β Scribed by Andrzej Staruszkiewicz
No coin nor oath required. For personal study only.
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*. The dependence on e* arises because the complete set of states is generated in a nonlinear, e* 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*/fic< I[. !Q 1989 Academic Press, Inc.
Mechanical units are those of duration, length, and mass. We use mechanical units such that fi = c = 1. We use metric such that (ds)2 = (dx0)2 -(dx')Z -(dx2)' -(dx3)2.
Electric units comprise additionally the unit of charge. We use the Gauss electric units in which the action of the electromagnetic field is -& j-d4x FpYFpv, F,,=a,A,-&A,.
When there is an external electric current jr, the action is -& j d4x FpYFp, -j d4x A, jfl.
The relation between mechanical units and electric units follows from the Hamilton-Jacobi equation for a classical particle of mass m and charge e moving in the potential A,: 354 ooo3-4916/89 87.50
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