Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution
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.
Keywords
Cite
@article{arxiv.0909.4434,
title = {Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution},
author = {Y. Strauss},
journal= {arXiv preprint arXiv:0909.4434},
year = {2015}
}
Comments
Accepted for publication in JMP. Supercedes arXiv:0710.3604. Discussion expanded to include the case of Hamiltonians with an infinitely degenerate spectrum