Thetas, tets, sixjays and such

[marcus]

got interested in Seth Major's "spin network primer" gr-qc/9905020

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 I've been reading. Physics Forums format may defeat this. I can't 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+ca!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 doesn't get too long I will put the actual definition of the Tet in the next post. It uses this number N just defined.

 

[marcus]

got interested in Seth Major's "spin network primer" gr-qc/9905020

and also "An efficient algorithm for the riemannian 10j symbols"
by JD Christensen and Greg Egan gr-qc/0110045

A continuation of previous post:

Here is how Tet[(a,b,e), (c,d,f)] is defined.
let m = min{a_i}
M = max{b_j}

Tet[(a,b,e), (c,d,f)] = N Σ (-1)^s (s+1)!/(Π(s - a_i)!Π(b_j - s)!)

Where the sum is over all integers s in the range [m,M]. These are the s for which the sum makes sense because the factorial is always of a non-negative number.

The sixjay symbol is now quick to define, using Tet, but I have
to go so will get back and define 6j symbol later. Seem amazing that nature should bother with these numbers? Running into them all over the place, especially 6j and 10j.
And Tet is used to define 10j as well.

 

[marcus]

Definition of the 6j symbol

Seth Major's "Spin Network Primer" has this Appendix A called
"Loops, Thetas, Tets, and All That" which I've been following. For the 10j I will have to go to another paper----Christensen and Egan. But now to finish up with the 6j:

It is written as a 2x3 matrix in curly braces, so in two rows, but here I write it with both parts on one row separated by comma. The definition of the 6j symbol uses both Tet and the theta function:

{(a,b,i), (c,d,j)} = Tet[(a,b,i), (c,d,j)] Δ_i/(θ(a,d,i)θ(b,c,i))

there are some identities that 6j symbols satisfy. You can find them in Seth Major's article.

Δ_n stands for the simple loop labeled by the integer n
and also for the number (-1)ⁿ(n+1)

Seth Major's integer labels (Penrose "colours") are twice the conventional half-integer spin labels used, e.g., by Christensen and Egan. While following Major's primer it seemed best to stick with his usage.

More later.

 

[marcus]

my take on it is that these combinatorial functions somehow do what the Einstein equation of General Relativity does on large scale, except they do it on a microscopic scale. They are how spacetime appears to work microscopically.

what corresponds in 4D to the spin network in 3D is a spin network with everything extended by one dimension.
Where we had a vertex we know have a line
Where we had a labeled edge, we have a labeled triangle or some other patch of area.

the 6j symbol is a numerical function of 6 spins because it is about the basic chunk of 3D space, the tetrahedron, and the tetrahedron has 6 edges----each one labeled with a spin.

In 4D the corresponding thing is a "4-simplex".
this is like a tetrahedron except that it has 5 vertices.
As for triangles, it has 10-----each one labeled with a spin.

There is the old combinatorial "5-choose-3" which equals 10.
Each choice of 3 vertices gives a 2-face of the the 4-simplex.

So people write the 10j symbol as a "pentagram"----the 5 pointed star in a pentagon that children often know how to draw.
The pentagram has 10 lines and each one is labeled with a spin, or as Penrose and others might sometimes do it, with an integer n that is twice the conventional half-integer spin (n+1 being the dimension of vectorspace acted on by the group)

All these group representation things constantly going on behind us and in the bushes. Never a dull moment. "Irrep" is emerging as slang for irreducible representation.

Christensen and Egan say: ...the 10j symbol is a function taking ten input spins and producing a complex number. It is at the heart of the calculation of the partition function, and thus an algorithm for calculating the 10j symbols is quite important..."

they describe 4 different approaches
" direct contraction, staged contraction, 3cut, and 2cut"
The time is order j⁹ or more for any of these if programmed naively. They do something clever with the 2cut and get it down to order j⁵ where j is the upper limit of the spins-----it can make the difference between 5 minutes (on a
300MHz microprocessor) and 30 years.

Still, 5 minutes seems like a bit of a wait for just one labeled pentagram. One spinfoam history must involve computing many 4-simplices.

the formula for the 10j is on page 8 of christensen/egan
and it doesn't LOOK all that bad----Tet is used twice and the Theta function is used twice: producing a matrix----then 5 of those matrices are multiplied together.

They put their C++ program online for anyone to download.
the paper is
www.arxiv.org/gr-qc/0110045[/URL]

trying to improve on this and speed it up still further might be a good computerscience research topic.

More recently, Baez, Christensen, Halford, Tsang got time on a supercomputer to calculate some spinfoams,
[PLAIN]www.arxiv.org/gr-qc/0202017[/URL]
in that paper (a year later) they were talking milliseconds or at most seconds to compute a 10j symbol
I could imagine computer science grad students getting competitive about spinfoam calculations. Will be interesting to see what happens.

 

[marcus]

A short exerpt from the Baez et al paper in which they did a lot of computing of partition functions for spinfoam sum-over-histories
"Spin Foam Models of Riemannian Quantum Gravity"

<<Spin foam models are an attempt to describe the geometry of spacetime in a way that takes quantum theory into account from the very start. A spin foam is a 2-dimensional analogue of a Feynman diagram. Abstractly, a Feynman diagram can be thought of as a graph with edges labelled by group representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional cell complex with polygonal faces labelled by representations and edges labelled by intertwining operators. Like Feynman diagrams, spin foams serve as a basis of ‘quantum histories’: the actual time evolution of the system is described by a linear combination of these quantum histories, weighted by
certain amplitudes. Feynman diagrams are 1-dimensional because they describe quantum histories of collections of point particles; spin foams are 2-dimensional because in loop quantum gravity, the gravitational field is described not in terms of point particles but 1-dimensional ‘spin networks’.
An ordinary quantum field theory provides a recipe for computing the amplitude for any Feynman diagram in terms of amplitudes for edges and vertices. Similarly, a spin foam model consists of a recipe to compute an amplitude for any spin foam as a product of face, edge and vertex amplitudes. The partition function in a spin foam model is computed as a sum or integral of these spin foam amplitudes. Using suitably weighted sums and normalizing by dividing by the partition function, one can also compute expectation values of observables. A number of spin foam models have been developed for both Lorentzian and Riemannian quantum gravity. By ‘Lorentzian quantum gravity’, we mean any quantum theory whose partition function is, at least morally speaking, given by

&int; e^iS ,

where S is the Einstein–Hilbert action for a Lorentzian metric on spacetime, or some closely related action. If all goes well, a theory of this sort should reduce in a suitable limit to the classical Einstein equations for Lorentzian metrics. ‘Riemannian quantum
gravity’ is the same sort of thing, but for Riemannian metrics.>>

I'm thinking that being told about spin foams in 2003 is a bit like being told about Feynman diagrams and path integrals in 1960 or earlier, when? 1955? My impression is that the QED calculations based on Feynman diagrams are comparably hairy but the alternatives are likely to be worse.

Anyway, people are starting to calculate spin foams and at each step there seems to be a 10j symbol----a little pentagonal surprise nature had in store for us----that may take several milliseconds or even seconds to compute.