Canonical Transformation for an Electron-Positron Field Coupled to a Time-Independent Electromagnetic Field. II

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Abstract
A prescription for constructing well-defined field operators from a single particle operator is given in a form in which electrons and positrons appear in a symmetric way. Such a prescription removes certain trivial infinities arising from the conventional "hole" theory approach. The canonical transformation derived in a previous paper is used to calculate the vacuum expectation values of various operators to the second order in electron charge. It is shown that, despite the fact that the scattering operator does not exist when the electromagnetic field is constant in time, one obtains the usual results for current and charge densities. In addition, the striking result is proved that for certain purely electrostatic fields the expectation values of the number operator are finite but that when a magnetostatic field is also present the expectation values are infinite.