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# on which Bohm-Sarfatti Back-Reaction acts

has

## Symmetric Space Geometry of MacroSpace

and has a Hydrodynamical Formulation that is useful in

## David Bohm's Quantum Theory

Some of the characteristics of Bohm's Quantum Potential Q and its corresponding Quantum Field PSI are described by Peter R. Holland in his book The Quantum Theory of Motion (Cambridge 1993):

• Q does not act within the 4-dimensional geometry of SpaceTime, it acts BEYOND the 4-dimensional geometry of SpaceTime on a particle of Matter/Energy to tell it how to move. Since Q acts BEYOND SpaceTime, it can and does establish NonLocal connections between different regions of SpaceTime.
• In Bohm's original formulation, there is NO reciprocal Back-Reaction of the particle of Matter/Energy on Q.
• Classical Potentials V (of Fields such as Electromagnetism) have Quantum Effects because the Quantum Force ( - grad Q ) depends on V, so that V acts on particles of Matter/Energy through the Quantum Potential Q.
• The amplitude of PSI = R exp( i 2 pi S / h ) is a Complex number, not a Real number.
• The influence of PSI = R exp( i 2 pi S / h ) on particles of Matter/Energy is INDEPENDENT of the intensity I = R^2 of PSI, because PSI acts through the Quantum Potential Q, and, if R is scaled to aR, the NonRelativistic Schrodinger Q remains unchanged:
Q = - ( h^2 / 2 m ) grad^2( aR ) / aR = - ( h^2 / 2 m ) grad^2( R ) / R
• If two Fields PSI1 and PSI2 do not overlap, their superposition PSI is the linear sum PSI1+PSI2, BUT IN GENERAL their superposition is NOT the linear sumPSI1+PSI2. Addition of a SMALL PSI1 to a pre-existing PSI with Quantum Potential Q can cause a LARGE nonlinear change in Q.
• In a Schrodinger picture of Relativistic Quantum Field Theory, the Wave Equation
d'Alembertian PSI = - d Q[PSI(x),t] / d PSI(x) |at PSI(x)=PSI(x,t)
The Energy E of the Quantum Field is continuously variable and NOT CONSERVED in general and
d E / d t = d Q / d t |at PSI(x)=PSI(x,t)
• The Relativistic Schrodinger Picture Wave Equation is NOT Lorentz Covariant.
Lorentz Covariance is only Statistically Valid, and is broken by Individual Quantum Processes.
• The Relativistic Schrodinger Picture Wave Equation is NonLinear and NonLocal.
Locality is only Statistically Valid, and Individual Quantum Processes are NonLocal.
• Although the Wave Equation for a Massless Quantum Field
d'Alembertian PSI = - d Q[PSI(x),t] / d PSI(x) |at PSI(x)=PSI(x,t)
is in general NonCovariant and NonLocal, there exist some particular Quantum Force States such that the Massless Quantum Field behaves as if it were a Classical Field with Mass. The Mass is not associated with a localized object, but is a property of the entire Field, so that, if the extent of the Field is the interior of a Kerr-Newman Black Hole, it might represent a Massive Particle such as an Electron.
• Bohm and Hiley, in their book The Undivided Universe (Routledge 1993) at pages 40-41, say that: "... R^2 has two interpretations, one through the quantum potential and the other through the probability density. It is our proposal the the more fundamental meaning [of R^2] ... is that it determines the quantum potential. ... its meaning as a probability is only secondary. ...|PSI|^2 has no necessary relationship to probability. ... however ... under typical ... conditions ... probability distribution P will approach and remain equal to |PSI|^2, the latter being an equilibrium distribution. ... a stochastic model ... gives an additional possible explanation of why P approaches |PSI|^2. ..."

Richard Feynman said, in his book QED, Princeton 1985 at page 129: "... If we make the minimum possible distance between two points as small as 10^(-100) centimeters ... the infinities disappear ... but the total probability of an event adds up to slightly more or less than 100%, or we get negative energies in infinitesimal amounts. It has been suggested that these inconsistencies arise because we haven't taken into account the effects of gravity ...".

Feynman discussed Negative Probabilities in his article of that title in the book Quantum Implications (Routledge and Kegan Paul 1987, pages 235-248), in which Feynman says "... all the results of quantum statistics can be described in classical probability language, ... provided we accept negative values for these probabilities. ...". However, Feynman preferred to formulate such things in amplitude language rather than probability language, saying: "... the equations with amplitudes are simpler and one can get used to thinking with them just as well. ...".

The Wigner distribution, devised by Wigner in 1932, can be used to visualize quantum trajectories. In their article in Physics Today, April 1998, pages 22-28, Liebfried, Pfau, and Monroe have some nice images of experimentally produced Wigner distributions showing negative probabilities. They describe an experiment using a single trapped ion performed by Wineland's group at the National Bureau of Standards in Boulder, and they also discuss a double-slit atomic beam experiment done at the University of Konstanz. Their figure 5,

from the experiment of Wineland's group, clearly shows that in some regions of position-momentum phase space there is negative probability.

An interesting description of Bohm's Theory
has been written by Deotto and Ghirardi.

The equivalence of David Bohm's approach to
the Many-Worlds approach has been noted by David Deutsch,
in his book The Fabric of Reality (Penguin 1997),
in which he says:

"... Bohm's theory is often presented as a single-universe variant of quantum theory. ... Working out what Bohm's invisible wave will do requires the same computations as working out what trillions of shadow photons will do. Some parts of the wave describe us, the observers, detecting and reacting to the photons; other parts of the wave describe other versions of us, reacting to photons in different positions. ... in his theory reality consists of large sets of complex entities, each of which can perceive other entities in its own set, but can only indirectly perceive entities in other sets. These sets of entities are, in other words, parallel universes. ..."

As Creon Levit says: "... Bohm's quantum potential (or quantum force, if you prefer to think with forces) is the integrated effect of all other universes on our own. ..."

One way to visualize Bohm's model, as well as the Q* = B correspondence of Jack Sarfatti, is to:

Let Q, the unfolding explicate order going from past to future, correspond to an ovulated egg cell;

Let Q*, the enfolding Super Implicate order from the future, correspond to the many sperm cells coming up to meet the egg cell; so that

the many sperm cells, corresponding to Q*, also correspond to the many beable possibilities B,

while

only one sperm cell in each World of the ManyWorlds would fertilize the egg and create new life.

Levit and Sarfatti have studied the possible equivalence of
the Bohm Quantum Potential
to
the phenomenological Bader Laplacian that describes
electronic charge density in small molecules.

The effectiveness of the

## constitutes Experimental Support for Bohm's theory.

Bohm's Hidden Variable papers I and II, published in Phys. Rev. 85 (1952) 166-93, written before QCD was known, do NOT explain WHY quarks are confined inside hadrons.

However, IF you ASSUME quark confinement by QCD,

THEN Bohm's Theory MAY explain why the NonRelativistic model of light-quark hadrons works so well

Consider paper II, section 5, (reprinted at page 387 of Quantum Theory and Measurement, edited by Wheeler and Zurek (Princeton 1983)):

"... A more striking illustration ... is afforded by the problem of a "free" particle contained between two impenetrable and perfectly reflecting walls, separated by a distance L. For this case, the spatial part of the PSI-field is

PSI = sin( 2 pi n x / L ),

where n is an integer and the energy of the electron is

E = ( 1 / 2 m ) ( n h / L )^2

Because the PSI-field is real, we deduce that the particle is at rest. Now, at first sight, it may seem puzzling that a particle having a high energy should be at rest in the empty space between two walls. Let us recall, however, that the space is not really empty, but contains an objectively real PSI-field that can act on the particle. Such an action is analogous to (but of course not identical with) the action of an electromagnetic field, which could create non-uniform motion of the particle in this apparently "empty" enclosure. We observe that in our problem, the PSI-field is able to bring the particle to rest and to transform the entire kinetic energy into potential energy of interaction with the PSI-field. To prove this, we evaluate the "quantum-mechanical potential" for this PSI-field

__^2                  __^2               /       \^2
- h^2    \/   R       - h^2    \/   PSI       1   |  n h  |
U = ______ _________  =  _______ ___________ =  _____ | _____ |
|       |
2 m      R            2 m      PSI         2 m  |   L   |
\       /

and note that it is precisely equal to the total energy, E. ..."

Conversely:

### the effectiveness of the NonRelativistic model of Light-Quark Hadrons may be considered to be experimental support for Bohm's theory.

The NonRelativistic model of Light-Quark Hadrons is described in many textbooks, including Gauge Field Theories (John Wiley 1991) by Mike Guidry , who says:

"By uncertainty principle agruments the momentum of a quark confined to the radius of one fermi is [about 200 MeV] ... For u or d quarks [the currrent mass is about 10 MeV or less and the constituent mass is about 300 MeV] ... and a nonrelativistic approximation is questionable ... relativity effects should be significant. Nevertheless, nonrelativistic models of quark structure for hadrons have been found to work surprisingly well, even for light hadrons. ..."

Perhaps, in addition to explaining why the nonrelativistic model of confined light-quark hadrons works so well,

## Bohm-Sarfatti Back-Reaction

Jack Sarfatti proposes combining Bohm's Theory with Back-Reaction, an idea useful in developing a Quantum Theory of Consciousness.

That idea seems to have been anticipated by Bohm, who, in his paper A new theory of the relationship of mind and matter, Philosophical Psychology, vol.. 3, no. 2, 1990, pp. 271-286, reprinted on The Neo-Noetics Page web site, said "... Let us now return to a consideration of the quantum theory. What is its relationship to the interweaving of the physical and the mental ...? First, let us recall that because the quantum potential may be regarded as information whose activity is to guide the "dance" of the electrons, there is a basic similarity between the quantum behaviour of a system of electrons and the behaviour of mind. But if we wish to relate mental processes to the quantum theory, this similarity will have to be extended. ... one could begin by supposing, for example, that as the quantum potential constitutes active information that can give form to the movements of the particles, so there is a superquantum potential that can give form to the unfoldment and development of this first order quantum potential. ...

[ My comment: Bohm's "superquantum potential" is Jack's "back-reaction", which can be realized physically from the Many-Worlds point of view by States of Consciousness using the Quantum Zeno effect and the Quantum Anti-Zeno effect to prune the branches of Worlds in the MacroSpace of the Many-Worlds. ]

... This latter would no longer satisfy the laws of the current quantum theory, which latter would then be an approximation ...

[ My comment: As Jack Sarfatti has pointed out, based on papers (Physics Letters A 156, No.1-2, June 1991 and Physics Letters A 158, No.1-2, August 1991) by Antony Valentini, conscious back-reaction could violate the assumption of equilibrium that current ordinary quantum theory uses to obtain the Born approximation that the square of the amplitude of the wave function gives the probability of experimental results. As Antony Valentini says in quant-ph/0203049, NonEquilibrium Quantum Processes can "... be used to send instantaneous signals, to violate the uncertainty principle, to distinguish non-orthogonal quantum states without disturbing them, to eavesdrop on quantum key distribution, and to read all the results of a parallel quantum computation. ... pilot-wave theory indeed allows ... one to consider arbitrary 'nonequilibrium' initial distributions ...". ]

... that which we experience as mind ... will ... move the body by reaching the level of the quantum potential and of the 'dance' of the particles. ... It is thus implied that in some sense a rudimentary mind-like quality is present even at the level of particle physics, and that as we go to subtler levels, this mind-like quality becomes stronger and more developed. ... there is no real division between mind and matter ... Extending this view, we see that each human being similarly participates in an inseparable way in society and in the planet as a whole. What may be suggested further is that such participation goes on to a greater collective mind, and perhaps ultimately to some yet more comprehensive mind in principle capable of going indefinitely beyond even the human species as a whole. ...".

### Jack Sarfatti's basic idea is:

Einstein's theory of General Relativity is the "two-way" physics of Matter/Energy and the 4-dimensional geometry of SpaceTime:

Geometry acts on Matter/Energy telling it how to move,

while Matter/Energy has a reciprocal Back-Reaction on Geometry telling it how to bend.

This bending of our four-dimensional SpaceTime Geometry explains the phenomena of Gravity, such as why an apple falls to the Earth and why there are Black Holes.

David Bohm's Quantum Field PSI of the Quantum Potential Q also tells Matter/Energy how to move, but in a way that is very different from the way General Relativity tells Matter/Energy how to move.

Bohm's Quantum Field PSI does not act within the 4-dimensional geometry of SpaceTime, it acts

BEYOND the 4-dimensional geometry of SpaceTime

on a particle of Matter/Energy to tell it how to move. Since the Quantum Field PSI acts BEYOND SpaceTime, it can and does establish NonLocal connections between different regions of SpaceTime.

In Bohm's original formulation, there is NO reciprocal Back-Reaction of the particle of Matter/Energy on the Quantum Field PSI.

Sarfatii's Back-Reaction model INTRODUCES such a reciprocal Back-Reaction of the particle of Matter/Energy on the Quantum Field PSI.

Sarfatti also notes that

Bohm's Quantum Field can beconsidered to be a Thought-like Quantum Information Field,

and then Sarfatti asks the question:

Why isn't Bohm's Quantum Field the explanation of how Mind fits into the rest of the universe?

IF A reciprocal Back-Reaction of the particle of Matter/Energy on the Quantum Field PSI is allowed,

Bohm's Quantum Field becomes the physical field of Thought.

Just as Einstein identified gravitation as the direct back-action of mass-energy on its spacetime geometry, Sarfatti has identified mind as the direct back-action of matter-geometry on its guiding quantum field.

The implications of this hypothesis are profound. One can now communicate and directly influence distant events faster than the speeding photon. The mind can effectively travel in time. This fits reports not only by Shamans, but also by the US Military-Intelligence Community on "remote-viewing". There are also consequences for NASA's mission to develop "breakthrough propellantless propulsion" to allow us to reach the stars and beyond.

## How do you build a concrete realization of Jack Sarfatti's application of Back-Reaction to Bohm's Quantum Theory?

First, you need a concrete description of the

## MacroSpace

and whose structure is described in terms of

## Symmetric Space Geometry

Bohm and Hiley (The Undivided Universe, Routledge 1993, Chapter 15) describe the

### Implicate Order of the Bohm formulation in terms of Lie Sphere Geometry

as describing trajectories "... as a kind of enfolded geometric structure whose meaning can be seen all at once as a 'chain' of successively contacting spheres. ... The Lagrangian is thus, in our approach, a property of the implicate order which holds at any given moment. ... backward tracks in time are replaced by tracks in which the implication parameter is decreasing. "

In the Lie Sphere geometry of Lie and Klein, the Implicate Order trajectories are related to Huygens' Principle. As Bohm and Hiley say (at pages 355-356 of The Undivided Universe): "... The process of enfoldment and unfoldment was already well known implicitly in the Huygens' construction. Waves from each point unfold. But at the same time waves from many points are enfolding to give rise to a new wave front. ... The Huygens' construction is actually the basis of the Feynman graphs ... consider waves which start at a point P and arrive at a point Q ... In the first interval of time, DELTA t , a possible path is from P to P', and in the second interval from P' to P'' and so on. ... the path eventually arrives at Q. The Huygens' construction implies that the waves that arrive at Q from P are built up of contributions from every possible path. ... Originally Feynman wanted to regard these paths are representing actual trajectories of particles. But ... various paths can interfere destructively as well as constructively. ... each path represents a contribution to the final field amplitude as implied by the Huygens' construction. .."

### Each trajectory of the Implicate Order only describes light-cone correlations using the Spacetime of One World.

To write the Bohm Quantum Potential for the Many-Worlds Sum Over Histories of All Trajectories, you need the Super Implicate Order. As Bohm and Hiley say at pages 378-379: "... Thus far we have been considering the implicate order mainly in relation to particle theories ... But when this field is quantized, a further kind of implicate order is introduced ... the Super Implicate Order. The super implicate order is related to the implicate order as the implicate order, in particle theories, is related to the particles. ..."

In my version of the Many-Worlds view,

### the Super Implicate Order shows the overall structure of the Many-Worlds,

just as Lie Sphere geometry shows how light-cone structures are correlated and organized. In other words, the geometry of the Super Implicate Order geometry should show how the Worlds of the Many-Worlds are organized, with each point of the Super Implicate Order geometry being one World.

In order to write the Bohm quantum potential in full detail, you need to have a concrete geometrical model of the Super Implicate Order, analogous to the Minkowski geometry of Spacetime and the Lie Sphere geometry of trajectories correlated by lightcone structure.

Just as the Lie Sphere geometry is based on Minkowski lightcones, the geometry of the Super Implicate Order should be based on, or at least consistent with, the Lie Sphere geometry and Minkowski Spacetime.

Maybe I am not doing justice to David Deutsch, on whose Many-Worlds model my thinking is based, but I do not know of any description by him of a geometry of the Super Implicate Order of the Many-Worlds beyond his comment on page 285 of The Fabric of Reality (Allen Lane, Penguin, 1997) that "...unlike spacetime, the multiverse does not consist of the mutually determining layers I have called super-snapshots, which could serve as "moments" of the multiverse. It is a complex, multi-dimensional jigsaw puzzle ... which neither consists of a sequence of moments nor permits a flow of time ..."

In my view, the Super Implicate Order has,

at the Nearest Neighbor lowest level of Interconnectedness, a 27-dimensional geometry based on the exceptional Jordan algebra of 3x3 Hermitian Octonion matrices and the 27-complex-dimensional symmetric space E7 / (E6xU(1)), and

at the Correlation middle level of Interconnectedness, a 28-dimensional geometry based on the 28-quaternionic-dimensional symmetric space E8 / (E7xSU(2)).

The 27-dimensional and 28-dimensional geometries correspond to the Stri Yantra, which in turn corresponds to StanTenen's geometry of the 22 Hebrew letters plus 5 Finals.

Another concrete candidate description for the geometry of the Super Implicate Order is given by Saul-Paul Sirag in his paper Consciousness: A Hyperspace View that is an appendix to The Roots of Consciousness by Jeffrey Mishlove (Council Oak 1993). As I understand it, he begins with the McKay correspondence between Finite Groups and Lie Groups, and then he uses the example of the 48-dim binary Octahedral Group OD and the 133-dim exceptional Lie Group E7. The 48-dim Octahedral Finite Group OD describes the physics of particles and forces in our spacetime, the Physical World, and the 133-dim Lie Group E7 describes the geometry of the Super Implicate Order, the Mental World. The whole structure of both things is then 48+133 = 181 dimensional, and is called the Spliced Bundle SB^181.

## | Einstein SpaceTime | MacroSpace - E7 and E8 |

### Einstein's Back-Reaction General Relativity is based on SpaceTime as the Geometric Object and Scalar Curvature of SpaceTime as the Geometric Variable.

In the Feynman Lectures on Gravitation (Addison-Wesley 1995, pp. 135-137), Feynman says (in the following I set lambda^2 = k, scalar curvature = R, and Ricci curvature tensor = R_mn ): "... It was Einstein's first guess that the stress-energy tensor was simply [proportional] to the Ricci tensor, k T_mn = R_mn. ... However ... Einstein finally chose

R_mn - (1/2) g_mn R = k T_mn

There is a good reason why this choice is better. If we take the covariant divergence ... the answer is identically zero. This means that the law of conservation of energy is a consequence simply of the form of the equation [and the Bianchi identities]. If we had set the stress-energy tensor equal to the Ricci tensor alone, the law of energy conservation would have been ... an additional requirement ... we shall play with the equations for a while. First of all, we will try to understand the relation ... to variational principles. ... we need an integral which is a scalar invariant. We choose ...

S_g = -(1/2k) INTEGRAL d^4x R sqrt(-g)

... The curvature tensor appears when we take the variation of S_g with respect to g_mn.

d S_g / d g_mn = -(1/2k) sqrt(-g) ( R^mn - (1/2) g^mn R )

... because the stress tensor appears in this way, from a variational principle, its covariant divergence is necessarily zero ... We have seen the connection from the other direction - that we could deduce a variational principle provided that we started from a divergenceless tensor. ..."

Therefore, Einstein-Hilbert General Relativity is based on the Curvature of SpaceTime and the SpaceTime Metric. Further, Einstein-Cartan General Relativity is based on the Curvature of SpaceTime, the SpaceTime Metric, and the SpaceTime Spin Connection.

Although Einstein-Cartan has Torsion and Spin phenomena that are not present in purely Metric Einstein-Hilbert, present-day experiments cannot distinguish between Einstein-Cartan and Einstein-Hilbert.

In both Einstein-Cartan and Einstein-Hilbert, the basic geometric object in the Lagrangian is the SpaceTime Curvature, which is roughly a measure of how much SpaceTime (Wick-rotated to its Euclidean version) resembles the 4-sphere S4 instead of flat Euclidean R4.

S4 = Spin(5) / Spin(4)

where Spin(5) = Sp(2)

and

Spin(4) = Sp(1)xSp(1) = SU(2)xSU(2) = Spin(3)xSpin(3) = S3xS3

Note that Spin(4) is the (Wick-rotated Euclidean version of) the Lorentz group that is the local symmetry group of SpaceTime.

Note also that if you do not Wick Rotate to S4, but use H4 (hyperbolic 4-space) with -+++ signature, you wind up using non-compact symmetric spaces like H4 instead of their compact dual spaces like S4. I find it easier to visualize compact spaces, so I sometimes wind up writing things like S4 for H4, which in my mind is really OK to do if the underlying structure is complex (where signature does not matter). Such an underlying complex structure does exist in my D4-D5-E6-E7-E8 VoDou Physics model, where, for example, SpaceTime is the Silov boundary of a bounded complex domain.

Note also that Spin(5) is the (Wick-rotated Euclidean version of) the de Sitter group from which the Einstein-Hilbert action can be derived by the MacDowell-Mansouri mechanism.

## For Sarfatti Back-Reaction with Bohm Quantum Theory,

the Geometric Object should be the Super Implicate Order, or

## MacroSpace,

and the range of the Geometric Variable should be a Generalized Curvature of the

## 27-complex-dimensional Symmetric Space E7 / E6xU(1)

whose Algebra is described by E8 / E7xSU(2)

The Symmetric Space E7 / (E6 x U(1)), and its related Bounded Complex Domain and Shilov Boundary, seems to me to be "natural" for several points of view:

Since the Lorentz Group Spin(4) corresponds to global rotations in flat SpaceTime, and since E6 is the global symmetry group of the D4-D5-E6 physics model, let E6 play the role of Einstein's Spin(4).

Since SpaceTime is the Geometric arena for General Relativistic Gravity, and since the MacroSpace of the Super Implicate Order of Bohm's Theory is the Geometric arena (at the Nearest Neighbor level of Interconnectedness), let 27-dimensional MacroSpace Geometry play the role of Einstein's SpaceTime S4 = Spin(5) / Spin(4) curvature.

MacroSpace is approximated by continuous 27-dimensional 3x3 Hermitian Octonion Matrices and by the 27-complex dimensional E7 / (E6 x U(1)) which is the 133-78-1 = 54-real-dimensional set of complexified Octonion projective planes (CxO)P2 that are in the octonionified Octonion projective plane (OxO)P2.

Now we have E7 / (E6 x U(1)) playing the role of Spin(5) / Spin(4) and E6 playing the role of Spin(4).

Since the U(1) of E7 / (E6 x U(1)) just means that MacroSpace is a space of 27 complex dimensions while SpaceTime is a space of 4 real dimensions:

The group E7 should play the role of Einstein's Spin(5), the (Wick-rotated Euclidean version of) the de Sitter group from which the Einstein-Hilbert action can be derived by the MacDowell-Mansouri mechanism.

Therefore:

### "... What I think Deutsch's multiverse really is, is ... a stack of snapshots ... all connected together by Threads ..."

To me, the Geometric MacroSpace Structure shows how the Threads connect the snapshots.

### The Threads can be thought of as Strings in a 26-dimensional subspace of the 27-dimensional MacroSpace.

At short distances, there are two fundamental types of Threads:

Timelike Link between two successive Points of a World-Line of one Fermion Particle; and

Null Link of a Massless Gauge Boson between a Source Particle and a Sink Particle.

Note that a Spacelike | Link can be made of two Null Links, |\ one / Future and one \ Past |/ Timelike __ Links and Null / Links ___ can form a 2-dim Loop with signature 1+1 /__/

The (1+1)-dim Loop encloses a String Theory World-Sheet. To specify the location of the Loop in Physical SpaceTime requires 4 more dimensions. To specify Internal Symmetry Space requires 4 more dimensions. To specify Fermion Particle and Antiparticle type requires 8+8 = 16 more dimensions. The total dimensionality of String Theory space is 26, with signature (1, 1+4+4+8+8) = (1,25).

At longer distances, Threads can merge, and bifurcate, and intersect. This can produce World-Sheets that are not Smooth Manifolds, but containing Singularities.

V. I. Arnold (Remarks on the Stationary Phase Method and Coxeter Numbers, Russian Math. Surveys 28 no. 5 (1973) 19-48; and Catastrophe Theory, 2nd ed., Springer-Verlag (1986)) has shown that Simple Singularities are classified by the A-D-E Classification. Arnold also points out that the A-D-E classification appears in such apparently (but not really) diverse areas as critical points of functions, Lie algebras, categories of linear spaces, caustics, wave fronts, regular polyhedra in 3-dimensional space, and Coxeter crystallographic reflection groups.

The A-D-E Classification. also appears in superstring theory. Michio Kaku (Strings, Conformal Fields, and Topology, An Introduction, Springer-Verlag (1991)) describes how the A-D-E classification appears in superstring conformal field theory, being in 1-1 correspondence not only with the modular invariants of SU(2)k, but also with the special solutions of solutions of c=1 theory for two continuous classes and the three discrete solutions. Kaku says that this is because of the correspondence between the simply laced groups and the finite subgroups of SU(2).

The A-D-E Classification also classifies Quivers of Arrows. If the Multiverse Snapshots are thought of as Points, and short Threads from Point to Point are thought of as Arrows from a Point at the Tail of the Arrow to a Point at the Head of the Arrow, then you can form Quivers of Arrows. In 1972, Peter Gabriel represented a quiver (P, A, t, h) by representing the Points as Complex vector spaces and the Arrows A as matrix maps from the Complex vector space representing the Tail to the Complex vector space representing the Head, and proved Gabriel's theorem:

If a connected Quiver has only finitely many non-isomorphic indecomposable representations, its graph is a Coxeter-Dynkin diagram of one of the Lie algebras An, Dn, E6, E7, or E8, and there is a 1-1 correspondence between the classes of isomorphic indecomposable representations and the positive roots of that Lie algebra.

Represent the 27-dimensional MacroSpace E7 / E6xU(1) by 3x3 Hermitian Octonionic matrices

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

which form the exceptional Jordan algebra J3(O).

In the D4-D5-E6-E7-E8 VoDou physics model, Jordan algebras correspond to the matrix algebra of quantum mechanical states, that is, from a particle physics point of view, the configuration of particles in spacetime upon which the Lie algebra gauge groups act.

Look at the traceless subalgebra J3(O)o that is 26-dimensional.

### The mathematical structure of 26-dimensional String Theory describes the Structure of the MacroSpace,

which is the Space of Possible Outcomes in Jack Sarfatti's Back-Reaction model.

Since

the String Theory action is similar to the Einstein-Hilbert action, and

Jack Sarfatti's Back-Reaction in the space of possible outcomes is motivated by analogy with Einstein's gravitation theory, and

if you look at quantized relativistic Strings in 26-dimensional space of signature (25,1), you have, as stated by Kaku in his book Strings, Conformal Field Theory, and Topology (Springer-Verlag 1991 page 13), "... Lorentz covariance is manifest in the Gupta-Bleuler formalism but unitarity is not ... [in the Light-Cone] quantization scheme ... only the physical states are present and unitarity is manifest. ...",

it seems that

### 26-dimensional Unoriented Closed Bosonic String Theory gives a unitary model of Jack Sarfatti's Back-Reaction model of MacroSpace phenomena including Quantum Consciousness.

Resonant Connections in MacroSpace may be useful in making a Conscious Selection of Fates among the Many Fates of the Many Worlds.

As discussed by Kaku in his books Strings, Conformal Field Theory, and Topology (Springer-Verlag 1991) and Introduction to Superstrings (Springer-Verlag 1988), in 26-dimensional String Theory:

• the action is the area of the world-sheet swept out by the string, and can be written as
S = - (1 / 4 pi a) INT d^2x sqrt(g) g^ab (d_aX_m) (d_bX_n) N^mn
where a is 1/2 for open strings and 1/4 for closed strings, g^ab is the metric tensor on the world-sheet surface, N^mn is the flat metric in 26-dimensional space with signature (25,1), x coordinates are world-sheet coordinates, and X coordinates are 26-dimensional space coordinates;
• the action can be invariant under 2-dimensional general coordinate transformations because the sqrt(g) factor cancels against the transformation of the 2-dimensional measure;
• the action is classically invariant under local scale transformations, and, after quantization, the conformal anomaly resulting from breakdown of the classical scale invariance disappears (for the Bosonic String) only in 26 dimensional space;
• after quantization, which can be done by Gupta-Bleuler, Light-Cone, or BRST methods, 26-dimensional Bosonic String Theory is seen to be Lorentz invariant, Conformal invariant, and Unitary with no non-physical states;
• Bosonic String interactions can be represented as an S matrix for which the Euler Beta Function is the lowest order term in a perturbation series that is a path integral summed over all conformally inequivalent Riemann surfaces;
• Light-Cone coordinates can be used, with twists, string lengths, and propagation times, to find specific moduli for high genus Riemann surfaces, thus solving the problem of triangulation of moduli space (whose dimension is 6g - 6 + 2N, where g is the genus and N is the space dimension so that here N = 26);
• the Unoriented Closed Bosonic String spectrum
contains Tachyons with imaginary mass, massless (at tree-level, but not necessarily after dynamical processes are considered) scalar Dilatons, and massless spin-2 MacroSpace Gravitons);
• Since Bosonic Unoriented Closed String Theory describes the physics of MacroSpace, there is no need to put in supersymmetry to make superstrings to get fermions - the World-Line Strings of MacroSpace are not fermionic; and
• Since the 26-dimensional subspace of MacroSpace is naturally 26-dimensional, there is no need to go to second quantization (string field theory) in order to reduce its dimensionality.

Bosonic String Theory is related to the Large N limit of the AN Lie Algebras.

For a nice introductory discussion of the mathematics of Bosonic Closed Strings, see Week 126 and Week 127 and other relevant works of John Baez.

## Here is how I got to E7, and then on to E8:

E7 comes from E6:

I want a space that has local E6 symmetry, so I am looking for an irreducible symmetric space G / K such that K is E6 itself, or E6 times another Lie group.

The irreducible symmetric spaces have been completely classified, and the only such non-trivial irreducible symmmetric space is

E7 / E6xU(1)

The U(1) means that the 54-real dimensional symmetric space E7 / E6xU(1) is really a complex space of 27 complex dimensions.

E6 comes from my D4-D5-E6-E7-E8 VoDou Physics model:

Since I think that Lagrangian formulations are fundamentally nice, I want a structure that has all the parts that I need to build a Lagrangian for the Standard Model plus Gravity. There are at least 3 parts:

Gauge Group, Fermions, and SpaceTime

The Gauge Group should have at least 12 dimensions for the Standard Model U(1)xSU(2)xSU(3) plus 16 dimensions for a U(2,2) group from which I can get Gravity by gauging its 15-dim SU(2,2) conformal subgroup (this is well known, if not widely known, being described for example in the textbook by Mohapatra). The 12+16-dim = 28-dim Lie group that I use is Spin(8), which has octonionic structure. Note that Spin(8) is NOT simply the Cartesian product of U(1)xSU(2)xSU(3) and U(2,2). They come out of Spin(8) in a complicated way that is NOT standard, but is related to dimensional reduction of spacetime in the D4-D5-E6-E7-E8 VoDou physics model.

The first-generation Fermion particles should be represented by at least an 8-dim space, whose basis I identify in the D4-D5-E6-E7-E8 VoDou physics model with octonions as follows:

• electron; E
• red, blue, green up quark; i, j, k
• red, blue, green down quark; I, J, K
• electron neutrino 1

The particles correspond to (left-handed) Spin(8) half-spinors. Anti-particles come from (right-handed) mirror-image Spin(8) half-spinors. The 7 right-handed particle states and 7 left-handed antiparticle states come from the tree-level mass of the 7 fermions with tree-level mass. (The neutrino is massless at tree-level. A small neutrino mass can come from non-tree-level effects.) The second and third generations (but no further generations) come from dimensional reduction of spacetime in the D4-D5-E6-E7-E8 VoDou physics model.

4-dim SpaceTime is a dimensionally reduced version of the 8-dim spacetime of the D4-D5-E6-E7-E8 VoDou physics model that is also octonionic, and corresponds to the 8-dim vector representation of Spin(8). The 4-dim spacetime is acted on by the U(2,2) that gives gauge gravity. The other 4 dimensions form the internal symmetry space that is acted on by the Standard Model U(1)xSU(2)xSU(3).

So far, we have:

• 28 - dim Gauge Group
• 8 - dim fermion particles
• 8 - dim fermion antiparticles
• 8 = 4-dim spacetime plus 4-dim internal symmetry space

52 - dim total "structure" for building the Lagrangian.

Now, I want this 52-dim stuff to have Lie group structure. There is in fact a 52-dim exceptional Lie group F4 and it does have such structure and it was the basis of my first attempt at a physics model. However, my F4 model was deficient, because my calculations of particle masses and force strength constants required that spacetime have an underlying complex structure (that is one reason that I like Wick rotations back and forth between Euclidean and Minkowski spacetime).

If I give 4-dim spacetime complex structure, then I should give complex structure to the 4-dim internal symmetry space, since they both come from the same 8-dim parent "spacetime".

If I give the 8-dim "spacetime" complex structure, then I should also give complex structure to the two 8-dim fermion representation spaces, because all three 8-dim spaces are isomorphic by the triality automorphism of the three 8-dim representations of Spin(8).

This does NOT require me to give complex structure to the Gauge Groups, so now I have

• 28 - dim Gauge Group
• 32 = 16+16 dim fermion particles and antiparticles, "complexified"
• 16 = 4-dim spacetime plus 4-dim internal symmetry space, "complexified"

76 - dim total "structure" for building the Lagrangian.

If you add in a U(1) for the complex symmetry of the 16-dim complex "spacetime" and a U(1) for the complex symmetry of the 32-dim fermion representation space, then you get the

76 + 1 + 1 = 78 - dim Lie group E6.

That is where E6 comes from. Since the E6 now contains all the parts of the Lagrangian, including the needed complex structure, I see E6 as the symmetry of the D4-D5-E6-E7-E8 VoDou hysics model.

If you get a quantum theory by Many-Worlds path-integral-sum, you see that each world or snapshot has E6 symmetry, so that a Geometric Structure of all the worlds should have local E6 symmetry.

Therefore, the E7 global Geometric MacroSpace Structure ought to be

### MacroSpace according to Wheeler and Thorne:

In Wheeler's picture, as described by Kip Thorne in his book Black Holes and Time Warps (Norton 1994), MacroSpace (called superspace by them) is a collection of 3-dim spatial spaces. As Thorne says on page 476: "... Quantum Gravity then [at the Planck energy] radically changes the character of spacetime. It ruptures the unification of space and time into spacetime. It unglues space and time from each other, and then destroys time as a concept and destroys the definiteness of space. Time ceases to exist ... Space, the sole remaining remnant of what was once a unified spacetime, becomes a random, probabilistic froth, like soapsuds. ..."

I do not like the Wheeler/Thorne picture, because its 3-dim space remains a coherent (although perhaps multiply connected) entity above Planck energy, while time is done away with. I think that the 3 spatial dimensions would cease to exist as much as time would cease to exist, and in support of my view I note that, at black hole horizons, time dimensions can become spatial, etc.

### MacroSpace according to Deutsch:

David Deutsch's multiverse is made up of snapshots that are each 3-dim spaces. Deutsch, like Wheeler and Thorne, uses 3-dim spaces as basic building blocks. Deutsch says on page 278 of Fabric of Reality (Allen Lane - Penguin 1997): "... there is no fundamental demarcation between snapshots of other times and snapshots of other universes. ... Other times are just special cases of other universes ... distinguished from other universes ... only in that they are especially related to ours the laws of physics ... the rest of the multiverse ... impinges on us very weakly by comparison, through interference effects. ..." On page 283, Deutsch says: "... in some regions of the multiverse, and in some places in space, the snapshots of some physical objects do fall, for a period, into chains, each of whose members determines all the others to a good approximation. ... In those regions and places, the multiverse does indeed look as ... a collection of spacetimes, and .. [one] can distinguish approximately between different times and different universes, and time is approximately a sequence of moments. But that approximation always breaks down if one examines the snapshots in more detail, or looks far forwards or backwards in time, or far afield in the multiverse. ..."

My problem with the Deutsch multiverse is not that is incorrect, but that it is incomplete.

When Deutsch says "... if one ... looks ... far afield in the multiverse ...", Deutsch does not define what "far afield" means. His agglomeration of snapshots does not have a mathematically well-defined overall geometric structure that allows him to define "far afield".

In my opinion, Bohm's Quantum Field, with its Implicate Order and Super Implicate Order is what is needed to give the multiverse an overall mathematical geometrical structure on which Sarfatti Back-Reaction can be formulated.

### E7 / E6xS1 MacroSpace:

The overall mathematical geometrical multiverse structure (which I call a MacroSpace, from the macrosphere in Greg Egan's sci-fi novel Diaspora) that I use is the 27-complex-dimensional symmetric space E7 / E6xS1 = E7 / E6xU(1). Each Point of the MacroSpace E7 / E6xU(1) is one Universe.

Each Point-Universe is NOT just a 3-dimensional spatial physical space, or a 4-dimensional physical SpaceTime, but is one Possible Configuration of elementary particle Fermions and Gauge Bosons on all the Points of an entire SpaceTime.

Fundamentally, I regard the SpaceTime as a Planck-length HyperDiamond Lattice, but what I am describing here is a continuous manifold approximation.

Two given Point-Universes may overlap each other not only by having coincident SpaceTime points, but also by having similar configurations of Fermions and Gauge Bosons.

How are these configurations and their Bohm Quantum Force interactions to be described mathematically?

MacroSpace E7 / E6xU(1) has two local symmetry groups: U(1) for its complex structure; and E6 for its physical local symmetry group.

Just as a Standard Model gauge group acts as local symmetry group independently at each point of SpaceTime and the Standard Model forces arise from the differences between the gauge group states, or phases, at different SpaceTime points, so the Bohm Quantum Forces should arise from the differences between the E6 states, or phases, at different multiverse MacroSpace points.

The 78-real-dimensions of E6, and therefore the E6 states or phases, have the following physical content in my D4-D5-E6-E7-E8 VoDou physics model:

• 33 real dimensions give an 8-complex-dimensional space of Fermion particles an 8-complex-dimensional space of Fermion antiparticles, and a 1-real-dimensional U(1) for the complexification;
• 17 real dimensions give an 4-complex-dimensional physical SpaceTime, a 4-complex-dimensional Internal Symmetry Space, and a 1-real-dimensional U(1) for the complexification;
• 28 real dimensions give 12-real-dimensional Standard Model SU(3)xSU(2)xU(1); a 15-real-dimensional SU(2,2) = Spin(4,2) Conformal Gravity; and a 1-real-dimensional U(1) for complex propagator phase.

Therefore:

each Point-Universe differs from another Point-Universe by their E6 state/phase difference, and all the Point-Universes, taken together, have the overall geometric structure of the MacroSpace E7 / E6xU(1).

27 = 8+8 8 3

The 27-Complex-dimenisonal space has 3 Octonionic 8-dimensional subspaces, corresponding to:

• first generation fermion particles,
• their antiparticles, and
• physical spacetime plus internal symmetry space

The remaining 3 dimensions are the E7 MacroSpace Embedding Dimensions in which the 24-dimensional subspaces are embedded in the full 27-dimensional space.

### Click here to see more detailed Geometric and Algebraic Structure of the MacroSpace of Many-Worlds,

both of which are combined in E8 by the fibration E8 / E7 x SU(2)

and describable in terms of Bosonic Unoriented Closed String Theory.

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