# Book 3 - Getting the exact value of fine structure constant, *α*

10 May 2017 AD; Feast of St John of Avila - 16 May 2017 AD; Feast of St Simon Stock and St Brendan the Navigator

There are three formulas for getting fine structure constant -

- first in agreement with approximation from Harvard University from 2008^{(1)}

- second with agreement with approximation from Kobayashi Institue (Nagoya University) from 2012^{(2)}

- third with agreement with approximation from 2014 CODATA

The formulas are slightly different, time will tell which one is better, once the work on this blog is completed.

Let us start with second case, as it gives the exact theoretical value of fine structure constant, *α*.

The calculation is done in couple of steps as follows (using FORTRAN):

**1** - main exponent (expm) is calculated so the value of the transcendental function at that point may be obtained.

**expm = ( C_{0} ⁄ ((C_{16} ⁄ 10)^{(11 ⁄ 3)}))^{(}^{C}_{16}^{ ⁄ 3) }(Eqn. 1)**

As you can see, two transcendental constants *C*_{0} and C_{16} are used and some integers: 10, 11, 3.

Constants C_{0} and C_{16} are the borders of our Universe, I mean they represent forces and elements at these spaces.

If someone wants to calculate this result, here are the numerical values of two constants ruling electromagnetic interactions

*C*_{0 = }**0.986 976 350 384 356 956**

and

*C*_{16} = 9.999 838 797 804 880 93

Plugging the numbers into Eqn. 1 gives result:

**expm = 0.957 432 928 678 624 41**

**2** - Now the value of the Transcendental Constant at point x = (16.0 + expm) may be calculated as follows:

From “Book1 – Transcendental Function - Introduction” we use Eqn. 11

**FT(x) = (C _{0})*(π ⁄ e)^{x } (Eqn. 11)**

**FT(16.0 + 0.957 432 928 678 624 41) = (0.986 976 350 384 356 956)*(1.155 727 349 790 9217) ^{(16.95743292867862441)}= 11.486 106 001 091 650**

Notice that fine structure constant, *α *is almost at integer = 17, but not quite, suggesting that the fine structure constant may be variable.

**3** - The third step is to calculate second exponent - “partial exponent one”, i.e. exppo.

For reasons explained in the next Book this value equals to:

**exppo = 17 / expm = 17.0 / 0.957 432 928 678 624 41 = 17.755 812 956 487 822**

**4** - Now we can calculate the exact value of fine structure constant, *α *as follows:

*α*^{( − 1 ⁄ 2) = }**( FT(x) ⁄ 10.0)^{exppo }(Eqn. 2)**

where FT(x) = value of Transcendental Function at the point x = (16.0 + 0.957 432 928 678 624 41), the result we just calculated a moment before.

Again plugging the numbers into Eqn. 2 we get:

**(11.486106001091650 ⁄ 10.0) ^{17.755812956487822}= 11.706 237 618 540 260 = α^{( − 1 ⁄ 2)} **

So, the above result equals to *α*^{ − }^{1 ⁄ 2}

when squared, it is equal to

*α*^{( − 1)} = 137.035 999 181 727 13 (Res. 1)

For comparison, the value calculated by Japanese team from Nagoya University from 2012^{(2) }obtained experimental result as follows:

*α*^{( − 1)} = 137.035 999 174 with precision (35) i.e. [0.25 ppb]

**My result is theoretical, and, I claim, exact.**

Relative Error, *ε*, in comparison to experimental result is:

*ε* = -5.638 749 727 × 10^{ − }^{11}

This value of fine structure constant, *α*, gives much better result, than another formulas, which agree with the results from Harvard University from 2008^{(1)}

and from 2014 CODATA for official value of fine structure constant for 2014.

The formulas are similar to the one in Eqn. 1, except the main exponent, which is slightly different. As I said before, once this blog is completed, it will be clear which result is better.

Anyway, for case three - 2014 CODATA, the modified Eqn. 1 is:

**expm = ( C_{0} ⁄ ((C_{16} ⁄ 10)^{(11 ⁄ 3)}))^{(10 ⁄ 3) }(Eqn. 3)**

This equation gives result (using Microsoft Mathematics high precision calculator):

*α*^{( − 1)} = 137.035 999 123 487 2 (Res. 2a)

and from TI-nspire CX CAS calculator:

*α*^{( − 1)} = 137.035 999 124 (Res. 2b)

and experimental result is:

*α*^{( − 1)} = 137.035 999 138 154 5

with relative error:

*ε* = 1.070 324 592 971 6 × 10^{ − }^{10}

The variation of the 2nd case is calculation in FORTRAN, with has slightly different inner exponent:

**expm = ( C_{0} ⁄ ((C_{16} ⁄ 10)^{((1+}**

^{C16) ⁄ 3)}))^{(}^{C}_{16}^{ ⁄ 3) }(Eqn. 4)

giving result:

*α*^{( − 1)} = 137.035 999 181 500 63 (Res. 3)

with relative error with respect to the experimental result from Nagoya University:

*ε* = -5.473 462 234 × 10^{ − }^{11}

It is obvious, that either Equation 1, Equation 3 or Equation 4 gives the best results, and Japanese team got the closest score

to the theoretical value of fine structure constant, *α.*

*However, the result from Harvard University is also quite good.*

*There is a fourth way of calculating fine structure constant, α, in agreement with Harvard University experiment.*

*The main exponent is again different and not as simple as in previous cases:*

**expm = ( C_{0} ⁄ ((C_{16} ⁄ 10)^{(11 ⁄ 3)}))^{(100 ⁄(}**

^{C16 * 3)) }(Eqn. 5)

with the result

*α*^{( − 1)} = 137.035 999 068 341 5 (Res. 4)

and experimental result is:

*α*^{( − 1)} = 137.035 999 084 [0.37ppb]

with relative error bigger than in the first case:

*ε* = 2.881 520 203 738 × 10^{ − }^{10}

Future articles will address this issue of main exponent, and final equation will be derived, hopefully.

It is quite possible, that numerical value of fine structure constant, α, may be depending on method of measurement(?)

and all values are correct(?), or maybe there is only one value of α, we'll see.

**Right now it seems that Nagoya University result and 2014 CODATA official numerical value of fine structure constant are the closest approximations to theoretical values (given by Eqn. 1 and Eqn. 3).The main exponents are different,****either (C _{16} / 3.0) or (10.0 / 3.0) respectively, and these issues will be sorted out later**.

Since the speed of light may not be constant after all, the experimental results introduce some slight error.

If the more precise result (Res. 1) is taken into consideration, and relative error is calculated from speed of light

(c = 299 792 458 m / s - just to get an idea how small is the difference) and then equals to:

** ε = 0.0169045464181731 [m/s] = 1.690 [cm/s]**, (centimeters not meters!) which it is really minuscule.

Speed of light not so constant after all

Wikipedia - fine structure constant

^{(1)}New Measurement of the Electron Magnetic Moment and the Fine Structure Constant

^{(2)}Tenth-Order QED Contribution to the Electron

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