Toward an energy-conserving model of spontaneous wavefunction collapse

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 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.

arXiv:1408.6446Scientific ReportsAug 5, 201561 citations

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

Aritro Mukherjee, Jasper van Wezel

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. Coloured-noise driven collapse theories extending the equilibrium description of spontaneous symmetry breaking (SSB) to spontaneous unitary violation (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 when collapsing in a basis of spatially localized 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 (FDR), which results in ensemble averaged probability densities following a linear quantum semi-group guaranteeing the absence of superluminal signalling.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. Coloured-noise driven collapse theories extending the equilibrium description of spontaneous symmetry breaking (SSB) to spontaneous unitary violation (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 when collapsing in a basis of spatially localized 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 (FDR), which results in ensemble averaged probability densities following a linear quantum semi-group guaranteeing the absence of superluminal signalling.

arXiv:2405.01077Proceedings of the Royal Society A Mathematical Physical and Engineering SciencesOct 1, 20251 citation

Colored collapse models from the non-interferometric perspective

Matteo Carlesso, Luca Ferialdi, Angelo Bassi

Abstract Models of spontaneous wave function collapse describe the quantum-to-classical transition by assuming a progressive breakdown of the superposition principle when the mass of the system increases, providing a well-defined phenomenology in terms of a non-linearly and stochastically modified Schrödinger equation, which can be tested experimentally. The most popular of such models is the continuous spontaneous localization (CSL) model: in its original version, the collapse is driven by a white noise, and more recently, generalizations in terms of colored noises, which are more realistic, have been formulated. We will analyze how current non-interferometric tests bound the model, depending on the spectrum of the noise. We will find that low frequency purely mechanical experiments provide the most stable and strongest bounds. Graphical abstractAbstract Models of spontaneous wave function collapse describe the quantum-to-classical transition by assuming a progressive breakdown of the superposition principle when the mass of the system increases, providing a well-defined phenomenology in terms of a non-linearly and stochastically modified Schrödinger equation, which can be tested experimentally. The most popular of such models is the continuous spontaneous localization (CSL) model: in its original version, the collapse is driven by a white noise, and more recently, generalizations in terms of colored noises, which are more realistic, have been formulated. We will analyze how current non-interferometric tests bound the model, depending on the spectrum of the noise. We will find that low frequency purely mechanical experiments provide the most stable and strongest bounds. Graphical abstract

arXiv:1805.10100The European Physical Journal DSep 1, 201830 citations

Quantum Wave Function Theory: A Covariant, Dissipative and Non-Markovian Framework for objective Wave-Functional Localization

Shreyansh Biswal

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 $\Psi$ to an exact physical ontology. QWF introduces a non-Markovian colored noise memory kernel $K(t - \tau)$ 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 $\hat{D}_{\text{diss}}$ into the stochastic differential equation, strictly enforcing energy conservation $\frac{d}{dt} \langle \hat{H} \rangle = 0$ 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 $\epsilon$ , without breaking density matrix linearity. Extending beyond non-relativistic domains, I formulate a fully covariant quantum field description on arbitrary spacelike hypersurfaces $\Sigma$ using a Tomonaga-Schwinger stress-energy framework, ensuring causality and preventing superluminal signaling. Finally, I prove via martingale analysis that standard Born-rule statistics ( $P = |\Psi|^2$ ) 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.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 $\Psi$ to an exact physical ontology. QWF introduces a non-Markovian colored noise memory kernel $K(t - \tau)$ 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 $\hat{D}_{\text{diss}}$ into the stochastic differential equation, strictly enforcing energy conservation $\frac{d}{dt} \langle \hat{H} \rangle = 0$ 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 $\epsilon$, without breaking density matrix linearity. Extending beyond non-relativistic domains, I formulate a fully covariant quantum field description on arbitrary spacelike hypersurfaces $\Sigma$ using a Tomonaga-Schwinger stress-energy framework, ensuring causality and preventing superluminal signaling. Finally, I prove via martingale analysis that standard Born-rule statistics ($P = |\Psi|^2$) 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.

May 2, 20260 citations

A Novel Approach to Parameter Determination of the Continuous Spontaneous Localization Collapse Model

Kristian Piscicchia, Alessio Porcelli, Angelo Bassi +23 more

Models of dynamical wave function collapse consistently describe the breakdown of the quantum superposition with the growing mass of the system by introducing non-linear and stochastic modifications to the standard Schrödinger dynamics. Among them, Continuous Spontaneous Localization (CSL) was extensively investigated both theoretically and experimentally. Measurable consequences of the collapse phenomenon depend on different combinations of the phenomenological parameters of the model-the strength λ and the correlation length rC-and have led, so far, to the exclusion of regions of the admissible (λ-rC) parameters space. We developed a novel approach to disentangle the λ and rC probability density functions, which discloses a more profound statistical insight.Models of dynamical wave function collapse consistently describe the breakdown of the quantum superposition with the growing mass of the system by introducing non-linear and stochastic modifications to the standard Schrödinger dynamics. Among them, Continuous Spontaneous Localization (CSL) was extensively investigated both theoretically and experimentally. Measurable consequences of the collapse phenomenon depend on different combinations of the phenomenological parameters of the model-the strength λ and the correlation length rC-and have led, so far, to the exclusion of regions of the admissible (λ-rC) parameters space. We developed a novel approach to disentangle the λ and rC probability density functions, which discloses a more profound statistical insight.

EntropyFeb 4, 20233 citations

Dissipative extension of the Ghirardi-Rimini-Weber model

Andrea Smirne, Bassano Vacchini, Angelo Bassi

In this paper, we present an extension of the Ghirardi-Rimini-Weber model for the spontaneous collapse of the wave function. Through the inclusion of dissipation, we avoid the divergence of the energy on the long-time scale, which affects the original model. In particular, we define jump operators, which depend on the momentum of the system and lead to an exponential relaxation of the energy to a finite value. The finite asymptotic energy is naturally associated to a collapse noise with a finite temperature, which is a basic realistic feature of our extended model. Remarkably, even in the presence of a low-temperature noise, the collapse model is effective. The action of the jump operators still localizes the wave function and the relevance of the localization increases with the size of the system, according to the so-called amplification mechanism, which guarantees a unified description of the evolution of microscopic and macroscopic systems. We study in detail the features of our model, at the level of both the trajectories in the Hilbert space and the master equation for the average state of the system. In addition, we show that the dissipative Ghirardi-Rimini-Weber model, as well as the original one, can be fully characterized in a compact way by means of a proper stochastic differential equation.In this paper, we present an extension of the Ghirardi-Rimini-Weber model for the spontaneous collapse of the wave function. Through the inclusion of dissipation, we avoid the divergence of the energy on the long-time scale, which affects the original model. In particular, we define jump operators, which depend on the momentum of the system and lead to an exponential relaxation of the energy to a finite value. The finite asymptotic energy is naturally associated to a collapse noise with a finite temperature, which is a basic realistic feature of our extended model. Remarkably, even in the presence of a low-temperature noise, the collapse model is effective. The action of the jump operators still localizes the wave function and the relevance of the localization increases with the size of the system, according to the so-called amplification mechanism, which guarantees a unified description of the evolution of microscopic and macroscopic systems. We study in detail the features of our model, at the level of both the trajectories in the Hilbert space and the master equation for the average state of the system. In addition, we show that the dissipative Ghirardi-Rimini-Weber model, as well as the original one, can be fully characterized in a compact way by means of a proper stochastic differential equation.

arXiv:1408.6115Physical Review ADec 31, 201430 citations

\Psi

to an exact physical ontology. QWF introduces a non-Markovian colored noise memory kernel

K(t - \tau)

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

\hat{D}_{\text{diss}}

into the stochastic differential equation, strictly enforcing energy conservation

\frac{d}{dt} \langle \hat{H} \rangle = 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

\epsilon

, without breaking density matrix linearity. Extending beyond non-relativistic domains, I formulate a fully covariant quantum field description on arbitrary spacelike hypersurfaces

\Sigma

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 = |\Psi|^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.