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Derivation II: The Feynman Phase from Foliation Holonomy

Yuri Alcocer

Published Feb 14, 2026• 0 citations
Publisher

Abstract

We derive the Feynman path integral weight not as a postulate of quantum mechanics, but as the geometric holonomy of parallel transport on the Update Bundle. The "sum over histories" emerges as the regulator's integration over its sub-ℏ blind spot—the fiber volume it cannot resolve. Key Result: The path integral is the volume calculation of geometric indistinguishability, with interference arising from rank-sorting (constructive for stable geometries, destructive for rank-deficit configurations).

DOI
10.5281/zenodo.18646180

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We derive the Feynman path integral weight not as a postulate of quantum mechanics, but as the geometric holonomy of parallel transport on the Update Bundle. The "sum over histories" emerges as the regulator's integration over its sub-ℏ blind spot—the fiber volume it cannot resolve. Key Result: The path integral is the volume calculation of geometric indistinguishability, with interference arising from rank-sorting (constructive for stable geometries, destructive for rank-deficit configurations).
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.
This paper examines the geometric structure underlying amplitudes, interference, and probability in quantum mechanics. Building on the interpretation of quantum phase as holonomy, amplitudes are analyzed in terms of parallel transport in complex line bundles. Interference is interpreted as arising from the composition of alternative transport paths, and probability emerges from the invariant comparison of amplitudes under phase transformations. Under general assumptions of linearity and phase invariance, the squared-norm structure associated with probability is shown to arise naturally from consistency conditions on amplitude composition. The analysis focuses on kinematic structure and does not assume specific dynamical models.