# Electron (mathematical)

Mathematical electron as a geometry of the Planck units

In the mathematical electron model , the electron is a geometrical formula fe that embeds the Planck unit objects for charge, length and time, yet itself is dimensionless (there is no 'physical' electron). The dimension-ed parameters associated with the electron (mass, wavelength, frequency ...) are derivatives of the Planck units themselves, the dimension-less fe dictating the frequencies of these embedded Planck units.

$f_{e}=4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23}$ ## Geometrical objects

Base units for mass $M$ , length $L$ , time $T$ , and ampere $A$ are constructed as geometrical objects in terms of 2 dimensionless physical constants, the fine structure constant α and Omega Ω

Being independent of any numerical system and of any system of units, these MLTA units qualify as "natural units";

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck 
$M=(1)$ $T=(2\pi )$ $L=(2\pi ^{2}\Omega ^{2})$ $A=({\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }})$ ### Relation

A mathematical relationship between the objects;

$(A)\;u^{3}\;$ $(L)\;u^{-13}\;$ $(M)\;u^{15}\;$ $(T)\;u^{-30}\;$ ### Attribute

Each object is assigned a unit (for example object L is 'length')

Geometrical units
Attribute Geometrical object Relation
mass $M=1$ $unit=u^{15}$ time $T=2\pi$ $unit=u^{-30}$ length $L=2\pi ^{2}\Omega ^{2}$ $unit=u^{-13}$ ampere $A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}$ $unit=u^{3}$ The following un groups (and so their associated MLTA units) cancel

${\frac {u^{3*3}u^{-13*3}}{u^{-30}}}={\frac {u^{-13*15}}{u^{15*9}u^{-30*11}}}=\;...\;=1$ ## Mathematical electron

The electron function (the mathematical formula for the electron) fe incorporates these geometrical base units yet itself is unit-less; units = 1.

$f_{e}=4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23},\;units=1$ For example, fe can be defined in terms of σe, where AL as an ampere-meter (ampere-length) are the units for a magnetic monopole.

$T=2\pi ,\;u^{-30}$ $\sigma _{e}={\frac {3\alpha ^{2}AL}{\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}},\;u^{-10}$ $f_{e}={\frac {\sigma _{e}^{3}}{T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }},\;units={\frac {(u^{-10})^{3}}{u^{-30}}}=1$ ##### Discussion

Associated with the electron are dimension-ed parameters, these parameters however deriving from the base units, fe is a mathematical function that dictates how these units are applied, but it does not have dimension units of its own, consequently there is no physical electron. By setting MLTA to their Planck unit equivalents;

electron mass $m_{e}={\frac {M}{f_{e}}}$ (M = Planck mass)

electron wavelength $\lambda _{e}=2\pi Lf_{e}$ (L = Planck length)

elementary charge $e=A.T$ We may interpret this formula for fe whereby for the duration of the electron frequency (0.2389 x 1023 units of the universe clock-rate) the electron is represented by AL magnetic monopoles, these then intersect with time T, the units then collapse (units (A*L)3/T = 1), exposing a unit of M (Planck mass) for 1 unit of the universe clock-rate, which we could define as the mass point-state. Wave-particle duality at the Planck level can then be simulated as an oscillation between an electric (magnetic monopole) wave-state (the duration dictated by fe) to this unitary mass point-state. The magnetic monopoles are analogous to quarks (by adding the exponents of u) but due to the symmetry and so stability of the geometrical fe this may not be observed (as the monopoles are equivalent there is no fracture point by which an electron could decay).

By this artifice, although the 'physical' universe is constructed from particles, particles themselves are not physical, they are mathematical. Consequently this approach is applicable to simulation argument modeling at the Planck scale.

Note: at the Planck scale, quantum effects would not apply, rather quantum events such as the electron are a measure of these discrete Planck events spread over time (the duration of a particle wave-state to point-state oscillation). The quantum world (of probabilities) thereby emerges from the discrete Planck world.

##### Electron Mass

If the particle point-state is a unit of (Planck) mass then we have a model for a black-hole electron (the electron function fe centered around a Planck size black-hole). When the wave-state (A*L)3/T units collapse, this black-hole center is exposed (for 1 unit of time). The electron now is mass M. Mass would not then be a constant property of the particle, rather the measured particle mass m would be the average mass, the average occurrence of the mass point-state as measured over time. As for each wave-state then is a corresponding point-state, and as E = hv is a measure of the frequency of the wave-state and m a measure of the frequency of the point-state, then E = hv = mc2.

If the scaffolding of the universe includes units of (Planck) mass M, then it is not necessary for the particle to have a mass M, instead the point state becomes the absence of particle.