Approximately Macroscopically Unique States and Quantum Mechanics

We show that Mumford’s Approximately Macroscopically Unique (AMU) states exist for quantum systems consisting of unbounded self-adjoint operators when the commutators are small. In particular, AMU states always exist in position and momentum systems when the Planck constant $\mathrm{\mid }\mathrm{\hslash }\mathrm{\mid }$ is sufficiently small. However, we show that these standard quantum mechanical systems are far away from classical mechanical (commutative) systems even when $\mathrm{\mid }\mathrm{\hslash }\mathrm{\mid }\to 0.$