Use of Retarded Green's Functions in Exciton Theory

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Abstract
It is pointed out that the usual treatment of excitons in interaction with radiation with spatial dispersion does not obviously satisfy causality in all cases, and it is shown explicitly that certain paradoxes and ambiguities can appear even in simple cases. In particular, it is shown for a simple special case that an apparently reasonable choice of boundary condition leads to a reflection coefficient greater than unity; and also that in some cases one must make rather arbitrary and inconsistent requirements in order to get the right number of boundary conditions to determine a solution. The theory is reformulated in a way that is manifestly causal, with both field and polarization being generated by retarded Green's functions. Umklapp processes are also included. This formulation confirms the results of the usual treatment in the cases in which the latter are unambiguous and acceptable, but avoids the difficulties mentioned above. The model used is essentially classical and one dimensional, but nevertheless sufficiently general for the treatment of these problems.