n B(n,q)

So we have the nice property that the V_q(m) are just the entries
of Pascal's triangle, multiplied by m^n!

The alternating sum of any row of Pascal's triangle is zero.
(Because (1-1)^m = 0)

So, the generalized Euler characteristic of an n-dimensional
hypertorus is zero. However, if we used not a hypertorus, but
for example an open hypercube, we would get something more complicated. For the 2 dimensional case:

 V_0 - V_1 + V_2 = m^2 - 2m(m-1) + (m-1)^2
                 = 1.

This checks with a single square (m=1): The number of edges
equals the number of vertices, and there is 1 face.

Hey, I just realize that the outcome is always 1, for any
dimensional hypercube. If you expand

 1 =  (m + (1-m)) ^n

you get just the right alternating sum for the Euler characteristic. 

So OK, the (generalized) Euler characteristic of an n-dimensional
cell complex seems to be well-behaved.

if we take some connected piece of space (a set of tetrahedrons, say, where there's a path from any one to any other through adjoining triangles) then it has a Euler characteristic (V_0 - V_1 + V_2 - V_3) which may be any integer. If we now, again just for fun, assume some kind of local deterministic law that determines how this spacelike hypersurface evolves into another spacelike hypersurface, there will be a local flow of Euler characteristic from the first surface to the second.

To see the meaning of of "evolving", maybe we could add time as an extra dimension of our lattice,
producing a space-time lattice. On my web site I show
that a space-time cell complex works just as well as a "space only"
cell complex. The trick is to use negative impedance on the cells
for the time-like direction.


My question is: might this conserved quantity be related to any that we know and love? Charge, perhaps? (And, if charge: would charge conjugation then have something to do with going to the Poincare dual?)


This is a bit related to something I'v been working on lately.
I call it "Cut functions". But maybe I'll think of a better name later.

[ I use the words 'voltage' and 'current' here and there for
convenience instead of 'scalar field' etc. ]

Anyway, the idea of a Cut function is a bit like the lattice analog
of a p-form, in the case that the p-form is an exterior product
of 2 other ones.

The p-forms (I call them n-chunks) that are easy to make on a lattice
are :
a. potentials defined on n-cells. For example the scalar potential
on the vertices, the vector potential on the edges.

b. From the potential of an n-cell, we can build a form on an
(n+1)-cell, that is the "gradient" of the potential. For example,
a voltage gradient over an edge, the curl of A around a face.
This would be the analog of taking the exterior derivative.

c. Using Ohm's law, we can make a dual of the (n+1)-chunks. In the case
of edges, we convert voltage gradients to currents. In the case
of electromagnetic fields, we convert (B,E) fields to (H,D) fields.
The 2 sides of each dual multiply to produce a "chunk" of Action
on that n-cell.

So far, so good.
It gets a bit trickier to see what happens to things like Poynting
vectors, that are the exterior product of "simple" forms. The simplest example of a non-simple form is perhaps the product of
a scalar (phi) with a current (d phi). In the continuous case,
we have the convience that phi varies smoothly, so that there
is only one value for

phi (d phi)

at each point. However, on a lattice, we have to choose from which
vertex we choose our voltage to multiply with our current. So at
first sight, there seems to be a problem. This also seems to get worse
as we increase the grade of our forms. If we want to multiply
a quantity on a face with one on an edge, we have 4 edges to
choose from (in the case of a square lattice).

But once we have decided to completely "cut" a part of the lattice
so that it becomes disjoint from the rest, we suddenly see a
meaningful way to form an exterior product. As an example, consider 3D space (no time), with electric fields on the
edges, and magnetic fields on the faces. The Poynting vector
would be the exterior product ExB. The meaning of this product
is that it gives the energy flowing out of a region, if
we integrate it over the surface bounding that region. Now this
is something that works also for a lattice. If we carefully
cut the lattice into 2 regions A and B, such that all edges
are either completely in A or completely in B, then some
Faces will be completely in A, some completely in B, and
some will be at the "cut", linking A with B. Now if we sum
all combinations of Faces that are in cut(A,B) and Edges that
are in A, multiplying the B-field of the face with the
E-field on the edges, we get the net power flowing outof region A!
So, in the analog case, to use a p-form when integrating
over a p-brane, we only need to specify where the p-brane
is in terms of coordinates. But on the lattice, we also have to specify how our surface ("cut") cuts up our region,
and on which side of the cut we want to integrate.

I think these cut-functions correspond to conserved
quantities like probability current, energy flux, etc.

The Euler characteristic itself need not be conserved in time,
because time is just a direction in our lattice, and we can
mess things up in the time direction if we want to.

I realize this didn't answer you question. Hope I can get back
on this soon.


Gerard



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Relevant Pages

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