## Technical Notes 2006

### Discrete Self-Similar Scaling For Charge And Angular Momentum [October 2006]

###### I. Scaling For Electromagnetic Charge

Given that

e^{2} = a ħ c,

where a is the fine structure constant (≈ 1/137.036), ħ is Planck’s constant divided by 2π (≈ 1.054 x 10^{-27} erg sec) and c is the velocity of light (≈ 2.99 x 10^{10} cm/sec), the dimensionality of e^{2} is

(ML^{2}/T^{2})(T)(L/T),

or ML^{3}/T^{2}.

According to the SSCP, e^{2} would scale as Λ^{D}Λ^{3}/Λ^{2} ≈ Λ^{D+1}.

Therefore, e would scale as Λ^{(D+1)/2}, and since D ≈ 3.174,

e_{Ψ} ≈ Λ^{2.087} e_{Ψ}_{-1}

e_{Ψ} ≈ 9.40 x 10^{36} e_{Ψ}_{-1}

Note that whereas mass scales as Λ^{D} ≈ 1.70 x 10^{56} between neighboring Scales, charge increases by the much smaller factor of Λ^{(D+1)/2 }≈ 10^{37} when we go “up” one cosmological Scale. This implies that the ratio of unbalanced to balanced **Ψ = -1** charges in a **Ψ = 0** system is about 10^{37}/10^{56} ≈ 5.4 x 10^{-20}. So only a __very small fraction__ of the **Ψ = 0** mass has a net charge, but this appears to be enough to produce **Ψ = 0** electromagnetic phenomena that is self-similar to **Ψ = -1** electromagnetic phenomena.

###### II. Scaling For Angular Momentum

The dimensions of angular momentum (J) are: ML^{2}/T.

According to the SSCP, J should scale as Λ^{D}Λ^{2}/Λ ≈ Λ^{D+1} ≈ Λ^{4.174} ≈ 8.84 x 10^{73}.

Therefore:

J_{Ψ}_{} ≈ 8.84 x 10^{73} J_{Ψ}_{-1}

for discrete self-similar analogues on neighboring Scales.

###### III. The Stellar Scale Planck's "Constant"

Angular momentum in the quantum realm is usually found in multiples of h or ħ, which have values of 6.626 x 10^{-27} erg sec and 1.054 x 10^{-27} erg sec, respectively.

The SSCP predicts that when we have enough high-quality observational data, the Stellar Scale will manifest a comparable discreteness in angular momentum phenomena with:

h_{Ψ}_{=0} ≈ Λ^{D+1} h_{Ψ}_{=-1 }≈ 5.860 x 10^{47} erg sec ,

and

ħ_{Ψ}_{=0} ≈ Λ^{D+1} ħ_{Ψ}_{=-1} ≈ 9.325 x 10^{46} erg sec.

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