The Fine Structure Constant and Discrete Calculus

The Fine Structure Constant and Discrete Calculus – GKO2018 To initiate this discussion, we kindly refer to our paper: The Elliptic Matrix and Discrete Calculus1) which among other things, introduces a shorthand for the Binomial Coefficient: k n n!  k1!  n  (n  1)  (n  2)⋯ (n  k  1)  n  1  n k 1 nk      k  (n  k )!  k !

Our aim here is to solve a special 1st order inhomogeneous recursive equation with two independent variables (, z) and two indices (n, m) where the former is our primary recursive index while the latter is a class-label to fully employ the discrete calculus at hand:

n

f ( , z ) 1  m n

z

m

n m1

 f nm1 ( , z )

;  ℝ ; z ℂ

Observe a first order recursion equation with a constant inhomogeneity we set to unity without loss of generality, as the function fn can absorb any non-unity constant. With a simple multiplicative transform to a new function gn with auxiliary property g0 = f0, the coefficient to fn+1 can be made unity which delivers our solution as a simple sum:

f ( , z )  m n

g ( , z )  m n

zn



m 2

n m1

n z

 g ( , z ) m n

f ( , z ) 



m n 1

m1

n m 2

 g ( , z )  m n 1

z n1

 g nm1 ( , z )

m1

 n1  k

g ( , z )   m 0

m 2

k 0

z

k

m1

 f 0m ( , z )

m 2

For the trivial case m = 0, the function on the right surfaced about 13 years ago in the works of Hans de Vries2) on the Fine Structure Constant  = 1/137.035999139(31) with present relative standard uncertainty of 0.23 ppb3). Furthermore, he asserted that the said sum for m = 0 is equal to a scaled Gaussian Kernel as follows:

k  f 00 ( , 2 )    k2 k 0 (2 ) 

1

 k 0

2

k



2

k ( k 1)



k ( k 1)

  e 4

To put his assertion to a numerical test, we solve this equation for Alpha () using a calculation engine with 100 decimal digits which gives the following value for the Fine Structure Constant to 27 significant digits: 1



137.035999095829700489647400   ;   1024

If Hans de Vries’s assertion holds, continued measurements of the Fine Structure Constant should replicate all our 27 decimal digits above, so time will tell. Still we have to wait a long time for all 27 digits to manifest, as we expect the precision of the Fine Structure Constant to increase by 1-2 digits every 10 years. However, for us impatient, we know that a Jacobi’s Theta Function4) solves the special case  = 1: 

f 00 (1, 2 )   (2 ) k 0



 k 2



 1  12  8 2  2 0,

1 2





 1  8 2  

1 2

 k  12 

2

j 0

So we go right ahead and try to evaluate the sum analytically - which will be the ultimate judge of the truth. 1) 2) 3) 4)

https://www.researchgate.net/publication/260480783_The_Elliptic_Matrix_and_Discrete_Calculus_-_GKO-2014 http://www.physics-quest.org/fine_structure_constant.pdf https://pml.nist.gov/cgi-bin/cuu/Value?alphinv|search_for=Fine+Structure+Constant http://mathworld.wolfram.com/JacobiThetaFunctions.html

Guðlaugur Kristinn Óttarsson – Academy of Industry & Arts – 11.07.2018 – [email protected]