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Some Applications of Semigroups

  • A. Verbeure
Part of the NATO ASI Series book series (NSSB, volume 144)

Abstract

In physics quantum dynamical semigroups show up when we consider a system in interaction with its surroundings, also called reservoir. The Hamiltonian of the total system is then
$$ H = {H_S} + {H_R} + \lambda {H_{{SR}}} $$
where HS is the Hamiltonian of the system, HR of the reservoir; HSR represents the interaction of the system with reservoir. The Heisenberg evolution of an observables X is then
$$ \alpha_t^{\lambda }(X) = \exp (i t {H_{\lambda }}) X exp ( - i t {H_{\lambda }}) $$
The dynamical semigroup {Γτ}τ≥0 is obtained in the weak coupling limit [1,2] i.e. λ→0, τ→∞ such that λ2t = τ and
$$ {\Gamma_{\tau }} = \mathop{{\lim }}\limits_{{\mathop{{\lambda \to 0}}\limits_{{t \to \infty }} }} \quad \alpha_t^{\lambda } $$
The dynamical semigroups obtained in this way have different characteristics depending on the type of state of the reservoir. However from a mathematical point of view they are all of the same type. The set of maps {Γτ}τ≥0 is a one parameter semigroup of completely positive unity preserving maps of the algebra A of observables.

Keywords

Detailed Balance Correlation Inequality Quantum Dynamical Semigroup Real Time Parameter Parameter Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Press, New York 1986

Authors and Affiliations

  • A. Verbeure
    • 1
  1. 1.Instituut voor Theoretische FysicaUniversiteit LeuvenLeuvenBelgium

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