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.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.

arXiv:0706.0268Jun 2, 20073 citations

Stochastic Processes and Quantum Theory

John R. Klauder

Springer eBooksJan 1, 19750 citations

Stochastic processes and the quantum evolution of states

R. Collins, F. G. HALL

Beginning with the statistical description of the results of a duplicated experiment as a set of configurations of some observables tagether with frequencies of occurrence of each configuration it is shown that an evolution of the system by a stochastic process is obtained when the conditioning parameters of the experiment are altered.It is then shown that any such stochastic process has a representation in terms of linear operators in an abstract vector space with a state vector evolving by an isometric operator S and commuting Hermitian operators representing observables which evolve by another, unitary, operator U.This has the structure of conventional quantum theory as in the interaction, or Dirac picture, but here S is not unitary as in conventional theory.This is shown to yield a matrix of transition probabilities that is not doubly stochastic as in conventional theory and hence the Pauli master equation does not follow.All implications of this completely irreversible quantum theory have not yet been fully explored, but it points to a new viewpoint in irreversible thermod ynamics.Beginning with the statistical description of the results of a duplicated experiment as a set of configurations of some observables tagether with frequencies of occurrence of each configuration it is shown that an evolution of the system by a stochastic process is obtained when the conditioning parameters of the experiment are altered.It is then shown that any such stochastic process has a representation in terms of linear operators in an abstract vector space with a state vector evolving by an isometric operator S and commuting Hermitian operators representing observables which evolve by another, unitary, operator U.This has the structure of conventional quantum theory as in the interaction, or Dirac picture, but here S is not unitary as in conventional theory.This is shown to yield a matrix of transition probabilities that is not doubly stochastic as in conventional theory and hence the Pauli master equation does not follow.All implications of this completely irreversible quantum theory have not yet been fully explored, but it points to a new viewpoint in irreversible thermod ynamics.

Pure and Applied ChemistryJan 1, 19700 citations

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.