The Theory of Quantized Fields. VI

This paper treats the effect of a time-independent external electromagnetic field upon a Dirac field by constructing the transformation function in a representation adapted to the external field. In addition to the alteration of the Green's function, the structure of the transformation function differs from that of the zero field situation by a factor which describes the energy of the modified vacuum state. A formula for the vacuum energy is obtained and expressed in a form appropriate to a localized field, in terms of the energy eigenvalues of discrete modes, and of the phase shifts associated with continuum modes. Determinantal methods are then introduced, and the class of fields is established for which a certain frequency-dependent modified determinant is an integral function of the parameter measuring the strength of the field. The properties of the determinant are investigated in the two frequency regions $\left|p_0\right|<m$ and $\left|p_0\right|>m$, with regard to the zeros of the real determinant in the former region, which are the frequencies of the discrete modes, and to the phase of the complex determinant in the latter region. In the second situation, a connection is established with a unitary matrix defined for modes of a given frequency, and the phase of the determinant is expressed in terms of the eigenphases of this matrix. Following a discussion of the asymptotic behavior of the determinant as a function of $p_0$, the modified determinant is constructed in terms of the discrete mode energies and of the eigenphases. This yields a more precise version of the vacuum energy formula, in which a single divergent parameter is exhibited, for a suitable class of fields.