© N. V. Kosinov

Contact: kosinov@unitron.com.ua

The number π together with the fine-structure constant α would give the opportunity to receive the important dimensionless fundamental constants, which it was not possible to receive in any other way. It is shown, that the constants π and α would be constants of the same class. Within the framework of this class of constants the physical equivalent of number π is found, and the number α₂=7,49648184638205•10^-3 or its inverse value α₂^-1 =133,395907639344 acts like it. This new constant has the status of the fine-structure constant-2. A geometrical equivalent of fine-structure constant α is the number π₂= 1,83360822(13) which together with π = 3,14159265358979… enables to receive simple numerical multipliers before the Dirac's large numbers.

The fine-structure constant was entered into physics by Sommerfeld in 1916 at creation of the theory of the fine-structure of energy of the atom. Originally the fine-structure constant (α) was determined as the ratio of the speed of the electron in lowest Bohr's orbit to the speed of light. With the development of the quantum theory it became clear, that such simplified representation does not explain its real sense. Till now the nature of the origin of this constant is not discovered. Except for the fine structure of energy of the atom this constant is shown in the following combination of fundamental physical constants: α = μ₀ce²/2h. Concerning the fact that the constant (α) appears in the ratio, connecting the Planck's constant, the charge and speed of light Dirac wrote [1]: "It is not known why this expression has just this value instead of any other. In this connection physics put forward various ideas however there is no standard explanation so far". The similar statement about number (α) belongs to Freyman [2]: "Since it was discovered... It was a riddle. This number put in impasse and by that caused anxiety to all the most experienced phisicals-theorists. Directly you would like to know, from where the connection of this constant appeared: whether it is connected to number π or it can be connected to the natural logarithms? Nobody knows". Concerning the value of the fine-structure constant the authors of the Berkeley's course of physics write [3]: "We have no theory, which would predict the value of this constant".

The problem of the fine-structure constant is one of 10 major problems of physics, which are called as “problems of the Millenium” [4,5]. Among these problems it is formulated in this way: whether all dimensionless parameters, which characterize the universe, can be calculated in principle or they are the consequence of quantum-mechanical accident and are impossible to be colculated? ”.

There is a question so far: is there the connection between this constant and the major geometrical constants, for example with the number π? The number π, known since the antique mathematics, belongs to the fundamental mathematical constants and has unique features. In spite of the fact that π is a mathematical number, it is included into many formulas of physics. It is conformable with the scientists’ statements about connection of physics and geometry. Here it is appropriate to refer to A.Poincare’s opinion that physics and geometry complement one another. According to Poincare, by experience we always notice a certain "sum" of physics and geometry [10]. The similar "sum" of physics and geometry is shown on the pattern of the primary constant basis of fundamental constants (tab.1) as the independent groups of physical and geometrical constants [7]. In this five-constant ontological basis there are three superconstants having dimensions, and two - dimensionless. Five primary superconstants are enough to receive a set of other constants on their basis by calculating [6,7,9].

Tab.1.

Except for the fine-structure constant α in physics there are also other dimensionless constants. Frequently occuring in physical equations, the large numbers about 10³⁹ -10⁴⁴ belong to the important dimensionless constants. Considering the coincidence of the large numbers as not casual, P.Dirac has formulated the following hypothesis of the large numbers [15]: “As a general principle it is possible to accept, that the increasing numbers about 10³⁹,10⁷⁸ etc., occurring in the general physical theory, calculated to within simple numerical multipliers, equal t, t² etc., where t is time in modern period, expressed in nuclear units. The mentioned simple numerical multipliers should be determined theoretically, when the complete theory of cosmology and atomism will be created.” In theoretical physics this mysterious problem of the concurrence of large numbers is still not solved. It was not possible to create “the complete theory of cosmology and atomism”, what P.Dirac hoped for [15]. It was not possible to deduce the large numbers mathematically, as P.DSavies wanted it [16]. So far the scientists have not cleared up the real values, mentioned by P. Dirac, “simple number multipliers” before large numbers.

In [6,11] there formulated a principle, according to which all dimensionless constants come from numbers π and α. For example, the combination of these constants gives the large number D_o [11]:

. Proceeding from symmetry of the formula, we shall designate . From here we shall receive: The value opposite to it has the following value: α₂^-1 = 133,395907639344... A new constant α₂ is a physical equivalent of the number π. This constant would occur in ratios together with the fine-structure constant. Let's call this physical constant as the second fine-structure constant (Fine-structure constant - 2). Using the new physical constant α₂ ontological basis of fundamental physical constants will look like (tab. 2):

Using the fine-structure constant α, we shall receive a new constant, which is its geometrical equivalent: . The number π₂ is a geometrical characteristic of the curved space [8]. This number is similar to number π because of the geometrical characteristic of the flat Euclidean of space. The values of equivalents of the constants π and α are given in the table 3.

Probably there are converging numbers for π₂ similar to numbers bringing to number π. The discovery of such numbers would allow to receive the exact value π₂.

The ratios for the numbers π and α and their physical and geometrical equivalents α₂ and π₂ are completely symmetric:

, , , .

The constants π and π₂ characterize the flat pseudoeuclidean space. Thus, π is the geometrical characteristic of the flat space, and α₂ is its physical characteristic. The constants π₂ and α₂ also characterize the curved space. Thus, π₂ is the geometrical characteristic of the curved space, and α is its physical characteristic. For these four constants the following ratio is fair:

The two of new constants α₂ and π₂ together with numbers α and π give new opportunities for the physical theory. Some examples are given below.

1. The ratio α/α₂ gives the constant of structurogenesis [13]: k_s=0,9734369645 (30). In the fractal law of structurogenesis this constant of the proton allows to receive the value of the weight of the proton with the help of the calculation.

2. The constants α and α₂ give the opportunity to receive by theoretical calculation the constant m_p/m_e, the value of which had no theoretical substantiation till now. The formula for the calculation of the value m_p/m_e follows from the fractal of the proton [14]:

The constant of structurogenesis k_s sets the value of the defect of weights for particles participating in structurogenesis of substance [13].

3. The product (π π₂)^-1/2 gives the value of the simple multiplier before the large number, the necessity to search it was specified by P.Dirac. [15]:

.

4. The product α^-1 on α₂^-1 in the tenth degree gives the large number D₀ [15]:

5. The combination of constants π, α, π₂, α₂ as the products (π α₂^-20) and (π₂ α^-20) gives the value of the large number D_u:

,

.

6. The products of four constants π, α₂^-20, π₂, α^-20, give the value of the large number D_u²:

.

7. The constants π and π₂ together with constants m_p , m_e, D_u, allow to receive with the large accuracy the value of the large number D, to which P.Dirac paid attention for the first time [15]:

.

Thus, it was possible to receive a large cosmological number, what P.Devies specified [16], including also "simple numerical multipliers", what P.Dirac specified [15].

In the table 4 the formulas for the calculation of values of large numbers and their values are given.

Below there are given values of combinations of constants which occured in formulas of structurogenesis of substance, in formulas of large numbers, and also in factors before the large numbers. The values of constants, which are caused by numbers π and π₂, are given in the table 5.

Values of constants, which are caused by numbers α и α₂, are given in the table 6.

1. For number π the physical equivalent α₂ =7,49648184638205•10^-3 is discovered.

2. For the fine-structure constant the geometrical equivalent π₂=1,83360822(13) is discovered.

3. In the family of physical constants the second fine-structure constant α₂ is a characteristic of the flat pseudoeucledian space.

4. In the family of mathematical constants the number π₂ is the geometrical characteristic of the curved space.

5. The group of dimensionless constants α, π, α₂, π₂ gives new opportunities in the physical theory. In particular, there appeared opportunities to receive the constant m_p/m_e through theoretical calculation, to receive simple numerical multipliers before Dirac's large numbers, to receive the exact values of the large numbers.

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