6 - Liouville Formalism

Summary

The Fano theory, described here, does not adopt the impact approximation, and is not confined to the line core. The main concession, implicit in an impact theory like that of Baranger or Anderson, is the neglect of initial correlations between the states of radiator and bath, allowing a separate average to be taken over the bath variables at the initial time. For Fano, the correlation function describes the linear response to the driving field and is governed dynamically by the Liouville operator, which, in Baranger’s line space, has a role similar to that of the Hamiltonian in the original state space. The response, in its Fourier transform, provides the line shape, and this is governed here by a relaxation operator that looks, formally, like a transition operator in quantum scattering. This is a very general theory that will, nevertheless, reduce to that of Baranger as soon as the impact approximation is imposed. Although the neglect of initial correlations will invalidate the fluctuation–dissipation theorem, this will only affect the line far-wing, where, unless remedied, it will cause an imbalance between the radiative processes induced by the field.