Polarization propagator moment expansion

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

When the perturbing potential V is an external electric field and the property of interest A is the dipole moment, the coefficients, indicated in the expansion above by the double angle brackets, are referred to as polarization propagators. These functions are equivalent to the tensor components... 

Making use of this superoperator formalism, the moment expansion of the polarization propagator can be written as... 

Comparing this with the classical expansion of a time-dependent dipole moment in Eq. (7.18) we can identify the frequency-dependent polarizability tensor as a linear response function or polarization propagator... 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

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