Lévy Models for Collapse of the Wave Function

Abstract: Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schr\"odinger equation. The present paper considers energy-based stochastic extensions of the Schr\"odinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the L\'evy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for L\'evy models the decoherence rate depends on the overall scale of the energy. Thus, in L\'evy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.