The point form of quantum dynamics and a 4-vector coordinate operator for a spinless particle

Abstract

We construct the analog in the quantum mechanics of a free spinless particle, of Dirac’s formula for the generators of space–time translations in his point form of classical dynamics, where one takes as fundamental variables the generators of homogeneous Lorentz transformations and the coordinate 4-vector of the point where the world line of the particle meets one sheet of a two-sheeted hyperboloid in space–time. A 4-vector coordinate operator is determined for such a particle, with commuting Hermitian components. The corresponding observable is the analog of the coordinate 4-vector of the point on the hyperboloid. This operator bears the same relation to such a surface as the Newton–Wigner operator does to an instant.