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

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