Of course, our formalism is just the first step to describe the phenomenon of consciousness on the
basis of a model of information reality. However, even this formalism implies some consequences
which might be interesting for neurophysiology, psychology, artificial intelligence complex infor-
mation systems, evolutionary biology and social sciences. Here we present briefly some of these
consequences.
Flows of cognitive information in the brain and other cognitive systems can be described
mathematically in the manner which is similar to the classical Newton mechanics for motions of
material systems. Therefore the motion of ideas notions, images in the brain has the determinis-
tic character of course, such a motion is per- turbed by numerous information noises, see
Dubischar et al., 1999, for the details. This mo- tion in mental space is not an evolution with
respect to physical time t
phys
, 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 mate- rial structures for example, groups of neurons. It
is induced by groups of evolving ideas. These dynamical groups of ideas produce a new infor-
mation field, conscious field, which induces a new information force, conscious force, which is the
direct analogue of quantum force in the pilot wave theory for quantum material systems. This
conscious force plays the great role in the infor- mation dynamics in the human brain and other
conscious cognitive systems. As in the classical cognitive mechanics, in quantum cognitive me-
chanics conscious potentials can connect thinking processes in different cognitive systems even in
the absence of physical potentials and forces. Therefore it is possible to speak about a collective
consciousness for a group of cognitive systems in particular, human individuals. We also note that
different conscious potentials conscious C-fields induce conscious forces f
C
of different informa- tion strength. The magnitude of the conscious-
ness can be measured at least theoretically. Thus different cognitive systems in particular, different
C
control or at least change their cogni- tive behaviours. From this point of view human
individuals and animals differ only by the behav- iors of their conscious C-fields.
As one of applications of our formalism to psychology, we try to explain Freud’s psychoanal-
ysis on the basis of our model as the process of the reconstruction of the conscious field of an
individual i having some mental decease via an information coupling of a psychoanalytic p on
the level of a collective C function of the system i, p.
2. Classical cognitive mechanics
First we recall some facts from Newton’s classi- cal mechanics.
In Newton’s model motions of material systems are described by trajectories in space X
mat
of localization of material systems
2
. Thus starting with the initial position q
a material object A evolves along the trajectory qt in X
mat
where t is physical time. The main task of Newton’s
mechanics is to find the trajectory qt in space X
mat
. 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 pt of A is equal to the velocity 6t
of motion. In the mathematical model the velocity
6 t can be found as 6t =
dqt dt
q ; t. The ve-
locity need not be a constant. Thus it is useful to introduce an acceleration at of A which is the
velocity of the velocity . The second Newton law says:
at = ft, q 1
2
In the mathematical model X
mat
= R
3
or some real mani- fold.
p ; t=ft, q, p0=p
, t, q, p X
mat
. 2
By integrating this equation we find the momen- tum pt at each instant t of time if the initial
momentum p is known. Then by integrating the
equation q
; t=pt, q0=q ,
t, q, p X
mat
3 we find the position qt of A at each instant t of
time if the initial position q is known.
We develop the formalism of the classical cog- nitive mechanics by analogue with Newton’s me-
chanics. Instead of the material space X
mat
, we consider mental space X
men
see Section 5 for the mathematical model. A cognitive system t is a
transformer of information: an information state q X
men
the collection of all ideas of t is in the process of continuous evolution; t makes trans-
formations q q q¦ ....The time parameter of this evolution is also an information parameter
mental time of t, t X
men
. Thus the activity of t generates the trajectory qt in mental space X
men
. Our deterministic cognitive postulate which is a
generalization of Newton’s deterministic mechani- cal postulate is that the trajectory qt of the
evolution of ideas is determined by initial condi- tions and forces. As in Newton’s mechanics, we
introduce the velocity 6t of the changing of the idea qt. This is again an information quantity a
new idea. It can be calculated as the derivative in mental space X
men
of qt with respect to mental time t. We start with development of the
formalism for a cognitive system t with the infor- mation 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 qt. We postulate that the cognitive dynamics in X
men
is described at least for some cognitive processes by an information analogue of Newton’s second
law. Thus the trajectory pt of the motivation of t
is described by equation p
; t=ft, q, p0=p ,
t, q, p X
men
, 4
where ft, q is an information force generated by external flows of information; in particular, by
then the motivation pt can be found at each instant of mental time t by integration of Eq. 4.
The trajectory qt of the evolution of ideas can be found by the integration of equation
q ; t=pt, q0=q
, t, q, p X
men
5 if the initial idea q
is known. We recall that in Newton’s mechanics a force
fq, q X
mat
, is said to be potential if there exists a function Vq such that fq = − dVqdq. The
function Vq 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 X
men
. The potential Vq, q X
men
, is an information potential, infor- mation field, which interacts with a cognitive sys-
tem t. Such fields are classical unconscious cognitive fields.
As we have already mentioned mental time t need not coincide with physical time t
phys
. Mental time corresponds to the internal scale of an infor-
mation process. For example, for a human indi- vidual t, the parameter t describes ‘psychological
duration’ of mental processes. Our conscious ex- perience demonstrates that periods of the mental
evolution which are quite extended in the t
phys
- scale 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 evo-
lution of a cognitive system. We remark that t
phys
can be also interpreted as a chain of ideas about counts n = 1, 2, ..., for discrete t
phys
and about
counts s R, for continuous t
phys
. In principle, physical time t
phys
, can be considered as the spe- cial representation for mental time t. However, it
is impossible to reduce all mental times to physi- cal time even if t
phys
is defined up to a transfor- mation, t
phys
= us
phys
. Different mental systems, t
1
, …, t
N
, and even different mental processes in a cognitive system t have different mental times,
t
1
, …, t
N
. The use of physical time t
phys
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
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 inter-
rupts p for periods of different duration. Denote by q the state of information of t on E. In
principle, the information evolution of t can be considered as an evolution with respect to physi-
cal time s = t
phys
mechanical clocks: q = fs. However, the physical parameter s is not directly
related to the information process p. For example, the velocity 6
s
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, 6
s
is not well defined. Denote by D
1 r
= [s
, s
1
, D
1 i
= [s
1
, s
2
, ... , intervals of reading and interruption of reading. Thus the information process p induces
the following
split of
physical time
s: D
1 r
, D
1 i
, ... , D
M r
, D
M i
, .... The intervals D
1 i
, ... , D
M i
, ... must be eliminated. New time parameter s¯ = fs
is defined as s¯ = s on D
1 r
, s¯ = s on D
1 i
, .... The parameter s¯ can be considered as one of possible
mental information scales for the process p. The use of time s¯ essentially improves the mathemati-
cal description of p. However, there is still no large difference with the standard physics
3
. Suppose now that intervals D
k r
, D
k i
depend on information that t obtains in the process p:
D
k r
a
k
, D
i r
b
k
, where a
k
, b
k
X
men
are information strings, ‘ideas’. Here s¯ = fs, c and q = hs, c,
where s R, c X
men
. The next natural step is to eliminate the real parameter s from the descrip-
tion 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 parame- ter t. This is information on E which is obtained
by t from the corresponding part of B. In the simplest model we can describe t as the text of the
men
depends on the initial information stage q
on E of t, the initial motivation p
of t to perform the informa- tion process p and an information force Ft, q
that changes the motivation. For example, if F0 and p
= 0, then qtq
. Thus the reading of the book B does not change the state of knowledge of
t on ancient Egypt.
This example demonstrates that the informa- tion force Ft, q which ‘guides’ the information
state q of t could not be reduced to external information forces ft, q for example, informa-
tion from radio, TV and other books. There exists some additional information force, f
C
t, q, which changes crucially the trajectory qt X
men
. If even p
= 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 qt q
the conscious force f
C
t, q can generate nonzero motivation to study this subject.
The concrete mathematical representations for mental time t by so called m-adic integers
branches of trees as well as some other examples will be given in Section 8.
3. Quantum cognitive mechanics, conscious field
First we recall some facts on quantum mechan- ics for material systems. The formalism of quan-
tum mechanics was developed for describing motions of physical systems which deviate from
motions described be Newton’s Eqs. 2 and 3. For example, let us consider the well known two
slit experiment. There is a point source of light O and two screens S and S. The screen S has two
slits h
1
, and h
2
. 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 mo-
tion Eqs. 2 and 3 could not explain the interference phenomenon: ‘classical forces’ f in-
volved in this experiment could not rule photons
3
Of course, as f is non-invertible, there are some differ- ences with the standard formalism.
Q
, quantum force, which must be taken into account in New-
ton’s equations. Thus instead of Eq. 2, we have to consider perturbed equation
p ; t=ft, q+f
Q
t, q, p0, t, q, p X
mat
. 6
It is natural to assume that this new force, f
Q
t, q, is induced by some field Ct, q. This field C
t, q can be found as a solution of Schro¨dinger equation
h i
c t
t, q = h
2
2
2
c q
2
t, q − Vt, qct, q. 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 wave
4
. The C-field associated with a quan- tum system has some properties which imply that
C q could not be interpreted as the ordinary
physical field as, for example, the electromagnetic field. The quantum force f
Q
q is not connected with Cq by the ordinary relation f =
dCq dq
. The ordinary relation between a force f and a poten-
tial V implies that scaling V cV, where C is a constant, implies the same scaling for the force,
namely, f cf. In the opposite to such a classical relation quantum force f
Q
is invariant with respect to the scaling C cC the C-function. Thus the
magnitude of the C-function ‘quantum poten- tial’ is not directly connected with the magnitude
of quantum force f
Q
. According to Bell 1987 and Bohm and Hiley 1993, Cq is merely an infor-
mat
in Bohm and Hiley 1993 Cq is compared with a radio signal which rules a large ship with the aid
of an autopilot. Here the amplitude of the signal is not important, only information carried by this
signal is taken into account.
5
From the introduction to this paper it is clear how we can transform the classical cognitive me-
chanics to quantum cognitive mechanics, con- scious mechanics. The main motivation for such a
development of the classical cognitive mechanics is that behavior of conscious systems cannot be
described by a ‘classical information force’ f. Be- havior of a conscious cognitive system strongly
differs from behavior of unconscious cognitive system even if both these systems are ruled by the
same classical information force f . Thus our information generalization 4 of the second New-
ton law is violated for conscious cognitive sys- tems. As in the case of material systems, it is
natural to suppose that there exists some addi- tional information force f
C
q, conscious force, associated with a cognitive system. This force
changes the trajectory of a cognitive system in the space of ideas X
men
. A new ‘quantum’ conscious trajectory is described by equation
p ; t=ft, q+f
C
t, q, p0 = p ,
t, q, p X
men
. 8
The conscious force f
C
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 Schro¨dinger equation see Section 10 describes the evolution of the conscious C-
field.
5
The pilot wave theory does not give a clear answer to the question: Is some amount of physical energy transmitted by
the C-field or not? The book of Bohm and Hiley 1993 contains an interesting discussion on this problem. It seems
that, despite the general attitude to the information interpreta- tion 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.
4
The pilot wave theory does not give the standard interpre- tation of quantum mechanics, namely, the orthodox Copen-
hagen interpretation. By the latter interpretation it is impossible to describe individual trajectories of quantum parti-
cles. Probably an analogue of the orthodox Copenhagen inter- pretation could be also interesting to quantize the classical
cognitive mechanics. However, in the present paper we shall concentrate on an analogue of the pilot wave formalism.
6
In the pilot wave formalism it is supposed that the C-field is created simulta-
neously with a quantum particle this field is only formally treated as separated from the particle.
In quantum cognitive theory the assumption on a complex internal structure of a quantum con-
scious cognitive system is quite natural. In princi- ple we may suppose that the conscious field Cq,
q X
men
, is generated by classical information pro- cesses in a cognitive system t. Moreover, it is
natural to suppose that higher information com- plexity of t implies that the C-field of t induces
the information force f
C
of larger information magnitude. We recall that in the pilot wave for-
malism both for material and mental systems the magnitude of C is not directly related to the
magnitude of f
C
. At the present stage of knowl- edge on cognitive phenomena the idea that the
C -field is generated by t seems to be the most
natural.
7
4. Collective unconscious and conscious cognitive phenomena
In the previous two sections we have studied the classical and quantum mechanical formalisms
We start with the classical unconscious cogni- tive mechanics. Let t
1
, …, t
N
be a family of cogni- tive systems with mental spaces X
men,1
, …, X
men.N
. We introduce mental space X
men
of this family of cognitive systems by setting X
men
= X
men,1
× … × X
men,N
. Elements of this space are vectors of information states q = q
1
, …, q
N
of individual cognitive systems t
j
. We assume that there exists an information potential V
q
1
, …, q
N
which in- duces information forces f
j
q
1
, …, q
N
. The poten- tial V is generated by information interactions of
cognitive systems t
1
, …, t
N
as well as by external information fields. The evolution of the motiva-
tion p
j
t and the information state q
j
t of the jth cognitive system t
j
is described by equations: p
;
j
t = f
j
t, q
1
, ... , q
N
, p
j
0 = p
0j
9 q
;
j
t = p
j
t, q0 = q
0j
, t, q, p X
men
10 In general for different j these evolutions are not
independent.
4
.
1
. Example
4
.
1
Let Vq
l
, q
2
= aq
1
− q
2 2
, where a is some in- formation constant given by a p-adic number in
the mathematical model. Motions of cognitive systems t
1
and t
2
in mental space are not indepen- dent; the information magnitude of the constant
of coupling a gives the strength of this depen- dence. On the other hand, if, for example,
Vq
1
, q
2
= q
1 2
+ q
2 2
, then motions of t
1
and t
2
are independent.
This model can be used not only for the de- scription of collective cognitive phenomena for a
group of different cognitive systems t
1
, …, t
N
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, p
1
, …, p
N
which produce ideas qt, …, q
N
t, related to dif- ferent domains of human activity.
8
We can apply
6
Even the Bohm – Hiley considerations on the complex in- ternal structure of quantum particles do not go so far to
assume that a quantum particle is a generator of the C-field. The Bohm – Hiley complexity is merely complexity of a re-
ceiver of radio signals on a ship.
7
On the other hand, if we try to generalize ideas of material quantum mechanics to the cognitive phenomena, then we have
to suppose that 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.
8
For example, p
1
produces ideas on food, p
2
produces ideas on sex, p
3
writes poems.
1
, …, p
N
.The main consequence of our model is that ideas q
1
t, …, q
N
t and motivations p
1
t, …, p
N
t do not evolve independently. Their simultaneous evolution is controlled by the infor-
mation potential Vq
1
, …, q
N
. It must be under- lined that an interaction between thinking modules
p
1
, …, p
N
has the purely information origin. The potential Vq
1
, …, q
N
need not be generated by physical field for example, the electromagnetic
field. A change of the information state q
j
q
j
or motivation p
j
p
j
of one of thinking processors p
j
will automatically imply via the information inter- action Vq
1
, …, q
N
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 Vq
1
, …, q
N
which give different types of connections between thinking blocks p
j
.
4
.
2
. Example
4
.
2
Let thinking processors p
1
, p
2
and p
3
be respon- sible for science, food and sex, respectively. Let
Vq
1
, q
2
, q
3
= a
1
q
1 2
+ a
2
q
2 2
+ a
3
q
3 2
+ a
12
q
1
− q
2 2
+ a
23
q
2
− q
3 2
+ a
13
q
1
− q
3 2
. 11
If the information constant a
1
, strongly dominates over all other information constants, then the
scientific thinking block p
1
works practically inde- pendent from the blocks p
2
and p
3
. If a
12
or a
13
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 de-
pend on the mental time of a cognitive system, V = Vt, q
1
, …, q
N
. Thus at different instances of mental time t a cognitive system can have different
information connections between thinking blocks p
1
, …, p
N
. We are now going to describe the collective
quantum conscious phenomena. Let t
1
, …, t
N
be a family of cognitive systems. The classical infor-
9
f
j
t, q
1
, …, q
N
by cognitive second Newton law 5. However, as in
the pilot wave formalism for many particles, for any family t
1
, …, t
N
of cognitive systems, there exists a C-field, Cq
1
, …, q
N
, of this family. This field is defined on the mental space X
men
= X
men,1
× … X
men,N
. This field generates additional information
forces f
j
t, q
1
, …, q
N
conscious forces and the Newton’s classicalunconscious
cognitive dynamics must be changed to quantum conscious cognitive dynamics
p ;
j
t = f
j
t, q
1
, ... , q
N
+ f
j,C
t, q
1
, ... , q
N
, j = 1, 2, ... , N.
12 In
general the
conscious force
f
j,C
= f
j,C
t, q
1
, …, q
N
depends on all information coor- dinates q
1
, …, q
N
information states of cognitive systems t
1
, …, tN. Thus the consciousness of each individual cognitive system t
j
depends on informa- tion processes in all cognitive systems t
1
, …, t
N
. The level of this dependence is determined by the
form of the collective C-function. As in the ordi- nary pilot wave theory in our cognitive model the
factorization
C t, q
1
, ... , q
N
= 5
N j = 1
C
j
t, q
j
of the C-function implies that the conscious force f
j,C
depends only on the coordinate q
j
. Thus the factorization of C eliminates the collective con-
scious effect. As in the classical cognitive mechanics, the
above considerations can be applied to a system of thinking blocks p
1
, …, p
N
of the individual cogni- tive system t for example, the human brain. The
conscious field C of t depends on information states q
1
, …, q
N
of all thinking blocks.
9
Throughout this paper we use ‘classical’ and ‘quantum’ as synonyms of ‘unconscious’ and ‘conscious.’ In fact, it would
be better to use only the biological terminology. But we prefer to use also the physical terminology to underline the parallel
development of mechanical formalisms for material and men- tal systems.
Classical and quantum fields, Vq
1
, …, q
N
and C
q
1
, …, q
N
, induce dependence between individ- ual thinking blocks p
1
, …, p
N
of a cognitive sys- tem t or individuals t
1
, …, t
N
belonging to a social group G.
In particular, this implies that all physiological systems of the organism are closely connected on
the information level. Therefore a decease in one of these systems may have an influence to other
systems even if they have no close connection on the physiological level. Of course, this is not a
new fact for medicine. But we now have the mathematical model see Sections 7 – 9. And, in
principle, we could at least after development of the model compute some effects of the informa-
tion influence on physiological processes. More- over, purely mental processes in the brain which
are not directly related to physiological processes are connected on the information level with phys-
iological processes. For example, let the mental block p
1
, control functioning of the heart and the block p
2
controls some psychological process for example, relations with some person and let the
classical information potential Vq
1
, q
2
= aq
1
q
2
, where a is a coupling information constant given
by a p-adic number in our mathematical model. Then purely mental process in p
2
has an influence to functioning of the heart. The classical informa-
tion force fq
1
, q
2
applied to the p
1
is equal to −
a q
2
. Thus it depends on the evolution q
2
t of the psychological process.
The presence
of the
conscious field
C q
1
, …, q
N
makes the connection between phys- iological and purely mental process more compli-
cated. There is the possibility of the conscious control of human physiological systems. In princi-
ple, if a person could change its conscious field C
q
1
, …, q
N
, shehe could change by just an information influence the functioning of some
physiological systems. Our model explains well the origin of homeopa-
thy. In fact, by a homeopathic treatment it is possible to change the information potential
Vq
1
, …, q
N
of the organism. Microscopic quan- tities of medicines which are used in the home-
k
described by the information state q
k
. The information concentrated in the home- opathic medicine could be applied to some other
information state q
j
, j k. The change of q
j
, q
j
q
j
, will imply the change of the trajectory q
k
t via
the change
of the
information force
f
k
t, q
1
, …, q
k
, …, q
j
, … q
N
f
k
t, q
1
, …, q
k
, …, q
j
, …, q
N
.
6. Freud’s psychoanalysis as a reconstruction of conscious field
By Freud’s theory, Freud 1933, mental space X
i
of a human individual i is split in two domains: 1 a domain of conscious ideas X
i c
; 2 a domain of unconscious ideas X
i u
. Thus X
i
= X
i c
X
i u
. In our information model Freud’s idea is repre-
sented in the following way. Let f: X
i
X
i
, be some function. As usual, we define a support of f
as the set supp f = {x X
i
: fx 0}. Let C be the conscious field generated by the individual i and
f
C
be the corresponding conscious force. Then supp f
C
is the set of conscious ideas ideas which can interact with the C-field, X
i c
= supp f
C
. The set X
i
¯ supp f
C
is the set of unconscious ideas X
i u
ideas which cannot interact with the C- field.
10
The motion of i in the space of ideas X
i
is described by the dynamical system:
p ;
i
t = ft, q
i
+ f
C
t, q
i
, q
i
X
i
13 where f = −
V
i
q
i
is the classical unconscious force generated by the classical information po-
tential V
i
of i and f
C
= − C
i
q
i
is the quantum conscious force generated by the conscious in-
formation potential C
i
of i. In the subspace X
i u
of
10
We remark that the sets of ideas, supp f
C
and supp C, do not coincide. It can be that supp f
C
is a proper subset of supp C.
p ;
i
t = ft, q
i
, q
i
X
i u
. 14
Let D be some domain in X
i u
and let a classical information potential V
i
t, q
i
, q
i
X
i u
, 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 q
X
i u
the information state q
i
t of i will always evolve to D. The dynamical system 14 is located in the space of
unconscious ideas. Here the conscious force f
C
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 q
the individual i will always arrive to aggression.
The aim of psychoanalysis is to extend the domain of conscious ideas X
i c
= supp f
C
. This ex- tension will perturb dynamics 14 by the action of
a conscious force f
C
. 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
i u
. Starting with q
X
i u
the i can have trajectories q
i
t which will be never attracted by the domain of ‘bad ideas’
D. The pair, a cognitive system i and a psychoana-
lytic p, can be considered as a coupled system of transformers of information. The information cou-
pling between i and p will generate a new informa- tion classical potential V
i,p
t, q
1
, q
2
which is defined on mental space X = X
i
× X
p
, where X
i
and X
p
, are spaces of ideas of the individual i and the psychoanalytic p, respectively.
Dynamics of the conscious field C
i
t, q
i
of i is described by the Schro¨dinger equation
h i
C
i
t t, q
i
= h
2
2
2
C
i
q
i 2
t, q
i
− V
i
t, q
i
Ct, q
i
15 Dynamics of the conscious field C
h i
C
i,p
t t, q
i
, q
p
= h
2
2
2
q
i 2
+
2
q
p 2
C
i,p
t, q
i
, q
p
− V
i,p
t, q
i
C
i,p
t, q
i
, q
p
. 16
If now the conscious force f
C
t, q
i
, q
p
= − C
i,p
t, q
i
, q
p
q
i
17 for some ideas q
i
X
i u
at least for some ideas q
p
X
p
, then the motion of i in the domain of unconscious
ideas X
i u
can be controlled consciously here C
i,p
t, q
i
, q
p
is the conscious potential induced by C
i,p
t, q
i
, q
p
. In fact, this means that X
i u
is reduced and X
i c
is extended. The aim of the psychoanalytic p
is to find ideas q
p
X
i
such that 17 takes place for unconscious ideas q
i
X
i u
of i. As the process of psychoanalysis is a conscious process at least for
p , it is natural to assume that ideas q
p
, used by p to induce condition 17 are conscious: q
p
X
p c
. 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
i u
in that condition 17 is satisfied, then in this domain dynamics 14 is
transformed in the conscious dynamics p
;
i
t = f 0 t, q
i
, q
p
+ f
C
t, q
i
, q
p
, q
i
O. 18
Under some circumstances this dynamical system can be free from ‘pathological features’ of dynam-
ical system 14. Of course, even the change of the classical
potential V
i
t, q
i
to a new classical potential V
i,p
t, q
i
, q
p
changes the motion of i: a new dynam- ics is ruled by the classical force f
0 t, q
i
, q
p
instead of the classical force ft, q
i
. However, it is not easy to change strongly the classical force f t, q
i
on the domain of unconscious ideas X
i u
by just the change of the classical potential. Typically V
i,p
t, q
i
, q
p
= V
i
q
i
+ V
p
q
p
+
Gq
i
, q
p
, where the informa- tion magnitude of Gq
i
, q
p
is small for ideas q
i
X
i u
. On the other hand, this minor change of the
6
. .
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 exten- sion is the extension of the domain of conscious
ideas X
i c
and the reduction of the domain of unconscious ideas X
i u
. This extension is realized by the information coupling between an individ-
ual i and a psychoanalytic p. By minor change of the classical information potential the p strongly
changes the conscious force acting on the i. Dy- namics of ideas in the unconscious domain 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 Schro¨inger equation Eq. 16 between an indi-
vidual i and a hypnotizer p the conscious dynam- ics 13 is changed in such a way that the
conscious force f
C
t, q
1
is practically totally elimi- nated by the action of the conscious force
f
C
t, q
i
, q
p
. For example, let f
C
t, q
i
, q
p
= − f
C
q
1
+ f
C
q
2
. Then information behavior of the i is ‘ruled’ by the conscious force f
C
q
2
of the p.
7. Mathematical models of material and mental spaces; real and p-adic numbers
From our viewpoint real spaces Newton’s ab- solute space or spaces of general relativity give
only a particular class of information spaces. These real information spaces are characterized
by the special system for the coding of informa- tion and the special distance on the space of
vectors of information. Any natural number m \ 1 can be chosen as the basis of the coding system.
Each x [0, 1] can be presented in the form:
x = a a
1
... a
n
... , 19
where a
j
= 1, …, m − 1, are digits. We denote the
set of all sequences of the form 19 by the symbol X
m
. For example, let us fix m = 10. One of the main properties of the real cording system is the
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 neighbor-
hoods; there exist arbitrary small right and left shifts. The identification 20 is connected with the
description of left hand side neighborhoods.
7
.
1
. Example
7
.
1
Let x = 10 … 0 … . Then x can be approxi- mated from the left hand side with an arbitrary
precision by numbers of the form y = 09 … 90 … . The following description of right hand side
neighborhoods will be very important in our fur- ther considerations.
AS Let x = a
… a
m
… . Then the numbers vectors of information which are close to the x
from the right hand side have the form y = b
… b
m
…, where a =
b , …, a
m
= b
m
for suffi- ciently large m.
This nearness has a natural information cogni- tive interpretation: AS implies the ability to
form associations for cognitive systems which use this nearness to compare vectors of information.
By AS two communications two ideas in a model of human thinking, Khrennikov 1997
which have the same codes for sufficiently large number of first the most important positions in
cording sequences are identified by a comparator of a cognitive system. Numbers vectors of infor-
mation which are close to x from the left hand side could not be characterized in the same way
see Section 7.1, there x and y are very close but their codes differ strongly.
7
.
1
.
1
. Conclusion The system of real numbers has been created as
a coding system for information which the con- sciousness receives from reality. The main proper-
ties of this coding system are the order structure on the set of information vectors and the re-
stricted ability see AS to form associations.
Let us ‘modify’ the real coding system. We eliminate the identification 20. Since now, there
is no order structure on the set X
m
. of information vectors. We consider on X
m
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 Z
m
see Schikhov, 1984. Therefore it is natural to use m-adic numbers
for a description of information at least cognitive processes. Mathematically it is convenient to use
prime numbers m = p \ 1 see Schikhov 1984. We arrive to the domain of an extended mathe-
matical formalism, p-adic analysis. We present some facts about p-adic numbers.
The field of real numbers R is constructed as the completion of the field of rational numbers Q with
respect to the metric rx, y = x−y, where · is
the usual valuation given by the absolute value. The fields of p-adic numbers Q
p
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 = 2
r
2
3
r
3
… p
r
p
…, and we define n
p
=p
− r
p
, writ- ing
p
= 0 and −
n
p
= n
p
. We then extend the definition of the p-adic valuation
· to all rational numbers by setting
nm
p
= n
p
m
p
for m 0.
The completion of Q with respect to the metric r
x, y = x−y
p
is the locally compact field of
p-adic numbers Q
p
. The number fields R and Q
p
are unique in a sense, since by Ostrovsky’s theo- rem see Schikhov 1984
Unlike the absolute value distance ·, the p-adic
valuation satisfies the strong triangle inequality x+y
p
5 max[
x
p
, y
p
], x, y Q
p
Write U
r
a = {x Q
p
: x−a
p
5 r} and U
r −
a =
{x Q
p
: x−a
p
B r}, where r = p
n
and n = 0, 9 1, 9
2, … . These are the ‘closed’ and ‘open’ balls in
Q
p
while the sets S
r
a = {x Q
p
: x−a
p
= r} are
the spheres in Q
p
of such radii r. These sets balls and spheres have a somewhat strange topological
structure from the viewpoint of our usual Eu- clidean intuition: they are both open and closed at
the same time, and as such are called 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 U
r
0 is an additive subgroup of Q
p
, while the ball U
1
0 is also a ring, which is called
the ring of p-adic integers and is denoted by Z
p
.
Any x Q
p
has a unique canonical expansion which converges in the
·
p
-norm of the form x = a
− n
p
n
+ ...a
+ ... + a
k
p
k
+ ...
where the a
j
{ , p − 1} are the ‘digits’ of the
p-adic expansion. The elements x Z
p
have the expansion
x = a +
... + a
k
p
k
+ ...
and can thus be identified with the sequences of digits
x = a ... a
k
... The p-adic exponential function
n = 0
x
n
n . The
series converges in Q
p
if x
p
5 r
p
, where r
p
= 1p,
p 2 and r
2
= 14
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 r
p
as the exponential series.
11
Thus here all information is considered from the view- point of associations.
m
by complet-
ing Q with respect to the m-adic metric r
m
x, y = x−y
m
which is defined in a similar way to above.
8. Hamiltonian dynamics on p-adic mental space
The rings of p-adic integers Z
P
can be used as mathematical models for mental spaces. Each ele-
ment x =
j = 0
a
j
p
j
can be identified with a se- quence x = a
a
1
… a
N
…, a
j
= 0, 1, …, p − 1. Such
sequences are interpreted as coding sequences in the alphabet A
p
= {0, 1, …, p − 1} with p letters
for some amounts of information. The p-adic metric r
p
x, y = x−y
p
on Z
p
corresponds to the nearness AS for information sequences. We
choose the space X = Z
p
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 dynam-
ics on mental spaces As usual, we introduce the quantity pt = q
; t =
d dt
qt which is the in-
formation analogue
of the
momentum, a
moti6ation.
The space Z
p
× Z
p
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
Hq, p I-Hamiltonian on the phase mental space which determines the motion of t in the phase
mental space:
q ; t=
H p
qt, pt, qt
= q ,
p ; t= −
H q
qt, pt, pt = p
. 22
The I-Hamiltonian Hp, 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 H
f
p = a
p 2
, a Z
p
describes the motion of a free cognitive system t, i.e. a cognitive system which uses only self-motiva-
tions for changing of its I-state qt. Here by solving the system of the Hamiltonian equations we
tivation p is the constant of this motion. Thus the free cognitive system ‘does not like’ to change its
motivation p in the process of the motion in the
mental space. If we change coordinates, q = q − q
k, k = 2ap , then we see that the dynamics of
the free cognitive system coincides with the dynam- ics of its mental time.
In general case the I-energy is the sum of the I-energy of motivations H
f
= a
p
2
which is an analogue of the kinetic energy and potential I-en-
ergy Vq: Hq, p = ap
2
+ Vq
The potential Vq 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 informa- tion string: x = a
, a
1
, …, a
N
, …, a
M
. This string
can be identified with an element of Z
m
, by setting a
j
= 0,
j ] M + 1: x = a
, a
1
, …, a
N
, …, a
M
, 0, …, 0, …. Counts of such a mental time are given
by blocks of x: t = a , a
1
, … , a
k
, 0, 0, …. Sup- pose now that the information state knowledge q
is coded
in the
following way:
q = b
, b
1
, …, b
j
, …, b
j
= 0, 1, …, k − 1, where b
is dynasty, b
1
is number of wars during dynasty b …
The symbol 0 is again used to denote zero knowledge
12
. Dynamics qt, b
j
= c
j
a , ..., a
N
, …, of knowledge of t is described by Hamiltonian
equations
13
.
12
In this example, the use of the homogeneous k-adic tree is not so natural. It would be more natural to use a nonhomoge-
neous tree in that the number k
j
of branches depends on the information characteristic b
j
, 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 Z
m
, q Z
k
, k m, we need more complicated mathematical analysis.
Fig. 2. The factorial tree Z
M
for m
1
= 2, m
2
= 3, m
3
= 4, …
for interactions
between arbitrary
individuals. There must be a process of learning for the group t
1
, …, t
N
which reduces mental times t
1
, …, t
N
to the unique mental time t. Thus, let us consider a group t
1
, …, t
N
of cog- nitive systems with the internal time t. The dy-
namics of I-states and motivations is determined by the I-energy; Hq, p, q Z
p N
, p Z
p N
. It is natu- ral to assume that
Hq, p =
N j = 1
a
j
p
j 2
+ Vq
1
, ... , q
N
, a
j
Z
p
. Here H
f
p =
j = 1 N
a
j
p
j 2
is the total energy of mo- tivations for the group t
1
, …, t
N
and Vq is the potential energy. As usual, to find a trajectory in
the phase mental space Z
p N
× Z
p N
we need to solve the system of Hamiltonian equations: q
j
= H
p
j
, p
j
= − H
q
j
, q
j
t = q
, p
j
t = p
.
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 cog-
nitive systems. As a consequence of such interac- tions cognitive systems produce new information.
The only distinguishing feature is that material objects are not involved in our formalism. Ac-
cording 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 phys-
ical 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.
8
.
2
. Example
8
.
2
, e6olution of scientific psychology
We introduce a mental time t = t
ps
which is used for describing the evolution of the psycho-
logical state
of a
scientist t
. Let
t = a
, a
1
, …, a
N
, …, where a
j
= 0, 1, …, m − 1, is a
number of publications of t in journals of the weight t. Journals with j = 0 are the most impor-
tant, 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 l
1
, = 2, 0, … = 2,
l
2
= 0, 8, 0, … = 80,
l
3
= 1, 1, 2, 0, … = 211. These instances of mental
time could not be ordered according their impor- tance. The evolution of the psychological state
qt of t is described by trajectory in the mental space in the simplest case X
men
= Z
m
,. If this trajectory is continuous, then t will have similar
psychological states qt
1
, qt
2
for close instances of mental time t
1
, t
2
. In the Hamiltonian framework we can consider
interactions between cognitive systems t
1
, …, t
N
These cognitive
systems have
mental times
t
1
, …, t
N
and I-states q
1
t
1
, …, q
N
t
N
. 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
t
1
, …, t
N
as a trajectory in the mental space Z
p N
= Z
p
× ... × Z
p
qt = q
1
t, ... , q
N
t. We think that the condition of consistency
t
1
= t
2
= ... = t
N
= t
23
Bohm and Hiley discussed a difference be- tween ‘active’ and ‘passive’ information. In fact,
our model supports their conclusion that ‘all in- formation is at least potentially active and that
complete passivity is never more than an ab- straction …’, Bohm and Hiley 1993, p. 37. If a
cognitive system t moves in the field of forces C classical or quantum, then the information
x supp V is active for t and the information x Z
p m
¯ supp V is passive for t. Let 6 = Vt, x be
a time p dependent potential. Then the set of active information Xt = supp Vt evolves in
mental space. Thus some passive information becomes active and vice versa.
9. Inertia of information
