The recurrence relations between Green functions are derived for the two and three particle potentials. The time variation of the field operator is used for the derivation of these recurrence relatiions.
Abstract Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
We investigate the dynamics of a spin-1/2 particle in a constant magnetic field driven by two transverse magnetic noise fields. These noise fields are exponentially correlated in time. By introducing suitable marginal averages and suitable expansions of these averages it is shown that the problem reduces to the solution of two independent tridiagonally coupled vector recurrence relations with vectors of dimension 3. One of these relations describes the decay of the spin component parallel to the constant field, the other the decay of the spin component transverse to the constant field. The equations are solved by a direct time integration and by determining the eigenvalues. It is shown that the Green's functions are determined by ordinary continued fractions. These continued fractions agree with those obtained by Shibata and Sato and are thus rederived by a completely different method. The long-time dependence is governed by the eigenvalue with the lowest real part and by its weight. The eigenvalues are found by determining the poles of the Green's function, the weights of the eigenvalues by evaluating the Green's function near the poles. The relaxation times T1 and T2 as well as a frequency shift of the perpendicular component are thus obtained. For certain parameters a single decay constant is not sufficient.