In this chapter, the authors consider the formal description of the response and of the correlations displayed by the nuclear many-particle system at finite temperature. They work out the theoretical tools which allow to describe mean field properties, of both single-particle and collective motion, as well as their interweaving in highly excited systems. The authors show that the analysis of the response of the system to a weak external probe, and the treatment of correlations proceed in much the same way as they did at zero temperature. There is, however, a main difference to be kept in mind. At finite temperature, one no longer finds a unique state which describes the system, and a statistical description is called for. To study a quantum mechanical system, it is generally convenient to use the grand canonical ensemble, for which the energy of the system and the number of particles are not required to have definite values but only particular average values.
In this chapter, the authors consider the formal description of the response and of the correlations displayed by the nuclear many-particle system at finite temperature. They work out the theoretical tools which allow to describe mean field properties, of both single-particle and collective motion, as well as their interweaving in highly excited systems. The authors show that the analysis of the response of the system to a weak external probe, and the treatment of correlations proceed in much the same way as they did at zero temperature. There is, however, a main difference to be kept in mind. At finite temperature, one no longer finds a unique state which describes the system, and a statistical description is called for. To study a quantum mechanical system, it is generally convenient to use the grand canonical ensemble, for which the energy of the system and the number of particles are not required to have definite values but only particular average values.
This chapter aims to present the essence of the general theory in the particularly simple and convenient case of systems at finite temperature which may be treated in the grand canonical ensemble. It describes how thermal averages of time-ordered products of Heisenberg operators in imaginary time emerge naturally from path integrals and how to evaluate them in perturbation theory. The chapter considers at the outset what level of mathematical rigor one can actually expect and how in practice one may select and assess an approximation scheme. In view of the general lack of mathematical control on the convergence properties of the original series and the obvious ambiguity associated with regrouping divergent series, any such resummations must be understood ultimately on physical grounds. The ultimate objective of the quantum theory of many-particle systems is to understand experimentally observable properties of a diverse range of physical systems.
A variational derivation of the Liouville-von Neumann equation of quantum-statistical mechanics is presented, in order to formulate mean~field approximations appropriate to mixed states. The Hartree-Fock and the RPA at finite temperatures are particular cases of the general formalism. A thennal boson expansion is defined, which allows us to describe anhannonic motion around a thennal excited state. In a numerical application on the basis of the Lipkin model, temperature-dependent phase transitions are observed. I. Introduction Studies of the decay of the compound nuclei formed in heavy-ion reactions have added a new dimension to the field of nuclear structure. The temperature of the compound nucleus at a given excitation energy rises from zero at the yrast state to a few MeV at angular momentum zero. Recently the decay of high-excited states above the yrast line, like the giant dipole resonance, has been observed 1,2). These measurements were carried out making use of sum-spectrometer multiplicity techniques, thus allowing to determine the average angular momentum of the system and from this knowledge the temperature of the nucleus. One can then study experimentally the nuclear structure at finite values of both the angular momentum and temperature, and the corresponding theoretical analysis of giant resonances above thermal excited states is now the object of intensive research 3-7). On the other hand, the observation of deep-inelastic reactions has been an important sourCe of information on the behaviour of nuclei under strong external fields. In these reactions the two interacting ions emerge after the collision with approximatley the same charge and mass, but having lost a large fraction of the energy and angular momentum of relative motion they had in the entrance channel (cf. e.g. ref. 8) and references therein). Different mechanisms play a role in these reactions, like the transfer of particles and the excitation of giant resonances (cf. ref. 9) and references therein). Because these vibrations have a damping width the energy and angular momentum absorbed through them into the nuclei lead to a finite temperature and to rigid rotation. This
In the first three chapters we have considered problems related to the physical behaviour of one or a few particles (or some other physical objects which can be effectively described by one or a few (quasi)particles as, e.g., the random walk model in polymer physics considered in chapter 1). Though quantum field theory describes systems with an arbitrary number of degrees of freedom (an arbitrary number of (quasi)particles), in chapter 3 we actually applied it to systems with a restricted and small number of particles (e.g., for the description of the scattering process for a few particles). Another way to express this fact is to say that in chapter 3 we have considered field theories at zero temperature. However, the majority of realistic systems contain many identical (indistinguishable) particles such as atoms, electrons, photons etc. An attempt to describe these systems in terms of the individual trajectories of all particles is absolutely hopeless. Instead, we are interested in the collective behaviour of systems and describe them in terms of partition functions, mean values, correlation functions, etc. The methods of derivation, analysis and calculation of such collective characteristics constitute the subject of classical and quantum statistical mechanics.
Preparation, I have often said, is rightly two-thirds of any venture. Amelia Earhart Summary Correlation functions provide a direct way to characterize and analyze many-body systems, both theoretically and experimentally. In this chapter we review the properties of one- and two-body correlation functions in quantum systems, with emphasis on several key quantities: static density correlations that determine the energy and thermodynamic potentials, dynamic correlation functions such as response functions that describe excitations of the system, and Green's functions that are basic tools in the theory of interacting many-body systems. Correlation functions are central quantities in the description of interacting many-body systems, both in the theoretical formulation and in the analysis of experiments. In contrast to single numbers like the total energy, correlation functions reveal far more information about the electrons, how they arrange themselves, and the spectra of their excitations. In contrast to the many-body wavefunctions that contain all the information on the system, correlation functions extract the information most directly relevant to experimentally measurable properties. Dynamic current–current correlation functions are sufficient to determine the electrical and optical properties: one-body Green's functions describe the spectra of excitations when one electron is added to or removed from the system, static and dynamic correlations are measured using scattering techniques, and so forth. In this chapter we present the basic definitions and properties of correlation functions and Green's functions that are the basis for much of the developments in the following chapters. In general, a correlation function quantifies the correlation between two or more quantities at different points in space r , time t , or spin σ . Very often the correlation function can be specified as a function of the Fourier-transformed variables, momentum (wavevector) k , and frequency ω . It is useful to distinguish between a dynamic correlation function which describes the correlation between events at different times and a static or equal-time correlation function, by which we mean that of a property measured or computed with “snapshots” of the system. Also, the different correlation functions can be classified by the number of particles and/or fields involved.
A number of powerful methods exist in statistical mechanics to describe the interweaving of the variety of elementary modes of excitation taking place in a quantal many-body system. Among them, one can mention the Static Path approximation (cf., e.g., Feynmann and Hibbs ( 1965 )), the Monte-Carlo Shell Model Method (Koonin et al. ( 1997 )), and the thermal Green-function formalism (cf., e.g., Mahan ( 1981 )). The first two approaches have not yet been systematically used to describe nuclear motion at finite temperature. We shall briefly discuss them in the present chapter. The Matsubara techniques, which have been applied to the study of a large variety of phenomena, are discussed in Ch. 9 .
Ensembles and expectation values Equations of motion provide predictions of time-evolving variables of a dynamical system, and thereby enable us to predict the changing behavior. These equations are characteristic of a particular type of system, say a classical harmonic oscillator or a two-state quantum system. To complete the description of behavior one must specify initial conditions that select, from the many possible solutions to the equations of motion, a particular realization. In the example of a classical point particle the equation is second order in time, and so we need two initial conditions, of position and velocity, to specify a particular solution. These two numbers, taken with the classical equation of motion, provide a complete description of the behavior of a specific harmonic oscillator. Similarly the two-state quantum system requires two initial values to complete its definition. The initial conditions are never defined with infinite precision, even for classical mechanics. More generally we deal with an ensemble, that is, a collection of similar dynamical systems (e.g. oscillators) that differ in their preparation. The individual cases share some common attributes, but not necessarily all. Thus we have only incomplete information about any single system. We might, for example, consider planetary motion in which we know only the mean energy and the orbital plane. The loci of positions of the planets then form ellipses that make up the ensemble. Although it is possible to prepare a single two-state quantum system in which the initial state is known (apart from an overall phase), often one deals with ensembles of such systems, each of which may differ in some uncontrollable property.