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The geometric and the atomic world views

Contributed by Ben R. Mottelson, August 20, 2008 (received for review June 29, 2007)
Related Article
 In This Issue Nov 11, 2008
Abstract
The atomic world view is based on the notion that matter is built of elementary constituents called atoms, and quantum mechanics was created in the pursuit of this view with probabilistic events caused by atomic particles. This conception involves unresolved ambiguities linked to the notion of an elementary quantum of action. The resolution of these problems in quantum mechanics requires a new, geometric, world view, which recognizes the occurrence of events, clicks in counters, coming without a cause, referred to as fortuitous. The possibility of a rational theory of probabilities for such events is based on the assignment to the individual click of a proper value of an element of (flat) space–time symmetry. Thereby, the distributions of uncaused clicks can be endowed with a geometric content in terms of the irreducible representations of space–time symmetry. Through fortuity, space–time invariance itself thus acquires a hitherto unrecognized role. Departing from the norms of physical theory, the uncaused click is not a measurement of something, and the reality mirrored in the distributions is the geometry of space time itself, and not a property of an imagined object. The geometric world view involves only the dimensions of space and time, and the absence of an irreducible dimension of mass is seen as the result of the discovery of new physical phenomena. Accordingly Planck's constant has no place in fundamental theory and is seen as a relic of dimensions that have become superfluous.
The origin of probabilities in quantum mechanics continues to be enigmatic and controversial. Thus, quantum mechanics contends that events occur beyond causal analysis and yet are caused by a particle. Such contradictory use of concepts is unavoidable as long as one holds to an atomic world view that starts by assuming that matter is composed of atomic particles. Within this view it is a foregone conclusion that the events referred to are observations of particles.
The present article faces the challenge to develop an alternative, geometric, world view based on events, clicks in counters, that come without a cause.* Such a click is a macroscopic change that occurs as a discontinuity in an object referred to as a counter. Thus, the click has an onset, a beginning from which it develops. Consequently, the event is without history prior to the onset and, in particular, cannot be attributed to an amplification of something, because before the onset there was nothing to be amplified. An event of this novel character will be referred to as fortuitous. Quantum mechanics does not admit fortuitous events. Readers may object that the actually observed clicks in quantal phenomena are not of this character, but as far as we are aware, there is no evidence in conflict with the characterization of the clicks as fortuitous.
The occurrence of fortuity is the basic axiom from which the geometric world view is developed. In the resulting conception, the theory of fortuity, the click is thus recognized as the basic discrete element of nature replacing the role that has been attributed to atom or quantum. The theory of fortuitous events deals with distributions of clicks and expresses its laws in terms of probabilities.^{†} However, in contrast to the use of probability in situations, where the lack of knowledge prevents a complete analysis, a fortuitous event cannot, as a matter of definition, be analyzed in terms of precursors.
The elimination of any reference to particles, and thus to causes, is made possible by the recognition that the geometry of space time provides a framework for a theory of probability distributions of fortuitous clicks.
In the creation of quantum mechanics, space–time symmetry played a minor role, but its profound significance for the formalism has gradually emerged. However, fortuity, which eliminates particles as carriers of symmetry, is required for the recognition of geometry itself as the exclusive content of the formalism.
A crucial point is that the click is not a measurement of something and that the theory, thereby, departs from the norms of physical theory that have hitherto been taken for granted. The departure has profound implications because structure in the distribution of fortuitous clicks can no longer be thought of as an imprint of something that is. In contrast, structure appears when counter arrangements are invented that make possible the assignment of a proper value of an element of space–time symmetry to individual clicks. In this way, distributions of clicks are endowed with a geometric content and structure is created from the chaos of causeless clicks.
The absence of something underlying implies that the theory exclusively deals with probability distributions for the clicks. Each click comes without a cause and, therefore, it makes no sense to raise questions concerning a history of an individual event, as in interference, or concerning connections between two clicks, as in entanglement for coincident pairs of clicks. The geometric world view is by its conception complete and yet probabilistic.
Quantum mechanics is an attempt to explore an imagined reality behind the clicks and accordingly conforms to the norms by which the click is interpreted as a measurement. This embrace of the world view of atoms, which fails to come to grips with fortuity, leads to the persistent problems of reality in confronting interference and entanglement. These problems arise in quantum mechanics because the focus has been shifted from fortuity to a wholeness embodied in the quantum of action. Thereby, Planck's constant becomes a hallmark for the atomic world view. This constant has no place in the geometric world view.
The following paragraph provides an overview of the subsequent presentation.
The occurrence of reproducible click rates is the first step in the establishment of the geometric theory for the fortuitous events (see Counting of Fortuitous Clicks). The organization of the clicks involves an assignment to the individual click of a proper value of an element of space–time symmetry [see Assignment of proper values, and supporting information (SI) Appendix, Proper Values]. The assigment is established by symmetry waves (SI Appendix, Waves in space–time carrying Poincaré symmetry) which, at the same time, provide an encoding of the click distributions into symmetry space (see Encoding of rates). Thereby the nonAbelian space–time symmetry imposes structure that can be revealed by counter arrangements designed for the purpose. Elementary structure can be associated with irreducible representations describing uncorrelated clicks. Coincidences between pairs of clicks reveal that overlapping symmetry waves may affect each other (see Temporal Correlations in Click Distributions). The new perspectives that the theory gives to central issues in the understanding of quantum mechanics are illustrated by examples (see Box 1, Atom, and Photo Effect). Planck's constant is a hallmark of quantum mechanics. It must, therefore, come as a surprise that the theory of fortuity develops in the total absence of ħ. To elucidate this enigma, Planck's Constant and Basic Dimensions of Physical Law takes up the broader significance of dimensions in physics. A summary contrasting the world views is provided in The Atomic World View and The Geometric World View.
The theory of fortuity has its mathematical basis in the quantification of space–time invariance expressed by its irreducible representations. The aim of making the full case for the geometric world view motivates a somewhat detailed exposition of mathematical results. Such an exposition is provided in SI Appendix. The presentation is especially tuned to the geometric world view in that it focuses, not on the standard notion of symmetry as a quality carried by objects, but on symmetry as a subject in itself (intrinsic symmetry as a subgroup, parallelity and chirality as properties of Minkowski space, Lorentz transformations and translations as conjugate in the Galilean corner, and symmetry waves providing a local mapping of vectors in symmetry space into space time).
Endowment of Click Rates with Geometric Structure
It is the vision of the geometric world view that counter arrangements can be invented that result in click distributions exhibiting a structure of an exclusively geometric content.
Counting of Fortuitous Clicks. Rates.
The individual click is without law, but if there occurs a sequence of clicks with a Poisson distribution, a rate is defined. The occurrence of rates marks the first step from the individual fortuitous event to the establishment of the geometric theory.
The Poisson distribution characterizes a sequence of uncorrelated clicks. However, fortuitous clicks can occur in coincidence in several different locations, which may have timelike or spacelike separations. In the following we will first focus on uncorrelated clicks.
The occurrence of rates is by convention attributed to a source. The notion of a source belongs to a situation in which a distribution of fortuitous clicks can be observed. However, the word is a misleading remnant from classical usage, because it evokes the notion that something has been emitted in an event that would be a precursor to the click. Probing the structure of the source cannot further illuminate the concept of fortuity, because such probing can only lead to further click distributions. Therefore, the nature of sources is of no concern in the following.
Assignment of proper values.
The quantification of space–time symmetry, as described in SI Appendix, expresses itself in terms of the proper values of the symmetry elements. The endowment of geometric structure to click distributions is based on the assignment of a proper value u of a symmetry element U to the individual click. (It will be clear from the context whether a symmetry element or its generator is involved and whether several different symmetry elements are involved.)
Fortuitous clicks are observed in counter arrangements, i.e., the context created by counters and apparatus, and the individual click is characterized by the counter in which it occurs and the time of its onset. The task, therefore, is to establish a link between the space–time point of the click, and the proper values of symmetry elements.
This link is provided by a stationary symmetry wave φ_{u}(x) by which the proper vector u〉 of the symmetry element U in an irreducible representation is mapped into space time (see the discussion SI Appendix, Waves in space time carrying Poincaré symmetry). For example, U may designate translational symmetry with proper vectors that can be labeled by the proper value,
The role of the counter arrangement is to provide a controlled break in the symmetry which makes possible the establishment of the link referred to (see SI Appendix, Local symmetry breaking). By the controlled break of symmetry, new wave components are added coherently to the symmetry wave φ_{u}(x). By the resulting interference, intensity is shifted from one part of space to another, in a controlled manner. Assigment of u is thus based on coherently modifying the symmetry wave that maps u〉 in such a manner that there results a new wave component directed to a particular counter. For example, the assignment of translational symmetry can be based on a grating spectrometer, which adds a definite wavenumber component perpendicular to the axis of the spectrometer (z axis). Thereby, a symmetry wave with definite translational symmetry k_{z} acquires a diffracted wave component resulting in constructive interference in directions determined by k_{z} and the grating. A click in a counter in such a direction is, therefore, assigned the proper value k_{z} that corresponds to the direction (and the grating).
Intrinsic rotational symmetry belonging to a wavenumber
Encoding of rates.
The linking of the click to the proper value u by means of the symmetry wave φ_{u}(x) that maps the vector u〉 provides a basis for encoding the counting rate for the clicks assigned the value u into the intensity of this wave. The click rates are incorporated, thereby, into the formalism in symmetry space, and the partnership between fortuity and geometry is founded.
The identification of the click rates with wave intensities implies that the relative click rates for different proper values u, in a counter arrangement assigning these proper values can be encoded in a superposition of waves φ_{u}(x) with different u. Thereby the phases between the components φ_{u}(x) are arbitrarily chosen.
The superposition of waves φ_{u}(x) can be seen as a mapping of a vector C〉 in the space carrying the irreducible representation where the phases of 〈uC〉 are fixed by the choices referred to above. The relative counting rates for different u, are thus encoded in the vector C〉, for the relative rates normalized to unity, when summed over u.
When click distributions for another symmetry element, v, observed in a counter arrangement specifically constructed for the assignment of proper values, v, of this element, are explored, the issue of a relation between the two click distributions arises. The program of endowing geometric structure to the click distributions suggests the possibility that both distributions may be derived from a single vector C〉.
In the geometric world view, structures in the click distributions are not something preexisting, but are created by the experimenter. Thus, in general, the distributions in u and v cannot be derived from a single vector. However, by the purposeful restriction of the conditions under which clicks are recorded it has been possible to produce situations in which a single vector is a viable approximation. A vector C〉 that characterizes such patterns of click distributions is referred to as a click vector. It goes without saying that the click vector is not a state of an object.
For a single vector C〉, the components in different bases are related by in terms of geometric coefficients, 〈vu〉, specified by the representation. The relation 3 implies constraints between rates involving interference terms created by the purposeful selection referred to. In the geometric world view, interference is exclusively a feature of the distribution function for fortuitous clicks.
The possibilities of encoding click rates into symmetry space as above are significantly restricted by a mathematical theorem concerning possible measures on a vector space (see ref. 5). Thus, the irreducible unitary representations admit a unique positive measure that is additive on orthogonal subspaces, such as the onedimensional subspace u〉 belonging to a proper value of a symmetry element. These unique measures are given by Eq. 2, for all symmetry elements U, v, etc. (see SI Appendix, Eqs. A20 and A21).
Wave Equation.
The symmetry waves obey a wave equation (given in SI Appendix, Box 3), which also describes the effect of symmetry breaking (SI Appendix, Box 4), as in the link establishing assignment of proper values. For example, if the intrinsic symmetry can be ignored, the breaking of translational symmetry by the spectrometer is described by the Klein–Gordon equation (SI Appendix, Eq. A41), with V_{μ} = 0, where the solutions are determined by boundary conditions. The breaking of intrinsic rotational symmetry in a medium (
The effect of external fields can lead to solutions of the wave equation that are confined to a limited region of space. The connection of these confined solutions to click distributions is discussed in Atom.
Embodiment of Symmetry.
The probabilities p(u) encoded in a click vector define the mean value Relation 5 in turn specifies the individual p(u) in terms of the mean values of U and its powers, also elements of the symmetry group. By relation 5, the click distributions appear as an embodiment of the symmetry of empty space time, expressed in terms of an identity between the mean value of a symmetry element U in the click distribution and the diagonal matrix element of U in the click vector in symmetry space. The embodiment 5, which is a consequence of Eq. 2, implies that the linear relations between the symmetry matrices, which specify an irreducible representation, are mapped into the same linear relations for the mean values in the click distributions. Conversely (see ref. 2), the requirement that the linear relations between the symmetry elements be fulfilled for the mean values 〈U〉 implies the result 2 (or its generalization into a density matrix).
Temporal Correlations in Click Distributions.
A symmetry wave mapping a vector in an irreducible representation gives the rate of clicks as a function of a single epoch. The clicks are accordingly uncorrelated in time and exhibit a temporal Poisson distribution. However, the experimental study of the temporal sequence of clicks has revealed that multiplets of clicks may occur disproportionately in coincidence in two or more counters.
The occurrence of such correlations reveals that two symmetry waves, each belonging to an irreducible representation, can affect each other when in the same space–time region. Thus the presence of one wave breaks the symmetry in the wave equation of the other, and conversely.
The resulting interlocking (entanglement) of the symmetry waves leads to a click vector that belongs to two (or more) irreducible representations, as in Eq. 6 in Box 1. This vector is mapped into a symmetry wave, which is a sum of products of two symmetry waves, each linking a wavenumber to the space–time point,
The entangling described above in terms of a mutual influence of symmetry waves is conventionally referred to as resulting from an “interaction” in the time evolution F(τ) in symmetry space. More generally, this nonlinearity may lead to symmetry waves belonging to new representations. For example, a vector i〉 in one irreducible representation may be coupled to a superposition of products of the form 6 involving two (in general, different) representations. Thereby, the symmetry waves originating from a localized region can exhibit strong temporal correlations in the occurring click distributions at differerent locations.
The sorting of clicks into coincident pairs makes it possible to observe correlations involving other proper values beyond
The discovery of uranium radioactivity was seen as the emission of an α particle from a source. However, it is the occurrence of coincident pairs of α and thorium clicks with proper values correlated as in Eq. 8 that constitutes the unique constellation of events that is the content of a decay.
In the geometric world view a temporal correlation is exclusively a feature of the distribution functions, and there is no connection between the individual events in a multiplet of (approximately) coincident clicks. Indeed, the notion that two clicks belong together is entirely based on the occurrence of a distribution exhibiting coincident pairs in disproportionate measure.
Atom.
The effect of external fields can lead to solutions of the wave equation that are confined to a limited region of space. The normal modes for an electronic wave in an external electromagnetic potential
The confined electronic waves are coupled to a product of electronic and photonic waves (see the discussion in Temporal Correlations in Click Distributions and Box 2). This coupling has only a small effect (of order e^{10}) on the spectrum 10. However, for n ≠ 0, the coupling implies that photonic waves are present. The photonic waves (see Eq. 11) have frequencies that match the differences between the frequencies 10 to within the line width. The photonic waves are not confined and belong to distributions of clicks that can be assigned frequency. From these spectra, the basic constants k_{0} and e can be determined.
In this manner, rates of different photonic wave numbers from a continuum are interwoven through the intermediary of confined electronic waves into a discrete spectrum. These confined waves are purely formal (mathematical) and do not themselves describe click distributions. They are integrated into a click vector through the coupling to the photonic waves (see Eq. 11), and the spectrum 10 thereby emerges in the click distributions. This system of lockedup waves is the only viable conception of what has been called an atom. In fact, fortuity is incompatible with the notion of an atom as an object underlying the click.
In contrast, the development of quantum mechanics involved the notion of a quantization of the objects assumed to constitute the building blocks of atomic structure. This quantization invokes an elementary quantum of action, which is totally absent in the present derivation of Eq. 10. This theme is discussed in Planck's Constant and Basic Dimensions of Physical Law.
Photo Effect.
The changed world view demanded by fortuity is strikingly illustrated by the photo effect. Thus, the coupling of a photonic wave to a confined electronic wave implies the presence of unconfined electronic waves that belong to click distributions in electronic counters. The frequency spectrum of the electronic clicks is thereby quantitatively related to the frequency of the photonic wave. This relation concerns the statistical distribution, and the notion of a history for the individual event (an electronic click) does not apply in the geometric world view. In contrast, the development of quantum mechanics involved the notion of a history for the individual event. Thus, it appeared necessary to appeal to a particlewave duality, which invoked a new constant of nature.
The system of confined waves that only manifests itself through couplings to unconfined waves is the only viable conception of what has been called the atomic structure of matter.
Planck's Constant and Basic Dimensions of Physical Law
Dimensions as a Language for Physical Qualities.
It is the core of the program of assigning numerical values to observed phenomena that it is necessary to isolate and define the distinguishable qualities in terms of dimensions, such as intervals of length and time, mass, force, temperature, electric charge, etc. Before relations between qualities are established, the amount of a given quality can only be given in terms of a comparison with a standard unit of its own kind. Each quality, therefore, has its own dimension.
The discovery of relationships between different qualities leads to the formulation of equations involving dimensional constants (physical laws). An example is provided by Boltzmann's constant, which was introduced to express the relationship between temperature and energy, and the constant that connects candlepower with energy flux. It is emphasized that qualities belong to the human effort to invent language appropriate to discourse on environment.
Quantification of Force in Terms of Mass, Space, and Time.
It has been the result of the development of classical physics, including relativity, to establish relations between qualities to such an extent that all measurable qualities can be expressed in terms of dimensions of mass (M), length (L), and time (T), as basic irreducible qualities. The reduction to the three dimensions M, L, and T, means that all qualities can be traced back to objects moving according to Newton's second law. The great discovery that is contained in Newton's “laws of motion” is that force is proportional to mass times acceleration, which could have been expressed by the equation where the force f_{2} has the dimension F with a unit based, for example, on a standard spring or the pull of the earth on a standard mass, just as m_{1} has the dimension M, with a unit, such as the gram. In Eq. 12, the constant ɛ has the dimension FT^{2}(ML)^{1}. In fact, the distillation of Principia's message, which appears in every text book, reads without a dimensional constant. The dimension F has thereby become superfluous in fundamental theory, reflecting Newton's vision that, in the mechanical world view that he was developing, force was identified with the acceleration imparted to a given mass. This identification implies that force has a composite dimension MLT^{2} and can be measured without reference to another force.
Quantification of Mass in Terms of Space and Time.
A new manifestation of force and mass as well as other macroscopic qualities is observed in the effect produced by objects on click distributions, in the theory of fortuity (see Assignment of proper values). This probing of objects goes beyond what can be seen in experiments based on Eq. 13. The click distributions are functions of the proper values of the elements of space–time symmetry involving only geometric dimensions (space and time) with no place for the dimension of mass. The extension of the geometric world view to macroscopic qualities reveals that the dimension of mass, hitherto irreducible, is in fact a composite of space and time.
The result can be found for example, by probing an electric field by its action on the wavelength of the electron symmetry waves produced by including the electric field in the wave equation (SI Appendix, Eqs. A30 and A43) for a static field that varies slowly over a wavelength. In Eq. 14, the dimensionless coupling constant, e, is determined from the atom (see Eq. 10). It is seen that the field ∈ measured by the change of the wavelength has the dimension L^{–1}T^{–1}. In turn, the mass m of a macroscopic body carrying a dimensionless charge q, determined by measuring its electric flux, is quantified through the acceleration produced by the field, Hence, mass is seen to have a composite dimension, L^{–2}T, which means that mass can be measured without reference to another mass, and a separate dimension for mass is superfluous. In this manner, all classical variables are determined in the geometric dimensions of space and time through the new quantification of mass.
The ratio between a mass m_{1} measured in its own dimension and the value m given by Eq. 15 defines a constant of dimension ML^{2}T^{–1}. For mass m_{1} measured in grams, η is found to have the value The constant η in Eq. 16 is the counterpart of the constant ɛ in Eq. 12, which can be expressed as the ratio f_{2}/f_{1}.
The equation 15 between mass and force, with mass having a separate dimension, implies that the same constant η gives the ratio between the electric force q_{1}∈_{1} quantified in M, L, and T, and the electric force q∈ quantified (as above) in L, T, The absence of an irreducible dimension of mass implies a simplification of fundamental theory avoiding superfluous causes.
Absence of Planck's Constant.
The constant η of value 17 is seen to be identical to Planck's constant ħ, which can, therefore, be compared with the constant ɛ in the above discussion of Newton's law. In the historical process, appeal to ħ, which was seen as the establishment of new relations between physical quantities (particle wave duality), can be recognized as an attempt to endow imagined quantal objects with qualities inspired by the atomic world view.
Seemingly Unavoidable Particles
The Atomic World View.
The atomic world view is based on the notion that matter is built of particles. The idea of atoms goes back to antiquity, and the striking developments in physics and chemistry in the nineteenth century gave strong support to the atom (law of multiple proportions, dynamics of gases, Brownian motion, heat and entropy).
The further exploration indicated that an atom of an element is itself composite (cathode ray, electron, and atomic nucleus). Effects were discovered (clicks) that were attributed to the action of individual atomic particles. These discoveries were seen as overwhelming evidence for the reality of the particles (radioactivity, cloud chamber track, and photo effect).
The atomic world view acquired a new content with the interpretation of experiments in terms of an elementary quantum of action. The introduction of the quantum inspired new approaches to the atom and the development culminated in the creation of the quantum formalism.
The new theory involves a major conceptual extension of the atomic world view. The particle is retained but is endowed with enigmatic properties and is referred to as indeterminate. Thus, the particle is associated with a wave that gives probabilities pertaining to the particle. The ambiguity of the conceptions are dramatically illustrated by the interpretation of interference. The click, by which the effect is established, is said to be caused by an indeterminate particle that, however, can be attributed to no well defined path in passing through the interferometer, but is said to interfere with itself in a superposition of alternative paths. These problems are further aggravated in the interpretation given to entanglement, which involves an interlocking of the indeterminacies of two individual particles at separate locations.
Despite these profound problems, the interpretation of experiments based on indeterminate particles has been maintained. Indeed, the world view itself has not been questioned. The notion of an event coming by itself without a cause has appeared to be beyond the scope of rational theory.
The Geometric World View.
As motivated in the article, we believe that a proper understanding of the consequences of fortuity compels an exclusively geometric world view.
A fortuitous event is characterized by the location of the counter and the epoch. As argued earlier, laws governing distributions of fortuitous clicks emerge by linking the individual click to a proper value of an element of space time symmetry. The link is established through symmetry waves that map the representation of symmetry into space time and that obey a wave equation that incorporates the symmetry breaking due to external fields, such as produced by the apparatus in the assignment of a proper value.
By the mapping, the click distributions of proper values of uncorrelated clicks are encoded in a vector in the symmetry space of an irreducible representation. In this process, distributions of proper values emerge reflecting the experimenters constructive exploitation of nonAbelian space–time symmetry. This project involves the interference of the probability amplitudes (symmetry waves). Thus, interference in the distribution functions is an inherent part of the structure that is created by the assignment of proper values to fortuitous clicks.
The patterns of fortuituos clicks are greatly enriched by the inclusion of the epochs of the clicks that can reveal the occurrence of approximately coincident pairs in disproportionate measure. In the analysis of coincident pairs, the individual event is assigned two proper values, involving two symmetry waves that must be entangled to encompass coincidence. The notion of coincident pairs exclusively concerns the probability distribution, because each click comes without a cause and therefore is unconnected to any other click.
Entangling leads to the notion of interactions (couplings) between representations, which can be seen as a symmetry breaking occurring when symmetry waves overlap in space time. It is beyond the scope of the present article to pursue these concepts further.
The theory of fortuity addresses the challenge of developing a world view based on events that come without cause. In this theory, therefore, there is no atomic world underlying the click and the discourse on environment takes an unexpected turn. In adopting the new world view of geometry, we realize that the structure created in the exploitation of fortuity stems from our ability to assign proper values of space–time symmetry to fortuitous clicks with no place for material building blocks (atom). Far from being unavoidable, the particle is seen to have been standing in the way for a world view based on the recognition of an inextricable partnership of fortuity and geometry. In this geometric world view Planck's constant is only of historical significance as a relic of dimensions that have become superfluous.
Acknowledgments
We thank K. Gottfried, J. Kalckar, B. Lautrup, D. Mermin, H.B. Nielsen, and W.J. Swiatecki for stimulating discussions.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: mottelson{at}nbi.dk

Author contributions: A.B., B.R.M., and O.U. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0807786105/DCSupplemental..

↵* Preliminary steps in the development of these ideas were discussed in recent articles (1, 2). In these references, an event coming “entirely without a cause” was said to be “genuinely fortuitous.” We find, however, that the unqualified terminology “without a cause” and “fortuitous” is more appropriate.

↵† We have used the word probability, although it will become clear that the theory exclusively refers to relative rates of clicks. We must admit, however, that it is not clear to us whether the resulting theory conforms or not to the axioms of probability theory defined, for example, in refs. 3 and 4.
 © 2008 by The National Academy of Sciences of the USA