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On Amplitudes, Interference, and Probability in Quantum Mechanics

Fredrik Jensen

Published Feb 12, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18623240

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Scientific Inquiry and Review•Sep 29, 2025•1 citation
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.
This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.
This paper examines measurement and representational limits within a geometric formulation of quantum mechanics based on connection geometry and holonomy. Measurement is interpreted as projection onto invariant quantities associated with comparison of states under parallel transport. This projection preserves invariant phase relations while discarding non-invariant information. As a result, measurement outcomes appear discrete and probabilistic at the representational level. Correlations between subsystems arise from shared geometric structure and exhibit nonlocal consistency without introducing additional dynamical mechanisms. The analysis clarifies how measurement and classical behavior can be understood within a geometric account of quantum kinematics.
Quantum mechanics is traditionally formulated in terms of complex Hilbert spaces, wavefunctions, and linear evolution equations. While extraordinarily successful, this framework leaves open several conceptual questions concerning the geometric origin of the wavefunction, the meaning of quantum phase, and the status of Hilbert space as a fundamental structure. In this work, we investigate a coherence-geometric reformulation in which the fundamental object is a coherence coordinate , where represents radial coherence and represents phase coherence associated with geometric transport. We show that the standard structures of quantum mechanics admit a natural interpretation within a differential-geometric framework. The wavefunction emerges from coherence coordinates, probability density is generated by radial coherence, probability flow is generated by phase coherence, and Hilbert spaces appear as spaces of square-integrable functions defined on a coherence manifold. The resulting formulation provides a unified geometric language connecting quantum mechanics, geometric phases, fiber bundles, holonomies, gauge theory, and differential geometry
Quantum mechanics is formulated in terms of complex phase and action-dependent amplitudes. In this work, quantum phase is identified with the holonomy of a geometric connection, and Planck's constant appears as the conversion scale between geometric transport and physical action. This provides a geometric formulation of quantum kinematics in which phase accumulation, noncommutativity, and quantization arise naturally from connection geometry and topology. This paper is Part I of a series which develops a geometric framework for quantum mechanics based on holonomy and transport structure. Subsequent papers develop operator representations and dynamical structure. This Zenodo record establishes priority and archival timestamp. Public release via arXiv is in progress.
This paper analyzes composite quantum systems from the perspective of bundle connections and holonomy. A geometric framework for composing subsystems is developed, and the behavior of phase information under restriction to subsystems is examined. It is shown that when access is limited to a subsystem, interference terms associated with relative phase can be suppressed in the reduced description, leading to additive weights for distinguishable components. This provides a geometric account of decoherence and the emergence of effectively classical behavior in composite systems. The analysis is formulated in terms of connection geometry and holonomy structure.
In this paper, we extend the framework introduced in [1], which proposed a two-dimensional model of time as a means to reconcile relativistic invariance with quantum discreteness. Building on this foundation, we derive key principles of quantum theory such as the uncertainty and exclusion principles and also introduce a fourth postulate: “The probability of an event is the same in all inertial frames of reference (IFR), independent of the observer's position in space.” This postulate enables us also to derive the probability amplitude interference from a purely relativistic time-phase geometry. We show that trajectories in the two-dimensional time plane, governed by relativistic constraints and discrete frequency-phase dynamics, naturally give rise to quantized measurements and path-based superposition. Using the Mach-Zehnder interferometer as an illustrative case, we demonstrate that the square of the proper time trajectory corresponds to quantum probability and interference phenomena emerge from the structure of phase evolution in time. Our formulation also aligns with Feynman’s path integral framework, showing that quantum mechanics can be interpreted as a direct geometric consequence of extended relativistic principles.