Using the Green's-function technique of Martin, Schwinger, and Puff, a calculation is made of the total linear response of nuclear matter, as measurable, for instance, in an inelastic electron or muon scattering experiment. In the independent-pair approximation used, two contributions stand out: the quasielastic response of properly dressed particles with well-defined lifetimes, and the contribution due to the scattering between those particles. An extensive analysis is made of the importance of the transverse component of the electromagnetic interaction as well as of the two mentioned contributions. The properly defined quasielastic scattering is predicted to dominate the scattering. A comparison is made with previous calculations by Czy\ifmmode \dot{z}\else \.{z}\fi{} and Gottfried.
Using the Green's-function technique of Martin, Schwinger, and Puff, a calculation is made of the total linear response of nuclear matter, as measurable, for instance, in an inelastic electron or muon scattering experiment. In the independent-pair approximation used, two contributions stand out: the quasielastic response of properly dressed particles with well-defined lifetimes, and the contribution due to the scattering between those particles. An extensive analysis is made of the importance of the transverse component of the electromagnetic interaction as well as of the two mentioned contributions. The properly defined quasielastic scattering is predicted to dominate the scattering. A comparison is made with previous calculations by Czy\ifmmode \dot{z}\else .{z}\fi{} and Gottfried.
The Martin-Schwinger-Puff theory for a many-fermion system is applied to a calculation of the appropriate density propagator. The real part of this function, the system's polarizability, is a measure for the lowest order change in energy due to a harmonic density ripple enforced on the infinite system. The imaginary part, the linear response function, yields the inelastic cross section for the scattering of a weak external agent. The linear response contains additional information on excited states and, in particular, on collective states, excited by an external field. The formalism necessitates the calculation of the self-energy of a particle and from it the optical model for nucleon scattering can be derived. The relation of the theory to perturbative and other nonperturbative approaches is discussed and a comparison is made with a calculation of the polarization energy of nuclear matter by Koltun and Wilets.
Linear density response functions are calculated for symmetric nuclear matter of normal density by time-evolving two-time Green's functions in real time. Of particular interest is the effect of correlations. The system is therefore initially time-evolved with a collision term calculated in a direct Born approximation until fully correlated. An external time-dependent potential is then applied. The ensuing density fluctuations are recorded to calculate the density response. This method was previously used by Kwong and Bonitz for studying plasma oscillations in a correlated electron gas. The energy-weighted sum-rule for the response function is guaranteed by using conserving self-energy insertions as the method then generates the full vertex-functions. These can alternatively be calculated by solving a Bethe -Salpeter equation as done in works by Bozek et al. The (first order) mean field is derived from a momentum-dependent (non-local) interaction while 2 nd order self-energies are calculated using a particle-hole two-body effective (or residual) interaction given by a gaussian local potential. We show results of calculations of the response function S(ɷ, q 0 ) for q 0 = 0.2, 0.4 and 0.8 fm -1 . Comparison is made with the nucleons being un-correlated i.e. with only the first order mean field included. We discuss the relation of our work with the Landau quasi-particle theory as applied to nuclear systems by Babu and Brown and followers.
Introduction and guide to the chapter Linear response functions contain a wealth of information about the physical properties of a many-body system. In the case of the electron liquid, for example, the density–density response function provides a unified framework for the understanding of different phenomena such as static screening, effective interaction, collective modes, electron energy loss spectra (inelastic scattering of electrons), and Raman spectra (inelastic scattering of photons). The spin–spin response function provides the corresponding information for the spin density fluctuations as probed, for example, by cross-polarized Raman scattering experiments, in which the incident and scattered photon have perpendicular polarizations, or by spin-flip electron energy loss spectroscopy, in which the incoming and outgoing electrons have opposite spin orientations. As we have seen, the Hartree–Fock approximation describes each electron as an independent particle moving in a self-consistent field generated by all the other electrons. There is no correlation between this self-consistent field and the instantaneous position of the electron. Reality is however quite different: whenever an electron moves, it acts on the surrounding electrons, causing a collective disturbance which eventually feeds back on its own motion. In order to study these effects we need, first of all, to learn how the electron liquid as a whole responds to disturbances caused by the charge, the spin, and the current of a single electron. The linear response theory of Chapter 3 provides the natural framework for this description. Strictly speaking, the linear response functions describe the readjustments of the electronic density, spin, or current, in response to externally controlled fields.
Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.
It is shown that the usual single-particle Green's function does not give direct information about the lowest excitation energies if there is a pairing energy (of the type familiar in nuclear physics). An alternate Green's function is suggested which will provide such information. (C.J.G.) It is noted that owing to the similarity existing between electromagnetic interaction and the vector part of the weak interaction proposed by Feynman and Gell-Mann, the contribution of the vector part of the weak interaction to mu /sup -/ capture by a proton is closely related to the electromagnetic form factors of the nucleon. Formula for the capture probability calculated with the renormaalized V-A, weak interaction is given. The correction to the contribution of the vector part of the weak interaction due to the charge and magnetic moment distribution of the nucleon is estimated on the basis of the data from electron-nucleon scattering experiments. (auth)
We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\ensuremath{\Lambda}$ for two-particle intermediate states in the T matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\ensuremath{\Lambda}}_{00}\ensuremath{\sim}{{G}_{1}}^{0}{{G}_{1}}^{0}$ have been noted previously and are not treated here. We show that the choice ${\ensuremath{\Lambda}}_{10}\ensuremath{\sim}{{G}_{1}}^{0}{G}_{1}$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\ensuremath{\Lambda}}_{00}$ approximation. (Numerical calculations using the ${\ensuremath{\Lambda}}_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\ensuremath{\Lambda}$ approximations (${\ensuremath{\Lambda}}_{00}$, ${\ensuremath{\Lambda}}_{10}$, and ${\ensuremath{\Lambda}}_{11}\ensuremath{\sim}{G}_{1}{G}_{1}$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\ensuremath{\Lambda}}_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${{k}_{F}}^{0}c\ensuremath{\lesssim}1$) than are the other $\ensuremath{\Lambda}$ approximations or Brueckner theory.