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APPENDIX 1: Pregeometry and F4.

"'Physics as manifestation of logic' or 'pregeometry as the calculus of propositions' is ... an idea for an idea." is proposed by Misner, Thorne, and Wheeler in their classic book Gravitation55 (p.1212). It is possible to build up Boolean set theory, classical logic, and distributive lattices from the null set 0 and the bracket operation { } forming the set containing a preexisting set:

0; {0}; {0,{0}}; {{0}}; ...

If the depth of brackets used is called the rank N, there are

2^...(N-1 arrows)...^2^2 sets of rank at most N. Even though there are only two rank 1 sets {0} and 0, a lot (2^2^2^2^2 = 265,536) of sets exist at the rank 5 level of construction of set theory. From this point of view, sets of points are fundamental and the logic is classical.

A different viewpoint that leads to the F4 model is to start with the empty set 0 as before, but to consider the other sets, starting with {0}, as rays rather than points. The ray {0} is the half-line going from the origin 0 in the direction {0}.

The rank 0 stage has only the origin 0.

The rank 1 stage has only the ray from 0 in the direction {0}.

The rank 2 stage has the ray in the direction {0} and the ray in the direction {{0}}. The rank 2 element {0,{0}} is a ray from the origin 0 to the ray from 0 to {0}, and is therefore equivalent to {0}.

The rank 3 stage has the ray in the direction {{{0}}} in addition to the rays in the directions {0} and {{0}}.

The rank 3 element {{0},{{0}}} is the ray space, or vector space, spanned by the rays in the independent directions {0} and {{0}}.

To compare independent rays such as the rays in the {0} and {{0}} directions, define xthem to be orthogonal to each other and by defining a point on each ray as the unit vector in that direction. Then define in {{0},{{0}}} a set of unit rays, the coherent superpositions of the states of the rays {0} and {{0}}, by marking a unit ray on each ray in the ray space. The set of unit rays is a circle S1, or norm 1 elements of the complex numbers C, or the interval [0, 2] of the real numbers R with end points identified.

It is at this point that the concept of nondenumerable real numbers R appears, and, R having been introduced, rays can be identified with real half-lines R+ in particular directions.

The rank 2 rays {0} and {{0}} are transformed into each other, and into any of their coherent superpositions {{0},{{0}}}, by the transformation group Spin(2) = U(1).

Higher rank elements are constructed in the same manner.

The rank N stage has N independent rays, in the directions

{0}, {{0}}, ... , { ...N braces... {0} ... }.

The transformation group of the rank N rays is Spin(N).

Spin(N) is represented by the Yang Hui triangle of the binomial expansion of (1 + 1)^N :

Rank N:

0 1

1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1

8 1 8   28 56 70 56 28 8 1

N 1 N N(N-1)/2 ...

The vector representation of Spin(N) has dimension N, the adjoint representation has dimension N(N-1)/2, and the spinor representation has dimension (1+1)^N if N is even and (1+1)^(N-1) if N is odd. For even N, the spinor representation reduces into two irreducible mirror image half-spinor representations.

The complex numbers C are defined from the unit complex numbers by Spin(2) = U(1) = S1.

The quaternions H are defined from the unit quaternions by

Spin(3)= SU(2) = Sp(1) = S3.

The octonions O are defined from the unit octonions S7, but S7 is not a Lie group. Octonionic structure is related to Spin(8), whose adjoint, vector, and two half-spinor representations can be combined at the Lie algebra level to form the exceptional Lie algebra F4 that is the basis of the F4 model.

 

The exceptional nature of Spin(8) is due to the triality automorphisms among the 8-dimensional vector, +half-spinor, and -half-spinor representations.

For N > 8, the spinor and half-spinor representations of Spin(N) are larger than the vector representations.

For N < 8, the only isomorphisms between vector representations and spinor representations are trivial ones for Spin(0) and for Spin(1) or between vector representations and reducible full spinor representations for:

Spin(2), related to the complex numbers C and the 2-dimensional Feynman checkerboard and Dirac equation, and for

Spin(4), related to the quaternions H, the 4-dimensional Dirac equation, and 4-dimensional self-dual (and anti-self-dual) Yang-Mills structures.

 

From my viewpoint, the exceptional nature of the transformation group Spin(8) of the rank 8 rays is the reason for building the F4 model.

 

The approach of building everything from 0, or nothing, has been around at least since the Tao Te Ching, and the exceptional structure of Spin(8) is reflected in the such traditions as the I Ching, the Hannukah Menorah, and the game of Wei-Ch'i. Whether or not these traditions indicate a predisposition in human thought toward representing physics by the F4 model, I think they are interesting enough to be discussed in the next few pages.

 

The Yang Hui triangle of the binomial expansion of (1 + 1)^N was discovered by Yang Hui in China about 1261 A.D.

Blaise Pascal rediscovered it in Europe about 1650 A.D.

 

The Yang Hui triangle starts at N = 0 , representing Spin(0), or the origin, or the empty set 0.

 

About 500 B.C., Lao-Tze wrote in the Tao Te Ching:

 

40: "existence was produced from nonexistence"; and

 

42: "The Tao produced the One. The One produced the Two.

The Two produced the Three. The Three produced All Things."

 

The Yang Hui triangle at N = 8 represents Spin(8):

1 7 21 35 35 21 7 1

1 8   28 56 70 56 28 8 1

 

The Hanukka Menorah has 8 + 1 = 9 candles, and is derived from the 7-candle Menorah of the Tabernacle of Moses about 1300 BC.

The 7-candle Menorah represents the S¶ [ S7 ] of the unit octonions. The 8 + 1 candles of the Hanukka Menorah represent the 8-dimensional vector representation of Spin(8) and the 1-dimensional scalar representation of Spin(8).

On the kth night of each of the 8 nights of Hanukka, 1 + k candles are burned. At the 8th night, 1+8 candles are burned, and 7+28 candles have been burned. 1+8+7+28 = 44 candles are burned in all.

 

The I Ching is based on the yin line - - and yang line -- , and the trigrams of 3 lines. There are 23 = 8 trigrams, and they correspond to octonions:

1 = 3 -- ;

i,j,k = 2 -- + 1 - - ;

ie,je,ke = 1 -- + 2 - - ; and

e = 3 - - .

They can be generated by casting 3 coins. The trigrams are said to have been originated by Fu Hsi about 2400 B.C..

Conventionally, pairs of trigrams are used to create 64 hexagrams of 6 lines each. Of the 64 hexagrams, 8 are made up of the same trigram taken twice. Of the remaining 56, each of 28 pairs of trigrams are the inverse of one of the other 28 pairs, being made up of the same two trigrams in the inverse order. The hexagrams are said to have been originated by Wen Wang about 1200 B.C..

I prefer to consider all the 28 = 1+8+28+56+70+56+28+8+1 = 256 subsets of the 8 trigrams.

The 28 = 8x7/2 pair subsets of the trigrams correspond to the 28-dimensional adjoint representation of Spin(8). The 28 = 24+4 pair subsets correspond to the 24 vertices of a 24-cell and the 4 dimensions of the 4-dimensional space of a 24-cell.

The 8 singleton subsets of the trigrams, the 8 trigrams, correspond to the 8-dimensional vector representation of Spin(8).

Since 16 = 256, all 256 subsets of the 8 trigrams can be represented by a 16 x 16 matrix, and the Clifford algebra Cl(8) of Spin(8) is the 16 x 16 real matrix algebra R(16). A minimal left ideal of Cl(8) is a 16 x 1 column vector, which is reducible into two 8 x 1 column vectors. A minimal right ideal of Cl(8) is a 1 x 16 row vector, which is reducible into two 8 x 1 row vectors.

The Spin(8) Lie algebra is the rank two part of the even subalgebra of Cl(8), under the [,] product. The even subalgebra of Cl(8) has dimension 1+28+70+28+1 = 128 = 64+64 = 8 x 8 + 8 x 8.

 

The game of Go originated in China as Wei-Ch'i about 2356 B.C.. Go is played with white or black stones placed at vertices of a square 19 x 19 lattice, effectively a 2-dimensional Feynman checkerboard, although the Go rules of play are not equivalent. There are 17 x 17 interior vertices, large enough for the two diagonal 8 x 8 blocks of the even Spin(8) Clifford algebra, the two off-diagonal 8 x 8 blocks of the odd Spin(8) Clifford algebra, a 1 x 17 separating row, and a 17 x 1 separating column.

 

 


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