Abstract
A concise study of ternary and cubic algebras with Z3 grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, S3, and its abelian subgroup Z3 may play an important role in quantum physics. We show then how most of important algebras with Z2 grading can be generalized with ternary composition laws combined with a Z3 grading. We investigate in particular a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, , one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra.An analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator, and some properties of its eigenfunctions are briefly discussed.
Highlights
More than fifty years ago Bogdan Mielnik has published several fundamental papers devoted to the analysis of physical and mathematical bases of quantum mechanics
The essence of Bohr’s approach was that neither a quantum object nor the classical object serving as measuring device can be totally separated from each other; any measurement describes the interaction between the quantum object and the observer
One is tempted to say that the so-called complex numbers deserve to be called “real”, because they reflect the deepest properties of quantum physics, while the real numbers with which we describe our perceptible world appear to be the result of averaging over great number of microscopic events, and deserve better the adjective “synthetic” or “imaginary”
Summary
More than fifty years ago Bogdan Mielnik has published several fundamental papers devoted to the analysis of physical and mathematical bases of quantum mechanics. The conjugate matrices span an inequivalent representation of the SL(2, C) group, labeled by dotted indeces; the antisymmetric 2-form αβ ̇ leads to the same result when its invariance is required: 1 ̇2 ̇ = − 2 ̇1 ̇ = 1, 1 ̇1 ̇ = 0, 2 ̇2 ̇ = 0; 1 ̇ 2 ̇ = Sαα ̇ S2β ̇ αβ ̇ Spinors transforming with the SL(2, C) group do not represent observable physical quantities; measurable quantities are formed by their quadratic and hermitian combinations, like e.g. a four-vector ψ† γμψ, transforming under the vector representation of the Lorentz group This can suggest that the very origin of the Lorentz transformations resides in the discrete properties of elementary particles, and not vice versa; more precisely, SL(2, C) appears as the invariance group of the non-trivial action of the Z2-group. The Z3-graded analogue of Grassman algebra Let us introduce N generators spanning a linear space over complex numbers, satisfying the following cubic relations [19, 20]: θAθBθC = j θBθCθA = j2 θCθAθB,.
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