The Axiom of History Dependence: A Self-Contained Geometric Framework for Quantum Foundations

Abstract

We present a self-contained geometric framework for the foundations of quantum mechanics based on a single axiom: the History Fiber Bundle postulate. Unlike conventional approaches that treat quantum postulates as co-equal rules, we demonstrate that this postulate is not an arbitrary assumption but the minimal necessary modification forced upon us when the Markovian (single-pendulum) paradigm is confronted with established experimental facts. The Markovian assumption—where physical evolution depends solely on instantaneous states—faces fatal contradictions from three classes of experiments: history-dependent thermal relaxation (Mpemba effect) [3], non-trivial geometric phases in quantum transport (Aharonov-Bohm effect) [32], and fractional exchange statistics in the fractional quantum Hall effect [29,30]. These phenomena share a common signature: a system traversing a closed loop in parameter space does not return to its original internal state. In mechanical terms, this is the signature of a double-pendulum structure, not a single pendulum. Formalizing this as a complex vector bundle over spacetime equipped with a connection possessing non-trivial holonomy (the history bundle), we prove that the following structures are forced consequences rather than independent postulates: complex Hilbert space (from the necessity of recording phase memory), unitary evolution (from the preservation of probability under parallel transport), canonical commutation relations, Schrödinger's equation, Born's probability interpretation, Bose-Fermi statistics, the spin-statistics theorem, Dirac's equation, and Feynman's path integral. The classical limit (ℏ→0 projection) emerges as the ℏ→0 projection of the history bundle dynamics. We further demonstrate that the fractional quantum Hall effect provides a decisive experimental refutation of the Markovian world: the observation of anyon statistics with exchange phases θ=π/m (m=3,5,…) is strictly forbidden in any single-pendulum framework, where particle exchange twice must rigorously equal the identity (H2=1). The existence of fractional holonomy demands the history bundle's non-trivial curvature. This transforms the Hall effect family from a mere “application” of the framework into the experimental evidence that compels its existence. The framework extends naturally to gauge field theory, the Yang-Mills mass gap, and a unified dynamical origin for gravity and particle masses via the thermal-history field u(x)—the coarse-grained

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