Minkowski and Abraham Momenta Revisited from the Perspective of Photon Dynamics

Technical Summary: Minkowski and Abraham Momenta Revisited from the Perspective of Photon Dynamics

Problem Statement
The distinction between Minkowski ($S_{Min} = D \times B$) and Abraham ($S_{Abr} = E \times H$) momentum densities in material media has been a subject of long-standing debate regarding their physical interpretation and applicability. While previous resolutions, such as those by Barnett and coworkers, utilized the point-dipole approximation for atoms to argue that both momenta are physically meaningful (corresponding to canonical and kinetic momenta, respectively), this approach relies on the dynamics of atoms interacting with fields. This paper addresses the problem from a distinct perspective: the dynamics of photons within a medium, utilizing a field-theoretic formulation to determine whether the identification of these momenta as canonical or kinetic is representation-dependent.

Methodology
The authors employ a photon wave-function formalism, specifically utilizing a four-component spinor representation analogous to the Dirac equation. The methodology involves:

1. Dual Representations: The study defines two distinct wave-function representations:
   • The $\tilde{\psi}$ representation, constructed from the electric displacement field ($D$) and magnetic induction ($B$).
   • The $\psi$ representation, constructed from the electric field ($E$) and magnetic field ($H$).
2. Quantization: The electromagnetic field and the medium are quantized as a coupled whole within the $\tilde{\psi}$ framework. The photon wave function is expanded using creation and annihilation operators for photons and antiphotons in helicity space.
3. Operator Analysis: The authors analyze the momentum operators in both representations. They demonstrate that in the $\tilde{\psi}$ representation, the Minkowski momentum arises from an ordinary derivative ($\nabla$), while the Abraham momentum arises from a covariant derivative ($D_i = \partial_i - ieA_i$) involving a medium-induced effective gauge field.
4. Effective Hamiltonian: An optical Dirac-like evolution equation is derived for the medium described by a specific dielectric tensor. This allows for a semiclassical analysis of the Hamiltonian to identify coupling terms.

Key Contributions and Results

• Representation Dependence: The paper establishes that the identification of Minkowski and Abraham momenta as canonical or kinetic is not absolute but depends on the chosen wave-function representation.
  • In the $\tilde{\psi}$ (D, B) representation: Minkowski momentum corresponds to the canonical momentum (ordinary derivative), and Abraham momentum corresponds to the kinetic momentum (covariant derivative).
  • In the $\psi$ (E, H) representation: The roles are interchanged; Abraham momentum is canonical, and Minkowski momentum is kinetic.
• Gauge Field Connection: The relationship between the two momenta is encoded through a medium-induced effective gauge field ($A$). In an inhomogeneous or anisotropic medium, the transformation between the two wave-function representations involves a gauge transformation of the form $\psi \propto \exp[-i \int A_i dx_i] \tilde{\psi}$. The total momenta remain equivalent upon spatial integration, though local momentum densities differ.
• Spin–Zeeman Coupling: Within the derived optical Dirac equation, the authors identify a spin–Zeeman-type coupling term ($-\sigma_z B^{eff}_z / k_z$) in the effective Hamiltonian. This term emerges even in the absence of an external magnetic field, arising from the interaction between the photon spin and the effective magnetic field generated by the medium's anisotropy (specifically the torsion in a helical waveguide context).
• Microscopic Origin of the Optical Hall Effect: The spin–Zeeman coupling is identified as the microscopic origin of the optical (photonic) spin Hall effect. The coupling leads to helicity-dependent phase velocities and propagation constants. The paper demonstrates that the accumulation of photons with different helicities results in a macroscopic transverse displacement of the beam centroid, consistent with the photonic spin Hall effect.
• Dual Description: The effective two-dimensional photon Hamiltonian is shown to be formally dual to the Hamiltonian of an electron in an external gauge potential, with Minkowski and Abraham momenta naturally mapping to canonical and kinetic momenta, respectively.

Significance
The paper claims to provide a resolution to the Abraham–Minkowski controversy by shifting the perspective from atomic dipole dynamics to photon dynamics within a quantized field-theoretic framework. By showing that the distinction between the two momenta reflects different quantization schemes (specifically, whether the field is quantized with or without the medium as a coupled whole), the work offers a natural explanation for the coexistence of both momenta. Furthermore, it links the fundamental properties of electromagnetic momentum in media to the microscopic mechanism of the photonic spin Hall effect via a spin–Zeeman coupling derived from a Dirac-like equation for photons. This unifies the understanding of momentum conservation, gauge theory in optics, and spin-orbit interactions of light.