# Physics/Essays/Martin Gibson/Dimensional Analysis of the Fine Structure Constant

While the fine-structure constant is generally described as a dimensionless number, it is still best understood in terms of a dimensional analysis of its usage.

## Dimensionality of the Fine Structure Constant

We can remove some of the mystery and the mysticism surrounding the fine structure constant, ${\displaystyle \alpha }$ by employing the following dimensional and functional analysis. As a coupling constant for the electromagnetic interaction, ${\displaystyle \alpha }$ is held to be a pure number that defines the strength of the interaction. As shown here, the dimensionless context of ${\displaystyle \alpha }$ is due to the fact that it is a ratio of two measurements/computations of like dimensionality. It is therefore itself a factor in the coefficient of one of those dimensional structures, which we might surmise includes the dimension of time as in an impulse or force. As an alternative it is a component of a statement of the ratio between the square of a unit of current and the square of a unit of elementary charge. As such it is a measure of frequency squared which would be the same in any system that uses the second as the time standard. Both of these approaches are developed below. While there is no new experimental data in this presentation, to the author's knowledge the relationships as to the underlying dimensionality of the constant and the significance of the inertial constant are novel. They point to a classical wave foundation basis for an emergent quantum particle phenomenology which is recognized in the Standard Model, as suggested below and developed more fully in the attached links.

## Development

### With respect to a Unit of Charge

Using the CODATA definition of the fine structure constant

• ${\displaystyle {e_{0}}}$ is elementary charge;
• ${\displaystyle \hbar =h/2\pi }$ is the reduced Planck constant, ${\displaystyle h}$;
• ${\displaystyle c}$ is the speed of light in vacuo;
• ${\displaystyle \varepsilon }$0 is the vacuum permittivity or the electric constant;
• ${\displaystyle \mu }$0 is the vacuum permeability;

${\displaystyle (1.1)}$ ${\displaystyle \alpha ={\frac {{e_{0}}^{2}}{\hbar c4\pi {\varepsilon _{0}}}}={\frac {c{e_{0}}^{2}{\mu _{0}}}{\hbar 4\pi }}}$

where the permeability constant, ${\displaystyle \mu }$0, is expressed (with dimensions shown in curly brackets) as

${\displaystyle (1.2)}$ ${\displaystyle {\mu _{0}}={\frac {1}{{c^{2}}{\varepsilon _{0}}}}=q\left\{{\frac {\rm {N}}{{\rm {A}}^{\rm {2}}}}\right\}}$

${\displaystyle {\mu _{0}}}$ is defined both quantitatively, as a number ${\displaystyle q}$, and qualitatively (dimensionally) in terms of the magnetic force in newton {N} induced by the product of 2 moving charges or electrical currents in ampere {A}. We note that the ampere is one of the fundamental units of the International System of Units (SI) along with the three units of mass, length and time, i.e. kilogram {kg}, meter{ m}, and second {s}, which define the newton as

${\displaystyle (1.3)}$ ${\displaystyle {\rm {N}}={\frac {m\cdot l}{t^{2}}}}$ ${\displaystyle \left\{{\frac {{\rm {kg}}\cdot {\rm {m}}}{{\rm {s}}^{\rm {2}}}}\right\}}$

Thus in dimensional terms, ${\displaystyle \mu }$0 is the ratio, here equal to ${\displaystyle q}$ , of a force to the square of a unit current, (technically the cross product of the field strength of one unit current operating on a parallel unit current at a distance of one unit of length.) Here the naught subscripts indicate a normalized or unit value of the respective variables, or as in the case of ${\displaystyle \mu }$0 or ${\displaystyle \varepsilon }$0 , a universal constant.

### With respect to a Unit of Current

Specifically, for two parallel wires, ${\displaystyle a}$ and ${\displaystyle b}$, of indefinite length, one meter apart, ${\displaystyle d_{0}}$, in vacuo, each carrying a unit of current, ${\displaystyle i_{0}}$, of one ampere or coulomb {${\displaystyle C}$} per second, a magnetic force, ${\displaystyle F_{ba}}$, of ${\displaystyle 2x10^{-7}{\rm {N}}}$ is generated on one of the wires by the other for each meter length, ${\displaystyle l_{0}}$, of the two, toward each other or positive if the currents are parallel and away from each other or negative if they are anti-parallel; thus expressed,

${\displaystyle (1.4)}$ ${\displaystyle {\frac {F_{ba}}{l_{0}}}={\frac {{\mu _{0}}{i_{0}}_{a}{i_{0}}_{b}}{2\pi {d_{0}}}}={\frac {2{\rm {x}}{{10}^{-7}}}{1{\rm {}}}}\left\{{\frac {\rm {N}}{\rm {m}}}\right\}}$

With transposition for the value of ${\displaystyle F_{ba}}$ as stated, we have the defined value of ${\displaystyle \mu }$0 as,

${\displaystyle (1.5)}$ ${\displaystyle {\mu _{0}}={\frac {2\pi {F_{ba}}}{{i_{0}}^{2}}}={\frac {4\pi {\rm {x}}{{10}^{-7}}}{{\left({{{n_{A0}}^{}{e_{0}}^{}}/{t_{0}}}\right)}^{2}}}\left\{{\frac {\rm {N}}{{\left({{\rm {C}}/{\rm {s}}}\right)}^{2}}}\right\}={\frac {4\pi {\rm {x}}{{10}^{-7}}}{1^{2}}}\left\{{\frac {\rm {N}}{{\rm {A}}^{\rm {2}}}}\right\}\therefore q=4\pi {\rm {x}}{10^{-7}}}$

In fact it is the value of ${\displaystyle \mu }$0 that defines ${\displaystyle {i_{0a}}={i_{0b}}=1{\rm {}}A{\rm {}}=1{\rm {}}C/s{\rm {}}}$ and ${\displaystyle {l_{0}}={d_{0}}=1{\rm {}}m}$ . We note that ${\displaystyle i_{0}}$ and ${\displaystyle d_{0}}$ are orthogonal to each other.

### Dimensionless Relationship of Charge to Current

Since a coulomb is deemed to be made up of a finite number, ${\displaystyle n_{A0}}$, of fundamental unit charges, ${\displaystyle e_{0}}$, for each ampere or

${\displaystyle (1.6)}$ ${\displaystyle {n_{A0}}{e_{0}}=1{\rm {}}C}$

an ampere can be written as in the divisor of the middle term of ${\displaystyle (1.5)}$. Substituting this into ${\displaystyle (1.1)}$, gives

${\displaystyle (1.7)}$ ${\displaystyle \alpha ={\frac {c{e_{0}}^{2}{\mu _{0}}}{\hbar 4\pi }}={\frac {{e_{0}}^{2}{\rm {x}}{{10}^{-7}}}{{\frac {\hbar }{c}}{{{n_{A0}}^{2}{e_{0}}^{2}}/{{t_{0}}^{2}}}}}\left\{{\frac {{{\rm {C}}^{2}}{\rm {N}}}{{\textstyle {{{\rm {kg}}\cdot {{\rm {m}}^{2}}{s^{-1}}} \over {m{s^{-1}}}}}\left({{\rm {C}}^{2}}\right){\rm {/}}{{\rm {s}}^{\rm {2}}}}}\right\}={\frac {{10}^{-7}}{{{(tav)}{n_{A0}}^{2}}/{{t_{0}}^{2}}}}\left\{{\frac {\rm {N}}{{\rm {kg}}\cdot {\rm {m/}}{{\rm {s}}^{\rm {2}}}}}\right\}}$

In the last term, the constants of action and the speed of light are reduced and the unit charge squared terms are canceled. As a change in the relative placement of two charges produces a force, and a force is a momentum differential, i.e. an impulse, per unit of time, a fundamental quantum or unit of charge can be viewed as a quantum of momentum or a unit impulse, potential or kinetic, i.e. static or moving. In the next to last term, the number, ${\displaystyle n_{A0}}$, is the number of such impulses in an ampere of current, or when divided by one of the time dimensions, an expression of the frequency of such impulses. Canceling the fundamental charges leaves a measure of force in the antecedent, and in the consequent a frequency squared, times Planck’s quantum of action divided by the speed of light, which resolves dimensionally to a measure of mass-length. Thus we have the quotient of two units of force, resulting in the non-dimensionality of the fine structure constant.

### Inertial Constant and Mass as Wave Number

In the last term, we introduce the inertial constant, ת (tav), as a time independent, fundamental mass–length unit, where for any rest mass quantum, ${\displaystyle m_{q}}$,

${\displaystyle (1.8)}$ ${\displaystyle {m_{q}}={\frac {\hbar }{c^{2}}}{\omega _{q}}={\frac {\hbar }{c}}{\kappa _{q}}=(tav){\kappa _{q}}{\rm {}}\therefore {\frac {\hbar }{c}}=(tav)}$

Omega and kappa are angular wave frequency and number, respectively, indicating that mass is essentially a proxy for wave number. The inertial constant derivation is the time integral of a quantum impulse, ${\displaystyle (tav)=J(t)\int _{0}^{1}{dt}={m_{q}}({\lambda _{C,q}}/2\pi )}$, which is invariant. Here ${\displaystyle {\lambda _{C}}}$ is the Compton wavelength of a quantum rest mass particle, the inverse of the corresponding wave number. The inertial constant is the invariant quantity of impulse present in each radian of quantum oscillation. This figure times the frequency gives the angular wave transverse momentum and times the frequency squared gives the angular wave force.

### Ratio of Current to Charge

Equation ${\displaystyle (1.7)}$ expresses the fine structure constant as a dimensionless number. It is dimensionless in the sense that a ratio of two like qualities is dimensionless, yet such dimensionless number can also be seen as a coefficient, whole or partial, in this case as a quantifier of the consequent (divisor) of such ratio. [It is the factor required, in product with the inertial constant times the square of its frequency found in two 1 ampere currents as figured above, to produce an induced force of ${\displaystyle 10^{-7}}$ Newton.]

Some rearrangement of ${\displaystyle (1.7)}$ gives

${\displaystyle (1.9)}$ ${\displaystyle {{{n_{A0}}^{2}{e_{0}}^{2}}/{{t_{0}}^{2}}}\left\{{\frac {{\rm {C}}^{\rm {2}}}{{\rm {s}}^{2}}}\right\}={\frac {{e_{0}}^{2}{\rm {x}}{{10}^{-7}}}{\alpha (tav)}}\left\{{\frac {{\rm {N}}{{\rm {C}}^{\rm {2}}}}{{\rm {kg}}\cdot {\rm {m}}}}\right\}={i_{0}}^{2}\left\{{{\rm {A}}^{\rm {2}}}\right\}}$

Here we have an expression of the current at one ampere, squared. Once again canceling the fundamental unit charges gives

${\displaystyle (1.10)}$ ${\displaystyle {\left({{n_{A0}}/{t_{0}}}\right)^{2}}\left\{{\textstyle {{{\rm {\#}}^{2}} \over {{\rm {s}}^{2}}}}\right\}={\frac {{10}^{-7}}{\alpha (tav)}}\left\{{\textstyle {{\rm {N}} \over {{\rm {kg}}\cdot {\rm {m}}}}}\right\}={\frac {i_{0}^{2}}{{e_{0}}^{2}}}\left\{{\textstyle {{{\rm {A}}^{\rm {2}}} \over {{\rm {quantumcharg}}{{\rm {e}}^{2}}}}}\right\}={i_{e}}^{2}=3.895644...x{10^{37}}\left\{{{\left({\textstyle {{\rm {chargeimpulses}} \over {\rm {s}}}}\right)}^{2}}\right\}}$

where it is apparent that the left hand term is the square of a frequency, ${\displaystyle f_{A0}^{2}}$ , perhaps periodic, semi-periodic, or some other duration. This frequency, in fact angular, gives the number of instances of the inertial moment or constant, ת, a time-independent quantum of (wave) momentum and force, as

${\displaystyle (1.11)}$ ${\displaystyle (tav){\left({{n_{A0}}/{t_{0}}}\right)^{2}}\left\{{{{\rm {kg}}\cdot {\rm {m}}}/{{\rm {s}}^{2}}}\right\}=(tav)f_{A0}^{2}\left\{{\rm {N}}\right\}={\frac {{10}^{-7}}{\alpha }}\left\{{\rm {N}}\right\}=\alpha '\left\{{\rm {N}}\right\}=137.035999...{\rm {x}}{10^{-7}}\left\{{\rm {N}}\right\}}$

## Significance

It is immediately clear that the value of ${\displaystyle \alpha }$ is dictated by the dividend at ${\displaystyle 10^{-7}}$, since the presumed invariant is their quotient,${\displaystyle \alpha '}$ , and a change in the dividend necessitates a corresponding change in the divisor, ${\displaystyle \alpha }$, the fine structure constant. In practice, ${\displaystyle 10^{-7}}$ sets the length of a meter in terms of ${\displaystyle c}$ in ${\displaystyle (1.4)}$ and ${\displaystyle (1.5)}$, so that a nominal change in the dividend would result in a nominal change in the speed of light and in the various wavelengths found in the fine structure series, and therefore in ${\displaystyle \alpha }$ as well. What would change the value of ${\displaystyle \alpha '}$ is a change in the value of a unit of time, ${\displaystyle t_{0}}$, so that a nominal lengthening of a time unit (to include more ${\displaystyle n_{A0}}$ per second) would nominally increase the force on the right. It is the duration of the second that determines the frequency, which determines ${\displaystyle \alpha '}$ , and given a nominal ${\displaystyle 10^{-7}}$, determines ${\displaystyle \alpha }$.

If we take a dimensional look at ${\displaystyle (1.2)}$ in the context of ${\displaystyle (1.4)}$, it appears that ${\displaystyle {\mu _{0}}}$ , the permeability of the vacuum, is converting the two current flows into a force component. It is equally correct to think that each current flow constitutes a force that interacts to produce a magnetic field force between them, the cross product given by ${\displaystyle F_{ba}}$, which gives ${\displaystyle {\mu _{0}}}$ the dimensions of an inverse force. If current is expressed in units of force, then charge becomes a count of momenta or impulse as the time integral of the current, and the fundamental unit of charge becomes a fundamental unit of momentum as discussed above.

### Discrete Rotational or Spin Wave Model

This discussion is facilitated by the conceptualization of rest mass quanta as local rotational oscillations of an inertially dense spatial continuum. Such quantum oscillations are instances of simple harmonic motion which can be expected to exhibit certain fundamental wave characteristics such as transverse wave force and transverse wave momentum, represented by a fundamental or resonant angular frequency, ${\displaystyle {\omega _{0}}}$.

Initial local torsion of space fabric twists X axis into Y-Z plane. Cross-product of restorative force and orthogonal torsion stress in Y-Z plane results in sustained rotational oscillation or spin.
Rotation of orthogonal torsion oscillation in the Y-Z plane Initiated and sustained by cosmic expansion. Such expansion produces spin, quantum gravity, nuclear congregation and over time beta decay.

This is not a body force or momentum of a particle, but rather a wave force and momentum of the density field oscillation, of which there are two simultaneous and opposed instances, one for each half of the cycle. In the quantum context, using the inertial constant as a quantum of mass-length, the fundamental transverse wave force, ${\displaystyle (tav)\omega _{0}^{2}}$, and wave momentum, ${\displaystyle (tav)\omega _{0}}$, are represented as here. These travel around with the oscillation, with force leading momentum by ${\displaystyle \pi /2}$. If we assume for the sake of argument, that the neutron represents this fundamental oscillation, then using the mass of the neutron in light of ${\displaystyle (1.8)}$ gives an expression and an evaluation of the wave force and momentum respectively of that particle as

${\displaystyle (1.12)}$ ${\displaystyle (tav){\omega _{0}^{2}}={\frac {\hbar }{c}}\omega _{n}^{2}={m_{n}}c\omega _{n}^{}=716,766.8351{\rm {N}}}$

${\displaystyle (1.13)}$ ${\displaystyle (tav){\omega _{0}}={\frac {\hbar }{c}}{\omega _{n}}={m_{n}}c=5.02130545...x{10^{-19}}{{{\rm {kg}}\cdot {\rm {m}}}/{\rm {s}}}}$

The above is a tremendous amount of force, especially for a single quantum, but it pales in comparison to the stress found by figuring the small cross sectional area upon which the force operates, which are on the order of ${\displaystyle 10^{37}}$ pascals.

Note the following ratio, in which the ${\displaystyle \pi }$ converts the angular frequency to a semi-periodic frequency

${\displaystyle (1.14)}$ ${\displaystyle {\frac {(tav){\omega _{0}}/{\pi }}{e_{0}}}={\frac {1.59833...x{{10}^{-19}}}{1.60217...x{{10}^{-19}}}}=0.997599940...=1-0.002400060...}$

where both antecedent and consequent terms are figured in units of momentum. That is, electron charge is neutron wave momentum observed as a result of beta decay. In the above referenced wave conceptualization, the fundamental frequency is driven by cosmic expansion, and the fundamental wave force itself is a function of the expansion stress, an isotropic stress that pervades space, isotropic that is, except at the boundary of the oscillation where the cross product effects of microscopic torsion prevent dispersion of the wave energy and make themselves felt as particle spin.

### Cosmic Expansion produces Beta Decay and Charge

The mechanical impedance, ${\displaystyle Z_{0}}$ of such inertial space, (not to be confused with the SI impedance of the vacuum), is the quotient of the fundamental characteristic wave force and the speed of wave propagation, evaluated here and compared with the last term of ${\displaystyle (1.14)}$

${\displaystyle (1.15)}$ ${\displaystyle {Z_{0}}={\frac {(tav)\omega _{0}^{2}}{c}}=(tav){\omega _{0}}{\kappa _{0}}=0.002390877...}$

Thus

${\displaystyle (1.16)}$ ${\displaystyle {\frac {(tav){\omega _{0}}({1+{Z_{0}}})/\pi }{e_{0}}}={\frac {(tav){\omega _{0}}({1+{(tav){\omega _{0}}{\kappa _{0}}}})/\pi }{e_{0}}}=.999985079...=1-.000014921...}$

where the subtrahend of the last term represents the difference of the antecedent on the left from a theoretically precise value of ${\displaystyle e_{0}}$. As the value of antecedent depends on a measured observation of the neutron mass, which presumably uses Newton’s constant, G, for its evaluation in terms of some mass standard, and as the relative uncertainty of that constant is ${\displaystyle 10^{-4}}$ and ${\displaystyle (1.16)}$ is precise to approximately ${\displaystyle 10^{-5}}$, these results recommend pursuit of this line of reasoning.

Combining ${\displaystyle (1.16)}$, ${\displaystyle (1.11)}$ and ${\displaystyle (1.4)}$ we have an expression relating fundamental charge and electromagnetic induced force in terms of the inertial constant and the resonant frequency and wavelength of the vacuum, and in light of the above uncertainty,

${\displaystyle (1.17)}$ ${\displaystyle 2(tav){\left({\frac {\pi }{(tav){\omega _{0}}\left({1+(tav){\omega _{0}}{\kappa _{0}}}\right)}}\right)^{2}}\simeq {\frac {2(tav)}{e_{0}^{2}}}=2(tav)f_{A0}^{2}={\frac {{F_{a0}}_{b0}}{\alpha }}={\frac {2{\rm {x}}{{10}^{-7}}}{\alpha }}\{{\rm {{N}\}}}}$

In terms of a model of three spatial dimensions and one dimension of time gauged to cosmic expansion, that expansion leads to a drop in the inertial density of space, and a related drop in its impedance. This produces a discontinuity at the boundaries, i.e. nodes of quantum oscillations, and results in the transmission of power and energy at those nodes which is registered as beta decay. The results are precisely determined by geometry, accounting for the observed neutron/electron, proton/electron, and neutron/proton mass ratios. Classical linear wave analysis, taken to three and four dimensions explicates the basics of quantum dynamics, in which it is seen that the lepton and quark phenomenology of the Standard Model are the nodal/antinodal structures of a multi-dimensional bound wave system.

## References

For a clear statement of classical wave mechanics see: Elmore, William C. and Heald, Mark A. (1969). Physics of Waves (Dover Edition, 1985). Mineola, N.Y.: Dover Publications, Inc. ISBN 0-486-64926-1.