It is emphasized that the collapse postulate of standard quantum theory can violate conservation of energy-momentum and there is no indication from where the energy-momentum comes or to where it goes. Likewise, in the Continuous Spontaneous Localization (CSL) dynamical collapse model, particles gain energy on average. In CSL, the usual Schrödinger dynamics is altered so that a randomly fluctuating classical field interacts with quantized particles to cause wavefunction collapse. In this paper it is shown how to define energy for the (...) classical field so that the average value of the energy of the field plus the quantum system is conserved for the ensemble of collapsing wavefunctions. While conservation of just the first moment of energy is, of course, much less than complete conservation of energy, this does support the idea that the field could provide the conservation law balance when events occur. (shrink)

We interpret the probability rule of the CSL collapse theory to mean to mean that the scalar field which causes collapse is the gravitational curvature scalar with two sources, the expectation value of the mass density (smeared over the GRW scale a) and a white noise fluctuating source. We examine two models of the fluctuating source, monopole fluctuations and dipole fluctuations, and show that these correspond to two well-known CSL models. We relate the two GRW parameters of CSL to fundamental (...) constants, and we explain the energy increase of particles due to collapse as arising from the loss of vacuum gravitational energy. (shrink)

The problem of getting a relativistic generalization of the CSL dynamical reduction model, which has been presented in part I, is discussed. In so doing we have the opportunity to introduce the idea of a stochastically invariant theory. The theoretical model we present, that satisfies this kind of invariance requirement, offers us the possibility to reconsider, from a new point of view, some conceptually relevant issues such as nonlocality, the legitimacy of attributing elements of physical reality to physical systems and (...) the problem of establishing causal relations between physical events. (shrink)

The continuous spontaneous localization (CSL) model modifies Schrödinger's equation so that the collapse of the state vector is described as a physical process (a special interaction of particles with a universal fluctuating field). A consequence of the model is that an electron in an atom should occasionally get “spontaneously” knocked out of the atom. The CSL ionization rate for the 1s electrons in the Ge atom is calculated and compared with an experimental upper limit for the rate of “spontaneously” generated (...) x-ray pulses in Ge. This gives, for the first time, an experimental constraint on the parameters which characterize this model (the GRW parameters and the relative collapse rate of electrons and nucleons). It is concluded that the values assigned to the GRW parameters by GRW may be maintained only if the coupling of electrons to the fluctuating field is 0.35% or less than the coupling of nucleons, suggestive of a mass-proportional (and therefore gravitational) collapse mechanism. For other allowed values of the GRW parameters, it is still argued that nucleons should collapse more rapidly than electrons. (shrink)

Wavefunction collapse models modify Schrödinger's equation so that it describes the rapid evolution of a superposition of macroscopically distinguishable states to one of them. This provides a phenomenological basis for a physical resolution to the so-called “measurement problem.” Such models have experimentally testable differences from standard quantum theory. The most well developed such model at present is the Continuous Spontaneous Localization (CSL) model in which a universal fluctuating classical field interacts with particles to cause collapse. One “side effect” of this (...) interaction is that the field imparts energy to the particles: experimental evidence on this has led to restrictions on the parameters of the model, suggesting that the coupling of the classical field to the particles must be mass-proportional. Another “side effect” is that the field imparts momentum to particles, causing a small blob of matter to undergo random walk. Here we explore this in order to supply predictions which could be experimentally tested. We examine the translational diffusion of a sphere and a disc, and the rotational diffusion of a disc, according to CSL. For example, we find that the rms distance an isolated 10−5 cm radius sphere diffuses is ≈(its diameter, 5 cm) in (20 sec, a day), and that a disc of radius 2 ⋅ 10−5 cm and thickness 0.5 ⋅ 10−5 cm diffuses through 2πrad in about 70 sec (this assumes the “standard” CSL parameter values). The comparable rms diffusions of standard quantum theory are smaller than these by a factor 10−3±1. It is shown that the CSL diffusion in air at STP is much reduced and, indeed, is swamped by the ordinary Brownian motion. It is also shown that the sphere's diffusion in a thermal radiation bath at room temperature is comparable to the CSL diffusion, but is utterly negligible at liquid He temperature. Thus, in order to observe CSL diffusion, the pressure and temperature must be low. At the low reported pressure of 5 ⋅ 10−17 Torr, achieved at 4.2°K, the mean time between air molecule collisions with the (sphere, disc) is ≈(80, 45)min. This is ample time for observation of the putative CSL diffusion with the standard parameters and, it is pointed out, with any parameters in the range over which the theory may be considered viable. This encourages consideration of how such an experiment may actually be performed, and the paper closes with some thoughts on this subject. (shrink)

It is shown that data on the dissociation rate of deuterium obtained in an experiment at the Sudbury Neutrino Observatory provides evidence that the Continuous Spontaneous Localization wavefunction collapse model should have mass–proportional coupling to be viable.

In the problem of the gambler's ruin, a classic problem in probability theory, a number of gamblers play against each other until all but one of them is “wiped out.” It is shown that this problem is identical to a previously presented formulation of the reduction of the state vector, so that the state vectors in a linear superposition may be regarded as “playing” against each other until all but one of them is “wiped out.” This is a useful part (...) of the description of an objectively real universe represented by a state vector that is a superposition of macroscopically distinguishable states dynamically created by the Hamiltonian and destroyed by the reduction mechanism. (shrink)

Radiationless motion of a charge distribution is reviewed, and the necessary condition conjectured by Goedecke is proved. Then it is shown that a nonrotating, uniformly charged, spherical shell (as seen from its own rest frame) which moves in a relativistically invariant fashion does not have any bounded radiationless motions, unlike its nonrelativistic counterpart.

A classical point electron radiates when it accelerates. However, there are classical electron models with extended charge distributions which can accelerate and/or deform without radiating. Can a model be contrived that will undergo radiationless motion while accelerating (on the average) over a distance large compared to its size? The answer is no: we prove that the “center” of the electron is always closer than the electron “diameter” to a fictitious point undergoing constant-velocity motion, if the electron's motion is radiationless.

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 can 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 given previously, 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. (shrink)

The propositions, that what we see around us is real and that reality should be represented by the statevector, conflict with quantum theory. In quantum theory, the statevector can readily become a sum of states of comparable norm, each state representing a different reality. In this paper we present the Continuous Spontaneous Localization (CSL) theory, in which a modified Schrodinger equation, while scarcely affecting the dynamics of a microscopic system, rapidly "reduces" the statevector of a macroscopic system to a state (...) appropriate for representing individual reality. (shrink)

We dedicate these papers to the memory of John Bell, whose contributions to, support for, and encouragement of the research program described here has meant more than words can say to those involved in it.In Schrödinger’s “cat paradox” example, a nucleus which has a 50% probability of decaying within an hour is coupled to a cat by a “hellish contraption” which, if it detects the decay, will kill the cat. If we take the point of view that what we see (...) around us is real, what occurs in reality is one of the two following evolutions :(—> means “evolves in an hour into“).According to Schrödinger’s equation, the evolution of the statevector describing this situation isThe right hand side of Eq. (1.2) does not correspond to either the reality on the right hand side of (1.1a) nor to the reality on the right hand side of (1.1b). (shrink)

In this part we will consider recent attempts to get a relativistic CSL theory. The problem of getting such a generalization, or at least of making plausible that it exists, is of great interest. J. Bell (1990), after having expressed his dissatisfaction with the fundamental lack of precision of the standard formulation of quantum mechanics and the opinion that the only available acceptable alternatives are the Pilot Wave and the Spontaneous Localization schemes has stated: The big question, in my opinion, (...) is which, if either, of these two precise pictures can be redeveloped in a Lorentz invariant way. The point of view of this paper is that what is most interesting is not a detailed exposition of the formal aspects of the generalization, but some crucial conceptual points which have come into focus in this attempt. (shrink)

A simple quantum model describing the onset of time is presented. This is combined with a simple quantum model of the onset of space. A major purpose is to explore the interpretational issues which arise. The state vector is a superposition of states representing different “instants.” The sample space and probability measure are discussed. Critical to the dynamics is state vector collapse: it is argued that a tenable interpretation is not possible without it. Collapse provides a mechanism whereby the universe (...) size, like a clock, is narrowly correlated with the quantized time eigenvalues. (shrink)

The formulation of a relativistic theory of state-vector reduction is proposed and analyzed, and its conceptual consequences are elucidated. In particular, a detailed discussion of stochastic invariance and of local and nonlocal aspects at the level of individual systems is presented.

A simple model is presented in which the statevector evolves every ε seconds in one of two ways, according to a particular probability rule. It is shown that this random walk in Hilbert space results in reduction of the statevector. It is also shown how the continuous spontaneous localization (CSL) theory of statevector reduction is achieved as a limiting case of this model, exactly as Brownian motion is a limiting case of ordinary random walk. Finally, a slightly different but completely (...) equivalent form of the CSL equations suggested by the simple model given here is discussed. (shrink)

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 can 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 given previously, 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. (shrink)