Physical unambiguity of the definition of the photon position operator and its special eigenstates

The paper provides three definitions of the photon position operator based on: the Poincare group generator, the transversality condition and the helicity operator. In each case, the correctness of the definition and Hermitianness of the operator in the domain of physical states are proven. All considered definitions lead to the same form of the position operator in the domain of physical states. The components of the photon position operator do not commute, but in analogy to the problem of eigenangular momentum, its eigenstates do exist. Three three-dimensional types of eigenstates of the position operator of a localized photon are given: on a straight line, on a plane close to a circle and on a plane close to a point. These states are eigenvalues of two or one component of the photon position operator.Comment: 14 pages, 16 references, 173 equations, 1 theorem, 1 lemma, 3 statement