We set up a general formalism for models of spontaneous wave function collapse with dynamics represented by a stochastic differential equation driven by general Gaussian noises, not necessarily white in time. In particular, we show that the non-Schrodinger terms of the equation induce the collapse of the wave function to one of the common eigenstates of the collapsing operators, and that the collapse occurs with the correct quantum probabilities. We also develop a perturbation expansion of the solution of the equation with respect to the parameter which sets the strength of the collapse process; such an approximation allows one to compute the leading order terms for the deviations of the predictions of collapse models with respect to those of standard quantum mechanics. This analysis shows that to leading order, the ``imaginary'' noise trick can be used for non-white Gaussian noise.
We set up a general formalism for models of spontaneous wave function collapse with dynamics represented by a stochastic differential equation driven by general Gaussian noises, not necessarily white in time. In particular, we show that the non-Schrodinger terms of the equation induce the collapse of the wave function to one of the common eigenstates of the collapsing operators, and that the collapse occurs with the correct quantum probabilities. We also develop a perturbation expansion of the solution of the equation with respect to the parameter which sets the strength of the collapse process; such an approximation allows one to compute the leading order terms for the deviations of the predictions of collapse models with respect to those of standard quantum mechanics. This analysis shows that to leading order, the ``imaginary'' noise trick can be used for non-white Gaussian noise.
We continue the analysis of models of spontaneous wave function collapse with stochastic dynamics driven by non-white Gaussian noise. We specialize to a model in which a classical "noise" field, with specified autocorrelator, is coupled to a local nonrelativistic particle density. We derive general results in this model for the rates of density matrix diagonalization and of state vector reduction, and show that (in the absence of decoherence) both processes are governed by essentially the same rate parameters. As an alternative route to our reduction results, we also derive the Fokker-Planck equations that correspond to the initial stochastic Schrödinger equation. For specific models of the noise autocorrelator, including ones motivated by the structure of thermal Green's functions, we discuss the qualitative and qantitative dependence on model parameters, with particular emphasis on possible cosmological sources of the noise field.
Spontaneous collapse models (SCM) employ an imaginary noise term in the modification of Schrödinger's equation in order to achieve a stochastic collapse process. Such noise term is typically assumed to be white, and therefore uncorrelated in time, which results in the dynamics being Markovian. In the first part of this paper I discuss the reasons why physicists are exploring the possibility to instead employ non-white stochastic noise terms in the dynamics of SCM, and I then briefly introduce these models. In the second part, my aim is to provide an evaluation of the major conceptual and philosophical consequences of the generalised collapse models with colored noise. In particular, I will focus on the broad class of implications having to do with the non-Markovian nature of the dynamics, with its resulting temporal non-locality, and on the possibility to consider the noise field as a real physical entity.
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 abstract
Table 1: Table of the collapse constants used in the thesis.L and T denote respectively "length" and "time".α is the inverse of the square root of the correlation length of the GRW model (p.18).m is the mass of the particle considered in the QMUPL model and m 0 is a reference mass (p.30).function undergoes a sudden, spontaneous, and random localization.Such a collapse process is implemented by modifying the linear structure of the Schrödinger equation, introducing suitable non-linear and stochastic terms.Changes of this kind are necessary if one wants to avoid linear superpositions, reproduce quantum indeterminism, and, at the same time, achieve the localization of macroscopic objects.Actually, these requirements imply the structure of the modified Schrödinger equation to be well defined [4]; of course it is possible for the parameters to take very different values, identifying in this way models with physically different features.All the models afore-mentioned are Markovian models: they involve a white noise, i.e. a noise with Dirac-delta temporal correlation function, which implies the absence of memory effects [3].Since it is natural to identify the stochastic field driving the collapse with a physical field in Nature, which is likely to have a cosmological origin, one then needs to go beyond Markovian models since white noises are only mathematical idealizations which do not correspond to any physical noise.For this reason, it is necessary to study non-Markovian collapse models, i.e. models which involve noises with general correlation functions in time [11].The aim of this thesis is to study the non-Markovian generalization of the QMUPL model, both regarding its mathematical features and its physical properties, and to compare them with the known features of the Markovian
The basic strategy underlying models of spontaneous wave function collapse (collapse models) is to modify the Schrödinger equation by including nonlinear stochastic terms, which tend to localize wave functions in space in a dynamical manner. These terms have negligible effects on microscopic systems—therefore their quantum behaviour is practically preserved. On the other end, since the strength of these new terms scales with the mass of the system, they become dominant at the macroscopic level, making sure that wave functions of macro-objects are always well-localized in space. We will review these basic features. By changing the dynamics of quantum systems, collapse models make predictions, which are different from standard quantum mechanical predictions. Although they are difficult to detect, we discuss the most relevant scenarios, where such deviations can be observed.
Abstract Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small.