Abstract
Publisher Summary This chapter discusses the quantum dynamical semi-group. An axiomatic approach to quantum dynamics of non-Hamiltonian systems gives the most general dynamical laws for quantum systems. These laws describe the most general linear mapping of the set of states into itself. One can introduce the semi-group laws as the fundamental dynamical postulate for non-Hamiltonian systems. The dynamical semi-groups are special examples of strongly continuous contractive semi-groups. The necessary and sufficient conditions for non Hamiltonian operator (Λ) to be the generating superoperator of a dynamical semi-group can be formulated by semi-scalar product. G. Lumer and R.S. Phillips have described contractive semi-groups by virtue of the notion of semi-scalar product. The infinitesimal generator of such a semi-group is dissipative with respect to this product. It is suggested that the necessary and sufficient conditions for generating superoperators of a dynamical semi-group can be described by using a system of orthogonal projections. One can define a dynamical semi-group for observables to describe the time evolution of the quantum observables. The axioms for a dynamical semi-group can be considered in the so-called “Heisenberg dynamical representation.”
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