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.

The scattering description is introduced by an evaluation of the Green's function, for a sufficiently large time interval, in terms of the discrete modes, and of linear combinations of free particle modes expressed by a unitary matrix which is an extension of that referring to modes of a single frequency. Transition probabilities are derived and summarized in a generating function that serves to evaluate occupation number expectation values for the final state, upon which is based the definition of differential and total scattering cross sections. A discussion is presented of various symmetry operations and the resulting properties of cross sections. Then, a determinantal formula for the individual transition probabilities is used to examine the probability for the persistence of a state, in its dependence upon occupation numbers. An incidental result of this analysis is a qualitative upper limit to total cross sections in relation to the character of the angular distribution. A section is devoted to the properties of eigenphases, including the demonstration of equivalence between phase shifts and eigenphases, and the discussion of alternative procedures for their evaluation in terms of quantities exhibited as convergent power series in the potential. Finally, the determinantal asymptotic behavior is used to obtain a high-energy approximation to the eigenphases for an isotropic scalar potential. The resulting high energy, small angle, form of the scattering cross section is discussed in the extreme quantum and classical limits. An alternative derivation of the high-energy scattering formula is provided in terms of an approximate construction of the Green's function.