GEOMETRY OF STOCHASTIC STATE-VECTOR REDUCTION

The state space of a quantum mechanical system is a complex projective space, the space of rays in the associated Hilbert space. The state s pace comes equipped with a natural Riemannian metric (the Fubini-Study metric) and a compatible symplectic structure. The operations of ordi nary quantum mechanics can thus be reinterpreted in the language of di fferential geometry. It is interesting in this spirit to scrutinize th e probabilistic assumptions that are brought ill at various stages in the analysis of quantum dynamics, particularly in connection with stat e vector reduction. A promising approach to understanding reduction, s tudied recently by a number of authors, involves the use of nonlinear stochastic dynamics to modify the ordinary linear Schrodinger evolutio n. Here we use stochastic differential geometry to give a systematic g eometric formulation for such stochastic models of state vector collap se. In this picture, the conventional Schrodinger evolution; which cor responds to the unitary flow associated with a Killing vector of the F ubini-Study metric, is replaced by a more general stochastic flow on t he state manifold. In the simplest example of such a flow, the volatil ity term in the stochastic differential equation for the state traject ory is proportional to the gradient of the expectation of the Hamilton ian. The conservation of energy is represented by the requirement that the actual process followed by the expectation of the Hamiltonian, as the state evolves: should be a martingale. This requirement implies t he existence of a nonlinear term in the drift vector of the state proc ess, which is always oriented opposite to the direction of increasing energy uncertainty. As a consequence, the state vector necessarily col lapses to an energy eigenstate, and a martingale argument call be used to show that the probability of collapse to a given eigenstate, from any particular initial state, is, in fact, given by precisely the usua l quantum mechanical probability.