Photon quantum mechanics with a position observable
arXiv:2510.20049
The paper develops a Lorentz‑invariant quantum theory for photons, defining a photon position operator and showing how the probability of finding a photon in a region relates to the Fourier transform of plane‑wave amplitudes.
Abstract
We second quantize an explicitly Lorentz invariant lagrangian density and derive a theory of photon quantum mechanics. The one photon Hilbert space is the vector space of normalizable positive frequency four-potentials. Observables are described by the Poincare operators augmented with a photon position operator. It is found that the probability amplitude to observe a photon in a bounded region of space, defined as the projection of the four-potential onto the basis of position eigenvectors, equals the inverse Fourier transform of the probability amplitude for a plane wave. A continuity equation that describes photon propagation in free space and an optical circuit is derived.
Replacement of Photon Quantum Mechanics and Covariant Photon Current. The treatment is covariant and the discussion of the position observable is new