A STOCHASTIC HAMILTONIAN APPROACH FOR QUANTUM JUMPS, SPONTANEOUS LOCALIZATIONS, AND CONTINUOUS TRAJECTORIES

We give an explicit stochastic Hamiltonian model of discontinuous unit ary evolution for quantum spontaneous jumps like in a system of atoms in quantum optics, or in a system of quantum particles that interacts singularly with 'bubbles' which admit a continual counting observation . This model allows one to watch a quantum trajectory in a photodetect or or in a cloud chamber by spontaneous localizations of the momenta o f the scattered photons or bubbles. Thus, the continuous reduction and spontaneous localization theory is obtained from a Hamiltonian singul ar interaction as a result of quantum filtering, i.e. a sequential tim e continuous conditioning of an input quantum process by the output me asurement data. We show that in the case of indistinguishable particle s or atoms, the a posteriori dynamics is mixing, giving rise to an irr eversible Boltzmann-type reduction equation. The latter coincides with the non-stochastic Schrodinger equation only in the mean-field approx imation, whereas the central limit yields Gaussian mixing fluctuations described by a quantum state reduction equation of diffusive type.