Electron (Mathematical)

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The mathematical electron model

In the mathematical electron model [1], the electron is a geometrical object denoted by the formula (ψ). Although dimensionless, this formula ψ encodes the information required to make the physical electron (which we observe as the electron parameters; wavelength, frequency, mass, charge). It does this by embedding within its geometry the objects MLTA (these are geometrical objects, analogues of the Planck units for mass

\[ m_{P}\]

, length

\[ l_{p}\]

, time

\[ t_{p}\]

and charge

\[ A\]

). The MLTA objects are themselves the geometry of 3 numbers; the fine structure constant alpha, a mathematical constant Omega, and pi (Omega itself is a construct of pi and e).

The electron formula ψ not only embeds the Planck units but also dictates their frequency;

\[ \psi =4\pi ^{2}(2^{6}3\pi ^{2}\alpha _{inv}\Omega ^{5})^{3}=.23895453...\times 10^{23},\;unit=1\]

(unit-less)

\[ \Omega ={\sqrt {\left(\pi ^{e}e^{(1-e)}\right)}}=2.0071349543...\]

For example;

electron wavelength

\[ \lambda _{e}=2\pi l_{p}\psi\]

electron mass

\[ m_{e}={\frac {m_{P}}{\psi }}\]

Thus it can be argued that this formula ψ, which resembles the volume of a torus or surface of a 4-D hypersphere, is itself a complex geometry that is the construct of simpler Planck unit geometries.

Planck objects MLTA

Main page: User:Platos Cave (physics)/Simulation_Hypothesis/Physical_constant_(anomaly)

The base units MLTA are geometrical objects, the geometry of 2 dimensionless constants (the inverse fine structure constant alpha = 137.035 999 139 (CODATA 2014) and a mathematical constant Omega).

table 1. Geometrical units

Attribute	Geometrical object	Unit
mass	\(\displaystyle M=(1)\)	(kg)
time	\(\displaystyle T=(\pi )\)	(s)
velocity	\(\displaystyle V=(2\pi \Omega ^{2})\)	(m/s)
length	\(\displaystyle L=(2\pi ^{2}\Omega ^{2})\)	(m)
ampere	\(\displaystyle A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha _{inv}}})\)	(A)

Geometrical objects (when compared to a numbering system) have the advantage in that the function (attribute) can be embedded within the geometry (even although the geometry itself is dimensionless); for example, the geometry of the time object T embeds the function 'time', the geometry of the length object L embeds the function 'length' ... and being geometrical objects they can combine to form more complex objects, from electrons to apples ... and so the apple has mass because embedded within it are the mass objects M, complex events thus retain all the underlying information.

This however requires a relationship between the Planck unit geometries that defines how they may combine, this can be represented by assigning to each attribute a unit number θ (i.e.: θ = 15 ⇔ kg).

Geometrical units

Attribute	Geometrical object	unit number (θ)
mass	\(\displaystyle M=1\)	kg ⇔ 15
time	\(\displaystyle T=2\pi\)	s ⇔ -30
length	\(\displaystyle L=2\pi ^{2}\Omega ^{2}\)	m ⇔ -13
velocity	\(\displaystyle V=2\pi \Omega ^{2}\)	m/s ⇔ 17
ampere	\(\displaystyle A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha _{inv}}}\)	A ⇔ 3

As alpha and Omega can be assigned numerical values (α = 137.035999139, Ω = 2.0071349496), so too the MLTA objects can be expressed numerically. We can then convert these objects to their Planck unit equivalents by including a dimensioned scalar. For example,

\[ V=2\pi \Omega ^{2}\]

= 25.3123819353... and so we can use scalar v to convert from dimensionless geometrical object V to dimensioned c.

scalar vSI = 11843707.905 m/s gives c = V*vSI = 25.3123819 * 11843707.905 m/s = 299792458 m/s (SI units)

scalar vimp = 7359.3232155 miles/s gives c = V*vimp = 186282 miles/s (imperial units)

Scalars

attribute	geometrical object	scalar (unit number)
mass	\(\displaystyle M=(1)\)	k (θ = 15)
time	\(\displaystyle T=(\pi )\)	t (θ = -30)
velocity	\(\displaystyle V=(2\pi \Omega ^{2})\)	v (θ = 17)
length	\(\displaystyle L=(2\pi ^{2}\Omega ^{2})\)	l (θ = -13)
ampere	\(\displaystyle A=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha _{inv}}})\)	a (θ = 3)

As the scalar incorporates the dimension quantity (the dimension quantity for v = m/s or miles/s), the unit number relationship (θ) applies, and so we then find that only 2 scalars are needed. This is because in a defined ratio they will overlap and cancel, for example in the following ratios;

scalar units for ampere a = u3, length l = u-13, time t = u-30, mass k = u15 (uΘ represents unit)

\[ {\frac {({u^{3}})^{3}{(u^{-13}})^{3}}{(u^{-30})}}={\frac {{(u^{-13})}^{15}}{{(u^{15})}^{9}{(u^{-30})}^{11}}}=1\]

For example if we know the numerical values for a and l then we know the numerical value for t, and from l and t we know k … and so if we know any 2 scalars (α and Ω have fixed values) then we can solve the Planck units (for that system of units), and from these, we can solve (G, h, c, e, me, kB).

\[ {\frac {a^{3}l^{3}}{t}}={\frac {m^{15}}{k^{9}t^{11}}}=1\]

and so

\[ {a^{3}l^{3}}=t\]

and

\[ m^{15}}={k^{9}t^{11}\]

In this table the 2 scalars used are r (θ = 8) which is related to momentum, and v (θ = 17). A further attribute is included, P = the square root of (Planck) momentum. This gives us 3 primary (Planck) units MTP; L, V and A can thus be considered composite objects.

Geometrical objects

attribute	geometrical object	unit number θ	scalar r(8), v(17)
mass	\(\displaystyle M=(1)\)	15 = 8*4-17	\(\displaystyle k={\frac {r^{4}}{v}}\)
time	\(\displaystyle T=(\pi )\)	-30 = 8*9-17*6	\(\displaystyle t={\frac {r^{9}}{v^{6}}}\)
sqrt(momentum)	\(\displaystyle P=(\Omega )\)	16 = 8*2	r2
velocity	\(\displaystyle V={\frac {2\pi P^{2}}{M}}=(2\pi \Omega ^{2})\)	17	v
length	\(\displaystyle L=VT=(2\pi ^{2}\Omega ^{2})\)	-13 = 8*9-17*5	\(\displaystyle l={\frac {r^{9}}{v^{5}}}\)
ampere	\(\displaystyle A={\frac {2^{4}V^{3}}{\alpha _{inv}P^{3}}}=({\frac {2^{7}\pi ^{3}\Omega ^{3}}{\alpha _{inv}}})\)	3 = 17*3-8*6	\(\displaystyle a={\frac {v^{3}}{r^{6}}}\)

Further information: User:Platos Cave (physics)/Simulation_Hypothesis/Planck units (geometrical) § Scalars

Mathematical electron

The mathematical electron formula ψ incorporates the dimensioned Planck units but itself is dimension-less (units = scalars = 1). Here ψ is defined in terms of σe, where AL is an ampere-meter (ampere-length = e*c are the units for a magnetic monopole).

\[ T=\pi ,\;unit=u^{-30},\;scalars={\frac {r^{9}}{v^{6}}}\]

\[ \sigma _{e}={\frac {3\alpha _{inv}^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha _{inv}\Omega ^{5}},\;unit=u^{(3\;-13\;=\;-10)},\;scalars={\frac {r^{3}}{v^{2}}}\]

\[ \psi ={\frac {\sigma _{e}^{3}}{2T}}={\frac {(2^{7}3\pi ^{3}\alpha _{inv}\Omega ^{5})^{3}}{2\pi }},\;unit={\frac {(u^{-10})^{3}}{u^{-30}}}=1,scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1\]

\[ \psi =4\pi ^{2}(2^{6}3\pi ^{2}\alpha _{inv}\Omega ^{5})^{3}=.23895453...x10^{23},\;unit=1\]

(unit-less)

Both units and scalars cancel.

Electron parameters

We can solve the electron parameters; electron mass, wavelength, frequency, charge ... as the frequency of the Planck units themselves, and this frequency is ψ.

\[ v=11843707.905...,\;units={\frac {m}{s}}\]

\[ r=0.712562514304...,\;units=({\frac {kg.m}{s}})^{1/4}\]

electron wavelength λe = 2.4263102367e-12m (CODATA 2014)

\[ \lambda _{e}^{*}=2\pi L\psi\]

= 2.4263102386e-12m (L ⇔ Planck length)

electron mass me = 9.10938356e-31kg (CODATA 2014)

\[ m_{e}^{*}={\frac {M}{\psi }}\]

= 9.1093823211e-31kg (M ⇔ Planck mass)

elementary charge e = 1.6021766208e-19C (CODATA 2014)

\[ e^{*}=A\;T\]

= 1.6021765130e-19 (T ⇔ Planck time)

Rydberg constant R = 10973731.568508/m (CODATA 2014)

\[ R^{*}=({\frac {m_{e}}{4\pi L\alpha _{inv}^{2}M}})={\frac {1}{2^{23}3^{3}\pi ^{11}\alpha _{inv}^{5}\Omega ^{17}}}{\frac {v^{5}}{r^{9}}}\;u^{13}\]

= 10973731.568508

From the above formulas, we see that wavelength is ψ units of Planck length, frequency is ψ units of Planck time ... however the electron mass is only 1 unit of Planck mass.

Further information: User:Platos Cave (physics)/Simulation_Hypothesis/Physical_constant_(anomaly)

Electron Mass

Particle mass is a unit of Planck mass that occurs only once per ψ units of Planck time, the other parameters are continuums of the Planck units.

units

\[ \psi ={\frac {(AL)^{3}}{T}}\]

= 1

This may be interpreted as; for ψ units of Planck time the electron has wavelength L, charge A ... and then the AL combine with time T (A3L3/T) and the units (and scalars) cancel. The electron is now mass (for 1 unit of Planck time). In this consideration, the electron is an event that oscillates over time between an electric wave state (duration ψ units of Planck time) to a unit of Planck mass point state (1 unit of Planck time). The electron is a quantum scale event, it does not exist at the discrete Planck scale (and so therefore neither does the quantum scale).

As electron mass is the frequency of the geometrical Planck mass M = 1, which is a point (and so with point co-ordinates), then we have a model for a black-hole electron, the electron function ψ centered around this unit of Planck mass. When the wave-state (A*L)3/T units collapse, this black-hole center (point) is exposed for 1 unit of (Planck) time. The electron is 'now' (a unit of Planck) mass.

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 over time. The formula E = hf is a measure of the frequency f of occurrence of Planck's constant h and applies to the electric wave-state. As for each wave-state there is a corresponding mass point-state, then for a particle hf == mc2. Notably however the c term is a fixed constant unlike the f term, and so the m term is the frequency term, it is referring to an average mass (mass which is measured over time) rather than a constant mass (mass as a constant property of the particle at unit Planck time). Thus as noted, when we refer to mass as a constant property, we are referring to average mass at the quantum scale, and the electron as a quantum-state particle.

If the scaffolding of the universe includes units of Planck mass M, then it is not necessary for a particle itself to have mass, what we define as electron mass could be the absence of electron [2].

Quarks and Spin

The charge on the electron derives from the embedded ampere A and length L, the electron formula ψ itself is dimensionless. These AL magnetic monopoles would seem to be analogous to quarks (there are 3 monopoles per electron), but due to the symmetry and so stability of the geometrical ψ 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 embedded within the electron suffice [3].

Quarks

Electron formula

\[ \psi =2^{20}\pi ^{8}3^{3}\alpha _{inv}^{3}\Omega ^{15},\;unit=1,scalars=1\]

Time

\[ T=\pi {\frac {r^{9}}{v^{6}}},\;u^{-30}\]

AL magnetic monopole

\[ \sigma _{e}={\frac {3\alpha _{inv}^{2}AL}{2\pi ^{2}}}={2^{7}3\pi ^{3}\alpha _{inv}\Omega ^{5}},\;u^{-10},\;scalars={\frac {r^{3}}{v^{2}}}\]

\[ \psi ={\frac {\sigma _{e}^{3}}{2T}}={\frac {(2^{7}3\pi ^{3}\alpha _{inv}\Omega ^{5})^{3}}{2\pi }}=2^{20}3^{3}\pi ^{8}\alpha _{inv}^{3}\Omega ^{15},\;unit={\frac {(u^{-10})^{3}}{u^{-30}}}=1,scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1\]

If

\[ \sigma _{e}\]

could equate to a quark with an electric charge of -1/3e, then it would be an analogue of the D quark. 3 of these D quarks would constitute the electron as DDD = (AL)*(AL)*(AL).

We would assume that the charge on the positron (anti-matter electron) is just the inverse of the above, however there is 1 problem, the AL (A; θ=3, L; θ=-13) units = -10, and if we look at the table of constants, there is no 'units = +10' combination that can include A. We cannot make an inverse electron. However we can make a Planck temperature Tp AV monopole (ampere-velocity).

\[ T_{p}={\frac {2^{7}\pi ^{3}\Omega ^{5}}{\alpha _{inv}}},\;u^{20},\;scalars={\frac {r^{9}}{v^{6}}}\]

\[ \sigma _{t}={\frac {3\alpha _{inv}^{2}T_{p}}{2\pi }}={\frac {3\alpha _{inv}^{2}AV}{2\pi ^{2}}}=({2^{6}3\pi ^{2}\alpha _{inv}\Omega ^{5}}),\;u^{20},\;scalars={\frac {v^{4}}{r^{6}}}\]

\[ \psi =(2T)\sigma _{t}^{2}\sigma _{e}=2^{20}3^{3}\pi ^{8}\alpha _{inv}^{3}\Omega ^{15},\;unit=(u^{-30})(u^{20})^{2}(u^{-10})=1,scalars=({\frac {r^{9}}{v^{6}}})({\frac {v^{4}}{r^{6}}})^{2}{\frac {r^{3}}{v^{2}}}=1\]

The units for

\[ \sigma _{t}\]

= +20, and so if units = -10 equates to -1/3e, then we may conjecture that units = +20 equates to 2/3e, which would be the analogue of the U quark. Our plus charge now becomes DUU, and so although the positron has the same wavelength, frequency, mass and charge magnitude as the electron (both solve to ψ), internally its charge structure resembles that of the proton, the positron is not simply an inverse of the electron. This could have implications for the missing anti-matter, and for why the charge magnitude of the proton is exactly the charge magnitude of the electron.

\[ D=\sigma _{e},\;unit=u^{-10},\;charge={\frac {-1e}{3}},\;scalars={\frac {r^{3}}{v^{2}}}\]

\[ U=\sigma _{t},\;unit=u^{20},\;charge={\frac {2e}{3}},\;scalars={\frac {v^{4}}{r^{6}}}\]

Numerically:

Adding a proton and electron gives (proton) UUD & DDD (electron) = 2(UDD) = 20 -10 -10 = 0 (zero charge), scalars = 0.

Converting between U and D via U & DDD (electron) = 20 -10 -10 -10 = -10 (D), scalars =

\[ \frac {r^{3}}{v^{2}}\]

Spin

Relativity at the Planck scale can be described by a translation between 2 co-ordinate systems; an expanding (in Planck steps at the speed of light) 4-axis hyper-sphere projecting onto a 3-D space [4]. In this scenario, particles (with mass) are pulled along by the expansion of the hyper-sphere, this then requires particles to have an axis; generically labeled N-S, with the N denoting the direction of particle travel within the hyper-sphere. Changing the direction of travel involves changing the orientation of the particle N-S axis. We can link that external N--S axis to the internal monopole (DDD) geometry, and from this show how the three internal phases produce the spin-½ transformation law under spatial rotations about the N-S direction.

Geometry coded universe

Modelling a Planck scale simulation universe using geometrical forms. For AI analysis, the following links have been compiled into a single file [5].

• User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical): Planck units MLTPA as geometrical objects
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly): Anomalies within the physical constants
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals: Gravity as a function of atomic orbitals
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity: Relativity as a translation between 2 co-ordinate systems
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding: CMB and a Planck unit universe scaffolding
• User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum: Link between charge and mass
• User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer): Introduction to a Planck scale Programmer God Simulation Hypothesis model

External links

• Programming at the Planck scale using geometrical objects -Malcolm Macleod's website
• Simulation Argument -Nick Bostrom's website
• Our Mathematical Universe: My Quest for the Ultimate Nature of Reality -Max Tegmark (Book)
• Dirac-Kerr-Newman black-hole electron -Alexander Burinskii (article)
• The Matrix, (1999)
• Pythagoras "all is number" - Stanford University
• Simulation Hypothesis
• Mathematical universe hypothesis
• Philosophy of mathematics
• Philosophy of physics
• Platonism

References

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.
2. ↑ Macleod, Malcolm J.; "1. Planck unit scaffolding correlates with the Cosmic Microwave Background". SSRN. Feb 2011. doi:10.2139/ssrn.3333513.
3. ↑ Macleod, M.J. "7. Geometric Origin of Quarks and Spin, the Mathematical Electron extended". RG. doi:/10.13140/RG.2.2.21695.16808.
4. ↑ Macleod, M.J. "2. Relativity as the mathematics of perspective in a hyper-sphere universe". SSRN. doi:10.2139/ssrn.3334282.
5. ↑ https://codingthecosmos.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf Compilation of journal articles 1-7