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Calculation of PlanckMass:

In the D4-D5-E6-E7 model, a Planck-massblack hole is not a tree-level classical particle such as an electronor a quark, but a quantum entity resulting from the Many-Worldsquantum sum over histories at a single point in spacetime.

Consider an isolated single point, or vertex in the latticepicture of spacetime. In the D4-D5-E6-E7 model, fermions live onvertices, and only first-generation fermions can live on a singlevertex. (The second-generation fermions live on two vertices that actat our energy levels very much like one, and The third-generationfermions live on three vertices that act at our energy levels verymuch like one.)

At a single spacetime vertex, a Planck-mass black hole is theMany-Worlds quantum sum of all possible virtual first-generationparticle-antiparticle fermion pairs permitted by the Pauli exclusionprinciple to live on that vertex.

Once a Planck-mass black hole is formed, it is stable in theD4-D5-E6-E7 model. Less mass would not be gravitationally bound atthe vertex. More mass at the vertex would decay by Hawkingradiation.

In the D4-D5-E6-E7 model, a Planck-mass black hole can be formedin two ways: either as the end product of Hawking radiation decay ofa larger black hole; or

by vacuum fluctuation, creating either a graviton4-pair virtual Planck-mass black hole or as part of a cosmologicalquantum conformal fluctuation that could create a newuniverse.

Since Dirac fermions in 4-dimensional spacetime can be massive(and are massive at low enough energies for the Higgs mechanism toact), the Planck mass in 4-dimensional spacetime is the sum of massesof all possible virtual first-generation particle-antiparticlefermion pairs permitted by the Pauli exclusion principle.

There are 8 fermion particles and 8 fermion antiparticles for atotal of 64 particle-antiparticle pairs. A typical combination shouldhave several quarks, several antiquarks, a few colorlessquark-antiquark pairs that would be equivalent to pions, and someleptons and antileptons.

Due to the Pauli exclusion principle, no fermion lepton or quarkcould be present at the vertex more than twice unless they are in theform of boson pions, colorless first-generation quark-antiquark pairsnot subject to the Pauli exclusion principle. Of the 64particle-antiparticle pairs, 12 are pions.

A typical combination should have about 6 pions.

If all the pions are independent, the typical combination shouldhave a mass of .14x6 GeV = 0.84 GeV. However, just as the pion massof .14 GeV is less than the sum of the masses of a quark and anantiquark, pairs of oppositely charged pions may form a bound stateof less mass than the sum of two pion masses. If such a bound stateof oppositely charged pions has a mass as small as .1 GeV, and if thetypical combination has one such pair and 4 other pions, then thetypical combination should have a mass in the range of 0.66 GeV.

Summing over all 2^64combinations, the total mass of a one-vertex universe should givemPlanck = 1.217-1.550 x 10^19 GeV.

Since each fermion particle has a corresponding antiparticle, aPlanck-mass Black Hole is neutral with respect to electric and colorcharges.

The value for the Planck mass given in the ParticleData Group's 1998 review is 1.221 x 10^19 GeV.


Perhaps a Planck-Mass BlackHole could be built using a pion laser.

The combinatorial basis of the Planck mass is related to theI Ching.


Aether, Gravity, and the Higgs Mechanism:


The Higgs Mechanism gives mass to theWeak Force gauge bosons, so that

the Weak Force has a MassFactor 1 / MW^2 = 1 / (mW+^2 +mW-^2 + mW0^2) = 1 / (150GeV)^2.

The Weak Force Mass Factor, abour 150 GeV, is determined by theVacuum Expectation Value ofthe Higgs Mechanism, about 250 GeV,

which also gives the linearcompressibility of the Aether.

The Mass Me of the smallest charged Elementary Particle, theFirst-Generation Fermion Electron ComptonRadius Vortex Particle, is about 0.5 MeV, so that

the Higgs VEV is about 250 x 2,000 Me = 5 x 10^5 Me.

If the Higgs VEV gives the linear compressibility of the Aether,the Gravitational VEV should be given by the 4-volumecompressibility of the Aether, so that the Gravitational VEV isabout ( 5 x 10^5 )^4 Me = 6 x 10^22 Me = 3 x 10^22 MeV = 3 x 10^19GeV.

Since the Gravitational VEV shouldcorrespond to a pair of Planck Mass Black Holes,

the Planck Mass is about 1.5 x 10^19 GeV.


The value for the Planck mass given in the ParticleData Group's 1998 review is 1.221 x 10^19 GeV.



Both physical Spacetime (MinkowskiRP1xS3 and Euclidean S4) and Internal Symmetry Space (CP2)have

Quaternionic structure.

From a discrete lattice point of view, the Quaternionic structureis represented by Integral Quaternions.

Integral Quaternions have a D4 lattice structure. Using the squarenorm of distance from the origin, the numberof vertices in each shell of the D4 lattice can be calculated. Ifyou look at shells whose square norm is a power of 2, you see thatall of them have exactly 24 vertices. Here are the number of verticesin a few of the layers of a D4 lattice:

Square Norm of Layer       Number of Vertices         1                          24         2                          24         3                          96         4                          24         5                         144         6                          96         7                         192         8                          24         9                         312         ...                       ...       127                       3,072       128                          24

The powers of 2 are adjacent to the Mersenne Primes, of the form2^k - 1 for prime k.

There is a sequence of Mersenne Primes of at least length 4:

 Power of 2                     Mersenne Prime  2^2 = 4                      2^2 - 1 = 4-1 = 3  2^3 = 8                      2^3 - 1 = 8-1 = 7  2^7 = 128                    2^7 - 1 = 128-1 = 1272^127 = about 1.7 x 10^38    2^127 - 1 = about 1.7 x 10^38

(It is not known whether or not 2^(2^127 - 1) - 1 is prime.)

MattiPitkanen has studied relationships between PrimeNumbers and Physics. He has noticed that, since the PrimeQuaternions are the Integral Quaternions whose square norm isrational prime, and since the ordinary distance is the square root ofthe square norm,

if you look at the fourth Mersenne Prime in the sequence, M127 =2^127 - 1 = about 1.7 x 10^38, and the power of 2 to which it isadjacent, 2^127 = also about 1.7 x 10^38,

you see that its ordinary distance from the origin, the squareroot of the square norm,

is about 1.3 x 10^19, which is a good estimate of theratio of the Planck mass to the Proton mass, and is about 1.22x 10^19 GeV.

The value for the Planck mass given in the ParticleData Group's 1998 review is 1.221 x 10^19 GeV.

I don't have a clear physical intuition as to why 2^127should be the square of the ratio of the Planck mass to the Protonmass, but I note that:

Maybe the Mersenne Prime M127 and its adjacent power of 2set the relative scale between Spacetime and Internal SymmetrySpace,

with the 24 vertices of the Quaternionic D4 shell at2^(128 - 1) of Internal Symmetry Space corresponding to the 24vertices of the first Quaternionic D4 shell at 2^0 = 1 ofSpacetime.


The Planck Mass, about 10^(-5) grams, or about 10^19GeV, or about 10^22 MeV,

is about 10^22 times

the Mass of the smallest charged Elementary Particle, theFirst-Generation Fermion Electron Particle, about 0.5MeV.

For clarity of discussion, I am considering order-of-magnitudeestimates and ignoring factors like 2 in considering 0.5 MeV to beabout 1 MeV, etc., so I note that

10^22 is about 2^72 = 2^(3x24) = (2^3)^24 = 8^24= 8^(8x3) = (8^8)^3.

Is there any significance to these numbers?

As to 72:

In the D4-D5-E6-E7 physics model,the Lie algebra E6 describes the Gauge Bosons, the SpaceTime, and theFermion Particles and Antiparticles. E6is 78-dimensional, and can be constructed from a 72-vertex E6Root Vector Diagram in 6-dimensional Euclidean space.


As to 24:

In the D4-D5-E6-E7 physics model,you can see that E6 can be constructedusing the traceless Jordan Algebra J3(O)o. In the notation that Iuse in the 3x3 matrix representation below, O1, O2, and O3 are threeOctonions.

Re(O4) O1 O2 O1* Re(O5) O3 O2* O3* -

The off-diagonal dimensionality of J3(O)o is 24, thedimensionality of the LeechLattice. that is J3(O)o itself is 26-dimensional, as isthe Lorentx Leech Lattice. Suchlattices are useful in building codesand physics models.


As to 8^8:

8^8 is the set of all maps from 8 things to 8 things, and so isrelated to the Reflexivity ofOctonions.


As to (8^8)^3:

Since 8^8 is related to the Reflexivityof Octonions and three Octionions (off-diagonal matrix elementsof the traceless Jordan AlgebraJ3(O)o) are fundamental to the D4-D5-E6-E7physics model,

the cube of 8^8 is related to both,

which may be its connection to the ratio of the Planck Mass to theElectron Mass.


Could (8^8)^3 be more Universal?


Our Universe has Large-ScaleStructure that looks like a set of superimposed egg-carton structureswith the Octonionic structure of nested Onarhedrallattices.

Saul-Paul Sirag has noted that:

the Mass of the of the Universe, about 10^80 hydrogenatoms, or about 10^80 GeV, or about 10^61 Planck Masses, or about10^56 grams;

is about 10^22 = (8^8)^3 times

the Mass of the smallest Black Hole Star, about 10 solarmasses, or about 10^34 grams.



Three Octionions ( off-diagonal matrix elements of thetraceless Jordan Algebra J3(O)o ) are fundamental to theD4-D5-E6-E7 physics model.

The 24-dimensional Leechlattice

can be regarded as a 3-dimensional Octonion lattice, with 720 ofits 3x240 + 3x16x240 + 3x16x16x240 = 196,560 units regarded as threesets of 240 vertices of three E8 lattices.

In the D4-D5-E6-E7 physics model,the 3 Octonion dimensions correspond to

Split the 8-dimensional spacetime into 1 time dimension and 7space dimensions.

Then, for any Timeline, you have

for a total of 23 space-fermion dimensions.

The lowest energy state for any Single tTmeline would have a zerotree-level mass neutrino (antineutrino) fermion particle(antiparticle).

Since second-generation and third-generation fermions decay, inthe D4-D5-E6-E7 physics model, intofirst-generation fermions by Standard Model processes; and since, inthe D4-D5-E6-E7 physics model,first-generation quarks decay intoelectrons or positrons plus neutinos or antineutrinos by VirtualBlack Holes; therefore: in the D4-D5-E6-E7physics model, the lowest energy fermion particle (antiparticle)is the electron (positron), which is stable, and the lowest non-zeroenergy state for any Single Timeline would have an electron(positron) fermion particle (antiparticle).

If a Single Timeline were Islolated from the rest of theuniverse,

so that nothing could decay from it, then any of the 23space-fermion dimensions could play the role of any spatial dimensionor any fermion particle or antiparticle dimension. Therefore

the Single Timeline could support a superposition of allstates represented by all 23! permutations of the 23 space-fermiondimensions


each such state could have the stable minimum non-zero mass-energyof an electron or positron

so that

the mass-energy of the Isolated Single Timeline would be23! me,

where me is the electron mass. Since 23! = 2.585 x 10^22 and me =0.51 x 10^(-3) GeV,

the mass-energy of the Isolated Single Timeline would be1.3184 x 10^19 GeV,


an Isolated Single Timeline represents a Planck massBlack Hole,

which is roughly consistent with the value for the Planck massgiven in the Particle Data Group's1998 review of 1.221 x 10^19 GeV.

Since each fermion particle has a corresponding antiparticle, sucha Planck-mass Black Hole is neutral with respect to electric andcolor charges because opposite charges would be represented equallyin the superposition.


The idea of looking at the number 23! came to me from studying thetopological model ofMarco Spaans. I do not necessarily agree with his physicalinterpretations, but he did use 23! as the ratio between the Planckmass and a mass related to the electron mass.



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