Homework Help: 2-3 Pachner move and Biedenharn-Elliot identity

1. May 22, 2008

KOSS

1. The problem statement, all variables and given/known data
Show that up to "fudge factors" (such as a few theta-nets an a loop) the 2-3 Pachner move is just the Biedenhard-Elliot identity between 6j symbols.

If you go to the Quantum Gravity Seminar notes of Baez and Alvarez, you can see this problem here: :math.ucr.edu/home/baez/qg-fall2000/QGravity/QGravity.pdf: (sorry can't make that an url since I don't have the 15 post privilege yet) (anyway, it's Ch.21, Ex 16 of that pdf).

2. Relevant equations
Roughly (imagine this "English" translated into diagrams), given the known relation between the 6j-symbols and the spin network tetrahedron (with a few theta-nets and a loop), this isomorphism,

("the 3-tet-net"*Oj)/(theta-net*theta-net*theta-net) = "the 2-tet-net"/theta-net

is a diagrammatic representation of Biedenhard-Elliot, namely,
$\sum_k \left\{\begin{array}{ccc}a&b&k\\ c&f&e\end{array}\right\}\left\{\begin{array}{ccc}a&k&i\\ g&d&f\end{array}\right\}\left\{\begin{array}{ccc}b&c&j\\ d&i&k\end{array}\right\} = \left\{\begin{array}{ccc}e&c&j\\ d&g&f\end{array}\right\}\left\{\begin{array}{ccc}a&b&i\\ j&g&e\end{array}\right\}$

3. The attempt at a solution
If I start with the Biedenhard-Elliot identity and work out all the triangular faces of the equivalent the faces of the tet-net I can obtain the above diagrammatic isomorphism with cancellation of two theta-nets and two loops on each side. But my labels are not the same as written in the text by Alvarez (cited to above). also, what confuses me is that the Alvarez-Baez diagram equation seems like it should have a further theta-net cancelled in the denominators on each side. Is this just a misleading typo, or do theta-nets not "cancel" that naively?

I'd appreciate any help on this, and since I'm not a student and just an amateur dabbler in physics if anyone wants to take the opportunity to expound on at length about this problem and similar puzzles then please feel free.

Last edited: May 22, 2008