Path Integral Approach to Many-Body Systems and Classical Quantization of Time-Dependent Mean Field

Abstract

We present the path integral formalism of many-body systems based on the representation of generalized coherent states. We give the path integral representation for the transition amplitude by utilizing the characteristic property of the coherent state and discuss its classical limit through the method of stationary phase. Applying to the bound state problem, we arrive at the classical quantization rule for the periodic mean field solutions, the utility of which is briefly discussed. § l. Introduction The time-dependent Hartree-Fock (TDHF) or mean field method has been known to provide with a basic formalism in the investigation of the large amplitude collective phenomena which is closely connected with the non-linear nature inherent to strongly interacting many-body systems. The framework of the mean field theory is, as is often recognized, essentially of classical nature, namely, the quantum nature originally inherent to many-body systems is smoothed by averaging process through the trial wave packet labeled by the parameters of classical nature. Hence the naive mean field theory does not give a prescription for extracting the quantum mechanical informations such as energy spectra of bound states for many-body systems. Thus it be­ comes a crucial problem to construct a proper quantum mechanical treatment of the time-dependent mean field so as to deduce the informations of physical importance. In the quantum mechanical treatment of mean fields one encounters a serious difficulty of handling the non-linearity of the mean-field equations. For such a system that the non-linearity plays a dominant role, the conven­ tional perturbative treatment does not work well and one is forced to rely upon an essentially non-perturbative approach. Path integral method is con­ sidered to provide a promising device for such a non-perturbative approach, the utility of which has been gradually recognized in nuclear many-body problems.D~s>