## Technical Notes 2006

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

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

Given that

e2 = 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 1010 cm/sec), the dimensionality of e2 is

(ML2/T2)(T)(L/T),

or ML3/T2.

According to the SSCP, e2 would scale as ΛDΛ32 ≈ ΛD+1.

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

eΨ Λ2.087 eΨ-1

eΨ 9.40 x 1036 eΨ-1

Note that whereas mass scales as ΛD ≈ 1.70 x 1056 between neighboring Scales, charge increases by the much smaller factor of Λ(D+1)/2 ≈ 1037 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 1037/1056 ≈ 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: ML2/T.

According to the SSCP, J should scale as ΛDΛ2/Λ ≈ ΛD+1 ≈ Λ4.174 ≈ 8.84 x 1073.

Therefore:

JΨ ≈ 8.84 x 1073 JΨ-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:

hΨ=0 ΛD+1 hΨ=-1 5.860 x 1047 erg sec ,

and

ħΨ=0 ΛD+1 ħΨ=-1 9.325 x 1046 erg sec.