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Physical Review A

Spontaneous radiation of free electrons in a nonrelativistic collapse model

Qijia Fu

Published Sep 1, 1997• 72 citations
Publisher

Abstract

One way to solve the long-standing ``measurement''problem in quantum mechanics is to modify Schr\"odinger's equation so that the collapse of a state vector is the result of some intrinsic physical dynamics. The introduction of such collapse dynamics often results in energy nonconservation. In this paper, we first derive a general expression for the rate of change in the expectation value of energy in a general collapse model that supports a linear evolution of the density matrix. In particular, we show, under certain plausible assumptions, that energy nonconservation is an inevitable consequence of collapse. We then work with a specific model, namely, the nonrelativistic continuous spontaneous localization (CSL) model, to derive further consequences of collapse. In CSL, in a nonunitary evolution, a particle interacts with a fluctuating scalar field, leading to the collapse of the state vector. One consequence of this interaction is that a free electron can radiate spontaneously. We calculate the spectrum of this radiation. The result is then compared with the observed upper bound on spontaneous radiation in Ge and a constraint on the parameters of CSL is obtained.

DOI
10.1103/physreva.56.1806

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One way to solve the long-standing ``measurement''problem in quantum mechanics is to modify Schr"odinger's equation so that the collapse of a state vector is the result of some intrinsic physical dynamics. The introduction of such collapse dynamics often results in energy nonconservation. In this paper, we first derive a general expression for the rate of change in the expectation value of energy in a general collapse model that supports a linear evolution of the density matrix. In particular, we show, under certain plausible assumptions, that energy nonconservation is an inevitable consequence of collapse. We then work with a specific model, namely, the nonrelativistic continuous spontaneous localization (CSL) model, to derive further consequences of collapse. In CSL, in a nonunitary evolution, a particle interacts with a fluctuating scalar field, leading to the collapse of the state vector. One consequence of this interaction is that a free electron can radiate spontaneously. We calculate the spectrum of this radiation. The result is then compared with the observed upper bound on spontaneous radiation in Ge and a constraint on the parameters of CSL is obtained.
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.
I suggest a simple alteration of my CSL (Continuous Spontaneous Localization) theory, replacing the mass density collapse-generating operators by relativistic energy density operators. Some consequences of the density matrix evolution equation are explored. First, the expression for the mean energy increase of free particles is calculated (which, in the non-relativistic limit, agrees with the usual result). Then, the density matrix evolution is applied to photons. The mean rate of loss of photon number from a laser beam pulse, the momentum distribution of the photons "excited" out of the laser beam pulse, and the alteration of the cosmic blackbody spectrum are all treated to first order in the collapse rate parameter λλλ. Associated possible experimental limits on λλλ are discussed.
Models of spontaneous wave function collapse modify the linear Schr\"odinger equation of standard quantum mechanics by adding stochastic nonlinear terms to it. The aim of such models is to describe the quantum (linear) nature of microsystems along with the classical nature (violation of superposition principle) of macroscopic ones. The addition of such nonlinear terms in the Schr\"odinger equation leads to nonconservation of energy of the system under consideration. Thus, a striking feature of collapse models is to heat nonrelativistic particles with a constant rate. If such a process is physical, then it has the ability to perturb the well-understood thermal history of the universe. In this article we will try to investigate the impacts of such heating terms, according to the continuous spontaneous localization model, on the standard evolution of nonrelativistic matter and on the formation of cosmic microwave background radiation. We will also put constraints on the continuous spontaneous localization collapse rate $\ensuremath{\lambda}$ by considering that the standard evolution of nonrelativistic matter is not hampered and the observed precise blackbody spectrum of cosmic microwave background radiation would not get distorted (in the form of $\ensuremath{\mu}$-type and yyy-type distortions) so as to violate the observed bounds.
A model is discussed where all operators are constructed from a quantum scalar field whose energy spectrum takes on all real values. The Schr\"odinger picture wave function depends upon space and time coordinates for each particle, as well as an inexorably increasing evolution parameter sss which labels a foliation of spacelike hypersurfaces. The model is constructed to be manifestly Lorentz invariant in the interaction picture. Free particle states and interactions are discussed in this framework. Then, the formalism of the continuous spontaneous localization (CSL) theory of dynamical collapse is applied. The collapse-generating operator is chosen to be the particle number space-time density. Unlike previous relativistically invariant models, the vacuum state is not excited. The collapse dynamics depends upon two parameters, a parameter $\mathrm{\ensuremath{\Lambda}}$ which represents the collapse rate/volume and a scale factor $\ensuremath{\ell}$. A common example of collapse dynamics, involving a clump of matter in a superposition of two locations, is analyzed. The collapse rate is shown to be identical to that of nonrelativistic CSL when the GRW-CSL choice of $\ensuremath{\ell}=a=1{0}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{cm}$, is made, along with $\mathrm{\ensuremath{\Lambda}}=\ensuremath{\lambda}/{a}^{3}$ (GRW-CSL choice $\ensuremath{\lambda}=1{0}^{\ensuremath{-}16}{s}^{\ensuremath{-}1}$). The collapse rate is also satisfactory with the choice $\ensuremath{\ell}$ as the size of the Universe, with $\mathrm{\ensuremath{\Lambda}}=\ensuremath{\lambda}/\ensuremath{\ell}{a}^{2}$. Because the collapse narrows wave functions in space and time, it increases a particle's momentum and energy, altering its mass. It is shown that, with $\ensuremath{\ell}=a$, the change of mass of a nucleon is unacceptably large but, when $\ensuremath{\ell}$ is the size of the Universe, the change of mass over the age of the Universe is acceptably small.
Collapse models introduce stochastic and nonlinear modifications in the quantum dynamics, predicting observable effects, such as spontaneous radiation from charged particles, which can be used to constrain their parameters. Recently, attention has focused on the 1-100 keV energy range, where the wavelength of the emitted photons becomes comparable to atomic dimensions, making the emission sensitive to atomic structure and leading to model-dependent behaviors that enable their discrimination. Here, we derive a general expression for the spontaneous emission rate for arbitrary noise, providing a framework that systematically incorporates the atomic structure through the radial distribution of the emitters, modulated by the specific collapse model. The formalism recovers previous results in the appropriate limits and naturally includes new low-energy effects, such as cancellation mechanisms arising from charge correlations. We evaluate the rates for germanium and xenon within the Diósi-Penrose and Continuous Spontaneous Localization models, showing how these correlations modify the predicted emission rates. This approach provides a unified framework to account for atomic effects and enables more robust, material-dependent experimental constraints on collapse model parameters.
A natural generalization of the CSL (Continuous Spontaneous Localization) theory of dynamical collapse is applied to a relativistic quantum scalar field ϕ(x,t)ϕ({\bf x},t)ϕ(x,t). It is shown that the modified Schrödinger equation is relativistically invariant, that the probabilities associated to all possible values of the classical scalar random field w(x,t)w({\bf x},t)w(x,t) (which determines the eventual state of collapse) add up to 1, that there is no energy production out of the vacuum and, in the limit of large time, the collapse is toward eigenstates of ϕ(x,0)ϕ({\bf x},0)ϕ(x,0).