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Applying the CSL Stratonovich Equation to the Harmonic Oscillator Undergoing Collapse in Position

Philip Pearle

Published Jan 30, 2024• 0 citations
Publisher

Abstract

Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to a harmonic oscillator particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was not previously obtained. It is quite complicated, and provides a demonstration of the power of the SDE approach to CSL.

DOI
10.1093/oso/9780198901372.003.0018

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Applying the CSL Stratonovich Equation to the Free Particle Undergoing Collapse in Position

Philip Pearle

Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to the free particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was previously obtained in Chapter 10 by using the linear Schrödinger equation/probability rule formulation of CSL (which was used in the first seven chapters of this book). This chapter shows how the same result is obtained from the stochastic differential equation formulation of CSL.

•Jan 30, 2024•0 citations

Operational Distances Between Collapse Models: A Quantitative Comparison of CSL and the Schrödinger–Newton Model

Dewan Sajid Islam

We present a quantitative operational comparison of the Continuous Spontaneous Localization (CSL) model and the deterministic Schrödinger–Newton model within a specific operational context. Using a common operational pseudometric, we compare standard quantum mechanics, CSL, and the Schrödinger–Newton model for a harmonic oscillator with a position measurement. Within the Gaussian-closure and Gaussian-harmonic approximations adopted here, the three models yield Gaussian position distributions. The CSL sector is treated using Gaussian-closure moment dynamics, while the Schrödinger–Newton sector is evaluated numerically within the Gaussian harmonic approximation. We derive the corresponding total-variation distances and prove two structural properties of the operational pseudometric: monotonicity under coarse-graining and a data-processing inequality. We also establish a perturbative scaling result for the quantum-mechanical versus Schrödinger–Newton distance for a specific protocol. For representative parameters, the CSL effect is extremely small, with a QM–CSL distance of approximately 7.6 × 10⁻¹⁸, whereas the Gaussian-harmonic Schrödinger–Newton calculation gives a substantially larger QM–SN distance of approximately 0.75. We investigate the mass dependence of these distances and identify protocol-dependent crossover scales. The results provide a quantitative operational framework for comparing collapse models and for assessing their distinguishability in prospective experiments.

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CSL Collapse Model And Spontaneous Radiation: An Update

Philip Pearle, Ring, James, J. I. Collar +1 more

A brief review is given of the Continuous Spontaneous Localization (CSL) model in which a classical field interacts with quantized particles to cause dynamical wavefunction collapse. One of the model's predictions is that particles "spontaneously" gain energy at a slow rate. When applied to the excitation of a nucleon in a Ge nucleus, it is shown how a limit on the relative collapse rates of neutron and proton could be obtained, and a rough estimate is made from data. When applied to the spontaneous excitation of 1s electrons in Ge, by a more detailed analysis of more accurate data than previously given, an updated limit is obtained on the relative collapse rates of the electron and proton, suggesting that the coupling of the field to electrons and nucleons is mass proportional.

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Continuous Spontaneous Localization (CSL) Theory

Philip Pearle

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).

•Jan 30, 2024•0 citations

Introduction

Philip Pearle

Abstract This chapter introduces the problem (that the wave function of standard quantum theory (SQT) cannot correspond to reality in nature) and its resolution (change Schrodinger’s equation). The chapter begins by citing Schrodinger’s argument as to how the collapse postulate of SQT leads to the cited problem. It then presents a resolution embodied in the continuous spontaneous localization (CSL) theory‘s altered Schrodinger equation. The chapter shows how the “Gambler’s Ruin” game, where two gamblers bet on a coin toss until one wins all the money, is a precise and intuitively appealing analogy to the needed collapse behavior for a superposition of two states evolving under random influence to one or another of those states. Following this, the chapter ends by presenting the structure of the book. The first seven chapters show how CSL works, discusses experimentally testable and already tested predictions, and covers interpretive features of CSL. The purpose of the remaining eleven chapters, called “Supplementary,” is summarized as well.

•Jan 30, 2024•1 citation
Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to a harmonic oscillator particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was not previously obtained. It is quite complicated, and provides a demonstration of the power of the SDE approach to CSL.
Abstract This chapter addresses the problem of applying the continuous spontaneous localization (CSL) Stratonovich equation to the free particle undergoing collapse in position, by finding the solution in Gaussian form. This solution was previously obtained in Chapter 10 by using the linear Schrödinger equation/probability rule formulation of CSL (which was used in the first seven chapters of this book). This chapter shows how the same result is obtained from the stochastic differential equation formulation of CSL.
We present a quantitative operational comparison of the Continuous Spontaneous Localization (CSL) model and the deterministic Schrödinger–Newton model within a specific operational context. Using a common operational pseudometric, we compare standard quantum mechanics, CSL, and the Schrödinger–Newton model for a harmonic oscillator with a position measurement. Within the Gaussian-closure and Gaussian-harmonic approximations adopted here, the three models yield Gaussian position distributions. The CSL sector is treated using Gaussian-closure moment dynamics, while the Schrödinger–Newton sector is evaluated numerically within the Gaussian harmonic approximation. We derive the corresponding total-variation distances and prove two structural properties of the operational pseudometric: monotonicity under coarse-graining and a data-processing inequality. We also establish a perturbative scaling result for the quantum-mechanical versus Schrödinger–Newton distance for a specific protocol. For representative parameters, the CSL effect is extremely small, with a QM–CSL distance of approximately 7.6 × 10⁻¹⁸, whereas the Gaussian-harmonic Schrödinger–Newton calculation gives a substantially larger QM–SN distance of approximately 0.75. We investigate the mass dependence of these distances and identify protocol-dependent crossover scales. The results provide a quantitative operational framework for comparing collapse models and for assessing their distinguishability in prospective experiments.
A brief review is given of the Continuous Spontaneous Localization (CSL) model in which a classical field interacts with quantized particles to cause dynamical wavefunction collapse. One of the model's predictions is that particles "spontaneously" gain energy at a slow rate. When applied to the excitation of a nucleon in a Ge nucleus, it is shown how a limit on the relative collapse rates of neutron and proton could be obtained, and a rough estimate is made from data. When applied to the spontaneous excitation of 1s electrons in Ge, by a more detailed analysis of more accurate data than previously given, an updated limit is obtained on the relative collapse rates of the electron and proton, suggesting that the coupling of the field to electrons and nucleons is mass proportional.
Abstract The Continuous Spontaneous Localisation (CSL) theory in the cosmological context is subject to uncertainties related to the choice of the collapse operator. In this paper, we constrain its form based on generic arguments. We show that, if the collapse operator is even in the field variables, it is unable to induce the collapse of the wavefunction. Instead, if it is odd, we find that only linear operators are such that the outcomes are distributed according to Gaussian statistics, as required by measurements of the cosmic microwave background. We discuss implications of these results for previously proposed collapse operators. We conclude that the cosmological CSL collapse operator should be linear in the field variables.
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).
Abstract This chapter introduces the problem (that the wave function of standard quantum theory (SQT) cannot correspond to reality in nature) and its resolution (change Schrodinger’s equation). The chapter begins by citing Schrodinger’s argument as to how the collapse postulate of SQT leads to the cited problem. It then presents a resolution embodied in the continuous spontaneous localization (CSL) theory‘s altered Schrodinger equation. The chapter shows how the “Gambler’s Ruin” game, where two gamblers bet on a coin toss until one wins all the money, is a precise and intuitively appealing analogy to the needed collapse behavior for a superposition of two states evolving under random influence to one or another of those states. Following this, the chapter ends by presenting the structure of the book. The first seven chapters show how CSL works, discusses experimentally testable and already tested predictions, and covers interpretive features of CSL. The purpose of the remaining eleven chapters, called “Supplementary,” is summarized as well.