The standard text book Green's function possesses a self-energy that is known to be an optical potential for elastic scattering. The introduction of an optical potential reduces the complex many-body scattering problem into a tractable one-body problem. In this paper inelastic Green's functions are introduced and discussed which possess self-energies that are optical potentials for inelastic scattering. If the projectile is indistinguishable from particles comprising the target, intriguing aspects arise even for noninteracting particles.
The completely antisymmetric solution ${\ensuremath{\Psi}}^{A}$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}{\ensuremath{\phi}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}}$, where ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ is a complete set of antisymmetric states for the target and ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are one-particle functions. Coupled equations for the ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\ensuremath{\Psi}}^{A}$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\ensuremath{\psi}}_{0}$ is shown to obey an inhomogeneous differential equation, rather than a Schr\"odinger equation. The homogeneous solution is identical to the elastic wave function obtained when the projectile is distinguishable, while the inhomogeneous solution is due entirely to exchange effects. The function ${\ensuremath{\psi}}_{0}$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\ensuremath{\psi}}_{0}$ obeys a Schr\"odinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\ensuremath{\psi}}_{0}$ yields the exact elastic amplitude including direct and exchange contributions, and a phase-shift analysis of the exchange term is given. The more standard form of solution ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}\mathcal{a}{{\ensuremath{\phi}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}}$, where
a is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.
The recent technical advances in the creation of rapidly rotating light polarizations are shown to provide new tools to explore elastic and inelastic scattering phenomena. In the case of inelastic scattering, the light fluoresced by atoms or molecules coherently excited by a rapidly rotating linear polarization is expected to exhibit new resonances. Such resonances, due to the transfer of coherence from light to atoms or molecules, are theoretically studied and experimentally observed. They provide simple methods to probe atomic internal constants, such as energy level life-times. The analogy between those resonances and the Hanle effect suggests new experiments in atomic physics. In the case of elastic scattering such as the one encountered during propagation through a dense turbid medium, the rotating linear polarization is shown to be an ideal tool to extract the ballistic photons from the large background of diffused light. This permits to image tiny objects embedded in a dense turbid medium, with an experimental resolution similar to the one obtained in usual homogeneous optical system.