Heisenberg代数的三值推广
Ternary generalization of Heisenberg's Algebra
摘要：
对具有$Z_3$分次的三次和三次代数进行了简明的研究。我们讨论了一些基本的想法，导致的结论是，三个对象，$S_3$，和它的阿贝尔子群$Z_3$的置换离散对称群可能在量子物理中发挥重要作用。然后，我们展示了如何与$Z_2$分级的最重要的代数可以推广与三元复合法结合$Z_3$分级。我们调查特别是三元，$Z_3$ -分级推广的海森堡代数。事实证明，引入一个非平凡的单位立方根，$j = e^{\frac{2 πi}{3}}$，可以定义两种类型的创建操作，而不是一个，伴随着通常的湮灭操作。这两个创造算符是非厄米算符，但它们是相互共轭的。这三个算符合在一起形成一个三元代数，它们的某些三次组合生成通常的海森堡代数。通过与通常谐振子的类比，构造了哈密顿算符的一个类比，并简要讨论了它的本征函数的一些性质。
英文摘要：
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, $j = e^{\frac{2 πi}{3}}$, 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.
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