Wave-particle duality in complex space

Shiang-Yi Han, Ciann-Dong Yang

Wave particle duality remains a central interpretational challenge in quantum theory. In this work, we develop a trajectory-based description of photon dynamics formulated in an extended complex space within the relativistic quantum Hamilton Jacobi framework. In this approach, photon motion is represented by complex trajectories whose real projections describe propagation, while imaginary components encode oscillatory structure. We show that momentum eigenstates correspond to straight line trajectories with uniform propagation at the speed of light, whereas superposition states lead to nontrivial quantum potentials and oscillatory motion in the complex plane. Extending the analysis to complex two dimensional space reveals richer dynamical behavior, including propagating wave like trajectories and standing wave like patterns in real projections. Energy momentum consistency is verified through an internal coherence analysis based on projected standing wave wavelengths. Rather than introducing new dynamical laws or additional physical dimensions, the complex space is employed as an interpretational framework that renders wave like and particle like aspects as complementary projections of a single underlying motion. The results suggest a unified geometric perspective on wave particle duality, while remaining fully compatible with standard quantum mechanics.

Bruce R. Wheaton

Paul Bracken

The equation of motion for a particle moving in complex space is obtained and then solved. The interaction between the real and imaginary components of motion can produce what can be regarded as a wave motion in real space. The formation of interference patterns and polarization of a matter wave can be accounted for by the particle’s motion in complex space. The deterministic particle view of a free particle can then be reconciled with its dual probabilistic wave picture but the formulation is entirely in complex space.

F. Selleri

Alireza Moallemzadeh Lake

This paper introduces a field-theoretic framework for wave–particle duality in which matter is described by a complex scalar field ψ carrying a conserved U(1) chargeIn the particle phase, the field condenses into a compact non-topological soliton (a Q-ball) with effective radius L ≈ r₀. In the wave phase, the same energy delocalizes in space and the field equation approaches the linear Klein–Gordon regime