This preprint introduces field-depth (w) as an intrinsic resonance coordinate associated with spatial extension, providing a geometric origin for key quantum phenomena. In this framework, the wavefunction is defined over (x, y, z, w), where w represents the depth of standing-field resonance rather than an additional spatial axis. A localized particle state is then the 3D projection of a stable standing-wave node in this extended field structure. This formulation yields direct geometric interpretations of several quantum-mechanical features that are typically treated as abstract or axiomatic: Wave–particle duality emerges naturally: all quantum entities are extended waves, while particle detections correspond to nodal projection into 3D space. Spin arises as a phase-orientation condition in the (r, w) resonance plane, producing the known 4π periodicity of spin-½ systems. Quantum entanglement is explained as shared field-depth node structure, enabling correlated outcomes without requiring signal propagation across 3D space. This model does not modify the Schrödinger or Dirac equations; instead, it clarifies the geometric context in which their solutions are realized. The approach is consistent with the SU(2) fiber bundle structure of spinor fields and with experimental tests of nonlocal quantum correlations. This geometric extension provides a unified physical interpretation of localization, spin holonomy, and entanglement without invoking wavefunction collapse, hidden variables, or new force laws. It suggests that quantum states are extended structures of space itself, rather than probabilistic abstractions. A follow-up preprint will address implications for the electron as a stable field-depth resonance node and its role in chemical bonding, atomic structure, and coherence phenomena.
Nilsson, Henrik (2025) studied this question.