The Theory of Quantized Fields. V

The Dirac field, as perturbed by a time-dependent external electromagnetic field that reduces to zero on the boundary surfaces, is the object of discussion. Apart from the modification of the Green's function, the transformation function differs in form that of the field-free case only by the occurrence of a field-dependent numerical factor, which is expressed as an infinite determinant. It is shown that, for the class of fields characterized by finite space-time integrated energy densities, a modification of this determinant is an integral function of the parameter measuring the strength of the field and can therefore be expressed as a power series with an infinite radius of convergence. The Green's function is derived therefrom as the ratio of two such power series. The transformation function is used as a generating function for the elements of the occupation number labelled scattering matrix $S$ and, in particular, we derive formulas for the probabilities of creating $n$ pairs, for a system initially in the vacuum state. The general matrix element of $S$ is presented, in terms of the classification that employs a time-reversed description for the negative frequency modes, with the aid of a related matrix $\Sigma$, which can be viewed as describing the development of the system in proper time. The latter is characterized as indefinite unitary, in contrast with the unitary property of $S$, which is verified directly. Two appendices are devoted to determinantal properties.