Quantum Mechanics of a Single Photon and the question of its Localizability in Space

We use the dynamical Maxwell equations to determine the space of state vectors of a single photon. We endow this space with a relativistically invariant positive-definite inner product to make it into a Hilbert space. We identify the Hamiltonian operator with the generator of time translations, construct momentum and helicity operators, and introduce a chirality (direction-of-time) operator. Next, we construct a position operator that has commuting components. These also commute with the helicity and chirality operators. We obtain the eigenstates of the position operator, which we identify with the localized states of the photon, and use them to determine photon’s position wave function. The position wave function for the localized states has a delta-function singularity at a single point in space, but their electromagnetic fields diverge on a particular plane containing this point. This behavior turns out to be related to an implicit freedom in the choice of the position operator. Each choice for the position operator determines the plane at which the electromagnetic fields of a given localized state diverge.