A framework to describe the real-time evolution of interacting fermion-scalar field models is set up. On the basis of the general dynamics of the fields, we derive formal equations of kinetic-type to the set of one-body dynamical variables. A time-dependent projection technique is used then to generate a nonperturbative mean-field expansion leading to a set of self-consistent equations of motion for these observable, where the lowest order corresponds to the Gaussian approximation. As an application, we consider an uniform system of relativistic spin-1/2 fermion field coupled, through a Yukawa term, to a scalar field in 3+1 dimensions, known as quantum scalar plasma. The renormalizability for the Gaussian mean-field equations, both static and dynamical, are examined and initial conditions discussed. We also investigate solutions for the gap equation and show that the energy density has a single minimum.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
The article outlines key topics related to deriving kinetic equations and, in general, to the relationship between the evolution equations governing quantum many-particle systems. The main issues being considered are the mathematical origin of quantum kinetic equations, novel methods for deriving kinetic equations from hierarchies of evolution equations for quantum systems, specifically focusing on kinetics within the framework of the evolution of observables, and the generalization of the quantum kinetic equation specified by non-separable initial states.
Introduction Mathematically, given a linear differential operator L x , one encounters the solution to the inhomogeneous differential equation Here ρ ( x ) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let G ( x, y ) is the Green's function. This is a function of x with y a parameter. Take G ( x, y ) to satisfy the same boundary conditions as φ ( x ). Then since An example of L x is, of course, the Schrödinger operator We take ρ ( x ) = V ( x ) ψ ( x , t ), ψ ( x , t ) being the wave function and V ( x ) the potential operator. We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al. , 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000). One- and two-time quantum Green's functions and their properties Let us introduce the creation operator ψ † ( r, t ) and annihilation operator ψ ( r, t ) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles: The Hamiltonian operator for the particles is and the number density of particles at r t is the operator Note that r = ( r 1 … r N ) for (1, 2, 3, …, N ) and V (| r − r ′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this thesis a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of relativistic quantum field theories in 3+1 space-time dimensions. In a first step, for simplicity we consider a real scalar Phi^4 quantum field theory and in a second step we generalize our results to a chirally invariant Yukawa-type quantum field theory including fermions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Apart from quantitative discrepancies, on a qualitative level the universality respected by Kadanoff-Baym equations is severely restricted in the case of Boltzmann equations. Furthermore, Kadanoff-Baym equations strongly separate the time scales between kinetic and chemical equilibration. This separation of time scales is absent for Boltzmann equations.