Why has de Broglie's theory been rejected?

0039-3681(94)EOOll-V

Why has de Broglie’s Theory been Rejected? Thomas Bank* Abstract - The arguments

offered for and against various interpretations of mechanics at the Solvay Conference on physics in 1927 are examined, and the framework of ‘Logical Learning’ is applied in an attempt to elucidate the widespread acceptance of the ‘Copenhagen interpretation’ in the contemporary community.

quantum Bayesian fast and scientific

1. Introduction AFTER A long period of proliferating interpretations of quantum theory, a view on matters microphysical came to dominate competitors like the ‘pilot wave’ approach, the ensemble and Schriidinger’s interpretation: namely, the so-called ‘Copenhagen interpretation’.’ The conference of the Solvay Institute held in Brussels 24-29 October 1927 under the topic filectrons et Photons marks a turning point in this process. The proceedings document a critical deliberation on the pros and cons of each alternative, and a conscious effort to eliminate the untenable ones. It is this unique historical process to which I want to draw attention. The focus on the Copenhagen interpretation as the ‘outcome’ of the Fifth Solvay Conference, should not imply that it was created or took final shape there, nor that the ‘verdict’ was uniform, nor that there were no sincere considerations on matters of interpretation in the years after, rather that it alone passed the ‘polarising filter’ of an open, critical collective deliberation between (prominent) members of the scientific community, with all the opponents present with the best chance to present their ideas. The process of acceptance of the Copenhagen interpretation in the late twenties and thirties may be divided roughly into two stages. In the second, the post-1927 stage, a new orthodoxy emerged with the rapid publication of *Presently FSP Philosophie und Geschichte der Wissenschaften, JIgerstr. 10/l 1, 10117 Berlin, Germany. Received 4 January 1993; injinalform 15 November 1993. ‘The quotation marks on ‘Copenhagen interpretation’ will be omitted from now on, with the understanding that the label does not stand for an immutable, homogeneous ‘block’ of postulates, in the Period under consideration at least. Pergamon

Stud. Hist. Phil. Sri., Vol. 25, No. 3, pp. 375-396, 1994.

Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0039-3681/94 $7.00+0.00

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authoritative texts on quantum mechanics, von Neumann’s proofs, ill-advised defences in popular contexts, etc. To many, second thoughts on ‘reactionary’ realist alternatives and even reflections on the finer points of the Copenhagen interpretation appeared to be nothing but untimely distractions from the task of putting a powerful new formalism to work. The complaint, that the fast and widespread acceptance of the Copenhagen interpretation was not entirely a ‘rational process’, voiced by Popper and Bohm, may be taken to refer to this stage. The Solvay Conference, on the other hand, in many respects the culminating point of the short, initial period (beginning with Heisenberg’s work on the uncertainty paper early in 1927) and preparatory for the second stage, had a quite different character. The conference created the opportunity for a reasonable fair and critical evaluation of various alternative views on the physical meaning of the formalism, in the elementary sense that no major view was excluded from being discussed and the exchange of arguments was uninhibited and open. In the following sections I will apply Bayesian models to describe the collective deliberation process. Various factors which might have reduced the effectiveness of the ‘filter’ immediately come to mind. A ‘romantic’ bias towards indeterminism, among the participants or, in the ‘Zeitgeist’, may have induced a lack of ‘interest’ in developing a proposed interpretation further if it was found to be in trouble. Factors such as these can be taken into account as ‘amplifying factors’, influencing the relative weights attached to various objections and theoretical virtues. Kuhn has emphasized different ideologies as a possible source of deep misapprehensions between members of a scientific community. However, differences in goals, methods and ‘philosophy’ between the participants play a less prominent role, because the arguments offered are generally designed so as to target shared assumptions. In Section 2, the historical process of deliberation is described (covering some familiar facts), and the objections brought forward are discussed in some detail, commenting on each lecture in turn. The conference’s proceedings contain in addition a version of Bohr’s Como Lecture, which replaces a remark made by Bohr in the main discussion session, reproduced in parts by Bohr (1985), p. 103, but not in the official proceedings. It is skipped for the purposes at hand. In Section 3, the possibilities for a Bayesian evaluation of some of the arguments and objections are examined in the framework of ‘logical learning’. It is shown how the learning of a new implication can affect the credibility of a theory, if the deduction proceeds from premises including auxiliary assumptions besides the theory itself. This is a desirable extension of Bayesian learning theory if it is to be applied to ‘real life’ situations. The proofs for the propositions are placed in an Appendix. A full Bayesian understanding of the outcome needs to include the set of theoretical alternatives. This complementary aspect of ‘elimination’ is the subject of Section 4. The ‘landslide’ type of acceptance of the

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Copenhagen interpretation, I suggest, has four basic causes: (1) a working ‘positivist’ theory in the Copenhagen interpretation, (2) defunct ‘realist’ alternatives, (3) a conservative attitude towards certain assumptions in the ‘background knowledge’, and (4) a relatively homogenous attribution of weights to particular objections, theoretical ‘virtues’, etc. The epilogue finally argues that the probabilistic reconstruction should not be taken to provide a normative basis for the rejection of de Broglie’s theory. 2. The Debates “In contrast to the conventional meeting in which reports are given on the successful solution of scientific problems, the Solvay Conferences were conceived to help directly in solving specific problems of unusual difficulty” [Heisenberg, foreword in Mehra (1975)]. The Brussels meeting in 1927 was dedicated to the development of quantum mechanics since the last conference in 1924. The first two talks, by W. L. Bragg and Compton, reviewed experimental findings in scattering experiments with X-rays on bound and unbound electrons. Compton, two years after the Bothe-Geiger experimentthe first experiment rigorously establishing that in scattering events with free individual electrons energy and momentum are strictly conserved-took painstaking care in arguing in favour of the photon hypothesis by first expounding the difficulties of the ether hypothesis and electrodynamics (the latter being the basis of Bragg’s lecture on L’lntensitk de Reflexion des Rayons X), and then discussing the experimental findings from the point of view of the particle picture of light. The chairman of the conference H. A. Lorentz promptly opened the discussion with a long elaboration of why he thought the difficulties are not nearly as grave as Compton had argued. It was noted that Wentzel had succeeded in deriving the Compton effect from Schrodinger’s equation, that is from a premise associated with wave-like behaviour. Judging from the proceedings, the discussion was one of the longest and liveliest at the conference, indicating among other things, that the degree of consensus and certainty regarding these basic matters was not quite as high as one might have expected in these circumstances. The essential results and ideas of de Broglie’s lecture La Nouvelle Dynamique des Quanta, which fittingly followed Compton’s review, had already been published in May 1927. They had received a cool reception [compare de Broglie’s remark on Einstein in de Broglie (1962), p. 1501.The lecture is divided into three parts of roughly equal length. In the first part he reviewed the work of Schrodinger, especially the attempt to represent particles as wave packets, Born’s probabilistic interpretation of the wave function, as well as the way on which de Broglie himself was led to the Einstein-de Broglie relations between a particle’s momentum and its wavelength, energy and frequency in 1923.

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As the main problem of quantum theory, he identified the question of how to relate particle mechanics and the existence of interference phenomena within the formalism. In de Broglie’s original ‘double solution’ theory, a singular solution besides the continuous solution of a wave equation, i.e. a solution with an appropriate singularity in the amplitude a of the ansatz ~=acos(y7), represents the particle [de Broglie (1929) pp. 5,6; (1962) pp. 163-167; Belinfante (1973) p. 921. In the lecture he emphasized the double solution theory as an alternative to Schrbdinger’s ‘wave packet representation’ of particles, but did not develop the difficult singularity aspect in any detail. de Broglie proposed instead an interpretation of the continuous solution of Schrodinger’s equation as onde pilote (Fiihrungsfeld de M. Born) and onde de probabilit.4 for a material particle. Drawing on the Hamilton-Jacobi formalism of classical mechanics, with its relation between mechanical action and particle velocity, and geometric optics, he reinterpreted the action (‘Jacobi function’) as the phase of the wave function associated with the particle in the quantum domain. The velocity defined in this way determines a class of possible particle trajectories in space and time, or trajectories of an n-particle system’s representative point in configuration space, since the particle’s location at one instant is unknown to us. de Broglie argued that a2 should play the role of a local ‘probability density’ for the presence of the particle (or the representative point) in an infinitesimal volume element at a given instant. [No mention is made at the conference of Madelung’s 1926 Quantentheorie in hydrodynamischer Form, and related attempts early in 1927 [Jammer (1974) p. 341, in which a quantum ‘conservation equation’ serves as a starting point, derived from Schrodinger’s equation and the complex analogue of de Broglie’s ansatz above.] This approach implies that the particle’s trajectory is subject to a new, non-local, fast-varying force beside any external potential. The ‘quantum force’ and the particle’s velocity at each space-time point are calculated from the continuous solution of Schrodinger’s equation in the non-relativistic approximation. de Broglie appeared to vacillate between

attributing

to the wave function

objective

physical

reality,

since it

supposedly ‘governs’ the particle’s motion, and taking it as ‘fictive’ for an n-particle system. The framework in the lecture’s second part is intended to be thoroughly relativistic, with an occasional approximation for small v/c to make contact with results of Born, Fermi or Schrodinger. Aspects of this theory were conceived previously by Einstein [Jammer (1974) p. 411 and Brillouin. In the last part de Broglie discussed recent defraction experiments with electrons by C. Davisson and G. H. Germer, G. P. Thompson and A. Reid, as well as others, which he took to support his view more or less directly. One difficulty for his theory had surfaced only recently from an unexpected quarter in the February issue of the journal Zeitschrifr der Physik 1927. The distinguished experimentalist Bothe had completed a series of experiments on the Compton effect in ‘standing waves’, whose results, Bothe claimed, refute the

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hypothesis of continuous photon trajectories, referring explicitly to de Broglie. de Broglie in his lecture just waived the problem and it played no role in the ensuing debates. In the discussion section following the lecture several objections were raised: the ‘strange’ consequence of velocity zero for an s-state electron [see also Belinfante (1973) p. 1931, the possibility of ‘trajectories’ depending on certain continuity equations typical for many field theories, the resulting general complexity of atomic electron ‘orbits’, and the arbitrariness in the construction of orbits in the presence of degeneracies in the atomic energy levels [&/ectrons et Photons pp. 135, 1361. More considered arguments and objections were brought forward during the main discussion period at the end of the Solvay Conference (by Pauli, Kramers, Langvin, Einstein); and Pauli may well have been the driving force behind the numerous attempts to come to terms with de Broglie’s theory [klectrons et Photons p. 1351. Leon Brillouin had illustrated de Broglie’s ideas with the description of the reflection of a photon under a given angle of incidence 0 at a perfectly reflecting mirror, pointing out features like the particle’s never ‘touching’ the surface but moving parallel to it for a certain amount of time. The photon’s total velocity during this interval is given by V=C . sin(e) and is smaller than the speed of light. The setting allows for the existence of a standing wave (e=O), in which the associated photon is at rest, as was pointed out by Einstein [klectrons et Photons p. 2661. If the mirror is partially transparent the hypothetical vacuum velocity of a photon can even be greater than c in the interference zone; compare with de Broglie (1929) pp. 112, 115. Whether this consequence, not mentioned in the proceedings, was already known to de Broglie, Brillouin or others, has to remain open. Kramers complained, that if the ‘fictitious’ photon trajectory does not give any information about the pressure at a given surface point, but only a pressure mean, the concept becomes ‘superfluous’. Focusing on what happens in regions of interferences, Fermi’s calculation of the inelastic scattering of an electron by a quantized rotator in the plane was redone by Pauli, to demonstrate that according to the guiding wave picture the rotator after the impact never ‘settles down’ into a stable final energy eigenstate [see Jammer (1974) p. 111 for technical details]. Pauli had a reputation for penetrating, crushing criticism of new physical ideas, but more than once he changed his mind later on. On the affirmative side, there are remarks by de Donder concerning the easy fit with Rosenfeld’s and his own theory of gravitation; and by Brillouin, who pointed out that the approach solves a ‘paradox’ discussed by R. C. Tolman, G. N. Lewis and S. Smith in 1926. How are these objections to be evaluated? Part of the answer is given in the next section. We know of Einstein’s general support for the program, but possibly for reasons other than its ‘truthlikeness’. At one point the ever-helpful Brillouin reminded Born about the empirical adequacy of de Broglie’s theory:

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“II me semble qu’aucune objection serieuse ne peut &tre faite au point de vue de L. de Broglie. . . . 11n’y a aucune contradiction entre le point de vue de L. de Broglie et celui des autres auteurs, puisque, dans son rapport . . ., L. de Broglie nous montre que ses formules sont en concordance exacte avec celles de Gordon, actuellement admises par tous les physiciens” [klectrons et Photons (1928) p. 1341. Yet, over the next decade at least, no scientist present at the conference seemed to have engaged in the kind of constructive research necessary to carry the ideas further. The view of Born and Bohr as expressed by Born at the end of the conference involves an objection of an entirely different kind to the ones offered by Pauli or by de Broglie later on: “Contrairement a l’opinion de M. Schrodinger, que c’est un non-sens que de parler de situation et de mouvement de l’electron dans l’atome, M. Bohr et moi nous sommes d’avis que cette facon de dire a toujour un sens, quand on peut indiquer une experience permettant de mesurer les coordonntes et les vitesses avec une certaine approximation” [klectrons et Photons (1928) p. 2861. For de Broglie himself we can gauge the impact of the criticism from his later writings, especially from his treatise on wave mechanics, published in 1929. Two features demand attention. His ideas on particles and guiding fields are wielded into the fabric of the book for illustrative purposes and properly criticized each time they appear-but there is no mention of the ‘fatal’ rotator example (which could have been treated at the level of the text). Furthermore, Fermi’s paper is cited already in de Broglie’s expository paper from May 1927 [de Broglie (1928) p. 1341, and mentioned in the lecture itself [Zbctrons et Photons (1928) p. 1191.Therefore Pauli’s objection at the conference could have been anticipated by de Broglie. This is also suggested by the ‘spontaneous’ way in which de Broglie, by way of answering the objection, came up with exactly the right suggestion to work with finite wavetrains instead of plane waves [“Dam le probleme de Fermi, il faut supposer l’onde w limit&e lattralement dans l’espace de configuration” (&lectrons et Photons (1928) p. 282)], which later was successfully employed by Bohm. de Broglie, nevertheless, appears to have abandoned this idea and never referred to it in the book. The power of Pauli’s argument is diminished in that it equally applies to the Copenhagen interpretation; in fact it illustrates the ‘measurement problem’. The objection addresses therefore not a clash with ‘facts’-as Pauli seemed to think. However, it has more bite in the guiding wave picture, which aims at providing an objective (observer-independent) space-time account of microphysics. The second remarkable feature to which I want to draw attention is de Broglie’s argument of 1929, that w is not a ‘real quantity’ and cannot therefore ‘guide’ the particle [de Broglie (1929) p. 1041. This is because of features like the ‘reduction of the wavefunction’, an instantaneous, global change in the wavefunction the very instant certain measurements are performed on the physical system. The ‘reduction’ was taken to be evidence for the ‘subjective’

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nature of the wavefunction. In a similar vein, the particle’s trajectory depends on the field configuration at t =O and therefore on the experimenters ‘knowledge’ of the particle’s initial position [de Broglie (1929) p. 1031. These are objections directed against the semantic consistency of the interpretation [and they are the ones de Broglie raised again in his later writings, see de Broglie (1962)]. However, all of them use as premises ‘properties’ of the wavefunction which depend on Copenhagen ideas. Another indication for de Broglie’s failure to entrust full explanatory power to his approach is his statement that energy and momentum are not conserved [de Broglie (1929) p. 1041,which is true only if the ‘quantum potential’ is not properly taken into account. But why is that? A possible explanation is that he simply did not believe there is anything corresponding to the term in nature, or that there will ever be independent experimental indication of the existence of such forces. He cautiously described it as “une sorte de force d’un genre nouveau qui existe seulement quand la masse propre est variable, c’est-a-dire la ou il y a des interferences” [&ectrons et Phofons (1928) p. 2791. The only argument in the book not enmeshed in this kind of conceptual mismatch, which instead involves a clash with a firmly established theory, is exemplified with the help of Brillouin’s refraction example: the case in which the vacuum velocity of a photon is different from c. He commented about this consequence with the words sehr unwahrscheinlich (very improbable) [de Broglie (1929) p. 1121 and schwer vereinbar (hard to reconcile) with the special theory of relativity [de Broglie (1929) p. 1151. Yet, granted this consequence, why did he not strive to show that this consequence does not lead to faster-than-light telegraphs etc? It is again not out of the question that he was aware of the difficulty beforehand; compare to de Broglie (1928) p. 136. This objection will be studied from a Bayesian perspective in Section 3. Do the arguments discussed so far make intelligible the apparent high degree of consensus in the scientific community regarding the refutation of de Broglie’s Nouvelle Dynamique des Quanta?

The lecture by Born and Heisenberg was the longest one, judging from the proceedings, and was followed by the shortest of all discussions (no comments from Einstein, Lorentz, Schriidinger or de Broglie). It differs characteristically from the other talks in that it placed strong emphasis on an axiomatic treatment [mainly due to Jordan, compare to Jordan (1927)]. The continuity with the earlier statistical line of work (Einstein, Born) is emphasized throughout the talk, and there is not much of a claim that anything radically new in this respect is presented [for shifts in Born’s positions see Beller (1990)]. The first part was a review of material covered in the joint work by Jordan, Born and Heisenberg of 1925, followed by an exposition of the probabilistic aspects of the framework and the ‘transformation theory’. Mainly due to Dirac and Jordan, the transformation theory proved to be a general and fairly rigorous formalism,

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which allowed one to switch between various convenient representations: for example between the matrix scheme and Schriidinger’s, taking different sets of physical observables as ‘fundamental’. Schrodinger’s wavefunction was discovered to be just an instance of a class of ‘transformation functions’ and Born’s probabilistic interpretation was generalized correspondingly. This development finally led to the Hilbert space formalism of present day quantum mechanics. In the lecture’s last section, a thought experiment derivation of the uncertainty relations for position and momentum was presented. Missing altogether is any mention of the ‘reduction of wavepackets’ issue or the concept of ‘complementarity’. Bohr had introduced the latter concept at the Como conference a month before in order to systematize the continued use of two ‘modes of description’ borrowed from classical physics in the quantum domain, namely ‘space-time co-ordination’ and the ‘claim of causality’. Their mutually exclusive relationship is illustrated by an application of Heisenberg’s uncertainty relation for a particle’s position and its linear momentum in an experimental arrangement. Following Bohr, classical notions are presupposed by the new formalism insofar as observations and measurements are concerned. The participants could have learned before about important aspects of the Copenhagen interpretation (if not by private communication or lectures like Bohr’s) through Heisenberg’s famous paper on the uncertainty relations and a paper by Kennard (1927). One important lacuna was exposed in the main discussion period (Discussion G&kale Des Id&es Nouvelles kmises) in an exchange about how, in a scattering experiment, the different ‘channels’ become objectively separated after the scattering event. Both Dirac and Pauli (see above) believed this happens independent of any subsequent measurement, by a kind of spontaneous probabilistic process as in the emission of a photon by an excited atom from the viewpoint of the older quantum theory. The denial of this marks one of the central tenets of the Copenhagen interpretation, openly embraced in the discussion by Heisenberg. This denial is bound to run into difficulties. Since it relates w, the wavefunction, to an individual system, the introduction of instantaneous ‘reduction processes’ and action-at-a-distance must have been seen as inevitable; as exemplified by the case of the emission of an a-particle as a spherical ‘wave’ from a nucleus, or the impinging of a single particle on a semicircular screen (both examples devised by Einstein, the former relayed in a remark of Born). The latter case is a thought experiment in which an electron passes through a small hole in a plane and then is recorded on a screen in the shape of a large hemisphere centred at the hole. Einstein argued that the assumption, the hemispherical wave emitted from the hole is the complete description of the electron’s state (so-called conception II), has annoying consequences. If the particle is ‘potentially’ present in the wave all over the screen’s surface, it should consequently interact with the photographic

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emulsion on more than one point, since the information about an observable localization event at one point on the screen cannot travel faster than the speed of light to any other point on the screen. Of course, ‘multiple localisation’ is incompatible with there being only one particle at a time in the apparatus. This objection is highlighted by the existence of a prima facie plausible statistical interpretation of the ‘wavefunction’, an alternative which does not present any of these problems at all: Einstein’s conception Z was put forward by him at the main discussion section [incidentally, he was invited by Lorentz to give an official address on the ‘statistical’ interpretation, see Fine (1986)]. Apart from Bohr’s direct reply of a rather general character [Bohr (1985) p. 1031, no further objection to conception Z is documented in the published conference proceedings. The ‘ensemble interpretation’ (in the sense of conception Z) had been evaluated critically long before the conference took place, as far as the difficulty of reconciling interference phenomena with the assumption of independent particles moving along continuous space-time trajectories is concerned (Heisenberg’s recollection). Duane’s hypothesis (proposed in 1923 in the context of the debate over the nature of light) may point to a way out and lends some credibility to this conceptual possibility. Duane considered the reflection of monochromatic light from an infinite line grating with spacing a. He proposed that changes in the grating’s momentum (in the grating’s plane perpendicular to its lines) are quantized. The quantum h/a, whose multiples are supposed to be transferred in the scattering of a photon, is derived from the Bohr-Sommerfeld quantization rule. Duane proved from his premise that one can deduce a well-known formula for the intensity pattern of the reflected light. This theoretical device, in the 1920s methodologically an ad hoc change in the auxiliary hypotheses, is still employed today by proponents of the ensemble interpretation [Ballentine (1970); Belinfante (1973)]. However, it is rather implausible for a double-slit set-up [see Zeilinger (1990)], and Duane’s strategy, investigated by Epstein and Ehrenfest in 1924, does not account in any obvious way for other types of diffraction phenomena. Since these considerations are highly relevant, and Ehrenfest and Heisenberg were present at the conference, they are likely to have surfaced during the conference. The second objection appears at the beginning of the exposition of Einstein’s famous remark: conception Z leads to a violation of energy conservation in individual microprocesses. It is hard to say if this was new to him (though his phrasing suggests it), but for many of the other participants it probably was. Indeed, from quantum mechanics one only finds O=d/dt(y\HI v/) for a closed system. Nothing can be inferred from the theorem-in an ensemble interpretation-about an individual system’s energy. But the validity of the conservation laws for individual microprocesses was taken to be established by the coincidence experiments of Bothe and Geiger (see above) and Compton.

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[On the other hand, the argument rests on the premise that the kinetic energy is non-negative, which certainly meshes well with the picture and our background knowledge, but the formalism doesn’t determine a distribution of values between potential and kinetic energy, see Redhead and Brown (1981).] The consequence typically is illustrated by a finite probability to find a system inside a classically forbidden region (‘tunnelling’); this is discussed for the H-atom in Heisenberg’s book of 1930, an example possibly at the source of Einstein’s remark. The famous Bohr-Einstein debate began in Brussels and centred, at this first stage, around attempts to refute by way of thought experiments one definite consequence of the Copenhagen interpretation: the impossibility of simultaneously measuring conjugate quantities. If these thought-experiments had been successful, the interpretation would have been rejected. But Einstein’s failure in this respect-whether at the centre of what today is thought to be an important issue or not-had tremendous ‘corroborating’ impact on the conference’s participants’ beliefs regarding the Copenhagen interpretation [Heisenberg (1962-l 963)]. Schrodinger, the last speaker, presented the non-probabilistic theory of a continuous distribution of charge and matter in space-time, the dynamics of which, given by his linear second order equation, he derived from a variational approach. He emphasized its attractive features and prospects for further development: which balance the objections, levelled against it by Bohr and Heisenberg, with whom he had discussed the matter a year before in Copenhagen. In the discussion section following the lecture, both Heisenberg and Schrodinger agreed that there was no hope for a nai’ve ‘reduction’ of the complex many-particle wavefunction in configuration space to a set of waves in physical space-time; but they disagreed about how utopique this program was, with Schrodinger citing the new techniques of Hartree to justify his hopes [,!&ctrons et Photons (1928) p. 2111. Recent years have seen a serious attempt to reevaluate these possibilities in the light of later developments [Dorling (1987) pp. 16401. According to de Broglie’s recollections, Schrddinger had several ‘shrewd objections to the probability approach’. Alas, none have found their way into the proceedings. At one point, Schrodinger dismissed a remark (by Bohr) about a result obtained by Dirac’s methods, for reason of its lack of explanatory power: “11 ne m’est pas encore possible, pour le moment, de voir une reponse a une question physique dans l’assertion que certaines grandeurs sont soumises a une algebre non commutative, surtout quand il s’agit de grandeurs qui doivent rep&enter des nombres d’atomes” [&~tvons et Photons (1928) p. 2081. Because of its dogmatic or ‘habitual’ character, the desideratum voiced here presumably enjoyed less power than the various attempts to produce a ‘reductio proof of a point of view in question by derivation of unwanted consequences. Contrary to de Broglie’s case, the final discussion did

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Why has de Broglie’s Theory been Rejected? not see further the main representing response, Broglie’s

substantial

discussion

arguments

section,

an electron

against

the chairman

by a wavepacket,

the interpretation. pointed

At the end of

out various

and Schrodinger

difficulties

conceded,

in

in his

that a naive identification is out of the question. He added, that de association d’ondes et d’tilectrons ponctuels has to be regarded as a

provisional solution at the very best. This short overview on the exchange of arguments and objections at the meetings shows a conscious attempt to compare the various proposals for a physical interpretation of quantum mechanics. They provide a remarkably exhaustive palette of conceptual possibilities -in relation to contemporary categories, including a pure wave picture, a statistical mechanics analogue, and two interpretations featuring individual particles. Of course, no claim that all logical possibilities have been taken into account can be made; but this is, I believe, as close as one ever gets to that Earmanian ideal in practice [Earman (1992)]. It is one thing to construct objections by drawing surprising consequences which clash with other beliefs-and all of the proposed interpretations, have difficulties of one sort or another. It is another thing to weigh objections and compare and assess their power. If there is no tentative agreement on relative weights, the construction of objections is of no help for theory comparison. The next two sections address this question.

3. Learning In simulating the confirmational impact of various objections to the theories offered at the Solvay Conference, two devices in the Bayesian framework prove to be helpful: 1. the interpretation of P(HIE) - P(H)>0 as saying the hypothesis H is confirmed by the new evidence E, where P(.) measures the coherent agent’s beliefs; 2. the

extension

of the

framework

to include

the

possibility

of ‘logico-

mathematical learning’. One way to incorporate this latter concept, following Garber (1983) is to expand a given ‘local’ language L consisting of independent atomic sentences and their truthfunctional compounds, to a language L* which includes in addition atomic sentences of the form al-b with a, b in L, and which is closed under truthfunctional operations. It can be shown that a probability function defined on L may be non-trivially expanded to L* while satisfying a certain condition, that makes it plausible to think of the primitive connective k as ‘logico-mathematical’ implication. The ‘rational agent’ knows all the logical truths of L*; but the agent may be initially ignorant to those consequences derivable in his language, with the help of calculus for instance. The upshot is: the ‘discovery’ of a logico-mathematical

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of‘ Science

implication can raise the ‘credibility’ of the hypothetical premise consideration. For details the reader is referred to Garber (1983).

under

Propositions 1 and 2 in the present section generalize these results in order to treat cases in which the implication newly learned is one from a theory plus auxiliaries, T&f?, to a hypothesis H of which the scientist has a definite opinion, expressed by a ‘high’ or ‘low’ prior probability assignment to H different from 1 or 0. This pattern of reasoning appears to be widely employed in theory comparison, when no ‘hard data’ are at hand or available. By way of example: the hypothesis that light in vacuum can travel with a velocity different from c, would have a very low but non-zero probability for almost every researcher (outside certain anti-Semitic circles in Germany) after, say, 1920. Learning that a theory implies this consequence, will generally count against it, depending on various factors as, for instance, the prior probabilities of the auxiliaries used in the derivation. This type of reductio ad absurdum argument forms an important subclass of ‘plausibility arguments’ used in theory appraisal. The formal problem is to estimate the impact on Tin isolation from auxiliary assumptions, and to do it in a transparent and useful way. Propositions ! and 2, proved in the Appendix and applied in the rest of this section, give estimates of the change of ‘credibility’ in Tin different ‘epistemic contexts’, defined by assumptions (i) through (iii), if new conclusions have been learned to follow from an application of T along with auxiliary hypotheses. Proposition 1 Let A denote the implication T&HtH. T&H+ 7H and assume: (i) O
Let

1

denote

the

implication

(ii) 1 = P(AVA) P( T&a&A) = P( T&H&A&H) (iii) ( P(T&&~)=P(T&H&&~H). Then the following holds true: (1) P(T&HlA)P(T&H)0 if P(TH)IP(T&H) - P(T)P(A). A couple of remarks are in order. We are interested in the first place in confirmational effects on T alone and not on, say, T&H. The positive sign in (2) and (4) expresses an increase in relative confirmation of T or T&n. Sufficient criteria for confirmation and disconfirmation on learning the implication A, are given for both cases. Reference to the background information K is dropped for convenience. Premise (iii) is Garber’s ‘modus ponens’ rule for the identification of C-as ‘logico-mathematical’ implication, generalized appropriately to preserve symmetry between A and 2. The criteria given are stronger than necessary, and

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refinements and variations are easy to obtain; they are stated in a form facilitating the problem of deciding whether they apply in the historical case under consideration. Note, it is not necessary to know precise values for the various quantities in the criteria, only ranges and ratios. If the consequence is virtually an observation sentence and P(H)= 1, one finds from (2) an unconditional confirmation effect for T&H as expected. This result is equivalent to the corresponding theorem of J. Earman, which he argues is somewhat more ‘realistic’ than R. C. Jeffrey’s [Earman (1992) pp. 127, 1281. From condition (4) one has a confirmation effect for T, if P(lH)I P( T&H) - P( T)P(A). Assuming probabilistic independence between theory and auxiliary assumption, the inequality amounts to P(Fj) 2 P(A). This condition appeals to intuition, since in case of a ‘high enough’ initial confidence in B, one tends to think of T as more relevant for the deduction of the correct result than is H. What ‘high enough’ means is specified in the relation above. Scientists sometimes are interested in what can be deduced from a theory, regarding a certain effect or theoretical hypothesis, etc. if T is conjoined with either one of two possible alternative auxiliary hypotheses. This case occurs when a scientist is confident that his theory gives a ‘wrong’ result, if applied together with an auxiliary hypothesis drawn from conventional background knowledge; and later learns that a specific different hypothesis would have given instead the ‘correct’ consequence. This situation is addressed in the following Proposition, which does not deal with the effect of learning one of two a priori exclusive consequences like A and 2, but with the learning of a specific implication, when 17 or TH along with T is taken into account. Proposition 2 Let A denote

the implication

T&Z&H.

Let 2 denote

the implication

T&lHtlH and assume (i) O
Then the following holds true: (1) P(TIA) - P(T)<0 if P(lH)
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implies the correct consequence regarding a specific question; while learning, that the theory he favours does so, only when applied with a distinct ‘non-standard’ auxiliary. How is his belief in the theory changed? This situation can be treated quite similarly to Propositions 1 and 2 in a simple and uniform manner, as can a number of other possible epistemic situations; the corresponding sufficient conditions will not be given in this paper. The existence of a (non-trivial) distribution P(.), when subjected to the double constraints in condition (iii) of Propositions 1 and 2, under the assumption P(H) #O, 1, is not evident. The proof of existence, in the framework of linear programming, for the case covered by Proposition 1, is rather tedious and omitted here. The case of Proposition 2 can be treated as a variant thereof. Proofs for Propositions 1 and 2 are given in the Appendix. I will now apply the results to some objections mentioned in Section 1. The first case is the velocity of light anomaly in the pilot wave picture. Let H denote: either the consequence of a photon having a velocity smaller than the speed of light in the interference zone in front of a perfect mirror (see Section 2); or the hypothesis of v*>c* for a photon in vacuum near the surface of a partially transparent mirror. Let H be the setting of a perfect (or imperfect) mirror, an incoming plane wave at a certain angle of incidence, etc.; and let T denote de Broglie’s theory. It is a-not altogether implausible-idealization that the conference’s participants granted the consistency of T&H, in the sense that Hand 1H are within the range of application of T, and either T&HkH (A) or T&HI-H (A), but not both (at least, there is only mention in the relevant sources of physical objections). Hence (ii) of Proposition 1 holds. Newly learned by most of the participants at the Solvay Conference is A, the implication of H, a consequence ‘highly improbable’, following de Broglie. Granting the probabilistic independence of T and Z? for the sake of simplicityhere and in the following example-and an initial high degree of belief in Z? (a piece of background information), one verifies P(H) I P(q(P(H) - P(A)), if the scientist is initially not too confident about the possibility of A. Consequently from Proposition 1: P(ZJIA) - P(T) 10, in accordance with the records and intuition. Let us apply the schema to an objection against the ensemble interpretation (conception Z), an early alternative to the Copenhagen interpretation. As pointed out in Section 2, there was a prima facie problem in accounting for well-established interference phenomena with the assumptions of particle swarms on continuous trajectories in space-time and strict conservation of energy and momentum, in the absence of any ‘new’ forces of the kind (half-heartedly) proposed by de Broglie. Duane had shown for the scattering of light at an infinite line grating, that the situation need not be hopeless. Learning that Duane’s hypothesis allows for the derivation of the correct experimental law boosts the confidence in quantum theory-cum-ensemble interpretation T, if

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the agent is initially fairly certain about the alternative implication. Let YH assert Duane’s hypothesis for the scattering interaction with an infinite diffraction grating, along with other assumptions necessary for the derivation of ‘Bragg’s law’. Let B differ from TA only by the absence of Duane’s hypothesis. Experiments like those by Davisson, Germer, Thompson, Reid (1927) suggested P(H) -0 if H describes a pattern of blackening on the absorbing screen different from an interference pattern. Premise (ii) in Proposition 2 holds, because a participant at the Solvay Conference was likely to know about the problem, that is P(T&HtH)= 1 [relaxing premise (i)]. If the relationship P(H) 2 P( T&~H~~H) held for a typical agent, which is a plausible assumption since H was relatively uncontroversial, his confidence in T should have grown. To sum up: the criteria given in this section provide a glimpse of the network of beliefs that manifests itself in the evaluation of scientific arguments. The framework may lend itself to a classification of plausibility arguments as they frequently occur in contexts of theory comparison. It was originally developed by Garber and others to take care of the ‘old-evidence’ problem. An alternative framework is Howson’s. However, while Garber’s approach perhaps is not an entirely satisfying solution to the old evidence problem, Howson’s seems to be beset by more fundamental problems (existence of counterfactual degrees of beliefs, etc.), if it is to be applied to historical cases. For references and a critical evaluation of both approaches, see Earman (1992). Finally, I should note that while the present framework formally generalizes Garber’s treatment, its application here involves learning the deducibility of a theoretical belief. And admittedly it is problematic from an empiricist point of view, to allow, say, a hypothesis partially expressed in theoretical vocabulary with a probability far from certainty, to play the role of confirming ‘evidence’. 4. Elimination The above as it stands cannot account for the striking uniformity in the acceptance of the Copenhagen interpretation. None of the conference’s participants, who did not belong to the early ‘pioneers’ of quantum theory, later elaborated constructively on the alternative proposals in the light of the objections put forward, or investigated systematically and sympathetically the space of theoretical alternatives. It is therefore time to embrace a complementary set of considerations. This section will see still more idealization than the previous one. A participant at the Fifth Solvay Conference had to make up his mind for one of four alternative interpretations: an ensemble interpretation with uncertainty relations as scattering relations: Schriidinger’s interpretation, the ‘guiding wave’ picture (with or without a moving singularity) and the Copenhagen SHIPS 25:3-H

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interpretation. The first, second and third are members of what I will dub the ‘realist programme’. The realist family members fared badly in the conference’s debates: the objections which actually surfaced, or were likely candidates for informal discussions, mark real difficulties, even if the prospects for remedies were evaluated slightly differently. The Copenhagen view finally succeeded, I suggest, because a few, but highly representative, theories in the ‘realist’ programme had been ‘eliminated’, discrediting strongly the whole programme. This in turn effectively blocked further pursuit of realist alternatives for some time. The fast and widespread acceptance of the Copenhagen interpretation thus has four causes: (1) a working ‘positivist’ theory in the Copenhagen interpretation; (2) defunct ‘realist’ alternatives, (3) a conservative attitude towards certain assumptions in the ‘background knowledge’, and (4) a relatively homogenous attribution of weights to particular objections, theoretical ‘virtues’, etc. I hesitate to call this ‘induction by elimination’, because of the scarcity of hard evidence, but something akin may well have been effective in determining the outcome of the Solvay Conference. Before proceeding with a (necessarily crude and preliminary) Bayesian model for the epistemic changes that took place, a few remarks about the two ‘programmes’. Planck’s well-known contemporary attack on what he took to be Mach’s philosophy, fought out in lectures, pamphlets and distinguished journals, makes it likely that the scientific community was aware of two scientific outlooks (world views, programmes): a ‘positivist’ and a ‘realist’ one (terms introduced into probabilistic matters by J. Dorling). The two programmes define scientific aims and methods by successful examples in scientific history or explicit demands. The realist programme is characterized from a contemporary point of view, among other things, by a preference for scientific explanation by means of continuous motion in space-time, and a (flexible) demand for Anschaulichkeit. Let me hasten to add that I do not think scientists (like Heisenberg or Pauli for instance) pursued dogmatically, independently of current theoretical developments, one of the programmes as a philosophical doctrine. Rather, the programmes provided a handy set of ideas, whenever questions of theory construction or comparison came up-which certainly was not a common event-linking various branches of physics to each other and to ‘external’ questions, and motivating possible ways to progress. It would be interesting to know better in what proportion the scientific community was divided over this issue. I suggest that at least for most of those close enough to witness the developments, the signs in the winter 1926/1927 pointed towards the prospect that the ultimate picture of microphysics on the basis of the ‘new mechanics’ would be a thoroughly probabilistic version of the familiar classical explanations of phenomena in terms of continuous motions in space-time. For

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instance, even the insightful remarks of Pauli and Dirac on the possibilities joint measurements, at the end of 1926, were still couched in the language space-time mechanics enveloped appeared

physics,

as was

Jordan’s

‘statistical’

formulation

of of

of quantum

[Jordan (1927) p. 1091. The new, successful quantum theory, from the outset in a rather explicit ‘positivist’ outlook, finally to be more and more in harmony with demands of the realist

programme, and thereby to support it. These expectations (in parts of the scientific community) were set back with the discovery of the uncertainty relations as a consequence of the formalism, which-as Heisenberg and Bohr forcefully argued-are ‘realized’ in nature for individual particles. In addition to the considerable drawbacks of each alternative physical interpretation of quantum mechanics, the uncertainty relations raised the spectre of hidden variables; and this was emphasized by Einstein’s failure to devise thought experiments capable of undermining the relations, as recorded in Bohr’s well-known recollections of the evening discussions during the 1927 conference. The revolutionary change in the physical world view may be summarized in a Bayesian model as a finite succession of ‘learning events’. One of these ‘events’ is the discovery of difficulties in a naive ensemble approach to quantum mechanics (conception Z). Let R denote the realist programme and P the ‘positivist’ one. R was realized by the ensemble interpretation of quantum mechanics T, and by de Broglie’s theory T,. In the following, I will skip over Schriidinger’s interpretation, because its exclusion does not change the overall picture and allows for simpler formulae. Let TX stand for all other possible realizations of the realist world view unknown in 1927. The Copenhagen interpretation, taken as the only explicit realization of P is denoted by T4 [the ‘catch-all’

relative

to P is numerically

a negligible

contribution,

therefore

P(T,IP&K)=l]. The probability for T, is taken to be the one before its difficulties have been appreciated by a ‘typical’ participant of the conference, and we are interested in the increase or decrease of the ‘credibility’ of R when they become known. The prior probability for T2 is the one which already reflects the difficulties of the theory (purported conflict with the special theory of relativity, etc.) via the mechanism of learning discussed in Section 3. Its assignment moreover is the place where individual simplicity considerations enter (highly complicated electron ‘orbits’, etc.). Let K, the background information, comprise various, well-entrenched auxiliary hypotheses involved (not including Duane’s hypothesis, ‘new’ physical forces, etc.). From what has been said above, reasonable estimates for the conditional probabilities, given R and background information K, should obey (taking into account the familiarity of the Gibbsian type of statistical theory), the inequalities:

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Since it does not matter for the present purpose how the ‘learning event’ takes place, it is sufficient to think of it as a discovery of new evidence. Let 1E be an observation sentence implied by T,, and let E be the corresponding prediction of the theoretical alternatives under identical circumstances (the double-slit interference pattern for instance, which however at that time existed as a thought experiment only). Suppose E surprisingly turns out to be the case. Consequently: P(EI

T&R&K) = 0 and P(EI T&R&K) = P(EI T&P&K) = 1.

An estimate of the quantity P(RIE&K)IP(RIK) is needed, signifying a decrease of confidence in R if smaller than 1. The programmes and theoretical alternatives are supposed to form disjoint alternatives, such that 1 =P(RIK)+P(PIK), P(RIK),P(PIK)#O and: l=P(T,&RIK)+P(T,&RIK)+P(T,&RIK)+P(T,&PIK).

(*)

It is shown in the Appendix (Proposition 3) that a necessary and sufficient condition for a decrease in confidence in R in the present case reads:

leaving the likelihood factor involving T3 undetermined. The inequality holds, given the ‘unlikeliness’ of de Broglie’s theory T2 at this stage and the elusiveness of the realist ‘catch-all’. Notice that the rise of credence in P-induced by the decrease of confidence in R-for a ‘typical’ member of the conference is not the effect of the ‘small accident’ in predicting 7E which ‘knocks out’ T,. The effect rather results from the truth of E within a larger pattern comprising the existence of a credible (though still far from ‘perfect’) alternative in the Copenhagen interpretation without the difficulties the rivals have; differences in the a priori estimates for the realist alternatives; a certain degree of conformity in assigning weights to hypotheses like the law on the vacuum velocity of light; and the decision to keep well-entrenched auxiliary assumptions. While I have presented the initial conversion of an influential set of members of the contemporary scientific community to the Copenhagen interpretation and to a general non-realistic outlook as due mainly to a complex learning process, I do not suggest that the account is complete. Various sociological or psychological factors (persuasion, influence by authorities, a bandwagon effect) besides the ones mentioned in the Introduction may have smoothed the way for the Copenhagen interpretation. Insofar as these factors do not a priori render arguments and objections obsolete and ‘respect’ the conditions of fair discourse mentioned in the Introduction, there is justified hope that they may be included in a more sophisticated version of the model (via the assessment of relative

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weights, mechanisms like the one proposed by K. Lehrer, etc.). In any case, if the considerations of the last two sections are on the right track, a much simpler and more ‘rational’ mechanism is sufficient to account for the actual judgements made at the Solvay Conference. 5. A Plea for Modesty

My main concern has been the question how the Copenhagen interpretation gained initial support to the almost total exclusion of other options; the Bayesian apparatus has been employed to yield a tentative answer. The formulae and criteria derived capture an important feature of scientific inference when concerned with theory comparison: the deduction of consequences from a hypothesis which clash with other beliefs, and the balancing of their relative merits. Now, would it have been ‘irrational’ for a scientist to believe, to investigate and develop de Broglie’s alternative to the Copenhagen interpretation? There is a strong tendency in the case study literature to maintain that Bayesian changes in the probability assignments, together with an appropriate ‘rule of acceptance’, provide the context of justification for scientific practice. In the present case this point of view has the somewhat awkward consequence that de Broglie’s approach is ‘rationally rejected’. I find this hard to accept, since while de Broglie’s presentation is flawed, his (non-relativistic) theory is empirically equivalent to quantum mechanics-cumCopenhagen interpretation (and, after all, a wave limit&e latkralement duns l’espace de configuration is a more realistic assumption than a wave of unlimited extension). There is nothing wrong with the arguments and objections in themselves. It is not as if a judgement later turns out to be wrong and has to be revised. Rather, the judgements in de Broglie’s case were based prematurely on a not fully understood and ill-defended theory. If one wishes to posit ‘scientific rationality’, it should be of the sort that looks beyond contingent, ‘temporary’ shortcomings of this kind, instead of rationalizing them. A sceptic will doubt, however, that a situation like the one discussed can ever be excluded with certainty in any comparative evaluation, when keen theoretical competition is involved. A ‘phenomenological’ Bayesian account of course has no problem at this point. Without metaphysical baggage, the formal machinery employed in Sections 3 and 4 may be viewed with advantage as a simulation device, or an algorithmic model of an individual, input-fed evaluation process. Its purpose is attained if the toy model’s output is in rough qualitative correspondence with its targeted counterpart: the weighing of practical matters and hypotheses in the light of previous opinions and new insight or evidence. The preceding sections show such a coincidence between the output of a simulation program and the

394

community odological

members’ norms)2.

Acknowledgement - Support edged.

deliberation

Studies in History and Philosophy

of Science

(which may or may not conform

to meth-

by the Swiss National

Science

Foundation

is gratefully

acknowl-

References Ballentine, L. E. (1970), ‘The Statistical Interpretation

of Quantum Mechanics’, Review of Modern Physics 42, 358-38 1. Belinfante, F. J. (1973), A Survey of Hidden-Variable Theories (Oxford: Pergamon Press). Beller, M. (1990), ‘Born’s Probabilistic Interpretation’, Studies in History and Philosophy of Science 21, 563-588. Bohr, N. (1985), Collected Works 6, J. Kalckar (ed.), (Amsterdam: North-Holland). de Broglie, L. (1928) Selected Papers on Wave Mechanics (London: Blackie & Son). de Broglie, L. (1929), Einfiihrung in die Wellenmechanik (Leipzig: Hirzel). de Broglie, L. (1962), New Perspectives in Science (New York: Basic Books). Dorling, J. (1987), ‘Schrodinger’s original interpretation of the Schriidinger equation, a rescue attempt’, in C. W. Kilmister (ed.), Schriidinger (Cambridge: Cambridge University Press). Earman, J. (1992) Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory (Cambridge, Mass.: The MIT Press). Fine, A. (1986) The Shaky Game (Chicago: University of Chicago Press). Garber, D. (1983), ‘Old Evidence and Logical Omniscience in Bayesian Confirmation Theory’, in J. Earman (ed.), Minnesota Studies in the Philosophy of Science X (Minnesota Press). Heisenberg, W. (1962-1963) Interview with T. S. Kuhn, in Archive for the History of Quantum Physics. Jammer, M. (1974) The Philosophy of Quantum Mechanics (New York: J. Wiley & Sons). Jordan, P. (1927), ‘Kausalitat und Statistik in der Modernen Physik’, Naturwissenschaften 15, 105-l 10. Jordan, P. (1927), ‘Uber eine neue Begrtindung der Quantenmechanik’, Zeitschrift fur Physik 40, 809-838. Kennard, E. H. (1927), ‘Zur Quantenmechanik einfacher Bewegungstypen’, Zeitschrtft fur Physik 44, 326-352. Mehra, J. (1975), The Solvay Conferences on Physics (Dordrecht: D. Reidel). Redhead, M. L. G. and Brown, H. R. (1981) ‘A Critique of the Disturbance Theory of Indeterminacy in Quantum Mechanics’, Foundations of Physics 11, 1-19. LEiectrons et Photons’, (1928), Rapports et Discussions du Cinquieme Conseil de Physique (Paris: Gauthier-Villaers). Zeilinger, A. (1990) in M. Cini and J-M. Levy-Leblond (eds), Quantum Theory without Reduction (New York: Bristol). ‘The usefulness of the model, as a means of systematic representation of judgement patterns, may be called into question on empirical grounds: various psychological tests seem to indicate that the human being is a bad performer of Bayesian inferences. This objection, I believe, is not yet threatening. A model is not supposed to work equally well under all circumstances, and a long, open, sophisticated debate on the relative merits of a handful of alternatives, is of a different character than a typical test situation in a psychology laboratory. The question asked in the paper’s title is then, in a sense, left unanswered.

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Why has de Broglie’s Theory been Rejected? 6. Appendix

In the following proofs of the two claims made in Section 3 the connective & is omitted for brevity. (i), (ii) etc. refer to the assumption of the proposition that justifies the line in question.

1 To demonstrate the first of our four conditions, one finds from the premise of condition 1, setting p = P(A)

Proof of Proposition

P(TnA)

- P(T@p=P(TRAH)

- P(T@p

(iii)

I P(H) - P(T&‘p 10

and case 2 follows similarly. To demonstrate case 3 the following result is useful. From assumption ii, equivalent to 1= P(A) + P(A) - P(A&A) =p+a - p’, it follows that P(TA) - P(T)p=P(TRA)+P(T+A) - P(T)p =P(TfiA)+P(TlH) - P(TTHA)+P(T= P( TaA) - P( T@p [P(T+?A) - P(T+@ -p’}]+P(T

HAA)

- P(T)p

(ii)

-, RAA).

This expression along with the premise of 3 leads to P(TA) - P(T)pl[P(ZI) - P(T@PI- W’V+Pl+ [7P(T+%) - P(T+7)p’+P(T-QIAA] =P(H)+P(T)/?P(T@ - [P(T@+ -P(T+lp’ - [P(TlHA) - P(TlHAA)]

(iii)

SO,

and similarly for condition 4. This completes the proof of Proposition

Proof of Proposition

1.

2

From assumption ii, equivalent to 1 = P(A)+ P(A) - P(A&A) =p+p - p’, and P(T)=P(Tjj)+P(TlH) one derives: -_ P(TA) - P(T)p=P(TlHA)+P(THA) - P(T)p =[P(TlHA) - P(Tl@A+P(T@ - P(TsA)+ P(THA& - P(T@{l -p+p’} =[P(TTH~) - P(T%?)A - [P(TRA) - P(T@p]+ [P( THAI) - P( Tmp’].

This result along with iii proves condition 1:

(ii)

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396

-P(TA) - P(~)~I[P(T~ZL~~H)

- P(Tl@A

-

[P( TRd) - P( T@p] + [P( THAA) - P( Tjjp’] =[P(T+&H)~P(T%i)jjj+P(T@pP(T@p’
- [P(T) - P(T@I,B+P(T@I

- P(T@p’

- P(T)p+P(Tlj)

10. Case 2 follows in a similar manner. This completes the proof of Proposition

2.

Proposition 3

Given (*), I=P(RIQ+P(PIK) and P(E&T,IP&K)#O, sufficient condition for P(RJE&K’)IP(RIK)I 1 to hold is: P(EI T,&P&K)P(T,IP&K)

2 P(El T,&R&K)P(T,

a necessary

and

IR&K) +

P(EI T,&R&K)P(T2~R&K)+

P(EI T,&R&K)P(T,IR&K).

Proof of Proposition 3

From (*) and the constraints on P(RIK), P(PIK) above one finds P(EIK)= P(E&T,&RIK)+

P(E&T,&RjK)+

P(E&T,&RIK)+

P(E&T,&PIK)

and P(ElR&K)=P(EIT,&R&K)P(T,IR&K)+ P(EIT,&R&K)P(T,IR&K)+P(EIT,&R&K)P(T,IR&K).

Using Bayes’ rule and applying denominator one derives

=

the two expressions just derived to the

f’UV=W

= P(EIRW

f’@lK)

P(EIK)

P(EIRW P(EIR&K)P(RIK)+P(E&T,Ip&K)P(PlK)

’

If a+/?= 1, a, /+O, for real positive numbers x, y, y #O one has xl(xa+yfl I 1 if and only if x