Y. Strauss
Forward and backward time observables for quantum evolution and quantum stochastic processes-I: The time observables
(405K, pdf)

ABSTRACT.  Given a Hamiltonian $\mathbf H$ on a Hilbert space $\mathcal H$ it is shown that, under the assumption that 
$\sigma(\mathbf H)=\sigma_{ac}(\mathbf H)=\mathbb R^+$, there exist uniquely defined positive operators $\mathbf T_F$ and 
$\mathbf T_B$ registering the Schr\"odinger time evolution generated by $\mathbf H$ 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.