On the existence of Lyapounov variables for Schroedinger evolution

TL;DR

This paper demonstrates that Lyapounov variables, which indicate monotonic evolution, can be constructed within standard quantum mechanics, challenging prior assumptions that required Liouville space, and establishes a framework for irreversible quantum representations.

Contribution

It proves the existence of Lyapounov variables in Hilbert space, showing MPC assumptions are more restrictive than necessary, and introduces a natural time observable in the irreversible representation.

Findings

Lyapounov variables can be constructed in Hilbert space.

Existence of a transformation to an irreversible quantum representation.

A natural time observable splits the Hilbert space into past and future.

Abstract

The theory of (classical and) quantum mechanical microscopic irreversibility developed by B. Misra, I. Prigogine and M. Courbage (MPC) and various other contributors is based on the central notion of a Lyapounov variable - i.e., a dynamical variable whose value varies monotonically as time increases. Incompatibility between certain assumed properties of a Lyapounov variable and semiboundedness of the spectrum of the Hamiltonian generating the quantum dynamics led MPC to formulate their theory in Liouville space. In the present paper it is proved, in a constructive way, that a Lyapounov variable can be found within the standard Hilbert space formulation of quantum mechanics and, hence, the MPC assumptions are more restrictive than necessary for the construction of such a quantity. Moreover, as in the MPC theory, the existence of a Lyapounov 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 observable splitting the Hilbert space at each t>0 into past and future subspaces.