Abstract
The questions on the existence of the three color quark symmetry and three quark-lepton generations could have the origin associated with the new exotic symmetries outside the Cartan-Killing-Lie algebras/groups. Our long-term search for these symmetries has been began with our Calabi-Yau space classification on the basis of the n-ary algebra for the reflexive projective numbers and led us to the expansion of the binary n = 2 complex and hyper complex numbers in the framework of the n-ary complex and hyper-complex numbers with n = 3, 4, … where we constructed new Abelian and non-Abelian symmetries. We have studied then norm-division properties of the Abelian nary complex numbers and have built the infinite chain of the Abelian groups U(n–1) = [U(1) × … × U(1)](n–1). We have developed the n-ary holomorphic (polymorphic) analysis on the n-ary complex space NC{n}, which led us to the generalization of the quadratic Laplace equations for the harmonic functions. The generalized Laplace equations for the n-ary harmonic functions give us the n-th order homogeneous differential equations which are invariant with respect to the Abelian n-ary groups U(n–1) and with some new spatial properties. Further consideration of the non-Abelian n-ary hyper-complex numbers opens the infinite series of the non-Abelian TnSU(n)-Lie groups(n=3,4,…) and its corresponding tnsu(n) algebras. One of the exceptional features of these symmetry groups is the appearance of some new n-dimensional spinors that could lead to an extension of the concept of the SU(2)-spin, to the appearance of n-dimensional quantum structures -exotic “n-spinor” matter(n = 3, 4, … - maarcrions). It is natural to assume that these new exotic “quantum spinor states” could be candidates for the pra-matter of the quark-charge leptons or/and for the dark matter. We will be also interested in the detection of the exotic quantum ’n-spinor” matter in the neutrino and hadron experiments.
Highlights
In the search for the new geometric varieties and related groups symmetries and their representations in the framework of extensions of the well-studied theory of the binary n = 2 complex and hyper-complex numbers to the theory of the n-ary complex (Abelian) and hyper-complex numbers with n > 2
The symmetry properties of this continuum are based on the space-time Lorentz-Poincaré symmetry group P = S O(1, 3) R4 and the corresponding quantum theory is constructed on the basis of the groups of internal symmetries S U(3C) × S U(2)I × U(1)Y
We have taken the idea of considering the n − ary complex numbers in the Euclidean Rn- spaces as a tool to find new symmetries in connection with the Calabi-Yau space classification of any dimension CY(d), d = complex.dim. = 2p − real.dim., what we have done on the basis of the n − ary theory of the reflexive projective numbers [1, 2]
Summary
Geometrical basis of modern quantum physics of quarks and leptons - Standard Model can be represented as a space-time D = (3 + 1) - four-dimensional continuum. The symmetry properties of this continuum are based on the space-time Lorentz-Poincaré symmetry group P = S O(1, 3) R4 and the corresponding quantum theory is constructed on the basis of the groups of internal symmetries S U(3C) × S U(2)I × U(1)Y .
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