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E8 and D4 Lattices:

Below the Planck energy, the F4 model ceases to have a continuum 8-dimensional spacetime, and acquires a Planck-length lattice structure. The 8-dimensional E8 lattice18,19 has octonionic structure, but no nearest neighbor light-cone links. Therefore, the E8 lattice is reduced to a Planck-length 4-dimensional D4 lattice18,19 with quaternionic structure and nearest neighbor light-cone links. To see more details about how this works, this Appendix 3 describes the E8 and D4 lattices and applies the D4 lattice with quaternionic structure to a physically realistic generalization of the Feynman checkerboard49,50.

The E8 Lattice:

Begin with an 8-dimensional spacetime R8 = O, where a basis for O is {1,i,j,k,e,ie,je,ke} . The vertices of the E8 lattice are of the form

(a01 + a1e + a2i + a3j + a4ie + a5ke + a6k + a7je)/2 ,

where the ai may be either all even integers, all odd integers, four even integers, or four odd integers, with residues mod 2 in the four-integer cases being (1;0,0,0,1,1,0,1) or (0;1,1,1,0,0,1,0) or the same with the last seven cyclically permuted. E8 forms an integral domain of integral octonions.

The E8 lattice integral domain has 240 units:

±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1±ie±je±k e)/2, (±e±i±j±k)/2, and the last two with cyclical permutations of {i,j,k,e,ie,je,ke} in the order (e, i, j, ie, ke, k, je).

The cyclical permutation (e, i, j, ie, ke, k, je) preserves the integral domain E8, but is not an automorphism of the octonions since it takes the associative triad {i,j,k} into the anti-associative triad {j,ie,je}.

The cyclical permutation (e, ie, je, i, k, ke, j) is an automorphism of the octonions but takes the E8 integral domain defined above into another of Bruck's cycle of seven integral domains. Denote the integral domain described above as 7E8, and the other six by iE8 , i = 1, ... , 6.

The 240 units of the 7E8 lattice corresponding to the integral domain 7E8 represent the 240 lattice points in the shell at unit distance

(also commonly normalized as 2) from the origin

(points on the line with iE8, jE8 notation are common points with

the iE8 and jE8 lattices):

±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±ie ±je ±ke)/2 (±e ±i ±j ±k)/2

(±1 ±ke ±e ±k)/2 5E8, 4E8 (±i ±j ±ie ±je)/2

(±1 ±k ±i ±je)/2 (±j ±ie ±ke ±e)/2

(±1 ±je ±j ±e)/2 6E8, 2E8 (±ie ±ke ±k ±i)/2

(±1 ±e ±ie ±i)/2 3E8, 1E8 (±ke ±k ±je ±j)/2

(±1 ±i ±ke ±j)/2 (±k ±je ±e ±ie)/2

(±1 ±j ±k ±ie)/2 (±je ±e ±i ±ke)/2

The other six integral domains iE8 are:

1E8: ±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±je ±i ±j)/2 (±k±e±ie±ke)/2

(±1 ±j ±ie ±ke)/2 5E8, 6E8 (±i±k±e±je)/2

(±1 ±ke ±k ±i)/2 (±j±e±ie±je)/2

(±1 ±i ±e ±ie)/2 7E8, 3E8 (±j±k±je±ke)/2

(±1 ±ie ±je ±k)/2 2E8, 4E8 (±i±j±e±ke)/2

(±1 ±k ±j ±e)/2 (±i±ie±je±ke)/2

(±1 ±e ±ke ±je)/2 (±i±j±k±ie)/2

2E8: ±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±i ±k ±e)/2 (±j,ie,je,ke)/2

(±1 ±e ±je ±j)/2 7E8, 6E8 (±i±k±ie±ke)/2

(±1 ±j ±ke ±k)/2 (±i±e±ie±je)/2

(±1 ±k ±ie ±je)/2 1E8, 4E8 (±i±j±e±ie)/2

(±1 ±je ±i ±ke)/2 3E8, 5E8 (±j±k±e±ie)/2

(±1 ±ke ±e ±ie)/2 (±i±j±k±je)/2

(±1 ±ie ±j ±i)/2 (±k±e±je±ke)/2

3E8: ±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±k ±ke ±ie)/2 (±i±j±e±je)/2

(±1 ±ie ±i ±e)/2 7E8, 1E8 (±j±k±je±ke)/2

(±1 ±e ±j ±ke)/2 (±i±k±ie±je)/2

(±1 ±ke ±je ±i)/2 2E8, 5E8 (±j±k±e±ie)/2

(±1 ±i ±k ±j)/2 4E8, 6E8 (±e±ie±je±ke)/2

(±1 ±j ±ie ±je)/2 (±i±k±e±ke)/2

(±1 ±je ±e ±k)/2 (±i±j±ie±ke)/2

4E8: ±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±ke ±j ±je)/2 (±i±k±e±ie)/2

(±1 ±je ±k ±ie)/2 1E8, 2E8 (±i±j±e±ke)/2

(±1 ±ie ±e ±j)/2 (±i±k±je±ke)/2

(±1 ±j ±i ±k)/2 3E8, 6E8 (±e±ie±je±ke)/2

(±1 ±k ±ke ±e)/2 7E8, 5E8 (±i±j±ie±je)/2

(±1 ±e ±je ±i)/2 (±j±k±ie±ke)/2

(±1 ±i ±ie ±ke)/2 (±j±k±e±je)/2

5E8: ±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±j ±e ±i)/2 (±k±ie±je±ke)/2

(±1 ±i ±ke ±je)/2 2E8, 3E8 (±j±k±e±ie)/2

(±1 ±je ±ie ±e)/2 (±i±j±k±ke)/2

(±1 ±e ±k ±ke)/2 7E8, 4E8 (±i±j±ie±je)/2

(±1 ±ke ±j ±ie)/2 1E8, 6E8 (±i±k±e±je)/2

(±1 ±ie ±i ±k)/2 (±j±e±je±ke)/2

(±1 ±k ±je ±j)/2 (±i±e±ie±ke)/2

6E8: ±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±e ±ie ±k)/2 (±i±j±je±ke)/2

(±1 ±k ±j ±i)/2 3E8, 4E8 (±e±ie±je±ke)/2

(±1 ±i ±je ±ie)/2 (±j±k±e±ke)/2

(±1 ±ie ±ke ±j)/2 5E8, 1E8 (±i±k±e±je)/2

(±1 ±j ±e ±je)/2 7E8, 2E8 (±i±k±ie±ke)/2

(±1 ±je ±k ±ke)/2 (±i±j±e±ie)/2

(±1 ±ke ±i ±e)/2 (±j±k±ie±je)/2

The vertices that appear in more than one lattice are:

(±1±i±j±k)/2 and (±e±ie±je±ke)/2 in 3E8, 4E8, and 6E8 ;

(±1±i±e±ie)/2 and (±j±k±je±ke)/2 in 7E8, 1E8, and 3E8 ;

(±1±j±e±je)/2 and (±i±k±ie±ke)/2 in 7E8, 2E8, and 6E8 ;

(±1±k±e±ke)/2 and (±i±j±ie±je)/2 in 7E8, 4E8, and 5E8 ;

(±1±i±je±ke)/2 and (±j±k±e±ie)/2 in 2E8, 3E8, and 5E8 ;

(±1±j±ie±ke)/2 and (±i±k±e±je)/2 in 1E8, 5E8, and 6E8 ;

(±1±k±ie±je)/2 and (±i±j±e±ke)/2 in 1E8, 2E8, and 4E8 .

The unit vertices in the E8 lattices do not include any of the 256 E8 light cone vertices, of the form (±1±i±j±k±e±ie±je±ke)/2.

They appear in the next layer out from the origin, at radius Ã2, which layer contains in all 2160 vertices.

The D4 Lattice as a Dimensional Reduction of the E8 Lattice:

The 4-dimensional D4 or {3,3,4,3} lattice is the natural lattice for 4-dimensional spacetime. Begin with a quaternionic 4-dimensional spacetime R4 = H, where a basis for H is {1,i,j,k}.

The vertices of the D4 lattice are of the form

a01 + a1i + a2j + a3k,

where the ai may be either all integers or four halves of odd integers

(i.e., all ai Œ Z or all ai Œ Z + 1/2).

The first shell of the D4 lattice has 24 vertices, or nearest neighbors of the central vertex.

With central vertex 0, the 24 first shell vertices are:

±1, ±i, ±j, ±k, (±1 ±i ±j ±k)/2.

There are also 24 second shell vertices:

(±a ±b), where a,b Œ {1,i,j,k} and ab.

The 24 vertices of the first shell are the root vector vertices of the Spin(8) Lie algebra D4.

The 48 vertices of the first and second shells together are the root vector vertices of the exceptional Lie algebra F4.

(Therefore the lattice can also be called the F4 lattice.)

If the D4 lattice is identified as the lattice of integral quaternions, the 24 first shell vertices are the 24 unit quaternions.

The integral quaternion structure of the D4 lattice is the basis for using it as the natural 4-dimensional spacetime lattice for the theory.

The vertices of the D4 lattice (also called the {3,3,4,3} lattice) have a natural light-cone structure. In quaternionic notation, a general vertex can be considered to be the center of a hypercube with 16 vertices (±1±i±j±k)/2 and the centers ±1, ±i, ±j, ±k of the 8 neighboring hypercubes, or, equivalently, to be the center of a 24-cell (24 vertices). If any of the 24 neighbor vertices is chosen to be the future timelike direction +1, then

the future light-cone is {(+1±i±j±k)/2} (8 directions),

the spacelike directions are ±i, ±j, and ±k (6 directions),

the past light-cone is {(-1±i±j±k)/2} (8 directions), and

the past timelike direction is -1.

The quaternionic D4 lattice is derived from the E8 octonionic lattices 7E8, 1E8, ... , 6E8 by a projection ¹: O -> H defined by

¹: {1,ie,je,ke ; e,i,j,k}8 -> {1,ie,je,ke ; 1,ie,je,ke}8 -> {1,i,j,k}4 .

¹ is the identity on {1,ie,je,ke}8 and right-multiplication by -e on {e,i,j,k}8 , followed by a map of the 4-dimensional subspace {1,ie,je,ke}8 of the 8-dimensional octonionic space E8 onto the 4-dimensional quaternionic D4 space {1,i,j,k}4 .

¹ can be thought of as a rotation of {e,i,j,k}8 by 3¹/2, or -¹/2.

Every time ¹ is applied to {e,i,j,k}8, a phase factor of -e or 3¹/2 is introduced.

¹ is defined for convenience with respect to the 7E8 lattice, but is effective for 1E8, ... , 6E8 as well. The projections under ¹ of all seven of the E8 lattices is the same D4 lattice.

Fig. 18 is a diagram of a transition between a 1-dimensional lattice and a 2-dimensional lattice to give some idea about the transition between a 4-dimensional D4 lattice and an 8-dimensional E8 lattice:

Before dimensional reduction, the 7E8 vertices are, in octonionic notation with a basis for O of {1,i,j,k,e,ie,je,ke}:

±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke,

(±1 ±ie ±je ±ke)/2 (±e ±i ±j ±k)/2

(±1 ±ke ±e ±k)/2 (±i ±j ±ie ±je)/2

(±1 ±k ±i ±je)/2 (±j ±ie ±ke ±e)/2

(±1 ±je ±j ±e)/2 (±ie ±ke ±k ±i)/2

(±1 ±e ±ie ±i)/2 (±ke ±k ±je ±j)/2

(±1 ±i ±ke ±j)/2 (±k ±je ±e ±ie)/2

(±1 ±j ±k ±ie)/2 (±je ±e ±i ±ke)/2

The 64 underline vertices are the vertices in the 7E8 lattice that are taken by the projection ¹: 7E8 -> D4 to light-cone vertices in the D4 lattice. Similar vertices in the 1E8, ... , 6E8 lattices also go to light-cone vertices in the D4 lattice.

The 8 future light-cone vertices of the D4 lattice are images of the 32 vertices in 7E8 that are in ¹-1( (+1 ±i ±j ±k)/2).

There are 4 cases:

(+1 ±ie ±je ±ke)/2 -> (+1 ±i ±j ±k)/2

(+1 ±k ±i ±je)/2 -> (+1 ±i ±j ±k)/2

(+1 ±i ±ke ±j)/2 -> (+1 ±i ±j ±k)/2

(+1 ±j ±k ±ie)/2 -> (+1 ±i ±j ±k)/2

Assume the origin of the path is in the {1,ie,je,ke}8 subspace of 7E8.

In the first case, the 7E8 vertex is also in the {1,ie,je,ke}8 subspace of E8 and the map ¹: 7E8 -> D4 is the trivial identity map. This structure corresponds to the first generation fermions and weak bosons.

In the last 3 cases the 7E8 vertex is not in the {1,ie,je,ke}8 subspace of 7E8 and the map ¹: 7E8 -> D4 is not the trivial identity map, but involves right-multiplication by -e as well as the direct identity map from 7E8 to D4. ¹ can then be thought of as being the composition of two maps,

¹e: 7E8 -> 7E8 of right-multiplication and ¹8->4: 7E8 -> D4 of direct identification. A virtual path vertex can exist between the action of ¹e and ¹8->4, and a virtual fermion (of any type having the destination vertex as a common E8 lattice point) can exist at the virtual vertex. Therefore in the 4-dimensional D4 lattice, the D4 vertex will appear to carry a fermion that corresponds to two E8 fermions, or an element of O¥O rather than O. Such fermions are second generation fermions.

Now assume the origin of the path is not in the {1,ie,je,ke}8 subspace of 7E8.

Sometimes in the last 3 cases the destination 7E8 vertex may be in the {1,ie,je,ke}8 subspace of 7E8 and the map ¹: 7E8 -> D4 involves right-multiplication by -e with respect to the origin vertex but not the destination vertex. There can then be a virtual path vertex at the origin of the path, and a virtual fermion (of any type having the origin vertex as a common E8 lattice point) can exist at the virtual vertex. Therefore in the 4-dimensional D4 lattice, the D4 vertex will appear to carry a fermion that corresponds to two E8 fermions, or an element of OxO rather than O. Such fermions are also second generation fermions.

In the first case and sometimes in the last 3 cases the destination 7E8 vertex and the origin 7E8 vertex are both not in the {1,ie,je,ke}8 subspace of 7E8 and the map ¹: 7E8 ü D4 involves right-multiplication by -e at both the origin and destination vertices. There can then be a virtual path vertex at both the origin and the destination of the path, and a virtual fermion (of any type having the origin or destination vertex, as appropriate, as a common E8 lattice point) can exist at the virtual vertex. Therefore in the 4-dimensional D4 lattice, the D4 vertex will appear to carry a fermion that corresponds to three E8 fermions, or an element of O¥O¥O rather than O. Such fermions are third generation fermions.

Similar structure exists for the other six E8 lattices 1E8, ... ,6E8.

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