The Nth-order response, effected by an external scalar field coupled to a system initially in thermodynamic equilibrium, is expressed with the help of imaginary-time correlation function using the spectral representation of its Fourier coefficients.
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
We develop a general nth-order analytic continuation formalism for evaluating the nth-order response of a many-body system which is subject to an arbitrary time-dependent mechanical perturbation K(t) and which is initially in a state of thermal equilibrium (i.e. its initial density matrix is e-βH/Z). The formalism is a generalization of the linear analytic continuation response formalism as presented by Luttinger and Nozieres and similarly entails the analytic continuation of certain suitably defined correlation functions which may, in principle, be evaluated exactly by the techniques of temperature perturbation theory. In this way the time development of the system is, in principle, exactly prescribed. Finally, as an example of the use of the generalized formalism, we evaluate the Hall effect in a metal with impurity scattering by means of the second-order formalism and the result justifies the usual expression.
Abstract The response of a superconductor to external fields is considered in arbitrary order. The response is calculated for an imaginary time interval and then analytically continued to real times. In this way it is possible to make use of the theorem proved by Kadanoff and Baym on the identity of the correlation functions connected with the physical and the unphysical problem. The result justifies the method of Gorkov and Eliashberg in which the analytic continuation has been performed in the frequency representation.