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Journal of Mathematical Physics

On time evolution for large quantum systems in superspace

Nobuhiko Mishima, Tomio Yamakoshi Petrosky

Published May 1, 1978• 6 citations
Publisher

Abstract

A reformulation in a superspace is given for Van Hove and Janner’s perturbation theory based on the two-resolvent method for nonequilibrium quantum statistical mechanics. This is attained by introducing an ordered superoperator in the superspace. As an application of our formalism, the derivation of an asymptotic generalized master equation in a Markovian form is given. Further, it is investigated how a quantum subdynamics theory similar to the Brussels school’s one can be formally constructed. Our subdynamics theory is formulated on every eigenspace of a superoperator (the ss-Hamiltonian) of which the eigenvalue is related to the intrinsic parameter E of Van Hove and Janner’s generalized master equation.

DOI
10.1063/1.523754

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A reformulation in a superspace is given for Van Hove and Janner’s perturbation theory based on the two-resolvent method for nonequilibrium quantum statistical mechanics. This is attained by introducing an ordered superoperator in the superspace. As an application of our formalism, the derivation of an asymptotic generalized master equation in a Markovian form is given. Further, it is investigated how a quantum subdynamics theory similar to the Brussels school’s one can be formally constructed. Our subdynamics theory is formulated on every eigenspace of a superoperator (the ss-Hamiltonian) of which the eigenvalue is related to the intrinsic parameter E of Van Hove and Janner’s generalized master equation.
In the frame of the theory of subdynamics a general picture of the time evolution of a many-body system has been developed by the Brussels school of statistical mechanics. In this paper, we avoid the perturbation scheme so that we get compact notations and demonstrations. We derive here a general expression for the commutator of an arbitrary superoperator W with the " asymptotic " evolution superoperator Σ(t) = ΠU(t) where [formula] is the complete evolution superoperator and where Π is the usual superprojector of the theory of subdynamics. In the frame of relativistic non-equilibrium statistical mechanics, the application of this commutator to all the generators of the Poincaré group is then given and we get back in particular the fundamental property of commutation with Π of the Lorentz-Liouville superoperator.
There exists the well-known approximate expression describing the large time behavior of matrix elements of the evolution operator in quantum theory: < U(t)>≃ exp (at). This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf–Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: <U(t)>= exp (At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.
A theoretical framework for the nonequilibrium statistical mechanics of open systems is constructed. This is concerned with a formulation of a generalized master equation governing the evolution of an arbitrary system S in interaction with a ``large'' reservoir R. The dynamics of S are analyzed on the basis of a precise quantum-mechanical treatment of the microscopic equations of motion for the combined system S + R. On proceeding to the thermodynamical limit for R we obtain a generalized master equation for S, subject to specified conditions on the many-particle structure of R, its initial state, and its coupling to S. This master equation corresponds to a self-contained law of motion for S, in which the R variables appear only in the forms of certain thermal averages, taken over the initial state. This dynamical law is a generalization of the quantum-mechanical Liouville equation to a form appropriate to open systems.
A critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time-dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,BRα(t) =[exp(−Λdt)]Bα, where Bα is a Schrödinger operator and Λd is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.