标题： 从合理的公设到广义哈密顿系统 显示英文标题

标题： From reasonable postulates to generalised Hamiltonian systems

摘要： 哈密顿力学通过系统的哈密顿量描述其演化。 哈密顿量通常也代表能量可观测量，这是与演化定律的时间不变性相关的诺特定理守恒量。 在量子和经典力学中，哈密顿力学都要求时间演化与可观测能量之间有精确的关系，尽管使用了略有不同的术语。 我们提炼了量子和经典力学共同满足的基本条件，包括正则坐标对称性和内积不变性。 我们在广义概率理论的框架下表达这些条件，并通过哈密顿系统的时间不变性质推广了能量本征态的定义。 通过假设这些条件成立，我们推导出一个统一的哈密顿系统模型。 这个统一框架用一致的语言描述了量子和经典力学，便于它们之间的比较。 此外，我们还讨论了替代理论：由对易关系混合而成的运动方程、量子理论的信息限制版本以及斯佩肯的玩具理论。 这些发现加深了对量子和经典理论中哈密顿量的理解，并指出了几个潜在的研究课题。 显示英文摘要

摘要： Hamiltonian mechanics describes the evolution of a system through its Hamiltonian. The Hamiltonian typically also represents the energy observable, a Noether-conserved quantity associated with the time-invariance of the law of evolution. In both quantum and classical mechanics, Hamiltonian mechanics demands a precise relationship between time evolution and observable energy, albeit using slightly different terminology. We distil basic conditions satisfied in both quantum and classical mechanics, including canonical coordinate symmetries and inner product invariance. We express these conditions in the framework of generalised probabilistic theories, which includes generalizing the definition of energy eigenstates in terms of time-invariant properties of the Hamiltonian system. By postulating these conditions to hold, we derive a unified Hamiltonian system model. This unified framework describes quantum and classical mechanics in a consistent language, facilitating their comparison. We moreover discuss alternative theories: an equation of motion given by a mixture of commutation relations, an information-restricted version of quantum theory, and Spekken's toy theory. The findings give a deeper understanding of the Hamiltonian in quantum and classical theories and point to several potential research topics.