Boundary Condition and the Auxiliary Phase in Feynman Path

Chung-Ru Lee

V. Janiš

Nakka Gurappa

M. Asorey, Jesús Clemente-Gallardo, J. M. Munõz-Castañeda

The path integral approach to quantum mechanics requires a substantial generalisation to describe the dynamics of systems confined to bounded domains. Nonlocal boundary conditions can be introduced in Feynman's approach by means of boundary amplitude distributions and complex phases to describe the quantum dynamics in terms of the classical trajectories. The different prescriptions involve only trajectories reaching the boundary and correspond to different choices of boundary conditions of selfadjoint extensions of the Hamiltonian. One dimensional particle dynamics is analysed in detail.

Soon Hee@Bungsuh Yeo

Abstract Feynman's path integral formulation describes quantum propagation as a sum over all possible paths, weighted by a phase factor e^{iS/ℏ}. The classical path emerges via constructive interference of paths with stationary action. We reinterpret this mechanism within the Generalized Mass as Twisted Time (GMaTT) framework. The phase factor is identified as the accumulated phase difference Δφ of a G₂ root vector rotating around the √α axis. The condition for constructive interference is shown to be equivalent to the TUT (TwistUntwist Threshold) coherence condition |δφ| < √α. We clarify that the TUT window is not a dynamic laboratory filter but a static geometric boundary condition of the phase space. Classical paths are not "chosen" by the particle; they are the trajectories that maintain phase coherence within the TUT threshold during propagation. This provides a geometric origin for the path integral and the principle of least action, grounding Feynman's sum in the pregeometric phase geometry of GMaTT.

Sergio A Albeverio, Raphael J. Høegh-Krohn