Ternary generalization of Heisenberg's Algebra

TL;DR

This paper explores a novel ternary generalization of Heisenberg's algebra using $Z_3$ grading, introducing new creation operators and algebraic structures that extend traditional quantum mechanics frameworks.

Contribution

It introduces a $Z_3$-graded ternary algebraic framework for Heisenberg's algebra, including new creation operators and their properties, expanding the mathematical foundation of quantum physics.

Findings

Defined two types of non-hermitian, conjugate creation operators

Constructed a ternary algebra that reproduces the Heisenberg algebra

Proposed a Hamiltonian analogue with eigenfunctions similar to harmonic oscillator

Abstract

A concise study of ternary and cubic algebras with $Z_3$ grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, $S_3$, and its abelian subgroup $Z_3$ may play an important role in quantum physics. We show then how most of important algebras with $Z_2$ grading can be generalized with ternary composition laws combined with a $Z_3$ grading. We investigate in particular a ternary, $Z_3$-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, $j = e^{\frac{2 \pi i}{3}}$, 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.