# Calculation of Planck Mass:

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

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

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

Once a Planck-mass black hole is formed, it is stable in the D4-D5-E6-E7 model. Less mass would not be gravitationally bound at the vertex. More mass at the vertex would decay by Hawking radiation.

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

by vacuum fluctuation, creating either a graviton 4-pair virtual Planck-mass black hole or as part of a cosmological quantum conformal fluctuation that could create a new universe.

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

There are 8 fermion particles and 8 fermion antiparticles for a total of 64 particle-antiparticle pairs. A typical combination should have several quarks, several antiquarks, a few colorless quark-antiquark pairs that would be equivalent to pions, and some leptons and antileptons.

Due to the Pauli exclusion principle, no fermion lepton or quark could be present at the vertex more than twice unless they are in the form of boson pions, colorless first-generation quark-antiquark pairs not subject to the Pauli exclusion principle. Of the 64 particle-antiparticle pairs, 12 are pions.

A typical combination should have about 6 pions.

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

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

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

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

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

## Aether, Gravity, and the Higgs Mechanism:

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

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

The Weak Force Mass Factor, abour 150 GeV, is determined by the Vacuum Expectation Value of the Higgs Mechanism, about 250 GeV,

which also gives the linear compressibility of the Aether.

The Mass Me of the smallest charged Elementary Particle, the First-Generation Fermion Electron Compton Radius 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-volume compressibility of the Aether, so that the Gravitational VEV is about ( 5 x 10^5 )^4 Me = 6 x 10^22 Me = 3 x 10^22 MeV = 3 x 10^19 GeV.

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

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

## Quaternionic structure.

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

Integral Quaternions have a D4 lattice structure. Using the square norm of distance from the origin, the number of vertices in each shell of the D4 lattice can be calculated. If you look at shells whose square norm is a power of 2, you see that all of them have exactly 24 vertices. Here are the number of vertices in 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 form 2^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 = 127
2^127 = 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.)

Matti Pitkanen has studied relationships between Prime Numbers and Physics. He has noticed that, since the Prime Quaternions are the Integral Quaternions whose square norm is rational prime, and since the ordinary distance is the square root of the 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 is adjacent, 2^127 = also about 1.7 x 10^38,

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

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

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

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

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

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

For clarity of discussion, I am considering order-of-magnitude estimates and ignoring factors like 2 in considering 0.5 MeV to be about 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 the Fermion Particles and Antiparticles. E6 is 78-dimensional, and can be constructed from a 72-vertex E6 Root 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 constructed using the traceless Jordan Algebra J3(O)o. In the notation that I use in the 3x3 matrix representation below, O1, O2, and O3 are three Octonions.

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

The off-diagonal dimensionality of J3(O)o is 24, the dimensionality of the Leech Lattice. that is J3(O)o itself is 26-dimensional, as is the Lorentx Leech Lattice. Such lattices are useful in building codes and physics models.

As to 8^8:

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

As to (8^8)^3:

Since 8^8 is related to the Reflexivity of Octonions and three Octionions (off-diagonal matrix elements of the traceless Jordan Algebra J3(O)o) are fundamental to the D4-D5-E6-E7 physics model,

the cube of 8^8 is related to both,

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

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

Our Universe has Large-Scale Structure that looks like a set of superimposed egg-carton structures with the Octonionic structure of nested Onarhedral lattices.

Saul-Paul Sirag has noted that:

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

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

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

Three Octionions ( off-diagonal matrix elements of the traceless Jordan Algebra J3(O)o ) are fundamental to the D4-D5-E6-E7 physics model.

## The 24-dimensional Leech lattice

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

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

• 8-dim spacetime plus
• 8 first generation fermion particles plus
• 8 first generation fermion antiparticles.

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

Then, for any Timeline, you have

• 7 space dimensions plus
• 8 fermion particle dimensions plus
• 8 fermion antiparticle dimensions

for a total of 23 space-fermion dimensions.

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

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

### If a Single Timeline were Islolated from the rest of the universe,

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

### the Single Timeline could support a superposition of all states represented by all 23! permutations of the 23 space-fermion dimensions

and

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

so that

### the mass-energy of the Isolated Single Timeline would be 23! me,

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

and

### an Isolated Single Timeline represents a Planck mass Black Hole,

which is roughly consistent with the value for the Planck mass given in the Particle Data Group's 1998 review of 1.221 x 10^19 GeV.

Since each fermion particle has a corresponding antiparticle, such a Planck-mass Black Hole is neutral with respect to electric and color charges because opposite charges would be represented equally in the superposition.

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