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marcus

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and also "An efficient algorithm for the riemannian 10j symbols"

by JD Christensen and Greg Egan gr-qc/0110045

calculations in 4D quantum geometry seem to depend on the 10j symbol and whether or not you can calculate the partition function depends on how good a 10j algorithm you have

(that includes calculating a sum-over-histories amplitude for getting from one spatial geometry to another----feynmann style quantum spatial dynamic) it seems to excite people just now---a fair number of papers (ask for links if you want)

so I want to gather together some equations and recipes for

a few of the numbers that keep surfacing in the papers Ive been reading. Physics Forums format may defeat this. I cant draw graphs!

There is a two vertex, three link graph called a "THETA" with the links labeled with integers m,n,o that sum to an even integer

and satisfy the usual condition that a,b,c non-negative

a = (m+n-o)/2

b = (n+o-m)/2

c = (o+m-n)/2

Then θ(m,n,o) = (-1)

^{a+b+c}a!b!c!(a+b+c+1)!/((a+b)!(b+c)!(a+c)!)

Seth Major gives an example θ(N,N,2) for any N.

Would you like to calculate it? Note that a=N-1, b=1, c=1.

Then there is TET which is a tetrahedron number. The 6 edges of the tetrahedron are labeled a,b,e,c,d,f

Here's where I should draw a picture but imagine a square

with the sides labeled clockwise a,b,c,d starting on the left.

So the left side is a and the top is b.

Then make the rising diagonal e and the falling diagonal f.

The vertices are where these threesomes meet:

abf

ade

bce

cdf

The pairs of opposite edges are ac, bd, ef.

These pairs are not in contact. Edge a touches all save c, and so on.

So Seth defines 4 vertex numbers

a1 = (a+d+e)/2

a2 = (b+c+e)/2

a3 = (a+b+f)/2

a4 = (c+d+f)/2

And he defines 3 edgepair numbers

b1 = (b+d+e+f)/2

b2 = (a+c+e+f)/2

b3 = (a+b+c+d)/2

Then he says N = Π (b

_{j}- a

_{i})!/(a!b!c!d!e!f!)

The product is of 12 terms, all possible i and j.

This is what computers are good for in quantum gravity obviously.

Finally there is the Tet symbol, a number written:

Tet[(a,b,e), (c,d,f)]

actually Seth arranges it so it looks like Tet of 2 row-vectors

or Tet of a 2x3 matrix. But other places I've seen it typed like I just did. So this doesnt get too long I will put the actual definition of the Tet in the next post. It uses this number N just defined.