Directory UMM :Data Elmu:jurnal:B:Biosystems:Vol56.Issue3.2000:

Classical and quantum dynamics on

p

-adic trees of ideas

Andrei Khrennikov

Department of Mathematics,Statistics and Computer Sciences,Uni6ersity of Va¨xo¨,S-35195 Va¨xo¨, Sweden Received 11 June 1999; received in revised form 27 December 1999; accepted 17 January 2000

Abstract

We propose mathematical models of information processes of unconscious and conscious thinking (based onp-adic number representation of mental spaces). Unconscious thinking is described by classical cognitive mechanics (which generalizes Newton’s mechanics). Conscious thinking is described by quantum cognitive mechanics (which generalizes the pilot wave model of quantum mechanics). The information state and motivation of a conscious cognitive system evolve under the action of classical information forces and a new quantum information force, namely, conscious force. Our model might provide mathematical foundations for some cognitive and psychological phenomena: collective conscious behavior, connection between physiological and mental processes in a biological organism, Freud’s psychoanalysis, hypnotism, homeopathy. It may be used as the basis of a model of conscious evolution of life. © 2000 Elsevier Science Ireland Ltd. All rights reserved.

Keywords:p-Adic number representation; Classical information force; Quantum information force; Conscious evolution of life www.elsevier.com/locate/biosystems

1. Introduction

It seems that the modern physics can in princi-ple explain (or at least describe) all phenomena which are observed in reality: motion of classical and quantum systems, classical and quantum fields, …, physiological processes in biological or-ganisms. This incredible power of physics induced the common opinion that all biological processes could be reduced to some physical processes. This concerns not only primary physiological processes in biological organisms such as, for example, the

functioning of the blood system, but even biologi-cal processes of the highest level of complexity, namely, cognitive processes. The idea that by studying physiological processes in the brain we could explain (probably after many years of inten-sive research) the functioning of the brain quickly propagates throughout the biological community (see, for example, Skinner, 1953; Lorenz, 1966; Dawkins, 1976; Clark, 1980, for reductionist psy-chological theories). Hence it is widely supposed that the phenomenon of the consciousness can be reduced to some (probably still unknown) physi-cal phenomena. Such an idea seems natural and attractive (at least at the present time). However, I do not support this viewpoint. I think that the phenomenon of consciousness will be never

This investigation was supported by the grant ‘Strategical investigations’ of Va¨xo¨ University and visiting professor fel-lowships at University of Clermont-Ferrand and Tokyo Sci-ence University.

0303-2647/00/$ - see front matter © 2000 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 3 0 3 - 2 6 4 7 ( 0 0 ) 0 0 0 7 7 - 0

duced to ordinary physical phenomena. And the modern neurophysiological activity gives some ev-idences of this. Here numerous investigations were performed to study the processes of the exchange of electric signals in the brain (see, for example, Eccles, 1974; Amit, 1989) and to study localization of these processes in different do-mains of the brain (see, for example, Cohen et al., 1997; Courtney et al., 1997, for fascinating exper-iments (based on functional magnetic resonance imaging machine) for memory neurons configura-tions or Hoppensteadt, 1997, for the frequency domain models for the exchange of signals in the brain). Nevertheless, despite all of these investiga-tions and the large amount of new information on physical processes in the brain, we now do not understand the phenomenon of consciousness much better than 100 years ago.

In the present paper we propose a new physi-cal – mathematiphysi-cal model for the brain functioning (see Khrennikov, 1998a). This model is not based on the modern (Newton – Einstein) picture of physical reality (in particular, we do not use the real space R3

as the mathematical basis of our model). We consider a new type of reality, namely, reality of information. Cognitive systems are interpreted astransformers of information. For transformers of information we develop the for-malism of classical mechanics on mental space

(space of ideas). In particular, this theory de-scribes evolution of human ideas. The general formalism of classical cognitive mechanics is de-veloped by analogue to the formalism of the ordinary Newton mechanics which describes the motion of material systems. We propose cognitive analogues of Newton’s laws of the classical me-chanics. Mathematically these laws can be de-scribed by differential equations (on mental spaces)1

. Starting with the initial idea x0 we can

obtain the trajectory q(t) in mental space. How-ever, the classical cognitive mechanics is not ob-tained as just a copy of the ordinary classical mechanics. First of all in cognitive models the time t (a parameter of the evolution of ideas) could not be always identified with physical time

tphys which is used in ordinary physical models. This is internal time of a cognitive system (we can call it mental or psychological time). The velocity 6_{(t) of the evolution of an idea (calculated as in} Newton’s mechanics as the derivative:6_(t)=dq(t) dt

has the meaning of the motivation (to change the information state q(t) of a cognitive system). Forces f(t,q) and potentials V(t,q) are informa-tion (mental) forces and potentials which are ap-plied to information states of cognitive systems. An information force changes the motivation and this change of motivation implies the change of the information state q of a cognitive system.

The mathematical formalization of the classical cognitive mechanics cannot be done in the frame-work of the real analysis. The real line Rand the Euclidean space R3 _{(and even real manifolds) are} not directly related to cognitive information pro-cesses. We use another number system, namely, the system of so called p-adic numbers (integers)

Zp(see Borevich and Shafarevich, 1966; Schikhov,

1984; Khrennikov, 1994; Vladimirov et al., 1994) as the mathematical basis of our model. Here

p\1 is a prime number which is the parameter of the model. Mathematical details can be found in Section 7. This section contains also some biolog-ical motivations (namely, the ability to form asso-ciations) to choose Zp as a mathematical basis of

the model.

Geometrically we can imagine Z2 as a tree starting with some symbol (a root of the 2-adic

tree which can be interpreted as the signal to start the creation of the space of ideas of a cognitive system). This root-symbol generates two branches

0 and1 (the first level of the tree); each vertex

of the first level generates two branches to two new vertices (the second level of the tree). Thus there are now four branches00,01,10,11.

Such a process is continued by an infinite number of steps. As a result, there appears an infinite 2-adic tree with branches x=a₀… which are

1_{At first sight it is quite surprising that motions of material}

systems and mental systems (ideas) are described by the same mathematical equations (Newton or Hamilton equations). The only difference is that these objects evolve in different spaces (Newton real space and mental space, respectively). However, if we consider, instead of the motion of real material objects, the motion of information about these objects, then such a coincidence of equations of motion for material and mental systems would not seem so surprising.

identified with 2-adic numbers x=j=0

aj2 j

. The 2-adic algebraic structure on this tree gives the possibility for adding, subtracting and multiplying branches of this tree (Fig. 1).

We use p-adic trees for prime numbers p only by mathematical reasons (see Section 7). The same information model can be developed for any homogeneous tree withmbranches on each level. It is even possible to consider trees such that the number of brachesmj depends on the level.

The information processes in the brain de-scribed by the classical cognitive mechanics are closely connected with neurophysiological pro-cesses. Roughly speaking neurophysiology de-scribes ‘hardware’ of the brain and the classical mechanics on mental spaces describes ‘software’ of the brain. Some mathematical models of this software have been presented in Khrennikov, 1997; Albeverio et al., 1998; Khrennikov, 1998b; Albeverio et al., 1999; Dubischar et al., 1999. The models of Khrennikov, 1997; Albeverio et al., 1998; Khrennikov, 1998a; Albeverio et al., 1999. Dubischar et al., 1999, were discrete time models, namely, it was assumed that the time parameter t

for the evolution of ideas is discrete: t=0, 1, 2, ,… (thus chains of ideasx0,x1, … were studied in these models). In the present paper we study ‘continuous time’ evolution. On one hand, this gives the possibility to apply (at least formally) the scheme of the standard formalism of the classical mechanics. On the other hand, in the present model we can discuss carefully the mean-ing of ‘mental time’ (and ‘mental velocity’).

The classical cognitive mechanics describesun

-conscious cogniti6_{e processes. The phenomenon of}

consciousness cannot be explained by the formal-ism of the classical cognitive mechanics. To ex-plain this phenomenon, we develop a variant of quantum cognitive mechanics. In this model an idea moves in mental space not only due to classical information forces (which can be in prin-ciple reduced to the functioning of the brain’s ‘hardware’ (neurophysiological processes in the brain)), but also due to a new information force, namely, quantum information force. This quan-tum information force (which will be called a

conscious force and denoted by fC(q)) is induced

by an additional information potential (quantum potential on mental space or conscious potential C(q)). The C(q) could not be reduced to neuro-physiolocal processes in the brain. It is induced by mental processes. The conscious potentialC(q) is induced by a wave function C(q) of a cognitive system (by the same relation as in the ordinary pilot wave theory for material systems).

In our model this wave function C is nothing than an information field conscious field). In the mathematical formalism this field is described as a function C: Xmen“Xmen, where Xmen is a mental space. The evolution of the C-function is de-scribed by an analogue of the Schro¨dinger equa-tion on mental space.

In fact, our formalism of conscious forces and fields is a (natural) extension of the well known theory of pilot wave (developed by Bohm, 1951; De Broglie, 1964; Bell, 1987 and many others) to cognitive phenomena. Even in the theory of pilot wave for material systems (especially in its variant developed in the book of Bohm and Hiley (1993) the quantum wave function C is merely an infor-mation field, but defined on real space R3 _of localization of material systems, this field acts to material objects and the problem of information – matter interaction is not clear in this framework. In our model a conscious field (C-function) is associated with purely mental processes and it acts to mental objects, ideas.

By our model each classical (unconscious) in-formation state of a cognitive system (the collec-tion of ideas and mental processes in that these ideas are involved) produces a new (non-classical) field, conscious field C. This field induces a new information force fC which induces a permanent

perturbation of the evolution of an idea in the mental space. This C-function is nothing than a

human conscious.

Of course, our formalism is just the first step to describe the phenomenon of consciousness on the basis of a model of information reality. However, even this formalism implies some consequences which might be interesting for neurophysiology, psychology, artificial intelligence (complex infor-mation systems), evolutionary biology and social sciences. Here we present briefly some of these consequences.

Flows of cognitive information in the brain (and other cognitive systems) can be described mathematically in the manner which is similar to the classical Newton mechanics for motions of material systems. Therefore the motion of ideas (notions, images) in the brain has the determinis-tic character (of course, such a motion is per-turbed by numerous information noises, see Dubischar et al., 1999, for the details). This mo-tion in mental space is not an evolumo-tion with respect to physical time tphys, but with respect to mental timet. Information potentials can connect different thinking processes (in a single brain as well as in a family of brains). The consciousness cannot be induced by a physical activity of mate-rial structures (for example, groups of neurons). It is induced by groups of evolving ideas. These dynamical groups of ideas produce a new infor-mation field, conscious field, which induces a new information force, conscious force, which is the direct analogue of quantum force in the pilot wave theory for quantum material systems. This conscious force plays the great role in the infor-mation dynamics in the human brain (and other conscious cognitive systems). As in the classical cognitive mechanics, in quantum cognitive me-chanics conscious potentials can connect thinking processes in different cognitive systems (even in the absence of physical potentials and forces). Therefore it is possible to speak about a collective consciousness for a group of cognitive systems (in particular, human individuals). We also note that different conscious potentials (consciousC-fields) induce conscious forces fC of different (informa-tion) strength. The magnitude of the conscious-ness can be measured (at least theoretically). Thus

different cognitive systems (in particular, different human individuals) may have different levels of the conscious C-field. By our model we need not suppose that a consciousness is a feature of only the human brain. Other cognitive systems (in particular, animals and even nonliving systems) induce conscious C-fields which (via conscious forces fC) control (or at least change) their cogni-tive behaviours. From this point of view human individuals and animals differ only by the behav-iors of their conscious C-fields.

As one of applications of our formalism to psychology, we try to explain Freud’s psychoanal-ysis on the basis of our model as the process of the reconstruction of the conscious field of an individual i having some mental decease via an information coupling of a psychoanalytic p (on the level of a collective C function of the system (i,p)).

2. Classical cognitive mechanics

First we recall some facts from Newton’s classi-cal mechanics.

In Newton’s model motions of material systems are described by trajectories in space Xmat of localization of material systems2_{. Thus starting} with the initial position q0 a material object A evolves along the trajectory q(t) in Xmat (wheret is physical time). The main task of Newton’s mechanics is to find the trajectory q(t) in space

Xmat. Let us restrict our considerations to the case in thatAhas the mass 1 (this can always be done via the choice of the unit of mass). In this case the momentum p(t) ofA is equal to the velocity6_(t) of motion. In the mathematical model the velocity 6_{(t) can be found as} 6_(t)=dq(t)

dt q;(t). The

ve-locity need not be a constant. Thus it is useful to introduce an acceleration a(t) of A which is the velocity of the velocity . The second Newton law says:

a(t)=f(t,q) (1)

2_{In the mathematical modelX}

mat=R3or some real

wheref(t,q) is the force applied toA. As the mass

m=1 and the momentum, p=m6_{, is equal to the} velocity, we have

p;(t)=f(t,q), p(0)=p0, t,q,pXmat. (2) By integrating this equation we find the momen-tum p(t) at each instant t of time (if the initial momentump0is known). Then by integrating the equation

q;(t)=p(t), q(0)=q0, t,q,pXmat (3) we find the positionq(t) of Aat each instanttof time (if the initial positionq0 is known).

We develop the formalism of the classical cog-nitive mechanics by analogue with Newton’s me-chanics. Instead of the material space Xmat, we consider mental spaceXmen(see Section 5 for the mathematical model). A cognitive system t is a

transformer of information: an information state

qXmen (the collection of all ideas of t) is in the process of continuous evolution; t makes trans-formations q“q%“q¦“....The time parameter of this evolution is also an information parameter (mental timeof t), tXmen. Thus the activity oft generates the trajectoryq(t) in mental spaceXmen. Our deterministic cognitive postulate (which is a generalization of Newton’s deterministic mechani-cal postulate) is that the trajectory q(t) of the evolution of ideas is determined by initial condi-tions and forces. As in Newton’s mechanics, we introduce the velocity 6_{(t) of the changing of the} ideaq(t). This is again an information quantity (a new idea). It can be calculated as the derivative (in mental space Xmen) of q(t) (with respect to mental timet). We start with development of the formalism for a cognitive systemtwith the infor-mation mass 1. Here we can identify the velocity 6 _{with the momentum p=m}6_{. We shall call p a}

moti6_{ation to change the information state q(t).} We postulate that the cognitive dynamics inXmen is described (at least for some cognitive processes) by an information analogue of Newton’s second law. Thus the trajectoryp(t) of the motivation of tis described by equation

p;(t)=f(t,q),p(0)=p0, t,q,pXmen, (4) wheref(t,q) is an information force (generated by external flows of information; in particular, by

other cognitive systems). Thus if the initial moti-vationp0and information forcef(t,q) are known, then the motivation p(t) can be found at each instant of mental time tby integration of Eq. (4). The trajectory q(t) of the evolution of ideas can be found by the integration of equation

q;(t)=p(t), q(0)=q0, t,q,pXmen (5) (if the initial idea q0 is known).

We recall that in Newton’s mechanics a force

f(q), qXmat, is said to be potential if there exists a functionV(q) such thatf(q)= −dV(q)/dq. The function V(q) is called a potential. We use the same terminology in the cognitive mechanics. Here both a force fand potentialVare functions defined on the space of ideas Xmen. The potential

V(q), qXmen, is an information potential, infor

-mation field, which interacts with a cognitive sys-tem t. Such fields are classical (unconscious) cognitive fields.

As we have already mentioned mental time t

need not coincide with physical timetphys. Mental time corresponds to the internal scale of an infor-mation process. For example, for a human indi-vidual t, the parameter tdescribes ‘psychological duration’ of mental processes. Our conscious ex-perience demonstrates that periods of the mental evolution which are quite extended in the tphys -scale can be extremely short in thet-scale and vice versa. In general instances of mental time are ideas which denote stages of the information evo-lution of a cognitive system. We remark thattphys can be also interpreted as a chain of ideas (about counts n=1, 2, ..., for discrete tphys and about counts sR, for continuous tphys). In principle, physical time tphys, can be considered as the spe-cial representation for mental time t. However, it is impossible to reduce all mental times to physi-cal time (even if tphysis defined up to a transfor-mation, tphys=u(sphys)). Different mental systems, t1, …,tN, (and even different mental processes in a cognitive system t) have different mental times,

t1, …,tN. The use of physical time tphys can be considered as an attempt to construct the unique time-scale for all mental processes. However, as we have already mentioned this is impossible. In particular, we could not claim that in general there is an order structure for t. It can be that

instances t, and t2 of mental time can be incom-patible. Thus mental time set cannot be imagined as a straight line.

The notion of mental time can be illustrated by the following example.

2.1. Example 2.1, reading of a book

Suppose that a human individualtis reading a book B on the history of ancient Egypt, E_{. The}

process of reading, p, is not continuous; t inter-rupts p for periods of different duration. Denote by q the state of information of t on E_{. In}

principle, the information evolution of t can be considered as an evolution with respect to physi-cal time s=tphys (mechanical clocks): q=f(s). However, the physical parameter s is not directly related to the information processp. For example, the velocity 6

s of the information state q with

respect tos has nothing to do with the cognitive evolution of thet. Moreover, as a consequence of the jump-structure (with respect to s) of p, 6_{s is} not well defined. Denote by D1r=[s0,s1), D1i= [s1,s2), ... , intervals of reading and interruption of reading. Thus the information process p induces the following split of physical time s: D1

r

,D1

i

, ... ,DM r

,DM i

, .... The intervalsD1

i

, ... ,DM i

, ... must be eliminated. New time parameters¯=f(s) is defined as s¯=s on D1

r_,

s¯=s on D1

i_{, .... The}

parameters¯can be considered as (one of possible) mental (information) scales for the processp. The use of times¯ essentially improves the mathemati-cal description of p. However, there is still no large difference with the standard physics3_.

Suppose now that intervals Dkr, Dki depend on

information that t obtains in the process p: Dkr(ak), Dir(bk), whereak,bkXmenare information strings, ‘ideas’. Here s¯=f(s,c) and q=h(s,c), where sR, cXmen. The next natural step is to eliminate the real parameter s from the descrip-tion of the informadescrip-tion processp and to consider the evolution of the information state q (on the subjectE) with respect to a purely mental parame-ter t. This is information on Ewhich is obtained by t from the corresponding part of B. In the

simplest model we can describetas the text of the book: t=(Ancient Egypt …) (see Section 8 for mathematics). So q=q(t) is a transformation of the information tB into the state of knowledge of t on E. The trajectory g(t)Xmen depends on the initial information stage q0 (on E) of t, the initial motivation p0 oftto perform the informa-tion process p and an information force F(t,q) that changes the motivation. For example, if F0 and p0=0, then q(t)q0. Thus the reading of the bookBdoes not change the state of knowledge of t on ancient Egypt.

This example demonstrates that the informa-tion force F(t,q) which ‘guides’ the information state q of t could not be reduced to external information forces f(t,q) (for example, informa-tion from radio, TV and other books). There exists some additional information force, fC(t,q),

which changes crucially the trajectory q(t)Xmen. If even p0=0 and f0 t is totally isolated from external sources of information and initiallythas no motivation to change his information state on ancient Egypt), in general q(t)*_{q0(the conscious} force fC(t,q) can generate nonzero motivation to

study this subject).

The concrete mathematical representations for mental time t by so called m-adic integers (branches of trees) as well as some other examples will be given in Section 8.

3. Quantum cognitive mechanics, conscious field

First we recall some facts on quantum mechan-ics for material systems. The formalism of quan-tum mechanics was developed for describing motions of physical systems which deviate from motions described be Newton’s Eqs. (2) and (3). For example, let us consider the well known two slit experiment. There is a point source of light O

and two screens S and S%_{. The screen S has two} slitsh1, and h2. Light passes S(through) slits and finally reaches the screenS%_{, where we observe the} interference rings. Let us consider light as the flow of particles, photons. Newton’s equations of mo-tion (Eqs. (2) and (3)) could not explain the interference phenomenon: ‘classical forces’ f in-volved in this experiment could not rule photons

3_{Of course, as f is non-invertible, there are some}

in such a way that they concentrate in some domains of S% _{and practically cannot appear in} some other domains of S%_{. The natural idea (see} Bohm, 1951; De Broglie, 1964) is to assume that there appears some additional force fQ, quantum

force, which must be taken into account in New-ton’s equations. Thus instead of Eq. (2), we have to consider perturbed equation

p;(t)=f(t,q)+fQ(t,q),p(0), t,q,pXmat. (6) It is natural to assume that this new force,

fQ(t,q), is induced by some fieldC(t,q). This field C(t,q) can be found as a solution of Schro¨dinger equation

h i

(_c

(_t(t,q)=

h2 2

(2_c

(_q2(t,q)−V(t,q)c(t,q). (7)

Thus each quantum system propagates together with a wave which ‘guides’ this particle. Such an approach to quantum mechanics is called pilot wa6_{e theory. Formally there are two different} objects: a particle and a wave. Really there is one physical object: a particle which is guided by the pilot wave4

. The C-field associated with a quan-tum system has some properties which imply that C(q) could not be interpreted as the ordinary physical field (as, for example, the electromagnetic field). The quantum force fQ(q) is not connected

withC(q) by the ordinary relationf=dC(q)

dq . The

ordinary relation between a force f and a poten-tial V implies that scaling V“cV, where C is a constant, implies the same scaling for the force, namely, f“cf. In the opposite to such a classical relation quantum forcefQis invariant with respect

to the scaling C“cC the C-function. Thus the magnitude of the C-function (‘quantum poten-tial’) is not directly connected with the magnitude of quantum forcefQ. According to Bell (1987) and

Bohm and Hiley (1993), C(q) is merely an infor-mation filed on material space Xmat. For example, in Bohm and Hiley (1993) C(q) is compared with a radio signal which rules a large ship with the aid of an autopilot. Here the amplitude of the signal is not important, only information carried by this signal is taken into account.5

From the introduction to this paper it is clear how we can transform the classical cognitive me-chanics to quantum cognitive meme-chanics, con

-scious mechanics. The main motivation for such a development of the classical cognitive mechanics is that behavior of conscious systems cannot be described by a ‘classical information force’ f. Be-havior of a conscious cognitive system strongly differs from behavior of unconscious cognitive system (even if both these systems are ruled by the same classical information force f). Thus our information generalization (4) of the second New-ton law is violated for conscious cognitive sys-tems. As in the case of material systems, it is natural to suppose that there exists some addi-tional information force fC(q), conscious force,

associated with a cognitive system. This force changes the trajectory of a cognitive system in the space of ideas Xmen. A new ‘quantum’ conscious trajectory is described by equation

p;(t)=f(t,q)+fC(t,q),p(0)=p0, t,q,pXmen. (8) The conscious force fC(t,q) is connected with a C-field, aconscious field, by the same relation as in the pilot wave formalism for material systems. An information Schro¨dinger equation (see Section 10) describes the evolution of the conscious C -field.

5_{The pilot wave theory does not give a clear answer to the}

question: Is some amount of physical energy transmitted by the C-field or not? The book of Bohm and Hiley (1993) contains an interesting discussion on this problem. It seems that, despite the general attitude to the information interpreta-tion ofc, they still suppose thatCmust carry some physical energy. Compared with the energy of a quantum system, this energy is negligible (as in the example with the ship). Another interesting consequence of Bohm – Hiley considerations is that quantum systems might have rather complex internal structure (roughly speaking a quantum system must contain some device to transfer information obtained from theC-field).

4_{The pilot wave theory does not give the standard}

interpre-tation of quantum mechanics, namely, the orthodox Copen-hagen interpretation. By the latter interpretation it is impossible to describe individual trajectories of quantum parti-cles. Probably an analogue of the orthodox Copenhagen inter-pretation could be also interesting to quantize the classical cognitive mechanics. However, in the present paper we shall concentrate on an analogue of the pilot wave formalism.

In ordinary quantum mechanics the origin of the C-field is not clear. It seems natural for me that C(q) is generated by a quantum particle. However, this assumption is too speculative for material quantum systems, because there are no experimental evidences that a quantum particle has a complex internal structure such that it could generate theC-field.6_{In the pilot wave formalism} it is supposed that the C-field is created simulta-neously with a quantum particle (this field is only formally treated as separated from the particle).

In quantum cognitive theory the assumption on a complex internal structure of a quantum (con-scious) cognitive system is quite natural. In princi-ple we may suppose that the conscious fieldC(q),

qXmen, is generated by classical information pro-cesses in a cognitive system t. Moreover, it is natural to suppose that higher information com-plexity of t implies that the C-field of t induces the information force fC of larger information

magnitude. We recall that in the pilot wave for-malism (both for material and mental systems) the magnitude of C is not directly related to the magnitude of fC. At the present stage of

knowl-edge on cognitive phenomena the idea that the C-field is generated by t seems to be the most natural.7

4. Collective unconscious and conscious cognitive phenomena

In the previous two sections we have studied the classical and quantum mechanical formalisms

for individual cognitive systems. In this section we consider collective classical (unconscious) and quantum (conscious) cognitive phenomena.

We start with the classical (unconscious) cogni-tive mechanics. Lett1, …,tNbe a family of

cogni-tive systems with mental spacesXmen,1, …,Xmen.N.

We introduce mental space Xmenof this family of cognitive systems by setting Xmen=Xmen,1× …×Xmen,N. Elements of this space are vectors of

information states q=(q1, …,qN) of individual

cognitive systems tj. We assume that there exists

an information potential V(q1, …,qN) which

in-duces information forces fj(q1, …,qN). The

poten-tial Vis generated by information interactions of cognitive systems t1, …,tN as well as by external

information fields. The evolution of the motiva-tionpj(t) and the information stateqj(t) of thejth

cognitive system tj is described by equations:

p;j(t)=fj(t,q1, ... ,qN),pj(0)=p0j (9)

q;j(t)=pj(t),q(0)=q0j, t,q,pXmen (10) In general for different j these evolutions are not independent.

4.1. Example 4.1

Let V(ql,q2)=a(q1−q2) 2

, where a is some in-formation constant (given by a p-adic number in the mathematical model). Motions of cognitive systemst1andt2in mental space are not indepen-dent; the (information) magnitude of the constant of coupling a gives the strength of this depen-dence. On the other hand, if, for example,

V(q1,q2)=q21+q22, then motions oft1and t2are independent.

This model can be used not only for the de-scription of collective cognitive phenomena for a group of different cognitive systems t1, …,tN but

also for a family of thinking processes in one fixed cognitive system. For example, it is natural to suppose that the brain contains a large number of dynamical thinking processors (see Khrennikov (1997) for a mathematical model), p1, …,pN

which produce ideasq(t), …,qN(t), related to

dif-ferent domains of human activity.8 _{We can apply}

6_{Even the Bohm – Hiley considerations on the complex}

in-ternal structure of quantum particles do not go so far to assume that a quantum particle is a generator of theC-field. The Bohm – Hiley complexity is merely complexity of a re-ceiver of radio signals on a ship.

7_{On the other hand, if we try to generalize ideas of material}

quantum mechanics to the cognitive phenomena, then we have to suppose that theC-field is created simultaneously with the creation of a cognitive system t. Such a viewpoint on the origin of the conscious field implies the great mystery of the act of creation of a conscious cognitive system. Here the conscious field is ignited(by whom?) in a cognitive system. Thus it seems to be impossible to create artificial cognitive systems by just ‘mechanical’ increasing of their information complexity.

8_{For example,p}

1produces ideas on food,p2produces ideas

our classical cognitive (collective) mechanics to describe the simultaneous functioning of thinking modules p1, …,pN.The main consequence of our model is that ideasq1(t), …,qN(t) and motivations

p1(t), …,pN(t) do not evolve independently. Their

simultaneous evolution is controlled by the infor-mation potential V(q1, …,qN). It must be

under-lined that an interaction between thinking modules p1, …,pN has the purely information origin. The

potential V(q1, …,qN) need not be generated by

physical field (for example, the electromagnetic field). A change of the information stateqj“qj%(or

motivationpj“pj%) of one of thinking processorspj

will automatically imply (via the information inter-action V(q1, …,qN)) a change of information

states (and motivations) of all other thinking blocks. In principleno physical energyis involved in this process of the collective cognitive evolution. In some sense this is the process of the cognitive (but still unconscious) self-regulation. Different cognitive systems can have different information potentialsV(q1, …,qN) which give different types

of connections between thinking blockspj.

4.2. Example 4.2

Let thinking processorsp1,p2andp3be respon-sible for science, food and sex, respectively. Let

V(q1,q2,q3)

=a1q1 2

+a2q2 2

+a3q3 2

+a12(q1−q2) 2

+a23(q2−q3) 2_+a

13(q1−q3)

2_{. (11)}

If the information constanta1, strongly dominates over all other information constants, then the scientific thinking blockp1works practically inde-pendent from the blocksp2andp3. Ifa12 (ora13) dominates over all other constants, then there is the strong connection between science and food (or science and sex).

Moreover, the information potentialV can de-pend on the mental time of a cognitive system,

V=V(t,q1, …,qN). Thus at different instances of

mental timeta cognitive system can have different information connections between thinking blocks p1, …,pN.

We are now going to describe the collective quantum (conscious) phenomena. Lett1, …,tNbe

a family of cognitive systems. The classical infor-mation motion is described by classical (uncon-scious) information forces9 _f

j(t,q1, …,qN) by

cognitive second Newton law (5). However, as in the pilot wave formalism for many particles, for any family t1, …,tN of cognitive systems, there

exists a C-field, C(q1, …,qN), of this family. This

field is defined on the mental space Xmen=

Xmen,1×…Xmen,N. This field generates additional

information forces fj(t,q1, …,qN) (conscious

forces) and the Newton’s (classical/unconscious) cognitive dynamics must be changed to (quantum/

conscious) cognitive dynamics

p;j(t)=fj(t,q1, ... ,qN)+fj,C(t,q1, ... ,qN),

j=1, 2, ... ,N. (12) In general the conscious force fj,C=

fj,C(t,q1, …,qN) depends on all information

coor-dinates q1, …,qN (information states of cognitive

systemst1, …,tN). Thus the consciousness of each individual cognitive systemtjdepends on

informa-tion processes in all cognitive systems t1, …,tN.

The level of this dependence is determined by the form of the collective C-function. As in the ordi-nary pilot wave theory in our cognitive model the factorization

C(t,q1, ... ,qN)=5 N

j=1

Cj(t,qj)

of theC-function implies that the conscious force

fj,C depends only on the coordinate qj. Thus the

factorization of C eliminates the collective con-scious effect.

As in the classical cognitive mechanics, the above considerations can be applied to a system of thinking blocksp1, …,pNof the individual

cogni-tive systemt(for example, the human brain). The conscious field C of t depends on information states q1, …,qN of all thinking blocks.

9_{Throughout this paper we use ‘classical’ and ‘quantum’ as}

synonyms of ‘unconscious’ and ‘conscious.’ In fact, it would be better to use only the biological terminology. But we prefer to use also the physical terminology to underline the parallel development of mechanical formalisms for material and men-tal systems.

5. Information connection between mental and physiological processes

Classical and quantum fields,V(q1, …,qN) and C(q1, …,qN), induce dependence between

individ-ual thinking blocks p1, …,pN of a cognitive

sys-tem t or individuals t1, …,tN belonging to a

social group G.

In particular, this implies that all physiological systems of the organism are closely connected on the information level. Therefore a decease in one of these systems may have an influence to other systems (even if they have no close connection on the physiological level). Of course, this is not a new fact for medicine. But we now have the mathematical model (see Sections 7 – 9). And, in principle, we could (at least after development of the model) compute some effects of the informa-tion influence on physiological processes. More-over, purely mental processes in the brain (which are not directly related to physiological processes) are connected on the information level with phys-iological processes. For example, let the mental blockp1, control functioning of the heart and the block p2controls some psychological process (for example, relations with some person) and let the classical information potential V(q1,q2)=aq1q2, whereais a coupling information constant (given by a p-adic number in our mathematical model). Then purely mental process inp2has an influence to functioning of the heart. The classical informa-tion force f(q1,q2) applied to the p1 is equal to

−aq2. Thus it depends on the evolution q2(t) of the psychological process.

The presence of the conscious field C(q1, …,qN) makes the connection between

phys-iological and purely mental process more compli-cated. There is the possibility of the conscious control of human physiological systems. In princi-ple, if a person could change its conscious field C(q1, …,qN), she/he could change (by just an

information influence) the functioning of some physiological systems.

Our model explains well the origin ofhomeopa

-thy. In fact, by a homeopathic treatment it is possible to change the information potential

V(q1, …,qN) of the organism. Microscopic

quan-tities of medicines which are used in the

home-opathic treatment are just sources of infor-mation. In principle, homeopathic medicine need not be applied directly to an ill physiological system pk (described by the information state

qk). The information concentrated in the

home-opathic medicine could be applied to some other information stateqj,j"k. The change ofqj, qj“

qj%, will imply the change of the trajectory qk(t)

(via the change of the information force

fk(t,q1, …,qk, …,qj, …qN)

“fk(t,q1, …,qk, …,qj%, …,qN)).

6. Freud’s psychoanalysis as a reconstruction of conscious field

By Freud’s theory, Freud (1933), mental space

Xiof a human individualiis split in two domains:

(1) a domain of conscious ideas Xic; (2) a domain

of unconscious ideas Xiu. Thus

Xi=Xi c_@

Xi u

.

In our information model Freud’s idea is repre-sented in the following way. Let f: Xi“Xi, be

some function. As usual, we define a support of f

as the set suppf={xXi: f(x)"0}. LetC be the

conscious field generated by the individual i and

fC be the corresponding conscious force.

Then suppfCis the set of conscious ideas (ideas

which can interact with theC-field),Xic=suppfC.

The set Xi¯suppfCis the set of unconscious ideas

Xi u

(ideas which cannot interact with the C -field).10

The motion of i in the space of ideas Xi is

described by the dynamical system:

p;i(t)=f(t,qi)+fC(t,qi), qiXi (13)

where f= −(Vi

(_q

i

is the classical (unconscious) force generated by the classical information po-tential Vi of i and fC= −

(_C

i

(_qi is the quantum (conscious) force generated by the conscious in-formation potential Ciof i. In the subspaceXi

u_of

10_{We remark that the sets of ideas, suppf}

Cand suppC, do

not coincide. It can be that suppfC is a proper subset of

unconscious ideas this dynamical system is reduced to the system:

p;i(t)=f(t,qi), qiXi

u_{. (14)}

Let D be some domain in Xi

u _{and let a classical}

information potentialVi(t,qi),qiXiu, have a form

such that the dynamical system (14) has the domain

Das a domain of attraction of trajectories. Thus starting with any initial ideaq0Xiuthe information

state qi(t) of i will always evolve to D. The

dynamical system (14) is located in the space of unconscious ideas. Here the conscious forcefC is

equal to zero. Therefore the i could not change consciously the dynamics (14).

Suppose now that theDis some domain of ‘bad ideas’. For example, if D is a domain of ‘black ideas’, thenihas a depression; ifDis a domain of ideas connected with alcohol, thenihas problems with alcohol; ifDis a domain of aggressive ideas, then i will demonstrate aggressive behavior (this behavior looks as totally unmotivated: starting with an arbitrary unconscious ideaq0the individual

i will always arrive to aggression).

The aim of psychoanalysis is to extend the domain of conscious ideas Xic=supp fC. This ex-tension will perturb dynamics (14) by the action of a conscious forcefC. This perturbation may change

the evolution of ideas in such a way that the domain

Dwill not be anymore a domain of attraction for the whole space of unconscious ideasXi

u_{. Starting}

withq0Xi

u _{theican have trajectories q}

i(t) which

will be never attracted by the domain of ‘bad ideas’

D.

The pair, a cognitive systemiand a psychoana-lytic p, can be considered as a coupled system of transformers of information. The information cou-pling betweeniandpwill generate a new informa-tion classical potential Vi,p(t,q1,q2) which is defined on mental spaceX=Xi×Xp, whereXiand

Xp, are spaces of ideas of the individuali and the psychoanalyticp, respectively.

Dynamics of the conscious fieldCi(t,qi) of i is

described by the Schro¨dinger equation

h i

(_Ci

(_t (t,qi)=

h2 2

(2

Ci

(_qi2(t,qi)−Vi(t,qi)C(t,qi)

(15)

Dynamics of the conscious fieldCi,p(t,qi,qp) of the system (i,p) is described by the Schro¨dinger equa-tion

h i

(_Ci,p

(_t (t,qi,qp)

=h

2

2

(2

(_q

i

2+

(2

(_q

p

2

Ci,p(t,qi,qp)

−Vi,p(t,qi)Ci,p(t,qi,qp). (16) If now the conscious force

f0C(t,qi,qp)= −

(_C

i,p(t,qi,qp)

(_q

i

"0 (17)

for some ideasqiXi

u_{at least for some ideas}

qpXp, then the motion ofiin the domain of unconscious ideas Xi

u _{can be controlled consciously (here}

Ci,p(t,qi,qp) is the conscious potential induced by Ci,p(t,qi,qp)). In fact, this means thatXi

u_{is reduced}

andXi

c_{is extended. The aim of the psychoanalytic}

p is to find ideas qpXisuch that (17) takes place

for unconscious ideasqiXiuofi. As the process of

psychoanalysis is a conscious process (at least for p), it is natural to assume that ideasqp, used byp to induce condition (17) are conscious: qpXpc. Typically such ideas are represented in the form of special questions toi. In some sense this is a kind of conscious intervention of the psychoanalytic p in the unconscious domain of the individuali. Ifp finds a domain O¦Xi

u _{in that condition (17) is}

satisfied, then in this domain dynamics (14) is transformed in the conscious dynamics

p;i(t)=f0(t,qi,qp)+f0C(t,qi,qp), qiO. (18)

Under some circumstances this dynamical system can be free from ‘pathological features’ of dynam-ical system (14).

Of course, even the change of the classical potential Vi(t,qi) to a new classical potential

Vi,p(t,qi,qp) changes the motion ofi: a new dynam-ics is ruled by the classical force f0(t,qi,qp) instead of the classical forcef(t,qi). However, it is not easy

to change strongly the classical forcef(t,qi) on the

domain of unconscious ideasXi

u_{by just the change}

of the classical potential. TypicallyVi,p(t,qi,qp)=

Vi(qi)+Vp(qp)+G(qi,qp), where the (informa-tion) magnitude of G(qi,qp) is small for ideas

qiXi

classical potential may induce the strong change of the quantum potential.

6.0.1. Conclusion

Freud’s psychoanalysis is nothing than the change of information dynamics of an individuali

(having some mental decease) via an extension of the support of the quantum force. Such an exten-sion is the extenexten-sion of the domain of conscious ideas Xic (and the reduction of the domain of

unconscious ideas Xiu). This extension is realized

by the information coupling between an individ-ualiand a psychoanalyticp. By minor change of the classical information potential the p strongly changes the conscious force acting on the i. Dy-namics of ideas in the unconscious domain ofi is changed. This change eliminates the mental decease.

In the same way we can describe information processes which take place inhypnotism. Here by the conscious information coupling (described by Schro¨inger equation (Eq. (16))) between an indi-viduali and a hypnotizerpthe conscious dynam-ics (13) is changed in such a way that the conscious forcefC(t,q1) is practically totally elimi-nated by the action of the conscious force

f0C(t,qi,qp). For example, let f0C(t,qi,qp)= −

fC(q1)+fC(q2). Then information behavior of thei is ‘ruled’ by the conscious force fC(q2) of the p.

7. Mathematical models of material and mental spaces; real and p-adic numbers

From our viewpoint real spaces (Newton’s ab-solute space or spaces of general relativity) give only a particular class of information spaces. These real information spaces are characterized by the special system for the coding of informa-tion and the special distance on the space of vectors of information. Any natural numberm\ 1 can be chosen as the basis of the coding system. Each x[0, 1] can be presented in the form:

x=a0a1...an... , (19)

where aj=1, …,m−1, are digits. We denote the

set of all sequences of the form (19) by the symbol

Xm. For example, let us fix m=10. One of the

main properties of the real cording system is the identification of the form:

10 ... 0 ...=09 ... 9 ...; 010 ... 0 ...=009 ... 9 ...; ... (20) In fact, this identification is closely connected withthe order structureon the real lineR(and the metric related to this order structure). For eachx, there exist ‘right’ and ‘left’ hand sides neighbor-hoods; there exist arbitrary small right and left shifts. The identification (20) is connected with the description of left hand side neighborhoods. 7.1. Example 7.1

Let x=10 … 0 … . Then x can be approxi-mated from the left hand side with an arbitrary precision by numbers of the formy=09 … 90 … . The following description of right hand side neighborhoods will be very important in our fur-ther considerations.

(AS) Let x=a0…am… . Then the numbers

(vectors of information) which are close to the x

from the right hand side have the form y=

b0…bm…, where a0=b0, …,am=bm for suffi-ciently large m.

This nearness has a natural information (cogni-tive) interpretation: (AS) implies the ability to form associations for cognitive systems which use this nearness to compare vectors of information. By (AS) two communications (two ideas in a model of human thinking, Khrennikov (1997)) which have the same codes for sufficiently large number of first (the most important) positions in cording sequences are identified by a comparator of a cognitive system. Numbers (vectors of infor-mation) which are close to x from the left hand side could not be characterized in the same way (see Section 7.1, there x and y are very close but their codes differ strongly).

7.1.1. Conclusion

The system of real numbers has been created as a coding system for information which the con-sciousness receives from reality. The main proper-ties of this coding system are the order structure

on the set of information vectors and the re-stricted ability (see (AS)) to form associations.

Finally, we pay attention to the ‘universal cod-ing property’ of the real system: any natural num-berm\1 can be used as the basis of this system. Thus any information process can be equivalently described by using, for example, 2-bits coding or 1997-bits coding. All these properties of the real coding system were incorporated in every physical model. I do not think that all information pro-cesses (especially cognitive) have an order struc-ture. On the other hand, the scale of coding system

m\1 may play the important role in a description of an information process.

Let us ‘modify’ the real coding system. We eliminate the identification (20). Since now, there is no order structure on the setXm. of information

vectors. We consider on Xm the nearness defined

by (AS)11_{. This nearness can be described by a} metric. The corresponding (complete) metric space is isomorphic to the ring of so called m-adic integers Zm (see Schikhov, 1984).

Therefore it is natural to use m-adic numbers for a description of information (at least cognitive) processes. Mathematically it is convenient to use prime numbers m=p\1 (see Schikhov (1984)). We arrive to the domain of an extended mathe-matical formalism, p-adic analysis. We present some facts about p-adic numbers.

The field of real numbersRis constructed as the completion of the field of rational numbersQwith respect to the metricr(x,y)=x−y, where · is the usual valuation given by the absolute value. The fields ofp-adic numbersQpare constructed in

a corresponding way, but using other valuations. For a prime numberp, the p-adic valuation ·_p is defined in the following way. First we define it for natural numbers. Every natural number n can be represented as the product of prime numbers,

n=2r2₃r3_…prp_{…, and we definenp=p}−rp_,

writ-ing 0p=0 and −np=np. We then extend the definition of thep-adic valuation·to all rational numbers by setting n/mp=np/mp for m"0. The completion of Q with respect to the metric r(x,y)=x−yp is the locally compact field of

p-adic numbers Qp. The number fields Rand Qp

are unique in a sense, since by Ostrovsky’s

theo-rem (see Schikhov (1984)) · and·_p are the only possible valuations on Q, but have quite distinc-tive properties.

Unlike the absolute value distance·, thep-adic valuation satisfies the strong triangle inequality

x+y_p5max[x_p,y_p], x,yQp

Write Ur(a)={xQp: x−ap5r} andUr−(a)=

{xQp:x−apBr}, wherer=pnandn=0,91,

92, … . These are the ‘closed’ and ‘open’ balls in

Qpwhile the setsSr(a)={xQp:x−ap=r} are the spheres inQpof such radiir. These sets (balls

and spheres) have a somewhat strange topological structure from the viewpoint of our usual Eu-clidean intuition: they are both open and closed at the same time, and as such are called clopensets. Another interesting property ofp-adic balls is that two balls have nonempty intersection if and only if one of them is contained in the other. Also, we note that any point of ap-adic ball can be chosen as its center, so such a ball is thus not uniquely characterized by its center and radius. Finally, any

p-adic ball Ur(0) is an additive subgroup of Qp, while the ball U1(0) is also a ring, which is called the ring of p-adic integersand is denoted by Zp.

Any xQp has a unique canonical expansion

(which converges in the ·p-norm) of the form

x=a−n/p n

+...a0+... +akp k

+...

where the aj{0,p−1} are the ‘digits’ of the

p-adic expansion. The elements xZp have the expansion

x=a0+...+akp k_+...

and can thus be identified with the sequences of digits

x=a0...ak...

The p-adic exponential function n=0

x

n

n!. The series converges in Qp if

xp5rp, where rp=1/p, p"2 and r2=1/4 (21)

p-adic trigonometric functions sinxand cosxare defined by the standard power series. These series have the same radius of convergence rp as the

exponential series.

11_{Thus here all information is considered from the}

for theI-stateq(t) oftcons. One of possibilities to perturb classicalI-equation, (4), ofI-motion is to use an analogue with material objects (which can be considered as a particular class of transformers of information, see Khrennikov 1997, 1999) and introduce a new (conscious) force via thec -func-tion which satisfies Schro¨dinger I-equation, (Eq. (26)). Of course, such an approach would be strongly improved if we could explain the mecha-nism of generation of the conscious c-field by

tcons. We note that such a mechanism is still unknown in the ordinary pilot wave theory for material objects, see Bohm and Hiley, 1993.

(S) It is impossible to perform a measurement of the I-state q as well as the motivation p or some otherI-quantity of a conscious system tcons without to disturb this system.22_{In particular, we} could not perform a measurement of bothqandp for the same tcons. As in the ordinary quantum mechanics, we have to use large statistical ensem-blesSof conscious systems and perform statistical measurements for such ensembles to find proba-bilities for realizations of I-quantities. One of possibilities is to use again an analogue with the ordinary quantum formalism and introduce a field of probabilitiesc(t,q) (which describes statistical properties of the ensembleS) such that the square of the amplitude of c

C(t,q)2_{=c(t,q)c( (t,q)}

gives the probability to findtconsSin the I-state q (at the instant tof mental time).

One of the main features of the pilot wave theory for material objects is that the pilot wave field coincides with the probability field:c(t,x)= cpilot(t,x)=c(t,x)prob (in fact, there is no clear explanation of such a coincidence). We also pos-tulate that the conscious c-field (which generates the conscious forcefCcoincides with the

probabil-ity field.

Let tcons be a conscious system. A performance of theI-stateq=q(t) oftcons can be viewed as a

performance of an image on some screen S. This screen continuously demonstrates ideas of tcons. The tconsis a self-observer for these ideas. As we have already noted, our conscious experience says that a new I-state q(t+Dt) (an ‘image onS’ at the instant t+Dt) is generated not only on the basis of the previousI-stateq(t) (an ‘image onS_’ at the instant t) with the aid of I-forces f(t,q) which are generated by external I-potentials. The main feature of tconsis that q(t+Dt) depends on all information which is collected in tcons.23 _Thus Newton’s I-equation (Eq. (4)) must be modified to describe such a unity of information in the process of the I-evolution of tcons.

Denote by D(t)Dt_cons(t) the domain in Xmen corresponding to information which is contained in the memory of tcons at the moment t. Then q(t+Dt)=F(q(t),D(t)). The main problem is to find a transformation F that can provide the adequate description of the conscious evolution of the I-state. We propose the following model.

For each idea xD(t), we define an I-quantity

c(t,x) which describes the I-activity of the idea x. In this way we introduce a newI-field c(t,x), xD(t), a field of memory acti6_{ation. It is}

postu-lated that the field of memory activation gives the pilot wave c(t,x), conscious field, which guides the I-state q(t) of tcons.

Let tand t% _{be two different instances of time.}

Schro¨dinger I-equation (Eq. (26)) implies that

c(t%_{,x), can be found as the result of integration}

of c(t,x) over the memory domain D(t):

c(t%_,x)=

&

D(t)

K(t%_{,t,x,y)c(t,y)dy (31)}

where the kernel K(t%_{,t,x,y) describes the time}

propagation of memory activation. K(t%_,t,x,y)

describes the influence of an idea yD(t) on the idea xD(t%_{). One of the main consequences of}

the quantum I-formalism is that memory activa-tion fields at different instances of time are con-nected by a linear transformation.

The Bohmian mechanics implies that the quan-tum forcefQdoes not depend on the amplitudeR

of a quantum field c. Roughly speaking the 22_{The process of anI-measurement is an interaction oft}

cons

with someI-potentialV(t,q). Anytconsis extremely

informa-tional unstable. Even low information potentialsV(t,q) dis-turb tcons. Such an I-instability is the common feature of

conscious and quantum systems.

23_{Hiley and Pylkka¨nen (1997) call such a property ‘the}

strength of fQ depends on the variation of R:

fQ= −Q%, where a quantum potential Q is

defined as Q= −R%%_{/R. For example, if R} Const, then fQ=0; ifR(x)=x (rather slow

vari-ation), then fQ is still equal to 0; if R(x)=x2,

then fQ:"0.

24

The differential calculus on the mental space Xmen=Zp, where p\1 is a prime

number, reproduces the same analytic properties (in particular, the form dependence) for con-scious potentials C(x). Moreover, we provide the explanation of this form (in fact, variation) dependence. In our model the pilot information

c-function is interpreted as a memory activation field. Our conscious experience says that the uniformly high activation of all ideas in the memory of tcons could not imply a conscious behavior. Uniform activation (C(x)=

Const,xDtcons) eliminates at all a memory ef-fect from the evolution of the I-state q(t) of

tcons. Only rather strong variation of the field C(x), xDtcons, of memory activation produces a conscious perturbation of the I-motion of

tcons.

12.0.1. Conclusion

The variation dependence of the conscious (quantum) force fC on the information pilot

wave c is a consequence of the fact that c is nothing else than a field of memory activation of a conscious system.

Of course, the information pilot wave theory predicts essentially more than we can extract from our vague conscious experience. In fact, (as in the Bohmian mechanics) fC=(C§C−

C¦C%_)/C2

. Thus the conscious force fC depends

not only on the first variation dC of the ampli-tude of the memory activation field, but also on the second and third variations, d2

C, d3 C.

These predictions must be verified experimen-tally.

12.1. Remark 12.1, neurophysiologic links

In the spatial neurophysiologic models of memory spatially extended groups of neurons represent human images and ideas. Thus it is possible (at least in principle) to construct a transformation Js: Bs“Bi, where Bs¦Xmat=R

3 is the spatial domain of a ‘physical brain’ and Bi¦Xmen=Zm

k _{is the}

I-domain of an ‘informa-tion brain’. Thus the memory activa‘informa-tion field

c(x), xBi, can be represented as an I-field on

the spatial domain Bs: f(u)=c(Js(u)). We now

consider the simplest model with the 0/1-coding (nonactivated/activated) of activation of ideas x in the memory. Roughly speaking the informa-tion pilot wave theory predicts that only large spatial variations of f(u) can imply conscious behavior. Of course, this is a vague application of our (information) pilot wave formalism. The transformation Js, can disturb the smooth

struc-ture of the mental space Zm

k _{and variations with}

respect to neuron’s location uBs¦R

3

may have no links to variations with respect to idea’s lo-cation xBi¦Zm

k_{. Moreover, the purely}

mathe-matical experience says that this is probably the case. The most natural transformations a: Zm“

R have images of fractal types (Vladimirov et al., 1994). This can be considered as a reason in favor of the frequency domain model, Hoppen-steadt, 1997, by that dust-like configurations of neurons in Bs, (oscillating with the same

fre-quency) seem to correspond to the same idea (image). In the latter model it is possible to con-struct a transformation Jf: Bf“Bi, where Bf¦

R3 _{is the domain of a ‘frequency brain’. Thus} the field c(x), xBi, can be represented as an

I-field on the frequency domain Bf: f(n)= c(Js(n)). The above experimental predictions

can be also tested for variations of f(n).

We now discuss briefly connection between the memory activation field and the probability field. Let us consider a large statistical ensemble S of conscious systems t having (at least ap-proximately) the same memory activation field 24_{We remark that the ordinary pilot wave theory could not}

provide a reasonable explanation of such a connection be-tween the field and force. In fact, D. Bohm and B. Hiley understood that this is due to the information nature of the pilot wave. However, they still tried to reduce such a new field to some ordinary physical field.

c(x). Suppose that initial I-states qt0 of tS are

uniformly distributed. After some period DT of the conscious evolution via (Eq. (8)), where fC is

defined by the stationary field c(x), the proba-bility P(qt(t)=q) to find tS in the I-state q

(at the instance t) is equal to C2

(q). Thus the memory activation field coincides with the prob-ability field due to the stationary (statistical) sta-bilization of information states of conscious systems tS. Intervals DT of such a stabiliza-tion to a stastabiliza-tionary distribustabiliza-tion are relatively small. Therefore only stationary distributions are observed.

12.2. Remark 12.2, a quantum particle as a complex I-system

The Bohmian mechanics cannot explain the origin of the pilot wave field. We can try to use an analogue between conscious and quantum systems to clarify this point. Let as consider (following Bohm and Hiley, 1993) a quantum particle s as a complex I-system. Suppose that such a system has a kind of memory. This memory contains not only information on con-temporary ‘properties’ of s, but also information based on the previous experience of s. If we use the anthropological principle and apply the con-scious I-model to s, then the Bohmian pilot wave is nothing else than a field of memory activation of s. The I-state q(t) of s is given by the coordinates of location of s in R3_{. Thus s is} guided not only by classical potentials (which can be considered as just a particular class of classical I-potentials), but also by the potential of the memory activation field of s. As a conse-quence of the statistical stabilization to a sta-tionary distribution, the memory activation field coincides with the field of probabilities. In fact, such an approach unifies the Bohmian and Copenhagen interpretations of quantum mechan-ics. By the Copenhagen interpretation s has no definite position before a position measurement; s is in the ‘superposition’ of different positions. By the Bohmian theory s has the definite posi-tion at each instant of time. In our model ‘su-perposition’ of positions is nothing else than

memory on these positions; the ‘real’ position of s is the I-state of s.

12.3. Example 12.1, a free conscious system Let us consider the memory activation field of a free conscious system tcons, see Section 10.1.25 Suppose that the memory of tcons is activated by some concrete motivation, p=a (which is not mixed with other motivations). Then ca(x)=

eiax/hp_{, where h}

p=1/p and p\1 is a prime

num-ber (the basis of the coding system of tcons). Such a field can be called a moti6_{ation wa}6_e.

This motivation wave propagates via I -Schro¨dinger equation: ca(t,x)=exp{i(ax−Eat)/

hp}, where Ea=a2/2m is the information

(‘psychical’) energy of the motivation p=a. Here S(t,x)=(ax−Eat) and C(t,x)1. Thus

the conscious force fC0.

26

Suppose now that two different motivations,

p=a and p=b activate the memory of tcons. The corresponding motivation waves are

ca(x)=eiax/hp _{and cb(x)=e}ibx/hp_{. Suppose that}

these waves have amplitudes da, dbQp. Suppose

also that there exists a phase shift, u, between these two waves of motivations in the memory of tcons. One of consequences of the quantum I-formalism is that the total memory activation field c(t,x) is a linear combination of these mo-tivation waves: c(x)=dae

iu/h_p

ca(x)+dbcb(x). The presence of the phase u implies that the motivation p=a started to activate the memory earlier than the motivation p=b (at the instant s= −u/Ea). If the motivation p=a has a small

I-energy, namely, EapB B1 then a nontrivial

phase shift u can be obtained for rather large time shift s. The I-motion in the presence of two different (‘competitive’) motivation waves in the memory of tcons is quite complicated. It is

25_{Such at}

consis an extremely idealized conscious system. A

conscious system could not be totally isolated from external

I-fields. In any casetconsmust continuously receive

informa-tion on physiological processes in his body.

26_{As we have already noticed, the uniformly strong}

activa-tion of all ideas in the memory of a conscious system implies unconscious behavior.

guided by a nontrivial conscious force fC(t,x).

We omit rather complicated mathematical ex-pression which formally coincides with the stan-dard expression (Holland, 1993). As the memory activation field coincides with the field of proba-bilities, probabilities to observe motivations

p=a and p=b are equal to da

2

and db

2 . As it was already mentioned, in general these are not rational numbers. Thus in general these proba-bilities could not be interpreted as ordinary lim-its of relative frequencies. There might be violations of the law of large numbers (the sta-bilization of frequencies) in measurements on conscious systems (see Khrennikov, 1999, for the details).

Complexity of the I-motion essentially in-creases if c is determined by k]3 different mo-tivations. Finally we remark that (in the opposite to the Bohmian mechanics) waves ca, and cb, a"b are not orthogonal. The covaria-tion Bca, cb\ "0. Thus all motivation waves in a conscious system are correlated.

12.4. Example 12.2, conscious e6_{olution of} complex biosystems

Let t be a biosystern having a high I -com-plexity and let l1, …,lM be different living forms

belonging to t. We consider the biosystem t as an I-object (transformer of information) with the I-state qt_{(t). It is supposed that t has a}

kind of collective memory. Let ct_{(t,x) be the}

memory activation field of t. I-dynamics of t

depends not only on ‘classical’ information fields, but also on the activation of the collective memory of t. Suppose that t can be considered (at least approximately) as an I-isolated biosys-tem. Each living form lj has a motivation aj (to

change the total information state qt_{). The pilot}

I-formalism implies that the total motivation pt

of the biosystem t could not be obtained via the summation of motivations aj. The mechanism of

generation of pt _{is more complicated. Each} aj

activates in the collective memory of t the moti-vation wave ca

j. A superposition of these waves

gives the memory activation field (‘conscious field’) of t.

ct

(t,x)=% j

dje iuj/hp_ca

j(t,x)

(compare with Eq. (30)).27 _{The c(t,x) induces} rather complicated conscious potential Ct_(t,x)

which guides the motivation pt_{and I-state q}t _of t.

Finally we discuss the correspondence between states of brain and states of mind. The thesis that to every state of brain there corresponds a unique state of mind is often called the materi-alistic axiom of cognitive science, see Bergson, 1919. In fact, this axiom is the basis of the modern neurophysiologic investigations. How-ever, there are some reasons to suppose that the brain does not determine the content of the mind (see Hautama¨ki, 1997, for the extended analysis of this question). Here we refer only to Putnam’s theory of meanings. According to Put-nam, 1988, ‘meanings are not in the head’, they are rather in the world, and reference is a social phenomenon. We shall prove that in our mathe-matical model for mental processes the material-istic axiom is violated.

Let us consider again a free conscious system

tcons. It will be shown that motivations of tcons could not be identified with waves of memory activation. Suppose that there exists a fixed mo-tivation p=a which activates the memory of

tcons. We know that the field ca(x)=e ixa/hp _is

the eigen-function of the position operator pˆ. Thus, for an ensemble Sa of free systems with

the same memory activation field ca(x),

obser-vations of the motivation will give the value

p=a with the probability 1. Let l(x) and u(x) be arbitrary differentiable functions, Zp“Zp,

having zero derivatives. Set cal,u

(x)=R(x)eiS(x)/ hp where R(x)=e

l(x) _{and S(x)=ax−u(x). The}

cal,u(x) is also an eigen-function of the

tion operator. Thus observations of the motiva-tion for an ensemble Sa

l,u _{of conscious systems}

27_{For some biosystems, amplitudesd}

j,j=1, … ,M, can be chosen as sizes of populations oflj, j=1, … ,M.

with memory activation fieldcal,u₍

x) will also give the motivation p=a. However, the fields cal,u₍

x) and ca(x) can have extremely different distribu-tions of activation of ideas in the memory. By any ‘social’ (external) observer all these fields (states of brain) are interpreted as the same state of mind, namely the motivationp=a.

Appendix A

A.1. p-adic differential calculus

The system ofp-adic numbers Qp is a number

field. Thus the operations of addition, sub-traction, multiplication and division are well defined. The derivative of a functionf:Qp“

Qp is defined (as usual) as limDxp“0 f(x+Dx)−f(x)

Dx . The main distinguishing feature of p-adic analysis is the existence of non-locally constant functions with zero derivative. We present the following well known example (see Schikhov, 1984), p.74. The function f: Zp“Zp is

defined as f(x)=n=0

anp

2n _{for x=} n=0

anpn.

This function is injective (f(x1)"f(x2) for x1" x2) and f%0.

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