Heisenberg dynamics. Covariant representations

The nonautomorphic Heisenberg dynamics proposed earlier by the author is studied. It is shown that the set of states for which the Cauchy problem for the Liouville equation is uniformly well posed forms a folium of states. The dynamical properties of covariant representations, i.e., representations defined by states satisfying the equation $L \omega_0 =0$, are examined. It is shown that on the set of normal states of such representations any direct or inverse Cauchy problem for the Liouville equation is uniformly well posed. If the state $\omega_0$ is physically pure or invariant under time reversal, both Cauchy problems are well posed and the dynamics in such representations is reversible. In this case, the dynamical transformations determine the group of automorphisms $\pi_{\omega_0}(\mathscr A)^{-}$.