A simple understanding of quantum electrodynamics using Bohmian trajectories: detecting non-ontic photons

1. Problem Statement

The paper addresses a fundamental conceptual challenge in quantum mechanics: the apparent incompatibility between Bohmian mechanics (which relies on deterministic particle trajectories in physical space) and Quantum Electrodynamics (QED) (which involves the creation and annihilation of photons).

• The Conflict: Standard interpretations often argue that a realist view based on trajectories cannot explain processes where particles (photons) are created or destroyed. In traditional QED, photons are treated as excitations of a quantized field, often requiring a "Fock space" formalism where the number of particles is not fixed.
• The Semantic Issue: The authors argue that the difficulty is largely semantic. If a photon is defined as an ontic (physically real) particle, its disappearance upon absorption is problematic for trajectory-based theories. However, if a photon is defined merely as a quantized amount of energy in an electromagnetic field, "creation" and "annihilation" are simply energy exchanges between matter and field modes, which can be described by a unitary Schrödinger equation without invoking particle ontology.
• The Measurement Paradox: How can a detector "click" (indicating a photon) if photons are not ontic entities? The paper seeks to explain how measurement outcomes (like photon partition noise) arise from the particle nature of the detector (matter) rather than an intrinsic particle nature of the light.

2. Methodology

The authors employ a mixed ontology within the Bohmian framework:

1. Ontic Matter: Electrons (fermions) are treated as point particles with well-defined trajectories ($r(t)$) in 3D physical space.
2. Emergent Fields: Electromagnetic fields are treated as non-ontic (emergent) quantities. They are described by well-defined field configurations evolving in time, represented by mode-decomposition coefficients ($q_\lambda(t)$).
3. Mathematical Framework:
   • Classical Hamiltonian: The system is modeled using a classical light-matter Hamiltonian with minimal coupling. The electromagnetic field is decomposed into modes (harmonic oscillators) using the Coulomb gauge.
   • Canonical Quantization: The system is quantized via the Dirac prescription. The matter coordinates ($r, p$) and field amplitudes ($q_\lambda, S_\lambda$) become operators acting on a Hilbert space.
   • Schrödinger Equation: The full system evolves according to a unitary Schrödinger equation in a high-dimensional configuration space ($r, q_1, \dots, q_M$).
   • Bohmian Guidance: The wave function $\Psi(r, q, t)$ is written in polar form ($R e^{iS/\hbar}$). The trajectories for both electrons and field modes are derived from the phase $S$ via guidance equations, ensuring the equivariance of the probability density $|\Psi|^2$ (preserving the Born rule).
   • Measurement Modeling: Measurement is modeled explicitly by coupling the system to "pointer" degrees of freedom (matter particles $y, z$) via a von Neumann interaction Hamiltonian. No "collapse" postulate is used; effective collapse emerges from the dynamical separation of wave function branches in the enlarged configuration space.

3. Key Contributions

• Rehabilitation of Early Proposals: The paper revives and formalizes earlier Bohmian proposals (e.g., by Bohm, Bell, and others) that treat fields as emergent rather than fundamental, demonstrating that QED can be modeled without ontic photons.
• Resolution of the "Particle" Paradox: It demonstrates that "photon creation/annihilation" is a semantic description of energy exchange. The unitary evolution of the Schrödinger equation in coordinate space is sufficient to describe these processes without Fock space operators, provided one accepts that the field variables are just coordinates.
• Explanation of Partition Noise: The paper provides a mechanism for photon partition noise (where a single photon is detected at one of two locations but never both) without attributing particle-like trajectories to the photon. The stochasticity arises from the initial conditions of the matter pointers (electrons in the detector), not the photon itself.
• Pedagogical Framework: It offers a computationally tractable way to visualize QED phenomena using standard non-relativistic quantum mechanics machinery, avoiding the abstraction of Fock space for practical simulations.

4. Results (Numerical Simulations)

The authors present numerical simulations of two electrons interacting with a single quantized electromagnetic mode in an optical cavity.

• Scenario A: Non-Measured System (Unitary Evolution)

  • Setup: Two electrons in quantum wells coupled to a cavity mode.
  • Dynamics: The system exhibits Rabi oscillations. Energy is coherently exchanged between the field and the electrons.
  • Observation: The energy of the "photon" ($\hbar\omega_c$) is distributed symmetrically between the two electrons in a superposition state ($\frac{1}{\sqrt{2}}(|100\rangle + |010\rangle)$). In the Bohmian picture, the field amplitude and electron positions evolve continuously, showing a "breaking" of the photon energy across both electrons simultaneously. This contradicts the naive particle view but is consistent with wave mechanics.

• Scenario B: Measured System (With Pointers)

  • Setup: Two measurement devices (pointers $y$ and $z$) are coupled to the electrons to measure their energy.
  • Dynamics: When the measurement interaction is activated, the wave function branches in the enlarged configuration space ($x_1, x_2, q, y, z$).
  • Outcome:
    • The Bohmian trajectory of the pointer selects one branch (e.g., $y$ moves, $z$ stays).
    • The conditional wave function for the system collapses effectively to a single state (e.g., $|100\rangle$ or $|010\rangle$).
    • Partition Noise: The simulation shows that the photon is detected at $x_1$ OR $x_2$, never both. This "anti-correlation" is an artifact of the particle nature of the detectors (the pointers), not the photon. The stochasticity of the outcome is determined by the initial position of the pointer particles (Quantum Equilibrium Hypothesis).

5. Significance

• Ontological Clarity: The paper argues for a "fermions-only" ontology in QED. Photons and electromagnetic fields are emergent properties arising from the dynamics of matter particles. This simplifies the conceptual foundation of QED by removing the need for ontic fields or particles of light.
• Measurement Theory: It resolves the measurement problem in QED without invoking wave function collapse. The "collapse" is shown to be an effective phenomenon resulting from the entanglement of the system with macroscopic matter pointers and the subsequent dynamical separation of branches.
• Experimental Interpretation: The findings clarify the interpretation of experiments measuring "weak values" of photon velocity. The authors argue that such experiments measure correlations between emitter and detector matter, not the velocity of an ontic photon (which does not exist in this framework).
• Computational Utility: By avoiding Fock space and using coordinate-space trajectories, the authors provide a framework (implemented in their software package QC-Slim) that makes QED phenomena more accessible for simulation and conceptual understanding, particularly for educational purposes.

In conclusion, the paper successfully demonstrates that quantum electrodynamics phenomena, including photon detection and partition noise, can be fully understood through a deterministic, trajectory-based model where only matter is ontic, and light is an emergent, quantized energy exchange.