论文标题
海森伯格代数的三元概括
Ternary generalization of Heisenberg's Algebra
论文作者
论文摘要
提出了对具有$ Z_3 $分级的三元和立方代数的简明研究。我们讨论了一些基本想法，得出这样的结论，即三个对象的离散对称置换组，$ s_3 $，其Abelian子组$ Z_3 $可能在量子物理学中起重要作用。然后，我们展示了如何使用$ Z_2 $分级的大多数重要代数使用三元构图法和$ Z_3 $分级的概括。 我们特别研究了海森伯格代数的三元，$ z_3 $的概括。事实证明，引入一个非平凡的立方根的统一根，$ j = e^{\ frac {2πi} {3}} $，一个人可以定义两种类型的创建操作员，而不是一个类型的创建操作员，并随附通常的歼灭操作员。两个创建运营商是非热的，但它们是相互共轭的。这三个操作员一起形成了三元代数，他们的一些立方组合产生了通常的海森伯格代数。 汉密尔顿操作员的类似物是通过与通常的谐波振荡器的类比来构建的，并简要讨论了其本征函数的某些特性。
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.
