Emergence of Classical Localization in a Quantum World

Abstract

We apply a canonical transformation Hubble’s law to turn it into a quantum equation and derive its solutions in a homogenous universe (assumptions analogous to the FLRW universe). The eigenfunctions of Hubble’s law are also stationary states (eigenfunctions of the Hamiltonian). The study of these solutions reveals many striking results, including: 1) By enforcing boundary conditions at the cosmic horizon, we derive a fundamental lower limit to the uncertainty in any rest mass (or measurement thereof) . This implies a lower limit also to the mean particle mass which we call the mass quantum . 2) We postulate that particles with finite mass can be regarded as a composition of a large number of mass quanta and deduce a relation between mass uncertainty and mass m 0 : . 3) This uncertainty leads naturally to localization of the composite mass, with the radius of localization proportional to the inverse square root of mass . We associate this localization with the classical localization of a massive particle. 4) We derive an expression for the critical mass where there is a crossover from quantum behavior to classical behavior , where is the material mass density. The classical sizes derived in 4) are consistent with empirical results for our universe. We note the theory described here has no free parameters, hence it points to a new fundamental equation of the universe, essentially defining the mass quantum. It is a pure quantum theory that does not invoke general relativity at any stage, and the derivation uses mathematics accessible to an upper level undergraduate student in physics.