The 4-dim HyperDiamond Feynman Checkerboard is based on the 4-dim HyperDiamond lattice.
Since the 4-dim HyperDiamond lattice is a 4-dim hypercubic lattice made up of two D4 lattices, one shifted by a glue vector (0.5,0.5,0.5,0.5) with respect to the other, one of the D4 lattices can be regarded as the even sites of the full 4-dim HyperDiamond lattice.
If fermions live only on the even D4 sublattice, then hep-lat/9508013 by Kevin Cahill in the xxx e-print archive shows that the fermion doubling problem is solved.
The 4-dim HyperDiamond Feynman Checkerboard is closely related to Spin Networks. (Gersch (Int. J. Theor. Phys. 20 (1981) 491) has shown that the 2-dim Feynman Checkerboard is equivalent to the Ising model.)
In quant-ph/9503015 "Square Diagrams" are used to represent the 4-link future lightcone leading from a vertex because the representation is written in LaTeX code to be printed in black-and-white on 2-dimensional paper. The "true" representation of the 4-link future lightcone would be as a 4-dimensional simplex, with the 4 future ends of the links forming a 3-dimensional tetrahedron. It is shown here, with a stereo pair showing 3 dimensions and color coding (green = present, blue = future) for the 4th dimension.
From the "true" representation, it is clear that the 4-link future lightcone leading from a vertex looks a lot like the Quantum Pentacle of David Finkelstein and Ernesto Rodriguez (Int. J. Theoret. Phys. 23 (1984) 887.
The symmetries of the 4 future lightcone vectors can be seen from different points of view:
My viewpoint is to look at the 3-dimensional tetrahedron formed by the ends of the 4 future lightcone vectors. Its symmetry group is the 12-element tetrahedral group (3,3,2). The double cover of (3,3,2) is the 24-element binary tetrahedral group {3,3,2}. {3,3,2} is the group of unit quaternions in the 4-dim quaternionic lattice. The 24 unit quaternions in the 4-dim quaternionic lattice are the root vectors of the D4 Lie algebra Spin(8), from which I construct my D4-D5-E6 model.
Another viewpoint is to look at the permutation group S4 of the 4 future lightcone vectors, then decompose S4 into subgroups, and then relate the subgroups of S4 to the groups of gravity and the standard model using a method of coherent states. This is the point of view of Michael Gibbs and David Finkelstein.
Still another viewpoint is to look at the permutation group S4 , then notice that S4 is the 24-element octahedral group (4,3,2), then use the McKay correspondence to get the Lie algebra E7, and then use E7 to build a physics model. This is the point of view of Saul Paul Sirag at the PCRG.
I hope, think, and believe that these 3 viewpoints are in a deep sense equivalent, and that we are really like 3 blind men trying to describe an elephant.
ANTIPARTICLES in the CHECKERBOARD:
In quant-ph/9503015 I considered only paths in which in each segment lies in the future lightcone, that is, in which time increased at each segment. Also, I used the Gersch convention of weighting changes in direction by -ime (where i is a quaternion imaginary).
Feynman also considered paths in which in each segment lies in the past lightcone, that is, segments going backward in time. He weighted the past lightcone changes in direction by the negative of the forward lightcone weight, which would be, using the Gersch convention, a weight by +ime (where i is a quaternion imaginary).
Feynman considered the path segments going backward in time to be antiparticle path segments. The following gif (300k) (from Schweber, Rev. Mod. Phys. 58 (1986) 449 at p. 482, (see box 13, folder 3, of Caltech's Feynman archives (Notes on the Dirac Equation))) shows Feynman's thinking:
see Conway and Sloane (Sphere Packings, Lattices, and Groups - Springer)
The complex Gaussian Z2 lattice, for which N(1)=N(2)=4, N(3)=0, N(4)=4, N(5)=8, ... and N(m)/4 is the number of distinguishable
(i.e., 2^2 +2^2 = 8 indistinguishable, so N(8) = 4, and 2^2 + 1^2 = 5 = 1^2 + 2^2 distinguishable, so N(5) = 8.)
ways m can be written as the sum of 2 squares.
The 8-dimensional HyperDiamond octonionic E8 lattice, for which N(m) is always less than N(m+1).
Each vertex of E8 has 240 nearest neighbors.
240 = 48 + 192 = 2x24 + 8x24
The E8 lattice is associated with the octonionic X-product of Cederwall and Preitschopf and a later paper of Dixon.
The 16-dimensional Barnes-Wall lattice /\16, for which each vertex has 4,320 nearest neighbors.
4,320 = 480 + 3,840 = 2x240 + 16x240
The /\16 lattice is associated with the octonionic XY-product of Dixon.
As described in a paper by Geoffrey Dixon, each vertex of the 24-dimensional Leech lattice /\24 has 196,560 nearest neighbors (norm(xx) = 4).
196,560 = 3x240 + 3x(16x240) + 3x(16x16x240)
Geoffrey Dixon is working on a book about the Leech lattice /\24.
The Conway group .0 (dotto) is the permutation group of the 196,560 vertices.
(See p. 295 of Conway and Sloane for connections among dotto, Fi24, M24, the binary Golay code C24, M12, and Suz.)
Note that the largest finite sporadic group, the Monster group, is the automorphism group of an algebra of dimension 196,884 = 196,560 + 300 + 24. (300 = symmetric tensor square of 24)
Here are some COMMENTS ON LORENTZ INVARIANCE, based on the properties of the D4 lattice, two copies of which make the 4HD HyperDiamond lattice. The D4 lattice nearest neighbor vertex figure, the 24-cell, is the 4HD HyperDiamond lattice next-to-nearest neighbor vertex figure.
Fermions move from vertex to vertex along links.
Gauge bosons are on links between two vertices, and so can also be considered as moving from vertex to vertex along links.
The only way a translation or rotation can be physically defined is by a series of movements of a particle along links.
A TRANSLATION is defined as a series of movements of a particle along links, each of which is the CONTINUATION of the immediately preceding link IN THE SAME DIRECTION.
An APPROXIMATE rotation, within an APPROXIMATION LEVEL D, is defined with respect to a given origin as a series of movements of a particle along links among vertices ALL of which are in the SET OF LAYERS LYING WITHIN D of norm (distance^2) R from the origin, that is, the SET OF LAYERS LYING BETWEEN norm R-D and norm R+D from the origin.
Conway and Sloane (Sphere Packings, Lattices, and Groups - Springer) pp. 118-119 and 108, is the reference that I have most used for studying lattices in detail.
(Conway and Sloane define the norm of a vector x to be its squared length xx.)
In the D4 lattice of integral quaternions,
layer 2 has the same number of vertices as layer 1, N(1) = N(2) = 24.
Also (this only holds for real, complex, quaternionic, or octonionic lattices), K(m) = N(m)/24 is multiplicative, meaning that, if p and q are relatively prime, K(pq) = K(p)K(q).
The multiplicative property implies that ( } means greater than):
K(2^a) = K(2) = 1 (a}0) and
K(p^a) = 1 + p + p^2 + ... + p^a (a}=0).
So, for the D4 lattice,
there is always an arbitrarily large layer (norm xx = 2^a, for some large a) with exactly 24 vertices, and
there is always an arbitrarily large layer(norm xx = p+1, for some large prime p) with 24(p+1) vertices, and
given a prime number P whose layer is within D of the origin, which layer has N vertices, there is a layer kP with at least N vertices within D of any other given layer in D4.
Some examples I have used are chosen so that the 2^a layer adjoins the prime 2^a +/- 1 layer.
The numbers of vertices in some of the layers are: m=norm of layer N(m)=no. vert. K(m)=N(m)/24 1 24 1 2 24 1 3 96 4 4 24 1 5 144 6 6 96 4 7 192 8 8 24 1 9 312 13 10 144 6 11 288 12 12 96 4 13 336 14 14 192 8 15 576 24 16 24 1 17 432 18 18 312 13 19 480 20 20 144 665,536=2^16 24 1 65,537 1,572,912 65,538
2,147,483,647 51,539,607,552 2,147,483,648 2,147,483,648=2^31 24 1
65,537 = 2^2^4 + 1 is the largest known Fermat prime. It is called F4, but is not likely to be confused with the exceptional Lie algebra F4.
2,147,483,647 = 2^31 - 1 is a Mersenne prime. It was shown to be prime by Euler. It is called M31, but is not likely to be confused with the Andromeda galaxy M31.
(see Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin, 1986)
If the D4 spacetime lattice length is taken to be the Planck length, about 10^-33 cm or, in terms of energy, about 10^19 GeV, then
the layer of norm 65,537 is at a distance sqrt(65,537) = 256.002 x 10^-33 cm or about 3.9 x 10^16 GeV, and
the layer of norm 2,147,483,647 is at a distance sqrt(2,147,483,647) = 46,340 x 10^-33 cm or about 2.2 x 10^14 GeV, and
THEREFORE
at energies below about 10^16 GeV, continuous rotations can be approximated by D4 lattice rotations to an accuracy of at least 2 PI^2 / 1,572,912 = about 1.3 x 10^-5 steradians(4-dim), and
at energies below about 10^14 GeV, continuous rotations can be approximated by D4 lattice rotations to an accuracy of at least 2 PI^2 / 51,539,607,552 = about 3.8 x 10^-10 steradians(4-dim).
The argument can be extended quite a long way by considering the Mersenne prime 2^859433-1 (258716 digits) found by Slowinski and Gage in 1994.
Prime numbers are interesting in many ways.
For example, consider the Riemann zeta function:
zeta(s) = SUM( 1 / (N^s) )
sum over N from 1 to infinity
It is also equal to
zeta(s) = PROD( 1 / (1 - P^(-s)) )
product over all prime numbers P
Note that zeta(1) = SUM(1/N) is the harmonic series.
The fact that the harmonic series diverges
shows that
the sum over all primes P SUM( 1/P ) also diverges,
which also shows that the number of primes is infinite.
(There is a theorem that if PROD( 1 + K ) converges,
then SUM( K ) converges.)
(see Introduction to Calculus and Analysis,
vol. 1, by Courant and John, Springer 1989)
You can also use zeta functions and generalizations
to calculate distributions of prime numbers, and
to do calculations for sum-over-histories path
integrals in quantum theory, and
for a lot of neat poorly understood stuff.
According to the 18 May 1996 issue of the New Scientist,
Michael Berry and Jon Keating have seen correspondences
between the spacing of the prime numbers
and the spacing of energy levels
of quantum systems that classically would be chaotic.
They would like to find a chaotic system that,
when quantized, would have energy levels that are
distributed exactly as the prime numbers.
Then they could verify the Riemann hypothesis
through "the harmonies in the music of the primes".
THEREFORE
IT IS UNLIKELY THAT PLANCK-LENGTH D4 LATTICE SPACETIME IS PRESENTLY EXPERIMENTALLY DISTINGUISHABLE FROM CONTINUOUS SPACETIME BY OBSERVATION OF ROTATIONS.
Thanks to Liam Roche at rochel@bre.co.uk for correcting some of my mistakes about prime numbers.
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