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Physics
Beyond the Standard Models
What is new with Koide sum rules?
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[QUOTE="CarlB, post: 6502134, member: 26624"] My new paper (out for review at Foundations of Physics) has a connection to Koide but it isn't mentioned: [URL]https://vixra.org/abs/2105.0146[/URL] There are a couple of relations. First, Marni Sheppeard recognized the way I redid the Koide formula as a Discrete Fourier Transform and supposed that what we needed was a Discrete Fourier Transform for a non Abelian (well she said non commutative) symmetry. My new paper is exactly about that. And the paper generalizes the Dirac / Weyl equation to one with more interesting Pauli spin matrices. But the underlying symmetry is a point group which implies that space is a lattice. For this the paper cites Iwo Bialynicki-Birula's paper on the Weyl / Dirac equation on a lattice: [URL]https://arxiv.org/abs/hep-th/9304070[/URL] That paper shows that you can get the special-relativity compatible Weyl / Dirac equation on a cubic lattice of quantum cellular automata provided you use a specific formula for updating the cellular automata. That formula is given by his equations (10) thru (12). But if you work out those equations, you'll find that both his paper and my old Koide paper [URL]https://arxiv.org/abs/1006.3114[/URL] in its equation (11) are about making steps in the +-x, +-y, and +-z directions. Except that while his considers all possible signs, mine is about +x, +y and +z only. The result is that where my paper is dedicated to the (1,1,1) direction where, over the long term, you have equal steps in the +x, +y and +z direction, his paper shows how to generalize it to steps averaging in any direction. And this tells precisely how to interpret the extra group of size three needed in my new paper; just as in my old Koide paper, the group of size 3 corresponds to assigning a factor of exp(2i k pi / 3) to steps in three directions. Such an assignment can be done 3 ways and still have phases cancel over different paths with the same beginning and end. Carl [/QUOTE]
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What is new with Koide sum rules?
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