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

Given a Hamiltonian H on a Hilbert space mathcal H it is shown that, under the assumption that σ(H)=σ_ac(H)=R^+, there exist unique positive operators T_F and T_B registering the Schrödinger time evolution generated by H in the forward (future) direction and backward (past) direction respectively. These operators may be considered as time observables for the quantum evolution. Moreover, it is shown that the same operators may serve as time observables in the construction of quantum stochastic differential equations and quantum stochastic processes in the framework of the Hudson-Parthasarathy quantum stochastic calculus. The basic mechanism enabling for the definition of the time observables originates from the recently developed semigroup decomposition formalism used in the description of the time evolution of resonances in quantum mechanical scattering problems.