Electron (mathematical)

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The mathematical electron model

In the mathematical electron model [1], the electron is a mathematical formula that describes a geometrical object (fe). Embedded within this object are base objects that embody the functions (attributes) of the Planck units. It is these Planck unit objects that provide the observed electron parameters of mass, wavelength, charge ... this electron formula (fe) itself is dimensionless, units = 1, it dictates the frequency of these Planck units.

fe is the geometry of 2 dimensionless physical constants, the (inverse) fine structure constant α = 137.035 999 139 (CODATA 2014) and Omega Ω = 2.007 134 9496

, units = 1


Planck objects[edit | edit source]

The electron formula fe embeds the geometrical objects for mass M, length L, time T and charge A ... and so these MLTA objects are also the geometry of the dimensionless constants alpha and Omega. However, although α and Ω are dimensionless, the MLTA objects are the basis of the dimensioned Planck units. Their respective unit attributes are relational, although the mass object would incorporate the function mass, the time object the function time ..., it is not necessary that there be an isolated physical mass or physical length or physical time ..., but only that in relation to the other units, the object must express that function (i.e.: the mass object has the function of mass when in the presence of the objects for space and time).

This overlap of function however requires that the Planck units (defined by kg, m, s, A, K) must be related geometrically, they cannot be independent of each other, and consequently evidence of such anomalies within the physical constants could be considered evidence that the universe is a simulation (the simulation hypothesis).

In table 1., the Planck objects are listed with a unit number and its associated SI unit. We then note there are combinations of the objects for which the unit numbers cancel, the combination is then itself dimensionless. For example, we can construct fe by combining the ampere object A with the length object L, and then divide by the object time T.

, θ = 3*3 -13*3 +30 = 0
table 1. Geometrical units
Attribute Geometrical object Unit number θ
mass (kg)
time (s)
velocity (m/s)
length (m)
ampere (A)


As α and Ω can be measured using numerical values the MLTA objects have numerical solutions, and so we may convert them to their SI equivalents (or any system of units) using dimensioned scalars. For example, object V = 2πΩ2 = 25.3123819;

scalar v = 11843707.905m/s gives c = V*v = 299792458m/s (SI units)

We can assign to each MLTA object a scalar, but as the scalars, being dimensioned, also follow this unit number relationship, we need only 2 scalars to define the others. Consequently if we know the precise values for α and Ω and for 2 dimensioned constants such as c and μ0, then we may solve the remaining dimensioned constants (G, h, e, me, kB ...) to equivalent precision.


Mathematical electron[edit | edit source]

The electron function fe incorporates these geometrical base units yet itself is unit-less; units = 1 (θ = 0), and the scalars cancel (scalars = 1), and thus is of itself a mathematical constant (as the scalars cancel, it will have the same numerical value regardless of which system of Planck units are used, SI or otherwise).

Here fe is defined in terms of σe, where AL is an ampere-meter (ampere-length = e*c are the units for a magnetic monopole). The scalars (v, r) are chosen as v can be derived from c and r from μ0.

(θ = -30)
(θ = 3 -13 = -10)
(unit-less)


Electron parameters[edit | edit source]

Associated with the electron are dimensioned parameters, these parameters however are a function of the base MLTA units, the formula fe dictating the frequency of these units. By setting MLTA to their SI Planck unit equivalents;

electron mass (M = Planck mass) = 0.910 938 232 11 e-30

electron wavelength (L = Planck length) = 0.242 631 023 86 e-11

elementary charge (T = Planck time) = 0.160 217 651 30 e-18

Rydberg constant = 10 973 731.568 508


We may interpret this formula for fe whereby for the duration of the electron frequency = 0.2389 x 1023 units of (Planck) time, the electron is represented by AL magnetic monopoles, these then intersect with time T, the units then collapse (θ = 3*3 -13*3 +30 = 0), exposing a unit of M (Planck mass) for 1 unit of (Planck) time, which we could define as the mass point-state. Wave-particle duality can then be represented at the Planck level as an oscillation between an electric (magnetic monopole) wave-state (the duration dictated by fe measured in Planck time units), to this discrete M = 1 mass point-state.

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 deep universe simulation hypothesis modeling at the Planck scale.


Electron Mass[edit | edit source]

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 this unit of Planck mass. When the wave-state (A*L)3/T units collapse, this black-hole center is exposed for 1 unit of (Planck) time. The electron is 'now' (a unit of Planck) mass M.

Mass in this consideration is not a constant property of the particle, rather the measured particle mass m would refer to the average mass, the average occurrence of the discrete Planck mass point-state as measured over time. The formula E = hv is a measure of the frequency v of occurrence of Planck's constant h and applies to the electric wave-state. As for each wave-state then is a corresponding mass point-state, then for a particle E = hv = mc2. Notably however the c term is a fixed constant unlike the v term, and so the m term is referring to average mass rather than constant mass.

If the scaffolding of the universe includes units of (Planck) mass M, then it is not necessary for the particle itself to have mass, only that in this state the electron has no dimensioned ALT parameters.


Quarks[edit | edit source]

The charge on the electron derives from the embedded ampere A and length L, the electron formula fe itself is dimensionless. These AL magnetic monopoles would seem to be analogous to quarks, but due to the symmetry and so stability of the geometrical fe there is no clear fracture point by which an electron could decay and so this would be difficult to test. We can however conjecture on what a quark solution might look like, the advantage with this approach being that we do not need to introduce new 'entities' for our quarks, the Planck units suffice.


The AL magnetic monopole 'quark'


An AV Planck temperature 'quark'


For example, at the base level a single AL monopole D = could equate to a quark with an electric charge of -1/3 e. 3 D quarks would constitute the electron as DDD = (AL)3. Conversely, we may consider a quark with an charge of 2/3 e, denoted here U = and this would be centered on Planck temperature Tp.

DDD:   

DDU:   

DUU:   


We see that the DDU formula (neutron) has no units or scalars. It corresponds to the formula fe. The DDD formula (electron) and DUU formula (proton) are separated from the DDU formula by a unit of T unit-30 and 1/T unit30 respectively. It may therefore be possible for the neutron to split into an electron and proton without breaking any mathematical rules.

The above formulas refer only to the 'quark' component. According to the mass of the electron, there is no binding energy required for the AL quarks, however the mass of the proton and neutron imply that a configuration that includes a U quark and a D quark requires significant binding energy, this has not been included in the DDU and DUU formulas.


Derivation via SI units[edit | edit source]

Magnetic monopole[edit | edit source]

A magnetic monopole is a hypothesized particle that is a magnet with only 1 pole. The unit for the magnetic monopole is the ampere-meter, the SI unit for pole strength (the product of charge and velocity) in a magnet (A m = e c). A proposed formula for a magnetic monopole σe;

The following gives a formula for an electron in terms of magnetic monopoles and Planck time ().

However, although this gives us the correct numerical value, this gives us incorrect units when solving the electron mass ( = Planck mass).

To resolve this we can consider the possibility that the units for A, m, s are related whereby units


Sqrt Planck momentum[edit | edit source]

The sqrt of Planck momentum is not a recognized constant (it has no SI designation) and so here is denoted as Q with units q whereby

Planck momentum = 2 π Q2, unit = kg.m/s = q2.

Replacing m with q;

Planck length

Speed of light

elementary charge

The excess electron mass units become;


The Rydberg constant R.

Vacuum permeability

This however now gives us 2 solutions for length m, if we conjecture that they are both valid, then there must be a ratio whereby the units q, s, kg overlap and cancel;

Which in terms of kg, m, s becomes


External links[edit | edit source]


References[edit | edit source]

  1. Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x.