Using density matrices we describe the time evolution of nonequilibrium states with a set of hierarchical of quantum kinetic equations. An analogous formalism to this equation is developed via the path integral method in a holomorphic representation of density matrices and related to the Green’s function formalism.
The problem of many-particle correlations in different approaches to the quantum kinetic theory is treated on the basis of Zubarev's method of the nonequilibrium statistical operator.It is shown that long-lived correlations can be incorporated through boundary conditions for reduced manyparticle density matrices and the nonequilibrium real-time Green functions.Within the method of Green functions the boundary conditions are conveniently formulated in terms of the "mixed" Green functions defined on a directed contour with the real-time evolution governed by the Hamiltonian of the system and the "imaginary-time" evolution governed by the entropy operator.The perturbation expansion of the mixed Green function is obtained in terms of the interaction part of the Hamiltonian and the correlation part of the entropy operator.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Abstract. Nonequilibrium Green’s functions represent underutilized means of studying the time evolution of quantum manybody systems. In view of a rising computer power, an effort is underway to apply the Green’s functions formalism to the dynamics of central nuclear reactions. As the first step, mean-field evolution for the density matrix for colliding slabs is studied in one dimension. Strategy to extend the dynamics to correlations is described. Keywords: Green’s functions, Kadanoff-Baym equations, time-dependent Hartree-Fock, nuclear reactions PACS: 24.10.-i, 24.10.Cn
By means of the time-smoothing method developed in the previous papers of this series, the quantum kinetic equation is derived from the principles of statistical mechanics. In this paper, we avoid the BBGKY hierarchy; instead we use the Liouville-von Neumann equation for the time-smoothed density matrix. The Boltzmann distribution function is defined in terms of this time-smoothed density matrix. The present scheme may be applied to strong-coupling systems even if they are not in near equilibrium.
Abstract Chapter 4 has introduced the functional integral representation of the quantum statistical operators and thus, formally, evolution in imaginary or Euclidean time. By contrast, to calculate the evolution operator and the scattering S-matrix elements, quantities relevant to particle physics, it is necessary to make a continuation from imaginary to real time. However, the representation of the S-matrix follows from additional considerations. To illustrate the power of the formalism, we show how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Chapter 4 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas, the evolution operator is then given by a holomorphic field integral. A parallel formalism leads to an analogous representation for the evolution operator of a system of non-relativistic fermions.