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An Ontological Interpretation of Photon Wave-Particle Duality via Complex-Space Trajectories

This paper proposes a unified ontological interpretation of photon wave-particle duality within the relativistic quantum Hamilton-Jacobi framework, demonstrating that complex-space trajectories—where real projections describe propagation and imaginary components encode oscillatory structure—reconcile wave and particle behaviors as complementary aspects of a single underlying motion without altering standard quantum mechanics.

Original authors: Shiang-Yi Han, Ciann-Dong Yang

Published 2026-01-30

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CC BY 4.0

Original authors: Shiang-Yi Han, Ciann-Dong Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Question: Is Light a Particle or a Wave?

For over a century, physicists have been puzzled by a strange fact: light (photons) acts like a tiny bullet (a particle) in some experiments, but like a rippling pond (a wave) in others. This is called wave-particle duality.

Standard quantum mechanics says, "We can predict what will happen, but we won't tell you what the light is actually doing between measurements." It treats the wave and the particle as two different descriptions of the same mystery.

This paper asks a different question: What if the light isn't switching between two modes? What if it is doing one single, complex thing that looks like a particle from one angle and a wave from another?

The Core Idea: The "Shadow" Analogy

Imagine you are holding a strange, 3D object in a dark room.

If you shine a light on it from the front, its shadow on the wall looks like a straight, solid line (a particle).
If you shine the light from the side, its shadow looks like a wiggly, oscillating curve (a wave).

The object itself hasn't changed. It hasn't turned into a line or a wave. It is a single, complex 3D shape, and the "wave" and "particle" are just different projections (shadows) of that same shape.

The authors of this paper propose that a photon is exactly like that 3D object. It is moving through a special kind of space called Complex Space.

Real Space: The world we see and measure (where the photon travels forward).
Imaginary Space: A hidden, mathematical dimension attached to the real one.

How the "Dance" Works

In this new framework, a photon doesn't just move in a straight line. It moves along a complex trajectory that has both a "real" part and an "imaginary" part.

The Straight Line (Particle Mode):
If the photon is in a simple state (like a pure beam of light), its path in this complex space is a straight line. When you project this onto our real world, it looks like a particle zooming forward at the speed of light. There is no "wiggling" because the imaginary part of its motion is zero.
The Wiggly Dance (Wave Mode):
When the photon is in a "superposition" (a mix of different states, like in a double-slit experiment), its path in complex space becomes a wiggly, spiraling dance.
- The Real part of the dance still moves forward.
- The Imaginary part of the dance oscillates back and forth.
- When we look at the "shadow" of this dance in our real world, that oscillation in the imaginary dimension shows up as the wave pattern (the ripples).

The Analogy: Think of a surfer riding a wave.

The surfer is the photon moving forward (the particle).
The wave they are riding is the oscillation in the imaginary dimension.
The paper suggests the surfer and the wave are actually the same thing: a single motion happening in a higher-dimensional space.

The "Quantum Potential" Trap

The paper introduces a concept called the Quantum Potential. You can think of this as an invisible landscape or a "force field" created by the photon's own wave nature.

In a straight line: The landscape is flat. The photon rolls straight through.
In a superposition: The landscape becomes bumpy and creates "valleys" and "hills."
- If the photon gets trapped in a valley in this complex landscape, it can't escape. It starts bouncing back and forth.
- When we look at this trapped motion from our real-world perspective, it looks like a standing wave (a wave that vibrates in place, like a plucked guitar string).

The authors ran simulations showing that if you trap the photon in this complex "valley," its shadow in the real world creates perfect standing-wave patterns.

Did the Math Check Out?

The authors wanted to make sure this idea didn't break the laws of physics. They asked: "If we calculate the energy of these weird, trapped, wiggly shadows, does it add up to the energy of the original light source?"

They tested this using different laser colors (like Ruby lasers and others).

They calculated the "wavelength" of the shadows in the complex space.
They converted those wavelengths back into energy.
The Result: The calculated energy matched the original laser energy almost perfectly (with an error of only 0.00026%).

This suggests that even though the photon is doing a complex dance in a hidden dimension, the total energy is conserved. The "particle" energy and the "wave" energy are just two sides of the same coin.

Summary

This paper doesn't invent new laws of physics or new particles. Instead, it offers a new geometric picture:

Light is always a single object moving in a complex space (Real + Imaginary).
Particles are what we see when the motion is simple and straight.
Waves are what we see when the motion is complex and oscillating in the hidden imaginary direction.
Standing waves are what we see when the photon gets trapped in a "potential well" in this complex space.

By viewing the photon's journey as a single, unified dance in a higher-dimensional space, the paper suggests that the confusing "duality" of light is actually just a matter of perspective.

Technical Summary: An Ontological Interpretation of Photon Wave-Particle Duality via Complex-Space Trajectories

Problem Statement
Wave-particle duality remains a central interpretational challenge in quantum theory. While standard quantum mechanics accurately predicts experimental outcomes via the probabilistic wave function, it deliberately refrains from describing the underlying physical reality between measurements. This instrumentalist stance leaves open fundamental ontological questions: does duality reflect a fundamental dualism of physical entities, or are wave and particle aspects merely different representations of a single underlying process? Recent experimental developments, particularly weak-measurement reconstructions, suggest that physically accessible quantities (such as weak values of position and momentum) can take intrinsically complex values. This motivates the exploration of whether extending configuration space into the complex domain can provide a coherent, realist description of quantum dynamics without altering the formal structure of standard quantum mechanics.

Methodology
The authors employ a relativistic quantum Hamilton–Jacobi (RQHJ) framework formulated in an extended complex space. In this approach:

Complex Action and Trajectories: The wave function $\Psi$ is related to a complex action function $S$ via $S = -i\hbar \ln \Psi$ . The space-time coordinates $q_\alpha$ and the action are treated as complex values.
Equations of Motion: The dynamics are governed by the RQHJ equation, which is shown to be formally equivalent to the Klein–Gordon equation. The equations of motion are derived from the canonical momentum $p_\alpha = \partial S / \partial q_\alpha$ , yielding complex trajectories $q_\alpha(\tau)$ parameterized by proper time.
Projection Mechanism: The framework posits that the real projection of a complex trajectory describes propagation, while the imaginary components encode oscillatory structure. The "wave" nature is not an independent field but a geometric feature of the trajectory in the complex plane.
Analysis Scope: The study analyzes the motion of a single free photon in:
- One-dimensional complex space ( $x \in \mathbb{C}$ ).
- Two-dimensional complex space ( $x, y \in \mathbb{C}$ ), considering separable superposition states.
Consistency Checks: The authors verify energy-momentum consistency by extracting effective wavelengths from projected standing-wave patterns in the real space and comparing the reconstructed total energy against the source photon energy using Einstein's $E=h\nu$ and de Broglie's $\lambda=h/p$ relations.

Key Contributions and Results

Unification of Eigenstates and Superpositions:
- Momentum Eigenstates: For a pure momentum eigenstate (no superposition), the complex quantum potential vanishes ( $Q=0$ ). The resulting trajectory is a straight line in the complex plane, corresponding to uniform propagation at the speed of light $c$ . This recovers the particle-like description.
- Superposition States: For superposition states (e.g., right- and left-propagating components), the quantum potential becomes non-zero. This induces complex velocities and non-straight trajectories. The motion in the complex plane becomes oscillatory.
- Classification of Motion: In 1D, the authors classify trajectories into three modes based on initial complex velocities: right-propagating, left-propagating, and bounded (closed loops). The bounded mode corresponds to a photon trapped in an effective potential well formed by the complex quantum potential.
Geometric Origin of Wave Phenomena:
- The study demonstrates that wave-like behavior (oscillations, standing waves) arises geometrically from the projection of bounded or oscillatory complex trajectories onto real coordinates.
- In 2D complex space, the interplay of propagation and boundedness in orthogonal directions yields four distinct trajectory morphologies:
  1. Propagation in both directions (weakly modulated straight line in real space).
  2. Propagation in one direction, bounded in the other (apparent wave-like trajectory in real space).
  3. Bounded in both directions (standing-wave-like patterns in real space).
- Crucially, the "standing wave" is interpreted not as a static field but as the projection of a photon moving in a bounded orbit within the higher-dimensional complex space.
Energy-Momentum Consistency:
- The authors perform an internal coherence analysis for the bounded (standing-wave) mode. By calculating effective wavelengths from the projected trajectories in various complex planes (e.g., $x_R-x_I$ , $x_R-y_R$ ), they reconstruct the total energy.
- Using a ruby laser source (694.3 nm), the sum of energies derived from six different projected standing-wave patterns yields a total energy of 1.7856795 eV. This differs from the source energy (1.78568342 eV) by only 0.00026%.
- Similar results are obtained for other light sources (IR diode, fiber diode, Yb:YAG, Nd:YVO4), confirming that the complex-trajectory description preserves energy-momentum constraints.

Significance and Claims
The paper claims to offer a unified geometric perspective on wave-particle duality that remains fully compatible with standard quantum mechanics.

Ontological Interpretation: The framework provides an ontological picture where wave and particle aspects are complementary projections of a single underlying complex motion. The distinction between the two is determined by the nature of the superposition: eigenstates yield straight trajectories (particle-like), while superpositions yield oscillatory complex paths (wave-like).
No New Physics: The authors emphasize that this is an interpretational framework, not a modification of physical laws. No new dynamical laws, hidden variables beyond the wave function, or additional physical dimensions are introduced. The complex space serves as a "bookkeeping device" to render experimentally observed complex weak values geometrically explicit.
Reconciliation of Relations: The study demonstrates that Einstein's photon energy formula ( $E=h\nu$ ) and the de Broglie relation ( $\lambda=h/p$ ) emerge as mutually consistent constraints within a single dynamical picture, rather than as separate postulates for distinct entities.
Limitations: The authors explicitly state that this work is a conceptual and structural investigation. It does not resolve the quantum measurement problem, account for state reduction, or address interactions with measurement apparatuses. The energy-consistency check is presented as an internal coherence verification rather than an independent experimental validation. Future work is needed to interface this approach with operational measurement schemes and extend it to multi-mode photonic states.