How far can you stretch classical mechanics?

Abstract

Classical and quantum theories are very different, but the gap between them may look narrow particularly if the notion of classicality is broadened. For example, if we do not impose all the classical assumptions at the same time, hidden variable theories reproduce the results of quantum mechanics. If a quantum system is restricted to Gaussian states, evolution and measurements, then classical phase space mechanics with a finite resolution fully reproduces its behavior. We discuss two examples of such extensions. In a version of the delayed-choice experiment we allow an otherwise classical system to exhibit two types of behavior ("P" or "W"), requiring, however, objectivity: the system is at any moment either "P" or "W", but not both. It turns out that the three conditions of objectivity, determinism, and independence of hidden variables are incompatible with any theory, not only with quantum mechanics. We then consider two harmonic oscillators with a Gaussian interaction between them. If one is treated as quantum and one is described by a classical theory with a finite phase space resolution, no consistent description of this interaction is possible. The lesson is that it is hard to be a little bit quantum: it is either pointless or quantumness takes over altogether.