Analysis of quantum mechanics with real-valued Schr\"{o}dinger equation,single-event quantum-path dynamics, Mauprtuis path in parameter space, and branching paths beyond semiclassics

TL;DR

This paper reconstructs the Schr"odinger equation from real variables, introduces a single-event quantum path dynamics, explores the Maupertuis principle in quantum parameter space, and examines the transition from semiclassical to full quantum behavior.

Contribution

It presents a novel real-valued formulation of quantum mechanics, a single-event path dynamics framework, and insights into quantum branching beyond semiclassical approximations.

Findings

Reconstructed Schr"odinger equation using only real numbers.

Formulated a single-event quantum path dynamics analogous to Langevin dynamics.

Demonstrated the necessity of path branching beyond semiclassical regimes.

Abstract

We analyze the Schr\"{o}dinger dynamics and the Schr\"{o}dinger function (or the so-called wavefunction) in the following four aspects. (1) The Schr\"{o}dinger 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 Schr\"{o}dinger 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 Schr\"{o}dinger 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 Schr\"{o}dinger equation is transformed. (3) To highlight the roles of the flux and energy conservation in the Schr\"{o}dinger 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.