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Kea
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Regarding This Week's Finds 214
http://math.ucr.edu/home/baez/week214.html
...I tried to post something to the SPR thread but I guess there were too many equations. Anyway, here's a PF thread. After reading TWF, I thought of the following quote:
Continuous geometries...are a generalization of complex projective geometry somewhat in the way that Hilbert space is a generalization of finite dimensional Euclidean space Halperin (1960)
This quote appears in the book Orthomodular Lattices by G. Kalmbach, on page 191. Just below this quote is the example of the Fano plane and its associated complete modular ortholattice on 16 points.
These are the sorts of lattices that quantum logicians like.
Tony Smith, on the page linked by John Baez, mentions the Golden Ratio in connection with the Fano plane. John has mentioned Fibonacci numbers. I thought it would therefore be interesting to bring up the following observations.
In the paper
Geometrical measurements in three dimensional quantum gravity,
J.W. Barrett, http://xxx.lanl.gov/abs/gr-qc/0203018
John Barrett discusses the following remarkable Fourier transform for [itex]6j[/itex] symbols
[tex]\frac{1}{N} \sum_{j_{1} \cdots j_{6}} [ j_{4}, j_{5}, j_{6};
j_{1}, j_{2}, j_{3} ]^{2} H(j_{1},i_{1}) \cdots H(j_{6},i_{6}) = [
i_{1}, i_{2}, i_{3}; i_{4}, i_{5}, i_{6} ]^{2}[/tex]
[itex]N[/itex] is a normalisation constant. Choosing [itex]q =
e^{\frac{i \pi}{5}}[/itex] as in
A modular functor which is universal for quantum computation,
M. Freedman M. Larsen Z. Wang,
http://arxiv.org/abs/quant-ph/0001108
gives allowable spin values [itex]j \in 0, \frac{1}{2}, 1, \frac{3}{2}[/itex]. The kernel function is given by the Hopf link invariant
[tex]H(j,i) = (-1)^{2i + 2j} \cdot \frac{\sin \frac{\pi}{5}(2j +
1)(2i + 1)}{\sin \frac{\pi}{5}}[/tex]
Observe that [itex]H[/itex] only takes values [itex]\pm \phi , \pm 1[/itex] where [itex]\phi = 1.61803399 \cdots[/itex] is the golden ratio
[tex]\phi = \frac{\sin \frac{2 \pi}{5}}{\sin \frac{\pi}{5}}[/tex]
The full kernel therefore takes values in powers of [itex]\phi[/itex], or in other words the truncated Fibonacci sequence
[tex]1 , \phi , \phi + 1 , 2 \phi
+ 1 , 3 \phi + 2 , 5 \phi + 3 , 8 \phi + 5[/tex]
of 7 terms.
Cheers
Kea
http://math.ucr.edu/home/baez/week214.html
...I tried to post something to the SPR thread but I guess there were too many equations. Anyway, here's a PF thread. After reading TWF, I thought of the following quote:
Continuous geometries...are a generalization of complex projective geometry somewhat in the way that Hilbert space is a generalization of finite dimensional Euclidean space Halperin (1960)
This quote appears in the book Orthomodular Lattices by G. Kalmbach, on page 191. Just below this quote is the example of the Fano plane and its associated complete modular ortholattice on 16 points.
These are the sorts of lattices that quantum logicians like.
Tony Smith, on the page linked by John Baez, mentions the Golden Ratio in connection with the Fano plane. John has mentioned Fibonacci numbers. I thought it would therefore be interesting to bring up the following observations.
In the paper
Geometrical measurements in three dimensional quantum gravity,
J.W. Barrett, http://xxx.lanl.gov/abs/gr-qc/0203018
John Barrett discusses the following remarkable Fourier transform for [itex]6j[/itex] symbols
[tex]\frac{1}{N} \sum_{j_{1} \cdots j_{6}} [ j_{4}, j_{5}, j_{6};
j_{1}, j_{2}, j_{3} ]^{2} H(j_{1},i_{1}) \cdots H(j_{6},i_{6}) = [
i_{1}, i_{2}, i_{3}; i_{4}, i_{5}, i_{6} ]^{2}[/tex]
[itex]N[/itex] is a normalisation constant. Choosing [itex]q =
e^{\frac{i \pi}{5}}[/itex] as in
A modular functor which is universal for quantum computation,
M. Freedman M. Larsen Z. Wang,
http://arxiv.org/abs/quant-ph/0001108
gives allowable spin values [itex]j \in 0, \frac{1}{2}, 1, \frac{3}{2}[/itex]. The kernel function is given by the Hopf link invariant
[tex]H(j,i) = (-1)^{2i + 2j} \cdot \frac{\sin \frac{\pi}{5}(2j +
1)(2i + 1)}{\sin \frac{\pi}{5}}[/tex]
Observe that [itex]H[/itex] only takes values [itex]\pm \phi , \pm 1[/itex] where [itex]\phi = 1.61803399 \cdots[/itex] is the golden ratio
[tex]\phi = \frac{\sin \frac{2 \pi}{5}}{\sin \frac{\pi}{5}}[/tex]
The full kernel therefore takes values in powers of [itex]\phi[/itex], or in other words the truncated Fibonacci sequence
[tex]1 , \phi , \phi + 1 , 2 \phi
+ 1 , 3 \phi + 2 , 5 \phi + 3 , 8 \phi + 5[/tex]
of 7 terms.
Cheers
Kea
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