Download Classical and quantum dynamics on p

BioSystems 56 (2000) 95 – 120
www.elsevier.com/locate/biosystems
Classical and quantum dynamics on p-adic trees of ideas
Andrei Khrennikov
Department of Mathematics, Statistics and Computer Sciences, Uni6ersity of Växö, S-35195 Växö, 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 on p-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
1. Introduction
It seems that the modern physics can in principle 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 organisms. 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
This investigation was supported by the grant ‘Strategical
investigations’ of Växö University and visiting professor fellowships at University of Clermont-Ferrand and Tokyo Science University.
functioning of the blood system, but even biological 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 intensive 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 psychological theories). Hence it is widely supposed
that the phenomenon of the consciousness can be
reduced to some (probably still unknown) physical 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 re-
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
96
A. Khrenniko6 / BioSystems 56 (2000) 95–120
duced to ordinary physical phenomena. And the
modern neurophysiological activity gives some evidences 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 domains of the brain (see, for example, Cohen et al.,
1997; Courtney et al., 1997, for fascinating experiments (based on functional magnetic resonance
imaging machine) for memory neurons configurations or Hoppensteadt, 1997, for the frequency
domain models for the exchange of signals in the
brain). Nevertheless, despite all of these investigations 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 physical–mathematical 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 as transformers of information. For
transformers of information we develop the formalism of classical mechanics on mental space
(space of ideas). In particular, this theory describes evolution of human ideas. The general
formalism of classical cognitive mechanics is developed 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 mechanics. Mathematically these laws can be described by differential equations (on mental
spaces)1. Starting with the initial idea x0 we can
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.
obtain the trajectory q(t) in mental space. However, the classical cognitive mechanics is not obtained 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
dq(t)
Newton’s mechanics as the derivative: 6(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 information (mental) forces and potentials which are applied 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 framework of the real analysis. The real line R and the
Euclidean space R3 (and even real manifolds) are
not directly related to cognitive information processes. 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 biological motivations (namely, the ability to form associations) 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 and 1 (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 branches 00, 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
A. Khrenniko6 / BioSystems 56 (2000) 95–120
j
identified with 2-adic numbers x =
j = 0 aj 2 . 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 with m branches on each level.
It is even possible to consider trees such that the
number of braches mj depends on the level.
The information processes in the brain described by the classical cognitive mechanics are
closely connected with neurophysiological processes. Roughly speaking neurophysiology describes ‘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 ideas x0, 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 meaning of ‘mental time’ (and ‘mental velocity’).
The classical cognitive mechanics describes unconscious cogniti6e processes. The phenomenon of
Fig. 1. The 2-adic tree.
97
consciousness cannot be explained by the formalism of the classical cognitive mechanics. To explain 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 principle 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 quantum 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 neurophysiolocal processes in the brain. It is induced by
mental processes. The conscious potential C(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 described by an analogue of the Schrödinger equation 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 information 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) information state of a cognitive system (the collection 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
98
A. Khrenniko6 / BioSystems 56 (2000) 95–120
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 information 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 deterministic character (of course, such a motion is perturbed by numerous information noises, see
Dubischar et al., 1999, for the details). This motion in mental space is not an evolution with
respect to physical time tphys, but with respect to
mental time t. 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 material structures (for example, groups of neurons). It
is induced by groups of evolving ideas. These
dynamical groups of ideas produce a new information 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 information dynamics in the human brain (and other
conscious cognitive systems). As in the classical
cognitive mechanics, in quantum cognitive mechanics 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 (conscious C-fields)
induce conscious forces fC of different (information) strength. The magnitude of the consciousness 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 cognitive behaviours. From this point of view human
individuals and animals differ only by the behaviors of their conscious C-fields.
As one of applications of our formalism to
psychology, we try to explain Freud’s psychoanalysis 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 classical 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 (where t
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 that A has the mass 1 (this can always be done
via the choice of the unit of mass). In this case the
momentum p(t) of A is equal to the velocity 6(t)
of motion. In the mathematical model the velocity
dq(t)
6(t) can be found as 6(t)=
q; (t). The vedt
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 model Xmat =R3 or some real manifold.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
where f(t, q) is the force applied to A. 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, p Xmat.
(2)
By integrating this equation we find the momentum p(t) at each instant t of time (if the initial
momentum p0 is known). Then by integrating the
equation
q; (t)=p(t),
q(0) =q0,
t, q, p Xmat
(3)
we find the position q(t) of A at each instant t of
time (if the initial position q0 is known).
We develop the formalism of the classical cognitive mechanics by analogue with Newton’s mechanics. Instead of the material space Xmat, we
consider mental space Xmen (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 transformations q “q% “q¦ “ ....The time parameter
of this evolution is also an information parameter
(mental time of t), t Xmen. Thus the activity of t
generates the trajectory q(t) in mental space Xmen.
Our deterministic cognitive postulate (which is a
generalization of Newton’s deterministic mechanical postulate) is that the trajectory q(t) of the
evolution of ideas is determined by initial conditions and forces. As in Newton’s mechanics, we
introduce the velocity 6(t) of the changing of the
idea q(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 time t). We start with development of the
formalism for a cognitive system t with the information mass 1. Here we can identify the velocity
6 with the momentum p =m6. We shall call p a
moti6ation to change the information state q(t).
We postulate that the cognitive dynamics in Xmen
is described (at least for some cognitive processes)
by an information analogue of Newton’s second
law. Thus the trajectory p(t) of the motivation of
t is described by equation
p; (t)=f(t, q), p(0) =p0,
t, q, p Xmen,
(4)
where f(t, q) is an information force (generated by
external flows of information; in particular, by
99
other cognitive systems). Thus if the initial motivation p0 and information force f(t, q) are known,
then the motivation p(t) can be found at each
instant of mental time t by 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 function V(q) such that f(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 f and potential V are functions
defined on the space of ideas Xmen. The potential
V(q), qXmen, is an information potential, information field, which interacts with a cognitive system t. Such fields are classical (unconscious)
cognitive fields.
As we have already mentioned mental time t
need not coincide with physical time tphys. Mental
time corresponds to the internal scale of an information process. For example, for a human individual t, the parameter t describes ‘psychological
duration’ of mental processes. Our conscious experience demonstrates that periods of the mental
evolution which are quite extended in the tphysscale can be extremely short in the t-scale and vice
versa. In general instances of mental time are
ideas which denote stages of the information evolution of a cognitive system. We remark that tphys
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 special representation for mental time t. However, it
is impossible to reduce all mental times to physical time (even if tphys is defined up to a transformation, 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
100
A. Khrenniko6 / BioSystems 56 (2000) 95–120
instances t, and t2 of mental time can be incompatible. 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 individual t is reading a
book B on the history of ancient Egypt, E. The
process of reading, p, is not continuous; t interrupts 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 physical time s =tphys (mechanical clocks): q =f(s).
However, the physical parameter s is not directly
related to the information process p. For example,
the velocity 6s of the information state q with
respect to s has nothing to do with the cognitive
evolution of the t. Moreover, as a consequence of
the jump-structure (with respect to s) of p, 6s is
not well defined. Denote by Dr1 =[s0, s1), Di1 =
[s1, s2), ... , intervals of reading and interruption of
reading. Thus the information process p induces
the following split of physical time s:
Dr1, Di1, ... , DrM, DiM, .... The intervals Di1, ... , DiM, ...
must be eliminated. New time parameter s̄ = f(s)
is defined as s̄= s on Dr1, s̄ =s on Di1, .... The
parameter s̄ can be considered as (one of possible)
mental (information) scales for the process p. The
use of time s̄ essentially improves the mathematical description of p. However, there is still no
large difference with the standard physics3.
Suppose now that intervals Drk, Dik depend on
information that t obtains in the process p:
Drk(ak ), Dri (bk ), where ak, bk Xmen are information
strings, ‘ideas’. Here s̄ = f(s, c) and q = h(s, c),
where sR, cXmen. The next natural step is to
eliminate the real parameter s from the description of the information process p and to consider
the evolution of the information state q (on the
subject E) with respect to a purely mental parameter t. This is information on E which is obtained
by t from the corresponding part of B. In the
3
Of course, as f is non-invertible, there are some differences with the standard formalism.
simplest model we can describe t as the text of the
book: t= (Ancient Egypt …) (see Section 8 for
mathematics). So q= q(t) is a transformation of
the information t B 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 of t to perform the information 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
book B does not change the state of knowledge of
t on ancient Egypt.
This example demonstrates that the information 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, information 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 initially t has
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 mechanics for material systems. The formalism of quantum 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
slits h1, and h2. Light passes S (through) slits and
finally reaches the screen S%, where we observe the
interference rings. Let us consider light as the flow
of particles, photons. Newton’s equations of motion (Eqs. (2) and (3)) could not explain the
interference phenomenon: ‘classical forces’ f involved in this experiment could not rule photons
A. Khrenniko6 / BioSystems 56 (2000) 95–120
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 Newton’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, p Xmat.
(6)
It is natural to assume that this new force,
fQ (t, q), is induced by some field C(t, q). This field
C(t, q) can be found as a solution of Schrödinger
equation
h (c
h 2 ( 2c
(t, q)=
(t, q) −V(t, q)c(t, q).
i (t
2 (q 2
(7)
Thus each quantum system propagates together
with a wave which ‘guides’ this particle. Such an
approach to quantum mechanics is called pilot
wa6e 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 quantum 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
dC(q)
with C(q) by the ordinary relation f =
. The
dq
ordinary relation between a force f and a potential 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 force fQ is invariant with respect
to the scaling C “cC the C-function. Thus the
magnitude of the C-function (‘quantum potential’) is not directly connected with the magnitude
of quantum force fQ. According to Bell (1987) and
4
The pilot wave theory does not give the standard interpretation of quantum mechanics, namely, the orthodox Copenhagen interpretation. By the latter interpretation it is
impossible to describe individual trajectories of quantum particles. Probably an analogue of the orthodox Copenhagen interpretation 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.
101
Bohm and Hiley (1993), C(q) is merely an information 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 mechanics to quantum cognitive mechanics, conscious 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. Behavior 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 Newton law is violated for conscious cognitive systems. As in the case of material systems, it is
natural to suppose that there exists some additional 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, a conscious field, by the same relation as
in the pilot wave formalism for material systems.
An information Schrödinger equation (see Section
10) describes the evolution of the conscious Cfield.
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 interpretation of c, they still suppose that C must 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 the C-field).
102
A. Khrenniko6 / BioSystems 56 (2000) 95–120
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 the C-field.6 In the pilot wave formalism
it is supposed that the C-field is created simultaneously 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 (conscious) cognitive system is quite natural. In principle we may suppose that the conscious field C(q),
qXmen, is generated by classical information processes in a cognitive system t. Moreover, it is
natural to suppose that higher information complexity 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 formalism (both for material and mental systems) the
magnitude of C is not directly related to the
magnitude of fC. At the present stage of knowledge 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
6
Even the Bohm–Hiley considerations on the complex internal structure of quantum particles do not go so far to
assume that a quantum particle is a generator of the C-field.
The Bohm – Hiley complexity is merely complexity of a receiver 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 the C-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.
for individual cognitive systems. In this section we
consider collective classical (unconscious) and
quantum (conscious) cognitive phenomena.
We start with the classical (unconscious) cognitive mechanics. Let t1, …, tN be a family of cognitive systems with mental spaces Xmen,1, …, Xmen.N.
We introduce mental space Xmen of 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 induces information forces fj (q1, …, qN ). The potential V is generated by information interactions of
cognitive systems t1, …, tN as well as by external
information fields. The evolution of the motivation pj (t) and the information state qj (t) of the jth
cognitive system tj is described by equations:
p; j (t)= fj (t, q1, ... , qN ), pj (0)= p0j
q; j (t)= pj (t), q(0)= q0j,
t, q, pXmen
(9)
(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 information constant (given by a p-adic number in
the mathematical model). Motions of cognitive
systems t1 and t2 in mental space are not independent; the (information) magnitude of the constant
of coupling a gives the strength of this dependence. On the other hand, if, for example,
V(q1, q2)= q 21 + q 22, then motions of t1 and t2 are
independent.
This model can be used not only for the description 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 ideas q(t), …, qN (t), related to different domains of human activity.8 We can apply
8
For example, p1 produces ideas on food, p2 produces ideas
on sex, p3 writes poems.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
our classical cognitive (collective) mechanics to
describe the simultaneous functioning of thinking
modules p1, …, pN.The main consequence of our
model is that ideas q1(t), …, qN (t) and motivations
p1(t), …, pN (t) do not evolve independently. Their
simultaneous evolution is controlled by the information potential V(q1, …, qN ). It must be underlined 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 state qj “q %j (or
motivation pj “p %)
j of one of thinking processors pj
will automatically imply (via the information interaction V(q1, …, qN )) a change of information
states (and motivations) of all other thinking
blocks. In principle no physical energy is 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
potentials V(q1, …, qN ) which give different types
of connections between thinking blocks pj.
4.2. Example 4.2
Let thinking processors p1, p2 and p3 be responsible for science, food and sex, respectively. Let
a family of cognitive systems. The classical information motion is described by classical (unconscious) information forces9 fj (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 coordinates q1, …, qN (information states of cognitive
systems t1, …, tN). Thus the consciousness of each
individual cognitive system tj depends on information 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 ordinary pilot wave theory in our cognitive model the
factorization
N
C(t, q1, ... , qN )= 5 Cj (t, qj )
j=1
V(q1, q2, q3)
= a1q 21 + a2q 22 + a3q 23 +a12(q1 −q2)2
+ a23(q2 −q3)2 +a13(q1 −q3)2.
103
(11)
If the information constant a1, strongly dominates
over all other information constants, then the
scientific thinking block p1 works practically independent from the blocks p2 and p3. If a12 (or a13)
dominates over all other constants, then there is
the strong connection between science and food (or
science and sex).
Moreover, the information potential V can depend on the mental time of a cognitive system,
V= V(t, q1, …, qN ). Thus at different instances of
mental time t a cognitive system can have different
information connections between thinking blocks
p1, …, pN.
We are now going to describe the collective
quantum (conscious) phenomena. Let t1, …, tN be
of the C-function implies that the conscious force
fj,C depends only on the coordinate qj. Thus the
factorization of C eliminates the collective conscious effect.
As in the classical cognitive mechanics, the
above considerations can be applied to a system of
thinking blocks p1, …, pN of the individual cognitive system t (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 mental systems.
104
A. Khrenniko6 / BioSystems 56 (2000) 95–120
5. Information connection between mental and
physiological processes
Classical and quantum fields, V(q1, …, qN ) and
C(q1, …, qN ), induce dependence between individual thinking blocks p1, …, pN of a cognitive system 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 information influence on physiological processes. Moreover, purely mental processes in the brain (which
are not directly related to physiological processes)
are connected on the information level with physiological processes. For example, let the mental
block p1, control functioning of the heart and the
block p2 controls some psychological process (for
example, relations with some person) and let the
classical information potential V(q1, q2) = aq1q2,
where a is a coupling information constant (given
by a p-adic number in our mathematical model).
Then purely mental process in p2 has an influence
to functioning of the heart. The classical information 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 physiological and purely mental process more complicated. There is the possibility of the conscious
control of human physiological systems. In principle, 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 of homeopathy. In fact, by a homeopathic treatment it is
possible to change the information potential
V(q1, …, qN ) of the organism. Microscopic quantities of medicines which are used in the home-
opathic treatment are just sources of information. 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 homeopathic medicine could be applied to some other
information state qj, j" k. The change of qj, qj “
q %,j 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, …, q %,j …, qN )).
6. Freud’s psychoanalysis as a reconstruction of
conscious field
By Freud’s theory, Freud (1933), mental space
Xi of a human individual i is split in two domains:
(1) a domain of conscious ideas X ci ; (2) a domain
of unconscious ideas X ui . Thus
Xi = X ci @X ui .
In our information model Freud’s idea is represented in the following way. Let f: Xi “ Xi, be
some function. As usual, we define a support of f
as the set supp f ={x Xi : f(x)" 0}. Let C be the
conscious field generated by the individual i and
fC be the corresponding conscious force.
Then supp fC is the set of conscious ideas (ideas
which can interact with the C-field), X ci = supp fC.
The set Xi ¯supp fC is the set of unconscious ideas
X ui (ideas which cannot interact with the Cfield).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 ),
qi Xi
(13)
(Vi
is the classical (unconscious)
(qi
force generated by the classical information po(C
tential Vi of i and fC = − i is the quantum
(qi
(conscious) force generated by the conscious information potential Ci of i. In the subspace X ui of
where f= −
We remark that the sets of ideas, supp fC and supp C, do
not coincide. It can be that supp fC is a proper subset of
supp C.
10
A. Khrenniko6 / BioSystems 56 (2000) 95–120
unconscious ideas this dynamical system is reduced
to the system:
p; i (t)= f(t, qi ),
qi X ui .
(14)
Let D be some domain in X ui and let a classical
information potential Vi (t, qi ), qi X ui , have a form
such that the dynamical system (14) has the domain
D as a domain of attraction of trajectories. Thus
starting with any initial idea q0 X ui the 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 force fC is
equal to zero. Therefore the i could not change
consciously the dynamics (14).
Suppose now that the D is some domain of ‘bad
ideas’. For example, if D is a domain of ‘black
ideas’, then i has a depression; if D is a domain of
ideas connected with alcohol, then i has problems
with alcohol; if D is a domain of aggressive ideas,
then i will demonstrate aggressive behavior (this
behavior looks as totally unmotivated: starting
with an arbitrary unconscious idea q0 the individual
i will always arrive to aggression).
The aim of psychoanalysis is to extend the
domain of conscious ideas X ci =supp fC. This extension will perturb dynamics (14) by the action of
a conscious force fC. This perturbation may change
the evolution of ideas in such a way that the domain
D will not be anymore a domain of attraction for
the whole space of unconscious ideas X ui . Starting
with q0 X ui the i can have trajectories qi (t) which
will be never attracted by the domain of ‘bad ideas’
D.
The pair, a cognitive system i and a psychoanalytic p, can be considered as a coupled system of
transformers of information. The information coupling between i and p will generate a new information classical potential Vi,p (t, q1, q2) which is
defined on mental space X = Xi ×Xp, where Xi and
Xp, are spaces of ideas of the individual i and the
psychoanalytic p, respectively.
Dynamics of the conscious field Ci(t, qi ) of i is
described by the Schrödinger equation
h (Ci
h 2 ( 2C i
(t, qi )=
(t, qi ) − Vi (t, qi )C(t, qi )
i (t
2 (q 2i
(15)
105
Dynamics of the conscious field Ci,p (t, qi, qp ) of the
system (i, p) is described by the Schrödinger equation
h (Ci,p
(t, qi, qp )
i (t
=
h2 (2
(2
+
Ci,p (t, qi, qp )
2 (q 2i (q 2p
− Vi,p (t, qi )Ci,p (t, qi, qp ).
(16)
If now the conscious force
f0 C (t, qi, qp )= −
(Ci,p (t, qi, qp )
"0
(qi
(17)
for some ideas qi X ui at least for some ideas qp Xp,
then the motion of i in the domain of unconscious
ideas X ui 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 that X ui is reduced
and X ci is extended. The aim of the psychoanalytic
p is to find ideas qp Xi such that (17) takes place
for unconscious ideas qi X ui of i. As the process of
psychoanalysis is a conscious process (at least for
p), it is natural to assume that ideas qp, used by p
to induce condition (17) are conscious: qp X cp.
Typically such ideas are represented in the form of
special questions to i. In some sense this is a kind
of conscious intervention of the psychoanalytic p
in the unconscious domain of the individual i. If p
finds a domain O¦ X ui 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 )+ f0 C (t, qi, qp ),
qi O.
(18)
Under some circumstances this dynamical system
can be free from ‘pathological features’ of dynamical 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 of i: a new dynamics is ruled by the classical force f0 (t, qi, qp ) instead
of the classical force f(t, qi ). However, it is not easy
to change strongly the classical force f (t, qi ) on the
domain of unconscious ideas X ui by just the change
of the classical potential. Typically Vi,p (t, qi, qp) =
Vi (qi )+ Vp (qp ) + G(qi, qp ), where the (information) magnitude of G(qi, qp ) is small for ideas
qi X ui . On the other hand, this minor change of the
106
A. Khrenniko6 / BioSystems 56 (2000) 95–120
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 individual i
(having some mental decease) via an extension of
the support of the quantum force. Such an extension is the extension of the domain of conscious
ideas X ci (and the reduction of the domain of
unconscious ideas X ui ). This extension is realized
by the information coupling between an individual i and a psychoanalytic p. By minor change of
the classical information potential the p strongly
changes the conscious force acting on the i. Dynamics of ideas in the unconscious domain of i is
changed. This change eliminates the mental
decease.
In the same way we can describe information
processes which take place in hypnotism. Here by
the conscious information coupling (described by
Schröinger equation (Eq. (16))) between an individual i and a hypnotizer p the conscious dynamics (13) is changed in such a way that the
conscious force fC (t, q1) is practically totally eliminated by the action of the conscious force
f0 C (t, qi, qp ). For example, let f0 C (t, qi, qp ) = −
fC (q1)+ fC (q2). Then information behavior of the i
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 absolute 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 information and the special distance on the space of
vectors of information. Any natural number m\
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
with the order structure on the real line R (and the
metric related to this order structure). For each x,
there exist ‘right’ and ‘left’ hand sides neighborhoods; 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 approximated from the left hand side with an arbitrary
precision by numbers of the form y=09 … 90 … .
The following description of right hand side
neighborhoods will be very important in our further 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 sufficiently large m.
This nearness has a natural information (cognitive) 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 information) 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 consciousness receives from reality. The main properties of this coding system are the order structure
on the set of information vectors and the restricted ability (see (AS)) to form associations.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
Finally, we pay attention to the ‘universal coding property’ of the real system: any natural number m\ 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 processes (especially cognitive) have an order structure. 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 set Xm. 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 mathematical formalism, p-adic analysis. We present
some facts about p-adic numbers.
The field of real numbers R is constructed as the
completion of the field of rational numbers Q with
respect to the metric r(x, y) = x −y, where · is
the usual valuation given by the absolute value.
The fields of p-adic numbers Qp are constructed in
a corresponding way, but using other valuations.
For a prime number p, 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 … p rp …, and we define np =p − rp, writing 0p =0 and −np =np. We then extend the
definition of the p-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 R and Qp
are unique in a sense, since by Ostrovsky’s theo11
Thus here all information is considered from the viewpoint of associations.
107
rem (see Schikhov (1984)) · and ·p are the only
possible valuations on Q, but have quite distinctive properties.
Unlike the absolute value distance ·, the p-adic
valuation satisfies the strong triangle inequality
x+ yp 5 max[xp, yp ],
x, yQp
Write Ur(a)= {xQp: x− ap 5 r} and U −
r (a)=
{x Qp: x− ap B r}, where r= p n and n= 0, 9 1,
9 2, … . These are the ‘closed’ and ‘open’ balls in
Qp while the sets Sr (a)= {x Qp: x− ap = r} are
the spheres in Qp of such radii r. These sets (balls
and spheres) have a somewhat strange topological
structure from the viewpoint of our usual Euclidean intuition: they are both open and closed at
the same time, and as such are called clopen sets.
Another interesting property of p-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 a p-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 integers and 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 x Zp 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
xn
. The
n!
series converges in Qp if
xp 5 rp, where rp = 1/p,
p" 2 and r2 = 1/4
(21)
p-adic trigonometric functions sin x and cos x are
defined by the standard power series. These series
have the same radius of convergence rp as the
exponential series.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
108
If, instead of a prime number p, we start with an
arbitrary natural number m \1 we construct the
system of so called m-adic numbers Qm by completing Q with respect to the m-adic metric rm (x, y)=
x − ym which is defined in a similar way to above.
8. Hamiltonian dynamics on p-adic mental space
The rings of p-adic integers ZP can be used as
mathematical models for mental spaces. Each elej
can be identified with a sement x= j = 0 aj p
quence x= a0a1 … aN …, aj =0, 1, …, p− 1. Such
sequences are interpreted as coding sequences (in
the alphabet Ap ={0, 1, …, p − 1} with p letters)
for some amounts of information. The p-adic
metric rp (x, y)= x −yp on Zp corresponds to the
nearness (AS) for information sequences. We
choose the space X = Zp (or multidimensional
spaces X =Zp N) for the description of information.
Everywhere below we shall use the abbreviation
‘I’ for the word information.
We use an analogue of the Hamiltonian dynamics on mental spaces As usual, we introduce the
d
quantity p(t)= q; (t) = q(t) which is the indt
formation analogue of the momentum, a
moti6ation.
The space Zp × Zp of points z =(q, p) where q is
the I-state and p is the motivation is said to be a
phase mental space. As in the ordinary Hamiltonian
formalism, we assume that there exists a function
H(q, p) (I-Hamiltonian) on the phase mental space
which determines the motion of t in the phase
mental space:
q; (t)=
(H
(q(t), p(t)),
(p
p; (t)= −
q(t0) = q0,
(H
(q(t), p(t)), p(t0) =p0.
(q
(22)
The I-Hamiltonian H(p, q) has the meaning of an
I-energy (or mental, or psychical energy, compare
Freud (1933)). In principle, I-energy is not directly
connected with the usual physical energy.
The simplest I-Hamiltonian Hf (p) = a 2p, a Zp
describes the motion of a free cognitive system t,
i.e. a cognitive system which uses only self-motivations for changing of its I-state q(t). Here by
solving the system of the Hamiltonian equations we
obtain: p(t)= p0, q(t)= q0 + 2ap0(t−t0). The motivation p is the constant of this motion. Thus the
free cognitive system ‘does not like’ to change its
motivation p0 in the process of the motion in the
mental space. If we change coordinates, q%= (q−
q0 )/k, k= 2ap0, then we see that the dynamics of
the free cognitive system coincides with the dynamics of its mental time.
In general case the I-energy is the sum of the
I-energy of motivations Hf = ap 2 (which is an
analogue of the kinetic energy) and potential I-energy V(q):
H(q, p)= ap 2 + V(q)
The potential V(q) is determined by fields of
information. We now consider examples which
illustrate the notion of mental time.
8.1. Example 8.1, reading of a book
We consider again the example of Section 3. Let
us enumerate words in the language of book B by
1, 2, …, m− 1 (including blank symbol). Denote by
0 words which have zero information value for t
(for example, special terms which are not known by
t). The text of B can be represented as an information string: x= (a0, a1, …, aN, …, aM ). This string
can be identified with an element of Zm, by setting
aj = 0, j] M+1: x= (a0, a1, …, aN, …, aM, 0,
…, 0, …). Counts of such a mental time are given
by blocks of x: t= (a0, a1, … , ak, 0, 0, …). Suppose now that the information state (knowledge) q
is coded in the following way: q=
(b0, b1, …, bj, …), bj = 0, 1, …, k− 1, where b0 is
dynasty, b1 is number of wars during dynasty b0…
The symbol 0 is again used to denote zero
knowledge12. Dynamics q(t), bj = cj (a0, ..., aN, …),
of knowledge of t is described by Hamiltonian
equations13.
12
In this example, the use of the homogeneous k-adic tree is
not so natural. It would be more natural to use a nonhomogeneous tree in that the number kj of branches depends on the
information characteristic bj, see, for example, Fig. 2.
13
In fact, the mathematical formalism developed in this
paper describes only the case m =k. To study dynamics for
t Zm, q Zk, k " m, we need more complicated mathematical
analysis.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
109
plays the crucial role in many psychological experiments. We can not obtain sensible observations for interactions between arbitrary
individuals. There must be a process of learning
for the group t1, …, tN which reduces mental
times t1, …, tN to the unique mental time t.
Thus, let us consider a group t1, …, tN of cognitive systems with the internal time t. The dynamics of I-states and motivations is determined
N
by the I-energy; H(q, p), q ZN
p , pZp . It is natural to assume that
Fig. 2. The factorial tree ZM for m1 = 2, m2 = 3, m3 = 4, …
N
H(q, p)= % aj p 2j + V(q1, ... , qN ),
8.2. Example 8.2, e6olution of scientific
psychology
We introduce a mental time t = tps which is
used for describing the evolution of the psychological state of a scientist t. Let t=
(a0, a1, …, aN, …), where aj =0, 1, …, m − 1, is a
number of publications of t in journals of the
weight t. Journals with j= 0 are the most important, journals with j =1 are less important and so
on. For example, let m = 10. Such a mental time
has no order structure. For example, take l1,=
(2, 0, …)= 2,
l2 =(0, 8, 0, …) = 80,
l3 =
(1, 1, 2, 0, …)=211. These instances of mental
time could not be ordered according their importance. The evolution of the psychological state
q(t) of t is described by trajectory in the mental
space (in the simplest case Xmen =Zm,). If this
trajectory is continuous, then t will have similar
psychological states q(t1), q(t2) for close instances
of mental time t1, t2.
In the Hamiltonian framework we can consider
interactions between cognitive systems t1, …, tN
These cognitive systems have mental times
t1, …, tN and I-states q1(t1), …, qN (tN ). By our
model we can describe interactions between these
cognitive systems only in the case in that there is
a possibility to choose the same mental time t for
all of them. In this case we can consider the
evolution of the system of the cognitive systems
t1, …, tN as a trajectory in the mental space ZN
p =
Zp × ... × Zp q(t) = (q1(t), ... , qN (t)).
We think that the condition of consistency
t1 =t2 = ... = tN =t
aj Zp.
j=1
(23)
2
Here Hf (p)= N
j = 1 aj p j is the total energy of motivations for the group t1, …, tN and V(q) is the
potential energy. As usual, to find a trajectory in
N
the phase mental space ZN
p × Zp we need to solve
(H
the system of Hamiltonian equations: qj = ,
(pj
(H
pj = − , qj (t0)= q0, pj (t0)= p0.
(qj
8.3. Remark 8.1, acti6e information
Our ideas about information and information
field are similar to the ideas of Bohm and Hiley
(1993) (see especially pp. 35− 38). As Bohm and
Hiley, we do not follow ‘Shannon’s ideas that
there is a quantitative measure of information
that represents the way in which the state of a
system is uncertain for us’, Bohm and Hiley
(1993). We also consider information as an acti6e
information. Such information interacts with cognitive systems. As a consequence of such interactions cognitive systems produce new information.
The only distinguishing feature is that material
objects are not involved in our formalism. According to Bohm and Hiley active information
interacts with material objects (for example, the
ship guided by radio waves). Bohm and Hiley
assume that information fields have nonzero physical energy that directs other (probably very large)
physical energy. However, physical energies are
not involved in our model. Thus we need not
assume that I-fields have some physical energy. In
particular, we need not try to find (as Bohm and
Hiley (1993), p. 38) an origin of such an energy.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
110
We also remark that Bohm and Hiley (1993)
discuss only quantum c-fields. We shall use
both classical and quantum I-fields.
Bohm and Hiley discussed a difference between ‘active’ and ‘passive’ information. In fact,
our model supports their conclusion that ‘all information is at least potentially active and that
complete passivity is never more than an abstraction …’, Bohm and Hiley (1993), p. 37. If a
cognitive system t moves in the field of forces C
(classical or quantum), then the information
xsupp V is active for t and the information
x Zm
p ¯supp V is passive for t. Let 6= V(t, x) be
a time p dependent potential. Then the set of
active information X(t) =supp V(t) evolves in
mental space. Thus some passive information
becomes active and vice versa.
plies also a variation dp of the motivation
p:dp = fdt.
The coefficient f of proportionality is called
an I-force. Thus any change of the motivation
is due to the action of an I-force f. If f= 0 then
dp= 0 for any variation dt of t. Thus a cognitive system cannot change its motivation in the
absence of I-forces. By analogue with the usual
physics we call the coefficient a of a proportion
between the variation dv of the I-velocity 6 and
the variation dt of the mental time t, d6 = adt,
an I-acceleration. Thus dp = amdt. This relation
can be rewritten in the form of an information
analogue of the second Newton law:
9. Inertia of information
f= −
We have considered dynamics of cognitive
systems of the unit mass. There the coefficient 6
of proportionality between the variation dq of
the I-state and the variation dt of mental time
t: dq = 6dt, was considered as a motivation. In
the general case the motivation p may not coincide with 6. Let us assume that the motivation p
is proportional to 6, p =m6, m Zp. This coefficient m of proportionality is called an I-mass.
p
We also call 6 an I-velocity. Thus dq = dt
m
Let t1 and t2 be two cognitive systems with
the I-masses m1, and m2 and let m1p \m2p.
Let t1, and t2 have the variations dt1, dt2 of
mental time of the same p-adic magnitude,
dt1p =dt2p, and let these variations generate
the variations dq1 and dq2 of their I-states of
the same p-adic magnitude, dq1p =dq2p. To
make such a change of the I-state, t1 needs a
larger motivation:
where V is called the potential, or potential energy. The total I-energy H is defined as the sum
of the kinetic and the potential I-energies,
p1p =
) )
) )
dq
dq
m1p \p2p =
m .
dt p
dt p 2 p
Thus the I-mass is a measure of an inertia of
information. We define a kinetic I-energy by
1 2
T=
p . A variation dt of mental time t im2m
ma=f or p; = f.
(24)
An I-force f is said to be a potential force if
there exists a function V(q) such that
(V
(q
H(q, p)=
1 2
p + V(q).
2m
The Hamiltonian equation
p; = −
(H
(q
coincides with the Newton equation p; = f.
In p-adic analysis the condition f0 does not
imply that a differentiable function f is a constant, see Schikhov (1984) or Escassut (1995).
There exist complicated continuous motions
(q(t), p(t)) in the I-phase space for cognitive
systems with zero I-energy (q; 0 or p; 0).
In psychological models these motions can be
interpreted as motions without any motivation.
Such motions do not need information force.
On the other hand, we can consider an I-poten(V
tial V(q) such that
= 0. Here the potential
(q
I-energy V(q) can have complicated behavior on
the mental space X= Zp. At the same time the
I-force f= 0. Thus there may exist I-fields which
do not induce any I-force.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
All mathematical pathologies can be eliminated
by the consideration of analytical functions. If
f % 0 and f is analytic then f = constant14.
10. p-adic model for the conscious (quantum)
mechanics
It is quite natural to quantize classical mechanics
on information spaces over Zp. We give the following reasons for such quantization. Observations
over I-quantities are statistical observations. We
have to study statistical ensembles of cognitive
systems (instead studying of an individual cognitive
system). Such statistical ensembles are described by
quantum states f. As usual in quantum formalism,
we can assume that a value l of an I-quantity A
can be measured in the state f with some probability Pf (A=l). This ideology is nothing than the
application of the statistical (ensemble) interpretation of quantum mechanics (see, for example, Ballentine (1989)) to the information theory. By this
interpretation any measurement process has two
steps: (1) a preparation procedure E; (2) a measurement of a quantity B in the states f which were
prepared with the aid of E.
Let us consider these steps in the information
framework. By E we have to select a statistical
ensemble f of cognitive systems on the basis of
some I-characteristics. Typically in quantum
physics a preparation procedure E is realized as a
filter based on some physical quantity A, i.e. we
select elements which satisfy the condition A= m
where m is one of the values of A. We can do the
14
In psychological models we can interpret analytical trajectories in the phase mental space as a ‘normal behavior’, i.e. an
individual needs a motivation for the change of a psychological state. Here we can observe some psychological (information) force that induces this change. There is a psychological
(information) field that generates this force. Trajectories (nonanalytical) with zero motivation are interpreted as abnormal
psychological behavior (probably such trajectories correspond
to mental diseases; on the other hand, they may explain
anomalous phenomena). Here an individual changes his psychological state without any motivation in the absence of any
psychological (information) force. Here, in fact, a p-adic generalization of the Hamiltonian formalism does not work. We
need to propose a new physical formalism to describe such
phenomena.
111
same in quantum I-theory. An I-quantity A is
chosen as a filter, i.e. cognitive systems for the
statistical ensemble f are selected by the condition
A=m where mZp is some information. For example, we can choose A= p, the motivation, and select
a statistical ensemble f= f(p=m) of cognitive
systems which have the same motivation mZp.
Then we realize the second step of a measurement
process and measure some information quantity B
in the state f(p = m). For example, we can measure
the I-state q of cognitive systems belonging to the
statistical ensemble described by f(p = m). We shall
obtain a probability distribution P(q = lp= m), l,
mZp (a probability that cognitive system has the
I-state q= l under the condition that it has the
motivation p=m). It is also possible to measure the
I-energy E of cognitive systems. We shall obtain a
probability distribution P(E = lp=m), l, mZp.
On the other hand, we can prepare a statistical
ensemble f(q = m) by fixing some information mZp
and selecting all cognitive systems which have the
I-state q= m. Then we can measure motivations of
these cognitive systems and we shall obtain a
probability distribution P(p= lq= m).
Another possibility is to use a generalization of
the individual interpretation of quantum mechanics. By this interpretation a wave function C(x),
xRn, describes the state of an individual quantum
particle. In the same way we may assume that a
wave function C(x), xZnp, on the mental space
describes the state of an individual cognitive system
t.15
In fact, a mathematical model for quantum
I-formalism has been already constructed. This is
quantum mechanics with p-adic valued functions,
see Khrennikov (1994, 1997) and Albeverio and
Khrennikov (1998). We present briefly this model.
The space of quantum states is realized as a p-adic
Hilbert space K (see Khrennikov (1994, 1997) and
Albeverio and Khrennikov (1998) for the theory of
such spaces). This is a Qp-linear space which is a
Banach space (with the norm · ) and on which
15
The problem of interpretations is the important problem
of ordinary quantum mechanics on real space. The same
problem arises immediately in our quantum I-theory. We do
not like to start our investigation with a hard discussion on the
right interpretation. We can be quite pragmatic and use both
interpretations by our convenience.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
112
is defined a symmetric bilinear form (·, ·):K ×
K“Qp. This form is called an inner product on
K. It is assumed that the norm and the inner
product are connected by the Cauchy–
Bunaykovski–Schwarz inequality: (x, y)p 5
x y, x, yK. By definition quantum I-state
f is an element of K such that (f, f) = 1; quantum I-quantity A is a symmetric bounded operator A: K“ K, i.e. (Ax, y) =(x, Ay), x, y K.16
We discuss a statistical interpretation of quantum
states in the case of a discrete spectrum of A. Let
{l1, …, ln, …} lj Zp be eigenvalues of A, Afn,=
lnfn, fn K, (fn, fn ) =1. The eigenstates fn of A
are considered as pure quantum I-states for A, i.e.
if the system of cognitive systems is described by
the state fn then the I-quantity A has the value
ln Zp with probability 1. Let us consider a mixed
state
f= % qnfn,
qn Qp,
(25)
n=1
17
2
where (f, f)=
By the statistical
n = 1 q n =1.
interpretation of f if we realize a measurement of
the I-quantity A for cognitive systems belonging
to the statistical ensemble described by f then we
obtain the value ln with probability P(A =
ln f)= q 2n.
The main problem (or the advantage?) of this
quantum model is that these probabilities belong
to the field of p-adic numbers Qp. The simplest
way is to eliminate this problem by considering
only finite mixtures (25) for which qn Q (the field
of rational numbers Q is a subfield of Qp ). In this
case the quantities P(A = ln f) =q 2n can be interpreted as usual probabilities (Kolmogorov
(1956)). Therefore we may assume that there exist
(can be prepared) quantum I-states f which have
the standard statistical interpretation: when the
number N of experiments tends to infinity, the
frequency nN (A= ln f) of an observation of the
information ln Zp tends to the probability q 2n.
16
In p-adic models we do not need to consider unbounded
operators, because all quantum quantities can be realized by
bounded operators (see Albeverio and Khrennikov, 1998).
17
As in the usual theory of Hilbert spaces, eigen-vectors
corresponding to different eigen-values of a symmetric operator are orthogonal.
However, we can use a more general viewpoint
to this problem. In the book Khrennikov (1994) a
(non-Kolmogorov) probability model with p-adic
probabilities has been developed. If we use a
p-adic generalization of a frequency approach to
probability (see von Mises, 1957), then p-adic
probabilities are defined as limits of relative frequencies nN with respect to the p-adic topology.
By using the p-adic frequency probability
model for the statistical interpretation of quantum
I-states we may assume that there exists I-states f
(ensembles of cognitive systems) such that the
relative frequencies nN (A=ln f) have no limit in
R, i.e. we cannot apply the standard law of the
large numbers in this situation. Hence if we realize measurements of an I-quantity A for such a
quantum I-state and study the observed data by
using the standard statistical methods (based on
real analysis), then we shall not obtain a definite
result. There will be only random fluctuations of
relative frequencies, see Khrennikov (1994).18
The evolution of a p-adic wave function is
described by an I-analogue of the Schrödinger
equation:
hp (C
h 2 ( 2C
(t, x)= p
(t, x)− V(t, x)C(t, x),
i (t
2m (x 2
(26)
where m is the I-mass of a quantum cognitive
system. Here a constant hp plays, the role of the
Planck constant. By pure mathematical reasons
(related to convergence of p-adic exponential and
trigonometric series) it is convenient to choose
hp = 1/p
We may also present some physical arguments
for such a choice. In ordinary quantum mechanics
the Planck constant is related to the measure of
discretization. The constant hp = 1/p is related to
the level of discretization of information.
18
Such a behavior can be related to psychological experiments. Here the possibility of the use of p-adic probability
models gives the important consequence for scientists doing
experiments with a statistical I-data: the absence of the statistical stabilization (random fluctuation) does not imply the
absence of an I-phenomenon. This statistical behavior may
have the meaning that this I-phenomenon cannot be described
by the standard Kolmogorov probability model.
A. Khrenniko6 / BioSystems 56 (2000) 95–120
The use of i implies the consideration of the
extension Qp (i ) =Qp ×iQp of Qp. Elements of
this extension have the form z =a +ib, a, b Qp.
This extension is well defined for p =3, mod 4. As
usual, we introduce a convolution z̄ =a −ib; here
we have zz̄= a 2 +b 2. In what follows we assume
that wave functions take values in Zp (i )=Zp×
iZp.
10.1. Example 10.1, a free conscious system
Let the potential V = 0. Then the solution of the
Schrödinger equation corresponding to the I-energy E=p 2/2m has the form19:
Cp(t, x)= e i(px − Et)/hp.
(27)
By the choice hp =1/p this function is well defined
for all xZp and t Zp. As CC( 1, this wave
function describes the uniform (p-adic probability)
distribution, see Khrennikov, 1994, on the ring of
p-adic integers Zp. Thus a cognitive system t in the
state C can be observed with equal probability in
any state x Zp. In this sense behavior of a free
cognitive system is similar to behavior of the
ordinary free quantum particle. On the other hand,
there is no analogue of oscillations: Cp(t, x)=
cos(px − Et)/hp + i sin(px −Et)/hp, and cos(px −
Et)/hp p =1, sin(px −Et)/hp p =(px −Et)/hp p.
We will consider this example again in Section
12.
11. The p-adic pilot wave theory for cognitive
systems
Let us consider a system of N cognitive systems,
t1, …, tN, with Hamiltonian
p 2k
+ k \ i Vki (xk −xi ).
2mj
The wave function C(t, x), x =(x1, …, xN ), xk Zm
p
(where m is the dimension of mental space which
H=k
19
We note that formal expressions for analytical solutions
of p-adic differential equations coincide with the corresponding expressions in the real case (in fact, we can consider these
equations over arbitrary number field, see Khrennikov, 1994).
However, behaviors of these solutions are different.
113
is used for the description of the I-state of tj )
evolves according to the Schrödinger equation
(Eq. (26)). A purely mathematical consequence of
this is that
(
(
r(t)+ %
jk (x, t)=0,
(t
k (xk
(28)
where r(t, x)= C(t, x)C(t, x) is a probability density on the configuration mental space ZmN
and
p
1
jk (x, t)= m −
k Im C(t, x)
(
C(t, x) .
(xk
As in the ordinary Bohm’s formalism, we assume
that a quantum cognitive system tk has at any
mental time20 well defined I-state xk and motivation pk. I-state xk evolve according to
x; k (t)=
jk (t, x)
.
r(t, x)
(29)
This model gives the natural description of an
evolution of the I-states of a system of cognitive
systems t1, …, tN.
We consider now a system S of brains t1, …, tN.
The wave function C(t, x) of the S depends on
I-states of all brains in the S. Thus motions of
are not
these brains in phase mental space Z2m
p
independent. At the same time there might be no
classical potential V which induces such a dependence. Of course, if (as in ordinary real formalism)
C(t, x)= N
j = 1Cj (t, x), then the I-motions of
brains tj are independent. There are no correlations between consciousness of different brains.
11.1. Remark 11.1, non-locality
This is the good place to discuss the problem of
non-locality of the pilot wave formalism. Often
non-locality is considered as one of the main
difficulties of the pilot wave formalism. However,
non-locality is not a difficulty in our pilot I-wave
formalism. This is non-locality in the mental
space. Such non-locality can be natural for some
I-systems. For cognitive systems, I-non-locality
20
Of course, we assume that mental times t1, … , tN the
cognitive systems t1, … , tN satisfy the condition of consistency (Eq. (23)).
A. Khrenniko6 / BioSystems 56 (2000) 95–120
114
means that ideas which are separated in a p-adic
space can be correlated. However, p-adic separation means only that there are no strong associations between ideas or groups of ideas. But this
absence of associations does not imply that these
ideas could not interact.
By our model each human society S has a wave
function C(t; x). The same considerations can be
applied to animals and plants. The only difference
is probably that here quantum I-potentials are
not so strong. Thus we get the conclusion that
there may exist a wave function Cliv(t, x) of
all living organisms. The wave function
C(t, x)liv(t, x) can be represented in the form:
C(t, x)liv(t, x)= % Cf (t, x),
(30)
f
where Cf (t, x) is a wave function of the living
form f. An observable F (a living form) can be
realized as a symmetric operator in a p-adic
Hilbert space, FCf =fCf, where f Zp is the code
of the living form f in the alphabet {0, 1, …, p−
1}. By Eq. (29) the evolution of the fixed form f0
depends on evolutions of all living forms f.
The process of the evolution of living forms is
not just a process based on Darwin’s natural selection. This is a process of a quantum I-evolution in
that the conscious field of all living forms plays
the important role. This model might be used to
explain some phenomena that could not be explained by Darwin’s theory. For example, the
beauty of colors of animals, insects and fishes
could not be consequence only of the process of
the natural evolution. This is a consequence of the
structure of the conscious field C(t, x)liv(t, x).21
By the same reasons we can explain some aspects
of relations between robbers and victims. It seems
that in nature there is a well organized system
which gives robbers a possibility to eat victims.
This system is nothing else than a result of the
evolution due to Eq. (29).
21
Of course, at the moment we cannot find such a function
C(t, x)liv(t, x) which induces the real distribution of colors. In
any case our model implies the existence of such a field.
Therefore, the process of colors’ evolution is a process of the
simultaneous evolution of colors of numerous living forms.
These colors do not serve only to the ‘convenience’ of concrete
forms (as it should be due to Darwin’s theory), but they were
produced by the correlated evolution of all living forms.
The C(t, x) can be considered as a new cognitive system, T (compare with Bohm and Hiley,
1993). The T gets information from cognitive
systems t1, …, tN (via the classical field V) and T
changes I-states of t1, …, tN (via Eq. (29)). The
only distinguishing feature is that the I-state of T
cannot be identified with a point in the mental
space Zm
p for a finite m. However, if we extend the
I-formalism by using infinitely dimensional mental spaces over Zp, then T can be considered as a
cognitive system. Thus each cognitive system or a
group of cognitive systems induce a new cognitive
system T (the consciousness) which evolves in
infinitely dimensional mental space.
In principle, T may induce a new field
F(t, C(·)). This field determine a quantum potential for T. The field F(t, C(·)) can be again considered as a cognitive system T 1(which evolves in
infinite dimensional mental space). It is the consciousness of (the consciousness T). If such a
construction can be repeated many (or infinitely
many?) times, then there is a ‘conscious tower’, T,
T 1, …, T n, … . We might speculate that the motion of a cognitive system in mental space is
determined by the hierarchic conscious system T,
T 1, …, T n, … . Of course, the effect of T 1 is not
present in the linear Schrödinger equation. If we
assume the hypothesis on the hierarchic conscious
structure, then the linear Schrödinger equation
has to be changed to nonlinear equation. Hence
the cognitive considerations support de Broglie’s
ideas
about
nonlinear
perturbations
of
Schrödinger’s equation (see De Broglie, 1964).
12. Conscious field as memory activation field
The reader may ask: ‘Why do we need to use an
I-analogue of quantum mechanics to describe
conscious phenomena? Why is it not sufficient to
use only the classical I-dynamics (with Newton’s
I-equation, (4))?’ There are two reasons to introduce a conscious field c(t, q):
(D) Our conscious experience (see Section 2.1)
demonstrates that I-motion of a conscious cognitive system tcons could not be described by classical Newton’s I-equation (4). Classical I-forces do
not provide the right balance of ‘guiding’ forces
A. Khrenniko6 / BioSystems 56 (2000) 95–120
for the I-state q(t) of tcons. One of possibilities to
perturb classical I-equation, (4), of I-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 the c-function which satisfies Schrödinger I-equation, (Eq.
(26)). Of course, such an approach would be
strongly improved if we could explain the mechanism 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 other I-quantity of a conscious system tcons
without to disturb this system.22 In particular, we
could not perform a measurement of both q and p
for the same tcons. As in the ordinary quantum
mechanics, we have to use large statistical ensembles S of conscious systems and perform statistical
measurements for such ensembles to find probabilities for realizations of I-quantities. One of
possibilities is to use again an analogue with the
ordinary quantum formalism and introduce a field
of probabilities c(t, q) (which describes statistical
properties of the ensemble S) such that the square
of the amplitude of c
C(t, q)2 =c(t, q)c( (t, q)
gives the probability to find tcons S in the I-state
q (at the instant t of 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 postulate that the conscious c-field (which generates
the conscious force fC coincides with the probability field.
Let tcons be a conscious system. A performance
of the I-state q= q(t) of tcons can be viewed as a
The process of an I-measurement is an interaction of tcons
with some I-potential V(t, q). Any tcons is extremely informational unstable. Even low information potentials V(t, q) disturb tcons. Such an I-instability is the common feature of
conscious and quantum systems.
22
115
performance of an image on some screen S. This
screen continuously demonstrates ideas of tcons.
The tcons is 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 on S’ at
the instant t+ Dt) is generated not only on the
basis of the previous I-state q(t) (an ‘image on S’
at the instant t) with the aid of I-forces f(t, q)
which are generated by external I-potentials. The
main feature of tcons is 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 new I-field c(t, x),
x D(t), a field of memory acti6ation. It is postulated 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 t and t% be two different instances of time.
Schrödinger 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)=
&
K(t%, t, x, y)c(t, y)dy
(31)
D(t)
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 activation fields at different instances of time are connected by a linear transformation.
The Bohmian mechanics implies that the quantum force fQ does not depend on the amplitude R
of a quantum field c. Roughly speaking the
23
Hiley and Pylkkänen (1997) call such a property ‘the
unity of consciousness’ (see also Searle, 1992).
116
A. Khrenniko6 / BioSystems 56 (2000) 95–120
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; if R(x) =x (rather slow variation), then fQ is still equal to 0; if R(x) = x 2,
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 conscious 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, x Dtcons) eliminates at all a memory effect from the evolution of the I-state q(t) of
tcons. Only rather strong variation of the field
C(x), x Dtcons, 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%)/C 2. Thus the conscious force fC depends
not only on the first variation dC of the amplitude of the memory activation field, but also on
the second and third variations, d 2C, d 3C.
24
We remark that the ordinary pilot wave theory could not
provide a reasonable explanation of such a connection between 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.
These predictions must be verified experimentally.
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 = R3
is the spatial domain of a ‘physical brain’ and
Bi ¦ Xmen = Zkm is the I-domain of an ‘information brain’. Thus the memory activation 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 information 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 structure of the mental space Zkm and variations with
respect to neuron’s location uBs ¦ R3 may have
no links to variations with respect to idea’s location xBi ¦ Zkm. Moreover, the purely mathematical 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, Hoppensteadt, 1997, by that dust-like configurations of
neurons in Bs, (oscillating with the same frequency) seem to correspond to the same idea
(image). In the latter model it is possible to construct a transformation Jf : Bf “ Bi, where Bf ¦
R3 is the domain of a ‘frequency brain’. Thus
the field c(x), x Bi, 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 approximately) the same memory activation field
A. Khrenniko6 / BioSystems 56 (2000) 95–120
c(x). Suppose that initial I-states q 0t of t S 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 probability P(qt (t)= q) to find t S in the I-state q
(at the instance t) is equal to C 2(q). Thus the
memory activation field coincides with the probability field due to the stationary (statistical) stabilization of information states of conscious
systems tS. Intervals DT of such a stabilization to a stationary distribution 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 contemporary ‘properties’ of s, but also information
based on the previous experience of s. If we use
the anthropological principle and apply the conscious 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 consequence of the statistical stabilization to a stationary distribution, the memory activation field
coincides with the field of probabilities. In fact,
such an approach unifies the Bohmian and
Copenhagen interpretations of quantum mechanics. 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 position at each instant of time. In our model ‘superposition’ of positions is nothing else than
117
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 hp = 1/p and p\ 1 is a prime number (the basis of the coding system of tcons).
Such a field can be called a moti6ation wa6e.
This motivation wave propagates via ISchrödinger equation: ca (t, x)= exp{i(ax − Eat)/
hp }, where Ea = a 2/2m 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 fC 0.26
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)= eibx/hp. Suppose that
these waves have amplitudes da, db Qp. 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 motivation waves: c(x)=dae iu/hpca (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, Ea p B B 1 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
Such a tcons is an extremely idealized conscious system. A
conscious system could not be totally isolated from external
I-fields. In any case tcons must continuously receive information on physiological processes in his body.
26
As we have already noticed, the uniformly strong activation of all ideas in the memory of a conscious system implies
unconscious behavior.
25
118
A. Khrenniko6 / BioSystems 56 (2000) 95–120
guided by a nontrivial conscious force fC (t, x).
We omit rather complicated mathematical expression which formally coincides with the standard expression (Holland, 1993). As the memory
activation field coincides with the field of probabilities, probabilities to observe motivations
p= a and p=b are equal to d 2a and d 2b. As it
was already mentioned, in general these are not
rational numbers. Thus in general these probabilities could not be interpreted as ordinary limits of relative frequencies. There might be
violations of the law of large numbers (the stabilization of frequencies) in measurements on
conscious systems (see Khrennikov, 1999, for
the details).
Complexity of the I-motion essentially increases if c is determined by k ] 3 different motivations. Finally we remark that (in the
opposite to the Bohmian mechanics) waves ca,
and cb, a " b are not orthogonal. The covariation B ca, cb \ "0. Thus all motivation waves
in a conscious system are correlated.
12.4. Example 12.2, conscious e6olution of
complex biosystems
Let t be a biosystern having a high I-complexity 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 q t(t). It is supposed that t has a
kind of collective memory. Let c t(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 biosystem. Each living form lj has a motivation aj (to
change the total information state q t). The pilot
I-formalism implies that the total motivation p t
of the biosystem t could not be obtained via the
summation of motivations aj. The mechanism of
27
For some biosystems, amplitudes dj, j= 1, … , M, can be
chosen as sizes of populations of lj, j=1, … , M.
generation of p t is more complicated. Each aj
activates in the collective memory of t the motivation wave caj. A superposition of these waves
gives the memory activation field (‘conscious
field’) of t.
c t(t, x)= % dj e iuj /hpcaj (t, x)
j
(compare with Eq. (30)).27 The c(t, x) induces
rather complicated conscious potential C t(t, x)
which guides the motivation p t 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 materialistic axiom of cognitive science, see Bergson,
1919. In fact, this axiom is the basis of the
modern neurophysiologic investigations. However, there are some reasons to suppose that the
brain does not determine the content of the
mind (see Hautamäki, 1997, for the extended
analysis of this question). Here we refer only to
Putnam’s theory of meanings. According to Putnam, 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 mathematical model for mental processes the materialistic 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 motivation p =a which activates the memory of
tcons. We know that the field ca (x)= eixa/hp is
the eigen-function of the position operator p̂.
Thus, for an ensemble Sa of free systems with
the same memory activation field ca (x), observations 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,
iS(x)/
having zero derivatives. Set c l,u
a (x)= R(x)e
l(x)
hp where R(x)= e
and S(x)= ax − u(x). The
c l,u
a (x) is also an eigen-function of the motivation operator. Thus observations of the motivation for an ensemble S l,u
of conscious systems
a
A. Khrenniko6 / BioSystems 56 (2000) 95–120
with memory activation field c l,u
a (x) will also give
the motivation p= a. However, the fields c l,u
a (x)
and ca (x) can have extremely different distributions 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 motivation p = a.
Appendix A
A.1. p-adic differential calculus
The system of p-adic numbers Qp is a number
field. Thus the operations of addition, subtraction, multiplication and division are
well defined. The derivative of a function f: Qp “
Qp is defined (as usual) as limDxp “ 0
f(x+ Dx)−f(x)
. The main distinguishing feature
Dx
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
2n
n
defined as f(x) =
for x = n = 0 anp
n = 0 anp .
This function is injective ( f(x1) " f(x2) for x1 "
x2) and f % 0.
References
Albeverio, S., Khrennikov, A.Yu., De Smedt, S., Tirozzi, B.,
1998. p-adic dynamical systems. Theor. Math. Phys. 114
(3), 349 – 365.
Albeverio, S., Khrennikov, A.Yu., 1998. A regularization of
quantum field Hamiltonians with the aid of p-adic numbers. Acta Appl. Math. 50, 225–251.
Albeverio, S., Khrennikov, A.Yu., Kloeden, P.E., 1999. Human memory as a p-adic dynamical system. Byosystems
49, 105 – 115.
Amit, D.J., 1989. Modeling Brain Functions. Cambridge
University Press, Cambridge.
Ballentine, L.E., 1989. Quantum Mechanics. Englewood
Cliffs, New Jersey.
Bell, J., 1987. Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press.
Bergson, H., 1919. Energie Spirituelle. Alcan, Paris.
Bohm, D., 1951. Quantum Theory. Prentice-Hall, Englewood Cliffs, New Jersey.
119
Bohm, D., Hiley, B., 1993. The Undivided Universe: an Ontological Interpretation of Quantum Mechanics. Routledge and Kegan Paul, London.
Borevich, Z.I., Shafarevich, I.R., 1966. The Number Theory.
Academic Press, New York.
Clark, A., 1980. Psychological Models and Neural Mechanisms. An Examination of Reductionism in Psychology.
Clarendon Press, Oxford.
Cohen, J.D., Perlstein, W.M., Braver, T.S., Nystrom, L.R.,
Noll, D.C., Jonides, J., Smith, E.E., 1997. Temporal dynamics of brain activation during working memory task.
Nature 386 (April 10), 604 – 608.
Courtney, S.M., Ungerleider, L.G., Keil, K., Haxby, J.V.,
1997. Transient and sustained activity in a disturbed neural system for human working memory. Nature 386
(April 10), 608 – 611.
Dawkins, R., 1976. The Selfish Gene. Oxford University
Press, New York.
De Broglie, L., 1964. The Current Interpretation of Wave
Mechanics. A Critical Study. Elsevier, Amsterdam.
Dubischar, D., Gundlach, V.M., Steinkamp, O., Khrennikov, A.Yu., 1999. A p-adic model for the process of
thinking disturbed by physiological and information
noise. J. Theor. Biol. 197, 451 – 467.
Eccles, J.C., 1974. The Understanding of the Brain. McGraw-Hill, New York.
Escassut, A., 1995. Analytic Elements in p-adic Analysis p.
80. World Scientific, Singapore.
Freud, S., 1933. New Introductory Lectures on Psychoanalysis. Norton, New York.
Hiley, B., Pylkkänen, R., 1997. Active information and cognitive science — a reply to Kieseppä. In: Pylkkänen, P.,
Pylkkö, P., Hautamäki, A. (Eds.), Brain, Mind and
Physics. IOS Press, Amsterdam.
Holland, R., 1993. The Quantum Theory of Motion. Cambridge University Press, Cambridge.
Hoppensteadt, F.C., 1997. An Introduction to the Mathematics of Neurons: Modeling in the Frequency Domain,
second ed. Cambridge University Press, New York.
Hautamäki, A., 1997. The main paradigms of cognitive
science. In: Pylkkänen, P., Pylkkö, P., Hautamäki,
A. (Eds.), Brain, Mind and Physics. IOS Press, Amsterdam.
Khrennikov, A.Yu., 1994. p-adic Valued Distributions in
Mathematical Physics. Kluwer Academic Publishers,
Dordrecht.
Khrennikov, A.Yu., 1997. Non-Archimedean Analysis:
Quantum Paradoxes, Dynamical Systems and Biological
Models. Kluwer Academic, Dordrecht.
Khrennikov, A.Yu., 1998a. Human subconscious as a p-adic
dynamical system. J. Theor. Biol. 193, 179 – 196.
Khrennikov, A.Yu., 1998b. p-adic information dynamics
and the pilot wave theory for cognitive systems. Report N. 9839, Växö University, Department of Mathematics.
120
A. Khrenniko6 / BioSystems 56 (2000) 95–120
Khrennikov, A.Yu., 1999. Interpretations of Probability. VSP International Scientific Publishers, UtrechtTokyo.
Kolmogorov, A.N., 1956. Foundations of Probability.
Chelsea, New York.
Lorenz, K., 1966. On Aggression. Harcourt, Brace and
World, New York.
von Mises, R., 1957. Probability, Statistics and Truth.
Macmillan, London.
Putnam, E., 1988. Representations and Reality. A Bradford
Book. MIT Press, Cambridge.
Schikhov, W., 1984. Ultrametric Calculus. Cambridge University Press, Cambridge.
Searle, J., 1992. The Rediscovery of the Mind. MIT Press,
Cambridge.
Skinner, B.F., 1953. Science and Human Behaviour.
Macmillan, New York.
Vladimirov, V.S, Volovich, I.V., Zelenov, E.L., 1994. p-adic
Analysis and Mathematical Physics. World Scientific
Publishers, Singapore.