Dirac Equation for Photons: Origin of Polarisation
arXiv:2303.18196 · doi:10.1016/j.heliyon.2024.e28367
Abstract
Spin is a fundamental degree of freedom, whose existence was proven by Dirac for an electron by imposing the relativity to quantum mechanics, leading to the triumph to derive the Dirac equation. Spin of a photon should be linked to polarisation, however, the similar argument for an electron was not applicable to Maxwell equations, which are already Lorentz invariant. Therefore, the origin of polarisation and its relationship with spin are not completely elucidated, yet. Here, we discuss propagation of coherent rays of photons in a graded-index optical fibre, which can be solved exactly using the Laguerre-Gauss or Hermite-Gauss modes in a cylindrical or a Cartesian coordinate. We found that the energy spectrum is massive with the effective mass as a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional () non-relativistic Schrödinger equation, which is equivalent to the space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the Dirac equation with spin. We applied the Bardeen-Cooper-Schrieffer (BCS)-Bogoliubov theory of superconductivity to a coherent ray from a laser and identified a radiative Nambu-Anderson-Higgs-Goldstone mode for recovering the broken symmetry. The spin expectation value of a photon corresponds to the polarisation state in the Poincaré sphere, which is characterised by fixed phases after the onset of lasing due to the broken symmetry, and it is shown that its azimuthal angle is coming from the phase of the energy gap.
References in corpus (30)
- Two-Dimensional Gas of Massless Dirac Fermions in Graphene
- Weyl and Dirac Semimetals in Three Dimensional Solids
- Correlated Electrons in High Temperature Superconductors
- Spin-orbit interactions of light
- The angular momentum controversy: What's it all about and does it matter?
- Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium
- Classical and quantum properties of cylindrically polarized states of light
- Reply to the Comment on "Spin and orbital angular momentum in gauge theories: Nucleon spin structure and multipole radiation revisited" [PRL 100:232002 (2008)]
- Spin and orbital angular momentum in gauge theories: Nucleon spin structure and multipole radiation revisited
- Spin and orbital angular momentum in gauge theories (II): QCD and nucleon spin structure
- Photon Mass Bound Destroyed by Vortices
- Quantum degrees of polarization
- Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium
- Geometric phases of light: insights from fibre bundle theory
- Solutions of the Maxwell equations and photon wave functions
- Quantum field theory for spin operator of the photon
- Invisibility cloaks in relativistic motion
- Granular Superconductor in a honeycomb lattice as a realization of Bosonic Dirac Material
- Quantum structured light: Non-classical spin texture of twisted single-photon pulses
- Relativistic quantum-mechanical description of twisted paraxial electron and photon beams
- Spin and Orbital Angular Momentum of Coherent Photons in a Waveguide
- Special Theory of Relativity for a Graded Index Fibre
- Quantum Commutation Relationship for Photonic Orbital Angular Momentum
- Dirac Hamiltonians for bosonic spectra
- Dirac cones and mass terms in bosonic spectra
- Fresnel-Fizeau drag: Invisibility conditions for all inertial observers
- SU(2) Symmetry of Coherent Photons and Application to Poincaré Rotator
- Topological Polarisation States
- Spin of Photons: Nature of Polarisation
- No Proca Photons