The 6j Symbol

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The 6j Symbol Math, Physics, and Asymptotics
Analee Miranda Dr. John Baez Department of
Mathematics
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• Wigner 6j Symbols
• 6j symbols simplify the calculations involving
  the coupling of angular momentum in many particle
  systems.
• This is especially useful in calculations
  involving atomic spectroscopy.
• Atomic Spectroscopy is a valuable tool in
  research laboratories.

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So how do we calculate this 6j Symbol?
There are many different methods that scientists
utilize to calculate the 6j Symbol.
Direct Calculation
Asymptotics
Tetrahedral Networks
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Direct Calculation
Where f (t) is described below and must iterate
over all integers for which the factorials in f
(t) all have nonnegative arguments
And where the triangle coefficient formula
calculates the triplets.
Eric W. Weisstein. "Wigner 6j-Symbol." From
MathWorld--A Wolfram Web Resource.
http//mathworld.wolfram.com/Wigner6j-Symbol.html
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Tetrahedral Networks
By using the properties of a tetrahedron and the
properties of the 6j symbols, we can calculate
the 6j-symbols through the following formula

Where TET(a,b,c,d,e,f) can be calculated using a
method that combines algebra, geometry, topology
and knot theory.
?f (-1)2q 2q 1 and T(x,y,z)
(-1)xyzxy-z!x-yz!y-xz!xyz1!
2x!2y!2z!
Cited From The Classical and Quantum 6j-Symbols.
Carter, J. Scott Flath, Daniel E. Saito,
Masahico. Princeton University Press 1995
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Tetrahedral Networks

• TET(a,b,c,d,e,f) is actually pretty fun to
  calculate.
• You must first draw a tetrahedron and label each
  edge.

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Tetrahedral Networks

• Immediately add each triplet. If a triple adds up
  to an odd number or if it does not satisfy the
  triangle inequality
• a b c
• then the answer is zero.
• If the first test is non-zero, then we can
  continue by using Functional Analysis, Group and
  Representation Theory, OR a diagrammatical
  method. I preferred the latter since I could
  handle the concepts without the graduate level
  knowledge.

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Tetrahedral Networks

• Going over the specifics of finding TET would
  take too long so an example will hopefully
  suffice.
• We want to find TET(1/2, 1/2, 1/2, 1/2,0,0)

1/2
1/2
The zeros can be viewed as areas without an edge
and with a little manipulation the tetrahedron
turns into
-2
0
1/2
0
1/2
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Tetrahedral Networks

• What I did was twist the remaining edges as if
  they were wire. This twisting of the edges has
  special rules for commuting and composing. For
  example

If you look at the picture long enough, you begin
to see that this is an equivalence relation that
essentially gives you the difference ways each
wire can be twisted.
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Tetrahedral Networks

• We assign a numerical value to these pictures by
  twisting them around, counting them, and
  remembering that anytime we can make a circle,
  the value of that circle is -2. Also, we must
  remember that we need to count the number of
  permutations we are able to make because a
  coefficient to all of these calculations in the
  inverse factorial of the permutations.

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Asymptotics

• Ponzano and Regge, two physicists, discovered
  asymptotic behavior for the 6j symbols. They
  devised a formula for this behavior which was
  later proven by mathematician Justin Roberts.
• The formula is quite simple. As k approaches
  infinity where the sum is over all edges and the
  theta is the exterior dihedral angle

v
S
ka kb kf ke kd kc

2 3pVk3
Te 2
p 4
COS
(Ka1)

Cited from Asymptotics and 6j Symbols. Roberts,
Justin. Geometry and Topology Monographs. Volume
4. Pages 245-261.
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Asymptotics
The pink curve represents the asymptotic formula
while the blue dots represent the actual 6j
symbols. As you can see, the formula becomes
more accurate as the k (x-axis) becomes larger.
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My Research

• I have been working with a pretty good 6j symbols
  program and have amassed a collection of data.
• I have written several programs to calculate the
  volume of a general and regular tetrahedron, the
  asymptotics formula, the general dihedral angles
  based on edge length, and I have modified the 6j
  program to give me only zero values.
• My proudest moment was working out a formula
  that corrects the asymptotics formula for a
  regular tetrahedron, getting closer to 6j symbols
  as it reaches its limit.

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FUN FACTS ABOUT 6j

• One of the more interesting applications for the
  6j symbols involve quantum gravity.
• No one has discovered a quantum theory of gravity
  for 3-1 space-time.
• As we have seen, 6j symbols involve tetrahedron .
  So if we think about 2-1 space time as a solid,
  and we break up this solid into tetrahedron, we
  can introduce the 6j symbols into the mix.

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FUN FACTS ABOUT 6j

• This picture is a tetrahedron. I drew it this way
  to prove a point. The two triangular faces
  connected by an edge can be seen as two planes in
  space. While the spring connecting the two faces
  can be seen as time.

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FUN FACTS ABOUT 6j

• Just for fun, lets imagine the edges of this
  tetrahedron are labeled as spins. Spins are
  intrinsic properties of an elementary particle.
• Now this tetrahedron can be tied to quantum
  mechanics and the 6j symbols. Ponzano and Regge
  saw their classical limit can also be describe in
  this form

Where the IRegge is a form of the
Einstein-Hilbert Action
Steven Carlip, "Quantum Gravity in 21
Dimensions The Case of a Closed Universe",
Living Rev. Relativity 8,  (2005),  1. URL
(cited on August 19, 2005) http//www.livingrevie
ws.org/lrr-2005-1
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FUN FACTS ABOUT 6j

• The result to all this mathematics is that we see
  elementary particles behaving in a classical way
  because the Einstein Hilbert Action can be seen
  as the action of general relativity.
• AMAZING!
• Unfortunately the same type of connection between
  simplexes (4-dimensional tetrahedra) and general
  relativity have not been found yet!

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Acknowledgements

• Dr. John Baez, Department of Mathematics
• Dr. Steven Angle, College of Natural and
  Agricultural Sciences, Dean
• UCR Department of Mathematics