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Given a Hamiltonian $H$ on a Hilbert space $\mathcal H$ it is shown that,under the assumption that $\sigma(H)=\sigma_{ac}(H)=R^+$, there exist uniquepositive operators $T_F$ and $T_B$ registering the Schr\"odinger time evolutiongenerated by $H$ in the forward (future) direction and backward (past)direction respectively. These operators may be considered as time observablesfor the quantum evolution. Moreover, it is shown that the same operators mayserve as time observables in the construction of quantum stochasticdifferential equations and quantum stochastic processes in the framework of theHudson-Parthasarathy quantum stochastic calculus. The basic mechanism enablingfor the definition of the time observables originates from the recentlydeveloped semigroup decomposition formalism used in the description of the timeevolution of resonances in quantum mechanical scattering problems.

Forward and Backward time observables for quantum evolution and quantum stochastic processes-I: The time observables