arXiv quant-ph/02071042002-07-18quant-ph

量子力学的自治形式

Towards an Autonomous Formalism for Quantum Mechanics

Tulsi Dass

备注：: 44 pages

摘要：

一种形式主义，其中量子粒子动力学可以开发自己的，而不是“量化”的基本经典理论。提出概率论与动力学的统一应作为量子理论的基本特征。参数给出表明，当这样的统一是试图在配置空间水平上，波函数的薛$\ddot{o}$丁格理论出现的自然候选人所需的统一。作用于这些波函数的适当稠密集上的线性算子的 *-代数$\mathcal{A}_{Q}$（不一定有界）似乎是量子运动学的舞台。一个简单的推广现有的形式主义在非交换几何发展的概念，广义代数辛结构（GASS），它可以容纳经典和量子辛结构的特殊情况。量子运动学和动力学是在一个非对易的哈密顿系统的框架内，采用适当的GASS基于$ \mathcal{A}_{Q}$。普朗克常量只在一个地方引入--以量子辛形式;它在常规地方的出现是自动的。酉维格纳对称性在非对易哈密顿系统中表现为正则变换。本文用适当的GASS给出了量子-经典对应的一种直接处理方法.

英文摘要：

A formalism is presented in which quantum particle dynamics can be developed on its own rather than `quantization' of an underlying classical theory. It is proposed that the unification of probability and dynamics should be considered as the basic feature of quantum theory. Arguments are given to show that when such a unification is attempted at the configuration space level, the wave funtions of Schr$\ddot{o}$dinger theory appear as the natural candidates for the desired unification. A *-algebra $\mathcal{A}_{Q}$ of (not necessarily bounded) linear operators acting on an appropriate dense set of these wave functions appears as the arena for quantum kinematics. A simple generalization of an existing formalism in noncommutative geometry is employed to develop the notion of generalized algebraic symplectic structure (GASS) which can accomodate classical and quantum symplectic structures as special cases. Quantum kinematics and dynamics is developed in in the framework of a noncommutative Hamiltonian system employing an appropriate GASS based in $ \mathcal{A}_{Q}$. The Planck constant is introduced at only one place -- in the quantum symplectic form; its appearance at conventional places is then automatic. Unitary Wigner symmetries appear as canonical transformations in the noncommutative Hamiltonian system. A straightforward treatment of quantum - classical correspondence is given in terms of appropriate GASSes.

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