Universal local symmetries and nonsuperposition in classical mechanics

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.