75 years of matter wave: louis de broglie and renaissance of the causally complete knowledge

8/2/2019 75 Years of Matter Wave: Louis de Broglie and Renaissance of the Causally Complete Knowledge

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arXiv:quant-ph/9911107v1

25Nov1999

quant-

ph/9911107

25Nov1999

75 Years of Matter Wave: Louis de Broglieand Renaissance of the Causally Complete Knowledge

A.P. KIRILYUK*

Institute of Metal Physics, Kiev, Ukraine 03142

______________________________________________________________________________________________

Mince, discret, secret, Louis de Broglie se tient toujours l'entre de la route qu'il ouvrit et, avec un sourirecourtois de gentilhomme, regarde l'avenir passer.

Maurice Druon, Prface Louis de Broglie (1973) [16]______________________________________________________________________________________________

ABSTRACT. A physically real wave associated to any moving particle and propagating in asurrounding material medium was introduced by Louis de Broglie in a series of short notes in 1923 and

in the most complete form in his thesis defended in Paris on the 25th November 1924. This result,

recognised by the Nobel Prize in 1929, gave rise to the whole field of the new physics known today as

quantum mechanics. However, although such notions as de Broglie wavelength and wave-particle

duality have become conventional in the standard quantum theory, it actually only takes for granted

(postulates) the formula for the wavelength (similar to other its formally postulated results) and totally

ignores the underlying causal, physically real and transparent picture of quantum/wave behaviour

outlined by Louis de Broglie in his thesis and further considerably developed in his later works, in the

form of double solution and hidden thermodynamics concepts (the first of them is only

mechanistically imitated within the over-simplified interpretation of Bohmian mechanics, now often

presented as causal de Broglie-Bohm theory). The payment for such crude deviation from the basic de

Broglian realism is the absolute domination of the fundamental science by purely abstract, detached from

reality and mechanistically simplified schemes made of formal symbols and rules that has inevitably led it

into the deep impasse judiciously described today as the end of science. However, an independent

approach of the quantum field mechanics (quant-ph/9902015,16, gr-qc/9906077) created recently

within the universal science of complexity and the related dynamic redundance paradigm

(physics/9806002) leads to confirmation and natural completion of unreduced de Broglie's resultseliminating all their difficult points and reconstituting the causally complete, totally adequate and

naturally unified picture of the objective reality directly extendible to all higher levels of the complex

world dynamics.

*Address for correspondence: Post Box 115, Kiev-30, Ukraine 01030.

E-mail: [email protected]


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TABLE OF CONTENTS

Page

1. Creation of causal wave mechanics by Louis de Broglie,

its imitation by the abstract (standard) quantum mechanics,and the deceptive results of the twentieth century scientific revolution 3

2. The universal science of complexity as the natural completionof the original ideas of Louis de Broglie . . . . . . . . . . . . . . . . . . . . . . .

2.1. The unreduced dynamic complexity and its imitations . . . . . . . . . .

2.2. Causally complete undulatory mechanics (quantum field mechanics)as complex dynamics of unreduced interaction between two primalfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3. Renaissance thinking against the medieval unitarity:

The lost fight still leads to victory of the unreduced complexity . . . . . . 31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

________________________________________________________________________________________________________________________________________________________

les progrs de la Science considrs indpendamment de leurs applications ont toujoursrsult d'efforts pour mieux comprendre. Le dsir de comprendre a t l'origine detoutes ses russites. Aussi suis-je aujourd'hui amen penser qu'il convient d'tre trs

rserv en face de l'affirmation, si souvent rpte par de nombreux physiciens depuis unequarantaine d'annes, suivant laquelle les phnomnes de transition quantiquetranscenderaient, pour employer un mot de Niels Bohr, toute description en termesd'espace et de temps et seraient par suite dfinitivement incomprhensibles. Il me parat

plus naturel et plus conforme aux ides qui ont toujours heureusement orient la recherchescientifique de supposer que les transitions quantiques pourront un jour tre interprtes,

peut-tre l'aide de moyens analytiques dont nous ne disposons pas encore, comme des processus trs rapides, mais en principe descriptibles en termes d'espace et de temps,analogues ces passages brusques d'un cycle limite un autre que l'on rencontre trs

frquemment dans l'tude des phnomnes mcaniques et lectromagntiques non linaires.Plus gnralement je dirai que, quand un processus physique nous parat impossible

comprendre et reprsenter, nous devons toujours penser que de nouveaux et vigoureuxefforts intellectuels nous permettrons un jour de le comprendre et de le reprsenter.

Louis de Broglie,Les ides qui me guident dans mes recherches (1965), In [40]

________________________________________________________________________________________________________________________________________________________


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1. Creation of causal wave mechanics by Louis de Broglie,its imitation by the abstract (standard) quantum mechanics,

and the deceptive results of the twentieth century scientific revolution

Since a truly revolutionary discovery changing qualitatively the existing basicnotions on the origin of things inevitably enters in antagonistic contradiction with the

established views, it often appears in an unexpected fashion, from a minor bywayand practically always needs much efforts and time for its full understanding, evenafter the formal acknowledgement by the scholar system of classified knowledge.This general rule is well illustrated by the destiny of the revolutionary discovery ofthe wave properties of particles (and thus the wave-particle duality) initiated byLouis de Broglie in 1923-24, then rather successfully confirmed and formallyrecognised, but remaining profoundly misunderstood and incomplete until the nowfading end of the twentieth century.

The wave associated to a (moving) material particle, or mobile, wasintroduced by Louis de Broglie in three short notes presented to the French Academy

of Sciences in September-October 1923 [1-3] (see also ref. [4] for a rsum inEnglish) and in complete form in his historical thesis defended on the 25th November1924 in Paris [5] and preceded by some development of the primary ideas [6-8]. Thethesis was defended with excellence according to the report signed by such prominentscientists as Jean Perrin and Paul Langevin [5]. Most remarkable, however, is the factthat the work of the young researcher contains the whole new outlook, physicallytransparent and confirmed mathematically, on the origin of matter and dynamicbehaviour of its elementary constituents incorporating the formidable synthesis of theemerging ideas of the new physics (relativity and the theory of quanta) and theunreduced conceptual realism of the main branches of the old, classical physics

(classical mechanics, optics, and thermodynamics).De Broglie shows in his thesis that the relativistic equivalence between massand energy, combined with the necessary fundamental (Planckian) quantization of anyform of energy, imposes the existence of an internal oscillation process within each(elementary) material particle which should naturally develop itself into an extendedwave when the particle moves in the surrounding material (physically real)continuum, so that the internal, localised oscillation of the particle-mobile shouldremain permanently in phase with the oscillations of the propagating wave (or else themotion of such compound wave-particle could not autonomously continue). Thisstatement, designated as the phase accord theorem and based on the profoundlycausal physical picture of the emerging wave-particle duality, leads to the famousmathematical expression for the length, , of that accompanying, or guiding, matterwave, if one demands that the differentrelativistic transformations of the frequency(or the related time flow) of the moving localised oscillation and extended wave donot violate the coherence between them (i. e. actually the physical integrity of thecompound object):

=h

mv, (1)


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where m is the (total, relativistic) mass and v is the classical velocity of the particle.The thoroughly realistic, physically transparent, but non-trivial, and everywherecontinuous synthesis of the hypothesis of quanta and canonical relativity is furtherconfirmed by the author within the independent, and equally causal, derivation basedon unification between the fundamental principles of the classical mechanics and optics(the principles of Maupertuis and Fermat, respectively), combined with the importantadditional involvement of the classical thermodynamics and relativisticelectrodynamics, so that the mathematical expression for the physical wave-particleduality, eq. (1), emerges from the whole unified picture only on page 111 of theoriginal thesis edition [5] (ending at page 128):

Si les vitesses sont assez faibles pour permettre de ngliger les termes de Relativit,la longeur d'onde lie au mouvement d'une molcule dont la vitesse est v, sera:

=

c

m0c2

h

=h

m0v...

It is therefore quite evident from the original foundation of the wave-particle dualityand derivation of its mathematical expression in de Broglie's work that what heproposes is not a formula, though experimentally verifiable and practically useful,but the qualitatively new level of understanding of the physical essence of reality andits detailed dynamical structure, produced by the non-trivial entanglement of variousfundamental approaches/principles which is absent in the old, classical physics, but inthe same time naturally, continuously emerges from it within the causal, realistic, andthus universal, basis of any truly scientific knowledge-understanding.

However, the intrinsic, extended causality of the original approach of Louis de

Broglie (mcanique ondulatoire [9]), profoundly related to the pronouncedhumanitarian quality of his thinking (see e. g. [10]) and aimed at the physicallycomplete understanding of reality and its consistent mathematical representation, wasthen quickly lost within the simplified, purely abstract adaptation of his results in thestandard (since 1927) quantum mechanics promoted especially by N. Bohr and hisfollowers (Copenhagen interpretation) and constituting a part of the generalmechanistic and totally formal approach of the actively imposed concept ofmathematical physics which, in order to survive, should be detached from thephysical reality it tries to imitate. Specifically, the scholar quantum mechanicsborrows from the results of de Broglie's undulatory (wave) mechanics only thefamous formula for the de Broglie wavelength of a moving particle, eq. (1), whilecompletely ignoring the fundamentally important, reality-based picture of dualisticdynamics underlying it, used by de Broglie for the very derivation of themathematical expression for the wavelength, and actually constituting, as we shall see,the basis of the unreduced complex dynamics of the underlying fundamentalinteraction process [11-14]. Even apart from the well-known physical incompletenessand logical contradictions of the mathematical quantum mechanics readily recognisedby the canonical sources (see e. g. [15]), the expression for the wavelength contains astrange, impossible mixture of the classical properties of the particle (velocity and


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mass, or momentum), its quantum (Planck's constant) and wave (wavelength)behaviour that can be but superficially hidden behind the formal postulates of thecanonical, math-physical approach.1

The well-organised, collective and ideological, rejection of de Broglianrealism by the international community of abstract-minded wunderkinds who didnot need any reality at all for their cabalistic exercises had taken the especiallypronounced form of a coup d'tat [10] during the Fifth Solvay Congress, in 1927,and since then the unreduced causality of the original wave-particle duality quicklybecame related to the practically total intellectual exclusion of any causal approach inquantum mechanics, and the whole new physics in general, despite some partialobjections from a number of prominent physicists (like those of E. Schrdinger andA. Einstein) and the formal highest recognition of de Broglie's results by the NobelPrize (in 1929) and other national and international recompenses. Because of thissharp divergence with the generally accepted values in the resulting totallyformalised science of the twentieth century obtained by the well-established(especially since Solvay Congresses) method of voting for the truth, Louis deBroglie was obliged to interrupt, for the next 25 years, his solitary search for the

causally complete understanding of the unique and unified reality and, contrary to hisnatural inclination, devoted his efforts in fundamental science mainly to teaching ofthe newly established principles of the canonical new physics, combining it withsome background investigation of a number of auxiliary problems [16] (the details ofthe dramatic development of de Broglie's life and work can be found in the exciting,but scientifically rigorous story of ref. [10]).

However, such is the force of the unreduced realism and causally completeknowledge that after 25 years of the forced suspension Louis de Broglie finds the newforces for development of his original ideas and since 1952, at his age of 60 (and untilhis retirement around 1975) produces an impressive number of fundamentally

important new concepts considerably completing and amplifying the causal picture ofwave-particle duality and concentrated around the famous double solution [17-19]and related hidden thermodynamics of the isolated particle [20-23] (see also refs.[16,24-26]).2

1Thus, the dualistic, complex dynamics of the physically real wave-particle object in de Broglianapproach, underlying the expression of eq. (1), is replaced in the canonical theory by artificialintroduction of the abstract operator of particle momentum (opening the whole series of equally formaloperators and other ever more abstract constructions) accompanied by thepostulatedexpression for itsformal, mathematical action on the formal state vector [15] simplistically imitating the real, dynamicwave-particle duality with the help of de Broglie's formula which is taken for granted in the scholar

quantum mechanics, despite its physically transparent, causal derivation in the thesis of 1924.2It is important to emphasize here the fundamental difference between the unreduced, essentiallynonlinearversion of de Broglie's undulatory mechanics, expressed by the double solution concept, andits mechanistic imitation within so-called Bohmian mechanics which is often referred to as de Broglie-Bohm theory and characterised as causal approach in quantum mechanics by its adherents (see e. g.[27-29]). In reality, however, the Bohmian mechanics is not more causal than any other interpretationof the conventional quantum mechanics: without changing the formal, postulated character of the theoryand finding the true, reality-based solution of any of the ensuing quantum mysteries, each of theinterpretations tries to accentuate one or another aspect of the canonical quantum enigmas bymechanistically playing with the same abstract symbols and trying to produce arbitrary, pseudo-philosophical guesses about what the resulting formal constructions couldmean. The reference to de


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In the meanwhile, the canonical, math-physical quantum mechanics hasremained at the same basic level quickly attained just after its creation during thetwenties, but having never changed since then despite the growing number of effortswithin ever more sophisticated abstractions of the canonical symbolism. Duringseveral tens of years after its establishment, this stagnation of the canonical quantumtheory was rather efficiently masked by its appearing new applications in atomic, thenparticle physics, chemistry, solid state physics, etc., even though it was quite clear thatthey have never brought anything new to the fundamentals of quantum mechanics assuch. However, contrary to some popular belief, the number of really meaningful,original applications of even a very fundamental theory is not infinite, and as far asthe stock of interesting problems in quantum mechanics was becoming the more andmore exhausted, in the second half of the century (not to mention the vanishinglysmall proportion of really useful solutions obtained), its conceptual incompletenessand gaping contradictions have come again to the foreground of fundamental science.This new, and ever growing interest in the most fundamental, substantiative problemsis closely related also to similar explicative difficulties in other branches of thefundamental micro-physics stemming from the same basis, such as field theory,

particle physics, (quantum) gravitation, and cosmology, that remain as explanativelydeficient and contradictory as the canonical quantum behaviour, but also basicallyseparated among them, despite the common nature of the objects of study and thelarge number of efforts put to attain the unification (such as various abstract schemesof unification of interactions within the canonical field theory). The practicalimportance of those fundamental problems also grows in an unexpected way, since thetechnical power of science has experienced in the same period the permanentdramatic, revolutionary increase, and now all that huge and sophisticated machineryfinds itself in the impasse of absent sensible, understandable and controlled,applications leading towards true discoveries (their absence proves the conclusion) and

involves instead quite real dangers created by the remaining purely empirical, trial-

Broglie in the designation of the Bohmian interpretation is the acknowledgement of the undeniable factthat its main results, before being rediscovered by D. Bohm in 1952 [30], have been elaborated byLouis de Broglie 25 years earlier, yet in 1927, under the name of pilot-wave description [9] (and thisexplains why Bohm's paper of 1952 gave a subjective impact to de Broglie permitting him to restart theresearch of the unreduced double solution [10]), but de Broglie considered it as an intentionallysimplified mathematical scheme obtained by simple algebraic (identical) transformation of the canonicalSchrdinger formalism and formally reproducing only an external part of the complete double solutionthat escaped the unreduced mathematical presentation, but remained nevertheless the main object of theveritable basic ideas of the undulatory mechanics [25] (see also [16-19,24,26]). Whereas the

unreduced de Broglie theory is totally devoted to the search for the adequate mathematical representationof the physically complete, realistic understanding of the dynamical micro-world structure, the adherentsof Bohmian mechanics persist in replacing the true causality by the same mechanistic, mathematicallybased manipulations with symbols, but presented as a physically consistent, causal version ofquantum mechanics (cf. [31]), while the unreduced double solution concept underlying the originalpilot-wave scheme is thoroughly excluded from their work. However, one can neither fool nature, norbribe the truth (contrary to the general public, science publishers and administrations), and theunresolved contradictions of the canonical quantum mechanics inevitably come out in its Bohmianinterpretation in the form of unexplained origin of the main features of quantum objects, such aspostulatedparticles and waves, their permanent, causeless and unpredictable transformation intoone another, and the intrinsically probabilistic character of all the occurring processes [11].


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and-error use of that immense, artificial, and most often destructive powersystematically exceeding the self-organised limits of the natural phenomena. Thefruitless shooting in all directions with the super-powerful guns provided by super-expensive super-accelerators represents a characteristic example of that moderntendency in the canonical science and the real dangers it involves (see [11] andespecially [14], section 4 for more details).

Those non-trivial, and ever growing, difficulties of the canonical fundamentalscience, readily acknowledged by many its prominent creators (see e. g. [32-37]) andnow taking the scale of the end of science [38] (see also [11]), are in reality butinevitable payment for the deviation from the road of causally complete, unreducedunderstanding of reality, that determined the development of the classical science,then was deliberately rejected by the international scientific community dominatedby the inherent mathematical physicists, with their characteristic inclination forabsolute power of low-level, pure calculations, justified by obscure, artificiallymystified speculations under the superficial slogans ofcompletely new physics [39],and later was maintained almost exclusively by the singular efforts of Louis deBroglie and his followers. Indeed, there is no wonder that the canonical fundamental

science cannot understand the veritable, ultimate origins of the observed dynamicalstructure of the world taking into account that the over-simplified, mechanisticabstractions absolutely dominating the canonical approaches are not intended for, andeven opposed to, such unrestricted, intrinsically complete understanding ofnature/reality, by their very origin. There is either no wonder that becoming themore useless the more their technical sophistication and speculative divergence fromreality grow with time, the abstract constructions of the canonical science provoke theoverflowing mistrust and disgust of the general public, younger generations andsociety in the whole, the attitude that is inevitably, though injudiciously, transposed tothe science in general, any kind of scientifically ordered knowledge, since the

canonical science, again readily describing in detail the growing mistrust (see [11]for the relevant references), puts at the same time all its well-organised efforts tosuppression of any dissemination ofanother, extended and fully adequate, creative andalways self-developing kind of scientific knowledge which in reality is much morerigorous (i. e. simply honest), both mathematically and physically, than the imitationsof the canonical formalism always based on incorrect approximations within a versionof perturbation theory, but also has, contrary to the latter, a human-friendly,causally complete and transparent character. It is difficult to overestimate the lossesproduced by such shameless substitution from the part of the educated mediocritythoroughly organised in a network of relations and mutual services permeating the

whole developed society. It is also clear that such huge deviation from the road ofcreation, involving in reality all aspects of life, cannot give anything else than the(already observed) rapid degradation of the quality of human living (always based onthe quality of human thinking) down to a series of more pronounced falls, orcatastrophes, so clearly felt within various modern prophecies and objectivelyprovoked precisely by that over-simplified, mechanistic approach to the intrinsicallycomplex natural phenomena. This link between causality in science, creativity ofthinking and the general development of society, as well as the critical character it


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acquires today, was unambiguously recognised and emphasised by Louis de Brogliewho devoted a large part of his activity and the whole series of publications to thefight against the growing dangers of the unlimited mechanistic simplification in thecontent and organisation of science (see e. g. [40] and other reference in [10,16]).

This conclusion about the true, unfortunately suppressed, and false, artificiallyimposed, directions of progress in knowledge development obtains the decisiveconfirmation from the recently appeared completion of the unreduced double solutionidea realised within the independent, new concept of universal dynamic complexity[11] and produced within the same type of reality-based, intrinsically causal approachas the de Broglian nonlinear wave mechanics (contrary to many recent imitationsaround science of complexity la Prigogine, Santa Fe & Co. fundamentally deficientby the same canonical limitations of one-dimensional reduction of reality to apositivistic classification of symbols, cf. [41]).

2. The universal science of complexity as the natural completionof the original ideas of Louis de Broglie

2.1. The unreduced dynamic complexity and its imitations

The unreduced, intrinsically universal dynamic complexity naturally emerges inthe unrestricted analysis of a generic interaction process within a closed system ofprimal entities [11] where no any usual perturbational cutting of inconvenient(non-separable) entanglement of the physically real participants is performed. Theapproach is realised within the generalised, universally nonperturbative version of theknown effective (optical) potential method (see e. g. [42,43]) that reveals the natural

structure of a problem, with the integrable part, giving the source of complexity assuch and determining the current level of the unreduced structure emergence incourse of interaction process development, and the related secondary interactionwhich is not rejected, however, but is further considered within the same analysis togive the next level of the emerging real structure, etc. (so that one finally obtains thedynamically fractal structure of the forming compound entity as the truly completesolution of a problem, representing the essential, causal extension of the conventional,purely mathematical imitations of the natural fractality).

The key result of this unrestricted, universally non-perturbative analysis of ageneric system with interaction, directly related to causal completion of the nonlinearundulatory mechanics, is the naturally emerging phenomenon ofdynamic redundance(multivaluedness) of the obtained solutions and real structures they describe.Physically, dynamic redundance results from the fact that each of the interactingentities exists, by definition, in the same reality as the result of their interactiveentanglement, but the unrestricted character of a generic interaction processdevelopment means that the number of combinations of the entanglement ofeverything with everything within the compound system will always be much greaterthan the general space in reality previously occupied by the non-interacting partners.Therefore, the unreduced, totally adequate analysis of the interaction process within


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the above method of effective dynamical functions shows that the complete problemsolution has many versions, each of them being complete in the ordinary sense (i. e.it totally occupies the accessible reality and is generally similar to the singlecomplete solution of the problem obtained within one of the canonical perturbationalmethods) and therefore incompatible with any other, equally real, version. Themultiplicity of the component solutions, called realisations of the system (problem), iseasily obtained from the simple analysis of the mentioned integrable part of theproblem, whereas its general nonintegrability manifests itself in a relatively weak, butirreducible, link of this level of structure formation to the next level of finer splittinginto another redundant set of solution-realisations, etc.

The only possible issue from the conflict among the realisations, at each levelof structure formation, is their permanent change, when the (closed) systemunceasingly and autonomously switches from one of its realisation to another.3 Sinceall the elementary realisations are strictly equal in their rights for existence, onecannot predict which is the exact realisation the system will take next, and this gives usthe causal, purely dynamic origin ofnaturally emerging randomness and the related

probability in a generic system with interaction. Namely, it is clear that the

probability, r, (corresponding to the experimentally measured frequency) ofemergence of a certain, r-th, elementary system realisation, constituting the causalextension of the canonical, postulatednotion ofeventin the mathematical theory ofprobability, is equal to 1/N, where N is the total number of realisations of the

current level, and now it is a dynamically determinedquantity that can be obtainedanalytically from the dynamic equation describing the interaction process. In thegeneral case, the elementary realisations can be dynamically grouped into denseagglomerates (or dynamical tendencies), so that the individual elementary realisationsare not discerned experimentally within each group that play therefore the role ofactual, combined system realisations, and their probabilities are not equal any more:

r(Nr) =Nr

N (Nr = 1,...,N) ,

r

Nr =N , r

r = 1 , (2)

where Nr is the number of elementary realisations within the r-th combined (actual)realisation (Nr is also determined dynamically), and the sum is taken over all suchactually observed system realisations. If, for the given system and level of its structure

formation, all the Nr (and thus r) tend to be small (r


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them, we have the limiting regime of uniform chaos characterised by the highestobserved irregularity in the system behaviour, while the opposite case of essentially

differing, relatively large Nr (r~ 1) corresponds to the characteristic regime of

(distinct) structure formation (causally extended self-organisation or self-organisedcriticality, SOC) and suppressed irregularity (though it is always present within theobserved quasi-regular patterns and indispensable for their very existence). Both

limiting regimes of causal randomness, or (dynamical) chaos (now rigorouslydefined), as well as all the possible intermediate situations occurring at all the levels ofinteraction development process, account completely for the whole diversity of theobserved structures of the world and their dynamics [11].

The quantitative measure of dynamic complexity, C, providing the unique,unifying characteristic of any part of that diversity, or regime of chaotic behaviour, isobtained as a growing function of either the corresponding number of realisations, orrate of their change, equal to zero for the unrealistically simple limiting case of onlyone system realisation (for example, ClnN), where the latter case constitutes theonly situation considered within the whole canonical, dynamically always single-valuedscience. This huge simplification of the canonical paradigm is the reason for allits unresolvable difficulties (like perturbation series divergence) and persistingmysteries (like those from the canonical quantum mechanics, see below); it is relatedto conventional perturbative cuttings of essential interaction links transforming theproblem from its original, nonintegrable quality to a deceptively convenient,integrable (or separable, exactly solvable) one, but simultaneously killing theuniversal mechanism of autonomous structure formation through the naturallyforming hierarchy of self-amplifying feedback loops of the nonintegrable, realinteraction process [11-14]. Note, by the way, that it is this unreduced, dynamicallyemerging hierarchy of self-amplifying, autonomously creative self-interactionprocesses that provides the causal extension of the notion ofnonlinearity and shows

that any real interaction process is intrinsically nonlinear, contrary to thefundamentally ambiguous ideas around nonlinearity in the canonical science (sincethe latter deals always with an effectively one-dimensional, and thus linear, processes,it tends to imitate nonlinearity by a strongly curved, intricately folded, or pointedline, always remaining, however, but a one-dimensional, and artificially produced,construction). The extended nonlinearity appears mathematically as generalisedeffective potential dependence on the eigen-solutions to be found, contrary to theperturbative equations of the canonical science, where the solutions to be found cannotoccur within the expression of interaction operator acting on them (which self-consistently leads to the formal canonical theorems about the uniqueness of

solution), and therefore the naturally emerging nonlinearity of operators isruthlessly cut by the invariably perturbational, one-dimensional logic of the canonicalscience [11-14] (see also the next section).

It is easy to understand also that unpredictably emerging realisations form thephysical structure (or texture) of space at the corresponding level of complexdynamics, while the facts of their appearance (i. e. the sequence of causally specifiedevents) provide the unceasing flow of time. This causally obtained, physicallytangible space is naturally discrete, while equally real and causal time is naturally


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irreversible (because of the causal randomness), but is nota tangible, material entityand therefore cannotbe considered as another physical dimension and be unifiedwith the spatial dimensions (contrary to the purely mathematical simplification of thecanonical relativity). One can see that causal time and space, together with the wholearborescence of dynamic complexity, include the hierarchical structure of levels(where the lowest level plays the role of the embedding, most fundamental space andtime, see the next section) and possess the natural property of relativity, resultingsimply from their dynamical, causal origin [11-14] and providing the realisticextension of the purely formal, postulated interpretation of the Lorentz-Poincarrelations within the canonical relativity. The intrinsic irreversibility of the causal timeflow, directly related to the dynamic redundance (causal randomness) phenomenon,provides also the causal understanding of the nonunitary character of any dynamicalsystem existence/evolution as the unceasing, qualitatively inhomogeneous alternationof (regular) motion within a realisation and highly irregular, unpredictable transitionto the next realisation, as opposed to the intrinsically unitary (i. e. qualitativelyuniform) character of any dynamical process considered within the dramaticallylimited framework of the canonical science, with its single realisation, arbitrarily

fixed once and forever. In this way one obtains also the extended, universalunderstanding of the internal structure, and the very physical sense, of any realinteraction which is represented by that permanent chaotic switch betweenqualitatively different phases of strong interaction between closely entangledinteraction partners (within each system realisation) and weak interaction within aquasi-free, irregular motion of transiently disentangled partners (during chaotictransitions between realisations), as opposed to the standard canonical idea ofqualitatively smooth (actually, averaged), regular distribution of interactionmagnitude.

Since the same, universal mechanism of dynamical system splitting into

incompatible realisations occurs at each emerging level of the interactive structureformation process, the resulting dynamical fractal that represents the whole multi-level arborescence of the system dynamics, has the intrinsicallyprobabilistic and self-developing character, which means that its branches permanently grow in anunpredictable (but partially, probabilistically ordered) fashion and in this way thesystem autonomously finds the (irregularly intricate) route of its intrinsicdevelopment, driven by the same interaction. This universal behaviour of a systemwith interaction provides the unified causal understanding of the observed property ofdynamic adaptability (or creativity) of the system (being another expression of theunreduced dynamic nonlinearity, see above) and considerable extension of the

canonical notion of fractal (Mandelbrot et al.), deprived from any intrinsic, dynamicalrandomness and autonomous development (creativity). In particular, our fundamentaldynamical fractal (of a problem/system), representing the unified, explicitly obtainedgeneral solution of a problem, can equally easy give rise, at different levels of itsstructure, not only to apparently fractal, fuzzy objects (like coast line, orcrystallisation patterns), but also to apparently non-fractal, quasi-smooth, distinctobjects (like a pebble reposing at the coast line, or an egg). We can say that anydynamical system with interaction can be characterised as alive, in a well-defined


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sense, since it shows such basic features of a living creature, now causallyunderstood, as autonomous dynamic creativity, adaptability, and growth towards themore and more involved structural elements (needless to say, the real life, in thenarrow sense, appears starting from a certain, high enough level of complexity of thesystem including, in particular, the ability to autonomously maintain the existence of awell-determined species of systems, by way ofcoding, or informational, reproduction[11]).

The general direction of life, or development, of fractal structure (and thedriving interaction process) is well defined by the growth of dynamic complexity ofits tangible, unfolded structure representing the causal, universal extension of thecanonical entropy, while the dual form of complexity, called (dynamical) informationand characterising the folded (or hidden) stock of the future, latent explicitcomplexity (entropy) at the start of the interaction process, correspondinglydiminishes, being directly transformed into the dynamical entropy, so that the totaldynamic complexity, equal to the sum of both its forms, remains always unchanged.4

This universal law of conservation (or symmetry) of complexity can be causally justified within the universal science of complexity, and being equivalent to the

causally extended (i. e. consistently derived/understood) forms of all the knowncanonical conservation laws, variational and various other principles (like thesecond law of thermodynamics or principle of relativity), is confirmed by thetotality of the existing experimental observations [11]. Providing the unified law, andcriterion, of (progressive) development/evolution of any system and the whole world,the universal dynamical symmetry of complexity avoids any simplistic symmetry ofidentical configurations of the canonical science and typically gives highly(dynamically) symmetric, but spatially irregular structures (contrary to the artificiallyimposed broken symmetry of the canonical, single-valued paradigm).

Similar to fractality, other over-simplified, imitating concepts of the unitary

science of complexity are basically deficient and therefore can only mechanicallyrepeat certain external, and irreducibly separated, properties, or signatures, ofcomplex behaviour (see e. g. [11,41]). In particular, the complex systems consideredwithin the canonical science are always some special systems, or configurations, orregimes of behaviour opposed, explicitly or implicitly, to other, non-complex(regular and simply structured) systems, without any clear criterion of distinctionbetween the two (to say nothing of other ambiguously intermixed notions, likenonlinearity, (non)integrability etc.), whereas the universal science of complexitydemonstrates, simply due to the unreduced consideration of an arbitrary driving

4This causal, reality-based, and universal extension of the notions of entropy and information permitsone to escape the multiple ambiguous speculations around these quantities within the canonical, unitaryscience (see [11] for more detail). Being capable to provide only single, qualitatively uniform realisation(object), the canonical science cannot consistently introduce any useful concept of entropy orinformation, and tries to imitate them by empirically counting the observed (or imaginary) objects andputting the found number under the sign of logarithm chosen by mechanical adjustment to observationsover some simple model systems. Being intrinsically unitary, the single-valued paradigm of thecanonical science cannot provide any consistent idea about qualitative development and thus thedifference (transformation) between information and entropy, and therefore it repeatedly, and arbitrarily,confuses one with another.


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interaction, that any really existing system is a complex one, which means that it hasmany (more than one) clearly defined (and consistently derived), permanentlychanging realisations. The characteristic, and most popular, imitations of the single-valued science of complexity are provided by the concepts of chaos that shouldresult from exponentially diverging trajectories and self-organisation (orsynergetics) considering only single-valued, exact solutions, both of these conceptsbeing deduced within the explicitly perturbative analysis, illegally (formally) appliedto the essentially nonperturbative case of complex dynamics. The general andparticular inconsistencies, resulting from such evident manipulations, immediatelyreveal themselves, and in particular the system states can diverge only according to apower law, and not exponentially (besides the evident fact that any regular dependencecannot serve as fundamental source of randomness by itself), whereas any self-organised structure can maintain its shape only due to the internal chaotic change of(sufficiently close) realisations organised in a fractal, nonintegrable hierarchy (see[11] for more details).

It is not difficult to see that it is precisely the characteristic features of theunreduced dynamic complexity, distinguishing it from the unitary science imitations

(signatures of complexity), that naturally show the inherent properties of quantumbehaviour, such as the causal, dynamic source of the true randomness (in the form ofredundant system realisations), dynamic duality between the localised, regularsystem state within each realisation and its delocalised, irregular state during chaoticjumps between the realisations, as well as the nonlinear dynamic reduction (orcollapse) of the latter, delocalised state when it is transformed into each localisedrealisation, determining system configuration, under the influence of the drivinginteraction. That is why it is especially important to avoid the imitations of the unitaryscience of complexity while considering the relation between dynamic complexityand quantum behaviour indeed representing, as we shall see, a situation with

pronounced, irreducible (though finally standard) manifestations of the dynamicredundance phenomenon within a system of interacting entities (so that the verypersistence of the quantum mysteries results from the total neglect of the unreduced,dynamically multivalued complexity within the effectively one-dimensional approachof the canonical science).

Equally important is the universal character of the characteristic features of theunreduced complex behaviour which leads to a straightforward generalisation ofpurely quantum properties to any system behaviour represented now in its complete,irreducibly complex (dynamically multivalued) form, which permits us to eliminatethe explicit or hidden inconsistencies of the canonical classical science (also single-

valued and unitary) and extend it to the ultimately universal and truly adequateknowledge exactly reproducing the observed unified diversity of the whole existingworld [11]. Without any coincidence, the same tendency towards ever growingunification of the basic principles of physics is clearly expressed in the work of Louisde Broglie [10,11,16].


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2.2. Causally complete undulatory mechanics (quantum field mechanics)as complex dynamics of unreduced interaction between two primal fields

____________________________________________________________________________________________________________________________

En conclusion, je pense que mes ides primitives, telles que je les ai repriseset dveloppes dans ces dernires annes, permettent de comprendre lavritable nature de la coexistence des ondes et des particules dont laMcanique quantique usuelle et ses prolongements ne nous donnent qu'unevue statistique exacte sans nous en rvler la vritable nature. Le postulatde l'accord des phases nous apprend, en effet, qu'il existe une Dynamique masse propre variable qui est sous-jacente toute propagationd'ondes, mme quand celle-ci s'effectue en dehors de l'approximation del'optique gomtrique. Et je crois que c'est l ce que la Mcaniquequantique actuelle n'a pas su voir.

Louis de Broglie, Sur les vritables ides de base de la Mcaniqueondulatoire, C.R. Acad. Sc. (srie B) 277 (1973) 71

____________________________________________________________________________________________________________________________

The fundamental dynamic multivaluedness was first revealed in the study ofcharged particle scattering in crystals performed within the Schrdinger formalism[44], and then was directly generalised to the description of the true quantum chaos ina generic Hamiltonian quantum system [45] and quantum measurement process [46](see also ref. [11]). Each of these situations is described as an unreduced interactionprocess within a quantum system described by the ordinary Schrdinger equation, butwhich is analysed then within the universally nonperturbative method of effectivedynamical functions (see the previous section) that gives redundant system realisations

and the ensuing causal randomness permitting one to understand the fundamentalorigin of the true quantum chaos in a closed (Hamiltonian) system (absent in theconventional approaches), in its agreement with the usual principle of correspondence(quasi-classical transition), and the causal, dynamical origin of quantum uncertaintyand wave reduction (collapse) in the process of quantum measurement considered asinteraction within a similar, totally quantum, but very slightly dissipative (open)system of object and instrument.

Although the latter result provides the causal solution of the old quantummysteries (quantum uncertainty and wave reduction) related to the quantummeasurement process involving explicit interaction between object and instrument,

the whole problem of (causally complete) foundation of quantum (wave) mechanicsdoes not stop there, since one needs also to understand, and rigorously describe, thephysical nature of a free elementary particle, including its corpuscular andundulatory properties, the duality between them, the causal meaning of the wavefunction, quantization, other related entities, and finally propose a consistentderivation of the Schrdinger equation and the accompanying canonical postulates,such as Born's probability rule. All these results, as well as many others, have indeedbeen obtained and generalised to any level of dynamic (complex) behaviour within thesame dynamic redundance paradigm, proving explicitly its universality [11-14], but


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the critically important starting point for the whole consideration is the definition ofthe corresponding system with interaction, necessary for any structure emergence, inthe considered case of the simplest system of the world, the free (isolated)elementary particle. The irreplaceable criterion that should govern the choice of thefundamental system of interaction, giving rise eventually to all observed structures ofthe world, is the demand for its simplest possible construction involving practically noartificial postulates, principles, or rules, since even the most fundamental of themshould causally result from the natural process of world structure formationconsidered here, by definition, from the really first principles.

Since there are two, and only two, universal, omnipresent, and extended realentities in the observed dynamical structure of the world, the electromagnetic (e/m)and gravitational fields/phenomena, it is natural to assume and this assumptionconstitutes the only one, unavoidable and reality-based, postulate of the universalscience of complexity that the simplest system with interaction giving rise to allother structures of the world is represented by the a priori absolutely homogeneoussystem of two interacting, physically real protofields of the e/m and gravitationalorigin, respectively, which are (homogeneously) attracted to one another. Since the

protofields are physically real field-like entities, it is clear that they possess a finitecompressibility which eventually gives rise to the forces of reaction compensatingtheir attraction in the situation of maximal approach (compression); these forces ofrepulsion are related to the necessarily existing fine structure elements of theprotofields (because they are physically real entities), but the details of this structurecannot be perceived from within this world, since its smallest emerging elements are(slightly) larger than the protofield constituents, and therefore we can consider theprotofields to be practically homogeneous. This large enough separation between thesuccessive structural levels of being is the consequence of the intrinsic discreteness(quantization) of complex dynamics, causally resulting from the unreduced, holistic

character of the driving interaction process development [11-14].We obtain thus indeed the simplest imaginable system of only two interacting

entities without any additional imposed structure. The structure appears as a result ofunreduced development of the protofield attractive interaction and is observed in theform of waves and particles of the world (and the further emerging more involvedstructures), whereas the unperturbed, free protofields cannot be observed fromwithin this world, but remain nevertheless physically real and provide the causalextension of the classical notion of ether [11-14], superficially rejected by themechanistic, purely formal approach of the new physics. As a matter of fact, theobserved world structure is asymmetrically displaced towards its e/m component,

which is related to somewhat different mechanical properties of the protofields,where the e/m protofield plays the role of rapid (light) and nondissipativecomponent that quickly conforms to the relatively slow (heavy) and dissipative(viscous) dynamics of the gravitational medium, so that any gravitational structure isalmost immediately covered with an e/m coating. Note that it is still absolutelynecessary to explicitly create (or obtain as a result of some previous development) thissimplest system with interaction in order it can then autonomously produce theobserved world structure (contrary to the dominating ideas of the unitary cosmology,


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formally justified by the mechanistic energy conservation law, about the possibilityof the universe emergence from nothing). From the other hand, if the attraction ofthe protofields with generic parameters is considered within the canonicalperturbative (and therefore single-valued) analysis, then one obtains a single, more orless homogeneous, fall of the protofields one onto another without any furtherstructure development that could resemble the observed universe. The unreducedinteraction analysis, leading to the universal concept of complexity and brieflyreviewed below (see refs. [11-14] for more details), gives a qualitatively differentresult, the chaotic quantum beat dynamics providing the causal wave-particleduality of the isolated elementary particle, causally complete interpretation of thewavefunction, inertial (and gravitational) mass and physically real de Broglie waveof a moving particle. Further autonomous development of the same interactionprocess towards higher levels of complexity (structure formation) proceeds by thesame mechanism of dynamical splitting into multiple incompatible realisations andgives all the existing structures and phenomena including, first, gravitational, e/m andother fundamental interactions (intrinsically unified from the beginning), thennatural emergence of classical (trajectorial) behaviour provided with the intrinsic,

causally extended relativity, and so on, up to the highest forms of human activity [11].

We express the protofield interaction by the existence equation that simply fixesthe fact of existence of a compound system consisting from the interactionparticipants, while the details of their resulting (physical) entanglement should bespecified within further (unreduced) analysis

[he(q) + hg () + Veg (q,)](q,) = E(q,) , (3)

where q and are continuous (but maybe also discretely structured), physically real

degrees of freedom of the e/m and gravitational protofields/media, respectively, he

(q)and hg() are the corresponding generalised Hamiltonians (or any other pertinentfunctions) describing the (unobservable) free state/dynamics of the protofieldswithout interaction, Veg (q,) is the (attractive) interaction potential, (q,) is thestate-function describing the (developing) state of the compound system, and Eis theeigenvalue characterising the property expressed by the generalised Hamiltonians inthis state (as the following analysis shows, it is always reduced to a measure ofdynamic complexity, expressed by the generalised energy in the resulting standarddescription [11]). We emphasize that eq. (3) does not involve any additionalassumption, apart from the fact of the compound system existence, and practically anyknown particular equation can be presented in this or equivalent form. Note also theabsence of time and usual space in this starting equation; the degrees of freedom, qand , represent the matter of the (unperturbed) protofields (or ether) that cannot bedirectly perceived in this world. Insisting on the first principles, emergent (andthus totally adequate) type of description, we have no right to specify, at this stage,any particular interaction configuration or other detail: all the details should self-consistently, dynamically emerge in course of interaction development exactlyreproduced by the unreduced description. An important by-product of suchunlimited generality is the absolutely universal character of the obtained results that


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can be further applied at any higher level of the world structure emergence, while thedifference between the appearing levels is determined by (growing) dynamiccomplexity (in the form of generalised entropy) represented by the number of thenaturally emerging realisations or rate of their change (see the previous section).

As we have explained above (section 2.1), the unreduced development ofinteraction processes leads to formation of multiple, incompatible states of thecompound system, called realisations, which are represented, in the case of interactingprotofields, by the locally squeezed (dynamically compressed, or reduced) states ofthe coupled (closely entangled) protofields centred around different points of thethus emerging,physical space. This dynamic redundance of equally possible reductioncentres necessarily leads to their permanent, causally random change by the systemthat performs permanent nonlinear, spatially chaotic oscillation/jumps betweensuccessive reduction centres always passing by the extended, decompressed phase oftransiently disentangled protofields (before they entangle around the next reductioncentre).

This physical picture results from the unreduced mathematical analysis of thestarting existence equation within the method of the effective dynamical functions [44]

(generalising the effective potential method) which will not be reproduced here indetail (see [11-14,45,46]). The causally complete general solution of eq. (3) isprovided by the intrinsically probabilistic sum of the observed (generalised) systemdensities, r(q,), for each of the obtained incompatible solution-realisations(numbered by index r) from the complete set ofN realisations:

(q,) = r= 1

N

r(q,) , (4)

where each r-th realisation is endowed with the dynamically derivedprobability r(cf. eq. (2)), so that the expectation value of(q,), obtained in a long enough (orrepeated) observation over the system, is given by

ex(q,) = r=1

N

rr(q,) , r

r = 1 . (5)

The details of the solution are specified by the following expressions of the effectiveformalism:

r(q,) = |r(q,)|2

, r(q,) =

ic

r

i

0(q)+n n(q)g

^r

ni()

r

0i() , (7a)

rni() = g

^rni()

r0i() =

dgrni(,)r0i(), g

r0i(,) =

i

0

ni()Vn0()0*ni ()

ri -

0

ni -n0,

(7b)


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where {0(q),n(q)} are the eigenfunctions (and {0,n} the eigenvalues) of the freee/m protofield (generalised Hamiltonian he(q)), constituting an ordinary complete setand actually chosen in the form of -like, localised functions representing(informally) the irresolvable structural elements of the e/m protofield (while 0,ncorrespond to coordinates of maxima of those functions), n0n - 0, n 0 takesinteger values;5

Vnn()

q

dq *n

(q)Veg (q,)n(q) ; (8)

{r0i(),

ri } is the complete set of eigenfunctions and corresponding eigenvalues of the

effective existence equation, numbered by index i within each realisation numbered byr(the redundant realisations are obtained just from this effective equation):

[hg () + Veff(;)]0() = 0() , (9)

with the operator of the effective (interaction) potential (EP), Veff(;), given by

Veff(;) = V00() + V^ (;) , V^ (;)0() =

dV(,;)0() , (10a)

V(,;) n,i

V0n()

0

ni()Vn0()0*ni ()

-0

n i - n 0 , (10b)

so that its action on the already found eigenfunction, r0i(), is expressed as

Veff(;ri)

r0i() =V00()

r0i() +

n,i

V0n()0

ni()

d0*ni ()Vn0()r0i()

ri -

0

ni - n 0 ;

(10c)

and finally {0ni(),0ni} are the eigenfunctions and corresponding eigenvalues of thetruncated problem characterising other levels of (fractal) complexity development(which renders the problem nonintegrable) and represented by the auxiliary systemof equations:

5Note that the matrix elements used in eqs. (7)-(11) and presented here in the canonical form ofoverlap integral, eq. (8), can have other particular expressions playing similar role in various problems;possible variation of such technical details cannot change the obtained results.


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[hg() + Vnn()]n() + nn

Vnn()n() = nn() . (11)

The coefficients cri in eq. (7a) are obtained from the dynamical boundary

(initial) conditions attained in the transient phase of disentangled, quasi-freeinteraction participants during their chaotic jump from one localised realisation to

another. They actually determine the realisation probabilities {r} [11], which has a

transparent physical interpretation: the protofield reduction probability is higher forthose realisations (centres of reduction) where the average of the chaotically changingprotofield magnitude is higher (the probabilities are proportional to squared modulusof the field amplitude expressed by the state-function). It is not difficult to understandthat the transitional phase of the system, forming a special, intermediate, or main,realisation of the system, provides the causal, physically real extension of thecanonical wavefunction, and therefore it becomes clear that the above relationbetween the irregularly changing, delocalised state-function (or wavefunction)magnitude and the probability of emergence of the highly localised, corpuscular state-realisation, physically established during chaotic system jumps between successivelocalised realisations, provides the causally complete substantiation for the Born'srule of probabilities stating that the probability of particle emergence in anyinteraction process is proportional to the squared modulus of the wavefunction (due tothe universality of our description, this causally substantiated rule is reproduced at allhigher levels of interaction process development). Therefore in practice one uses theequation for the wavefunction at a higher sublevel of complexity, such as Schrdingeror Dirac equation, that can be causally derived within the universal science ofcomplexity [11-14], in order to determine the distribution of probabilities of thelocalised (corpuscular) state emergence (cf. eq. (2)).

We have provided the main mathematical expressions of the unreduced

interaction analysis in order to demonstrate that the obtained qualitatively new resultshave not only a transparent physical interpretation, but also rigorous formalsubstantiation (contrary to the canonical fundamental science always obliged to use anumber of formally imposed, mysterious postulates quickly growing with the levelof system complexity). In particular, the key phenomenon of physically real,dynamical squeeze, or reduction (collapse), of a portion of extended interactingprotofields is a manifestation of the omnipresent instability of any unreducedinteraction, completely rejected by the conventional, perturbative approaches anddescribed, within our analysis, by the EP dependence, in the effective dynamicalequation, eqs. (9)-(10), on the eigenvalue to be found, . This dependence, arbitrarily

eliminated for convenience in any scholar formalism (e. g. [42,43]), corresponds tothe self-consistent local amplification of interaction through a dynamically emergingfeedback loop: a change of at the right-hand side of eq. (9) will provoke the EPchange at the left-hand side, which will further influence the value at the right, etc.The physical basis of this self-amplified instability is also quite transparent: if aroundcertain location an occasional slight approach between the formally homogeneousprotofields occur, then their attraction around that location should also slightlyincrease which will provoke further local approach/squeeze, etc., until the state of


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maximal compression where the opposed forces of reaction compensate theattraction. The localised, or corpuscular, character of the resulting state-realisationis described by the resonant terms in eqs. (7b) and (10c) among which only a small,concentrated group has sufficiently high magnitudes due to the cutting actions of theintegrals in the numerators. It is important that the same dependencies and relatedinstability mechanism occur for both state-function, eq. (7b), and EP, eq. (10c), whichmeans that the state-function dynamically squeezes around certain centre because theEP self-consistently forms a potential well (which was not there before and shalldisappear after the transient realisation emergence) just around the same, randomlychosen location. The causal randomness of this choice of the current reduction centreis determined mathematically by the same resonant denominators, effectivelycounting the number of possible state-realisations and increasing the maximal powerof the eigenvalue to be found () in the characteristic equation obtained from theeffective existence equation, eq. (9), up to the redundant number of realisations. Thecorresponding physical origin of redundance is quite evident from the aboveinstability interpretation: an initial, infinitesimal deviation from the averagehomogeneity, determining the location of the resulting reduction centre, cannot be

fixed or predicted and happens in one of many adjacent places with comparableprobabilities. In the case of strictly homogeneous general system configuration(isolated elementary particle at rest) the probabilities for all N realisations willevidently be equal, as it also follows from the causally extended Born's probabilityrule.

It is not difficult to see that the intrinsic instability of the protofield interactionacts in both directions and leads to the reverse phase of protofield extension after theself-amplifying squeeze stops at the state of maximal compression: the neighbouringportions of protofields start increasingly pulling the corpuscular state, notintrinsically amplified any more, to all sides, which induces protofield

disentanglement/extension and system transiently return to the delocalised state ofwavefunction, after which a new reduction to a randomly chosen centre occurs, etc.The periodic sequence of reduction events represents the most elementary, inherentclock of the world and gives rise to the causally determined time flow, which isevidently unceasing and naturally irreversible because of the causally random(intrinsically probabilistic) choice of consecutive centres of reduction, formingsimultaneously the tissue of the emerging, physically real (tangible) space (whichresolves also the problem of configurational space in the standard quantummechanics). This elementary space and time, resulting only from the unreducedinteraction in the a priori homogeneous system of two protofields, form the

corresponding embedding, most fundamental entities that serve as a basis foremergence, by the same universal mechanism, of the whole multilevel hierarchy ofchaotically switching realisations of the world structure, forming the respectivehigher levels of space and time, with a specific physical quality of the space tissuestructure at each level [11] (see section 2.1).

We call the spatially chaotic, time-periodic sequence of cycles of highlynonlinear reduction-extension in the system of interacting protofields quantum beat. Itrepresents the standard(universal) form of the unreduced interaction development, in


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the form of chaotic realisation change, and can also be described as chaotic spatialwandering of the virtual soliton, designating the system state in the phase of maximaldynamical squeeze and describing the causal version of the localised, corpuscular stateof the elementary particle as such. The latter has also the dynamically related, dualstate of extended field, or wave, so that such elementary field-particle ispermanently, naturally transformed, only due to the same a priori uniforminteraction, from one of these complementary, or dual, states to another performingits totally irregular internal pulsation, equivalent to the rest mass, or partiallyordered displacement, equivalent to global motion (see also below). In this way weobtain simultaneously the causally complete, intrinsically consistent interpretation ofelementary particle, its inherent wave-particle duality, wavefunction, quantization (=dynamical discreteness), and emergent space and time, which brings us much closer tothe causal completion of the approach initiated by Louis de Broglie 75 years ago.

In particular, it is evident why and how the above quantum beat process,described by eqs. (7)-(11), realises the causally complete version of the doublesolution by naturally, dynamically combining within the same, explicitly obtaineddynamics both the delocalised quantum field and the singular field (our virtual

soliton) that were heuristically introduced in the original de Broglie's concept [17-19]. The natural dynamic transformation between the two components of suchextended double solution with chaos, realising the universal causal duality ofcomplex dynamics (and providing the realistic extension of the speculative, ambiguouscomplementarity introduced by Niels Bohr), is evidently possible only due to thedynamic multivaluedness phenomenon revealed by the unreduced protofieldinteraction analysis, which constitutes therefore just the missing link for the originalnonlinear undulatory mechanics. The causal randomness in the choice of each nextreduction centre, coming from the same dynamic redundance of interaction process,provides the physically real basis for the entity of the wavefunction which is made

from the physically tangible matter, but behaves strangely just due to its permanentinternal chaotic (intrinsically random) change. The chaotically changing wavefunctionserves as a common blanket for all emerging corpuscular states (virtual solitonlocations) and thus provides the physically real link to the causally completed hiddenthermodynamics of the virtual soliton which is none other than the true, intrinsicdynamical chaos of the randomly emerging corpuscular realisations of the system,also prodigiously predicted by Louis de Broglie [20-23] well before the establishmentof even its unitary imitations.

Among many intrinsically unified aspects of the obtained double solution withchaos [12,13] note especially the qualitatively rich, totally realistic and rigorously

justified interpretation of the elementary particle as a complex-dynamical process (ofquantum beat) involving already all the intricate details of complex dynamics(intrinsic instability, discreteness, irregularity and unpredictability, fractalentanglement of components).6 It is sufficient to compare the obtained picture with the

6In particular, the unreduced fractal entanglement of the interacting protofields gives rise to the causallycomplete interpretation of the property ofspin emerging, due to the inevitable shear instability, as avortex motion of the e/m protofield matter performing its reductive interlacement with the gravitationalmedium components aroundthe emerging centre of the compressed, corpuscular state [11-14].


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series of abstract, absolutely detached from reality and therefore unexplainable,simplifying postulates of the canonical, unitary science leading to artificialmanipulations with state vectors in some purely mathematical spaces, instead of realobjects in real space, in order to understand the crucial advantages provided by theintrinsically realistic de Broglian approach naturally completed by the universalscience of complexity. We call the application of the latter to the lowest levels of theuniversal hierarchy of complexity, described here and corresponding to quantumbehaviour of micro-objects, quantum field mechanics [11-14]. It not only eliminatesthe difficulties of the nonlinear undulatory mechanics of Louis de Broglie (includingmatter wave, double solution, and hidden thermodynamics) with the help of thecausally complete concept of the true, internal dynamical chaos, but leads, as weshall briefly demonstrate below, to its natural unification with the causally extendedrelativity (both special and general) and field theory/particle physics (involvingthe intrinsically unified fundamental interactions).

____________________________________________________________________________________________________________________________________

La relation du quantum n'aurait sans doute pas beaucoup de sens sil'nergie pouvait tre distribue d'une faon continue dans l'espace, mais nousvenons de voir qu'il n'en est sans doute pas ainsi. On peut donc concevoir que

par suite d'une grande loi de la Nature, chaque morceau d'nergie de masseproprem0 soit li un phnomne priodique de frquence telle que l'on ait:

h0 = m0c2

0 tant msure, bien entendu, dans le systme li au morceau d'nergie. Cettehypothse est la base de notre systme: elle vaut, comme toutes les hypothses,

ce que valent les consquences qu'on en peut dduire.Devons-nous supposer le phnomne priodique localis l'intrieur du

morceau d'nergie? Cela n'est nullement ncessaire et il rsultera du paragraphe III qu'il est sans doute rpandu dans une portion tendue del'espace. D'ailleurs que faudrait-il entendre par intrieur d'un morceau isold'nergie? L'lectron est pour nous le type du morceau isol d'nergie, celuique nous croyons, peut-tre tort, le mieux connatre; or d'aprs lesconceptions reues, l'nergie de l'lectron est rpandue dans tout l'espace avecune trs forte condensation dans une rgion de trs petites dimensions dont les

proprits nous sont d'ailleurs fort mal connues. Ce qui caractrise l'lectroncomme atome d'nergie, ce n'est pas la petite place qu'il occupe dans l'espace, je rpte qu'il l'occupe tout entier, c'est le fait qu'il est inscable, nonsubdivisible, qu'il forme une unit.

Louis de Broglie, Recherches sur la thorie des quanta, Thse (1924), [5]

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Although the obtained picture of hidden complex dynamics of the elementaryfield-particle provides the uniquely consistent, realistic interpretation of the knownqualitative patterns of quantum behaviour, it cannot be observed in detail because itrepresents the lowest level of the world complexity whose internal structure is notresolved in principle (though remains quite real). What can be observed, however, isthe average effect of the chaotic quantum beat dynamics of the isolated field-particle,and we argue that the main observed global manifestation of the complex field-particle dynamics is none other than inertial mass, universally defined and causallyextendible to effects of both special and general relativity (see below). It is notdifficult to understand that the inertial mass of the isolated particle-process,intrinsically equivalent to its energy, can be consistently interpreted as the temporalrate, or intensity, of realisation change of the chaotic quantum beat process,representing a measure of its complexity. Due to the internal, purely dynamic originof the causal randomness of the jump-like wandering of the virtual soliton inside thefield-particle, such understanding of the property of mass directly provides the mainfeature of inertia (resistance to the change of state) which is due to the alreadyexisting (and irreducible) internal motion and can be compared to resistance to

compression of a gas of chaotically moving molecules. In addition, such interpretationof mass provides the natural extension and decisive clarification of de Broglie's ideasabout hidden thermodynamics of isolated particle and related variable mass of aparticle [20-23], by finding the necessary source of randomness within the a priorideterministic dynamics of the (basically simple) system and thus permitting one toavoid a formal imposition of some external, and inevitably ambiguous, source ofchaoticity, hidden thermostat which produces only variations of an already existing(and remaining unexplained) proper mass.

The mathematical expression of the proposed complex-dynamical definition ofthe elementary field-particle mass, compatible with the conventional definitions of

energy,E, and mechanical action, A, is [11,13,14]:

E0= m0c2 = -A t =

h

0= h0 , (12)

where index 0 refers to the simplest case of the state of rest, c2 is simply a coefficient,for the moment (to be properly specified later as the square of the velocity of light),-A = h (Planck's constant) is the quantum of action-complexity corresponding

physically to one cycle of the quantum beat (nonlinear reduction-extension of thecoupled protofields or one quantum jump of the corpuscular state, the virtual

soliton), t = 0 is the emerging quantum of time equal to one period, 0, of thesame cycle, and 0 1/0 is the corresponding quantum beat frequency, forming thebasis of the causal time concept (0 ~ 10

20 Hz for the electron). The obtained relation,m0c2 = h0, is heuristically introduced by de Broglie [1,5] as inevitable unification ofthe main (postulated) principles of relativity and quantum theory in the behaviour ofa physically real particle and makes the starting point for his causal derivation of the(matter) wave associated to a (moving) particle. Now this deceptively simple relationis causally substantiated by the above description of the holistic complex-dynamicalprocess within the elementary field-particle, eqs. (7)-(11), so that it is the formal


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postulates of the standard relativity and quantum mechanics that can be consistentlydeduced together, in their natural unity artificially disrupted by the canonical science,from the same causally complete picture of complex protofield interaction dynamics.As a result, such quantities as mechanical action A, its natural discrete portion(quantum) h, and the internal frequency of the particle (mobile) acquire quite anew, complex-dynamical sense containing the causally specified, dynamic meaning of

nonlinearity, instability, and physical protofield entanglement of the quantum beatprocess [11-14].Further advantages of this causally complete description of the quantum field

mechanics appear when we pass to the case of a moving field-particle. We can, first,objectively (and universally) define the states of rest and motion, using only the first

principles within the emergent concept of dynamic complexity that underlies anyreally occurring process. The state of restof an object is the state with the minimumpossible dynamic complexity (energy) of that object (this minimum always existsbecause complexity is a positively defined, finite quantity). It is not difficult tounderstand that the state of rest thus defined corresponds simply to the most uniform(totally uniform for the isolated field-particle) distribution of the system realisationprobabilities, which means that the quantum beat dynamics of the field-particle at restis represented by the totally irregular wandering of the virtual soliton within itswavefunction (any average, global tendency is absent). Any growth of the systemcomplexity-energy above this minimum corresponds to its transition to a state ofmotion, appearing as a finite average (global) tendency in the virtual solitonwandering (its preferred orientation within the thus emerging direction of motion),whereas each individual jump of the virtual soliton (i. e. system realisation change)always preserves its irreducibly probabilistic, unpredictable character. Therefore astate of motion is characterised by a certain order (increased inhomogeneity) in thedistribution of realisation probabilities that accounts eventually for all observed

structures of the world and is associated to the corresponding growth of complexity(in the form of its generalised kinetic energy representing entropian part ofcomplexity and increasing at the expense of the equal decrease of the informationalcomplexity form, or generalised potential energy [11], see also section 2.1).

The global order in virtual soliton wandering means that a state of motion ofthe field-particle appears as an observable spatial degree of freedom in its wave fieldoriented in its direction of (global) motion and obtained due to the correspondingincrease of its mass-energy-complexity over the minimum of the state of rest (eq.(12)), so that the simplest relation of eq. (12) is transformed, for the moving field-particle, into

E= mc2 = - A t +A

xt =h

+h

v = h+pv , (13)

where

E= -Atx=const =

h

= h , (14)

p =Ax t= const =

A

=h

, (15)


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v =xt

, (16)

(x)|t= const is the emerging quantum of space, an elementary directly measurable(regular) space inhomogeneity characterising the elementary quantum field withcomplexity-energy E (>E0) and resulting from its global (average) displacement

(motion), t = is the total period of nonlinear quantum beat of the field in thestate of motion with complexity-energy E(= 1/ is the corresponding quantum-beat frequency), x = is the total quantum of space, while = (t)|x = const is thequantum-beat oscillation period measured at a fixed space point (we thus obtain herethe naturally quantized and realistic versions of the canonical partial and totalderivatives and corresponding increments).

It is not difficult to understand that the elementary space inhomogeneity of thewave field of the moving field-particle, thus dynamically emerging in the underlyingprocess of quantum beat, is none other than the de Broglie wavelength of theparticle, = = h/p, which provides this rather ambiguous notion of the canonical

quantum mechanics with a causally complete interpretation directly completing itsmeaning introduced by Louis de Broglie in his thesis [5], but then superficiallyneglected by the conventional, postulated science. The global motion tendency of thefield-particle corresponding to its increased energy-complexity and described by thesecond summand in eqs. (13) appears as a real, corrugated (or undulatory)structure in the physically real wave field within the e/m protofield always coupled toits (hidden) gravitational partner. The spatial period of undulation equals to , butdespite its canonical presentation in the form of a linear, plane wave, this undulatorystructure of the wave field has essentially nonlinear, complex-dynamical nature, justproviding unified causal explanation for the associated mysterious peculiarities ofquantum (and in reality complex-dynamical) behaviour. This global wave fieldcorrugation with the period of has an internally dynamic, changing, but at thesame time fixed, character of a standing-propagating, nonlinear wave which isindeed fixed in its nodes, but permanently makes average, quantized displacements tothe distance of (in reality, it is always the virtual soliton that produces theundulation and performs the average displacement/motion superimposed on thebackground of its totally irregular wandering described by the first summand in eqs.(13)). In general, the emergence of the undulatory space inhomogeneity in the wavefield of the moving field-particle can be compared to appearance of waves at theformally homogeneous sea surface under the (uniform) influence of wind, where thegravitational (proto)field influence on the moving matter plays an essential role inboth cases, but in the case of elementary field-particle all the effects and motions arenecessarily quantized into indivisible quantum beat cycles, each of them correspondingto the increment of -h of the (extended) action-complexity.

In addition to the obtained causally complete interpretation of the matter waveand the related wavefunction, the quantum beat dynamics provides the causalinterpretation ofrelativistic effects, considerably extending their formal, mechanisticinterpretation within the canonical postulates and artificially imposed principles.Thus, the above intrinsic combination of irregular and global (regular) tendencies


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within the same causally probabilistic process provides physically completeexplanation for the universal equivalence between mass and energy, as well as causalderivation of the relativistic dispersion relation [13,14],

p =Ev

c2= mv , (17)

where mE/c2, now by rigorously substantiated definition. The physical sense of thisdeceptively simple relation is that it establishes the connection between the twotendencies within the same field-particle dynamics (regular global motion andirregular deviations from it): the explicitly observed global (regular) tendency,represented by the quantity of momentum (p), constitutes only a part (the relativeproportion of= v/c) of the whole dynamics represented by the total energy, E. Thiscausal substantiation of the familiar (but always postulated) relation of eq. (17) hasmany important consequences. In particular, Newton's laws, now in their causallycomplete (and relativistically extended) form, can be obtained from eq. (17) by takingits time derivative. Substitution of eq. (17) into expression for of eq. (15) gives the

detailed expression for the de Broglie wavelength,

B =h

mv. (18)

which coincides with the canonical expression, eq. (1), but implicitly includes thenontrivial complex dynamics of quantum beat providing the causally completesolution of all contradictions existing around this relation. Thus, one can avoid usingany fictitious wave of phase moving with the (variable) superluminal velocity andgiving the actual particle velocity v only as the group velocity [5], which contradicts,in particular, to the nonlinear character of the causal wave mechanics. The physical

picture of complex dynamics behind the dispersion relation of eq. (17) shows that inreality the part of the total energy leading to the superluminal motion of the regularmatter wave in the original interpretation is dispersed in the hidden, but irreducible,irregular wandering of the virtual soliton around the observed regular wave, so thatthe remaining part just gives the real, internally nonlinear matter wave propagatingwith the particle velocity v (though in the quantized, quasi-standing mode describedabove). This interpretation demonstrates again that the true dynamical chaos, based onthe dynamic redundance phenomenon, forms just the necessary missing link for theoriginal de Broglie's theory and perfectly completes it to the totally consistent andadequate description of quantum (and relativistic) behaviour.

Using the dispersion relation of eq. (17) in the complex-dynamical partition ofthe total energy, eq. (13), we obtain the relation between the different time measures,and , of the same quantum beat period:

=

1- v 2

c2. (19)

This expression can be further transformed to the more convenient form if we use anadditional relation between the participating time periods which also directly followsfrom the underlying picture of complex dynamics [11,13,14]:

8/2/2019 75 Years of Matter Wave: Louis de Broglie and Renaissance of the Causally Complete

75 years of matter wave: louis de broglie and renaissance of the causally complete knowledge

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