The dynamical evolution in quantum physics and its semi-group

Arno Böhm, Hai Viet Bui

Malkhaz Mumladze

In this article, we proposed a method for describing the evolution of quantum physical systems.We define the action integral on the functional space and the entropy of distribution of observable values on the set of quantum states.Dynamic of quantum system in this article is described as dynamical system represented by one parametric semi group which is extremal of this action integral.Evolution is a chain of change of distribution of observable values.In the closed system, it must increase entropy.Based on the notion of entropy of distribution energy, on the principle of maximum entropy production, we get a picture of evolution of closed quantum systems.

A. I. Solomon, Sonia G. Schirmer

We show that a quantum control procedure on a two-level system including dissipation gives rise to a semi-group corresponding to the Lie algebra semi-direct sum gl(3,R)+R^3. The physical evolution may be modelled by the action of this semi-group on a 3-vector as it moves inside the Bloch sphere, in the Bloch ball.

Holger Friedrich Hofmann

John Edward Gough, Tudor Stefan Ratiu, Oleg Georgievich Smolyanov

Noether’s theorem on constants of the motion of dynamical systems has recently been extended to classical dissipative systems (Markovian semi-groups) by Baez and Fong [J. Math. Phys. 54, 013301 (2013)]. We show how to extend these results to the fully quantum setting of quantum Markov dynamics. For finite-dimensional Hilbert spaces, we construct a mapping from observables to completely positive maps that leads to the natural analogue of their criterion of commutativity with the infinitesimal generator of the Markov dynamics. Using standard results on the relaxation of states to equilibrium under quantum dynamical semi-groups, we are able to characterise the constants of the motion under quantum Markov evolutions in the infinite-dimensional setting under the usual assumption of existence of a stationary strictly positive density matrix. In particular, the Noether constants are identified with the fixed point of the Heisenberg picture semi-group.