Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Sunday, March 15, 2009

Oxford Life

In the land of Monty Python, the Home Office still hasn't seen the error of its ways, and is procrastinating about issuing me with another work permit so that I can apply to extend my visa to a date after the expiry of the work period. Apparently, this problem is not unusual here.

Meanwhile, the University has efficiently issued cards, keys, computing accounts and numerous other useful things to help me get some research done. So I have been wondering a little about how our phenomenological MUBs might relate to, or extend, certain dagger symmetric monoidal structures studied by Coecke et al.

Aside: Apparently the referees did not like Carl's paper because it uses quantum information theory and not QCD, and they know that quantum information theory could not possibly be used to derive such patterns between particle masses.


Blogger L. Riofrio said...

The Silly Party is truly in charge of Britain.

March 18, 2009 8:21 AM  
Blogger CarlBrannen said...

I thought this was a particularly good PI talk, relevant to what we're doing: Dynamics in the Dark (Energy)by Andrew Tolley. Also, Phys Math Chem is reconsidering rejection of my hadron paper.

March 18, 2009 5:36 PM  
Blogger kneemo said...

Connecting Carl's work to QCD, or Yang-Mills theory in general, shouldn't be too difficult. The most elegant way to go about this is by invoking the D-brane interpretation where the low energy dynamics of the worldvolume theory for N-coincident branes is just Yang-Mills theory with gauge group U(N).

For leptons, the interpretation is straightforward, where one initially begins with three coincident D-branes with U(3) symmetry. As the D-branes separate, the strings connecting them are stretched and the U(3) symmetry is broken, giving rise to massive fields with masses proportional to the corresponding open string's tension.

This is all neatly encoded in a 3x3 matrix in the adjoint representation of the unbroken U(3) gauge group. This matrix describes the three D-brane positions in spacetime. To describe the leptons, however, Carl's insight is to use a 1-circulant matrix in the adjoint representation of U(3), i.e. a 3x3 1-circulant Hermitian matrix.

Diagonalizing such a 1-circulant gives rise to a spectral decomposition where each term (T) is just an eigenvalue multiplying a primitive idempotent (D-brane), e.g, T=c*P. Calculating the length squared of a given term T gives (using idempotency): ||T||^2=tr(c*P o c*P)=c^2*tr(P^2)=c^2*tr(P)=c^2, i.e., the eigenvalue squared.

Using the eigenvalue/mass relation given in Brannen 06, we see the lepton masses are the squared distances of the D-branes from the origin (zero matrix) in Hermitian matrix space. In summary, the three lepton generations can be shown to arise from a system of three coincident D-branes with unbroken U(3) worldvolume symmetry. The masses of the electron, muon and tau correspond to the length squared of each D-brane in an internal, noncommutative space.

March 21, 2009 10:47 AM  

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