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1. INTRODUCTION resentation), interation representation [27, 37, 38, 39, 40], AdS/CFT correspondence in non-linear quantum finance Econophysics is a relatively new interdisciplinary sci- [54] etc. are earning the spotlight. entific school, which tends to develop itself rapidly, hav- Among the authors, working purposefully and fruit- ing taken its shape and name in late 90-ies of the XX fully in the field of intersection of quantum physics and century [1]. According to our estimation the number of economics, we can mention Russian academician V.P. original works and articles on the Internet, surveys and Maslov ([20, 35, 36, 41, 42] and the literature quoted monographs has already exceeded thousands. Moreover there), researchers from distant foreign countries D. Sor- respective courses and special subjects are being intro- nette ([61] and the literature quoted there), B.E. Baaquie duced in the high schools of far and near abroad [1, 2, 3]. [14, 32], C. Pedro Goncalves [18, 43, 44, 45, 46, 47, 48, In Western countries young theoretical physicist, who 49], E. Guevara Hidalgo [21, 50, 51, 52, 53]. look for the application of their knowledge and abilities Although the first works, connected with the appli- not only in physical and technical fields, are employed by cation of quantum-mechanical models to economic phe- large corporations, banks, holding companies and other nomena appeared in the early 90-ies of the last century subjects of national and world financial and economical [17, 18, 55], it can be confidently contended that a new activity. scientific school in the socio-economic systems modeling In its classical part econophysics is working on the is being born. Not going beyond the emerged terminol- application of mathematical apparatus of statistical ogy, it will be the most logical to call this school – what physics, random systems physics and non-linear physi- by the way, most of the authors of the afore-mentioned cal dynamics included, to discover socio-economic phe- works, including the authors of this investigation, incline nomena, using one or another physical model and giving to do – quantum econophysics [18, 20, 21, 22]. the appropriate economical interpretation to physical no- We consider that the appearance of such a scientific tions, variables and parameters [1, 2, 3, 4, 5, 6, 7, 8]. direction is caused not only by search for the new ap- Though statistical physics can’t get along without plications of quantum mechanics mathematical appara- quantum-mechanical ideas and notions in its fundamen- tus and new quantum-mechanical analogies, but also tals, the main sphere of its interest is the macroscopic de- by the evidently shown problems of the socio-economic scription of systems with large number of particles, the modeling, which required deep conceptual analysis and dynamic behavior of which can’t be brought to micro- philosophical generalization, including probable change scopic dynamical equations of quantum mechanics fig- of the established mathematical [42] and economic [56] ured out for separate particles without use of respective paradigms. In the opinion of authors, the relativistic statistical postulates [9]. aspects in the conceptual fundamentals of the quantum During last years an increasing flow of works was ob- physics and philosophical reasoning of them, including served, in which detailed models of market process partic- critical analysis of measurement, state, memory, time and ipants interactions and quantum-mechanical analogies, space notions not only in physical, but also in psycolo- notions and terminology based on methods of describ- gycal and socio-economic contexts [56, 60] are gaining ing socio-economic systems are drawn to explain both great significance in the scope of the new quantum di- particular peculiarities of modern market dynamics and rection in econophysics [57, 58, 59]. The purpose of this economic functioning in whole ([10, 11, 12, 13, 14, 15, 16, work is the well-reasoned exposition of the totality of 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, the above-mentioned issues, which, as far as we can see, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, must be related to the competence of the special and 47, 48, 49, 50, 51, 52, 53] and quoted there literature). dedicated section of econophysics – relativistic quantum In spite of discord of names and key word combinations econophysics. – quantum economics [10],[11],[12], quantum finances [13],[14], quantum market games [15], quantum game theory [16],[17], quantum evolutionary game theory [18], 2. ABOUT ECONOPHYSICS, QUANTUM quantum economic theory [19], quantum econophysics ECONOPHYSICS AND COMPLEX SYSTEMS [20],[21],[22], quantum decision making [23, 24, 25, 26] etc., - the emphasis on using the mathematical appara- Econophysics, or physical economics, already men- tus, input equations and quantum-mechanical models is tioned as a relatively young scientific school, recently the common feature of all the above listed works. celebrated its tenth anniversary. Of course that doesn’t Schr¨odinger equation for the wave function [12], von mean that there were no works on the boundary of eco- Neumann equation for the density matrix [21], secondary nomics and physics before the econophysics was officially quantization for systems with variable number of parti- born, howewer the new direction is usually formed only cles [18, 27], the last modifications of which were called when the certain conditions appear and the necessity to the ultra-secondary and the ultra-tertiary quantization concentrate the scientific forces arises. Quantum econo- [10, 20], Ising spin model [28, 29, 30], Feinman path in- physics is not an exception. That is why, though the tegrals [31, 32, 33, 34] Bose condensation in quantium first work according to Gonsales [18], which can be re- liquids [35, 36], operator representation (Heisenberg rep- lated to the application of quantum mechanical ideas to 3 the economic phenomena, appeared in 1990 [55], we can years, the upper (socio-economic) levels of the matter or- speak about the birth of the new scientific direction called ganization are constantly getting more complex and de- econophysics only nowadays. velop in time, and in the last decades it happens beneath In short, quantum econophysics currently includes our eyes. As the time is irreversible – and this experimen- tal fact has not been disproved yet, – all the attempts to a) adaptation and usage of mathematical apparatus model or predict the behaviour of socio-economic or other of quantum mechanics in order to model processes complicated systems using straight “physical” methods in economics (linear operators in the Hilbert space, can be rather difficult due to the impossibility of the wave function and the Schr¨odinger equation, den- strict following one of the basic exact sciences principle sity matrix and the von Neumann equation, sec- – the principle of experiment and observation results re- ondary, ultra-secondary and ultra-tertiary quanti- producibility. zation apparatus, Feinman path integrals etc.); Of course, models of socio-economic systems should not contradict the physical and other processes of the b) application of quantum-mechanical models and lower level running in them; nevertheless not all pecu- analogies (the Ising spin glass model, evolutionary liarities of socio-economic systems can be derived from quantum game model, Bose condensation of quan- their physical qualities (known in the general system the- tum fluids etc.); ory emergent principle). Actually, this statement can be related to any pair from the model hierarchy existing on c) application of quantum mechanical ideology (the different levels and describing the world around us. uncertainty principle, the principle of complemen- The straight application of physical approaches and re- tarity, other elements of the quantum measure the- spective to them mathematical models in description of ory, probabilistic interpretation of the system dy- socio-economic systems is useful; though going beyond namics). the bounds of their applicability may lead to paradoxes already observed in the history of science. Mechanical But, in our opinion, complex analysis of the concep- determinism, based on the classic Newtonian mechanics; tual fundamentals of the modern theoretical physics, ba- heat death of the Universe, following from basic thermo- sic postulates of the systems theory and system analysis, dynamics postulates; persistent mathematics paradoxes, subject to results of the observations and investigations derived from the infinity notion etc. can be related to of the real socio-economic processes and systems, is of no the number of such paradoxes. less importance for progress in the correct statement and solving problems of mathematical modeling of complex systems. 3. THEORETICAL PHYSICS AS ONE OF THE In the contemporary comprehension complex systems REALITY MODELS AND MATHEMATICS AS are the problem in terms of formalization nonlinear sys- THE FORMALIZED LANGUAGE OF ITS tems, in the dynamics of which synergetic phenomena DESCRIPTION are observed, instabilities and poor predictability take place; the so-called aftereffect and “long memory” con- 21st century is the century of the triumph of the new nected with it act the significant part. First of all socio- theoretical physics – relativity theory and quantum me- economical, ecological and other, which are similar to chanics, which explained new phenomena, observed both them and depict the upper levels of an integrated, orga- in macro- and micro-world, as well as changed or filled nized and functioning in a complicated manner matter, well-established physical notions with the new sense. can be related to such systems. These notions were creating the basis of natural sciences, Using one or another physical analogy in complex sys- forming respective philosophic concepts and ideas in each tems modeling, or, as it is often shortly said, in model- and every science without exception, including the phi- ing the complexity, we must not forget that physics is losophy itself (the so-called metaphysical approach) for the experimental science first of all (in principle, as any ages. other science is). Each physical theory is based solely on Though new concepts became firmly established, first experimental facts, and its mathematical apparatus and of all, technologically, as a tool in physics, we consider respective mathematical model is just the tool, used to them to be not fully realized yet and used in modeling of describe the results of observations and/or experiments, socio-economic systems and processes running in them. which is always more or less approximate, and usually The reasons of it are hidden not only in the lack of suf- not the only one. ficient physical and mathematical models spectrum, but Models, describing physical processes, and models, also in the inertia, in the absence of the deeply integrated which claim to be adequately describing socio-economic analysis, concerning classical physics fundamentals, rela- processes, are on the essentially different and in some way tivity theory, quantum mechanics, theoretical and practi- opposite levels of hierarchy of models of the world around cal economics, as well as historical, psychological, social, us. If the physical picture of the world, at least in its fun- philosophical and other, strictly “humanitarian” aspects damental principles, does not change for about ten billion of the problem. 4

In connection with all afore-mentioned, the solution of axioms, in 1931 in his famous incompleteness theorem problems in mathematical modeling of complex systems [76, 77]; though the true meaning of the theorem, includ- must be sought on the intersection of various scientific ing the philosophical one, is getting fully appraised only schools, including not only mathematics, physics, cyber- now [78]. netics, computer science etc., but traditionally human- Though probability theory, as one of the chapters in itarian disciplines - philosophy, political science, sociol- mathematics, is developed to describe uncertainties, it ogy, psychology, linguistics and others as well - probably, also brings the problem definition to the formally deter- it will be effective to get the new ideas from them [61]. ministic state, bringing the notion of the probability of V.I.Vernadsky gave classical example of such complex ap- the event (a determined, strictly given number between proach to the problem of space and time that keeps un- zero and one) in, and gains substantial sense only for the til the present [62]. Synergetics [63], fractal theory [64], repeated number of phenomena. However not every un- chaos theory [65], econophysics [1], quantum informat- certainty, observed in real systems and processes, first of ics [66, 67, 68], neuroeconomics [69], p-adic mathemati- all socio-economic ones, can be described with the help cal physics [70, 71, 72] and others can be named as the of probability language. examples of new interdisciplinary directions. Quantum We should note that, in our opinion, both in method- econophysics, which is discussed in this work, can also ological and conceptual aspects, the language of discrete be related to such an interdisciplinary direction, which mathematics is gaining special meaning in description of we consider to have great perspectives [18, 20, 21, 22]. complex systems. Discrete mathematics is based on the On the one hand, development of physics and math- application of the algorithmic (discrete) models; it is con- ematics, appearance of electronic calculating machines structive in realization and gives an opportunity to get and, later, computers, which have performed an informa- rid of the number o philosophical paradoxes, which take tional revolution in all fields of human activity without place in the continuous (“infinite”) mathematics. exception, created the illusion of omnipotence of math- The famous Banach-Tarski paradox is a striking ex- ematics as the tool of description, modeling and solving ample of such a philosophical “deadlock” [79]. Accep- any tasks, connected with the intellectual activity. On tance of the so-called axiom of selection in the rigorous the other hand it revealed its shortcomings. set theory allows to split the sphere into the finite num- Let us take note that mathematics, as one of the lan- ber of parts so that it will be possible to make up two guages of reflection and description of the surrounding spheres, equivalent to the initial one. Non-acceptance of reality, substantially developed in the scope of exact sci- this axiom does not lead to contradictions [80], though it ences, first of all physics and its technical applications, considerably weakens investigation of the continuous ab- and only thereafter it was used to solve more “humani- stract structures in analysis, algebra, topology and other tarian” tasks. branches of the mathematics. But, as it has been already mentioned before, it is nec- essary to approach an application of physical analogies in modeling of the systems of “non-physical” origin, that 4. GENERAL SYSTEMS THEORY – occupy the highest (in complexity and time of the ap- LANGUAGE AND METHODOLOGY OF pearance) levels of models of the universe hierarchy with SOLVING HARDLY FORMALIZABLE care. PROBLEMS Mathematics is built on axioms, and one of its pecu- liarities is its determinacy, the “rigidity” of the language The creation of the general systems theory in 1951 by used. Unlike the usual language, mathematics bars from Bertalanffy [81, 82], which in modern interpretation in- explanations and contexts; both the strength and the cludes the systems theory itself, systems analysis as its narrow-mindedness come from it. methodology, and mathematical modeling as the techno- An economist, politician and thinker A.A. Bohdanov logical tool [83, 84, 85], was one of the attempts to go wrote about it in his ”organized science” (tectology) beyond the limits of the circle of tasks, being solved by [73, 74]. (Though as a thinker he treated philosophy the classical “accurate” mathematics. rather negatively). His ideas are starting to revive only The general systems theory, an integral part of which nowadays. includes mathematical tools, does not exist as the theory The writer of genius A.N. Tolstoy in his fairy-tale ”The in the strict mathematical understanding of this word. Golden Key, or the Adventures of Buratino” described We suppose it was that very peculiarity of the theory, the inadequacy, to say the least, of mathematical lan- Bertalanffy wanted to emphasize, adding the character- guage as the tool of reflection of the gorgeousness of sur- istic “general” to its definition and mentioning, that even rounding reality in the allegoric way (episodes of the ill- in bounds of the common classical mechanics mathemat- ness of Buratino, ”the patient is either alive or dead”, and ically unsolvable problems appear (three-body problem), of his method of solving the elementary sum, ”2-1=2”) not speaking about more complex systems and more “ad- [75]. vanced” models of modern theoretical physics. G¨odel was the first to prove the boundedness of math- The general systems theory can be considered as an ematics as the language, based on the closed system of empirical set of logically unprovable principles, concepts 5 and approaches, which are deduced from observations of theory of sets, for example Zermelo-Frenkel axiomatic, real complex systems, including those that function with this is one of the nine axioms [86, 87]), leads to physical the human participation, are common for objects of any paradoxes, and the systems theory discreteness principle, nature and appear to be useful when conducting obser- which is being realized and logically developed in discrete vations, investigation, and, above all, when solving prac- mathematics and theory of algorithms, is the most rea- tical tasks. sonable alternative to continuity and continuous mathe- There are various definitions of system. As one of matics, based on it. Likely, continuity isn’t the necessary the possible working and rather “integrated” definitions, link neither in the physical nor in the mathematical de- which take into account ontological, gnosiological and dy- scription of the reality [88, 89, 90, 91, 92]. namical aspects of the “system” notion, we can use the The continuity of the basic physical quantities, includ- following one. ing those of spatial coordinates and time – is merely a System is the totality of the interacting elements, into hypothesis and is likely to be an approximation, which which the subject divides the object according to some is not always appropriate for the tasks of representa- rules, in order to observe, describe, examine and, in the tion the world around us; therefore within the bounds end, solve one or another practical task, meanwhile the of the systematic approach, when realizing its principles interaction of the elements in the system when function- sequentially and to the end, these quantities must be also ing causes the new quality, which is not peculiar to the considered as discrete. (It should be mentioned that the separate element of the system. question of discontinuity and continuity of our time and This definition could be considered as the free “inte- space in physics is still controversial.) grated” interpretation of definitions [84, 85], although we The openness of any system is in certain sense the con- should mention, that considering the essence of the gen- sequence of its hierarchy principle, and the actually ob- eral theory, its statements and initial definitions must be served presence of memory (aftereffect) and the registra- neither only, nor “strict”, nor closed, as it in itself is one tion of time as one of the system-forming factors makes of the systems, and its own principles are applicable to it formally open even when from the very beginning of it. functioning the system is physically isolated. In the lat- To the most important principles and statements of the ter case the openness is imparted to the system by its general systems theory, which determine the gist of the history, the full description and registration of which are so-called systems approach the following must be related: just impossible (setting the history of the system as the totality of initial conditions – as it is done in the classical a) discreteness; physics – is a quite narrow and approximate way of its b) hierarchy; registration) c) emergence; 5. HIERARCHY OF CONCEPTIONS AND d) openness. MODELS IN MODERN THEORETICAL In spite of the absence of direct links between gen- PHYSICS eral systems theory, which is difficult to formalize, and modern theoretical physics, based on the usage of rather As it has been already mentioned, theoretical physics abstract mathematical models, both of them, being dif- of the last century fundamentally changed the view on ferent experimentally grounded ways to reflect the real the notions of time and space, measuring procedures and and only world, have deep and common roots. the achievable accuracy of the results, on the notion of First of all we should note that from the definition the predictability of system’s behaviour; it also put a of the system and fundamental statements of the gen- question of the time irreversibility problems, paid atten- eral systems theory follows that within the bounds of tion to presence of the aftereffect (memory) in real phys- the systematic approach the question about the objec- ical processes. tive, i.e. non-depending on the subject, existence of the One of the most important problems, which should world around us is insensible. Of course the world exists be related to the quantum econophysics’ competence, is regardless of us, but its description or reflection is sub- in tracing the influence, which was or will be exerted jective, and the “subject –object” couple is in compliance by these changes on problem statement in mathematical with system principles the new system, the properties of modeling of the socio-economic processes and interpreta- which under the emergence principle cannot boil down tion of its results. neither to the object’s properties, nor to the subject’s Instrumental approach to physics as to the means of ones taken apart. (In quantum mechanics such a philo- prediction of the results of the experiments prepared in sophical problem of a systematic nature appears when certain way is working perfectly in the physics itself, nev- analyzing the measuring procedure in the couple “gauge ertheless the transfer of its notions and mathematical ap- – measured object”.) paratus on systems of the other nature requires obliga- Continuity, based on the hypothesis of the existence of tory and in-depth analysis of its initial conceptions. infinity, which is unprovable in its essence (in the rigorous We should note that in modern understanding theo- 6 retical physics is the hierarchy of models of the physical (force f~12, exerted by the particle 1 on the other material qualities of the substance, starting with classical New- particle 2, is of the same magnitude and acts as the op- tonian mechanics and ending with the general relativity posite to the force f~21 direction, exerted by the particle theory and modern parts of relativistic quantum micro- 2 on the particle 1). and macro- (cosm-) theory, each of them having its own Differential and integral calculi serve as the mathemat- special postulates and own domains of applicability. In ical apparatus for solving the problems of classical me- this regard Newton’s laws are as much fundamental as chanics, time appears to be the independent variable, and the quark or superstrings theory, and those connections, system state is characterized by coordinates and veloci- which exist between the more and less general theories, ties of its material particles in Euclidean space, system as often as not are similar to the temporary “bridges”, dynamics is described by differential equations. functioning as the “scaffolding” on the theory develop- In modern physics instead of Newton’s equations are ment phase, the rigorous and full substantiation of which used formalisms equivalent to them and based on the usually fails. We will concern ourselves with analysis of principle of least action for Lagrangian function of the the conceptual states of the most important models men- system or on the Hamilton equations [93], though it does tioned above, making digressions to the general systems not change the essence of the concerned problems. theory and applying to the practices of real complex sys- Even within the bounds of classical physics assump- tems functioning. tions 1)-3) concerning physical quantities and relevant measuring procedures are approximations and must be considered as hypotheses, true only under certain condi- 5.1. Classical physics and its paradigms – critical tions. analysis Indeed, if we proceed not from the abstractions, but from the classical measure theory realities, the notion In classical physics it is supposed that basic physical of the physical quantity (and any other one) is insepara- quantities can be considered as the quantities, accept- bly connected with a certain measuring procedure, which ing a continuous value series and existing regardless the also includes the comparison with some kind of a stan- measuring procedures. Meanwhile: dard. As any measuring procedure takes finite time ∆t, it is 1) there are instantaneous values of physical quanti- assumed that during all that time values of the measured ties, describing the state of the system; physical quantity and essential standard’s characteristics 2) in principle there are procedures, which allow to (or the values of the physical quantity relative to the measure the instantaneous values of these physical standard) do not change. quantities; Is it really like that? If you think about it, is not quite like that, strictly speaking it is not like that at all. 3) the influence of the measuring procedure on the For example, the length of the bar under the tempera- value of the physical quantity being measured can ture oscillation of the component atoms (or, if the bar is be made arbitrarily (negligibly) small. under the temperature close to the absolute zero, under To such quantities (to make it easier we will confine so-called “zero-point” quantum oscillations unremovable ourselves to mechanics) we can relate – mass of a m in their essence) is constantly changing. particle, distance (position vector ~r with the orthogo- It means that the value of the measured bar length, nal coordinates x,y,z), force (vector f~ with projections attributed to the t moment of the procedure finishing, x (t), is a certain functional (in the simplest case it is a fx,fy,fz on the orthogonal axes of coordinates), which ′ ′ can change in time (the time t is absolute, continuous, mean value) of the x (t ) values when t

existing (“the immediate one”) x (t′) (on the right There are various and virtually equivalent formulae of side) and x (t) (on the left), which was really mea- the fundamental quantum mechanical regulations, never- sured (“the integral one”), while F [x (t′)] is an im- theless any mathematical formalism used must satisfy all plicitly defined functional of the implicitly defined the above-listed conditions and results of the experiments function x (t′) , t′

5.2. Non-relativistic quantum mechanics – c) physical quantities operators. experimental facts, postulates and consequences Such a scheme has a historical, psychological and The facts, found experimentally, which underlie the logical explanation. The problem, stated before non-relativistic mechanics are the evidence of the follow- the famous founders and ideologists of the quan- ing regulations: tum theory (M.Planck (1858-1947), A.Einstein (1979- 1955), N.Bohr (1985-1962), E.Schrodinger (1887-1961), a) the indeterminancy principle turns up, thus there Louis de Broglie (1892-1987), W.Heisenberg (1901-1976), is no conception of the particle path; W.Pauli (1900-1958), E.Fermi (1901-1954), P.Dirac (1902-1984), M.Born (1882-1970), V.Fock (1898-1974), b) physical quantities can possess not every value, the D.Blochinzev (1908-1979), L.Landau (1908-1968) and spectrum of the permitted values can be discrete; others), was not only in the development of the math- ematical apparatus, which would explain results of the c) as in the classical physics it is assumed that phys- physical experiments, not only in understanding the qual- ical quantities can have immediate values, but not itatively new ideology, based on the classical school they every set of them can be measured simultaneously; grew up on, but also in bringing it home to the minds of d) the eventual influence of the measuring procedure the physical society. on its result takes place, meanwhile system state Under such circumstances (inevitably) the conceptions becomes indeterminate in a varying degree after the formulated could not help having one foot in the “old” measuring; classical quantum physics, and the other foot – in the “new” one. However such a “half-hearted” approach was e) every system is an open one in its essence, because to become a brake on the noncontradictory philosophical the wave function, which helps to characterize the interpretation of its laws and wide spread occurrence of system state in quantum mechanics (the existence its conceptions sooner or later. of this function is postulated), is formally deter- As far back as 1974, when studying in the postgrad- mined and continuous in all the space. uate course of the Lomonosov Moscow State University 8 and preparing a paper, which dealt with philosophical (and therewith a time t function in the general case) problems of the quantum mechanics, one of the authors is juxtaposed with the operator: paid attention to the rapid and thoroughgoing nature of ⌢ the majority of discussions, applied to the differences in L ≡ L ~p, ~r, t . (6) quantum-mechanical notions and phenomena interpreta-   tions, done by different scientific schools, nevertheless he These rules reflect the so-calledb conformity principle. did not understand their essence. Thus the classical systems’ total energy E = H (~p, ~r, t) As we know, the discussions, connected with the prob- is associated with the systems’ total energy operator lems of interpretation of quantum physics, don’t abate (Hamiltonian): even now, and not only physics and philosophers take part in them, but, voluntarily or not, scientists from ⌢ the other fields get involved, in their attempts to use H = H ~p, ~r, t . (7) quantum-mechanical notions and analogies (quantum   psychology [94], quantum sociology [95], quantum logic As operator expressions (6)b cannot always have clear [96], [97] etc.). and definite interpretation, additional rules are brought As it has been already noted the approach to expound- in. ing the fundamentals of quantum mechanics, which es- Thus, for example, physical quantity xpx ≡ pxx can tablished in [59], is not a traditional one. In the fore- be formally associated with three different operators: word to the first edition of this book, its scientific editor ⌢ ~ ∂ ⌢ ~ ∂ academician N.N. Bogolyubov mentioned the following: x · p x ≡ i x ; pxx·≡ i x·; “the merit of this book is in the logical and consistent ∂x ∂x character of the exposition, based on the rules and regu- lations, formulated in explicit form”. However, it seems 1 ⌢ ⌢ i~ ∂ ∂ to us that the compact and explicit exposition of rules, x · px + pxx· ≡ x + x· , (8) which can be also called axioms or postulates, in their 2 2 ∂x ∂x     logical sequence, without superfluously looking back at however, only the last one (symmetric) expression is the classical physics, is exactly what gives an opportunity Hermitian operator and, consequently, the operator of to look at the conceptual fundamentals of quantum me- the physical quantity xp . chanics in a completely different way and make proper x If the function L = L (~p, ~r, t) is not polynomial to vari- conclusions of both physical and philosophical nature. able ~p, its formal expansion into the multidimentional Six postulates of non-relativistic quantum mechanics, Taylor series is used. Problems of the convergence of set out below, are the lecturing variant of exposition [59] infinite operational and functional series and interpreta- (V.D. Krivchenkov, 1970, MSU, physical faculty). tion of them, which occur meanwhile, are the subject of a A1. According to the first postulate any physical quan- special discussion, and correspondence of conducted the- tity L (except time t, which is not a physical quantity oretical calculations to results of the experiment serves in non-relativistic quantum theory and is considered as as the selection criterion for the operator representation. an independent parameter) is associated with the linear ⌢ A2. According to the second postulate the given phys- Hermitian operator L. ⌢ ical quantity L can possess only eigenvalues λ of itsL Rules of the juxtaposition are based on the classical i operator: expressions for physical quantities and formulated in the following way: ⌢ ⌢ Lϕ = λϕ; ⇒ λi, ϕi; Lϕi ≡ λiϕi (9)

• classical x,y,z coordinates are confronted with the ⌢ coordinate operators: which are always real under the Hermitian character of L

⌢ ⌢ ⌢ (standard λi eigenvalues and ϕi eigenfunctions problem x ≡ x·; y ≡ y·; , z ≡ z · ⇒ ~r → ~r ≡ ~r; (4) ⌢ for the linear Hermitian operator L). It arises from the afore-mentioned postulate that, un- • classical momentum projections px,pyb,pz are con- like the classical physics, not every value of the physical fronted with momentum projections operators: quantity can be allowed; particularly even the quantized (discrete) spectrum of its values is possible. The hydro- ⌢ ∂ ⌢ ∂ ⌢ ∂ p ≡ i~ ; p ≡ i~ ; p ≡ i~ ; → ~p ≡ i~∇~ (5) gen atom energy permitted values spectrum affords an x ∂x y ∂y z ∂z example of a discrete spectrum (it is the only mathe- (i is an imaginary unit, ~ = 1, 0546 · 10−27erg · s matical problem in non-relativistic quantum mechanics, is the Planck’s constant, in (4) and (5) coordinate related to the real system, which can be approximately representation of operators is used and postulated); solved). In the conceptual sense the first and the second pos- • arbitrary classical physical quantity L = L (~p, ~r, t), tulates of quantum mechanics actually give the first cor- which is the momentum and coordinate function roboration of a thesis, brought forward by us, about the 9 primacy of the procedure against its result, which is di- system, consisting of N particles the number of such mea- ametrically opposite to the conception accepted in the surements is twice as little (not taking into consideration classical physics. In the sequel we will repeatedly return purely quantum spin variables) as the number we get, to this thesis, weighing in with the arguments and proofs when defining the system state in the classical way, i.e. in its favour. 3N. The conformity principle can be considered as an il- As the wave function is formally defined in whole space lustration of genetic aspects, which characterize perpet- even for the single particle, than any real quantum- ual historical development of both theoretical physics mechanical system is virtually open. In order to describe and scientific cognition in whole, including the following such systems (i.e. to take into account system’s interac- phases: tion with its surroundings, if it is not deliberately small) the density matrix representation is used [58]. • filling the old formulae and statements with the new A4. The fourth postulate says that mathematical ex- meaning; pectation (the mean value) of the L physical quantity ⌢ • generation of the new formulae and statements as with the L operator, for the system, which is at the state a result of the conflict between the new and the old with the wave function ψ (x,y,z,t), is defined by the in- and mutations, which occur at that time; tegral:

• selection of the well-grounded theories among the = set of possible ones.

We find it important to note this aspect, because at- +∞ +∞ +∞ tempts to create the “single theory of everything”, to find ⌢ those universal “fundamentals”, which will give the op- = ψ ∗ (x,y,z,t) Lψ (x,y,z,t) dxdydz. (11) portunity to explain and band together everything that −∞Z −∞Z −∞Z happens in this world for good, occur very often, even on the modern level. Such attempts in our opinion have no It follows from this postulate that the result of any prospects even in the field of fundamental physics, not measurement has, actually, ambiguous character. (Phys- speaking of the theories, which claim to give the compre- ical quantity can possess a deterministic value as a hensive and timeless description of socio-economics phe- result of measurement only if ψ (x,y,z,t) agrees with ⌢ nomena. one of the eigenfunctions ϕi of the L operator.) The A3. According to the third postulate every physical |ψ (x,y,z,t)|2 dxdydz quantity is interpreted as the prob- system state is associated with the normalized wave func- ability of the particle detecting in the differential of vol- tion ψ: ume dxdydz. The probabilistic nature or, to be precise, the uncertainty of measurement result, is the fundamen- +∞ +∞ +∞ tal peculiarity of quantum-mechanical systems. ψ = ψ (x,y,z,t); ψ ∗ ψdxdydz = A5. The fifth postulate (the Schr¨odinger equation) de- fines system evolution (change of its wave function ψ) in Z Z Z −∞ −∞ −∞ time:

+∞ +∞ +∞ ∂ψ ⌢ 2 i · ~ = Hψ (12) = |ψ| dxdydz = 1 (10) ∂t −∞Z −∞Z −∞Z and plays the same part as the Newton’s second law in (to make it easier we consider the system which consists quantum mechanics does. of one particle, and use the coordinate representation A6. The sixth postulate concerns the identical mi- of its wave function in compliance with the coordinate croparticle system and comes to the statement, that par- representation for the physical quantities operators, ac- ticles are indistinguishable in such a system. The exis- cepted above). tence of a spin – a new, purely quantum (relativistic) In classical mechanics dimensioning of 3N coordinates variable, and division of all known particles into two and 3N momentum (or velocity) particle projections – types – fermions (antisymmetric wave function, parti- 6N phase coordinates, which presumably can be approx- cles with the half-integer spin) and boson (symmetric imately evaluated – for the system, which consists of N wave function, particles with the integer spin) are also particles, completely defines the system state. postulated. In quantum mechanics the system state is specified From the sixth postulate follows the existence of the by the wave function, which does not allow defining all specific quantum (exchange) interaction, which is imple- classical phase system coordinates both accurately and mented only in the collective of identical microparticles simultaneously. Set of the measurements, that allows and does not have a classical analog. In the conceptual defining of the wave function is called full, and for the aspect this postulate can be considered as an obvious 10 physical illustration of one of the fundamental principles where ∆x and ∆v (∆p) represent the root-mean-square in systems analysis – the emergence principle. errors of measuring the x coordinate and v =x ˙ velocity Briefly, touching upon the issue of mathematical as- (p = mx˙ momentum) of the particle of the m mass. pects and omitting the details, but emphasizing the con- From the ratio (13) five important for the future con- ceptual moments, six postulates of the non-relativistic ceptual conclusions follow in turn: quantum mechanics can be reformulated in the following way: • neither particle coordinate nor its velocity can have accurate values, because when ∆x = 0 the veloc- 1) Instead of the classical notion “physical quantity ity uncertainty ∆v, and therefore the velocity itself L” a new fundamental notion is being brought in turns into infinity, and when ∆v = 0 particle is ⌢ “operator of the physical quantity L”. totally delocalized, i.e. it can be detected in any point of the physical space; 2) Possible (permitted) values of the physical quantity L are the consequence (the result) of solving the • there is no notion of the immediate speed as the eigenvalues λ mathematical problem for the opera- Newtonian limit: ⌢ tor of the physical quantity L: x (t) − x (t − ∆t) v (t) =x ˙ (t) = lim ; (14) ⌢ ∆t→0 ∆t Lϕ = λϕ. • classical particle coordinate and velocity, defining 3) For the system performance a new notion is being its state in the classical mechanics in the t moment brought in – normalized wave function ψ: of time, can be determined only approximately, when ∆t is finite and big enough;

ψ ∗ ψdτ = |ψ|2 dτ =1. • in reality there is no continuous classical particle Z Z path – it is a rough notion, which is worthwhile only when ∆t intervals between adjacent measurements 4) Classical value of the physical quantity L in the of the particle’s location are big enough; state with the normalized wave function ψ is as- sociated with a new quantity – mean value of the • prediction of the particle’s behaviour, deliberately physical quantity , which is defined by the approximate, which is defined by the pair of clas- ratio: sical phase variables (x (t) , v (t)), is possible only when taking into account its history, i.e. afteref-

⌢ fect, since: = ψ ∗ Lψdτ. Z x (t) − x (t − ∆t) 5) System evolution in time is characterized by its nor- v (t) ≈ = ∆t malized wave function evolution, which is defined by solving the Schr¨odinger equation: 1 1 = x (t) − x (t − ∆t) (15) ∂ψ ⌢ ∆t ∆t i · ~ = Hψ. ∂t depends both on x (t) and x (t − ∆t). 6) In the identical particles system all particles are We can also approach to the conclusion about the pres- indistinguishable. ence of aftereffect on basis of analysis (15), from the other side. Juxtaposing the classical velocity definition (14) The postulates of quantum (non-relativistic) mechanics with the uncertainty ratio (13) we realize that in (15) (postulates A1-A6), which were mentioned above and neither x (t), nor x (t − ∆t), nor both of these quantities are in certain sense analogous to the laws of Newtonian simultaneously can be defined accurately (otherwise the classical mechanics, are that very basis, on which all its accurate value of the limit (14) would exist too), and the theoretical apparatus and practical applications are be- uncertainty depends on ∆t, and when ∆t → 0 (disap- ing constructed. Thus, using rather elementary calcu- pearance of the aftereffect) it formally becomes infinitely lations, it is possible to show, that from the postulates large (impossibility of the prediction). A1 A4 - follows the fundamental ratio of uncertainties for Thus, quantum mechanics eliminates the classical me- coordinates and velocities (or momenta): chanics paradox, connected with the absence of the after- effect in mathematical models, used by it. ~ ~ From quantum-mechanical analysis of the system and ∆x · ∆v ≥ ; ∆x · ∆p ≥ , (13) 2m 2 measuring “tool” interaction process (analysis, which was   11 based on the perturbation theory) also follows that the a linear space. But physical sense is peculiar only to uncertainty of the system energy value ∆E, acquired as normalized wave functions, i.e. to the solution of either a result of such an interaction, is connected with its du- the Schr¨odinger equation or the eigenfunction and op- ration ∆t in the ratio: erator eigenvalue equations with additional normalizing conditions. Though the set of normalized wave functions belongs to the linear space, it does not form the linear ~ ∆E · ∆t ∼ (16) space on its own account. From quantum-mechanical analysis of the particle mo- It is well-known that a great number of non-linear mentum measuring procedure, taking into account (16), problems exist within the bounds of classical mechanics, follows one more ratio, which is useful for the future which is considered as a particular extreme case of quan- and connects the minimal possible uncertainty of the ∆p tum mechanics. In terms of common sense it seems to momentum with the duration of its measuring ∆t and strange, how the more general and formally linear theory change of the particle velocity ∆v during the time of generates frequent non-linear problems. measuring [58]: Of course there are no paradoxes in it, and everything falls into place, if we take into account that mathemat- ~ ∆v · ∆p · ∆t ∼ . (17) ical formalism of quantum mechanics is, first of all, the It seems important to us to emphasize one more time, operator formalism, based on operation algebra with spe- that in the quantum-mechanical axiomatics, expounded cial commutation relations, which is not linear at all; and above, the measuring procedure, not values of the physi- the wave function is the secondary mathematical object cal quantities as it was in the classical physics, moves to derived from formalism. the first place. Meanwhile, as it follows from the postu- Generally speaking, nonlinearity, as a concept opposite lates, the result of the measurement in the general case to the linearity notion, is substantial for rather narrow has probabilistic nature, not every value of the physical mathematical model class, underlain by linear (vector) quantity can be permitted, and the system state turns space. Thus, for example, there is no point in speak- out to be more or less uncertain, because of the uncon- ing about nonlinearity of Boolean algebra and probabil- trollable interaction between the observed system and ity theory, they are just different mathematical models. measuring tool. However, the absolutization of the notion of value of the The fact that the existence of immediate values of the physical quantity, which is in the essence the natural el- physical quantities is actually conceded in non-relativistic ement of the natural linear space, has lead to the ab- quantum mechanics, allows bringing the wave function or solutization of the notion of nonlinearity, including ap- density matrix (for open systems) in as the characteristic pearance of the disorienting and therefore poor, in our of the current system state. Meanwhile the wave func- opinion, term: “nonlinear science” (i.e. science, which tion can have various representations (coordinate, mo- differs from the linear one). mentum, matrix representation in one or another total Thereby, here we, conducting historical and logical system of proper functions, in state occupation numbers analysis of notions, implicitly find arguments approv- within the secondary quantization apparatus etc.). ing thesis about the priority of the measuring procedure Though, as it follows from the premises, the analysis, against its result in quantum mechanics, i.e. value of the conducted even within the non-relativistic quantum me- physical quantity, which can be considered as the charac- chanics apparatus, is the evidence of the idea, that there teristics of the current system state, which is secondary are neither immediate nor accurate values of the phys- and deliberately subordinate. ical quantities for real systems and real measuring pro- And the last remark, it is well-known that time in cedures. Within the bounds of non-relativistic quantum quantum dynamic equations (e.g. the Schr¨odinger equa- mechanics existence of the immediate accurate values of tion for the wave function in the coordinate representa- the physical quantities is a hypothesis useful for theory tion) is formally reversible, but the specific character of and practice, but impossible to confirm for sure by logi- quantum-mechanical monitoring (measuring) procedures cal or experimental conclusions, as in the case of classical makes it irreversible. So long as in the reality time is mechanics. really irreversible, it will be natural to include time ir- Let us make a number of remarks that are important reversibility into the axiom scheme as an experimentally in our opinion. found fact. Thereto it is enough to change the emphases, It is normal to consider non-relativistic quantum me- taking the primary nature and necessity of the measur- chanics as a linear theory (e.g. [19]), since the carrier of ing procedure (i.e. the action) as the basic, and, nat- information on the current system state – its wave func- urally, accepting the presence of the aftereffect and the tion – is subject to the linear equation – the Schr¨odinger influence, which the measuring procedure has on the re- equation, and physical quantities operators are entirely sult. In this case the question of the time irreversibility linear operators. Nevertheless it is not quite so. and existence of sets of parameters or variables, with the The notion of a linear operator or linear transforma- help of which it is possible to describe the system state tion includes a superposition principle and at least as- and its evolution in time accurately, within the bounds sumes that the set of input and output elements form of properly formulated axiom scheme of non-relativistic 12 quantum mechanics, loses its philosophical currency. be obtained only when the time of measuring is equal to Thus, we think that even on the ground of the analy- infinity, and it means that only one free particle momen- sis of non-relativistic quantum mechanics we have reason tum can be accurately measured, when the particle is in to accept the afore-mentioned hypothesis (thesis) which such (free) state for an infinite amount time. states that the notion of state in quantum physics is nei- We should note that mathematical formalism based on ther primary, nor fundamental. If we consider this very the Lie groups and algebra (algebra of operators, which hypothesis as a “bridge” and rely on the emergence prin- follow certain commutation relations [103]) is used to ciple, it will be reasonable to found any theory of suffi- construct the vast majority of modern models in relativis- ciently complex dynamic systems on it. tic quantum mechanics, including the latest theories [99], The actual proof of this hypothesis can be found, [100], [101], [102]. Thereby in mathematical formalism through analyzing the real dynamics of real systems of relativistic quantum mechanics the dominating part of any nature. However the most valid arguments for of the procedure, of the action, is in fact ”legitimated”, such conclusions, and they are just from natural sciences and operator is its formal representation or mathematical (which is extremely important from historical, psycho- image. logical and philosophical points of view), are given by As we have already mentioned, non-relativistic quan- relativistic quantum physics. So let us proceed to the tum mechanics is created on the possibility of immediate analysis of its conceptions. measuring of quantities, which characterize the system, in principle. Just that very assumption gives an oppor- tunity to bring the wave function in as the means of an 5.3. Relativistic quantum mechanics. New unbiased description of system state and its evolution in paradigms in complex system modeling time [57], and consequently the notion of state as the fundamental system characteristic can be brought in as Relativistic quantum mechanics is considered to be not well. entirely complete yet because of the lack of the proper Within the bounds of relativistic quantum mechanics experimental basis. this assumption is rejected, therefore the so-called scat- We should note that full experimental substantiation tering matrix or S-matrix gains in the biggest impor- of one or another modern relativistic theory requires en- tance. This matrix allows, if the noninteracting particle 20 ergies up to 10 ev and more, which are yet inaccessible system (when t = −∞) states are known, predicting the under terrestrial conditions (particles with such energy probability of various free particle system states, which are relatively seldom registered with the help of exten- occur after the interaction, when t → +∞ [57]. sive air shower method in cosmic rays), though some of Such a “refined” statement of the problem of experi- the problems can be solved by the recently launched col- mental investigation of relativistic quantum effects can lider [98], which is able to give the interaction energy up 12 hardly correspond with the overwhelming majority of to 14 · 10 ev on the colliding electron beams). real physical processes which occur in nature, though it Nevertheless, the results, already achieved within its helps to get rather accurate and reproducible results and bounds of relativistic physics (and achieved rather long is rather useful for the elucidation of fundamental, but ago), corroborate the analysis conducted above and its only physical laws of nature. conclusions, giving it not only technical but also concep- As far as we know, relativistic effects in quantum tual character. econophysics in the aspects that were touched upon Among the new statements of relativistic quantum me- above have not been discussed till now. However it chanics is the fundamental one, which says that any doesn’t mean that there are no analogues of relativistic measuring procedure takes fundamentally finite time ∆t, effects or their consequences in socio-economic processes, therefore there are no immediate values of physical quan- so long as the quantity, playing the part of the maximum tities. The limiting error (terminologically, we think, it possible velocity in these processes, doesn’t have to be will be more accurately to say, limiting uncertainty) of connected with the physical light speed c. measuring any physical quantity is in this case increasing Thereby in terms of conceptual statements of relativis- with the decrease of the time of measuring and finite un- tic quantum mechanics, taking the conducted analysis der any finite ∆t, and the value itself can be attributed into account we have all reasons for accepting the hypoth- only to this time interval ∆t [57]. esis, which states that the particle measuring procedures Thus, if taking into account the relativistic constraint (applied to any type of particles) take finite time in socio- on the maximum possible change of velocity ∆v ∼ c (c economic systems as well, and the results of measure- represents the light speed) in the ratio (17), it is possi- ment depend on the chosen procedure and are secondary ble to get the relativistic quantum uncertainty principle, against the latter. It is also reasonable to accept the hy- expressed by the ratio [57]: pothesis, which says that there are no immediate values of economical and other quantities and indices, and the ∆p · ∆t ∼ ~/c. (18) accuracy of measuring decreases when the time of the measuring diminishes (or these quantities lose their pri- Thus the accurate value of a particle momentum can mary sense completely). The latter can be interpreted as 13 one of the corroborations for the hypothesis of occurrence the bounds of the hereinabove explained approach - any of the non-excludable aftereffect in system, i.e. memory. ”ping-pong” takes time, and , if this ”future” will have Let us proceed to the further “relativistic” conclusions. become the ”past” by the moment of prediction, the pre- The ultimate accuracy of measuring increases with the in- dictions will become pointless (In this regard the real ob- crease of its duration, but it is possible only when the sys- served dynamics of the real world can be interpreted as tem is in the constant state; therefore there is an optimal the real-time work of some kind of the ”utmost”, unique time of the measuring for real dynamic systems, which and inimitable gigantic ”supercomputer”, when it is of no means that the optimal observation (measurement) on importance whether it is a determinate one or has some the system presently must be of discrete nature in time. uncertainties. Our world is virtually such a one.) The stride parameter, of course, depends on what is mea- We should mention that the part of not abstract “SSS” sured, and on the way how it is measured, and optimality can be played by the possessor of a prediction technology has a subjective component in certain sense. (which is unique and rapid enough for that time) who has Formally (and not only formally in our opinion) any the necessary information content. It is he, who, being calculations in socio-economic systems, that involve the personally interested, can gain the local, in time or other totality of initial data, including the dynamics predic- financial and economical “coordinates”, profit. tion, must be labeled as complex indirect measurements It is evident that the abilities of such a materialized (observations) in compliance with some kind of an al- “SSS” depend on the historical experience, accumulated gorithm. Thereby the algorithm becomes a measuring by this civilization, and mastered mass prediction tech- procedure, which generates the quantity respective to it, nologies, therefore the socio-economic dynamics and re- while the realization of this procedure, as the realization ality of the ancient world, the Middle Ages, these days of any other one, can change the system state and its and of the more or less distant future – are different in future behaviour unpredictably. their essence. To make our conclusions even more convincing, we will Thus, the new paradigms arise from our analysis, carry out the following mental experiment. Let us as- which is considerably based on the conceptual fundamen- sume that some kind of an authoritative and personally tals of relativistic quantum mechanics. On our opinion, uninterested higher being (let us call him “SSS” - the “su- these paradigms must be accepted and taken as a prin- percomputer” with “super-memory” and “supermodel”), ciple of mathematical modeling of complex systems. In having all the information on our world (including the expanded form these conceptual statements can be for- information on its history), able to conduct any calcula- mulated in the following way: tions and predict the future arbitrarily accurate, is pre- • Priority of the measuring procedure against its re- dicting the rise of the dollar/euro cross-rate every other sult and its unavoidable influence on it; month roughly at 10% (dollar tumble). If this information is inaccessible for others, it is likely • Absence of the notion of immediate value of the to be like that. Not much will change in predictions, if physical quantity as a matter of principle, and, con- “SSS” brings this information home to one of the busi- sequently, absence of the notion of system state as nessmen, smart enough to carry out proper banking op- its fundamental characteristics; erations without any fuss and increase his capital every other month. If everybody gets this authoritative infor- • Discretness and approximate nature of the sys- mation, which is beyond any doubt (it comes from “SSS” tem time dynamics (the dynamics is considered as himself!), the dollar/euro cross-rate will rise not every the sequence of system definitionally approximate other month but every other day, and not at 10% but at states under review); dozen percents, if not times. • Presence of the irremovable aftereffect, i.e. mem- Let us assume that “SSS”, having imparted the first ory; variant of his forecast to all the interested participants, will consider the other variant of it, taking into account • Finite length and influence of any measuring proce- that everybody is acquainted with his first version (which dure, including observation and prediction, derived can be interpreted as expectations now) and has already from realization of the algorithmic procedure, on made a decision. If this variant is known to the public at system state and its future behaviour; large as well, everything will be repeated all over again. • Refusal of the infinity as the conceptual notion; Within the bounds of the hypothesis of continuous time and infinite (“untimely”) computation velocity, such • Time irreversibility. “ping-pong” between “SSS” and users of his information can go on endlessly, what leads to the insoluble paradox of both prognostication and real behaviour of the socio- 6. ALGORITHMIC MODELS WITH DISCRETE economic dynamics. TIME Within the bounds of the foregoing approach, if we re- ject the infinity (and, of course, continuity) as the concep- The statements, expounded above, seem quite obvious tual notion, such paradox simply will not appear within to us not only in terms of physics, but with a view to 14 the observation practice, research and real functioning of Determinate chaos models [63, 65, 105], neural net- socio-economics systems. Therefore they must be taken works [106, 107, 108] and continuous models, based on into account during the mathematical statement of rele- differential and integral equations (after being realized in vant problems. one or another difference scheme [109]) virtually come to Algorithmic models are gaining importance in connec- the models (19), (20). tion with it, being discrete in their essence and putting However, on our opinion, classical differential and in- the algorithm, i.e. the procedure, the action, with the tegral equations form a rather narrow model class, which help of which one or another process fulfils itself, on the does not involve all the problem spectrum of modern first place. complex system theory, since, as it has been already men- It is well-known that the algorithmic approach, was tioned, differential equation don’t include aftereffect and developed in due time by A.N. Kolmogorov (the Kol- the integral ones don’t take into account all possible non- mogorov complexity theory, 1956) [104], who foreknew linearities, that can occur in the system (the integrating the great future for it. It was he who made one of the first operation is linear by definition). In addition, both of indications of the priority and independence of the dis- them are based on the untestable hypothesis of the ex- crete approach (against the continuous one) in the mod- istence of infinities and assume the existence of limits, eling of complex systems [88]. which not always takes place. It seems to us that within the bounds of this very ap- Identification of the model (19) comes to the function proach, when using the algorithmic models extensively, f~i(~xi) definition, and the differences between determinate we can take into consideration and implement all the chaos models and neural networks are connected with above-listed conceptual statements, concerning the prob- the form and methods of defining these functions (in the lem definition and solving in mathematical modeling of neural network models a narrow, from the mathematical complex systems. Let us consider one of such opportuni- point of view, representation class fi(xi) is used). Gener- ties. ally speaking steadiness or convergence of the processes (19), (20) is not assumed, and either a single-stage xi vector component set or their time history can be of in- 6.1. General statement of the discrete modeling terest. problem A single-component model with the memory of the fol- lowing form: Sufficiently great algorithm class of models with dis- crete time can be specified by the recurrent process of the following form: xn+1 = f (xn; xn−1; xn−2; ...xn−k); k ≥ 1 (21) can be also brought down to the model (20) relative to (1) (2) (k) ~xn+1 = f~n f~n−1 ... f~0 (~x0) , n =0, 1, 2, ..., (19) the (k+1)-dimensional vector (xn; yn ; yn ; ...yn ), when the proper lag variables are being brought in:     where f~i(~xi) is representing the nonlinear mapping of the multidimensional vector ~xi, i is for discrete, real or ficti- (1) (2) (k) yn = xn−1; yn = xn−2; ... yn = xn−k. tious time, ~x0 is an input pattern which is considered to be set in every member function. In the particular case Thus, due to the finite time digitization models with it is possible for f~i(~x) not to depend on discrete time i, memory can be created on basis of the model (20), though f~i(~x) ≡ f~(~x) (autonomous models): it does not contain aftereffect (the future depends only on the present). The question on, whether it is possible to bring the ~xn+1 = f~(~xn) , n =0, 1, 2, .... (20) vector model without memory (20) with (k + 1) com- ponents to the model with memory (21) for one of the Autonomous models usually describe systems, which components (this procedure has a certain analogy with are considered to be isolated. Strictly speaking, the pro- the process of combining a system of the first-order dif- cess (19) can be considered a recurrent one only in this ferential equations into a single one of the higher order), case, although formally process (20) can be made au- requires separate consideration, which will be carried out tonomous, by giving discrete time the dependent variable later. status n ≡ yn and adding the ratio yn+1 = yn + 1 (but in this case the new process, formally autonomous, will have the deliberately unlimited amplitude). 6.2. On the time irreversibility. The Verhulst Within the bounds of the model (20) we will be inter- model ested in divergent, limited, nonperiodical sequences, since they can reflect complex processes, occurring in real sys- Within the bounds of the model (20) time irreversibil- tems without the participation of exogenous (external) ity can be concerned as the biunique correspondence be- factors. tween the vectors ~xn and ~xn+1 on the certain subset Xi 15 of the system (20) phase space X: 6.3. Aftereffect and “long” memory in discrete models with nonlinearities −1 ~xn+1 = f~(~xn); ~xn = f~ (~xn+1); Let us consider the problem of bringing the vector model (20) to the scalar model (21) for one of the com- ponents. Let us start from the case of a two-component ~xn, ~xn+1 ∈ Xi ⊆ X. (22) model:

In the general case Xi must include the system attrac- x = f (x ,y ); tor – the subset Xa, and belong to the subset X0, which n+1 x n n n =0, 1, ... (28) yn+1 = fy (xn,yn); is a subset of initial values, drawing the system up to the  attractor Xa: In order to exclude the yi variables, we will write down a system of three equations for 5 variables X ⊆ X ⊆ X ⊆ X. (23) xn,yn, xn+1,yn+1, xn+2, having temporarily equated n = a i 0 0 to simplify the notation: Let us consider the Verhulst model [110, 111, 112] as the simplest example. The model is a nonlinear logical x2 = fx (x1,y1); and single-component mapping in the following form: x = f (x ,y ); (29)  1 x 0 0  y1 = fy (x0,y0) . xn+1 = f (xn)= xn (1 + α (1 − xn)); Let us assume that the second equation of the system (29) can be definitely solved relative to the y0 variable, i.e. the function x = f (x ,y ) has an inverse one relative 1+ α 1 x 0 0 0 <α< 3; x ∈ 0; = X , (24) to this variable: 0 α 0   where α is a given numerical parameter. We chose the −1 y0 = fx0 (x0, x1) . (30) limits for α and x0 so that xn values would stay positive with any chosen n> 0. Substituting the third equation of the system (29) into The largest extremum xn+1 = xmax of the function its first one: xn+1 = f (xn) is reached in the point where xn =x ¯: x2 = fx (x1,y1)

1+ α (1 + α)2 x¯ = ; xmax = . (25) 2α 4α = fx (x1,fy (x0,y0)) ≡ f˜x (x1, x0,y0) (31) The inverse mapping x = f −1 (x ) is: n n+1 and substituting the expression (30) for y0 in the (31), we get:

2 ˜ 1+ α (1 + α) x x2 = fx (x1, x0,y0)= x = ± − n+1 ; n 2α s 4α2 α

˜ −1 = fx x1, x0,fx0 (x0, x1) ≡ Fx (x1, x0) . (32) (1 + α)2 xn+1 ∈ 0; = Xi ⊆ X0 (26) Such memory, the length of
which is determined by 4α ! the number of components in the initial vector model (28) (where the aftereffect is absent), can be called short and is a two-digit one, generally speaking. for convenience. Thereby, the Verhulst model is the one with the irre- If the inverse mapping (30) in the phase variables do- versible discrete time. However, if the following condition main of variation is ambiguous, for example it has two is fulfilled: branches:

2 (1 + α) 1+ α y = f −1 (x , x ); y = f −1 (x , x ) , (33) x ≤ x¯; ⇒ ≤ ; ⇒ α ≤ 1, (27) 0 1x0 0 1 0 2x0 0 1 max 4α 2α we should choose the branch, corresponding with the y0 and the interval (0;x ¯) is chosen for the Xi subset, the value, which is observed (given) within the initial model inverse mapping becomes a single-digit one. (28), for this pair of variables. 16

Thereby the mapping (32) becomes virtually not only the function of x0, x1, but also of y0: y2 = fy (x1,y1,z1) ≡ fy (~r1) ; (41)

x2 = F˜x (x1, x0,y0) . (34)

Similarly: z2 = fz (x1,y1,z1) ≡ fz (~r1) ; (42)

x3 = F˜x (x2, x1,y1)= F˜x (x2, x1,fy (x0,y0)) ≡ ˜ x = fx (x ,y ,z ) ≡ fx (~r ) ; (43) ≡ F˜x (x2, x1, x0,y0); 1 0 0 0 0 ˜ ˜ x4 = F˜x (x3, x2, x1,y1)= F˜x (x3, x2, x1,fy (x0,y0)) ≡ ˜ ≡ F˜x (x3, x2, x1, x0,y0); ... y1 = fy (x0,y0,z0) ≡ fy (~r0) ; (44) (35) It follows from the received correlation chain, that even in the two-component system (28) the “long” single- z1 = fz (x0,y0,z0) ≡ fz (~r0) . (45) component memory, determined by nonlinear and oblig- atory nonmonotonic interactions of the components, is Substituting the expressions ((41 and (42 into the right actually possible. Of course, everything afore-mentioned side of equation (39) can be considered to be merely necessary conditions for the realization of arbitrary “long” single-component memory in systems (20) with the limited quantity of com- x3 = fx (x2,y2,z2)= fx (x2,fy (~r1) ,fz (~r1)) ≡ ponents; however the wealth of trajectories and phase portraits, observed for such systems during numerical experiments, leaves us hoping for the existence of suffi- ˜ cient conditions. To reach these conditions a model with ≡ fx (x2, x1,y1,z1) , (46) more than two components will be, probably, required, and further expressions ((44,(45) into (46) we get: however it does not change the essence of the analysis ˜ ˜ conducted and conclusions made. The ternary nonlinear x3 = fx (x2, x1,y1,z1)= fx (x2, x1,fy (~r0) ,fz (~r0)) ≡ Lorenz’s mapping [113] can be considered to be one of ˜ the examples of the model, where it is possible to realize ≡ fx (x2, x1, x0,y0,z0) . (47) the “long” single-component memory. Let us briefly consider the scheme of reasoning and In order to exclude variables y0,z0 in (47) we use the computations for the ternary model (N = 3): ratio (40), having substituted expressions for y1,z1 (45, 46) and ratio (43) in it beforehand. xn+1 = fx (xn,yn,zn); yn+1 = fy (xn,yn,zn); n =0, 1, ... (36)  x2 = fx (x1,y1,z1)= fx (x1,fy (~r0) ,fy (~r0)) ≡ yn+1 = fy (xn,yn,zn);  ≡ f˜x (x1, x0,y0,z0); ⇒ We will write a set of k equations, x = f˜ (x , x ,y ,z ); 2 x 1 0 0 0 (48) k = N (N − 1)+1=3(3 − 2)+1=7, (37) x = f (x ,y ,z ) .  1 x 0 0 0 for p variables, If the mapping (48) is biunique relatively to the pair of variables y0,z0, i.e. if there is a single solution of the set (48): p = N 2 +1=32 +1=10, (38) lettered as x0,y0,z0, x1,y1,z1, x2,y2,z2,z3, having −1 y0 = fy (x2, x1, x0); equated n = 0 to simplify the notation as before (when −1 (49) z0 = f (x2, x1, x0) , N = 2):  z

then, substituting y0,z0 from (49) into (47), we will fi- nally receive: x = f (x ,y ,z ) ; (39) 3 x 2 2 2 ˜ −1 −1 x3 = fx x2, x1, x0,fy (x2, x1, x0) ,fz (x2, x1, x0) ≡

x2 = fx (x1,y1,z1) ≡ fx (~r1) ; (40) ≡ Fx (x2, x1, x0) . (50) 17

If the inverse mapping (x1, x2) in (y0,z0) for (48) is 7.1. About the nature of uncertainties and role of not a single one, it is necessary to carry out the rea- action in mathematical statement of a problem soning, similar to the one conducted in the case of two- component model, which leads to the possibility of ex- When the attempts to describe the mechanism of the istence of the “long” single-component memory in the evolutionary development of the Universe, which would mapping for the xn component. take into account the practical impossibility of an ac- Similar calculations and reasoning can be carried out curate future prediction, are taken the two paradigms for N = 4, 5, 6 etc., and the conclusions will remain the collide: same. It is also obvious that we can consider any other a) incompleteness of the information on the Universe, component instead of xn in any situation, what will lead only to the change of components indices; it is also pos- including its past, and rough character of any sible to consider groups of components, which form any model as a result; part of the initial component set. b) probabilistic nature of future against the present. The idea of bringing the set of equations for the mul- ticomponent model to one equation (a group of lower Both paradigms are virtually untestable though. equation count) for one of the components (group of com- Indeed, concerning the first paradigm, any informa- ponents) is, as it has been already mentioned, analogous tion on the system must have a material object, which to the idea of bringing the system of ordinary first-order is either a part of the system (and cannot contain the differential equations to one differential equation (group full description of it), or an external system, interact- of quations) of higher order for one of the initial (a group ing with it, i.e. a part of a new fuller system. In of the initial) unknown functions. However there is an this case the interpretation of the process uncertainty important difference – aftereffect, i.e. memory, does not is brought to different variations of hidden variables appear in the set of differential equations because of the model [18, 114, 115, 116, 117] within the bounds of this limiting process (the size of pace according to time ∆t paradigm. tents to zero). The second paradigm virtually comes from the hypoth- esis of existence of multiple, absolutely identical parallel Let us imagine for a moment, hypothetically, a dy- worlds (the quantum ensemble of worlds) in every mo- namic Universe model as the complex nonlinear au- ment of time, when each of them can develop itself ac- tonomous system, which started functioning within the cording to its own probabilistic scenario, but only one of bounds of a discrete model of the (20) type in some rea- them is realized in our world and observed by us [18] (the sonably distant initial moment of time t . 0 many-world interpretation was suggested first in [118] Taking into consideration the huge initial number of with the prehistory of it in [119]). Thus, according to components of such a model and complex, nonlinear char- this paradigm, the real world dynamics is a chain or a acter of their interactions, we can assume that sufficiently sequence of events, having a random component of the long observation of some limited part of its components quantum-mechanical nature. will show the “long” memory, the uncertainties, the ab- However, the notion of an accidental event and proba- sence of repetitions (creation of new information) etc. bility assumes a hypothetical possibility of infinite experi- At least, the analysis conducted above does not exclude ment repeatability under identical conditions, and, by the such a possibility, though the realization of it is likely reason of it, the probability theory must be considered to be a rather rare phenomenon in our Universe both in to be merely one of possible and deliberately approxi- time and space, demanding a number of specific circum- mate models of description of uncertainties observed in stances. Our Earth could serve as an example of such a the world. realization, having reached a noosphere (the highest for In fact there are no accurate procedures, which would today) phase of its development by now. give the opportunity to distinguish the “true” random se- quence of events or quantities from the “pseudorandom” one, i.e. the one similar to the arbitrary, such as gen- erated by any suitable determinate chaos model. Really and truly any “random” finite sequence cannot be ran- 7. NEW PARADIGMS AND PROBLEMS OF dom because of its finiteness, and any “nonrandom” finite COMPLEX SYSTEMS MATHEMATICAL sequence can be considered to be the one of possible and DESCRIPTION scarce samples of a true infinite random sequence. (Here we proceed from the idea, that the notion of infinity is Having conducted the afore-mentioned analysis, we the one of hypotheses, unverifiable on principle, which made some conclusions, and not claiming to make it uni- included as one of the postulates into the rigorous theory versal we will briefly dwell on some problems of philo- of sets [86].) sophic, conceptual and technical nature, that appear dur- Moreover, socio-economic phenomena don’t repeat ing mathematical modeling of real complex systems dis- themselves accurately, and quite low disturbances in real cussions and problem statement. systems can lead to rather big anomalies, which are hard 18 to predict (crises, crashes, bankruptcies and other phe- mentals of modern quantum theory, academician V. P. nomena of critical character, that usually show their in- Maslov. In his latest work, dealing with the mathemati- dividual and unique peculiarities). cal model of the world economical crisis of 2008 [37], he Both paradigms mentioned above proceed from the as- clearly shows that the probability theory and the theory sumption that there is a notion of system state and this of optimization, which form the fundamentals of mod- notion is the primary and fundamental one. However, ern economic science, are inadequate as the mathemati- repeating the above written, if take into consideration cal toolbox for dynamic description of modern economy. those conceptually new things the modern theoretical On his opinion, the Kolmogorov complexity theory [104], physics has brought into the world, including the rela- based on the algorithmic approach, should be used as an tivity theory and relativistic quantum mechanics, and be alternative. consistent in application of the general system theory, the And, finally, we can’t help mentioning the empiri- notions of measuring procedure and interaction between omonism of famous Russian politician, economist and the system and measuring tool, i.e. the result of the pro- thinker A. A. Bogdanov [120] and his organizational sci- cess, become primary and fundamental. It seems to us ence – tectology [73, 74]. His ideas are close to ideas of that with such statement of a question uncertainty of the the general system theory, having anticipated cybernet- state becomes merely a technical problem. Particularly, ics, had been wrongly forgotten because of the political within the bounds of quantum mechanics uncertainty of motives (both in the West and in Russia) for almost a the state, i.e. of the quantities characterizing it, is a con- century. These ideas have actually outstripped their time sequence of certain commutation relations of algebra of for century, and only now they start to enter the mod- operators of these quantities [59]. ern science. His interpretation of organization as the ac- For justice’ sake it is necessary to mention, that such tion, which is the fundamental element of the process of a point of view on the fundamental role of action, not functioning in any system, is rather similar to ours and the status, was upheld by the prominent world and na- other modern conceptions in philosophical sense. It cor- tive psychologist and philosopher S. L. Rubinshtein, who roborates the old conception one more time: any new is the author of the fundamental work “Fundamentals of thing or idea is a well-forgotten old one, having been General Psychology” [60], written more than forty years pulled out and rediscovered in the “right” time and in the ago, but still actual. A scientist of encyclopedic knowl- “right” place. Unfortunately he thought about the Ein- edge, educated in the field of natural sciences, math- stein relativity theory rather critically and wasn’t thor- ematics, psychology and philosophy, S. L. Rubinshtein oughly aware of quantum physics, arising at that time conducted a brilliant analysis of historical development and being beyond his scientific interests. of conceptual fundamentals of scientific world-view. The Thus the conceptions, not necessarily coincident with authors think that he consciously did not use mathemati- the traditional ones, should form the fundamentals of cal formalism, realizing that the language of mathematics mathematical modeling of complex systems dynamics of of “states” and “functions” known to him is not appropri- any nature. Relativistic quantum mechanics, as it has ate for the level and essence of problems, he was solving. been already mentioned, can serve as one of such sources, Economists involved in researches and discussing fun- however, a certain level of caution will be required in this damental problems of modern economical theory, use case. mathematical language carefully or don’t use it at all (even nowadays, in time of “informatization” and “com- puterization”), preferring to bring in their own, new and 7.2. About peculiarities, problems and correctness ex facte unusual notions, when the doubts in its adequacy of quantum mechanical socio-economic systems appear. Thus, the notion of coordination is brought in modeling to characterize the stable socio-economic system state in the monograph written by famous French scientist and Most of the researchers who use quantum-mechanical practitioner J. Sapir “Economic theory of heterogeneous models to explain socio-economic phenomena, market dy- systems: an essay on decentralized economies” [56]. It namics in particular, assume that state distribution of the is impossible to bring this notion to such mathemati- set of its agents (by state strategies are meant) conform cal or physical concepts as equality, identity, equivalence, to Bose-Einstein statistics (e.g. [18, 20, 21, 27, 37, 38, equilibrium, stationarity etc. This notion should rather 39, 40]). It means that at one state (one strategy) an be considered to be some kind of a specific character- arbitrary large amount of agents can coexist. Is it really istic of the non-stationary action, which secures stable like that? and steady structural existence for socio-economic sys- If analyze the real behaviour and interrelations of mar- tem. Here we find implicit “economical” arguments for ket (or any other socio-economic process) participants the thesis on the priority of the procedure in description thoroughly, it is possible to make a conclusion, that the of complex systems dynamics. equilibrium condition (“equilibrium” competition) is not Sufficiently persuasive evidences in favour of our po- a fundamental phenomenon, moreover it is a relatively sitions are present in works of the greatest specialist rare one. During any kind of interaction in real systems in both classical mathematics and mathematical funda- domination relations quickly get established, since they 19 are more constructive and stable – and that is, if speaking (synergetic effect, aftereffect, “long memory”, thresh- of quantum-mechanical analogies, rather the Fermi-Dirac old phenomena, conditioned by weak interactions with statistics (only one agent can be at each state). the environment etc.). On the other hand structureless From the microparticles identity principle and equa- “field” approaches, based on the ideas of the quantum tions of quantum mechanics comes a special quantum- field theory (the unified field theory), if developed, would mechanical exchange interaction, which is implemented possibly turn out to be not productive as well. in the group of identical particles and put into effect ac- On our opinion only measured, discrete by definition, cording to the ”each to every other” principle [58, 59]. data series, characterizing dynamic change of system However, this principle is of local nature and can get state during quite long time period T , can serve as the broken, if the size of the system considerably exceeds source of information about complex system. the product of the light speed and the time of observing In this case that problem statement becomes accept- the system (lagging effects). Mechanical transfer of the able and reasonable, where the approximate prediction interaction mechanism (according to the principle of in- of system behaviour, its informative characteristics and termeshed exchanges) to socio-economic systems, where algorithm design are considered. Such a statement is typ- agents play the part of microparticles, and the relativis- ical for the new scientific direction in socio-economic pro- tic interaction lagging effect analogues are not necessarily cesses – data analysis (developed since 1990) [125, 126]. connected with light velocity, is not quite competent. Concerning time irreversibility and discrecity, it can The sufficiently successful explanation of some statis- be added that time irreversibility must be considered as tic characteristics and dynamic peculiarities of mar- an experimentally found within the bounds of its appli- ket behavior, derived from quantum-mechanical calcu- cation, fact. Time characterizes duration of procedures, lations with the help of Bose statistics [18], may not be processes, phenomena, i.e. the duration of actions, and connected with choosing that very quantum-mechanical can be determined only with the help of various actions. model. Multicomponent nonlinear models, e.g. (20), can Minimal time interval is actually determined by the ob- give rise to rather rich and various scenarios of the dy- served action of minimal duration. However, according namic system behavior, even under the circumstances of to the special and general relativity theories this notion small quantity of varying parameters and variables (de- (i.e. the idea of duration) must be considered as the one, terminate chaos models [65]). Such models can be tan- which is relative, local in time and space and depending gential to equations of quantum mechanics, but let us on the coordinate system [[127]]. emphasize that fundamental quantum-mechanical prin- In theoretical physics, energy and momentum (angu- ciples are present in them and observed in their essential lar momentum) conservation laws are considered as fun- peculiarities. damental consequences of homogeneity of our time and Complex systems are usually synergetic systems with space (space isotropy) [[93]]. Hypotheses on their conti- “long memory” (information on their history), charac- nuity are the convenient, but not necessary component terized by intensive metabolism (constant “pumping” of for receipt of respective laws of conservations. For exam- energy and substance) and able to generate new informa- ple the energy conservation law can be considered as the tion. Formal quantum-mechanical problem statement, universal postulated technology of detection of new (or pretending to be the one to make a detailed “micro- already known) interactions and types of energy and sub- scopic” description of such a complex system, can turn stance transformation in physical systems. Thus the new out to be inadequate to the processes that really take elementary particle neutrino was discovered as the conse- place in the system, although it will reproduce some ex- quence of formally observed failure of conservation laws ternal peculiarities of its behavior. during experiments on β-decay of radioactive elements. By the same reason conservation laws that form the (The weak interaction connected with neutrino was so basis of equations of physical dynamics and must an- “weak” that this particle can fly through the Sun and swer physical processes can have no analogues in socio- experience no collision.) The other example is Einstein’s economic processes. Indeed, in such processes an infor- ratio of energy and mass E = mc2, which tied physical mational component is present (including informational quantities, considered to be heterogeneous before that. asymmetries of agents [121]), transaction costs are possi- On our opinion, during mathematical modeling of com- ble (the “fifth” market [122]) and memory occurs (insti- plex systems laws of conservation of various quantities, tutions, mentality [123, 124]), the energy and substance time irreversibility or reversibility, its discrecity and con- receipt and dissipation take place, other types of “rough” tinuity, homo- or heterogeneity etc. must be considered and “delicate” interaction between the environment and as the properties of this very mathematical model, first the past are also possible. of all appreciating the level of its adaptation to the de- Real non-linear interactions in the multi-component scription of real properties and real system dynamics, the socio-economic system can change the relations between history of which must be considered as unique experiment agents and generate a complex dynamics in the way, that data, not always possible to repeat. The level of system traditional analysis, conducted according to the scheme adequacy to the processes investigated, maximum possi- “structure-state-interaction-dynamics” would hardly ex- ble predictability and practical significance must serve as plain anything concerning dynamic system behaviour the basic criteria of the model. 20

8. CONCLUSION • Openness;

To resume we will briefly formulate new paradigms and • Hierarchy; main conceptual statements in complex systems model- ing, which come from the analysis we conducted. • Emergence. • Priority of the measuring (observing, action, inter- action) procedure against its result; Some of the afore-mentioned positions coincide with • Unoriginality and approximate nature of notions of positions of the general system theory, what is not “system state” and “immediate values of quanti- strange from the one side, and allows interpreting our ties” as characteristics of this state; analysis as the physical quantum-mechanical substantia- tion of system conceptions in modeling complex systems • Finite length and unremovable influence of any from the other one [22]. measuring procedure, including computer predic- In this analysis and conclusions facts and postulates of tion (indirect measurement), the state and future relativistic quantum physics and experience of observing behaviour of the system; and researching real socio-economic systems are consid- • Uncertainty principle and its fundamental connec- erably used, which gives us the reason to relate this work tion with the duration of the measuring procedure; to the new direction in physical economics, declared in the name – relativistic quantum econophysics. • Discreteness of time, space and any other quantity, We have begun specific research and development on connected with the notion of state and system dy- realization of the above-listed conceptions in modeling namics; and prediction of socio-economic processes, based on the observation data (history) of relevant time series [128, • Aftereffect (memory) as the fundamental quality of 129, 130]. One of the prediction technologies is based on any complex dynamical system; the use of complex Markov chains (Markov chains with • Refusal of infinity as the conceptual notion; memory), is implemented in Matlab 6.5 environment and is currently being tested on the stock fund indexes and • Time irreversibility; exchange rates data.

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