共轭线性对称性及其对PT-对称量子理论的应用

Conjugate linear symmetry and its application to PT-symmetry quantum theory

传统量子系统的哈密顿是自伴算子,哈密顿的自伴性不仅保证系统遵循酉演化和保持概率守恒,而且也保证了它自身具有实的能量本征值,这类系统称为自伴量子系统.然而,确实存在一些物理系统(如PT-对称量子系统),其哈密顿不是自伴的,这类系统称为非自伴量子系统.为了深入研究PT-对称量子系统,并考虑到算子PT的共轭线性性,首先讨论了共轭线性算子的一些性质,包括它们的矩阵表示和谱结构等;其次,分别研究了具有共轭线性对称性和完整共轭线性对称性的线性算子,通过它们的矩阵表示,给出了共轭线性对称性和完整共轭线性对称性的等价刻画;作为应用,得到了关于PT-对称及完整PT-对称算子的一些有趣性质,并通过一些具体例子,说明了完整PT-对称性对张量积运算不具有封闭性,同时说明了完整PT-对称性既不是哈密顿算子在某个正定内积下自伴的充分条件,也不是必要条件.

The Hamiltonians of classical quantum systems are Hermitian(self-adjoint)operators.The self-adjointness of a Hamiltonian not only ensures that the system follows unitary evolution and preserves probability conservation,but also guarantee that the Hamiltonian has real energy eigenvalues.We call such systems Hermitian quantum systems.However,there exist indeed some physical systems whose Hamiltonians are not Hermitian,for instance,PT-symmetry quantum systems.We refer to such systems as non-Hermitian quantum systems.To discuss in depth PT-symmetry quantum systems,some properties of conjugate linear operators are discussed first in this paper due to the conjugate linearity of the operator PT,including their matrix represenations,spectral structures,etc.Second,the conjugate linear symmetry and unbroken conjugate linear symmetry are introduced for linear operators,and some equivalent characterizations of unbroken conjugate linear symmetry are obtained in terms of the matrix representations of the operators.As applications,PT-symmetry and unbroken PT-symmetry of Hamiltonians are discussed,showing that unbroken PT-symmetry is not closed under taking tensor-product operation by some specific examples.Moreover,it is also illustrated that the unbroken PT-symmetry is neither a sufficient condition nor a necessary condition for Hamiltonian to be Hermitian under a new positive definite inner product.