Authors: J. S. Markovitch
The fine structure constant and the quark and lepton mixing angles are shown to arise naturally in the course of altering the symmetry of two algebraic identities. Specifically, the symmetry of the identity x2 = x2 is "broken" by making the substitution xn → xn - yp on its left side, and the substitution x → x - z on its right side, where p equals the order of the identity; these substitutions convert the above identity into the equation x2 - y2 = (x - z)2. These same substitutions are also applied to the only slightly more complicated identity (x/a)3 + x2 = (x/a)3 + x2 to produce this second equation (x3 - y3) / a3 + x2 - y3 = (x - z)3 / a3 + (x - z)2. These two equations are then shown to share a mathematical property relating to dz/dy, where, on the second equation's left side, this property helps define the special case (x3 - y3) / a3 + x2 - y3 = (103 - 0.13) / 33 + 102 - 0.13 = 137.036, an equation which incorporates a value close to the experimental fine structure constant inverse. Moreover, on the second equation's right side, this same special case simultaneously produces values for the sines squared of the mixing angles. Specifically, the sines squared of the leptonic angles φ12, φ23, and φ13 appear as 0.3, 0.5, and not larger than roughly 1/30 000, respectively; and the sines squared of the quark mixing angles θ12 and θ13 appear as 0.05, and close to 1/90 000, respectively. Despite closely mirroring so many experimental values, including the precisely-known fine structure constant, the above mathematical model requires no free parameters adjusted to fit experiment.
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