Wave Function Collapse Models Research Articles

The quantum theory of measurement within dynamical reduction models

We analyse in mathematical detail, within the framework of the QMUPL model of spontaneous wavefunction collapse, the von Neumann measurement scheme for the measurement of a 1/2 spin particle. We prove that, according to the equation of the model, (i) throughout the whole measurement process, the pointer of the measuring device is always perfectly well localized in space; (ii) the probabilities for the possible outcomes are distributed in agreement with the Born probability rule; (iii) at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding to the measured eigenvalue. This analysis shows rigorously how dynamical reduction models provide a consistent solution to the measurement problem of quantum mechanics.

The frictional Schrödinger–Newton equation in models of wave function collapse

Replacing the Newtonian coupling G by –iG, the Schrödinger--Newton equation becomes "frictional". Instead of the reversible Schrödinger–Newton equation, we advocate its frictional version to generate the set of pointer states for macroscopic quantum bodies.

The Conway-Kochen Argument and Relativistic GRW Models

In a recent paper, Conway and Kochen proposed what is now known as the “Free Will theorem” which, among other things, should prove the impossibility of combining GRW models with special relativity, i.e., of formulating relativistically invariant models of spontaneous wavefunction collapse. Since their argument basically amounts to a non-locality proof for any theory aiming at reproducing quantum correlations, and since it was clear since very a long time that any relativistic collapse model must be non-local in some way, we discuss why the theorem of Conway and Kochen does not affect the program of formulating relativistic GRW models.

A Relativistic Version of the Ghirardi–Rimini–Weber Model

Carrying out a research program outlined by John S. Bell in 1987, we arrive at a relativistic version of the Ghirardi-Rimini-Weber (GRW) model of spontaneous wavefunction collapse. The GRW model was proposed as a solution of the measurement problem of quantum mechanics and involves a stochastic and nonlinear modification of the Schrödinger equation. It deviates very little from the Schrödinger equation for microscopic systems but efficiently suppresses, for macroscopic systems, superpositions of macroscopically different states. As suggested by Bell, we take the primitive ontology, or local beables, of our model to be a discrete set of space-time points, at which the collapses are centered. This set is random with distribution determined by the initial wavefunction. Our model is nonlocal and violates Bell’s inequality though it does not make use of a preferred slicing of space-time or any other sort of synchronization of spacelike separated points. Like the GRW model, it reproduces the quantum probabilities in all cases presently testable, though it entails deviations from the quantum formalism that are in principle testable. Our model works in Minkowski space-time as well as in (well-behaved) curved background space-times.

A Model of Wavefunction Collapse in Discrete Space-Time

We give a new argument supporting a gravitational role in quantum collapse. It is demonstrated that the discreteness of space-time, which results from the proper combination of quantum theory and general relativity, may inevitably result in the dynamical collapse of the wave function. Moreover, the minimum size of discrete space-time yields a plausible collapse criterion consistent with experiments. By assuming that the source to collapse the wave function is the inherent random motion of particles described by the wave function, we further propose a concrete model of wavefunction collapse in the discrete space-time. It is shown that the model is consistent with the existing experiments and macroscopic experiences.

The Status of the Wave Function in Dynamical Collapse Models

The idea that in dynamical wave function collapse models the wave function is superfluous is investigated. Evidence is presented for the conjecture that, in a model of a field theory on a 1+1 lightcone lattice, knowing the field configuration on the lattice back to some time in the past, allows the wave function or quantum state at the present moment to be calculated, to arbitrary accuracy so long as enough of the past field configuration is known.

Completely quantized collapse and consequences

Promotion of quantum theory from a theory of measurement to a theory of reality requires an unambiguous specification of the ensemble of realizable states (and each state's probability of realization). Although not yet achieved within the framework of standard quantum theory, it has been achieved within the framework of the continuous spontaneous localization (CSL) wave-function collapse model. In CSL, a classical random field $w(\mathbf{x},t)$ interacts with quantum particles. The state vector corresponding to each $w(\mathbf{x},t)$ is a realizable state. In this paper, I consider a previously presented model, which is predictively equivalent to CSL. In this completely quantized collapse (CQC) model, the classical random field is quantized. It is represented by the operator $W(\mathbf{x},t)$ which satisfies $[W(\mathbf{x},t),W({\mathbf{x}}^{\ensuremath{'}},{t}^{\ensuremath{'}})]=0$. The ensemble of realizable states is described by a single state vector, the ``ensemble vector.'' Each superposed state which comprises the ensemble vector at time $t$ is the direct product of an eigenstate of $W(\mathbf{x},{t}^{\ensuremath{'}})$, for all $\mathbf{x}$ and for $0\ensuremath{\leqslant}{t}^{\ensuremath{'}}\ensuremath{\leqslant}t$, and the CSL state corresponding to that eigenvalue. These states never interfere (they satisfy a superselection rule at any time), they only branch, so the ensemble vector may be considered to be, as Schr\"odinger put it, a ``catalog'' of the realizable states. In this context, many different interpretations (e.g., many worlds, environmental decoherence, consistent histories, modal interpretation) may be satisfactorily applied. Using this description, a long-standing problem is resolved, where the energy comes from the particles gain due to the narrowing of their wave packets by the collapse mechanism. It is shown how to define the energy of the random field and its energy of interaction with particles so that total energy is conserved for the ensemble of realizable states. As a by-product, since the random-field energy spectrum is unbounded, its canonical conjugate, a self-adjoint time operator, can be discussed. Finally, CSL is a phenomenological description, whose connection to, or derivation from, more conventional physics has not yet appeared. We suggest that, because CQC is fully quantized, it is a natural framework for replacement of the classical field $w(\mathbf{x},t)$ of CSL by a quantized physical entity. Two illustrative examples are given.

Collapse models: analysis of the free particle dynamics

We study a model of spontaneous wavefunction collapse for a free quantum particle. We analyse in detail the time evolution of the single-Gaussian solution and the double-Gaussian solution, showing how the reduction mechanism induces the localization of the wavefunction in space; we also study the asymptotic behaviour of the general solution. With an appropriate choice for the parameter λ which sets the strength of the collapse mechanism we prove that: (i) the effects of the reducing terms on the dynamics of microscopic systems are negligible, the physical predictions of the model being very close to those of standard quantum mechanics; (ii) at the macroscopic scale the model reproduces classical mechanics: the wavefunction of the centre of mass of a macro-object behaves, with high accuracy, like a point moving in space according to Newton's laws.

Quasirelativistic quasilocal finite wave-function collapse model

A Markovian wave function collapse model is presented where the collapse-inducing operator, constructed from quantum fields, is a manifestly covariant generalization of the mass density operator utilized in the nonrelativistic Continuous Spontaneous Localization (CSL) wave function collapse model. However, the model is not Lorentz invariant because two such operators do not commute at spacelike separation, i.e., the time-ordering operation in one Lorentz frame, the "preferred" frame, is not the time-ordering operation in another frame. However, the characteristic spacelike distance over which the commutator decays is the particle's Compton wavelength so, since the commutator rapidly gets quite small, the model is "almost" relativistic. This "QRCSL" model is completely finite: unlike previous, relativistic, models, it has no (infinite) energy production from the vacuum state. QRCSL calculations are given of the collapse rate for a single free particle in a superposition of spatially separated packets, and of the energy production rate for any number of free particles: these reduce to the CSL rates if the particle's Compton wavelength is small compared to the model's distance parameter. One motivation for QRCSL is the realization that previous relativistic models entail excitation of nuclear states which exceeds that of experiment, whereas QRCSL does not: an example is given involving quadrupole excitation of the $^{74}$Ge nucleus.

Consequence for Wavefunction Collapse Model of the Sudbury Neutrino Observatory Experiment

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.

Simulating causal wavefunction collapse models

We present simulations of causal dynamical wavefunction collapse models of field theories on a 1 + 1 null lattice. We use our simulations to compare and contrast two possible interpretations of the models, one in which the field values are real and the other in which the state vector is real. We suggest that a procedure of coarse graining and renormalizing the fundamental field can overcome its noisiness and argue that this coarse grained renormalized field will show interesting structure if the state vector does on the coarse grained scale. We speculate on the implications for quantum gravity.

Problems and aspects of energy-driven wave-function collapse models

Four problematic circumstances are considered, involving models which describe dynamical wavefunction collapse toward energy eigenstates, for which it is shown that wavefunction collapse of macroscopic objects does not work properly. In one case, a common particle position measuring situation, the apparatus evolves to a superposition of macroscopically distinguishable states (does not collapse to one of them as it should) because each such particle/apparatus/environment state has precisely the same energy spectrum. Second, assuming an experiment takes place involving collapse to one of two possible outcomes which is permanently recorded, it is shown in general that this can only happen in the unlikely case that the two apparatus states corresponding to the two outcomes have disjoint energy spectra. Next, the progressive narrowing of the energy spectrum due to the collapse mechanism is considered. This has the effect of broadening the time evolution of objects as the universe evolves. Two examples, one involving a precessing spin, the other involving creation of an excited state followed by its decay, are presented in the form of paradoxes. In both examples, the microscopic behavior predicted by standard quantum theory is significantly altered under energy-driven collapse, but this alteration is not observed by an apparatus when it is included in the quantum description. The resolution involves recognition that the statevector describing the apparatus does not collapse, but evolves to a superposition of macroscopically different states.

Wavefunction Collapse and Random Walk

Wavefunction collapse models modify Schrodinger'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 fluctuating classical field interacts with particles to cause collapse. One "side effect" of this interaction 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 a disc of radius 2 cdot 10^{-5} cm and thickness 0.5 cdot 10^{-5} cm diffuses through 2 pi rad in about 70sec (this assumes the "standard" CSL parameter values). The comparable rms diffusion of standard quantum theory is smaller than this by a factor 10^-3. At the reported pressure of < 5 cdot10^{-17} Torr, achieved at 4.2^{circ} K, the mean time between air molecule collisions with the disc is approximately 45min (and the diffusion caused by photon collisons is utterly negligible). This is ample time for observation of the putative CSL diffusion over a wide range of parameters. This encourages consideration of how such an experiment may actually be performed, and the paper closes with some thoughts on this subject

On a model of wave-function collapse

A full analysis is made of van Kampen’s recent claim (Physica A 153 (1988) 97) to demonstrate an explicit model for wave-function collapse, and hence to show that measurement may be fully described using the Schrödinger equation. It is shown that van Kampen only describes the initial correlation between measuring and measured sub-systems at the time of measurement, and does not attempt to describe or model the actual collapse process as put forward by von Neumann. A full discussion of van Kampen’s model is given.

Is spontaneous localization compatible with the energy density of the universe?

We investigate a dynamical model of wavefunction collapse in the cosmological context by using the idea of continuous spontaneous localization of baryonic wavefunctions. By invoking a cosmological time dependence for the effective collapse rate from an appropriate epoch in the early universe up to the present time, we calculate the integrated total contribution of the liberated energy to the present energy density of the universe. This is found to be of an order of magnitude comparable with the critical density of the universe.

Standard quantum mechanics with environment induced decoherence and wavefunction collapse: Possibility of an empirical discrimination using neutron interferometry

We develop in detail the idea of a thought experiment with a view to differentiate between standard quantum mechanics incorporating environment induced decoherence effects and a formulation based on a dynamical model of wavefunction collapse. An appropriate variant of a neutron interferometric arrangement is used for this purpose.

Symmetric versions of explicit wavefunction collapse models

Two versions of the GRW “hitting” model for explicit wavefunction collapse, which are consistent with preserving the symmetry of the wavefunction, are considered. The predictions of the models for excitation of bound systems are calculated and compared with experiment and with the predictions of other similar models. It is shown that our preferred model strongly supports the idea that collapse, if it occurs, has gravitational origin.

Bound state excitation, nucleon decay experiments and models of wave function collapse.

We discuss bound state excitation in models where wave function collapse is a physical process. Consequences of a feasible photon counting experiment are spelled out, were an upper limit to be found on the rate of anomalous photon emission from solid matter that is better than 1 photon/sec cc. It is also pointed out that experiments already performed, which give an upper bound on the rate of nucleon decay, can have significant consequences for such models. We show how these experimental results could be considered to rule out the original version of the collapse models, and support a version in which the rate of collapse is proportional to the mass, indicative of a gravitational mechanism.

Comment on "Wave-function collapse by measurement and its simulation"

Namiki and Pascazio [Phys. Rev. A 44, 39 (1991)] propose a model of wave-function collapse associated with the nonfiring of a detector in one of two paths for which a particle is known to be present, and argue that the collapse has the same status as the collapse associated with a measurement involving an actual detection. I question this proposal on the following grounds: (i) it is based on a shift in detection probabilities in the two paths while there is no shift in wave-function amplitudes, and (ii) Namiki and Pascazio use an insufficient measure, loss of interference, as an indicator of collapse.

Wavefunction collapse and ultraviolet photons

The number of photons expected to be emitted from hydrogen on the basis of a specific model of wavefunction collapse is calculated. It seems to be sufficiently large that it ought to be detectable, perhaps by suitable modifications to some of the experiments designed previously to search for proton decay.