Philip Pearle

Relativistic dynamical reduction models: General framework and examples

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.

Toward explaining why events occur

The possibility is raised of adding a randomly fluctuating interaction term to the Schrödinger equation, so that the new equation “reduces” the state vector. The exact event that occurs is predicted by the equation and depends upon the precise time dependence of the interaction term. The uncertainty in nature is attributed to the random behavior of this term. A class of such terms is found. This class includes terms whose nonlinear dependence on the wave function is identical to that of terms introduced in a previous paper for a similar purpose. In the previous paper, the exact event predicted depends upon the initial phase factors in the superposition making up the state vector: the uncertainty in nature is attributed to random initial phase factors. Another derivation of the results in the previous paper is given in an appendix: the calculations in that paper and in this appendix are of second order in perturbation theory. On the other hand, the calculations in the present paper are exact. A possible answer is given to the question, raised in the previous paper, of the nature of the “observable” states to which the state vector reduces.

Continuous spontaneous localization wave function collapse model as a mechanism for the emergence of cosmological asymmetries in inflation

The inflationary account for the emergence of the seeds of cosmic structure falls short of actually explaining the generation of primordial anisotropies and inhomogeneities. This description starts from a symmetric background, and invokes symmetric dynamics, so it cannot explain asymmetries. To generate asymmetries, we present an application of the continuous spontaneous localization model of wave function collapse in the context of inflation. This modification of quantum dynamics introduces a stochastic nonunitary component to the evolution of the inflaton field perturbations. This leads to passage from a homogeneous and isotropic stage to another, where the quantum uncertainties in the initial state of inflation transmute into the primordial inhomogeneities and anisotropies. We show, by proper choice of the collapse-generating operator, that it is possible to achieve compatibility with the precise observations of the cosmic microwave background radiation.

Gravity, energy conservation and parameter values in collapse models

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.

Toward a Relativistic Theory of Statevector Reduction

“For each measurement, one is required to ascribe to the ψ-function a quite sudden change… The abrupt change by measurement… is the most interesting point of the entire theory.… For this reason one can not put the ψ-function directly in place of the physical thing… because from the realism point of view observation is a natural process like any other and cannot per se bring about an interruption of the orderly flow of events.”

Wavefunction Collapse and Random Walk

Wavefunction collapse models modify Schrödingers 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 spheres 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.

The proper role of clusters in mathematical models of continuous recall

Abstract Previous research has shown that the number of words cumulatively recalled ( N ) at time ( t ) is a negatively accelerated function that reaches an asymptote as t → ∞. Research has also shown that the increase in N with t occurs in bursts or clusters. Several models purport to account for this cumulative recall curve in terms of cluster characteristics. The present research shows that previous models have not in fact successfully linked continuous recall to cluster characteristics. This research demonstrates that cluster models need to employ three empirical characteristics of clusters: T b , the time between clusters; T w , the average time between words within a cluster; and W c , the number of words within a cluster. It is shown that these three quantities determine the cumulative recall curve, and these three quantities may in turn be characterized by four parameters. Of these four parameters, only three actually characterize the cumulative recall curve. Two parameters determine the initial slope and final asymptote of the curve, while a third parameter, which we introduce for the first time, characterizes the curves shape. This latter parameter may be interpreted as the ratio ofthe time spent in retrieving and discarding a cluster that has been previously recalled to the amount of time spent in retrieving and outputting a newly encountered cluster. It is pointed out that previous success in fitting the cumulative recall data with a two-parameter function may be explained by the fact that this parameter lies in a restricted range about unity. Further experimental work is suggested to elucidate the behavior of this new parameter. Two models are then proposed to account for these characteristics of clusters and the shape of the recall curve.

Absence of radiationless motions of relativistically rigid classical electron

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.

Relativistic collapse model with tachyonic features

A finite relativistic model for free particles, which describes the collapse of the statevector, is presented.

Wavefunction Collapse and Conservation Laws

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.