Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Friday, July 25, 2008

M Theory Lesson 211

Generalising the matrix $K$ further still by adding phases to $(2)$, one can offset the exact Koide eigenvalues for a phase $\theta = \frac{2}{9} + \frac{2 \pi n}{3}$ in the component $(1)$ by a very small phase $\delta \simeq 10^{-7}$. Note that the cyclic $\frac{2 \pi n}{3}$ factors from the Fourier transform have not yet been accounted for in the basic $6 \times 6$ eigenvalue problem, so they are included in $\theta$. The charged lepton $K$ operator eigenvalues for the democratic 6-vector then take the form

$\lambda = a + b \textrm{cos} \theta + c \textrm{sin} \delta$

where the last term is considered a small electric field term. That is, by making the amplitude $c$ small, $\delta$ need not be small and one could take $\delta = \theta$. This indicates a correspondence between, on the one hand $(1)$ and magnetic fields, and on the other $(2)$ and electric fields, which holds even if the factor $(2)$ becomes a 2-circulant matrix. Electric magnetic duality then swaps the two triangles making up a basic $S_{3}$ hexagon, as previously discussed. For the cube this may be viewed as the duality of a triangle and trivalent vertex in the plane. That is, a duality for Space and Time.


Blogger CarlBrannen said...

Ooooo. That's cool. I've got too much to think about right now to look more carefully at this. Maybe it will be the next order correction on things.

Since the 1-circulants give the weak hypercharge quantum numbers, and the 2-circulants give the weak isospin quantum numbers, one might think that there would be a natural way here of distinguishing the strengths of these fields (i.e. the electromagnetic versus the weak).

Must go to sleeppppppppppppp

July 25, 2008 6:01 PM  
Blogger L. Riofrio said...

That is, a duality for Space and Time.

Great conclusion!

July 26, 2008 8:17 AM  

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