Non-unitary Evolution of Quantum Logics

Abstract

In this work we present a dynamical approach to quantum logics. By changing the standard formalism of quantum mechanics to allow non-Hermitian operators as generators of time evolution, we address the question of how can logics evolve in time. In this way, we describe formally how a non-Boolean algebra may become a Boolean one under certain conditions. We present some simple models which illustrate this transition and develop a new quantum logical formalism based in complex spectral resolutions, a notion that we introduce in order to cope with the temporal aspect of the logical structure of quantum theory.

The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance

“…One important consequence of the non-unitary evolution described by non-Hermitian Hamiltonians is the fact that observables which initially are compatible (zero commutator), become non-compatible after some period of time (non-vanishing commutator). The opposite behavior can also happen, namely, a pair of non-commuting operators, can eventually commute after some time evolution [35]. As we have remarked before, in some situations, it is possible to recover the unitarity condition for Non-Hermitian operators if the P T symmetry is satisfied [6,7].…”

“…Non-Hermitian Hamiltonians describe non-unitary evolutions in general [35]. Then their eigenvalues are in general complex numbers.…”

Evolution of quantum observables: from non-commutativity to commutativity

“…This approach has been applied in the context of self-induced decoherence [1,2], enviroment induced decoherence [3,4,5], and for quantum maps [6]. Also, we have studied its formal aspects and its algebraic properties [6,7].…”

Classical Limit and Quantum Logic

“…From the point of view of the state operator, this means that its different nondiagonal components vanish at different characteristic times (Fortin, Holik and Vanni 2016).…”

“…Finally, it is important to mention that the approach to decoherence based in nonHermitian Hamiltonians was also applied to the study of the time evolution of the commutators (Fortin, Holik and Vanni 2016).…”