Dynamics of algebras in quantum unstable systems

Abstract

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is non-unitary. This allows to describe observables that are initially non-commutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples. * All authors contributed equally to this manuscript.

Evolution of quantum observables: from non-commutativity to commutativity

“…However, this produces a contribution in O(t) that grows exponentially, which cannot be cancelled out with any other term [6]. This suggests that (32) may not be a good definition for the time evolution operator in our case.…”

“…In [6], we have given an example based in the Gamow formalism for resonances. The space of states is formed by the Gamow vectors representing resonance states.…”

“…The algebra of operators is given by the linear mappings on the space of this resonance states. In order to get a suitable non-unitary time evolution, it is necessary to isolate the resonances from the environment, as described in [6]. This is summarized in Section 2 of this work.…”

“…In this paper, we give a further generalization of the non-unitary evolution presented in [6]. This is done by appealing to a formalism for quantum observables and states that was originally developed so as to include van Hove states, originally conceived to be applied in statistical mechanics of systems far from equilibrium.…”

Non-Deterministic Semantics for Quantum States

“…The logic-operational approach to quantum theory dates back to the 1930s, after the contribution of Birkhoff and von Neumann [23] (for further developments of the quantum logical approach, see [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]). More recently, a growing interest is put in describing logical structures associated to quantum computation [39][40][41][42].…”