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Non-Markovian collapse models

Luca Ferialdi

Published Mar 26, 2010• 0 citations
OpenAlex

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

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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
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.
Spontaneous collapse models and Bohmian mechanics are two different solutions to the measurement problem plaguing orthodox quantum mechanics. They have, a priori nothing in common. At a formal level, collapse models add a non-linear noise term to the Schrödinger equation, and extract definite measurement outcomes either from the wave function ( e.g. mass density ontology) or the noise itself (flash ontology). Bohmian mechanics keeps the Schrödinger equation intact but uses the wave function to guide particles (or fields), which comprise the primitive ontology. Collapse models modify the predictions of orthodox quantum mechanics, whilst Bohmian mechanics can be argued to reproduce them. However, it turns out that collapse models and their primitive ontology can be exactly recast as Bohmian theories. More precisely, considering (i) a system described by a non-Markovian collapse model, and (ii) an extended system where a carefully tailored bath is added and described by Bohmian mechanics, the stochastic wave-function of the collapse model is exactly the wave-function of the original system conditioned on the Bohmian hidden variables of the bath. Further, the noise driving the collapse model is a linear functional of the Bohmian variables. The randomness that seems progressively revealed in the collapse models lies entirely in the initial conditions in the Bohmian-like theory. Our construction of the appropriate bath is not trivial and exploits an old result from the theory of open quantum systems. This reformulation of collapse models as Bohmian theories brings to the fore the question of whether there exists `unromantic' realist interpretations of quantum theory that cannot ultimately be rewritten this way, with some guiding law. It also points to important foundational differences between `true' (Markovian) collapse models and non-Markovian models.
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
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.
Spontaneous collapse models provide a possible solution to the measurement problem by modifying standard quantum dynamics. The modification consists of adding non-linear and stochastic terms inducing wavefunction collapse in space. Non-Markovian versions of these models are motivated by physical reasons, phenomenological consistency, and potential for relativistic extensions. Here, we investigate a non-Markovian version of the Poissonian Spontaneous Localization (PSL) model, i.e., a model characterized by instantaneous and localized collapse events. We assume that our model is characterized by a typical time scale τCτ_CτC​ so that, given an initial state ρ0ρ_0ρ0​ of standard quantum matter at time t=0t=0t=0, we derive an effective long-time (t≫τCt\gg τ_Ct≫τC​) statistical dynamics in terms of a CPTP map ΦtΦ_tΦt​. We then show how ΦtΦ_tΦt​ can be made equal to that obtained by non-Markovian CSL models. Moreover, given ΦtΦ_tΦt​, we obtain the associated time-convolutionless master equation by means of a supercumulant expansion. Finally, we characterize the collapse events process (for events with t≫τCt \gg τ_Ct≫τC​) given an initial quantum state ρ0ρ_0ρ0​ at t=0t=0t=0.