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Quanta, Physicists, and Probabilities ... ?

Elemér Elad Rosinger

Published Oct 22, 2016• 0 citations
OpenAlex

Abstract

There seems to be nothing short of a {\it double whammy} hitting the users of probability, and among them physicists, especially those involved in the foundations of quanta. First is the instant instinctual reaction that phenomena which interest one do sharply and clearly divide into the {\it dichotomy} of {\it two and only two} alternatives of being {\it either} ``deterministic", {\it or} on the contrary, being ``probabilistic". However, there is also a second, prior and yet deeper trouble, namely, the ``probabilistic" case is strongly believed to be equally clear and well-founded as is the ``deterministic" one. And the only difference seen between the two is that the latter can talk also about ``individual" phenomena, while the former can only do so about large enough ``ensembles" for which, however, it is believed to be equally clear, precise and rigorous with the ``deterministic" approach. Or briefly, ``probabilistic'' is seen as nothing else but the ``deterministic'' on the level of ``ensembles" ... \\The fact, however, is that there is a {\it deep gap} between the empirical world of ``random" phenomena, and on the other hand, theories of ``probability". Furthermore, any attempt to bridge that gap does inevitably involve {\it infinity}, thus aggravating the situation to the extent that even today, and even if not quite realized by many, theories of ``probabilities" have a {\it shaky} foundation. \\This paper tries to bring to the awareness of various users of ``probabilities", and among them, to physicists involved in quanta, the fact that - seemingly unknown to them - they are self-inflicted victims of the mentioned double whammy. \\ \\

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There seems to be nothing short of a {\it double whammy} hitting the users of probability, and among them physicists, especially those involved in the foundations of quanta. First is the instant instinctual reaction that phenomena which interest one do sharply and clearly divide into the {\it dichotomy} of {\it two and only two} alternatives of being {\it either} deterministic", {\it or} on the contrary, being probabilistic". However, there is also a second, prior and yet deeper trouble, namely, the probabilistic" case is strongly believed to be equally clear and well-founded as is the deterministic" one. And the only difference seen between the two is that the latter can talk also about individual" phenomena, while the former can only do so about large enough ensembles" for which, however, it is believed to be equally clear, precise and rigorous with the deterministic" approach. Or briefly, probabilistic'' is seen as nothing else but the deterministic'' on the level of ensembles" ... \The fact, however, is that there is a {\it deep gap} between the empirical world of random" phenomena, and on the other hand, theories of probability". Furthermore, any attempt to bridge that gap does inevitably involve {\it infinity}, thus aggravating the situation to the extent that even today, and even if not quite realized by many, theories of probabilities" have a {\it shaky} foundation. \\This paper tries to bring to the awareness of various users of probabilities", and among them, to physicists involved in quanta, the fact that - seemingly unknown to them - they are self-inflicted victims of the mentioned double whammy. \ \
INTRODUCTION In his essay ‘Quantum Mechanics Without “the Observer”’ (in [20], 7–44), Popper criticizes the Copenhagen Interpretation and suggests construing the quantum theory as a ‘generalization of classical statistical mechanics’ ([20], 16). The uncertainty relations, he says, set ‘limits to the statistical dispersion … of the results of sequences of experiments’ ([20], 20) and not to what can be said about individual systems. Some strange features of orthodox microphysics are due either to a misinterpretation of probability, involving some ‘very simple mistakes’ ([20], 42); or else they are a straightforward consequence of the fact that the quantum theory is a statistical theory. For example, ‘the reduction of the wave packet … has nothing to do with quantum theory: it is a trivial feature of probability theory’ ([20], 37). The idea of probability that is presupposed in all these arguments, however, is Popper's propensity interpretation. Adopting this interpretation suffices for making clear what the quantum theory really is: a ‘generalization of classical statistical mechanics’ ([20], 16). This is Popper's position as I understand it. This position, it seems to me, is attractive enough as long as one does not carry the analysis too far. But it breaks down when confronted with all the facts which Bohr wanted to take into account. Altogether Popper's criticism and his alternative proposals once more reveal the strong points of the Copenhagen philosophy, and especially of the ideas of Niels Bohr. Today it is very difficult to find a satisfactory account of these ideas.
La théorie des probabilités et la théorie quantique, l’une mathématique et l’autre physique, s’apparentent dans la mesure où chacune admet plusieurs interprétations fort différentes. Certains ont proposé que les problèmes conceptuels de la théorie quantique puissent être, sinon résolus, du moins atténués par une interprétation adéquate des probabilités. Nous montrerons plutôt, à travers un survol historique et analytique des probabilités et de la théorie quantique, que si certaines interprétations de l’une et de l’autre s’agencent particulièrement bien, aucune ne s’impose de façon unique. * * * The theory of probability and the quantum theory, the one mathematical and the other physical, are related in that each admits a number of very different interpretations. It has been proposed that the conceptual problems of the quantum theory could be, if not resolved, at least mitigated by a proper interpretation of probabilities. We rather show, through a historical and analytical overview of probabilities and quantum theory, that if some interpretations of the one and the other go along particularly well, none follows in a unique way.
In our time, experimental physicists have obtained data on a very large number of phenomena and objects of the physical world. Very rarely there is a situation when theoretical physicists do not have enough experimental data to understand some known fundamental law of Nature. This situation arose almost a hundred years ago and sparked a discussion between A. Einstein and N. Bohr on the probabilistic nature of microcosm phenomena. From the time, it seemed that most physicists are inclined to believe that the proponents of a quantum explanation of the randomness of the phenomena of radioactive decay are right. Now this problem has been solved experimentally. The results of these measurements [1] show that A. Einstein and other proponents of determinism were right. In most cases, theoretical models are based on some already existing experimental data and are intended to explain them. At the same time, in the twentieth century, among microscopic, well-mathematically based models, there were several that raise doubts about their correctness, since they cannot explain a number of other experimental data that can be attributed to the fundamentally important properties of the studied objects [2] [3]. Therefore, the usual criterion for the correctness of the theory, which consists of its agreement with the measurement data, is ambiguous in this case. An additional criterion for the correctness of a microscopic theory can be formulated if it is assumed that the microscopic theory must be quantum one. The coefficients of quantum equations are world constants. Therefore, the solutions of these equations must be equalities made up of world constants only. For this reason, a correct microscopic model must rely on equalities consisting of world constants only. This criterion is shown to work successfully for models of superfluidity and superconductivity, for models of a number of particles, and models of the star interior.
In this note we argue there exist two force delivery schemes which are already well-known in classical mechanics namely: F(x)dx and F(t)dt. The former is associated with work and kinetic energy, is deterministic (yields x(t)) and requires information about velocity (or rest mass). It deals with F(x) where F=Force. The second is associated with an impulse which changes momentum. In this case, one does not need to know the rest mass or velocity. For example, a particle with momentum p (this allows for many mo,v pairs with the same p) strikes a surface and is reflected back with -p. It is also associated with probability as such a scenario exists for light reflecting and refracting from a surface. One does not know a priori the probabilities for these two phenomena so the problem is probabilistic. Classical mechanics primarily deals with the F(x) force delivery scheme, e.g. solving a differential equation(s) which arise directly from dp/dt = F(x) or using Lagrange’s or Hamilton’s formalism. Ultimately one obtains the deterministic result x(t). Although the notion of an impulse is well-known in Newtonian mechanics, it does not seem to be used much in calculations because an explicit form of F(t) is usually not known i.e. the force acts in a very tiny dx region in a very short period of time. As noted above, both schemes i.e. both deterministic and probabilistic situations appear in the macroscopic classical world, yet for some reason it is suggested that Newtonian mechanics is only deterministic. Newton considered a photon to be a particle with momentum p. A single photon reaching a surface with a different index of refraction has a certain probability to reflect and another to refract. This is a probabilistic, not deterministic, scenario governed by an impulse delivery. The probabilistic problem of light reflecting and refracting in a macroscopic situation may be solved by using continuity of the periodic electric field and its derivative from Maxwell’s solution for a photon from his electromagnetic equations. We have suggested in previous notes, however, that no knowledge of Maxwell’s result is needed. A steady state reflection/refraction problem removes time (unlike the x(t) deterministic picture), but seems to require some kind of balance on the right and left hand sides of the medium. Furthermore two equations are needed to solve for the two unknowns (i.e. reflection and refraction probabilities). Intensity/flux does not balance because the incident and reflected rays have positive flux. For n1<n2, it would be good to have a balance function which allows partial cancellation of the incident and reflected fluxes. The solution exp(ipx) allows exactly this and this solution only requires knowledge of p, momentum, not of rest mass (i.e. velocity). Thus there seem to be two force delivery mechanisms in the macroscopic classical world with very different solutions. The first is the F(x) delivery yielding the deterministic x(t) and requiring knowledge of mo (rest mass). The second is the impulse delivery which is probabilistic, makes use of exp(ipx), and does not require rest mass. If rest mass is known, however, exp(ipx) is linked with the average equation x = p/m t. At first the two delivery schemes seem mutually exclusive, but in the impulse situation, we try to show that a series of impulses may create the semblance of an F(x). Thus one has the exp(ipx) from the probabilistic force delivery scheme, but at the same time an F(x) emerges and so an x(t) type solution should also follow. How can this be? exp(ipx) is probabilistic and one may create averages with probabilities. Such an average or root mean square average may exhibit deterministic behavioiur. Thus exp(ipx) may be used together with a potential V(x) to create a solution W(x)= Sum over p a(p)exp(ipx) such that the prms(x) behaves like a Force(x) delivery result, yet at the same time one has the features of the impulse delivery as exp(ipx)s are used. Thus the exp(ipx) impulse approach may be used in strict impulse delivery problems and in F(x) delivery problems. This is exactly what occurs in quantum mechanics.