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Barykinsky Gennady Mikhailovich Institute of conceptual studies Moscow, Russia E-mail: k@genn.ru
UNIVERSAL LAW OF SCALING
Abstract: in this paper, on the basis of deductive logic and analysis of known mathematical expressions that determine some physical parameters of existing things in Nature, conclusions are obtained, the generalization of which corresponds to the status of the law, which determines the scaling of the fundamental properties of things in Nature in the widest range - from elementary particles to composite matter, including the ratio of the parameters of different Universes.
Keywords: elementary particles, monopole, universe, physical parameters, space, energy, scaling.
By getting into the swing of things, We gain understanding. And by realizing the meaning of things, We create consciousness [1]. there a law according to which the symmetry that determines the world of fundamental properties of elementary particles is scaled and respectively destructed?" It is still unclear under what law the fundamental properties of elementary particles are changed as they move in space from their centre to the periphery and further up to distances much larger than their own sizes. It is also still unclear in what ratio is the existence genesis of a huge variety of objects in the universe, and in particular of the objects that owe their existence to the action of various manifestations of forces of electromagnetic and gravitational origin.
The general analysis of the difficult problems of the theory of the Universe shows that many physical parameters characterizing the objects of a single or composite matter in the Universe are functionally linked. This linkage is a result of the Universal law of scaling, which is based on classical geometric progression: an = a1 gn-1, where: a1 and an - are the first and following terms of the sequence; g - is a ratio of the progression, n - is a quantum number of scaling.
The destruction of simple symmetries, due to which objects of a matter of increased complexity or new forms of interactions are created, occurs in points of a geometric mean, see Figure below:
The next part of this work is a continuation of the work cycle under the title: The relativity of the fundamental properties of the electron [3]. The comment of the authors of the works [4] highlighting relevance of the problem under discussion is of interest in this regard: "There is no doubt that the understanding of the structure of elementary particles will be as important as the discovery of the atomic and nucleus structure".
The vector defining the seek search directions and development of ideas related to the solution of such complex problems is represented by the outstanding physicist and philoso-
Current concepts of elementary particle physics and the universe are based on the absence of adequately acceptable models. For this reason, the theoretical description and further experimental development of physics in these directions has slowed down. Further attempts to develop particle physics at accelerators the continuously increasing capacity are becoming more futile. The logic of this futile is a simple aphorism: "You can't make an omelette without breaking eggs" (Literal translation: If you hew trees the chips must fly), and if the power output of the hewing process increases to "madness", then the whole forest turns into chips. The situation has reached a point where many hundreds of supposedly short-lived useless particles had been already registered as "chips". It should not be forgotten, however, that the cost of such experiments is commensurate with the budgets of many States participating in these projects. Therefore, the methods of non-accelerator and theoretical physics are becoming more and more important [2].
Desired models are those that could combine, for example, the properties of the internal structure of elementary particles with the properties of experimentally observed interactions of these particles with their environment. There is a need for more realistic models of the Universe, since the current standard models of the Universe is built on the "Big Bang" theory and is burdened with problems leading the theory of the Universe to a dead end. The futility of this theory has stand out a mile for a long time, because this theory does not correspond to one of the fundamental principle of the existence of everything - objectives definition; moreover, the idea of a singularity itself is absurd.
As one of the many approaches addressed to the solving of issues discussed above, it would be advisable to consider: "Is
pher Niels Bohr, who has repeatedly noted that it is important to create theories as insane as possible, but at the same time
he believed that these theories ought to lend themselves to simple the description [5].
Figure 1.
At this point, expressions for a number of spatial characteristics of elementary particles are well known, for example, for electrons such as: classical electron radius r, the Compton wavelength Xk, Radius of Bohr's Orbit rb, de Broglie wavelength mean square displacement of electron rv, determined by the impact on the electron of virtual vacuum particles, and some others. Separate links between these characteristics are also known. Despite the fact that the essence of these and some other parameters needs its clarification, it would be desirable that there was a mathematical expression, shedding light on the essence and meaning of the hierarchy of stepwise complexity of the existing forms of elementary matter.
The analysis of above mentioned and some other mathematical expressions describing the different spatial characteristics of the rn electron leads to a generalization that corresponds to the status of the law of scaling [6]:
Table
rn = hft2n, (1)
where: rh and Xk - are Compton radius and wavelength: Xk = 2nrk, ft - is a common ratio of geometric progression, which is determined by the equations: ft2a = 1 or eft = q, a -is a fine-structure constant, q - is an internal charge of the electrons, e - is an outer electric charge of electron, n - is a quantum number of scaling, which takes an integer or a halfinteger value.
1. Association of Electron Properties with the Universal law of scaling.
The following table No.1, based on the law defined by the formula (l), presents the expressions and corresponding values of the electron's spatial characteristics. Full compliance of the obtained data with the already known table proves that the proposed expression (l) can be used to describe the spatial scaling law of the fundamental properties of the electron. 1.
n Parameter (cm) Name of the parameter
-3 r = r, = e6/m c4h2 = e2/m c2 m k> e p 1.5-1017 Classical monopole radius
-2 r = r,ft~4 = e4/m c3h = h/m c p kr ' e p 2.06-10-15 Compton radius of the monopole
-1 r = r,8~2 = e2/m c2 e k> e 2.82-10-13 Classical electron radius
0 rk = = h/mc 3.86-10-11 Compton radius of the electron
1 rb = rkP2 = h2/me2 5.29-10-9 Min atomic size, Bohr radius
2 r = r.B4 = h3c/m e4 a k< e 7.25-10-7 Max atomic size
3 r = r,B6 = h4c2/me6 M k< e 9.94-10-5 Size of biomolecules
The table No. 1 shows that each point corresponding to the current discrete spatial parameter of the electron is the geometric mean of the two neighbouring:
2
r2 = rr
p em
2
r2 = r r = r r
e p k m b
2
r 2 = rr = r r = r r
k e b pa Mm
Moreover, these are the points where a certain part of the previous symmetries is destroyed.
The spatial scaling of the electromagnetic forces, which is based on the Coulomb's law F^ ~ 1/r2, is responsible for the decay of the seemingly simple Coulomb force into residual and more complex forces that determine the interatomic and
2
2
r, = r.r = r r
b k a Me
intermolecular interaction. For example, such residual forces as: dipole, quadrupole, ionic, metal, hydrogen, valence, co-valent, Van der Waal's, polarization, induction, dispersion, orientation and some others.
It should be noted that the electric force itself, as determined by the Coulomb's law, is also residual, since the outer electric charge of the electron e is ^ times less than the internal q, i.e. q = efí. For this reason, it is the internal charge, which is the primary source of all the others, determines the interaction of the electron with the external vacuum environment.
As a result of scaling of the fundamental properties of the electron there are functional spatial ranges with high specificity, between which, starting from the centre, the gradual complication of the created objects of matter occurs when moving, i. e. the formation of forms from simple to more complex, which is shown in table No. 2. It is obvious that not only the electron, but also more important particles such as proton and neutron are responsible for the formation of many forms, as they are also structurally made up of electric charges, and therefore have their own specific scaling ranges.
Table 2.
Discrete range Name of specifity
r_1 - r_2 = 2.82.10-13-2.06-10-15 Effect of monopole. K-capture. Muon.
r0 - r-1 = 3.86-10-11-2.82-10-13 Inner energy effect - mc2.
r1 - r0 = 5.29-10-9-3.86-10-11 Effect of the charge q, force field generation
r2 - r1 = 7.25-10-7-5.29-10-9 Range of electron shells of atoms.
r3 - r2 = 9.94-10-5-7.25-10-7 Water molecule, DNA, viruses.
r4 - r3 = 1.36-10-2-9.94-10-5 Bacteria, erythrocytes, cell nucleus.
r - r = 1.86-10_1-1.36-10"2 5 4 Skin cells, a large bacterium, amoeba.
r, - r =2.55-101-1.86-10-1 Ant, chicken egg.
r7 - r, =3.49-103-2.55-101 7 6 Human
Let's write down the expression for the force acting between the internal electron charges for n = 0 as follows: Fq = q2/r02. This expression is completely symmetrical in relation to the electric e and magnetic ^ of the electron charges. Write down the expression F for n = 1 and -1:
= q2/r^, for q2/e2 = e2, * f4a = e2^2. (2)
, = q2e2/-^ for q2e2 = ^ * ^ = F2/r-i2 (3)
It follows from (2) and (3) that the symmetry of the electron internal charge q has fallen into classical charges: electric e and magnetic y. Excluding from (2) and (3) в and considering that q2 = hc, we get:
ey = hc, (4)
In 1931, P.A.M. Dirac, who proposed the idea of the existence of magnetic monopoles, derived an equation for quantization of the electric charge as follows [7]:
ey = (n/2)hc, (5)
In the calculations (5) was used the condition that the electron orbital magnetic moment is quantized. However, in our case, n = 1, as for (4) in (2) and (3) was considered only the basic state, and the deuce should be absent, because in calculations of the electron magnetic moment (as part of a more modern electron model) the full area of current sheet is covered completely when turning not on 2n, two times less. Therefore, the formulas (4) and (5) are the same.
If the right side of equation (4) is multiplied and divided by в2, and considering that:
e2 = hc/p2, (6)
y2 = hcp2, (7)
the following system of equations is obtained:
y = (n/2)ep2, (8)
e = (n/2)y/p2. (9)
Solving the system of equations (8,9) referred to ^ or e, we get: n = ± 2, which proves the coincidence of formulas (4) and (5).
Solving the system of equations (6, 7) referred to p, and considering that q2 = hc, we get:
q2 = ey. (10)
The expression (10) shows that the internal charge of the electron q, is the geometric mean between the external electric charge of the electron e and its internal magnetic charge y.
With expressions and their corresponding values (see the table No. 1), characterizing the classical radius of electron and monopole, it becomes possible to calculate the mass of monopole. In order to do this we shall write down the ratio of the two radii in the following form and we get:
m = (r /r )m = 1.88-104m . (11)
y y e y' e e
my value in (11) is 4 times higher than the value of the monopole mass obtained in [2]. This non-coincidence is due to the fact that the calculations in [2, p.354] are based on the equation (5), while the present calculations are based on a more correct equation (4). Proof of this conclusion is the fact that m is also subject to the scaling law, i.e.:
m= fi4me = 1.88-104me, as according to the law (l) re = rjS4.
Let's write down (l) for n = -1/2:
r-1/2 = rv = rjp, (12)
The analysis shows that the expression (12) up to the weakly varying logarithm coincides with the known expression for the mean square displacement of rv electron in its interaction with the field of vacuum virtual photons [8], the author of this work has considered this phenomenon by the methods of the classical theory. It should be noted that this phenomenon in the scientific literature is more known as "Lamb shift". It should be added that rv obeying law (1) is connected with other important spatial characteristics of electron by the geometric mean ratio:
2
r2 = rr.
v k e
Association of properties of the universe with Universal law of scaling.
Examples of the application of the law of Universal scaling in different spheres of existence of matter in the Universe are represented in the following § § 5,6,7,8. Obviously that this list is not exhaustive, and in fact there are countless examples of this.
Up-to-date information about the distribution of matter in the Universe, that become standard, shows that the major global types of matter are approximately within the following percent range, the values ofwhich are specified in table No. 3, but for quite a long time, the results of the study of these ranges are increasingly converging to their geometric mean ratio, and therefore increasingly satisfy the law of Universal scaling:
(13) Table :
Type of matter in the universe % Average
M 0 ordinary matter 4-6 M 2 = mm, dm o de
dark matter 18-24
Mde dark matter 68-75
The following table No.4 shows the geometric mean ratio of the fundamental forces in the universe at the Planck's extreme values:
Table 4.
Fundamental forces at the Planck point Average
F g Gravity: Fg = m2/r2 ii F «F
F c Electromagnetic: Fe = q2/r2
F e Elastic: F = kr, where: k = mu2 e 1
DNA and blood are the most important components of living matter in the universe. Tables No. 5 and 6 indicate that the main constituents of these components are also connected by the geometric mean ratio. The value ranges of the
Table
constituents shown in table No. 5, as well as in table No. 3, are increasingly converging to their geometric mean ratio as the relevant studies develop.
Genetic variation in DNA % Average
G cp genes encoding proteins 2-5 G 2 = G G rs cp ep
G rs repetitive gene sequences 20-30
G ep genes that do not encode proteins 70-80
Table 6.
Blood corpuscles pcs/mm3 СреAнее
k A leukocytes 8-103
k m platelets 3-105 km KK
k 3 erythrocytes 5-106
Tables No. 7 and 8 indicate the values and their compliance to the law (l) the most important parameters of living and inanimate matter in the Universe. The tables show that the most important members of the matter in the Universe:
living-human, and inanimate - electron, hold their geometric mean positions in the Universe respectively, which is fully consistent with the law of Universal scaling.
Table 7.
Masses of objects and the universe g Average
m mu min mass of object in the Universe 10-48
m p human mass 104 2 m = m m p mu u
m u mass of the Universe 1056
Since the human mass accurately holds his average geometric position in the Universe, it is fair to conclude that the estimation of the mass of the Universe as a whole as mu = 1056g is realistic. If this is so, it is reasonable and logical to assume that the human size must also correspond to his geometric mean position in the Universe. However, recent developments
in astrophysics show that the size of the Universe is equal to ru = 1027cm, which is 1010 times less than the number specified in the table No. 8. This contradiction does not conform to the law of Universal scaling, which requires that the size of the universe correspond to the value specified in table No. 8, which means that the real size of the corresponds to ru = 1037cm.
Table 8.
Sizes of the objects and the universe cm Average
r mo min size of object in the Universe 10-33 2 r = r r p mo u
r p human size 102
r u size of the Universe 1037
Let us imagine that, according to the existing embedded universe theory, there are Universes that differ in size as they are embedding: the upper - U , "our" current - U0 and the lower- U . Assuming that the ratio of Universes sizes corre-
sponds to the law of Universal scaling, it is logical to assume that the ratio of these sizes also satisfies the geometric mean, as shown in table No. 9, where common ratio of the progression is chosen equal to 1021.
Table 9.
Sizes of the embedded universes cm Average
U: Size of the upper Universe 1058 U2 = U U 0 -1+1
U0 Size of the current Universe 1037
U+t Size of the lower Universe 1016
Thus, it should be considered to be proven that the fun- conclusively its universality. This universality is based on the
damental fact - the law of Universal scaling exists. 18 funda- principle of objectives definition, without which the Universe
mental examples in a wide range and the correspondence of cannot exist. the law to reality, which are represented in this paper, prove
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