Equation of Collective Submanifold for Mixed States

Abstract

With the aim of giving a microscopic description -of various nuclear phenomena observed in highly excited states, the equation of the collective submanifold for mixed states is presented. The basic idea is a possible TDHF-like variation in the enlarged space which is adopted in the thermo' field dynamics formalism. A set of equations, which determines the collective submasifold, is obtained. The form is analogous to that given in the conventional TDHF theory. At the small amplitude limit, an equation, the form of which is similar to the conventional RP A equation, can be derived in a natural way. One of the recent interests in study of the nuclear many-body theories may be to present a microscopic theory which enables us to describe nuclear phenomena ob­ served in highly excited states and interpreted in the language of the concept of thermal' equilibrium with the temperature T=f=.O. The references concerned with these phe­ nomena can be found in Refs. 1) and 2). In these phenomena, individual highly excited states are equally populated and usually the average properties of the system are measured. Therefore, the statistical approach is necessary. In contrast to the above phenomena, low lying states are related to T=O. We know a powerful' method for describing such states, i.e., the TDHF theory based on a single Slater determinant. The Slater determinant is an example of a pure state and the state of thermal equilibrium with T=f=.O can be regarded as a mixed state. The TDHF theory formulated in terms of the equation of the collective submanifold enables us to describe collective motion as a pure state not only in the linear but also in the non-linear type. 3 ) Then, it may be interesting for describing the mixed states to extend the TDHF theory in the statistical sense.