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 space comes equipped with a natural Riemannian metric (the Fubini-Study metric) and a compatible symplectic structure. The operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry. It is interesting in this spirit to scrutinize the probabilistic assumptions that are brought ill at various stages in the analysis of quantum dynamics, particularly in connection with state vector reduction. A promising approach to understanding reduction, studied recently by a number of authors, involves the use of nonlinear stochastic dynamics to modify the ordinary linear Schrodinger evolution. Here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse. In this picture, the conventional Schrodinger evolution; which corresponds to the unitary flow associated with a Killing vector of the Fubini-Study metric, is replaced by a more general stochastic flow on the state manifold. In the simplest example of such a flow, the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the Hamiltonian. 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 the existence of a nonlinear term in the drift vector of the state process, which is always oriented opposite to the direction of increasing energy uncertainty. As a consequence, the state vector necessarily collapses 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 usual quantum mechanical probability.