Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution

TL;DR

This paper constructs a self-adjoint Lyapunov variable in quantum mechanics, linking it to irreversible representations and a natural time ordering, thus providing a framework to treat time as a dynamical variable.

Contribution

It introduces a method to construct Lyapunov variables in quantum models and shows their connection to irreversible representations and temporal ordering.

Findings

Existence of Lyapunov variables in certain quantum models.

Construction of an irreversible representation from Lyapunov variables.

Identification of a natural time ordering observable.

Abstract

In non relativistic quantum mechanics time enters as a parameter in the Schroedinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable - i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admit a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proved that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t>0 into past and future subspaces.