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Bulletin de la Classe des sciences

On a relation between exact and asymptotic time evolution operators

R. Balescu, Paul Clavin, Paul Mandel, John Wm. Turner

Published Jan 1, 1969• 1 citation
Publisher

Abstract

A simple relation is shown to exist at all times between the exact evolution operator U(t) and the asymptotic evolution operator Σ(t) introduced by the Brussels school. Furthermore, it is shown that all invariants under U(t) are invariant under Σ(t) and that under certain conditions the converse is also true.
DOI
10.3406/barb.1969.62501

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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.
We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of SU(2) by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any SU(2) model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties.
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