Updates
Well, it took me quite a while, but I finally came across a Kapranov paper on the non-commutative Fourier transform! Meanwhile Carl Brannen has been updating his blog with fascinating calculations in snuark QFT. Thanks to Tommaso Dorigo for joining the Category Theory for Physics outreach program. Things here will probably be quieter next week, since my schedule is quite full with both latte serving and teaching.
posted by Kea | 1:45 PM
10 Comments:
Wow, what a paper! It's a shame that it deals with nilpotents instead of idempotents.
When I decided on learning geometric algebra, I concentrated on idempotents. This was going against tradition as nilpotents are much much more often used for QM.
The reason for using nilpotents is that due to the Pauli exclusion principle, creation and annihilation operators are nilpotent. My version of QFT avoids creation and annihilation, and the Pauli exclusion principle is reduced in importance when you do stuff in qubits (i.e. there is no need for equal time commutators at different positions in space that anticommute).
After I told David Hestenes that I was using idempotents instead of nilpotents, he told me that there is an intimate relationship between them, or something like that.
He didn't explain, but I suppose the relationship is this: You get nilpotents from idempotents by multiplying by something that anticommutes with one of the (square) "roots of unity" used to define the idempotent.
Example in SU(2): Let x,y,z be Pauli matrices. So xx = 1. Then x is a root of unity. This creates the idempotent (1+x)/2. Since z anticommutes with x, we have:
z(1+x)/2 = (1-x)/2 z.
But (1+x)/2 and (1-x)/2 are different primitive idempotents made from the same root of unity, and therefore annihilate each other (multiply to zero).
It follows that z(1+x)/2 is nilpotent. Representing the Pauli algebra with the Pauli matrices, this particular nilpotent 2x2 matrix is:
(+0.5 +0.5)
(-0.5 -0.5).
I've started writing the simulation for calculating sums of Feynman diagrams of products of Pauli projection operators. It will be very general and easy to use and will do both very simple and very complicated calculations. I will publish it on the net.
Also, I'd like to say that your post over at Tommaso's did great service for functors.
Your guest post on Tommaso's blog was very nice to see!
Yeah, isn't the paper great? Suggests a lot more stuff to put in your book!
Nice find, Kea. I was quite pleased to read that the Fourier transform of the Wiener measure on paths in R^n is the matrix Guassian series in Theorem (6.2.7). The Guassian series looks very much like the probability density on the space of all NxN Hermitian matrices, although it is missing a trace.
Probability distributions in Chern-Simons matrix models have been shown to agree with the probability distribution corresponding for the ν = 1 Laughlin wavefunction (http://arxiv.org/abs/hep-th/0106016). However, for filling fractions ν = 1/(2p+1) only the long distance behavior is in agreement.
While playing with these matrix probability distributions, I have found they take on a very convenient form when taken over the space of Hermitian primitive idempotents.
Cool, kneemo! Thanks, Louise: I'm really not sure what to say to anomolous cowherd, who has been very industriously recommending textbooks for me to read.
I'm really not sure what to say to anomolous cowherd, who has been very industriously recommending textbooks for me to read.
I guess you should feel honored. ;) anomolous cowherd is also throwing references at QFT buff "Guess Who" a.k.a. Lubos Motl.
Hi all,
Terence Tao has some interesting comments on idempotents, nilpotents and unipotents in two threads on his blog WN.
1 - Unipotent elements of the Lorentz group, and conic sections
2 - The quantitative behaviour of polynomial orbits on nilmanifolds
see September 29th, 2007 at 10:18 comment, “not much of a link between nilpotents ... and idempotents”, but “strong link between nilpotents and unipotents”.
http://terrytao.wordpress.com/
Kea: great guest post on AQDS blog.
I will be making a comment concerning an alternative perspective RE: cubic representation of categories.
Hi Kea,
T Tao referenced Madhu Sudan, “Reliable transmission of information“ [information and coding theory], in his post 'PCM article: The Schrodinger equation'.
Therein was reference to 'Hamming distance'.
http://en.wikipedia.org/wiki/Hamming_distance
This cubic representation reminds me of your cube at AQDS.
A paper on braid theory is out that is using angles of pi/3 and 2pi/3, which I think is an improvement.
...using angles of pi/3 and 2pi/3, which I think is an improvement.
Yes, in a few years time they might figure out that Fano plane diagrams (triangles) can be rotated by these angles.
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