Retardation and dispersive effects in the nuclear mean field

• Published in: Physics reports 1993-03, Vol.224 (5), p.237-360
• Main Authors: Mahaux, C., Davies, K.T.R., Satchler, G.R.
• Format: Article
• Language: English
• ISSN: 0370-1573; 1873-6270
• DOI: 10.1016/0370-1573(93)90017-8

The one-body potential which represents the average interaction between a particle and a medium depends upon energy. This is equivalent to a nonlocality in time. The energy dependence and the nonlocality in time are connectes by a Fourier transform. The temporal nonlocality is a retardation effect because it entails that the wave function at time t is influenced only by its values at other times which must be prior to t. This reflects a causality property. We investigate the temporal nonlocality of the nuclear mean field. A comparison is made with the temporal nonlocality of the relation between the electric displacement and the electric field in classical electrodynamics. We consider several parametrizations of the energy dependence of the imaginary part of the field, for nucleons as well as heavy ions. These parametrizations specify the energy dependence of the corresponding real part, because the real and imaginary parts are connected by a dispersion relation. The latter can be viewed as equivalent to the causality property. Since Hilbert transforms appear in the dispersion relation and since Fourier transforms give the correspondence between energy dependence and temporal nonlocality, we derive several properties of these transforms which are of particular interest in the present context. The most useful one is that the Fourier transform of a function F( E) which is analytic in the upper half of the complex E-plane can be expressed in terms of the Fourier transform of the imaginary part of F ( E) alone. We investigate several schematic models for the mean field. They fall into two main categories. These correspond to the two main definitions which have been proposed for the mean field, namely the self-energy and Feshbach's potential. Both of these definitions can be used for the nucleon-nucleus system, in which case they correspond to two different ways of handling the combined influence of ground state correlations and antisymmetrization. The resulting two mean fields have different energy dependences and, correspondingly, temporal nonlocalities. Feshbach's approach can also be applied to the nucleus-nucleus system. Our schematic models are semi-realistic, in the sense that they all take account of the “Fermi surface anomaly” for the nucleon-nucleus system or of the “threshold anomaly” for the nucleus-nucleus case. The temporal nonlocality is investigated for each model. A physical interpretation of this nonlocality is given in terms of a time delay o