19. Book 6 - Derivation of General Formula for Constants of the Cosmology and Quantum Mechanics - part II

 

23 September 2017 AD, the feast of St Pio and St Linus

Andrew Yanthar-Wasilik

 

We are finding General Formula for Equations from the previous book, "Book 5 Integer Formula for Dimensionless Coupling Constants of Fundamental Forces".

This time, a general term of the constant similar to the fine structure constant, alpha, α_E, let us name it just alpha, α, will be derived.

There may also be other constants we do not know yet, so the general term alpha, α, seems appropriate.

In Part I of Book 6, the Exponent Main, ExpM was derived:

 

ExpM = (A/B)^C

 

✠Now, continuing further derivation from Book 5, we get part D (Equation D is the exponent needed to calculate the value of the function at (x=D), represented by partial sequence:

 

D₁₆ = 16 + ExpM₁₆

D₁₇ = 17 + ExpM₁₇

D₁ + 1 + ExpM₁

 

General formula for exponent D_x is:

 

D_x = [ x + ExpM_x ]

Or just:

✠D_x = [ x + ExpM ]     (Eqn. D)

 

Having exponent D, we may get the value of the transcendental function at point x:

 

✠FT( x = D ) = ( C₀) * ( π/ e )^D    (Eqn. FT)

 

✠The next step is getting the Partial Exponent ExpP. We have three terms of the sequence:

 

ExpP₁₆ = ( 16 + (24 / 24) ) / ExpM₁₆

ExpP₁₇ = ( 17 + (27 / 24) ) / ExpM₁₇

ExpP₁ = ( 1 + (-21 / 24) ) / ExpM₁

...,1,...,16, 17, ... are just x

...,(-21 / 24),...,(24 / 24),(27 / 24),... are equal to exponent

y = ((3)/(24))*x + (-(24)/(24)) = (3*x − 24)/(24) = (x − 8)/(8)

 

of (Eqn. A)

A_x = ( C₀)^{((x − 8)/(8))}     (Eqn. A)

 

Adding these two terms

x + y = x + (x − 8)/(8) = (8x + x − 8)/(8) = (9x − 8)/(8)

This sum has to be divided by Exponent Main, ExpM, to get the value of Exponent Partial, ExpP:

 

✠ExpP = ((9x − 8))/(8ExpM)      (Eqn. EP)

 

✠The last sequence will be the denominator of

 

( α_E )^{( − 1 ⁄ 2)} = ( FT(x) / ( 8 + 2 * (24/24))) ^ExpP     (Eqn. α_E)

 

which is

For constant C ₁₆: ( 8 + 2 * (24/24))

For constant C ₁₇: ( 9 + 2 * (24/27))

For constant C ₁: ( -7 + 2 * (-24/21))

The first part of this sum gives a partial sequence:

...,-7,...,8,9,...

And this is equal to ( x - 8 )

The second part of the sum gives a partial sequence:

...,2 * (-24/21),...,2 * (24/24),2 * (24/27),...

We have already calculated this. It is equal to

E_partB = 2 * ((8)/(x − 8)) = ((16)/(x − 8))

 

Adding both terms gives:

( x - 8 ) + ((16)/(x − 8)) = (x² − 16x + 80)/(x − 8)

 

Taking reciprocal of this term, we can avoid quotient and use the product instead in (Eqn α_E)

✠Finally, the General Formula for ( α )^{( − (1)/(2))} is:

 

✠( α_x )^{( − (1)/(2))} = {[(C₀)(π ⁄ e)^(x + ExpM)]*[(x − 8)/(x² − 16x + 80)]}^{[(9x − 8)/(8ExpM)]}      (Eqn. α)

 

Where: 

ExpM = ( A / B )^C      (Eqn. EM)

 

And parts A, B, and C are:

 

✠A_x = ( C₀)^{((x − 8)/(8))}     (Eqn. A)

✠B_x = {[(C₀)(π ⁄ e)^x][(x − 8)/(x² − 16x + 80)]}^{[(11x − 88)/(24)]}     (Eqn. B)

✠C_x = ( ( C₀) ( π/ e )^x ) * ((x − 8)/(24))      (Eqn. C)

 

To get (α) ^− 1simply square the previous equation

To get ( α), take the reciprocal of the previous equation (for those not much in the math).

Where "x" may be any number: complex, transcendental, real, etc.

 

Getting the shorter formula for ExpM and αis probably possible but very difficult. I am unsure if I can do that; maybe some professional mathematicians can derive it.

Now, this General Equation for any x allows calculating any alpha value.

 

The next book, Book 7, will show the coupling constant of gravity force (two candidates) and switch to Quantum Mechanics Neutrino Mixing Angles and Weinberg Angle for weak interactions.

Quarks will follow later. 

 

Andrew Yanthar-Wasilik

Link for the next article   >>>   109. FORTRAN Source Code - calculation of the Theta Angles i.e. Mixing (Oscillation) Angles of Quark, Neutrino, Boson and Graviton.

 

Return to Syllabus   >>>   132. Syllabus of the course – “Voyage through God’s Universe (Cosmos and beyond) according to St. Hildegard von Bingen and others, with the help of Mathematics, Cosmology and Quantum Mechanics."