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

TL;DR

The paper shows that the white-noise limit of spontaneous unitarity violation (SUV) models is equivalent to a continuous spontaneous localization (CSL) description collapsing into symmetry-breaking energy eigenstates, while conserving energy and accommodating arbitrary initial states. By formulating a multi-scale noise homogenization framework, it derives a CSL-type stochastic Schrödinger equation in the white-noise limit and demonstrates that, upon noise averaging, the dynamics reduces to a linear GKSL master equation, with Born-rule statistics and no-signaling guaranteed by a fluctuation-dissipation relation. The continuum formulation introduces space-time white noise via a space-time Wiener process (Brownian sheet) and yields a corresponding continuum GKSL master equation, ensuring norm preservation and consistent probabilistic predictions for measurement outcomes. Collectively, the results connect SUV and CSL frameworks, clarifying their overlap in the white-noise limit and highlighting implications for quantum-to-classical transitions, phase transitions, and irreversible quantum dynamics.

Abstract

Objective collapse theories propose modifications to Schrödinger'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.

Figures (1)

• Figure 1: Venn diagram showing the relation between two classes of objective collapse theories. Any model for Spontaneous Unitarity Violation (SUV) falls in he central (brown) area, collapsing into a basis of symmetry-breaking states and coupling linearly to an arbitrary (possibly non-Gaussian) stochastic process. They may be extended to include uncorrelated white noise ($\tau=0$) as an idealized, mathematical limit of the physical correlated-noise scenario Wezel10Mertens22aritro2. A specific example of such a limiting model is given by Eq. \ref{['eq:two-state']} and indicated by the green star. Any model of Continuous Spontaneous Localization (CSL) on the other hand, falls into the pink shaded area on the left of the diagram. Models in this class do not necessarily possess any of the SUV constraints. Since they assume the time evolution itself to contain an inherent stochastic modification, CSL models are generally formulated in the white-noise limit. Their stochastic modification often couples linearly to the quantum state, but non-linear contributions are allowed. CSL models may be formulated with respect to any collapse basis. Common choices include the position and mass density bases Bassi_03_PhyRepBassi2013Review. Here, we show that the white-noise limit of SUV models coincides with CSL equations based on a linear coupling to a stochastic process and collapsing into a symmetry-breaking basis. For a two-state system, this particular CSL model is described by Eq. \ref{['eq:two-state_CSL']}, and since it is equivalent to Eq. \ref{['eq:two-state']}, it is also indicated by the green star.