Tony's Home 240 Thoughts

Correlations and LieSphere Geometry,

historical/cultural studies such as the Kabbala,

mental phenomena such as Archetypes,and

physical phenomena of

| Gravity| Vacuum| BEC| Sonoluminescence| WaterStructure |

| AtomicClusters | Cellsand Chips | QiField | DNAand Genetic Code |

| QuantumComputing and Superluminal Information |Communication|

of a World in the Many-Worlds is approximated by the27-dimensional MACROSPACE is the space of 3x3 Hermitian matrices with3 Octonion spaces O4 ,O5, and O6,forming the Jordan algebra J3(O) with the symmetric product AB =(1/2)(AB + BA),

Re(O1) O4 O5 O4* Re(O2) O6 O5* O6* Re(O3)

**One 8-dimensional set** of neighbors differs from the GivenWorld only by an element of the Octonion spaceO5.

If it differs only with respect to the timelike real axis plusspacelike associative 3-space of the 8 dimensions ofO5, it differs with respect to4-dimensional PHYSICAL SPACETIME, whose geometry is related toGravity.

If it differs only with respect to the coassociative 4-space ofthe 8 dimensions of O5, it differs withrespect to 4-dimensional Internal Symmetry MICROSPACE, related to theconfiguration of the Gauge Bosons of the Color Force, the Weak Force,and Electromagnetism.

**A second 8-dimensional set** of neighbors differs from theGiven World only with respect to an element of the Octonion spaceO4. It differs with respect to theconfiguration of Fermion Particles.

**A third 8-dimensional set **of neighbors differs from theGiven World only with respect to an element of the Octonion spaceO6. It differs with respect to theconfiguration of Fermion AntiParticles.

**The other three 1-dimensional sets** of neighbors differingonly with respect to the Real Parts of Octonions Re(O1), Re(O2), andRe(O3) do not differ with respect to the configurations of spacetime,gauge bosons, or fermions, but only with respect to the way theyinteract in the 27-dimensional Jordan algebra J3(O). In other words,these neighbors are identical with respect to the 24-dimensionalChevalley subalgebra of J3(O), but not identical with respect to thefull Jordan algebra J3(O).

a Particle-AntiParticle pair (say, Electron E and Positron P)created at a single point of lattice Physicial Spacetime:

EP

The properties of E and P are Correlated because E and P aremirror images of each other with respect to physical properties likeelectric charge and spin orientation.

As time goes on (up in the illustration), E may move in onedirection and P in another direction. At the first time step, E and Pwill no longer be at the same point, but will be NearestNeighbors.

E P\ /EP

As time goes on, E and P may become widely separated by manyintervening points in lattice Spacetime:

E P\ /\ /\ /\ /EP

If E and P do not interact with other particles during theirjourney apart, they remain COMPLETELY CORRELATED - the physicalproperties of E and P remain MIRROR IMAGES of each other.

Even if they undergo interactions changing some of theirproperties, other properties may remain Correlated.

In any of the spaces of physics, what is

Start with the future light-cone Nearest Neighbors, each and allof them denoted by X, of a point denoted by * at an event that wewill take to be the origin event of the Sphere X

X X\ /*

In the continuum approximation of N-dimensional spacetime, theyform a Sphere, also denoted by X, that is the intersection of the(N-1)-dimensional future lightcone with an (N-1)-dimensionalspacelike hyperplane, so that the dimension of the Sphere X isN-2.

As time goes on toward the future, the Sphere X expands as it getsmore distant in time from the point * of origin.

X X\ /\ /\ /*

If another event happens at a point o of the Sphere X, that eventcan be considered to be the origin event of another Sphere Y

X Y YX\ \ /\ \ /\ o\ /\ /\ /*

Sphere Y is a smaller sphere nested within Sphere X, and tangentto it at point YX, which is so designated because it belongs to bothSphere Y and Sphere X. Since the Sphere Y is inside the Sphere X, thegeometry is that of oriented spheres, with the sphere of oppositeorientation to Sphere Y being tangent at the same point YX and of thesame radius as Sphere Y, but outside Sphere X.

The lightcones originating at * and at o are also nested, and aretangent to each other on the lightlike line from the point o to thepoint YX.

of the Conformal Group of N-dimensional spacetime. A goodintroductory paper is ConformalTheories, Curved Phase Spaces Relativistic Waveletsand the Geometry of Complex Domains, by R.Coquereaux and A. Jadczyk,Reviews in Mathematical Physics, Volume 2, No 1 (1990) 1-44, whichcan be downloadedfrom the web as a 1.98 MB pdf file. Lie Sphere Geometry is, asdescribed by Amassa Fauntleroy in his papers Projective Ranks ofHermitian Symmetric Spaces, Mathematical Intelligencer 15 (no. 2,1993) 27-32 and On the Projective Rank of Compact Hermitian SymmetricSpaces, North Carolina State Math preprint 1993, bilholomorphic tothe Hermitian Symmetric Space of Lie groupcosets

If N = 4, we have Spin(6) / (Spin(4)xSpin(2)) = SU(4) /(SU(2)xSU(2)xU(1)), which is biholomorphic to the Klein quadric Q4 inCP^5. As described by S. G. Gindikin (The Complex Universe of RogerPenrose, Mathematical Intelligencer 5 (no. 1, 1983) 27-35), eachpoint of the 4-complex-dimensional Klein quadric corresponds to aline in CP^3. The line in CP^3 is such that, if it passes through z,it also passes through z* (here * denotes complex conjugate). Througheach point z of CP^3, there is one and only one line also passingthrough z*. Therefore, all of CP^3 is the union of suchnon-intersecting lines, and CP^3 is fibred into a base manifold M andfibres which are the 1-complex-dimensional lines. The dimension of Mis the 3 complex dimensions of CP^3 less the 1 complex dimension ofeach line, so that M is 2-complex-dimensional, or 4-real-dimensional.All this is fundamental for the theory of Penrose Twistors, in whichthe Klein quadric Q4 is complexified compactified spacetime, and isrelated to the isomorphism between Hermitian Symmetric Spaces oftypes BDI(4) and AIII(2,2) due to

Penrose and Rindler (Spinors and space-time, vol. 1, Cambridge1984, 1986, section 5.12) show how to describe massive, as well asmassless, fields in terms of the lightcone structures of Lie SphereGeometry.

Tittel, Brendel, Gisin, Herzog, Zbinden, and Gisin, have reportedon their Experimentaldemonstration of quantum-correlations over more than 10 kilometers inquant-ph/9707042.

As Widom, Srivastava, and Sassaroli describe in their paperAcuasal Behaviorin Quantum Electrodynamics, quant-ph/9802056, "... conventionalquantum electrodynamics allows for interactions to proceed forwardand backward in time as well as space-like in direction."

Timelike trajectories can be described by a series of lightconesegments with changes in direction,

\ \ / /*

and spacelike trajectories can be described by a series oflightcone segments, some of which go backward in time (as in theStueckelberg-Feynman description of antiparticles as particlespropagating backward in time).

/\ / \ *

In their paper, Widom, Srivastava, and Sassaroli discuss "...engineering problems involved in experimentally testing suchcausality violations on a macroscopic scale ..." Their examplesinclude:

sending a superluminal signal, with only a few electrons so thatthe decay exponent is less than or equal to 1, down a 50 Ohm cable;and

an experiment with two photon coherent sources which areguaranteed to fire off exactly two photons at a time so that thecounting statistics at counter 1 depends on how one sets the positionof counter 2, via counter 2 events that are possibly space-like orpossibly in the future of counter 1 events.

Jack Sarfatti has long advocatedstudying acausal phenomena in terms of the Bohmformulation of Quantum Theory.

Bohm and Hiley (The Undivided Universe, Routledge 1993, section15.9) describe the Implicate Order of the Bohm formulation in termsof Lie Sphere Geometry as describing trajectories "... as a kind ofenfolded geometric structure whose meaning can be seen all at once asa 'chain' of successively contacting spheres. ... The first step inour trajectory is represented by the point of contact P and a sphereof radius r. The corresponding null ray is in the direction of theradius of the sphere at point P. The next null ray will berepresented by a larger sphere contacting the first sphere at thepoint Q. The next null ray will correspond to a still larger spherecontacting the second sphere at yet another point R. This procedureis to be continued indefinitely so that we obtain a completedescription of the zigzag trajectory. ... a sphere of infinite radius... represents a plane wave. ...

In our model, the implication parameter is just the radius of thesphere. ... we could have the radii of spheres decrease for a whileand then increase again. ... this corresponds to pair production. ...The Lagrangian is thus, in our approach, a property of the implicateorder which holds at any given moment. ... backward tracks in timeare replaced by tracks in which the implication parameter isdecreasing."

The equivalence of David Bohm's approachto the Many-Worlds approach has been noted byDavidDeutsch (The Fabric of Reality, Penguin 1997, pp. 93-94), inwhich he says: "... Bohm's theory is often presented as asingle-universe variant of quantum theory. ... Working out whatBohm's invisible wave will do requires the same computations asworking out what trillions of shadow photons will do. Some parts ofthe wave describe us, the observers, detecting and reacting to thephotons; other parts of the wave describe other versions of us,reacting to photons in different positions. ... in his theory realityconsists of large sets of complex entities, each of which canperceive other entities in its own set, but can only indirectlyperceive entities in other sets. These sets of entities are, in otherwords, parallel universes. ..."

The D4-D5-E6-E7-E8 VoDou Physicsmodel is fundamentally based on the Conformalstructure of Lie Sphere Geometry. It not only contains aConformal Physical Spacetime

but also is based on the OctonionicConformal structures

- tzimtzum (contraction or withdrawal) in which God as the Infinite (En Sof) withdraws into himself in order to make room for the creation, which occurs by a beam of light from the Infinite into the newly provided space;
- shevirat ha-kelim ("breaking of the vessels") in which he divine light is enclosed in finite "vessels," most of which break under the strain, and the catastrophe of the "breaking of the vessels" occurs; and
- tiqqun ("restoration") in which the divine realm itself is reconstructed, the divine sparks returned to their source, and primordial harmony is restored.

(Reference - Encyclopaedia Britannica Online.)

ARCHETYPES in the human mind may help humans understand the relationships among the various levels.

A few Archetypes ( each of which may contain substructures thatare also Archetypes ) are:

- Real Clifford Algebra 8-Periodicity, its Octagonal Mandala, I Ching, and IFA
- J3(O) and Sri Yantra and Leech Lattice
- Octonions and E8 Lattice
- 7-sphere and D4 and Lie Sphere for D5 / D4xU(1)
- 3-sphere and its Clifford-Hopf Fibration
- Beethoven111
- VoDou Chile
- Grosse Fugue
- Es MussSein!

AT LEVELS BELOW THE PLANCK ENERGY, our genetic code uses the Double Helix

as do the quantum transactions of the D4-D5-E6-E7-E8 VoDou Physics model. Another archetype for D4-D5-E6-E7-E8 VoDou physics is the geometrical structure of a generalized torus whose equator is the 7-sphere S7. It is S7 x S1 , the product of a 7-sphere S7 and a circle S1. It is a generalized torus because it is NOT the 8-dim torus T8, but when projected into lower dimensions it looks like the 2-torus T2 = S1 x S1. Here is a stereo image of the 2-torus, generated by the program 3D-Filmstrip for Macintosh by Richard Palais.

You can see the stereo with red-green or red-cyan 3D glasses. The program is on the WWW at http://rsp.math.brandeis.edu/3D-Filmstrip. If the horizontal equatorial S1 is expanded to S7, you have S7 x S1, the octonionic structure used in the D4-D5-E6-E7-E8 VoDou Physics model to describe 8-dimensional spacetime. If the horizontal equatorial S1 is expanded to S3, you have S3 x S1, which can describe our 4-dimensional spacetime. If the vertical S1 is taken to be time, flowing upwards, then our 3-dimensional space evolves as indicated roughly by this (730 k) binhexed QuickTime movie. Played in loop mode, it shows evolution through a large-scale closed timelike loop. Another useful low-energy archetype may be the 24-cell, the 4-dimensional regular polytope shown in this stereo view of a 3-D projection(from C program by Michael Gibbs)The 4th dimension is color-coded by blue = +, green = 0, and red = -.

Click here for 742k quicktime.mov animation (It can be played in a loop.)NOTE that this projection of a 24-cell contains a 3-dimensional cuboctahedron

which Fuller called a vector equilibrium and regarded as a fundamental archetype. The 24-cell tiles 4-dimensional space with a tiling that is useful in understanding the Octonion Mirrorhouse. Professor Koji Miyazaki at Kyoto has done much work on such archetypes, including their long history in Asian civilizations. AT THE PLANCK ENERGY LEVEL, useful algebraic structures areDivision Algebras such as the octonions and also Clifford Algebras and the sedenions. The 240-vertex Witting polytope is the 8-dimensional counterpart of the 24-cell. The 7-dimensional exotic spheres have as 3-dimensional counterparts the Poincare Dodecahedral Space. Its visualization in terms of platonic solids

is a useful archetype. It has been illustrated (as above) by Richard Hawkins and by Gerald de Jong, who call it a Mayan Time Star.

The natural fractal structure of octonions has been studied by

Onar Aam and by Girish Joshi. AT LEVELS ABOVE THE PLANCK ENERGY, a useful archetype may be the n-dimensional simplex, the high-dimensional version of the 1, 2, and 3 dimensional line, triangle, and tetrahedron as well as

the 4 dimensional pentahedron. WITH RESPECT TO INFORMATION ON ALL LEVELS, a useful archetype may be the 24-dimensional Leech lattice, currently being studied by Geoffrey Dixon, that is related to the extended Golay code G24 and the Monster finite simple group, a group of order 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71, or about 8 x 10^53. The Monster is naturally represented on a space of 196,884 dimensions, which can be broken down into 196,560 + 300 + 24. Since 300 = 25x24/2 is the symmetric square of 24, and since the Leech lattice discrete nearest-neighbor sphere has 196,560 points, and since IF the 196,560 points form a group, just as the 240 of an E8 lattice form unit octonions and as the 24 of a D4 lattice form unit quaternions then it should be possible to form the 196,560 dimensional space of the group algebra of the Leech lattice nearest-neighbor group, and then add the 300-dim space of symmetric squared Leech lattice, and then add the 24-dim space of the Leech lattice itself, to get the 196,884-dim representation space of the Monster. By going down to the underlying 24-dim Leech lattice space, it should be possible to represent the Monster on the 24-dim space of the Leech lattice. Not having a 24-dim or 196,560-dim or 196,884-dim picture, here is a Penrose tiling that is a low-dimensional slice thereof:

Quantum Information Theory is not the same as classical information theory. For example, Calderbank, Rains, Shor, and Sloane have shown that whereas many useful classical-error-correcting codes are binary over the field GF(2), quantum-error-correcting codes are quaternary over the field GF(4). As Dixon has noted, GF(4) is related to quaternions as GF(2) is to complex numbers and GF(8) is to octonions.

UNIVERSAL TRANSFORMATIONS are interpreted by IFA

and by the I CHING.

The 4x4x4 = 64 genetic code, the 2x2x2x2x2x2 = 64 I Ching, and the 8x8 = 64 D4-D5-E6-E7 physics model are all just different representations of the same fundamental structure. This fundamental structure is not only shared by Golay codes and Leech lattice but also by Penrose tilings and musical sequences.

| Gravity| Vacuum| BEC| Sonoluminescence| WaterStructure |

| AtomicClusters | Cellsand Chips | QiField | DNAand Genetic Code |

| QuantumComputing and Superluminal Information |Communication|

| NanoMachinesand the Second Law | DARPA|

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