基于接触变换的 van Hove 幺正表示的经典系统与混合经典-量子系统
Classical and mixed classical-quantum systems from van Hove's unitary representation of contact transformations
AI总结：
本文基于 van Hove 接触变换群的幺正表示，提出了一种希尔伯特空间中经典力学的新颖表述，并扩展至混合经典-量子系统，应用于经典仪器对量子比特的测量。
摘要：
希尔伯特空间中对经典力学的描述可追溯至 1930 年代 Koopman 和 von Neumann 的工作。数十年后，van Hove 推导出了接触变换群的幺正表示，该表示近期被用于发展希尔伯特空间中经典力学的新颖表述。该表述在诸多方面不同于 Koopman-von Neumann 理论。经典可观测量由 van Hove 算符表示，这些算符满足与相空间中函数的 Poisson 代数同构的对易代数。此外，这些算符既是可观测量也是变换的生成元，这使得无需像 Koopman-von Neumann 理论那样引入不可观测的辅助算符。另外，为与经典力学保持一致，必须施加一个固定波函数相位的约束。该方法可扩展至希尔伯特空间中的混合经典-量子系统。该形式体系被应用于经典仪器对量子二能级系统（qubit）的测量。
英文摘要：
Descriptions of classical mechanics in Hilbert space go back to the work of Koopman and von Neumann in the 1930s. Decades later, van Hove derived a unitary representation of the group of contact transformations which recently has been used to develop a novel formulation of classical mechanics in Hilbert space. This formulation differs from the Koopman-von Neumann theory in many ways. Classical observables are represented by van Hove operators, which satisfy a commutation algebra isomorphic to the Poisson algebra of functions in phase space. Moreover, these operators are both observables and generators of transformations, which makes it unnecessary to introduce unobservable auxiliary operators as in the Koopman-von Neumann theory. In addition, for consistency with classical mechanics, a constraint must be imposed that fixes the phase of the wavefunction. The approach can be extended to hybrid mixed classical-quantum systems in Hilbert space. The formalism is applied to the measurement of a quantum two-level system (qubit) by a classical apparatus.
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