Calculating Dimensionless Physical Constants

by Dr. Bruce D. McLaughlin

An algorithm for computing dimensionless physical constants is illustrated by calculating the Fine Structure Constant. The calculation is based on algebra and does not employ the principles of physics.

Introduction

A dimensionless physical constant is an essential numerical characteristic of space, time, matter and energy expressed as a pure number. Such constants play a central role in physics because they are intrinsically invariant under changes in units of measurement. Historically, these constants were foundational because they could not be derived from fundamental principles. However, in the 1920’s, Arthur Eddington pursued the idea that dimensionless physical constants might be obtainable from pure mathematics. The workings of the universe might emerge solely from abstract, unit-free mathematics without need of empirical input, beyond establishing a starting point for calculations. Eddington’s efforts were generally not successful. This note is intended to revive his ideas and take them to the next level. The Fine Structure constant, denoted by $\alpha$, is employed for illustration. If the proposed conjecture is directionally correct, then Arthur Eddington will have been vindicated on this matter.

Analysis

The fundamental conjecture is that at least one infinite power series exists in the variable $\omega$ (dimensionless physical constant), where$1>\omega >0$, of the form

$\mathrm{\Delta }$($\omega$) = ${\sum }_{n=1}^{\infty }{F}_n$($\tau ,\phi ,\omega )$ = 0 (1)

This power series converges to zero and satisfies the Ratio Test for convergence. $\tau$and $\phi$ represent two of the four numbers $\pi$, 22-6$\sqrt{13}$, 2$\sqrt{3}$ and 22+6$\sqrt{13}$. The selection of these particular numbers will be addressed later but other pairs of numbers may be equally satisfactory.

The first term in the series is $-{\omega }^{-1}$. The second term is ${\tau }^x{\phi }^y$ where x and y are selected to make $-{\omega }^{-1}$ + ${\tau }^x{\phi }^y$ as close to zero as possible. The third term is a simple function of $\tau$ and $\phi$ multiplied by $\omega$ to make $-{\omega }^{-1}$ + ${\tau }^x{\phi }^y$ + $f(\tau ,\phi )\omega$ as small as possible. Each remaining term is of the form ${\tau }^{\theta }{\phi }^{\vartheta }{\omega }^{\mu }$ in which$\mu$ is a positive integer increasing by 1 with each successive term. If $\tau >1$then $\theta$ is a positive integer; if $\tau <1$then $\theta$ is a negative integer. If $\phi >1$then $\vartheta$ is a positive integer; if $\phi <1$then $\vartheta$ is a negative integer. By choice, $\tau >\phi$.

A value of $\omega$ to any known level of precision can be used as a starting point. For each term after the third, ${\omega }^{\mu }$ is viewed as a factor in ${\tau }^{\theta }{\phi }^{\vartheta }{\omega }^{\mu }$. After calculating ${\omega }^{\mu },\left|\theta \right|$ is increased until ${\tau }^{\theta +1}{\omega }^{\mu }(or{\tau }^{\theta -1}{\omega }^{\mu }$ if $\theta$ is negative) is barely greater than the magnitude of the previous sum. In other words, ${\tau }^{\theta }{\omega }^{\mu }isbarelylessthanthemagnitudeoftheprevioussum.$ Next, $\left|\vartheta \right|$ is increased until ${\tau }^{\theta }{\phi }^{\vartheta }{\omega }^{\mu }$ is barely less than the magnitude of the previous sum. In other words, ${\tau }^{\theta }{\phi }^{\vartheta +1}{\omega }^{\mu }$ is barely greater than the magnitude of the previous sum. Finally, a sign is affixed to the term to ensure that the magnitude of the next sum is smaller than the magnitude of its predecessor. This rule is unambiguous and produces a unique term for each index which makes the series mathematically legitimate. This rule also forces the series to satisfy the Ratio Test and converge to zero.

This framework, for calculating dimensionless physical constants, represents a self-consistent, quasi fixed-point model where a particular constant is defined implicitly by a series whose terms depend on the constant itself.

Fine Structure Constant

For the Fine Structure constant, Equation (1) can be written as:

$\mathrm{\Delta }$($\alpha$) = ${\sum }_{n=1}^{\infty }{F}_n$($b,\pi ,\alpha )$ = 0 (2)

which converges to zero and satisfies the ratio test for convergence. The variables b = 22+6$\sqrt{13}$ and $\pi$ = 3.141592...are independent parameters and $\pi b$ = 137.078… which is nearly equal to ${\alpha }^{-1}$= 137.035… The series in Equation (3), used to illustrate the conjecture, was constructed using a rough estimate for the Fine Structure Constant of $\alpha =0.007297$.

$\mathrm{\Delta }\left(\alpha \right)$ = ${-\alpha }^{-1}$ + $b\pi$ – $(2\pi -3/2)\alpha$ - ${\pi }^2{\alpha }^2$ + b${\alpha }^3$ - $b^2{\pi }^2{\alpha }^4$+ $b^2{\pi }^4{\alpha }^5$ ... (3)

Each remaining term in the series is of the form $b^{\theta }{\pi }^{\vartheta }{\alpha }^{\mu }$ and each successive term increases the value of $\mu$ by 1. The last term in the series is $b^{28}{\pi }^2{\alpha }^{35}$. Equation (2) does not apply directly because the series in (3) is not infinite. Instead, the equation

$\mathrm{\Delta }$($\alpha$) = ${\sum }_{n=1}^{37}{F}_n$($b,\pi ,\alpha )$ = $R_1$ (4)

was used where $R_1$ can be viewed as a residue. When this truncated series is evaluated, using the rough estimate $\alpha =0.007297$, the residue, $R_1$, is equal to -2.732$\times {10}^{-29}$. Now increase the precision of $\alpha$ by one digit to 0.0072973; the residue $R_2$ becomes 0.0056324888.... Next increase the precision of $\alpha$ by one more digit to 0.00729735; the residue $R_3$ becomes 0.0065711919.... This computation can be continued up to a precision level of about $\alpha$ = 0.007297352564311 which produces a residue of $R_{10}$= 0.0066193341.... This is near the current precision level for $\alpha .$

By the fundamental conjecture, the sequence $R_1$, $R_2$, $R_3$, $R_3$, $R_4$, ... has an upper limit, R. That limit was computed by Mathematica to be:

R = $0.00661934636627880927661760068759$

This corresponds to a Fine Structure Constant value of:

$$0.00729735256496044196828721304695258401433871295628262761655433$$

which compares favorably with the February 2023 value, by Fan et.al. of:

0.0072973525649(8)

An infinite series built on b, $\pi$ , and $\alpha$ may allow the Fine Structure Constant to be computed with arbitrary precision by algebraic manipulation. This might address some of the concerns of Feynman. If dimensionless physical constants have their origin in relatively simple infinite series and not in the deep recesses of quantum mechanics, string theory and relativity, then all mysteries emanate from mathematics.

Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans. You might say the "hand of God" wrote that number, and "we don't know how He pushed His pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out – without putting it in secretly! Feynman, R. P. (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 978-0-691-08388-9

Source of the Parameters

Although the author had a particular reason for selecting the numbers $\pi$, 22-6$\sqrt{13}$, 2$\sqrt{3}$ and 22+6$\sqrt{13}$ as potential values for $\tau$and $\phi$, the reader does not have to accept the legitimacy of this reason to accept the legitimacy of the conjecture presented herein. In other words, the reader can view these four numbers as serendipitous guesses without affecting the algorithm for determining dimensionless physical constants. However, in truth, these four numbers were extracted from the first nine characters of Genesis in the Hebrew Bible.

Let us begin with $\pi$ . First invert the character sequence, in the first chapter of Genesis, so the characters are read from left to right. Next connect the first three contiguous characters in the first verse of Genesis to form a string. Now represent each of the Hebrew characters by a base-3 triplet (column vector) according to a rule that starts with Aleph as (000)^T and ends with Tsadey Final as (222)^T. The three column vectors are (001^T,201^T,000^T). These three base-3 triplets create a 3 by 3 matrix having the elements 0, 1 and 2:

020

Ayz = 000 (5)

110

The first eigenvector of (Ayz)(Ayz)^T is ($\rho$,0,1) where $\rho$ is the Golden Ratio which is, in turn, related to $\pi$ by:

π = (5) cos^-1($\rho$/2) (6)

The Golden Ratio and therefore $\pi$ are therefore concealed in the first three Hebrew characters of Genesis.

A path to the remaining three numbers is a bit more tedious. First, introduce the column vectors (202^T,100^T,210^T) and (001^T,201^T,000^T) representing the second three and third three characters in the first verse of Genesis. These base three triplets create two additional 3 by 3 matrices:

212

Byz = 001 (7)

200

020

Cyz = 000 (8)

110

Note that Ayz and Cyz are identical, so we really have only two independent matrices with 0, 1 and 2 as elements. Now stack these three matrices together to form what looks like a Rubic Cube. Each of 15 slices through the cube produces a 3 by 3 array; three slices each parallel to the yz, zx and xy planes and two diagonal slices perpendicular to each plane (denoted by X). Each slice can be rotated about various axes to produce eight 3 by 3 matrices: dihedral group of order eight (D₄). In this trial, only one matrix is selected for each slice. These 15 matrices can then be combined by matrix multiplication to form

yz = Ayz.Byz.Cyz.Xyz.Xzy

zx = Azx.Bzx.Czx.Xzx.Xxz (9)

xy = Axy.Bxy.Cxy.Xxy.Xyx

Finally, to create symmetry,

x = yz.yz^T

y = zx.zx^T (10)

z = xy.xt^T

Now use the instructions in the publication “Characteristic polynomial and higher order traces of third order three dimensional tensors,” by Guimei Zhang and Shenglong Hu, Front. Math China 2019, 14(1): 225-237 to create a characteristic equation of degree 12. The non-zero solutions to this equation are 22-6$\sqrt{13}$, 22+6$\sqrt{13}$, -2+2$i\sqrt{3}$, and -2-2$i\sqrt{3}$.

By pure conjecture, the numbers 22-6$\sqrt{13}$, 22+6$\sqrt{13}$, 2$\sqrt{3}$ and $\pi$ were selected to create various infinite series that satisfied the Ratio Test and converged to zero.

What are the implications of all these findings? Do they represent a completely serendipitous concurrence of physics, mathematics and Hebrew text? Or did our transcendent, immanent, infinite, eternal and immutable God provide us with a user manual encrypted within His inspired, inerrant and infallible Word. The first verse of Genesis says: “In the beginning God created the essence of the heavens and the essence of the earth.” What a perfect place to encrypt a user manual revealing the physics and mathematics of this new creation from nothing (bara). But why would God bother to encrypt a message within a message? A partial answer might be found in the New Testament.

“For since the creation of the world, God’s invisible qualities – His eternal power and divine nature – have been clearly seen, being understood from what has been made, so that men are without excuse” (Romans 1:20).

Only the first nine characters of Genesis have been used in this article. The concatenated words of the Torah comprise a 304,805 string of Hebrew characters. What mysteries might be encrypted therein?