Abstract We study the Chernoff averages for random generalized shift operators in the case of noncanonical commutation relations between creation and annihilation operators. We introduce the concepts of shift-dual ladder operators and generalized shift operators. As an example, we consider a one-parameter family of commutation relations for which generalized shift operators are unitary and satisfy the semigroup property on straight lines passing through the origin. For this family, we prove that the sequence of expectations of Feynman–Chernoff iterations of random shift operators converges to a limit strongly continuous semigroup.
In this paper, we study the possibility of approximating some bi-continuous semigroup on a bi-admissible locally convex space
X using Chernoff iterations
(Ft)t≥0 of some class of operator-valued functions
(Ft)t≥0 . We obtain results that do not require a direct verification of the complicated condition of the Chernoff theorem, which consists in the fact that for some
λ>0 the subspace
(λI−L)D is bi-dense in the ambient space. However, we have to abandon the fact that the limit bi-continuous semigroup is defined on the
X and restrict ourselves to some bi-closed subspace that is constructed from the system
(Ft)t≥0 based on vectors with certain properties connected with the behavior of
(Ft)t≥0 . The formulation of the problem of approximation of a semigroup for bi-continuous semigroups is motivated by the dynamic systems existing in quantum mechanics, the evolution of which is described in terms of bi-continuous semigroups, but not strongly continuous ones. Here we can, for example, include the example of the evolution of a quantum system under a continuous measurement process, described by the Schrödinger–Belavkin equation. The study of approximations of such dynamics can be carried out by the proposed method.
Quantum mechanics can be formulated in terms of phase-space functions, according to Wigner's approach. A generalization of this approach consists in replacing the density operators of the standard formulation with suitable functions, the so-called generalized Wigner functions or (group-covariant) tomograms, obtained by means of group-theoretical methods. A typical problem arising in this context is to express the evolution of a quantum system in terms of tomograms. In the case of a (suitable) open quantum system, the dynamics can be described by means of a quantum dynamical semigroup 'in disguise', namely, by a semigroup of operators acting on tomograms rather than on density operators. We focus on a special class of quantum dynamical semigroups, the twirling semigroups, that have interesting applications, e.g., in quantum information science. The 'disguised counterparts' of the twirling semigroups, i.e., the corresponding semigroups acting on tomograms, form a class of semigroups of operators that we call tomographic semigroups . We show that the twirling semigroups and the tomographic semigroups can be encompassed in a unique theoretical framework, a class of semigroups of operators including also the probability semigroups of classical probability theory, so achieving a deeper insight into both the mathematical and the physical aspects of the problem.