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4-dim HyperDiamondLattice

Conway and Sloane (in their book Sphere Packings, Lattices, andGroups, Third Edition, Springer 199) say on page 119:

"... Formally we define Dn+ = Dn u ( [1] + Dn ). ... Dn+ is a lattice packing if and only if n is even. D3+ is the tetrahedral or diamond packing ... and D4+ = Z4. When n = 8 this construction is especially important, the lattice D8+ being known as E8 ...".] ...

The HyperDiamond FeynmanCheckerboard model is based on the 4-dimHyperDiamond lattice and is a generalization of the (1+1)-dimensionalFeynman Checkerboard.

The Planck length is the fundamental lattice link scale in theD4-D5-E6-E7-E8 VoDou Physicsmodel.

According to JohnC. Baez and S. Jay Olson in their paper at gr-qc/0201030:

"... Ng and van Dam have argued that quantum theory and general relativity give a lower bound delta L > L^(1/3) L_P ^(2/3) on the uncertainty of any distance, where L is the distance to be measured and L_P is the Planck length. Their idea is roughly that to minimize the position uncertainty of a freely falling measuring device one must increase its mass, but if its mass becomes too large it will collapse to form a black hole. ... Amelino-Camelia has gone even further, arguing that delta L > L^(1/2) L_P ^(1/2) ... Here we show that one can go below the Ng-van Dam bound [ and the Amelino-Camelia bound ] by attaching the measuring device to a massive elastic rod. ...

[ while the Ng-van Dam ] result was obtained by multiplying two independent lower bounds on delta L, one from quantum mechanics and the other from general relativity, ours arises from an interplay between competing effects. On the one hand, we wish to make the rod as heavy as possible to minimize the quantum-mechanical spreading of its center of mass. To prevent it from becoming a black hole, we must also make it very long. On the other hand, as it becomes longer, the zero-point fluctuations of its ends increase, due to the relativistic limitations on its rigidity. We achieve the best result by making the rod just a bit longer than its own Schwarzschild radius.

... Relativistic limitations on the rod's rigidity, together with the constraint that its length exceeds its Schwarzschild radius, imply that zero-point fluctuations of the rod give an uncertainty delta L > L_P . ...".

The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:

Volumes of Spaces ofSuperpositions of other given Sets of Basis Elements correspond toVolume of Physical SpaceTime and Volumeof Internal Symmetry Space representedby those Basis Elements.  

The D4-D5-E5-E7-E8 VoDou PhysicsHyperDiamond Feynman Checkerboard model can be constructedfrom the 8-fold Periodicity of RealClifford Algebras

Cl(8N) = Cl(8) x ..N.. x Cl(8)

where x ..N.. x denotes N-fold tensor product, by using a verylarge Clifford Algebra Cl(8N) as a starting point, and using 8-foldPeriodicity to factor Cl(8N) into N copies of Cl(8).

To see what to do with the N copies of Cl(8), look at howFeynman's 2-dimensional Checkerboardmight be constructed from N copies of Cl(2).

Given N copies of Cl(2), how can they be put in a useful order?

Look at Cl(2) = C(2), with graded structure

1 2 1

The Cl(2) vector space is 2-dimensional, so the N copies of Cl(2) should be put in a 2-dimensional array. The most natural such array would be the Complex Gaussian integers, which make a 2-dim Feynman checkerboard with all the Cl(2) at the vertices of the checkerboard connected to other Cl(2) in such a way as to make a 2-dim lattice that is consistent with complex number multiplication.

Cl(2) has 1-dim U(1) as its bivector Lie algebra, which is consistent with the electromagntism gauge group, and with complex number multiplication, and Cl(2) has full spinor space of dimension sqrt(4) = 2, and so half-spinor spaces that are 1-dimensional, one for the electron and one for the positron to move around on the checkerboard, so a Complex Gaussian lattice with a Cl(2) at each vertex defines a 2-dimensional Feynman Checkerboard with half-spinor particles moving around on it and a U(1) gauge group providing the i that Feynman used to weight changes of direction.

If you look at the N copies of Cl(8) the same way, you see thatCl(8) has graded structure

1 8 28 56 70 56 28 8 1

so that Cl(8) has an 8-dim vector space, so that the N copies ofCl(8) wants to be connected as an 8-dim checkerboard, so the Ncopies of Cl(8) should be ordered as an array of integral octonions,that is, they should live on an E8lattice, which is the Octonionic 8-dim correspondent of the2-dim Complex Gaussian lattice.

You can look at the natural 4-dim physical spacetime sublattice,and see that it is the 4-dim HyperDiamondlattice.

Since Cl(8) half-spinors are 8-dim, you get (where Cl(2) gives youthe electron and positron) for fermions to move on the latticethe 8 first generation fermionparticles

and their 8 antiparticles.The projection of an 8-dim E8lattice onto the 4-dim HyperDiamond lattice gives the second andthird generation fermions.

At each vertex, instead of the Cl(2) gauge group U(1), you get theCl(8) gauge group Spin(8), whichgives Gravity and the Standard Model from the point of view ofthe 4-dim HyperDiamond lattice, and which is consistent with octonionmultiplication.

The weighting of changes of directionin the 4-dm HyperDiamond lattice is given by Quaternion Imaginariesof the change of direction, in this way:

There are 7 different chargedfirst-generation fermion particles {electron, rgb up quarks, rgb downquarks}. There are 7 different E8 lattices in the full 8-dimensionalSpaceTime prior to Dimensional Reduction. All of the 7 E8 latticesproduce, by Dimensional Reduction, the same 4-dimensional PhysicalSpaceTime and 4-dimensional Internal Symmetry Space.

The HyperDiamond generalization has discrete lightcone directions.If the 4-dimensional Feynman Checkerboard is coordinatized by thequaternions Q:

In cylindrical coordinates t,r with r^2 = x^2+y^2+z^2, theEuclidian metric is

t^2 + r^2 = t^2 + x^2+y^2+z^2

and the Wick-Rotated Minkowski metric with speed of light c is

(ct)^2 - r^2 = (ct)^2 - x^2 -y^2 -z^2.

For the future lightcone links on the 4-dimensional Minkowskilightcone, c = sqrt3.

Any future lightcone link is taken into any other future lightconelink by quaternion multiplication by +/- i, +/- j, or +/- k.

For a given vertex on a given path, continuation in the samedirection can be represented by the link 1, and changing directioncan be represented by the imaginary quaternion +/- i, +/- j, +/- kcorresponding to the link transformation that makes the change ofdirection.

Therefore, at a vertex where a path changes direction, a path canbe weighted by quaternion imaginaries just as it is weighted by thecomplex imaginary i in the 2-dimensional case.

If the path does change direction at a vertex, then the path atthe point of change gets a weight of -im e, -jm e, or -km e wherei,j,k is the quaternion imaginary representing the change ofdirection, m is the mass (only massive particles can changedirections), and sqrt3 e is the timelike length of a path segment,where the 4-dimensional speed of light is taken to be sqrt3.

For a given path, let C be the total number of direction changes,c be the cth change of direction, and ec be the quaternion imaginaryi,j,k representing the cth change of direction.

C can be no greater than the timelike Checkerboard distance Dbetween the initial and final points.

The total weight for the given path is then m sqrt3 ec to the Cthpower times the product (c from 0 to C) of -ec

Note that since the quaternions are not commutative, the productmust be taken in the correct order.

The product is a vector in the direction +/- 1, +/- i, +/- j, or+/- k.

Let N(C) be the number of paths with C changes in direction. Thepropagator amplitude for the particle to go from the initial vertexto the final vertex is the sum over all paths of the weights, that isthe path integral sum over all weighted paths:

the sum from 0 to D of N(C)


the Cth power of m sqrt3 ec


the product (c from 0 to C) of -ec

The propagator phase is the angle between the amplitude vector inquaternionic 4-space and the quaternionic real axis. The plane inquaternionic 4-space defined by the amplitude vector and thequaternionic real axis can be regarded as the complex plane of thepropagator phase.




ManyWorlds of the FeynmanCheckerboard can be represented by SurrealNumbers.

The 4-dim HyperDiamond lattice isbased on the D4 lattice.


Conway and Sloane, in their book Sphere Packings, Lattices, andGroups (3rd edition, Springer, 1999).in chapter 4, section 7.3, pages119-120) define a packing

D+n = Dn u ( [1] + Dn )

[ where the gule vector [1] = (1/2, ... , 1/2) ]and say: "... D+n is a lattice packing if and only if n is even.

D+n is what David Finkelstein and I named a HyperDiamond lattice(although in odd dimensions it is technically only a packing and nota lattice).

Conway and Sloane also say in chapter 4, section 7.1, page 117)that the lattice Dn is defined only for n greater than or equal to3.

To see what happens for n = 2, note that D2 should correspond tothe Lie algebra Spin(4),which is reducible to Spin(3)xSpin(3) =SU(2)xSU(2) = Sp(1)xSp(1) = S3xS3, and is not an irreducible Liealgebra. The root lattice of D2 is two copies of the root lattice ofSU(2), which is just a lattice of points uniformly distributed on aline.

If you are to fit the two lines together, you have to specify theangle at which they intersect each other, and requiring "latticestructure" or consistency with complex number multiplication does NOTunambiguously determine that angle: it can be either

60 degrees, which gives you the A2 root lattice
*--* / \ * * \ / *--*
of the Lie algebra SU(3) and the Eisenstein complex integers
90 degrees, which gives you the C2 = B2 root lattice
*--*--*| |* *| |*--*--*
of the Lie algebra Spin(5) = Sp(2) and the Gaussian complex integers.

Since a Dn lattice for n > 3 is a checkerboard, or halfof a hypercubic lattice, it is natural to define D2 as acheckerboard, or half of a C2 = B2 square lattice. Then the2-dimensional HyperDiamond lattice D+2 = D2 u ( [1] + D2 ) isseen to be the Z2 square lattice C2 = B2

where the orginal D2 is made up of the centers of the yellowsquares, the glue vectors are the (1/2,1/2) represented by pairs ofarrows, and the ( [1] + D2 ) is made up of the centers of thewhite squares.

The total 2-dim hyperdiamond structure is the Z2 integer lattice,sort of analogous to the 4-dim case in which D4 u ( [1] + D4) = Z4, so that

Note that the basic D2 structure is consistent with Feynman's2-dimensional checkerboard in which the lines of the checkerboardare 2-dim light-cone lines. 

Since the 4-dim HyperDiamond lattice is a4-dim hypercubic lattice made up of two D4lattices, one shifted by a glue vector ( 0.5, 0.5, 0.5, 0.5 ) withrespect to the other, one of the D4 lattices can be regarded as theeven sites of the full 4-dim HyperDiamondlattice.

Since the 4-dim HyperDiamond lattice is a4-dim hypercubic lattice made up of two D4lattices, one shifted by a glue vector ( 0.5, 0.5, 0.5, 0.5 ) withrespect to the other, one of the D4 lattices can be regarded as theeven sites of the full 4-dim HyperDiamondlattice.

If fermions live only on the even D4 sublattice, then hep-lat/9508013by Kevin Cahill in the xxx e-print archive shows that the fermiondoubling problem is solved.

The 4-dim HyperDiamond Feynman Checkerboard is closely related toSpin Networks. (Gersch (Int. J.Theor. Phys. 20 (1981) 491) has shown that the 2-dim FeynmanCheckerboard is equivalent to the Ising model.)

In quant-ph/9503015"Square Diagrams" are used to represent the 4-link future lightconeleading from a vertex because the representation is written in LaTeXcode to be printed in black-and-white on 2-dimensional paper. The"true" representation of the 4-link future lightcone would be as a4-dimensional simplex, with the 4 future ends of the links forming a3-dimensional tetrahedron. It is shown here, with a stereo pairshowing 3 dimensions and color coding (green = present, blue =future) for the 4th dimension.

From the "true" representation, it is clear that the 4-link futurelightcone leading from a vertex looks a lot like the Quantum Pentacleof David Finkelstein and Ernesto Rodriguez (Int. J. Theoret. Phys. 23(1984) 887).

The future tetrahedron, not including the origin vertex, contains4 vertices and 4 bivector triangles. Those bivectors correspond tothe 4 translation bivectors of the 10-dimensional Spin(5) Liealgebra, which is based on the Cl(0,5) Cliffordalgebra with graded structure

1 5 10 10 5 1

so that there are: 1 empty set, 5 vector vertices, 10 bivectoredges, 10 triangles, 5 tetrahedra, and 1 4-simplex. By Hodge duality,the 10 bivector edges correspond to the 10 triangles.

The future lightcone edges leading from the origin are edges onthe 6 bivector triangles that include the origin vertex. Thosebivectors correspond to the 6 bivectors of the Spin(4) subalgebra ofthe Spin(5) Lie algebra. The 6-dimensional Spin(4) subalgebra isreducible, isomorphic to SU(2) x SU(2), so that it reduces to 3rotations and 3 Lorentz boosts.

The symmetries of the 4 future lightcone vectors can be seen fromdifferent points of view:

My viewpoint is to look at the 3-dimensional tetrahedron formed bythe ends of the 4 future lightcone vectors. Its symmetry group is the12-element tetrahedral group (3,3,2). The double cover of (3,3,2) isthe 24-element binary tetrahedral group {3,3,2}. {3,3,2} is the groupof unit quaternions in the 4-dim quaternionic lattice. The 24 unitquaternions in the 4-dim quaternionic lattice are the root vectors ofthe D4 Lie algebra Spin(8), from which I construct my D4-D5-E6-E7-E8VoDou Physics model.

Another viewpoint is to look at the permutation group S4 of the 4future lightcone vectors, then decompose S4 into subgroups, and thenrelate the subgroups of S4 to the groups of gravity and the standardmodel using a method of coherent states. This is the point of view ofMichael Gibbs andDavidFinkelstein.

Still another viewpoint is to look at the permutation group S4 ,then notice that S4 is the 24-element octahedral group (4,3,2), thenuse the McKay correspondence to get theLie algebra E7, and then use E7 to build a physics model. This is thepoint of view of SaulPaul Sirag at the PCRG.

Yet another viewpoint, motivated by SpinNetworks such as those described of Barnettand Crane, is to look at the dual to the 4-simplex illustratedabove, in which its 5 vertices correspond to 5 tetrahedral 3-faces,and vice versa:

In this dual picture, the 4 vector edges leading from the originbreak down into 3 green spacelike vector edges leading from theorigin and 1 green-to-blue timelike vector edge leading from theorigin. The 3 triangles with 2 spacelike sides correspond to the 3rotations, and the 3 triangles with 1 spacelike side and 1 timelikeside correspond to the 3 Lorentz boosts. As in the original lightconepicture, the 4 triangles that do not include the origin correspond tothe 4 translations of the Spin(5) Lie algebra.


I hope, think, and believe that all these viewpoints are in a deepsense equivalent, and that we are really like several blind mentrying to describe an elephant.


Feynman describes the geometry of Spin-2gravitons

with his characteristic clarity in his "Lectures on Gravitation"(Caltech 1971). The following 2 gifs (63k and 90k) show relevantparts of pages 41 and 42:

Antiparticles in the Checkerboards:

In quant-ph/9503015I considered only paths in which in each segment lies in the futurelightcone, that is, in which time increased at each segment. Also, Iused 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 thepast lightcone, that is, segments going backward in time. He weightedthe past lightcone changes in direction by the negative of theforward lightcone weight, which would be, using the Gerschconvention, a weight by +ime (where i is a quaternion imaginary).

Feynman considered the path segments going backward in time to beantiparticle path segments. The following gif (300k) (from Schweber,Rev. Mod. Phys. 58 (1986) 449 at p. 482, (see box 13, folder 3, ofCaltech's Feynman archives (Notes on the Dirac Equation))) showsFeynman's thinking:

Some Lattices in 2, 8,16, and 24dimensions

see Conway and Sloane (Sphere Packings, Lattices, and Groups -Springer)



The complex Gaussian Z2lattice, for which N(1)=N(2)=4, N(3)=0, N(4)=4, N(5)=8, ... andN(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 thesum of 2 squares;


the complex Eisensteinlattice.


The 8-dimensional HyperDiamond octonionic E8lattice is associated with the octonionic X-product of Cederwalland Preitschopf and a later paper of Dixon.For an E8 lattice N(m) is always less than N(m+1). Each vertex of anE8 lattice has 240 nearest neighbors and 2160 second-levelnext-nearest neighbors.

There are 7 distinct E8 lattices, denoted by iE8(i = 1, ..., 7). All 7 of the E8 lattices have some points incommon, and some subsets of three have some points in common, but notwo of the E8 lattices are identical.

There are 7 distinctassociative triangle 3-element subsets of

the 7 imaginary Octonions{i,j,k,E,I,J,K}.  

There are 7 charged Diracfirst-generation fermion particles:

At high energies, prior to DimensionalReduction of SpaceTime, there is only one generation of fermions,so the first generation is the only generation. Therefore, eachcharged Dirac fermion particle, and its antiparticle, correspond toone imaginary Octonion, to one associative triangle, and to one E8lattice:


red Down Quark red Up Quark green Down Quark Electron green Up Quark blue Down Quark blue Up QuarkrD gD bD E rU gU bU I J K E i j kj/ \i---k J j J I J K/ \ / \ / \ / \ / \ / \i---K I---K I---k E---i E---j E---k3E8 6E8 4E8 7E8 1E8 2E8 5E8
Each charged Dirac fermion  propagates in its own E8Generalized Feynman Checkerboard Lattice.

Since all the E8 lattices have in common the vertices { ±1,±i, ±j, ±k, ±e, ±ie, ±je, ±ke },all the charged Dirac fermions can interact with each other.Composite particles, such as Quark-AntiQuark mesons and 3-Quarkhadrons, propagate on the common parts of the E8 latticesinvolved.

The uncharged e-neutrino fermion, which corresponds to theOctonion real axis with basis {1}, propagates on any and all of theE8 lattices.


After Dimensional Reduction ofSpaceTime, the associative triangle of each E8 lattice is mappedinto the same {i,j,k} Quaternionic associative triangle of thespatial part of 4-dimensional PhysicalSpaceTime, and the co-associativesquare of each E8 lattice is mapped into the {E,I,J,K} structureof 4-dimensional Internal SymmetrySpace. There, the Generalized FeynmanCheckerboard game is played on a 4-dimensionalHyperDiamond Lattice.




The 16-dimensional Barnes-Walllattice /\16, for which each vertex has 4,320 nearestneighbors.

4,320 = 480 + 3,840 = 2x240 + 16x240

The /\16 lattice is associated with the octonionic XY-product ofDixon.





As described in a paper by Geoffrey Dixon, each vertex ofthe 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 aboutthe Leech lattice /\24.

The Conway group .0 (dotto) is the permutation group of the196,560 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, isthe automorphism group of an algebra of dimension 196,884 = 196,560 +300 + 24. (300 = symmetric tensor square of 24)



the 24-dimensional Leech lattice/\24 can be used to represent the24 non-Abelian of the 28 Spin(8) gauge bosons.


Since the D4-D5-E6-E7-E8VoDou Physics model is fundamentally a Planck ScaleHyperDiamond LatticeGeneralized FeynmanCheckerboard model, it does violate LorentzInvariance at the Planck Scale, affecting UltraHigh Energy Cosmic Rays.
The first person to propose Planck Scale Lorentz Invariance Violation as a solution to the problem of Ultra High Energy Cosmic Rays may have been L. Gonzalez-Mestres around 1995, prior to a proposal by Coleman and Glashow. Gonzalez-Mestres, in physics/0003080, gives his version of the history of the idea of Lorentz Invariance: "... Henri Poincare was the first author to consistently formulate the relativity principle, stating (Poincare, 1895): "Absolute motion of matter, or, to be more precise, the relative motion of weighable matter and ether, cannot be disclosed. All that can be done is to reveal the motion of weighable matter with respect to weighable matter". ... In his June 1905 paper (Poincare, 1905), published before Einsteins's article (Einstein, 1905) arrived (on June 30) to the editor, Henri Poincare explicitly wrote the relativistic transformation law for the charge density and velocity of motion and applied to gravity the "Lorentz group", assumed to hold for "forces of whatever origin". ... In 1921 , A. Einstein wrote in "Geometry and Experiment" (Einstein, 1921): "The interpretation of geometry advocated here cannot be directly applied to submolecular spaces... it might turn out that such an extrapolation is just as incorrect as an extension of the concept of temperature to particles of a solid of molecular dimensions". ...". Gonzalez-Mestres prefers a Quadratically Deformed Relativistic Kinematics ( QDRK )to the Linearly Deformed Relativistic Kinematics ( LDRK ) preferred by Amelino-Camelia and Piran in whose paper QDRK corresponds to the parameter value a = 2 and LDRK to a = 1. Gonzalez-Mestres says: "... QDRK naturally emerges when a fundamental length scale is introduced to deform the Klein-Gordon equa-tions. It is typical, for instance, of phonons in condensed-matter physics. ... LDRK can be generated by introducing a background gravitational field in the propagation equations of free particles. In the first case, the Planck scale is an internal parameter of the basic wave equations generating the "elementary" particles as vacuum excitations. In the second case, it manifests itself only as a parameter of the background gravitational field, similar to a refraction phenomenon. ...".

A December1998 eprint post from Laurent Siebenmannsays:

"... censorship ... may seem to you, gentle reader, quite out of place in modern science. However, consider the case evoked below, where the preprint archive mp-arc is cited as offering evidence of unjust practice ...

... Cordially

Laurent Siebenmann

%%%%%%%%%%%%%%%%%%%%%%%%% ...

  • From: Luis Gonzalez <>
  • Message-Id: <>
  • Subject: The politics of Lorentz symmetry violation?
  • To:
  • Date: Tue, 24 Nov 1998 15:29:54 MET

Dear Colleague,

You have perhaps noticed that a paper presented at TAUP 97, "Lorentz symmetry violation and high-energy cosmic rays" (author: Luis Gonzalez-Mestres), did not appear in the conference Proceedings recently distributed by Elsevier. The paper exists, indeed, and has electronic dates at Los Alamos, APS, mp_arc... (first week of December 1997). It was also registered at KEK on December 12, 1997 and was, of course, sent to the editors in due time.

The local organizers actually refused to publish it, five months after it had been accepted and allocated three pages in the Proceedings. As you know, all papers selected for oral presentation were to be published in the Proceedings according to the official announcements, so why was my paper rejected by a late decision?

Looking at the Proceedings, I have found a paper by Sheldon L. GLASHOW on Lorentz symmetry violation, presenting two of my results without any mention to my work. These results are the possible absence of GZK cutoff and the stability of unstable particles at very high energy. Actually, such results had already been published electronically, but also from and editorial point of view, before the TAUP 97 conference was held. An example is my paper physics/9705031 (26 May 1997) of the Los Alamos archive, which was published in the Proceedings of ICRC 97 distributed at the end of July 1997, before the ICRC 97 conference started.

In the relevant region, the mechanisms producing the above mentioned effects are essentially identical in both approaches (that of Harvard and mine). However, I prefer the one I put forward because it automatically preserves Lorentz symmetry in the limit where k (wave vector) vanishes, and is likely to make things much easier in order to incorporate gravitation in the model. ...

... With my best regards

Luis Gonzalez-Mestres ...".


Here are some further

Comments on Lorentz Invariance

based on the properties of the D4 lattice,two copies of which make the 4HD HyperDiamond lattice. The D4 latticenearest neighbor vertex figure, the 24-cell, is the 4HD HyperDiamondlattice 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 beconsidered as moving from vertex to vertex along links.

The only way a translation or rotation can be physically definedis by a series of movements of a particle along links.

A TRANSLATION is defined as a series of movements of a particlealong links, each of which is the CONTINUATION of the immediatelypreceding link IN THE SAME DIRECTION.

An APPROXIMATE rotation, within an APPROXIMATION LEVEL D, isdefined with respect to a given origin as a series of movements of aparticle along links among vertices ALL of which are in the SET OFLAYERS LYING WITHIN D of norm (distance^2) R from the origin, thatis, the SET OF LAYERS LYING BETWEEN norm R-D and norm R+D from theorigin.

Conway and Sloane (Sphere Packings, Lattices, and Groups -Springer) pp. 118-119 and 108, is the reference that I have most usedfor studying lattices in detail.

(Conway and Sloane define the norm of a vector x to be its squaredlength 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, oroctonionic 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:

K(2^a) = K(2) = 1 (for a greater than 0) and

K(p^a) = 1 + p + p^2 + ... + p^a (for a greater than or equal to0).

So, for the D4 lattice,

there is always an arbitrarily large layer (norm xx = 2^a, forsome large a) with exactly 24 vertices, and

there is always an arbitrarily large layer(norm xx = P, for somelarge prime P) with 24(P+1) vertices (note that Mersenne primes areadjacent to powers of 2), 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 Nvertices within D of any other given layer in D4.

Some examples I have used are chosen so that the 2^a layer adjoinsthe prime 2^a +/- 1 layer.

The D4 lattice canbe described:

with a minimum lattice distance of2 (as a sublattice of the 4-dim HyperDiamond lattice)or

with a minimum lattice distance of1 (as the lattice of integral quaternions):

If you consider the D4lattice to be the even sublattice of the 4-dimensionalHyperDiamond lattice of the D4-D5-E6-E7-E8physics model, then the minimal norm of the D4 lattice wouldbe 2, and you would have a table in which the entries of the firstcolumn (m=norm of layer) are each twice the entries below, so that,for example, the layer of norm 22 withrespect to the origin of the HyperDiamond lattice would have 288vertices. This is the first definition (equation 86) of the D4lattice in Chapter 4 of Sphere Packings, Lattices, and Groups, 3rdedition, by Conway and Sloane (Springer 1999). Conway and Sloanedenote the HyperDiamond structure of dimension n by "the packingDn+", because Dn+ is a lattice only if and if the dimension n is even(since Conway and Sloane define Dn only for n at least 3) .

The D4+ 4-dimensional HyperDiamond lattice is exactly the same asthe 4-dimensional cubic lattice Z4, the lattice in 4-dimensionalspace made up of 4-dim hypercubes.

Conway and Sloane (Chapter 4, eq. 49) give equations for the number of vertices N(m) in the m-th layer of the D4+ HyperDiamond lattice:
for m odd: N(m) = 8 SUM(d|m) d

for m even: N(m) = 24 SUM(d|m, d odd) d

where d is a divisor ( including 1 and m ) of m.

The coincidence between D4+ and Z4 is peculiar to 4-dimensions.For example, D3+ is the familiar 3-dim diamond lattice, and D8+ isthe E8 lattice, and they are not cubic lattices.

Conway and Sloane (Chapter 4, eq. 49, and eq. 102) give equations for the number of vertices N(m) in the m-th layer of the D8+ HyperDiamond lattice and of the E8 lattice:
for D8+: N(m) = 16 SUM(d|m) d^3

for E8 and m odd, N(m) = 0

for E8 and m even, N(m) = 240 SUM(d|(m/2)) d^3

where d is a divisor ( including 1 and m / 2 ) of m / 2. For E8, N(m) is the number of integral octonions of norm m / 2.

The HyperDiamond lattice D4+ is made up of two D4lattices. One of the D4 lattices in the D4+ is called the even D4of D4+. If you start from the origin of D4+, the even D4 contains thelayers that are even distances from the origin.

Since the D4 lattice is the lattice of integral quaternions, theeven D4 of D4+ is the integral quaternion lattice expanded by afactor of 2 so that each layer is twice as far from the origin (and,in particular, the closest layer is at distance 2 instead of 1 fromthe origin).


Here are the numbers of vertices in some of the layers of the D4+lattice. The even-numbered layers correspond ot the even D4sublattice:

m=norm of layer             N(m)=no. vert.   0                                 1   1                                 8  =    1 x 8   2                                24  =    1 x 24   3                                32  =  ( 1 + 3 ) x 8    4                                24  =    1 x 24   5                                48  =  ( 1 + 5 ) x 8   6                                96  =  ( 1 + 3 ) x 24    7                                64  =  ( 1 + 7 ) x 8   8                                24  =    1 x 24   9                               104  =  ( 1 + 3 + 9 ) x 8  10                               144  =  ( 1 + 5 ) x 24  11                                96  =  ( 1 + 11 ) x 8  12                                96  =  ( 1 + 3 ) x 24  13                               112  =  ( 1 + 13 ) x 8  14                               192  =  ( 1 + 7 ) x 24   15                               192  =  ( 1 + 3 + 5 + 15 ) x 8   16                                24  =    1 x 24  17                               144  =  ( 1 + 17 ) x 8  18                               312  =  ( 1 + 3 + 9 ) x 24  19                               160  =  ( 1 + 19 ) x 8  20                               144  =  ( 1 + 5 ) x 24  21                               256  =  ( 1 + 3 + 7 + 21 ) x 8  22                               288  =  ( 1 + 11 ) x 24  23                               192  =  ( 1 + 23 ) x 8  24                                96  =  ( 1 + 3 ) x 24  25                               248  =  ( 1 + 5 + 25 ) x 8  26                               336  =  ( 1 + 13 ) x 24  27                               320  =  ( 1 + 3 + 9 + 27 ) x 8  28                               192  =  ( 1 + 7 ) x 24  29                               240  =  ( 1 + 29 ) x 8  30                               576  =  ( 1 + 3 + 5 + 15 ) x 24  31                               256  =  ( 1 + 31 ) x 8   32                                24  =    1 x 24  33                               384  =  ( 1 + 3 + 11 + 33 ) x 8  34                               432  =  ( 1 + 17) x 24  35                               384  =  ( 1 + 5 + 7 + 35 ) x 8  36                               312  =  ( 1 + 3 + 9 ) x 24  37                               304  =  ( 1 + 37 ) x 8   38                               480  =  ( 1 + 19 ) x 24  39                               448  =  ( 1 + 3 + 13 + 39 ) x 8  40                               144  =  ( 1 + 5 ) x 24  41                               336  =  ( 1 + 41 ) x 8  42                               768  =  ( 1 + 3 + 7 + 21 ) x 24  43                               352  =  ( 1 + 43 ) x 8  44                               288  =  ( 1 + 11) x 24  45                               624  =  ( 1 + 3 + 5 + 9 + 15 + 45) x 8

How to visualize the 288vertices in layer 22.  


Note that layer 22 with 288 vertices follows layer 21 with (1 + 3 + 7+ 21 ) x 8= 256 = 16x16 =2^8 vertices.


The notation in the following table is basedon the minimal norm of the D4 lattice being 1,in which case the D4 lattice is the lattice of integralquaternions. This is the second definition (equation 90) of theD4 lattice in Chapter 4 of Sphere Packings,Lattices, and Groups, 3rd edition, by Conway and Sloane (Springer1999), who note that the Dn lattice is the checkerboard lattice in ndimensions.

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                  6  127                             3,072                128 128                                24                  1 65,536=2^16                         24                  165,537                       1,572,912             65,538 2,147,483,647           51,539,607,552      2,147,483,6482,147,483,648=2^31                  24                  1

Prime Numbers

127 = 2^7 - 1 is a Mersenne prime, calledM7.

65,537 = 2^2^4 + 1 is the largest known Fermat prime. It is calledF4, but is not likely to be confused with the exceptional Lie algebraF4.

2,147,483,647 = 2^31 - 1 is a Mersenne prime. It was shown to beprime by Euler. It is called M31, but is not likely to be confusedwith the Andromeda galaxy M31.

(see Wells, The Penguin Dictionary of Curious and InterestingNumbers, Penguin, 1986)

If the D4 spacetime lattice length is taken to be the Plancklength, 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 x10^-33 cm or about 3.9 x 10^16 GeV, and

the layer of norm 2,147,483,647 is at a distancesqrt(2,147,483,647) = 46,340 x 10^-33 cm or about 2.2 x 10^14 GeV,and


at energies below about 10^16 GeV, continuous rotations can beapproximated by D4 lattice rotations to an accuracy of at least 2PI^2 / 1,572,912 = about 1.3 x 10^-5 steradians(4-dim), and

at energies below about 10^14 GeV, continuous rotations can beapproximated by D4 lattice rotations to an accuracy of at least 2PI^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 theMersenneprime 2^859433-1 (258716 digits) found by Slowinski and Gage in1994.



HOWEVER, since the D4-D5-E6-E7-E8VoDou Physics model is fundamentally a Planck Scale HyperDiamondLattice Generalized FeynmanCheckerboard model, it does violate LorentzInvariance at the Planck Scale, affecting UltraHigh Energy Cosmic Rays.

Click Here to See more about PrimeNumbers.

Richard Feynman said, in his book QED, Princeton 1985 at page129:

"... perhaps the idea that two points can be infinitely closetogether is wrong - the assumption that we can use geometry down tothe last notch is false. ..."

Why did Feynman fail in his efforts to generalizeto higher dimensions his successful 2=(1+1)dimensional Feynman Checkerboard?

Feynman got a nice representation of Dirac physics in 2-dim byusing a 2-dim lattice that is really the Gaussian integer lattice inthe Complex plane. His generalization to 4-dim was:

Feynman's attempted generalization failed because,instead of following his own suggestion that "... the idea that twopoints can be infinitely close together is wrong ...", he thoughtin terms of continuous light-cones and spheres, in which the onlydiscrete thing is the 0-dim sphere S0.

Details of Feynman's unproductive line of thought are inSchweber's book, QED, Princeton 1994, pages 406-407: "... Feynmanshowed that one could derive the Dirac equation in one-space-one-timedimension if the amplitude for a path with R corners is taken to be(i m epsilon)^R. Stated differently, each time the electron reversesspatial direction, it acquires a phase factor e^(i pi / 2) ...Feynman encountered difficulties in extending the idea of loadingeach turn through e^(i theta / 2) which worked in one-space dimensionto higher dimensions because in those situations the angles theta arein different planes. He tried to use quaternions and octonions(quaternions representing euclidean 4-dimensional rotations) torepresent wave functions, but he was not able to obtain a naturalrepresentation of the Dirac equation as an integral over path....".

More details are in his letter to Welton of 10 February 1947 (backin 1986, a friend of mine got a copy from the Caltech archives), inwhich Feynman says: "... we have at the point ... in the path to keeptrack of a rotation in three space (around the axis of the tangent tothe path) ... when I studied quaternions which I knew were designedto represent rotations I realized that they were the mathematicaltool in which to represent my thoughts. ... My purpose now is toconsider a path as a set of four functions Xm(s) = X,Y,Z,T(s) of aparameter s. Thus the speed need not be that of light ... The ideanow is to consider a path is a succession of 4vectors Vm(1), Vm(2),Vm(3) .. representing successive proper velocities (V_m V^m = 1) onthe path. Then each path represents a net Lorentz-Rotationtransformation. (I mean a combination of a Rot. and a Lorentz.) ...We desire a symbolism, which by analogy works in 4(3+1)space like thequaternions do in 3. I shall show that the symbolism is furnished byquaternions using complex numbers for components! ... (lets call themoctonions) ... Thus, a octonion is a quaternion whose coeficients areof the form a + Rb. In 3+1 space R^2 = -1 and we can take R = i if wewish, so octonions for Lorenzt Trans. and Rotations are quaternionswith complex (a + bi) coefficients. QED. ..."

Feynman gives an octonion basis that looks like the conventionalone: if the quaternion basis is {1,i,j,k}, then, if you let R = E, Ri= I, Rj = J, and Rk = K, you get for his basis {1,i,j,k,R,Ri,Rj,Rk}which looks like a conventional {1,i,j,k,E,I,J,K}.

However, Feynman does NOT use the octonions to represent an8-real-dimensional vector space: rather,

Feynman uses octonions to represent Rotations (by the {i,j,k}) andLorentz transformations (by the {Ri,Rj,Rk}).

That is related to a technical problem with Feynman's statement"... we can take R = i if we wish ... so octonions ... arequaternions with complex (a + bi) coefficients ..."

What I (and most other people) call octonionsare NOT complexified quaternions, but are a division algebra over theREAL number field. Therefore, R is NOT to be identified with theComplex imaginary i.

In fact, the i in the basis {1,i,j,k,R,Ri,Rj,Rk} is the Compleximaginary i, because {1,i} is a basis for the Complex subalgebra ofthe quaternions and of the Octonions, so the role of the Compleximaginary i is already filled.

In conventional math, the complexified quaternions (biquaternions,tensor product CxQ) DO exist, but they are just the algebra of 2x2Complex matrices, GL(2,C), so Feynman reinvented in his own way theresult that the Lorentz transformations (including Rotations) are inGL(2,C), and since Feynman's discussion could be restricted to unitmodulus,

Feynman's reinvention is really that the Lorentz transformations(including rotations) are SL(2,C), or, in conventional math language,that SO(3,C) is isomorphic to SL(2,C).

His mathematical result is correct (even though histerminology is NOT consistent with normal math terminology), butit did not help him at all with his physics problem, which wasgeneralizing his 2-dim Feynman Checkerboard to higher dimensions.

What Feynman should have done is to have followed his ownsuggestion that "... the idea that two points can be infinitely closetogether is wrong ..." and looked at discretestructures:

He would have found that 4-dim Quaternions don't quite workfor all 4 forces of physics, but that the 8-dimensional Octonioniclattice will give you all 4 forces of physics, and it naturallybreaks down into a 4-dim HyperDiamond physical spacetime lattice andan internal symmetry lattice, which is the HyperDiamondlattice version of the D4-D5-E6-E7-E8VoDou Physics model.

I did not fully realize what Feynman should have done until abouta year after I saw (in 1986) a copy of his 10 Feb 1947 letter toWelton, and then I did not understand my own stuff as well as I donow. Then (in 1987) I went to Caltech to tell Feynman that he shouldlook at discrete 4-dim quaternionic and 8-dim octonionic integers asthe proper generalization of his Gaussian Complex FeynmanCheckerboard lattice. He did not know me, and I just went to hisoffice at Caltech. He was there, but Helen Tuck told me to go away,so I did. Later I realized that my visit was shortly before his 4thcancer operation and only about half a year before his death in early1988, so that he was really quite sick and in a good deal of pain, soshe was being very protective of him.



References, Acknowledgements, etc:

Thanks to Liam Roche at for correcting someof my mistakes about prime numbers.



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