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\title{The Standard Model and Beyond:\\ Interrelation between Theory and Reality}
\author{Vladimir Slobodenyuk \\{E-mail: vslob@yandex.ru}}
\date{September 2002}
\maketitle
\begin{abstract}
Numerous attempts of theoretical physicists to create absolute theory ---
Theory of Everything, which are realized in models GUT, SUSY, supergravity,
superstrings, and superbranes, are subject to critical analysis. Attention
is payed to the fact, that no, even very exact, theories are able to
give exhaustive picture of the phenomenon described. Any theories give
only approximate models of the object, reflecting more or less number of
its features. But always such its peculiarities remains, that are not
taken into consideration in the model because of principle impossibility
of absolute exact and complete picture of the nature. Roughness of
tools available to human for description of physical world is displayed,
particularly, in essential defects of theories describing the nature
at the most fundamental level --- models of quantum field theory
(divergences, renormalizations etc.). Because the models pretending on the
role of TOE are not founded on any experimental data contradicting to
the Standard Model, then really they even cannot pretend on the role of
generalization of the Standard Model.
\end{abstract}

\bigskip

\noindent {\bf Keywords}: fundamentality, quantum field theory, renormalization, Standard Model, TOE, Theory of Everything.
\newpage


\section{Introduction.}

There were such moments in the history of physics when it seemed, that
one was to do only some last and decisive steps to understand the
organization on the nature, and then complete and absolutely exact
physical picture of the world would be created, in which all physical
processes would have got exhaustive description and explanation. In XIX
century such hopes were connected with classical theories: mechanics,
electrodynamics, and thermodynamics. At present some of quantum field
models generalizing the Standard Model may pretend, in opinion of a lot of
theoretical physicists, on the role of such theory. At last decades a
number of directions of such generalization was created. That are
Grand Unification Theories (GUT), supersymmetric models (SUSY),
supergravity, and, at last, superstrings and superbranes. Adherents of one or
another approach from this sequence mean, that one can create in its
framework the model describing physics of fundamental interactions
absolutely exactly and absolutely completely. Such model is called the
unified theory of fundamental interaction or Theory of Everything
(there exists even special abbreviation TOE). The last term testifies
the level of pretension of that theory.

To illustrate typical relation of physicists to unified theories
we reproduce statement of Devies (1985) about one of candidates on this
role --- supergravity. He says, that till now the physical theories were
considered as models, which describe reality approximately. But some
physicists assert now, that supergravity {\em is} reality itself, and that
this model ideally corresponds with real world. Then he states, that
similar to many tempting images, unified theory may prove to be
mirage, but first for the hole history of science our notions about
{\em complete scientific theory of everything} are arising.

The idea of construction of TOE is so popular, that some last decades
scientific interests of majority of theoretical physicists investigating
the problems of fundamental interactions are connected in any
way with working out the unified theory. One should note, nevertheless,
that in spite of increasing level of abstraction and complication of
the theory apparatus with appearance of every new approach, till now
there was no achieved any successes in more deep in comparison with
the Standard Model understanding of construction of the nature. Even the
fact, that efforts of leading theoretical physicists during so long
time have not led to any distinguishable result, make us to think about
causes of such failure.

In a present paper the attempt is undertaken to make more clear
interrelation between the theory and the objects of nature been described
by it in implementation to the quantum field models, and on this basis
to put boundaries of opportunities of these models in understanding the
phenomena of the nature and, respectively, to put restrictions on their
pretensions. One is to notify from a very start, that the article is
addressed preferably to theoretical physicists. Its main purpose is
to carry to them some ideas, which are rather simple for philosophers,
but which are ignored by physicists in their researches.

Before starting to discuss the problem outlined let us consider clear and
useful example. Assume we wish to describe geometrical shape of the surface
of the Earth. If we apply to history, then we can see the sequence of the
models, where every consequent one is more precise then previous. First
the Earth was imagined as flat (in global scale, without taking into
consideration relief of a place). From the modern viewpoint we can say,
that this picture may be accepted as zero approximation for description
of a small domain of a surface. But in global scale it becomes perfectly
incorrect.

Later it has become clear, that the Earth did have the shape of a ball. That
model was essentially more exact, but, as it has turned out to be, even it
did not describe the shape of our planet sufficiently good. More exactly
the shape of the Earth may be represented as an ellipsoid of rotation,
because it is flattened out in direction of rotation axis. But this is not
all. It proves to be, that the Earth is not symmetric relative to the
equator, but it has the egg-like shape. And one cannot find some simple
equation for description of such shape even in global scale, contrary
to the cases of a ball and an ellipsoid, for which the equations are well
known and have a simple form. But let us assume, that we have found some
equation, which quite satisfactory describes the shape of the Earth. Can
one hope, that the shape of the real Earth surface will not deviate from
this equation, even if we will take into account distances only of any
large scale and will not consider details of smaller sizes? Naturally,
one cannot. Even in large scales the surface  has so improper shape,
that one is not able to describe it exactly with any even very complicated
equation. Note, that we do not mean here construction of simple level
lines directly according experimental measurements, as it is usually done
at geographic maps, but we mean creation of some mathematical model of
the surface shape permitting to calculate these lines.

Besides, there is once more cause which does not allow to describe the
Earth's shape exactly. Every description suppose presence of some scale,
such that the structures with sizes smaller then this scale are simply
ignored. Nevertheless, if we diminish this scale, then we are to consider
part of such structures. Other ones, which are smaller, will to be ignored
as before. But one can diminish the scale once more and introduce
into consideration new structures and so on. This process may be
continued infinitely. At any stage we will achieve microscopic scales.
But for these scales usual macroscopic notions loose their sense at all.
Moreover, at scales of order of atomic size one cannot say about static
picture and geometric surface. If we wish to obtain the model of the
Earth's shape taking into account local features of relief, then we are
to give up the idea of describing the surface by only one , may be,
very complicated equation. We have to use instead any interpolations
for separate domains of the surface, making ``sewing together" the
interpolation functions at boundaries of domains. Let the domain $A$ is
described by the function $f_A$, the domain $B$ --- by the function $f_B$
etc. And besides, the function $f_A$ itself is defined not only at the
domain $A$, but at the neighboring domains too, particularly, at the
domains $B$, $C$ and so on. The same is fair for other functions. But
their meanings at the other domains do
not carry any information about the shape of surface. So, the structure
of the model is such, that it includes a lot of functions describing
surface. But to obtain prediction directly for definite domain one
is to use only one of these functions.

The example discussed shows us that it is not possible to create
absolutely exact model of the Earth's surface. We can build even very
good model, but it will be correct only with definite accuracy and till
definite scales. It is obvious that one can do similar conclusions in respect
to any other object of the nature. The geometric shape of the Earth has been
taken here quite arbitrary only for the illustration.

We can build some models of the objects of phenomena investigated. These
models may be more or less rough. In the first case they allow us to
get predictions with more high accuracy and/or for more wide range of
phenomena or meanings of any parameters. But however, these predictions will
not be absolutely exact, because every even very complicated and refined
model is only some approximation to infinitely varied and complicated world,
which cannot be understood by human completely and finally.

After this affirmation it becomes clear, that there is no any sense to
try to create absolute theory, even if this theory should describe
construction of the nature at the most fundamental level. In any case
the model will be only more or less rough approximation to real world.
This statement is rather trivial from the viewpoint of philosopher.
But the problem is connected with following circumstances. Probably,
the most theoretical physicists do not realize this fact and consider
the models created by them as absolute models exactly reflecting
organization of the nature. Moreover, they often seriously think, that
the nature is organized according any equations, and task is only
to guess these correct equations and, correspondingly, throw away all
incorrect ones. But really the equations are only our human way of
description of the nature. So they on principle cannot describe the
nature exactly. They only allow us to obtain approximate estimates
for some parameters (maybe, many of them) related to the phenomenon
considered. But always there is something more, that is inaccessible for
human understanding currently and is not described by any models.


\section{Quantum field models.}

Let us apply to quantum field models the ideas discussed above.

\subsection{Divergences and renormalizations.}

Creating quantum field theory (see, e.g., Bogolubov (1980)) physicists
have run against the problem of divergences of the integrals, via
various observable values was calculated.
Nevertheless, it has turned out to be, that for some models one could carry
out the program of renormalization, which consists in following. One
can introduce any regularized functions, integrals of which are convergent,
instead of functions leading to divergences. These functions depend on
some additional parameters. When these parameters tend to definite
values, then regularized functions tend to their initial nonregularized
values (taking off regularization). After calculation of integrals with
regularized functions one is to carry out limit transition for the
parameters of regularization, i.e. take off regularization. But during
this transition infinite terms appear again. To eliminate them one make
renormalization, i.e. subtract infinite terms so that the result becomes
finite. Such an operation may be realized only for the models, in which
at all orders of perturbation theory only finite number of various
structures with divergences arises. If at high orders of perturbation
theory new and new types of structures with divergences appear, then
renormalization cannot be made and such models are nonrenormalizable.
Modern methods of quantum field theory does not allow to obtain any
sensible predictions with a help of nonrenormalizable theories when
the coupling constant is not small. Contrary to this, renormalizable
theories allow to calculate different values in the framework of
perturbation theory, and what's more, predictions for observable
physical values practically do not depend on the ways of regularization
and renormalization.

So, although renormalizations make the theory not so attractive
aesthetically and complicate a technique of operating with it, but they
do not make the theory unacceptable (if, naturally, it is renormalizable).

Infinite contributions and counterterms (terms that are to be subtracted
to eliminate divergences) in some  features are similar with the functions
$f_B$, $f_C$ etc. at the domain $A$ from the example with the Earth's
surface. They have not physical sense, but formalism of the theory is
such that one cannot do without them. If one could define the shape of
the Earth with unified function, which is determined through out the hole
surface by the only one analytical expression, then one would not have
these excessive functions. But because one cannot do this, then one is
to agree with presence at the model these senseless objects. Situation
at quantum field theory is the same.

We do not know how to construct the formalism of the theory so that it
would not include divergences from a very beginning. So, we construct
the theory with divergences and then eliminate them according definite
recipe. Because such procedure allows us to obtain predictions, which
quite good correspond with the experimental data, then we have the model,
describing definite region of natural phenomena. If we do not demand
from the theory that it would be absolute and would give exhausting
description of the object investigated, then we may consider, that our
purpose is achieved.

Nevertheless, one should take in the mind, that this model, perhaps,
has essentially restricted possibilities for extrapolation. It describes
definite class of phenomena at definite conditions, and there are no any
foundations to think, that this model will work satisfactory at conditions
essentially different from ones, for which it was constructed. E.g., the
Standard Model is fair at the energies achieved at modern accelerators.
However, at the more high energies some objects and phenomena, which
should not exist according current notions, really may appear. Then it
will be necessary for their description to create the new theory
generalizing the Standard Model.

One may note here, that ``inelegance" of the quantum field theory connected
with renormalizations is, in the main, due to the restrictions of our
means for description of the nature. When more simple phenomena were
studied, e.g. subjects of following disciplines: classical mechanics,
classical electrodynamics, thermodynamics, then one have been able to
create quite elegant theories (which, nevertheless, have some defects too).
When one have come to more detailed description, taking into account
quantum effects, then one have become not able to construct the theory
been free of unnecessary structures, which was not connected with any
features of the phenomena described. It is similar to situation, which we
have met at the example with description of the Earth's shape. If for
global description of the shape one can take a single equation (it is
a ``beautiful" theory), then for description of the local structure
one is already to introduce a lot of functions, every of which has a
physical sense only within the range of its ``own" domain of surface
(the theory with ``unnecessary" structures).


\subsection{Standard Model.}

At present the Standard Model is generally accepted and quite good
experimentally tested (in the domain, where it permits to obtain
predictions) model of interactions of elementary particles. It includes
quantum chromodynamics (QCD) describing strong interactions of quarks
and the Weinberg -- Salam model describing unified weak and electromagnetic
interactions. Review of the Standard Model and gauge theories lying on
its foundation may be found in Halsen (1984) and Huang (1982).

Let us pay a special attention to electroweak model. Before the Weinberg --
Salam model electromagnetic and weak interactions were described by separated
theories --- quantum electrodynamics (QED) and Fermi model with four-fermion
interaction. Electrodynamics described (and describes now)
electromagnetic processes quite satisfactory. This is renormalizable
theory with sufficiently small coupling constant, that allows successfully
operate in the framework of perturbation theory. Contrary to this, Fermi
model is not renormalizable because of presence four-fermion vertexes
within it. It permits only to make calculations at first order in the
weak coupling constant $G_F$ (the Fermi constant). Due to weakness of
this interaction such calculations gave predictions quite corresponding
with the experimental data available. However, there was not any theoretical
justification for such approach. Standard quantum field theory demands to
take into account contributions of more high orders in perturbation
theory. But during their calculations divergences, which cannot be removed,
appear.

So, it was necessary to construct renormalizable model of weak interaction.
It turned out to be, that it is not possible to make the separate model
of weak interaction to be renormalizable. But it is possible to construct
the unified model of the weak and electromagnetic interactions, satisfying
the demand of renormalizability. In this theory carriers of weak
interaction --- intermediate vector bosons ($W$- and $Z$- bosons) are
introduced. They are massive particles (presence of their mass is
connected with
a finite radius of interaction). Due to these bosons the four-fermion
vertexes of the old theory are separated on two three-particle vertexes.
The Weinberg -- Salam model belong to the class of gauge theories,
all of which have general property --- renormalizability. But the
carriers of interactions in guage theories should be massless, while
vector bosons have a mass and moreover their masses are over 100 times
larger then the mass of proton.

To overcome this difficulty the mechanism of spontaneous symmetry breaking
(the Higgs mechanism) was suggested. It permits to start from the
exactly symmetric Lagrangian containing massless gauge fields and to
obtain afterwards in the result of spontaneous symmetry breaking the
Lagrangian with the massive gauge fields. Because the theory with
the initial Lagrangian is renormalizable, then it remains renormalizable
and after symmetry breaking. The pay for the obtained in a such way
renormalizability is appearance in the theory of the new hypothetical
particles ---
the Higgs scalar bosons. Namely due to the Higgs fields with their unusual
Lagrangian of self-interaction the possibility of spontaneous symmetry
breaking appears, together with renormalizability of gauge
theory. However, serious problems arise instead, which are connected
with problematical character of existence of the Higgs bosons in the nature.

Because of nonstandard in comparison with the fields of really existing
particles Lagrangian the Higgs fields seem to be rather artificial
innovations. There are serious reasons to have doubts that such intricate
mechanism as the Higgs is one really embodied in the nature as if to make it
possible for physicists to construct renormalizable theory. Besides, the
mass of Higgs particles is not fixed at the model. From one hand, it makes
difficult their experimental searches, from other hand, it allows to
consider this mass so large, that these particles cannot be born at
modern accelerators. This closes the possibility to refute the hypothesis
about existence of Higgs particles in observed future (i.e. the model is not
falsified, and this is essential defect itself).

Experimental searches of Higgs particles have not given positive results.
And this, naturally, was explained by large mass of this hypothetic
particles. However, the most probable cause of these results is
absence of such particles in the nature.

On the example of the Higgs mechanism in the Standard Model we have run
against the situation, when for description of real natural phenomena
the formalism have been created, which gave sufficiently satisfactory
description in one sector, but did not correspond to anything in another
one. Namely, the Standard Model quite well describes the sectors of
real particles: quarks, leptons, photons, $W$- and $Z$- bosons, and gluons,
but all things related to  Higgs bosons are only some intrinsic structures
of the theory and do not correspond to any real phenomena.

Addressing to the example with the Earth's shape one can say, that the
sector of real particles in the Standard Model is analogous to the
function $f_A$ at the domain $A$, and the Higgs sector --- to the
function $f_A$ at the domain $B$. This function as mathematical object
exists at the domain $B$ too, but it does not carry any information
about structure of the surface at the domain $B$.

In principle, one could take only part of the Lagrangian of the Standard
Model with broken symmetry, in which Higgs fields are absent, and
consider it as the Lagrangian of the true theory. But such theory will
be nonrenormalizable and so one will not can operate with it in the
framework of perturbation theory. But one cannot effectively work with
quantum field theories in any other way. This is why one is to introduce
Higgs fields into the theory. But one should not be mistaken in respect to
its real status, and should not hope, that the nature is constructed in
correspondence with restricted opportunities of our mathematical apparatus.

Besides, there are other difficulties in the Standard Model. It is known,
that the series of perturbation theory are asymptotic, i.e. they are divergent,
but for the small coupling constant some first terms of the series give
sufficiently good approximation for the values calculated. If we take into
consideration a large number of terms of the series, the sum begin
strongly and strongly deviate from the truth value of the function, which
is represented by the series. In the Weinberg -- Salam model the coupling
constants are really small and so at this model one can obtain quite
exact predictions with the help of perturbation theory.

In QCD (other constituent of the Standard Model) the coupling constant
already is not sufficiently small and perturbation theory can be applied
only for description of very limited class of processes. It have been
introduced the notion of running constant. Its value depends on energy
transfer in the process. When large energy is transferred, the coupling
constant becomes quite small to calculate with a help of perturbation
series.

For processes with low energies the coupling constant is large and
perturbation theory is not applicable. There is no any regular theory
founded on the first principles of quantum field theory in this region.
Even in such models that, as it is asserted, are founded on the first
principals of QCD (e.g. lattice QCD, QCD sum rules) really some additional
suppositions are used. So, they cannot be treated as simply other methods
of working with the same theory. In realty they are {\em other} models,
different from perturbative QCD, and they cannot pretend on the same
power of generality as QCD can.

So, QCD in its modern state does not permit to obtain the predictions
for the most of processes, which it has to describe and which are connected
with so called nonperturbative effects. In particular, the process of
creation of hadrons from quarks are not described theoretically.

Physicists usually represent this situation as follows. There is the theory
--- QCD, which on principle gives complete description of strong interactions.
But we cannot make predictions for low energy processes with its help not
because the theory does not contain this information, but because we yet
cannot work with this theory in proper manner. Such a viewpoint is not
irreproachable, because if the regular ways of extraction of the information
about nonperturbative effects suddenly will be found, then there is no
any warranty, that these predictions
will be corresponded with that ones, which are being obtained now in the
framework of perturbative formalism. On principle, it may occur that
nonperturbative theory, founded on the same Lagrangian as perturbative one,
will have essentially different properties and will not be adequate
to existing experimental data.
This may be caused by following fact. Formalism of perturbative quantum
field theory is adapted exclusively for operating in the framework of
perturbative theory. But properties of the theory are due to not only
its Lagrangian, but to features of its formalism too. It, probably, will
be necessary to change the formalism itself for nonperturbative approach.
And this will lead, in fact, to construction of new model although founded
on former Lagrangian.

So, the Standard Model seems as some quite satisfactory approximation for
some sufficiently wide class of natural phenomena. But nothing more.
Defects of its formalism (impossibility to operate beyond the framework of
perturbation theory, renormalization, divergence of perturbation series,
necessity to introduce unbroken gauge symmetry, when it is not exact in the
nature, appearance of Higgs bosons at the theory, when they are absent in the
real world) do not allow us to hope that beyond the range of phenomena, for
description of which this theory have been constructed, it may give any
reliable predictions.


\section{Beyond the Standard Model.}


Outstanding success of the Standard Model in unification of weak and
electromagnetic interactions has inspired physicists for further search
of possible unifications of fundamental interactions. Investigations
connected with looking for the more high group of symmetry of gauge theories,
which would include as subgroup the group of symmetry of the Standard
Model $SU(3) \times SU(2) \times U(1)$ have begun to develop intensively.
This symmetry of high level is treated as exact one. At any very high
energy it is broken by means of the Higgs mechanism till the symmetry of
the Standard Model. Such models are called Grand Unification Theories.
Review may be found in Ross (1985).
It was proposed a number of models from this group. All of them
had any defects. But the matter is, that all these models, according their
construction, have all defects characteristic for their prototype ---
the Standard Model. Even this circumstance does not allow for these
models to have the status more high then the Standard Model has. However,
at times of boom of researches related to this approach many physicists
did hope that one of such models may occur correct model of the world,
describing it on fundamental level and, besides, absolutely exactly.

Later new directions of searching of TOE were arisen. These are
supersymmetric theories, supergravity (see, e.g., Ferrara (1983)),
superstrings (see, e.g., Green (1987)) and different
types of branes. The key idea of all of them is supersymmetry, i.e.
symmetry between bosons and fermions. It is supposed that there exist
hypothetical supersymmetric partners --- fermion for every really existing
boson, and boson for every fermion. E.g., photino is partner for photon
with spin 1/2 (photon has spin 1), selectron is partner for electron
with spin 0, squark is partner for quark with spin 0 (electron and quark
have spin $1/2$) and so on. Thus, according to these models, it
should exist in the world at least twice more of fundamental particles,
then we know. Where is the second half of the world? As no any
supersymmetric partner was observed till now, then it was supposed that
their masses are very large and power of modern accelerators is not
enough to produce these hard particles. I.e. here the same trick is used,
which has been used earlier for justification of absence of experimental
data about existence of Higgs particles in the Standard Model.

The most refined approaches, related to supersymmetry, is superstrings, and
besides, their generalization on the objects with more then one spatial
dimension --- branes. The fundamental objects in the theory of superstrings
are the one-dimensional lengthy objects instead of point-like particles ---
superstrings, possessing the fermionic degrees of freedom besides usual
bosonic ones. It is considered, that fundamental frequencies of these
strings' oscillations correspond to definite elementary particles.

This is characteristic of the place of this theory at modern physics
given by I. Aref'eva and I. Volovich (1990). ``Superstring theory is
the modern version of unified theory of fundamental interactions, sometimes
it is called the Theory of Everything. Aspiration to create such theory
penetrates all the history of science --- from Pythagoreans to Newton
and Leibniz, and further Reimann, Weyl, Einstein, and Heisenberg. It is
not obvious, that any successes are possible on the way to this purpose
at all. So much the striking the remarkable achievements of the superstring
theory seem. There are calculation of dimension of the space-time$^1$),
fixing of definite gauge group, including into unified scheme the
gravitation theory, and, probably, absence of divergences. The superstring
theory touches the most deep problems of the universe, and is the most
advanced current approach to the questions about nature of the fundamental
interactions. However, in spite of great interest to the theory and
remarkable achievements, one is to say, that basic problems remain opened
here. From the physical viewpoint the main problem is absence of the
experimental predictions. We may hope to enclose all information about
elementary particles into the superstring theory, but the superstring theory
itself have not given any experimental predictions. From mathematical
viewpoint the problem remains to prove the absence of divergences even
in the framework of perturbation theory".

We have cited so long quotation, because the position characteristic for
the most of physicists studying the superstring is expressed  here in
concentrated form. We see that they treat the superstring theory as a real
candidate for the role of absolute theory of everything (the possibility
of construction of absolute theory is implied by them as going without
saying). However, this theory does not give any new experimental predictions
(in comparison with the Standard Model), and, correspondingly, the
experimental data available do not demand for their interpretation any more
general, then the Standard Model, theory, and do not give any foundations to
choose one or another way of generalization.

What is the basis for belief of theoretical physicists that namely
supersymmetric theories may give exact picture of the world? Perhaps, there
are two the most important points: a hope to create the theory unifying all
four fundamental interactions and a hope to get rid of the divergences
usual for standard quantum field theory. It is known that there are no
divergences in one-loop approximation in the superstring theory, and
there are serious foundations to hope that divergences do not appear in
higher orders of perturbation theory too (but this fact is not proved till
now).

So, these reasons have quite speculative character and do not connected
with an experiment in any way. Here researchers attempt to create the general
theory with very cumbersome and complicated formalism, which does not
give in experimentally achieved for today range any predictions
different from given by partial theory --- the Standard Model. Usually hopes
are connected with a launch of new accelerators of elementary particles for
more high energies and discovering the new physics with their help.
Namely then, as adherents of this approach consider, predictions of
supersymmetric models will be tested! However, existence of such hopes is
due to overestimate of human ability of cognition of the nature without
any experimental premises.

To make situation more clear let us apply once more to the example with
description of the shape of the Earth's surface. Assume, that we have
chosen the accuracy, with which we desire to describe the surface
(let us determine it as minimal distinguished length $d$), and have
obtained on the basis of experimental measurements the set of the functions
$f_A$, $f_B$ etc., describing the domains $A$, $B$ etc., correspondingly.
Then we have set our mind on getting description with more high precision
$d'$ ($d'<d$). But we have not got any new, more exact, experimental
data and description with a help of the functions $f_A$, $f_B$ etc. have
been in a good
correspondence with the data available. However, such description does not
satisfy us in any theoretical aspect, and besides, we hope in close future
to obtain experimental data with a more high precision.

Proceeding from these speculative motives and from any beautiful theoretical
idea, we construct the new set of functions $f'_A, \; f'_B, \; f'_C$
and so on, which describes the shape of the surface with more high
precision then previous one, and besides, does not have many of unpleasant
(in theoretical sense) properties of old set. In limit transition to the
more rough approximation $f'_A$ transits into $f_A$, $f'_B$ --- into $f_B$
etc. Naturally, the new functions $f'$ themselves are organized more
complicated then old ones $f$. They describe various more fine detailes
within ``their native domains" (these are predictions of the model). But
outside them they describe some
highly intricate structures, which have not been seen whenever by anybody.
These properties are so unnatural as, perhaps, they cannot be discovered
whenever, but they are to exist obligatory according notions lying on the
basis of the new model.

The situation sketched seems as absurd. It is clear that there are unlimited
number of ways to build generalization of the functions $f$ such that the
structures of a scale less then $d$ would appear at new functions $f'$,
and in transition to former scale $d$ new functions $f'$ would transit
into corresponding old functions $f$. Because we do not have experimental
data describing the structures with scales less then $d$, we cannot choose
from infinite number of possible sets of functions $f'$ namely that one,
which corresponds to real situation. Moreover, there are no any
foundations to hope, that amoung the sets of the functions $f'$, which
we have chosen proceeding from some beautiful theoretical ideas, the
only one, which corresponds to real form of the surface, will be present.
And it is quite obvious that the structures been described by the functions
$f'$ outside the domains, for which they have been constructed, cannot have
any physical meaning.

One is to trace analogy between the situation, which have been described
just now, and situation taking place in theoretical physics in the field of
research beyond the Standard Model, i.e. in the approaches connected
with GUT, SUSY, supergravity, superstrings, and superbranes. From a very start
it become clear that attempts to anticipate structure of the
matter on inaccessible for current experiment level possessing nothing
experimental data, indicating that contemporary description in the
framework of the Standard Model is insufficient, and proceeding only from
desire to eliminate theoretical defects of the Standard Model are perfectly
absurd. This leads to huge number of various models and approaches, in spite
of initial desire to find only model, which would be determined unambiguously
(the question is about absolute theory!). This leads to impossibility
to give preference to one of the models in comparison with all other ones
(there is no critical experimental data!). One is to note that physicists
have invented till now only {\em some} guide ideas, on which their
generalizations of the Standard Model are been constructed. One can invent
yet a lot of other beautiful ideas and, being based on them, construct
even greater number of the models. And besides, the most probable
possibility is that nothing of them will describe any laws of the real world
which are not known now, but will be discovered later after
corresponding development of the experimental technique.  Really the
question is about chance to guess the unique correct answer when there is
infinite number of variants of choice. Probability of success is equal to
$1/\infty$.

It is possible to look at this situation from the historical viewpoint.
Always in the history of physics the new, more general, successive theories
were arisen as a result of analysis of the new experimental data, which
did not keep within the framework of the old theory. Never such precedents
were occur, when the new theory would be constructed exclusively
according speculative reasons, would explain just the same range of known
facts as the old theory, and would prove to be correct at the future
after discovering new experimental data successfully explained by it.

Thus, special relativity was been created to explain a number of facts
contradicting to the Newtonian mechanics. Similarly, quantum mechanics
was been built under force of a great number of new experimental data
in the field of microphysics, which could not be explained in any way
from a viewpoint of the classical physics.

General relativity (see, e. g., Moller (1972)), for the first sight, was
been created proceeding
exclusively from speculative motives. But this is only from the first
sight. Really, the situation with this theory is more delicate. From the one
hand, special relativity did not solve the problem of relativistic
description of the motion in noninertial reference frames and, besides,
did not describe gravity in relativistic manner. So, the same facts,
that did demand to create special relativity, considered in noninertial
frames (one is to note, that all real reference frames are
always noninertial; inertial systems are idealized ones, they appear only
as a result of approximate consideration of real systems), are, in
practice, the cause, although indirect, for generalization of this theory.
From the other hand, deficiency of experimental data inexplicable by
the Newtonian theory of gravity did not permit (and does not permit
till now) to build the theory of gravity unambiguously. At present
general relativity is the most popular relativistic gravity theory, but
it is not unique. At the same time there are tens other models, describing
the same range of phenomena. Tiny number of experimental data at this
field existing now allows to fix only so-called post-Newtonian approximation.
And all models, which reproduce correctly this post-Newtonian approximation
(there exist a large number of such models) are equivalent candidate for
the role of relativistic gravity theory. So, preference given to Einstein's
general relativity is caused rather by historical reasons, but not by
properly physical ones.

The example of general relativity once more displays that deficiency of
the experimental data does not allow to choose unambiguously the
generalizing model. Besides, it quite may occur, that when experimental
data for the next after post-Newtonian approximation will be obtained,
then no one among existing now models will reproduce them in adequate manner.

As we can see, historical facts also testify, that it is not possible to
create generalizing theory starting from nothing, i.e. without any
experimental data not corresponding with previous theoretical notions.


\section{Conclusion.}

The notions on correspondence between the theory and the natural phenomenon
described by it, discussed above, from the one hand, allow us to look with
more condescension at imperfections in formalism of quantum field models
(in particular, of the Standard Model) such as renormalizations,
divergences of perturbation series, presence of the Higgs sector etc.,
and from the other hand, completely deny a possibility of construction
the absolute Theory of Everything. This gives corresponding estimate
of activity of researchers been busy with creation of approaches beyond the
Standard Model. All these models are being built as speculative
generalizations of the Standard Model, which are not founded on any
experimental facts inexplicable in the framework of the Standard Model.
So, there are no any foundations to consider, that one of these models
may be corresponded with the real construction of the world at the
field inaccessible to experiment now. They are some highly refined
mathematical formalisms, that do not correspond to any physical
objects. Because of their specific form these formalisms (contrary
to many other mathematical formalisms) cannot be applied to any other
physical objects, excluding that ones, for which they were created. But
they cannot be applied to latter ones, because there are no any
reasons to hope seriously that, besides now experimentally known
elementary particles, many other particles exist at the nature, which
are predicted by GUT, supersymmetric theories, and, besides, superstrings
and superbranes from the theories of the same names. So, these formalisms are
fully useless. Because of indefiniteness of various parameters of
these models and possibility practically unlimitedly to increase the
masses of hypothetical particles to justify absence of their tracks
at the experimental data, these models effectively cannot be falsified,
since one may always say, that the energies achieved at accelerators
are not enough to bear these particles.

We see, that aspiration to create the absolute theory of everything
has led to huge embezzlement of power of researchers for useless
creation of useless models. This purpose cannot been achieved on
principle.

Nevertheless, it is quite possible to improve existing now theoretical
approaches even on the basis of available experimental data. We mean here
not any narrow special problems, but principal ones. Some of them are
mentioned below.

Quantum field theory at present exists, in fact, only in the perturbative
version. And its formalism is adapted only for calculations in the framework
of perturbation theory. It's unlikely, that one can hope to get the
description of nonperturbative effects without going out of the framework
of the first principles of the theory. In this connection one may think about
revision of this formalism to eliminate the features characteristic only for
perturbative approach.

Other directions of improvement of the theory may be related to the
following circumstance. There are really elementary (according to
modern viewpoint) particles: electron, quarks and others. And there are
constituent particles (which are also called according historical tradition
elementary): hadrons. So, we have here two different levels of the
matter organization. However, the formalism used for their description
is, in fact, just the same. The processes both on the quark and on the
hadron levels are described with a help of similar formalisms of
quantum field theory. Only different fields are used for different
particles. And what's more, the theory describing hadrons formally does
not follow from one describing quarks. Although at the nature properties
of hadrons are completely conditioned by properties of interacting
fundamental particles constituting them.

Probably, ``satisfactory" quantum field model is to describe really
elementary particles in such manner, that one could obtain from it the
information about creation of bound states --- hadrons, and about the
way of description of these more complicated particles. If the formalism
of quantum field theory may be used for them too, then it has to arise
as any approximation from the quantum field formalism for quarks.

Further, interacting nucleons may constitute coupled states --- atomic
nuclei. There is no complete general theory describing nuclear interaction
till now. There are only various special models describing one or another
feature of these processes instead. However, if sufficiently reliable
theory describing interactions  of hadrons existed, one might with its
help both describe the process of creation of nuclei and predict their
properties.

Nevertheless, all this problems have remained aside of the main interests
of theoretical physicists, which are busy with search  of beautiful mirage
--- TOE.



\newpage
\centerline{\Large \bf References.}
\bigskip

%\begin{thebibliography} {99}

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\end{quote}

%\end{thebibliography}


\newpage

\centerline{\Large \bf Footnotes.}
\bigskip

\begin{quote}

$^1$) Remarkable character of that achievement becomes completely clear
if one take into account, that the dimension of the space-time in the
superstring theory is $D=10$, and that one of the problems in this theory is
the problem how to get rid of six unnecessary dimensions (the
compactification problem), which on principle may be solved, but there
exist so many ways of its solution, as one cannot say about any
unambiguous choice.

\end{quote}



\end{document}


