Collapse models explain the absence of quantum superpositions at the macroscopic scale, while giving practically the same predictions as quantum mechanics for microscopic systems. The Continuous Spontaneous Localization (CSL) model is the most refined and studied among collapse models. A well-known problem of this model, and of similar ones, is the steady and unlimited increase of the energy induced by the collapse noise. Here we present the dissipative version of the CSL model, which guarantees a finite energy during the entire system's evolution, thus making a crucial step toward a realistic energy-conserving collapse model. This is achieved by introducing a non-linear stochastic modification of the Schrödinger equation, which represents the action of a dissipative finite-temperature collapse noise. The possibility to introduce dissipation within collapse models in a consistent way will have relevant impact on the experimental investigations of the CSL model, and therefore also on the testability of the quantum superposition principle.
Collapse models explain the absence of quantum superpositions at the macroscopic scale, while giving practically the same predictions as quantum mechanics for microscopic systems. The Continuous Spontaneous Localization (CSL) model is the most refined and studied among collapse models. A well-known problem of this model, and of similar ones, is the steady and unlimited increase of the energy induced by the collapse noise. Here we present the dissipative version of the CSL model, which guarantees a finite energy during the entire system's evolution, thus making a crucial step toward a realistic energy-conserving collapse model. This is achieved by introducing a non-linear stochastic modification of the Schrodinger equation, which represents the action of a dissipative finite-temperature collapse noise. The possibility to introduce dissipation within collapse models in a consistent way will have relevant impact on the experimental investigations of the CSL model, and therefore also on the testability of the quantum superposition principle.
Spontaneous unitarity violations were recently proposed as a cause of objective quantum state reduction. This complements proposals based on stochastic modifications of Schr\"odinger's equation, but also differs from them in several aspects. Here, we formalize the description of models for spontaneous unitarity violation, and show that they generically imply models of dynamical quantum state reduction driven by colored noise. We present a formalism for exploring such models as well as a prescription for enforcing explicit norm-preservation, and we show that the resulting pure state dynamics is described by a modified von Neumann-Liouville equation which in a particular limit reduces to the Gorini-Kossakowski-Sudarshan-Lindblad master equations. We additionally show adherence to Born's rule emerging in the same limit from a physical constraint relating fluctuating and dissipating components of the model.
Standard quantum mechanics relies on an observer-dependent measurement postulate and an artificial linear-nonlinear dichotomy to resolve the superposition of macroscopically distinct states. While objective collapse models---such as Continuous Spontaneous Localization (CSL), Ghirardi-Rimini-Weber (GRW), and Di\'{o}si-Penrose gravitationally induced models---introduce stochastic non-linearities to achieve objective state reduction, they suffer from three fundamental pathologies: continuous energy non-conservation (spontaneous heating), reliance on memoryless white-noise approximations, and unphysical infinite spatial tails. In this work, I present Quantum Wave Function (QWF) Theory, a comprehensive non-perturbative framework that elevates the state functional
Ψ to an exact physical ontology. QWF introduces a non-Markovian colored noise memory kernel
K(t−τ) governed by a finite high-frequency cutoff, capturing realistic vacuum and environmental back-action times. To resolve the long-standing spontaneous heating anomaly, I incorporate a momentum-dependent dissipative counter-term
D^diss into the stochastic differential equation, strictly enforcing energy conservation
dtd⟨H^⟩=0 in closed isolated systems. Furthermore, I eliminate the infinite tail pathology by introducing a state-dependent stochastic amplitude filtering protocol where the microscopic collapse coupling dynamically increases in the asymptotic regime to enforce exponential tail damping below a physical threshold
ϵ, without breaking density matrix linearity. Extending beyond non-relativistic domains, I formulate a fully covariant quantum field description on arbitrary spacelike hypersurfaces
Σ using a Tomonaga-Schwinger stress-energy framework, ensuring causality and preventing superluminal signaling. Finally, I prove via martingale analysis that standard Born-rule statistics (
P=∣Ψ∣2) emerge naturally as a dynamical attractor of the stochastic localization process. QWF provides a mathematically rigorous, falsifiable, and complete resolution to the quantum measurement problem with explicit, testable signatures for modern gravitational wave and optomechanical experiments.
Abstract This chapter introduces the two equations that completely define CSL, for the simplest case, that is, a state vector in a two-dimensional Hilbert space. One equation is the modified evolution equation for the state vector which depends upon a Brownian motion function of time, B(t), a Hermitian operator A called the “collapse-generating operator,” and a collapse rate parameter λ. The state vector evolution is linear, but not unitary, so its norm changes with time. The second equation is called the probability rule. It says the probability a particular B(t) occurs in Nature is proportional to the squared norm of the state vector, that is, high norm state vectors are most likely to occur. The chapter shows how an initial state vector that is a superposition of the eigenstates of A evolves to one eigenstate, with the Born probability, as predicted by standard quantum theory. Finally, it discusses the density matrix description of the ensemble of evolutions under all possible B(t).
Objective collapse theories propose a solution to the quantum measurement problem by predicting deviations from Schrödinger’s equation that can be tested experimentally. A class of objective theories based on spontaneous unitarity violation was recently introduced, in which the stochastic field required for obtaining Born’s rule does not depend on the state of the system being measured. Here, we classify possible models for the stochastic field dynamics in theories of spontaneous unitarity violation. We show that for correlated stochastic dynamics, the field must be defined on a closed manifold. In two or more dimensions, it is then always possible to find stochastic dynamics yielding Born’s rule, independent of the state being measured or the correlation time of the stochastic field. We show that the models defined this way are all isomorphic to the definition on a two-sphere, which we propose to be a minimal physical model for the stochastic field in models of spontaneous unitarity violation.