What question did this study set out to answer?

The aim is to explore how phase-coherent configurations organize geometrically in three dimensions without specific dynamical models.

May 16, 2026Open Access

Gauge Structure from Phase Coherence: A Kinematic Origin of the Hopf Sector

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Geometric analysis reveals a U(1) structure in phase-coherent configurations, indicating its importance in three dimensions.

The aim is to explore how phase-coherent configurations organize geometrically in three dimensions without specific dynamical models.
Investigated geometric organization of phase-coherent configurations around compact defects in 3D.
Analyzed the emergence of a U(1) connection structure and the phase field as a nontrivial fiber bundle.
Compared geometric frameworks with the localized relativistic Dirac wave packets.
Established a minimal globally regular realization via the Hopf fibration S3→S2, yielding a topological current from bundle geometry.
Identified that the minimal defect-supporting solution aligns with the topological sector revealed in the kinematic analysis.
Proposed that the Hopf structure is a fundamental principle for organizing globally regular phase coherence in three dimensions.

Ping Zhang (2026) studied this question.

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