Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, 34014 Trieste, Italy.
Phys Rev Lett. 2010 Oct 8;105(15):150604. doi: 10.1103/PhysRevLett.105.150604. Epub 2010 Oct 7.
In the Hilbert space formulation of classical mechanics, pioneered by Koopman and von Neumann, there are potentially more observables than in the standard approach to classical mechanics. In this Letter, we show that actually many of those extra observables are not invariant under a set of universal local symmetries which appear once the Koopman and von Neumann formulation is extended to include the evolution of differential forms. Because of their noninvariance, those extra observables have to be removed. This removal makes the superposition of states in the Koopman and von Neumann formulation, and as a consequence also in classical mechanics, impossible.
在经典力学的 Hilbert 空间表述中，由 Koopman 和 von Neumann 开创，潜在的可观测数量比经典力学的标准方法多。在这封信件中，我们表明，实际上，这些额外的可观测量中的许多在一组普遍的局部对称性下不是不变的，这些对称性一旦 Koopman 和 von Neumann 表述被扩展到包括微分形式的演化，就会出现。由于它们的不变性，那些额外的可观测量必须被移除。这种移除使得 Koopman 和 von Neumann 表述中的状态叠加，以及因此也使得经典力学中的状态叠加成为不可能。
