Quantum mechanics of phase and charge and quantization of the coulomb field

Abstract

A closed dynamical system is constructed which is linear and has the electric charge as one of its degrees of freedom. The system consists of the electric part of the infrared electromagnetic field and of two additional degrees of freedom. Quantization of this system gives quantization of the electric charge in units equal to the constant e. An evidence is indicated that the theory has a nontrivial dependence on the constant e 2. The dependence on e 2 arises because the complete set of states is generated in a nonlinear, e 2 dependent way. In particular, comparing the present theory with the theory of unitary representations of the Lorentz group by Gelfand, Graev, and Vilenkin, one finds the unitarity condition 0 < e 2 khc < π .