A DYNAMICAL THEORY OF QUANTUM MEASUREMENT AND SPONTANEOUS LOCALIZATION

We develop a rigorous treatment of discontinuous stochastic unitary ev olution for a system of quantum particles that interacts singularly wi th quantum ''bubbles'' in a cloud chamber at random instants of time. This model allows us to observe and track quantum particle trajectorie s as in a cloud chamber by sequential unsharp localization of spontane ous scatterings of the bubbles. Thus, the continuous reduction and spo ntaneous localization theory is obtained as the result of quantum filt ering theory, a theory describing the conditioning of the a priori qua ntum state by the measurement data. We show that in the case of indist inguishable particles, the a posteriori dynamics is a mixing dynamics, giving rise to an irreversible Boltzmannian reduction equation. The l atter coincides with the nonstochastic Schrodinger equation only in th e mean field approximation, whereas the central limit yields Gaussian mixing fluctuations described by stochastic reduction equations of dif fusive type.