THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS

Simon Diner

Institut de Biologie Physico-Chimique 13 rue P. et M. Curie 75005 PARIS

Der Begriff eines Noumenon ist also nicht der Begriff eines Objekts, sondern die un­vermeidlich mit der Einschrankung unserer Sinnlichkeit zusammenhangende Aufgabe, ob es nicht von jener ihrer Anschauung ganz entbundene Gegenstande geben mage.

E. KANT Kritik der reinen Vernunft

QUANTUM THEORY : A UNITARY THEORY FOR LIGHT AND MATTER

De Broglie's fundamental idea was to extend to particles the laws of the quantum theory of light. It was the idea of a Unitary Theo­ry, "une grande loi de la Nature" as he said himself.

Such a Unitary Theory is a general formulation of the energy and momentum concepts, independently of the physical system (light or matter) .

This requirement of unity seems a guarantee for success, opening the way to the concept of wave properties of matter, asserting its effectiveness in the general concept of quantization. This lead in return to the developement of quantum electrodynamics (and quantum field theory) along formal rules identical to those of quantum mechanics.

The great conceptions of Nature are those which are free from spe­cifical properties of objects, at the price of an abstract formu-

215

S. Dineretal. (ed,.). The Wfllle-ParticleDualism. 215-229. e 1984 by D. Reidel Publishing Company.

216 S.DlNER

lation which offers dreadful problems of inter~reta!ion. This is the case for the fundamental law of dynamics (F = my), for the

+ mm' law of gravitation (F = g ~), for the law of conservation of

energy, for the second principle of thermodynamics.

This is also the case for the renormalization techniques and the associated universal properties in the theory of critical and chaotic phenomena.

Such a unitary strategy is still at work with success in quantum field theory where for example a conception as supersymmetry allows to handle bosons and fermions together.

Quantum Theory discloses such universal principles which are at first sight unexpected.

The Kirchhoff theorem on the independance of the law of black-bo­dy radiation upon the material nature of the cavity, is some kind of "miracle" which has to be included in every theory, even if some explanation for it is not given. This is indeed the gateway through which Planck proceeded when he introduced abstract oscil­lators whose physical nature was left undefined. These oscillators are in some way present in Quantum Mechanics which appears in many circumstances as a general model of abs­tract harmonic oscillator. A striking example being given by the universality of the response theory of quantum systems : there is a unique general form for the fluctuation-dissipation theorem. This simplicity of the quantum case is in sharp contrast with the high specificities of the classical case. The universality of the wave-particle duality is another universal abstract principle. It works surprisingly in physical systems as different as electrons or neutrons, helium atoms or hydrogen mo­lecules, which exhibit typical wave phenomena. This universality is a challenge to the fundamental specificities of physical systems. As such it must be the manifestation of a kind of universal reality. The nature of this reality has to be discovered.

ACCESSIBILITY OF THE VARIOUS LEVELS OF REALITY

The debate upon the possibility and conditions of knowledge is as old as Science and Philosophy. The dispute over "occult" quali­ties, characterized as insensible, as opposed to "manifest" qua­lities, goes back to Aristotle and the Christian Aristotelianism, and has been a major epistemological impasse not surmounted until the seventeenth century. At that time occult qualities became fully and consciously accepted in natural philosophy, just as it became recognized that no qualities were ever directly perceived.

THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS 217

This evolution is a major component of the Scientific Revolution and has culminated in Kant's conception of the noumenon and the phenomenon. A new aspect of this debate, typical of contemporary Science, is the systematic inclusion of limitations to knowledge in the scien­tific doctrines themselves. One has to distinguish between expe­rimentally accessible and non accessible elements of reality. A philosophy expressed in the "weak anthropic principle" : "What we can expect to observe must be restricted by the conditions neces­sary for our presence as observers". The Carnot theorem, the second principle of thermodynamics and the conception of available energy where the first clear manifes­tations of such an operational limitation to knowledge. In the 20th century such manifestations appear everywhere : Special Relativity is built on a limitation of the speed of light, Quan­tum Mechanics is built on limitations upon certain types of mea­surement. A new type of limitation appears with Godel's theorem, where it is shown how there exist properties which are non-prova­ble in the framework of certain formal systems. To prove that certain questions are unsolvable has become a mood, even a fashion. The theory of algorithmic complexity provides a precise defini­tion of randomness (or chaos) through limitations in computabi­lity [1]. Theoretical physics describes many elements of reality which ex­perimental physics is unable to observe, and in many cases one witnesses the appearance of a theory of the non-observability. How many phantoms of that kin d ? Magnetic monopole, advanced action, gravity waves, quarks, black holes, multiple worlds .•. Myths or reality? What is reality? To the naIve conception of physics as an experimental science is opposed a time long neo-platonic tradition which accompanies to day a true metaphysics : the physics of the inaccessible. But while neo-platonism invaded physics by following the use (and abuse) of formalistic mathematics (represented at best by the Bourbaki group), among mathematics itself there is a renewal of empiricism (constructive mathematics, non standard analysis) which is also manifest in the concerns of information theory and compu­ting theory [2].

VACUUM : AN EVERYWHERE PRESENT INACCESSIBLE REALITY

The conception of the world which was shaped in the XVIIth centu­ry, in conclusion to the Renaissance period, is being deeply mo­dified nowadays. The rising modern atomism of Boyle and Gassendi was linked with a precise conception of the vacuum, supported by the experimental work of Pascal and Torricelli. The vacuum was considered as a

218 S. DINER

perfectly empty region with no matter and no shape at all. Such a conception of the vacuum, faithful to Democrite but opposed to the ideas of Aristotle or Leibniz, is naturally linked to a conception of infinite space and in someway of continuum. The con­cept of chance is rooted in such an interpretation of the vacuum. Chance as emptiness, as absence of determinations and causes. The MechanicaL PhiLosophy is a subtLe intertwining of Atomism, Empty Space and Chance. The contemporary revoLution comes from the fact that the discove­ry of chaotic (or stochastic) motion destroys the traditionaL an­tagonism between determinism and chance [3J and that atomism onLy survives if one introduces a new conception of the vacuum. A va­cuum not empty at all. A new kind of aether. As soon as relativity theory discarded the idea of an elastic mo­tionless ether, the vacuum was populated again. The quantum vacuum is no mope a medium but a state of a quantized system. It is the state of Lowest energy of a quantum fieLd. Due to the interaction between quantum fields, a unique Vacuum must exist, as the state of lowest energy of the Universe. Energy appears as one of the most fundamental attribute of reality. As such it is submitted to precise limitations. All energy is not dynamically available according to the s,econd law of thermodynamics. The third law of thermodynamics (Nernst principle) says that the state corresponding to a zero temperature cannot be reached. At that temperature matter is annihilated. So that there appears an energetic reality in a situation where no more "normal matter" exists. There is some reality beyond matter. This is the vacuum, an inaccessible reality, which nevertheless manifests itself in the behaviour of matter. The roots of the quantum behaviour seem to be in the very existence of the vacuum. The relationship between zero-temperature, vacuum and quanta appeared first in the work of Planck in 1911 and 1912. That Planck in his second theory of black-body radiation obtains the exact formula with the quantum term Yz~~ must be looked upon as purely coincidental [4J. But one must think much more about Planck's work on the Nernst theorem in 1912 [5J. He gives the new formulation showing the decrease of entropy to zero with tempera­ture, and establishes the link between this theorem and the Planck constant~. This constant determines the size of cells in phase space. Planck says: "the entropy of a state has a quite definite, posi­tive value, which, as a minimum becomes zero, while in contrast therewith the entropy may, according to classical thermodynamics, decrease without limit to minus infinity. For the present, I would consider this proposition as the very quintessence of the hypo­thesis of quanta". But the champion of the zero point energy has been Nernst, who considered it has an "ordered energy" of the radiation field or of the ether. A champion even not challenged by Einstein and Stern

TIlE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS 219

who disagree with themselves at the Congres Solvay of 1913, con­sidering that the arguments they have given for the existence of a zero point energy are not right [6]. In fact Stern was not really convinced of that, as he testifies in a letter written to C.P. Enz in 1960 [7]

"Pauli and I have continually discussed the question of the zero-point energy in Hamburg in the early twenties. We always went together from the Institute to the house for lunch, and thereby there were two problems which we talked over time and again ... I for my part always tried to convert Pauli to the zero-point energy against which he had the gravest hesitations".

Pauli was never convinced, and wrote in Wellenmechanik in 1933 [8]: "At this point it should be noted immediately that it is more consistent here, in contrast to the material oscillator, not to introduce a zero point energy of ~~~ per degree of freedom. For, on the one hand, the latter would give rise to an infini­tely large energy per unit volume due to the infinite number of degrees of freedom, on the other hand, it would be princi­pally unobservable since nor can it be emitted, absorbed or scattered and hence, cannot be contained within walls and, as is evident from experience neither does it produce any gravi­tational field".

Nevertheless in 1962, Stern wrote : "Wave mechanics is not only compatible with the third law of thermodynamics but I believe that it should be possible under quite general assumptions, to derive the content and formalism of wave mechanics with the help of the third law" [9]. "Electromagnetic vacuum" is an electromagnetic field without real photons. The requirements of a unitary theory, where the field becomes the fundamental physical reality and the particles are just excita­tions of the field, leads to associate a vacuum field to any mi­crophysical particle. The "particle vacuum" supports in some way the particle. It crea­tes the particle, as a singularity. But, as any singularity, the particle organizes the "mother field". One must without doubt consider the existence of a wave in the vacuum, resulting from the action of the moving particle on the supporting vacuum. The particle appears as a "dressed particle" or "quasiparticle", re­membering the properties of the inaccessible underlying vacuum. In absence of any measurement for localizing the particle, it is lost in the vacuum. The measurement breaks the umbilical cord between the particle and the vacuum. It gives a reality to a po­tentiality. It creates the observable reality. This idea of a quantum vacuum, out of reach aether, has been reasserted by Dirac in his famous small paper in Nature in 1951 [10] :

"We can now see that we may very well have an aether, subject to quantum mechanics and conforming to relativity, provided we are willing to consider the perfect vacuum as an idealized

220 s. DINER

state, not attainable in practice. From the experimental point of view there does not seem to be any objection to this. We must make some profound alterations in our theoretical ideas of the vacuum".

In fact Dirac modifies drastically the relationship between vacuum and chance, because he remarks that a complex random medium is not suitable as a reference frame, the same for an absolute vacuum. The non existence of an absolute reference frame implied by the Relativity theory leads either to an absolute vacuum or to a chao­tic vacuum. In 1920 Einstein remarked that from the point of view of General Relativity empty space must have physical properties, so that an aether must exist, but the concept of motion cannot be applied to it [11]. A chaotic aether fulfills such a requirement. Recovering the ancient myths, contemporary science is lead to con­sider that Vacuum is Chaos. A modern version of the Tohu-Bohu. But the microphysical chaos must be described by the quantum for­malism which is, as we shall see later, a theory of some category of chaotic behaviour. In that way the chaotic vacuum must be des­cpibed as a quantum state, a state of a quantum fieLd.

AN ELECTROMAGNETIC FIELD IN THE VACUUM. VIRTUAL PHOTONS.

According to classical electrodynamics systems of charged parti­cules emit radiation in a continuous way. But it is not logical to consider radiation from material systems and not to consider the presence of radiation in the space. One is lead to suppose that a universal electromagnetic field is present in space even in region where there is no matter or detectable field, even in the Vacuum. But this electromagnetic field of the vacuum cannot be detected directly, due to the impossibility of a permanent transfer of energy from the vacuum. In quantum electrodynamics this vacuum field is the lowest state of the quantized electromagnetic field. It must be a random field. The direct inaccessibility of this field corresponds to the ab­sence of actual associated photons. The photon is experimental evidence of the presence of the electromagnetic field. But if one considers this vacuum field as an actual electromagne­tic field, one must describe it as any other electromagnetic field, on a quantum basis, introducing the conception of photon, but in the present case a virtual photon. To speak of a virtual photon means that the associated field is not detectable, or that one cannot transfer energy from that field to a material detector in a permanent way. The concept of a viptuaL paPticLe extends the wave-papticLe dua­Lity to the quantum vacuum itseLf.

THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS

THE FAILURE OF A CLASSICAL STOCHASTICAL PICTURE OF THE VACUUM STOCHASTIC ELECTRODYNAMICS.

221

It has been thought very often that one can do without a quantum model of the electromagnetic field. This is the conception of all "semi-classical theories of radiation-matter interaction" where matter is described by the quantum theory whereas radiation is a pure classical electromagnetic field. It seems largely proved to day that such theories obtain only limited successes and that electromagnetic radiation has actually non classical properties, mainly as statistical properties are involved [12J[13J. Stochastic Electrodynamics belongs to such attempt of using a classical picture of the electromagnetic field. It describes the motion of classical charged particles, obeying the laws of clas­sical electrodynamics (Abraham-Lorentz-Dirac theory), subjected to the effect of a fluctuating classical electromagnetic field in the vacuum. The concepts and the results of Stochastic Electrody­namics (S.E.D.) have been largely reviewed elsewhere, so that we need only here recall and comment briefly the failure of S.E.D. [14 J. S.E.D. consider a stochastic electromagnetic field in the vacuum with spectral statiscal properties corresponding to the ("quantum mechanical") second Planck law·for black-body radiation. The energy density is given by

PE(w) = 1T~:3 [~~W + '¥1!W ~ e kT _ 1

where the first term is the zero-point energy, the only remaining term at T = O. Spectral density, and correlation function for the stochastic field are easily deduced according to :

4~2 c8E (w)

roo e -iwaE (e) de -OO

~E(e) = <E(t)E(t + e»

This zero point radiation spectrum is the unique spectrum of sto­chastic electromagnetic radiation which is Lorentz invariant. This invariance makes the field in the vacuum not detectable (by a Milchelson and Morley experiment for example). Let us remark that using a spectral density deduced from Planck law does not mean introduction of any quantization. We remain in a strictly classical Kolmogorovian probability calculus, where it is always possible to build stochastic processes reproducing any probability law (even of quantum origin) [15J.

222 S.DINER

Recall that in quantum mechanics the zero point energy is a con­sequence of the commutation relations. The fundamental equation of S.E.D. for a charged particle (charge e, mass m) in an external force field F is the Braffort-Marshall equation :

m; F(;) + mw; + e E(t)

2e 2

where T - 3mc 3 •

One eliminates the runamway solutions by using an integro diffe­rential form

+ mr = + e foo E(t+Ts)e-sds

L2-E(T)(t)~

;t;h) where ~ (t) is a new stochastic field, with a spectral density

~ (T) =<j (w)/l + T2W2 E E

A simple development gives

~ + + ~ ~ +(T) mr = F(r) + T(v~)r + e E (t)

+ + •

Intead of the mechanical moment p mr, which is a too irregular stochastic process, one uses the more regular elektro kinetic mo­mentum

to

P = +ek f

give +ek

dp dt

+

a phase space representation

m ~~ pek + T F(;) wherelh ) (t) is a new spectral density

~cri.T) =c:jE(w)/w 2 (1 + T2W 2).

The ultra-violet catastrophy has disappeared. Among the successes of S.E.D. one can list the following problems: harmonic oscillator, black body radiation law (Planck law, "photon" term in the fluctuations of the electromagnetic field), absorption of light by the plane and the spatial rigid dipolar rotator, Lamb shift, spontaneous emission, Vand der Waals forces. Some of these successes are superficial : if one uses non linear oscillators as model of the black body cavity one gets the Rayleigh Jeans law instead of Planck's, the Kirchoff law is not satisfied, absorption

THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS 223

curves for the rotator are too broad. These are forewarning signs of deeper failures. The three major failures of S.E.D. are :

1) No waves appear associated to a partic~e. No wave properties are to be deduced from the fundamental equations. Schrodinger equation is certain~y not deducib~e from the Braffort MaPsha~~ equation.

2) The absorption of ~ight gives rise to broad absorption bands. There is nothing like narrow line spectra.

3) There are major discrepancies between S.E.D. and Quantum Mechanics for non ~inear systems.

The study of non linear systems is made possible through an appro­ximation of markovian type. This is due to the presence in the equations of the very small factor T ~ 1/c 3 . Using the Kharsminski­Lax procedure one is able in principle to write a Fokker-Planck­Kolmogorov (FPK) equation for a markov process which is a very good approximation of the true solution. For the hydrogen atom (Kepler problem) where the external force field acting on the electron is defined by a potentiel V = - K/r, the F.P.K. equation can be reduced to only two variables choosen among the constants of motion. The probability density W is a function of these two variables. If we choose the energy E and the total angular momentum M as variables, the reduced FPK equa­tion reads :

81T2MT :~ = diV[WC ... ~ grad W}

where T, the orbital period, equal 2MK~2;13 and the drift vector

C and the diffusion matrix ~ are

C = leeMEI IeEE (C= eME

where

e E 161T3TmK2{~ M2 + ~ mK21

2 M

eM 161T 3mK 2 !. M

e EE 161T 31't1M I;~ 14 4>3 (E)

e MM 161T3T~M(2E) 4>1(E) 5/2

e ME e EM = 161T3~M ~ (mK2 )~ 4>2(E)

and 4>1(£)' 4>2(£)' 4>3(£) are Kapteyn series expressed in term of Bessel functions, where £ is the eccentricity

/1 2EM2 £ = + mK2

224 S. DINER

As Claverie, Pesquera and Soto have shown [16] a stationary solu­tion Wo = Constant is possible because the stationarity condition is

diV(W oC +(Cgrad W ~ = 0

and it is easy to show that div C = O. They don't know if there is any other stationary solution but if any, it must be non recurrent, which implies non ergodic. In any case this is in deep disagreement with the experimental stability and symetry of the hydrogen atom, both properties repro­duced by quantum mechanics. The origin of this instability is due to the damping force (Lorentz force) on which the drift coefficients depend exclusively. They do not depend on the stochastic force exerted by the vacuum. This is due mainly to the fact that we have not introduced any depen­dency of the stochastic force on the point of the phase space. The stochastic force is not position or velocity dependant. That means that having negLected any back reaction of the charged particLe on the vacuum, we are not abLe to prevent a continuous transfer of energy from the vacuum to the particLe. This is in principLe strictLy forbidden. Hence the failure of the model. We see that it must be necessary to study this problem further to introduce some kind of non Linear interaction between the particLe and the vacuum. According to modern conceptions this could open the road to wave phenomena in the vacuum (not directly detectable) associated to an intrinsic stochastic behaviour of the particle. But one has for the time being no physical arguments to write down this non linear interaction.

QUANTIZATION AND WAVE-PARTICLE DUALISM AS A MANIFESTATION OF SI­MULTANEOUS APPEARANCE OF ORDER AND CHAOS

Quantization is a mathematical procedure for defining a model satisfying some fundamental requirements of "physical observables". One thinks usually that the main virtue of this model is to allow a transition from a continuum to a discrete physics. Just look at the broad use of the word quantization instead of discretization. The initial purpose of quantization for classical electrodynamical systems was to provide a model with two fundamental properties:

1) stability of atomic and molecular systems 2) discrete properties of spectral responses (line spectra for

absorption and emission of light). In both cases one faces strong manifestations of Order. That a concept of probability, randomness, Chaos, slips into the quanti­zed model is considered as an unavoidable parasitic phenomenon. Probability looks as a by product of quantization, which is not clearly accounted for. One has really to make the best of it, all the more so since those are strange probabilities, not ruled by

THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS 225

the classical Kolmogorovian calculus of probability [17]. The contemporary progress in the field of classical dynamical systems on the mechanism of generation of randomness [3] sets a new intellectual demand for the justification of chaotic phenome­na. These phenomena appear in precise conditions as complex dyna­mical behaviour induced by some kind of interaction (in principle nonlinear interactions). It is no more possible just to say that quantum systems show an "irreducible randomness". One day or ano­ther it will be unavoidable to give the reasons for the appearance of the "microphysical chaos". In that spirit it seems essential to considep quantization as the asseption of the vepy existence of a micpophysical chaos, and of the ppobabilistic stpategy which gives a pegulapized descpiption of this chaos. Quantization is above all a way of mathematical correspondence between a classical non chaotic (or chaotic) dynamical system and some new kind of chaotic system. In any case a new kind of chaos appears. But in this chaos it appears simultaneously some kind of order, of organization, of auto-organization. Order and chaos are interwoven. Quantum phenomena set the ppoblem of "stpuctupe of chaos" [18]. Historically the wave-particle duality is born in such a problematic; it expresses a structure of the statistical properties of the electromagnetic field in black-body radiation. The quantum formalism is well adapted to this problematic intro­ducing both order and chaos through the mathematical properties of operators. For instance the discrete spectrum of operators expresses organization while eigenfunctions give a probabilistic measure (regularized measure) of the underlying chaos. The wave-particle structure of fluctuations arises from the non­commutation of operators, as the appearance of the zero-point energy in the harmonic oscillator [12]. One is forced to evoke those phenomena where order and chaos co­exist, as turbulence or more generally auto-organisation phenome­na. More precisely auto-organisation in turbulence is produced by non linear equations but the auto-organized state is described by equations which can be linear. This remembers also of the situa­tion for the soliton where the original non-linear equation can be solved through linear equations. From this point of view the wave function is just a mathematical device. ~s for de Broglie wave, the wave of the wave-particle dualism, it LS a peculiar wave, both order and chaos. A random wave? Such the same for the corpuscle which is at the same time an expression )f the statistical properties of the field (the wave) and of its )rganiZed properties (exchange of energy through quanta). 7haos is opganized and opganization is not just opdep. A duality )hich gives pise to Dualism.

226 S. DINER

NON KOLMOGOROVIAN CALCULUS OF PROBABILITY AS A REGULARIZED DES­CRIPTION OF THE MICROPHYSICAL CHAOS

We have already seen that quantum theory must be considered as a probability calculus which gives a regularized description of chaotic phenomena called "the microphysical chaos". It has been often stressed that the quantum probability calculus is not all equivalent to the classical Kolmogorovian probability calculus. In fact the latter considers implicitely a restricted class of chaotic phenomena, where all the possible observables are endowed with regularized properties (existence of measures). That chaotic phenomena exists which are not giving rise to Kolmogorov calculus, is clear from the theory of pseudo-random functions due to J. Bass [19]. In that theory it is possible to show the exis­tence of chaotic phenomena which though happening simultaneously and being individually described by asymptotic measures, do not allow any regularization for their simple product. Just the same situation as in quantum mechanics. We think that requiring a quantum description is fundamentally requiring the use of a non Kolmogorovian probability calculus. Quantum physics is the manifestation of a microphysical chaos which must be des­cribed probabilistically in a different way than a billiard or a game of dice. The actual nature of this chaos is completely unknown.

IS MICROPHYSICAL CHAOS BROUGHT ABOUT BY THE OBSERVATION ?

There is a wide agreement to say that Quantum Mechanics does not describe a system in itself but only deals with the results on actual observation on it. Quantum mechanics is a probability cal­culus for the results of measurement on microphysical systems. These affirmations imply in fact that observation plays an active role in the appearance of results. G. Lochak writes for example: "C'est pourquoi Piron (28.0 livraison 19. Lettres epistemologiques de l'Association Gonseth) a parfaitement raison de dire "C'est la mesure qui, perturbant Ie systeme, cree les valeurs trouvees". Cette phrase n'a aucun rapport avec Ie subjectivisme, l'indeter­minisme, la non objectivite des lois de la nature, ni avec d'au­tres problemes philosophiques. Elle ne fait que resumer un etat de chose experimental qui provient de ce que Ie seul procede de mesure que no us connaissions en microphysique consiste a etudier la reponse d'un systeme a un champ perturbateur et que cette re­ponse est extremement compliquee en raison des proprietes ondula­to ires de la matiere" [20]. In fact one can ask if the perturbation introduced by the measu­rement is not responsible for the appearance of the microphysical chaos (or a part of it). Such a concept of chaotic phenomena ge­nerated by perturbation of non-linear systems is becoming a current concept for classical dynamical systems and plays some role in the problem of "quantum chaos" [21][22].

THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS 227

TEN THESIS AS CONCLUSION

1_ There is a Vacuum, inaccessible by direct experiments.

2. Vacuum is chaotic.

3. There is a great diversity of chaotic phenomena.

4. Ko1mogorov probability calculus is not the only possible regu­larized description of chaotic phenomena.

5. Quantum mechanics is a non-kolmogorovian probability calculus.

6. All "microphysical chaos" including phenomena in the vacuum must be described by the quantum formalism on a probabilistic level.

7. Wave-particle dualism is a manifestation of an interplay bet­ween order and chaos in a physical system.

8. The actual physical nature of the microphysical chaos is unknown.

9. The observation could playa role in the generation of the mi­crophysical chaos.

10. Be patient ! Some Order will appear in all that Chaos.

REFERENCES

(1) The best accounts of this subject are : Zwonkin, A.K., Levin, L.A., Usp. Mat. Naouk 25,85, 1970 Dellacherie, C., Gazette des mathematiciens nOll, 1978 Alekseev, V.M., Yakobson, M.V., Phys. Reports, 75, 287, 1981 Popular accounts are given in Ford, J., Physics Today, 40, April 1983

(2) See for example the special issues of International J. of Theor. Physics (1982) devoted to the proceedings of a confe­rence on "Physics of Computation"

(3) Berry, M.V., in Jorna, S., ed. Topics in non linear dynamics Am. Inst. of Physics; 1978 HeIleman, R.H.G., in Cohen, E.G.D., ed. Fundamental problems in statistical mechanics vol.5, North-Holland 1980

(4) Planck, M., Verhandl. Dtsch. phys. Ges. 13, 138, 1911

228 S.DINER

(5) Planck, M., Uber neuere thermodynamische Theorien (Nernstsches Warme theorem und Quantenhypothese) Leipzig 1912, Ber. Deutsch. Chern. Ges. 45, 5, 1912. Phys. Z. 13, 165, 1912

(6) Einstein, A., Stern, 0., Ann. Phys. 40, 551, 1913

(7) Enz, C.P., in Enz, C.P., Mehra, J., eds Physical Reality and Mathematical description p. 125

(8) Pauli, W., "Die Allgemeinen Prinzipien der Wellenmechanik" in Handbuch der Physik vol. 24, part 1, 83, Springer 1933

(9) Stern, 0., Helv. Phys. Acta 367, 1962

(10) Dirac, P.A.M., Nature 168, 906, 1951

(11) Einstein, A., Ather und Relativitatstheorie, Julius Springer, Berlin 1920

(12) See the article by Milonni in this book

(13) Mandel, L., in Progress in Optics, Wolf, E., ed. vol.XIII p. 27, North-Holland 1976 Loudon, R., Rep. Progr. Phys. 43, 911, 1980

(14) Milonni, P.W., Physics Reports 25C, 1, 1976 Claverie, P., Diner, S., Int. J. Quant. Chern. 12, Supp. 1, 23, 1978 Boyer, T.H. in Barut, A.O., ed., Foundations of radiation theory and quantum electrodynamics, Plenum 1980, p. 49 Claverie, P., in Blaquiere, A., Fer, F., Marzollo, A., eds. Dynamical systems and microphysics, Springer 1980, p. 111

(15) For such a simulation of quantum mechanical probability dis­tributions, cf : Nelson, E., Dynamical theories of brownian motion, Princeton U.P. 1967 Erber, T., Rynne, T.M., Sklar, A., Acta Physica Austriaca 53, 145, 1981

(16) Claverie, P., Pesquera, L., Soto, F., Physics Letters 80A, 113, 1980

(17) See the article by Accardi in this book, p. 297.

(18) The problem of "structure of chaos" is the main recurrent thema of the self-organization phenomenon in chaotic or tur­bulent media. Cf. Haken, H., Synergetics, Springer, 1978

Hasegawa, A., Europhysics News, 1~, 1, 1982

THE WAVE-PARTICLE DUALITY AS AN INTERPLAY BETWEEN ORDER AND CHAOS 229

(19) Bass, J., J. of Math. analysis and applications 47, 354, 1974 Bass, J., Ann. Inst. H. Poincare, A, XXXIII, 301, 1980 Bass, J., in Diner, S., Fargue, D., Lochak, G., "LaPensee Physique Contemporaine" p. 391, A. Fresnel, 1982

(20) Lochak, G., Epistemological letters, Association F. Gonseth, 33 livr.

(21) Tomita, K., Physics Reports, 86, 113, 1982

(22) Zaslavsky, G.M., Physics Reports 80, 157, 1981.