Analysis of quantum mechanics with real-valued Schrödinger equation,single-event quantum-path dynamics, Mauprtuis path in parameter space, and branching paths beyond semiclassics

Abstract

We analyze the Schrödinger dynamics and the Schrödinger function (or the so-called wavefunction) in the following four aspects. (1) The Schrödinger equation is reconstructed from scratch in the real field only, without referring to Newtonian mechanics nor optics. Only the very simple conditions such as the space-time translational symmetry and the conservation of flux and energy are imposed on the factorization of the density distribution in configuration space, giving rise to a two-dimensional real vector. On returning to the original Schrödinger equation, the imaginary number arises naturally. (2) Like the Langevin equation in a Brownian dynamics, we formulate a single-event path dynamics in quantum mechanics, contrasting with the Schrödinger distribution function. The path thus attained is referred to as one-world path, which represents, for instance, a path of a singly launched electron in the double-slit experiment that leaves a spot at the measurement board, while many of accumulated spots give rise to the fringe pattern. We start from the Feynman-Kac formula to draw a relation between a stochastic dynamics and the parabolic differential equations, to one of which the Schrödinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schrödinger dynamics, we build that the quantum Maupertuis-Hamilton principle, which reveals the symplectic structure in the parameter space. (4) We track how the inherent quantum nature like the Huygens-principle-like properties is built. We show that classical trajectory components in semiclassics are demanded to branch into many coherent pieces beyond the semiclassical regime and dissolve into the deep dynamics of genuine full quantum dynamics.