Abstract
We investigate a ternary, Z3-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, j = e 2πi/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.A cubic analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator. A set of eigenstates in coordinate representation is constructed in terms of functions satisfying linear differential equation of third order.
Highlights
Our goal being a ternary generalization of Heisenberg’s algebra, let us start with recalling basic facts about ternary algebras [1], [2] The usual definition of an algebra involves a linear space A endowed with a binary constitutive relations
With the help of these structure constants all essential properties of a given algebra can be expressed, e.g. they will define a Lie algebra if they are antisymmetric and satisfy the Jacobi identity: fikj = −fjki, fikmfjml + fjkmflmi + flkmfimj = 0, (3)
When we speak of algebras, we mean binary algebras, understanding that they are defined via quadratic constitutive relations (2)
Summary
Our goal being a ternary generalization of Heisenberg’s algebra, let us start with recalling basic facts about ternary algebras [1], [2] The usual definition of an algebra involves a linear space A (over real or complex numbers) endowed with a binary constitutive relations. 4πi e3 can be used in place of the former one; this will define the conjugate algebra S, satisfying the following cubic constitutive relations:. The Z3-graded analog of Grassman algebra Let us introduce N generators spanning a linear space over complex numbers, satisfying the following cubic relations [3], [4]: θAθBθC = j θBθCθA = j2 θCθAθB,. The relation 23) may serve as the definition of ternary Clifford algebra Another set of three matrices is formed by the hermitian conjugates of Qa, which we shall endow with dotted indeces a , b, ... Ternary Z3-graded commutator In any associative algebra A one can introduce a new binary operation, the commutator, using the generator of the Z2 group in form of multiplication by −1 : X, Y ∈ A → [X, Y ] = XY + (−1)Y X = XY − Y X. Positive eigenvalue functions of the Hamiltonian Hare obtained by acting on f (x) with consecutive powers of a†. [9]
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