Photon Quantum Mechanics

We second quantize the Fermi Lagrangian in the Lorenz gauge to obtain a covariant theory of photon quantum mechanics. Number density is real so it is interpreted as position probability density. The Hilbert space is the vector space of fields with norm 1 describing physical photons and the Poincare operators are extended to include position to represent observables. A photon continuity equation is derived that describes creation, propagation and annihilation of photons in an optical circuit. The relationship to orthodox quantum mechanics is discussed.

💡 Research Summary

The paper presents a covariant formulation of photon quantum mechanics by second‑quantizing the standard electromagnetic Lagrangian while imposing the Lorenz gauge as a subsidiary condition. Starting from the classical Lagrangian density
(L_{\rm std}= -\frac{1}{4}\varepsilon_{0}c^{2}F_{\mu\nu}F^{\mu\nu}-J^{\mu}{e}A{\mu}),
the author enforces (\partial_{\mu}A^{\mu}=0). In this gauge the longitudinal component of the four‑potential satisfies (A_{\parallel}= \phi/c), which leads to the cancellation of the Gupta‑Bleuler scalar and longitudinal modes; only the two transverse helicity states ((\lambda=\pm1)) survive in free space.

The field operators are split into positive‑frequency ((A^{+})) and negative‑frequency ((A^{-}=A^{+\dagger})) parts:
\