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AIP conference proceedings

Dynamical reduction models and the energy conservation principle

Angelo Bassi

Published Jan 1, 2006• 2 citations
Publisher

Abstract

A common feature of dynamical reduction models is the violation of energy conservation principle which usually shows up as a constant increase in time of the energy of isolated systems. Anyway for typical values of the parameters of the models, such a violation is usually so weak that cannot be detected with present‐day technology. . Despite the reduction mechanism itself seems responsible for this behaviour, we show that this is not a intrinsic property of dynamical reduction models: we exhibit a collapse model such that the energy of isolated systems does not diverge for large times but reaches an asymptotic finite value. This result could be interesting in understanding how to work out relativistic extensions of dynamical reduction models.

DOI
10.1063/1.2219349

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A common feature of dynamical reduction models is the violation of energy conservation principle which usually shows up as a constant increase in time of the energy of isolated systems. Anyway for typical values of the parameters of the models, such a violation is usually so weak that cannot be detected with present‐day technology. . Despite the reduction mechanism itself seems responsible for this behaviour, we show that this is not a intrinsic property of dynamical reduction models: we exhibit a collapse model such that the energy of isolated systems does not diverge for large times but reaches an asymptotic finite value. This result could be interesting in understanding how to work out relativistic extensions of dynamical reduction models.
A typical feature of spontaneous collapse models which aim at localizing wavefunctions in space is the violation of the principle of energy conservation. In the models proposed in the literature the stochastic field which is responsible for the localization mechanism causes the momentum to behave like a Brownian motion, whose larger and larger fluctuations show up as a steady increase of the energy of the system. In spite of the fact that, in all situations, such an increase is small and practically undetectable, it is an undesirable feature that the energy of physical systems is not conserved but increases constantly in time, diverging for t→∞t \to \inftyt→∞. In this paper we show that this property of collapse models can be modified: we propose a model of spontaneous wavefunction collapse sharing all most important features of usual models but such that the energy of isolated systems reaches an asymptotic finite value instead of increasing with a steady rate.
A typical feature of spontaneous collapse models which aim at localizing wavefunctions in space is the violation of the principle of energy conservation. In the models proposed in the literature the stochastic field which is responsible for the localization mechanism causes the momentum to behave like a Brownian motion, whose larger and larger fluctuations show up as a steady increase of the energy of the system. In spite of the fact that, in all situations, such an increase is small and practically undetectable, it is an undesirable feature that the energy of physical systems is not conserved but increases constantly in time, diverging for ttoinftyt \\to \\inftyttoinfty. In this paper we show that this property of collapse models can be modified: we propose a model of spontaneous wavefunction collapse sharing all most important features of usual models but such that the energy of isolated systems reaches an asymptotic finite value instead of increasing with a steady rate.
A new energy-based stochastic extension of the Schrödinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all finite-dimensional quantum systems, is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs. A time-change technique is introduced to construct a “clock” variable that relates the asymptotic and the finite-time collapse models by means of a nonlinear transformation.
Spontaneous collapse models have been proposed as a possible realist solution of the quantum measurement problem. They assume that the standard unitary evolution is valid only in the microscopic limit, with stochastic corrections leading to a dynamical reduction of the wave function at macroscopic level. A remarkable feature of collapse models is that, unlike interpretations of quantum mechanics, they predict departures from standard theory that can be experimentally tested. A common prediction is a violation of energy conservation, which manifests as an universal force noise acting on mechanical systems. Here, we will discuss recent bounds on collapse models set by mechanical experiments, focusing on ultracold cantilevers and gravitational wave detectors. First notable results are a partial exclusion of the CSL parameters proposed by Adler and a full exclusion of an early model based on quantum gravity arguments.
A new energy-based stochastic extension of the Schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed. An exact closed-form solution to the dynamical equation, valid for all finite-dimensional quantum systems, is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs. A time-change technique is introduced to construct a "clock" variable that relates the asymptotic and the finite-time collapse models by means of a nonlinear transformation. (c) 2005 American Institute of Physics
We consider a scheme whereby it is possible to reconcile semi-classical Einstein's equation with the violation of the conservation of the expectation value of energy-momentum that is associated with dynamical reduction theories of the quantum state for matter. The very interesting out-shot of the formulation is the appearance of a nontrivial contribution to an effective cosmological constant (which is not strictly constant). This opens the possibility of using models for dynamical collapse of the wave function to compute its value. Another interesting implication of our analysis is that tiny violations of energy-momentum conservation with negligible local effects can become very important on cosmological scales at late times.