Continuous spontaneous localization as the white-noise limit of spontaneous unitarity violation

Objective collapse theories propose modifications to Schr\"odinger's equation that solve the quantum measurement problem by interpolating between microscopic quantum dynamics and projective evolution of macroscopic objects. Colored-noise driven collapse theories extending the equilibrium description of spontaneous symmetry breaking to spontaneous violations of unitarity (SUV) in quantum dynamics were recently shown to possess a Markovian white noise limit when applied to initial two-state superpositions. Here, we show that this limit coincides with a subclass of continuous spontaneous localization (CSL) models collapsing in a basis of spatially localised energy eigenstates. We show that the energy expectation value remains conserved throughout this process, and we also extend the model to a form that can be applied to any initial state. We furthermore show that, as for the SUV models, the emergence of Born rule statistics in the Markovian limit is enforced by a fluctuation-dissipation relation which results in ensemble averaged probability densities following a linear quantum semi-group guaranteeing the absence of superluminal signaling.