Complex Spacetime Geometry and the Geometric Origin of the Aharonov-Bohm Effect

The Aharonov-Bohm (AB) effect fundamentally demonstrates that electromagnetic potentials, rather than strictly classical fields, play a foundational, observable role in quantum mechanics. While standard quantum theory successfully accommodates this phenomenon via the axiomatic insertion of the minimal coupling principle, a purely geometric interpretation of the gauge potential has historically remained elusive. Building on a previously established complex spacetime framework—where conventional spacetime coordinates are extended into real and imaginary domains—this paper proposes a novel, localized geometric origin for the AB effect. We postulate that the imaginary spatial derivative of the complex wavefunction is physically equivalent to the electromagnetic gauge potential, thereby deriving the minimal coupling prescription directly from spacetime geometry rather than treating it as an independent axiom. Under this framework, the topological phase shift experienced by a charged particle in a field-free region arises naturally from its local evolution through the imaginary dimensions of complex spacetime. Furthermore, by treating the gauge potential as an affine connection or cross-metric term that links real and imaginary dimensions, we successfully redefine the complex metric tensor. This allows the classical electromagnetic field tensor to emerge naturally as the Riemann curvature of the complexified space. This approach provides a unified geometric foundation for both gauge potentials and quantum interference phenomena, resolving previous field-tensor metric limitations while offering a promising mathematical framework for future integrations into relativistic quantum mechanics and cosmic geometry.