论文标题
时间依赖的伪 - 温米顿汉密尔顿人和量子力学的隐藏几何方面
Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics
论文作者
论文摘要
在希尔伯特（Hilbert）空间中定义的非热门操作员$ h $，内部产品$ \ langle \ cdot | \ cdot \ rangle $ $可能是单一量子系统的哈密顿量，如果是$η$ -Pseudo-pseudo-hermitian，用于公制操作员（正置自动摩尔抗体）。后者定义了内部产品$ \ langle \ cdot |η\ cdot \ rangle $的物理希尔伯特空间的$ \ mathscr {h}_η$。对于某些$ h $的特征态取决于时间的情况，$η$变成时间依赖。因此，该系统具有非平稳的希尔伯特空间。这种量子系统在宇宙学背景下的量子力学研究中也遇到了这种量子系统，它遭受了时间演化的单位性与哈密顿量的不可观察性之间的冲突。他们的适当处理需要一个几何框架，该框架阐明了可观察到的能量的概念，并导致量子力学（GEQM）的几何扩展。我们提供对该主题的一般介绍，回顾一些最近的发展，对具有时间依赖性希尔伯特空间的量子系统的动力学的Heisenberg-picture公式提供了直接的描述，并概述了GEQM中动态的Heisenberg-Picture配方。
A non-Hermitian operator $H$ defined in a Hilbert space with inner product $\langle\cdot|\cdot\rangle$ may serve as the Hamiltonian for a unitary quantum system, if it is $η$-pseudo-Hermitian for a metric operator (positive-definite automorphism) $η$. The latter defines the inner product $\langle \cdot|η\cdot\rangle $ of the physical Hilbert space $\mathscr{H} _η$ of the system. For situations where some of the eigenstates of $H$ depend on time, $η$ becomes time-dependent. Therefore the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.
