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Register a new userTernary Z2 x Z3 graded algebras and ternary Dirac equation
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Authors
Richard Kerner
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The wave equation generalizing the Dirac operator to the Z3-graded case is introduced, whose diagonalization leads to a sixth-order equation. It intertwines not only quark and anti-quark state as well as the u and d quarks, but also the three colors, and is therefore invariant under the product group Z2 x Z2 x Z3. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry and of the SU(2) x U(1) that arise automatically in this model, leading to the full bosonic gauge sector of the Standard Model.
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We show how the discrete symmetries $Z_2$ and $Z_3$ combined with the superposition principle result in the $SL(2, {bf C})$-symmetry of quantum states. The role of Paulis exclusion principle in the derivation of the SL(2, C) symmetry is put forward as the source of the macroscopically observed Lorentz symmetry, then it is generalized for the case of the Z3 grading replacing the usual Z2 grading, leading to ternary commutation relations. We discuss the cubic and ternary generalizations of Grassmann algebra. Invariant cubic forms are introduced, and their symmetry group is shown to be the $SL(2,C)$ group The wave equation generalizing the Dirac operator to the Z3-graded case is constructed. Its diagonalization leads to a sixth-order equation. The solutions cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry.
We discuss the structure of the Dirac equation and how the nilpotent and the Majorana operators arise naturally in this context. This provides a link between Kauffmans work on discrete physics, iterants and Majorana Fermions and the work on nilpotent structures and the Dirac equation of Peter Rowlands. We give an expression in split quaternions for the Majorana Dirac equation in one dimension of time and three dimensions of space. Majorana discovered a version of the Dirac equation that can be expressed entirely over the real numbers. This led him to speculate that the solutions to his version of the Dirac equation would correspond to particles that are their own anti-particles. It is the purpose of this paper to examine the structure of this Majorana-Dirac Equation, and to find basic solutions to it by using the nilpotent technique. We succeed in this aim and describe our results.
A Z3 symmetric generalization of the Dirac equation was proposed in recent series of papers, where its properties and solutions discussed. The generalized Dirac operator acts on coloured spinors composed out of six Pauli spinors, describing three colours and particle-antiparticle degrees of freedom characterizing a single quark state, thus combining Z2 x Z_2 x Z_3 symmetries of 12-component generalized wave functions. Spinorial representation of the Z3-graded generalized Lorentz algebra was introduced, leading to the appearance of extra Z2 x Z2 x Z3 symmetries, probably englobing the symmetries of isospin, flavors and families. The present article proposes a construction of Z3-graded extension of the Poincare algebra. It turns out that such a generalization requires introduction of extended 12-dimensional Minkowskian space-time containing the usual 4-dimensional space-time as a subspace, and two other mutually conjugate replicas with complex-valued vectors and metric tensors. Representation in terms of differential operators and generalized Casimir operators are introduced and their symmetry properties are briefly discussed.
To simulate a quantum system with continuous degrees of freedom on a quantum computer based on quantum digits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to discrete observables. We consider this problem based on expanding quantum observables in series in powers of two and three analogous to the binary and ternary representations of real numbers. The coefficients of the series (digits) are, therefore, Hermitian operators. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary and ternary expansions of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values).
We construct ternary self-distributive (TSD) objects from compositions of binary Lie algebras, $3$-Lie algebras and, in particular, ternary Nambu-Lie algebras. We show that the structures obtained satisfy an invertibility property resembling that of racks. We prove that these structures give rise to Yang-Baxter operators in the tensor product of the base vector space and, upon defining suitable twisting isomorphisms, we obtain representations of the infinite (framed) braid group. We use these results to construct invariants of (framed) links. We consider examples for low-dimensional Lie algebras, where the ternary bracket is defined by composition of the binary ones, along with simple $3$-Lie algebras, and their applications to some classes of links.
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