Programming a Virtual (Mathematical) Universe

Our external physical reality is a mathematical structure. That is, the physical universe is mathematics in a well-defined sense, and in those [worlds] complex enough to contain self-aware substructures SAS [they] will subjectively perceive themselves as existing in a physically 'real' world (- Max Tegmark's MUH).

Summary: Sqrt of Planck momentum as link between mass and charge, electron formula as magnetic monopole,  reduces required number of dimension units to 2, solves physical constants in terms of c, μ0, R, alpha

Sqrt of Planck momentum:

1. A premise of this model is that the sqrt of Planck momentum can be used to link the mass domain (the mass constants) and the charge domain (the charge constants). I have defined (see article) the sqrt of Planck momentum as Q with its associated unit as q such that Planck momentum = 2piQ^2 = 6.524 kg.m/s (q^2 = kg.m/s).

The 5 principal SI units; kg, m, s, A(mpere), k(elvin).
The 6 principal dimensioned constants; G, h, c, e, me, kB

I use this sqrt of Planck momentum Q to derive formulas for elementary charge e and for vacuum of permeability μ0. From these, a formula for a magnetic monopole as an ampere-meter can be constructed which then gives a solution for the electron frequency function fe in terms of magnetic monopoles and time.

Mass formulas use Q^2 and thus exhibit integer dimensions (q^2 = kg.m/s).
Charge formulas use Q^3, Q^5, Q^15 and thus exhibit non-integer dimensions (q^3 = q.kg.m/s).

2. From analysis of the electron formula fe, I premise that the units for mass, space and time overlap and cancel in this ratio; units M^9T^11/L^15 = 1, this ratio is found in fe, thus the electron formula, although constructed from magnetic monopoles and time, is a dimensionless mathematical constant units (AL)^3/T = M^9T^11/L^15 = 1.

By linking M^9T^11 = L^15 we can for example define L in terms of M and T and thus reduce the number of required units from 5 (the 5 SI units) to 2 (for example mass and time).

With a reduced unit set we can replace Q with the Rydberg constant R and then redefine the dimensioned fundamental constants (G, h, e, me, kB)  in terms of the 4 most precise physical constants c (exact), μ0 (exact), R (12-13 digits), alpha (the fine structure constant). Results are consistent with CODATA (see calculator below);



The model is described in the following article (continued on black-hole electron, next page);

Alpha* as input (0 = default)




Solutions (α, c, μ0, R) :

Planck constant*

h = 6.626 070 040(81)e-34


elementary charge*

e = 1.602 176 6208(98)e-19


electron mass*

me = 9.109 383 56(11)e-31


Boltzmann constant*

kB = 1.380 648 52(79)e-23


Gravitation constant*

G = 6.674 08(31)e-11

μ0 = permeability of vacuum = 4π/10000000 (fixed) 
Rydberg = 10973731.568508 (CODATA 2014 mean)
alpha α = 137.035999139 (CODATA 2014 mean)

Maple code
pi:=3.14159265358979323846:
c:=299792458:  
R:=10973731.568508: 
mu0:=pi/2500000:  

Constants in terms of c, μ0, R, α
h:=(2*pi^10*mu0^3/(3^6*c^5*a^13*R^2))^(1./3):
e:=(4*pi^5/(3^3*c^4*a^8*R))^(1./3):
kB:=(pi^5*mu0^3/(2*3^3*a^5*c^4*R))^(1./3):
G:=(pi^3*mu0/(2^20*3^6*a^11*R^2))^(1./5):
Tp:=(2^10*3^3*c^15*a^3*R/(pi^4*mu0^3))^(1./5):
me:=(16*pi^10*R*mu0^3/(3^6*a^7*c^8))^(1./3):
Bm:=(pi^2/(2^7*3^3*c*a^14*R^4))^(1./3):
lp:=(pi^22*mu0^9/(2^35*3^24*a^49*c^35*R^8))^(1./15):
tp:=(pi^22*mu0^9/(2^20*3^24*a^49*c^50*R^8))^(1./15):
mP:=(2^25*pi^13*mu0^6/(3^6*c^5*a^16*R^2))^(1./15):
A:=(2^10*pi*3^3*c^10*a^3*R/mu0^3)^(1./5):