Tony's Home

on which Bohm-Sarfatti Back-Reactionacts

has

Symmetric Space Geometry ofMacroSpace

and has a Hydrodynamical Formulation that is usefulin

DavidBohm's Quantum Theory

Some of the characteristics of Bohm'sQuantum Potential Q and its corresponding Quantum Field PSI aredescribed by Peter R. Holland in his book The Quantum Theory ofMotion (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) |atPSI(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 isbroken by Individual Quantum Processes.
• The Relativistic Schrodinger Picture Wave Equation is NonLinear and NonLocal.
Locality is only Statistically Valid, and IndividualQuantum Processes are NonLocal.
• Although the Wave Equation for a Massless Quantum Field
d'Alembertian PSI = - d Q[PSI(x),t] / d PSI(x) |atPSI(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 variantof quantum theory. ... Working out what Bohm's invisible wave will dorequires the same computations as working out what trillions ofshadow photons will do. Some parts of the wave describe us, theobservers, detecting and reacting to the photons; other parts of thewave describe other versions of us, reacting to photons in differentpositions. ... in his theory reality consists of large sets ofcomplex entities, each of which can perceive other entities in itsown set, but can only indirectly perceive entities in other sets.These sets of entities are, in other words, parallel universes...."

As Creon Levitsays: "... Bohm's quantum potential (or quantum force, if you preferto think with forces) is the integrated effect of all other universeson our own. ..."

One way to visualize Bohm's model, as well asthe Q* = B correspondence of JackSarfatti, 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 cellscoming up to meet the egg cell; so that

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

while

only one sperm cell in each World of the ManyWorlds wouldfertilize 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

constitutesExperimental 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 explainWHY quarks are confined inside hadrons.

However, IF you ASSUME quark confinement by QCD,

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

Consider paper II, section 5, (reprinted at page 387 of QuantumTheory and Measurement, edited by Wheeler and Zurek (Princeton1983)):

"... A more striking illustration ... is afforded by the problemof a "free" particle contained between two impenetrable and perfectlyreflecting walls, separated by a distance L. For this case, thespatial 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 atrest. Now, at first sight, it may seem puzzling that a particlehaving a high energy should be at rest in the empty space between twowalls. Let us recall, however, that the space is not really empty,but contains an objectively real PSI-field that can act on theparticle. Such an action is analogous to (but of course not identicalwith) the action of an electromagnetic field, which could createnon-uniform motion of the particle in this apparently "empty"enclosure. We observe that in our problem, the PSI-field is able tobring the particle to rest and to transform the entire kinetic energyinto potential energy of interaction with the PSI-field. To provethis, we evaluate the "quantum-mechanical potential" for thisPSI-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-QuarkHadrons may be considered to be experimental support for Bohm'stheory.

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

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

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

Bohm-SarfattiBack-Reaction

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

That idea seems to have been anticipated by Bohm,who, in his paper A new theory of the relationship of mind andmatter, Philosophical Psychology, vol.. 3, no. 2, 1990, pp. 271-286,reprinted on TheNeo-Noetics Page web site, said "... Let us now return to aconsideration of the quantum theory. What is its relationship tothe interweaving of the physical and the mental ...? First, letus recall that because the quantum potential may be regarded asinformation whose activity is to guide the "dance" of the electrons,there is a basic similarity between thequantum behaviour of a system of electrons and the behaviour ofmind. But if we wish to relate mental processes to thequantum theory, this similarity will have to be extended. ... onecould begin by supposing, for example, that as the quantumpotential constitutes active information that can give form to themovements of the particles, so there is a superquantumpotential that can give form to the unfoldment and development ofthis 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 currentquantum 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 byreaching the level of the quantum potential and of the 'dance' of theparticles. ... It is thus implied that in some sense a rudimentarymind-like quality is present even at the level of particle physics,and that as we go to subtler levels, this mind-like quality becomesstronger and more developed. ... there is no real division betweenmind and matter ... Extending this view, we see that each human beingsimilarly participates in an inseparable way in society and in theplanet as a whole. What may be suggested further is that suchparticipation goes on to a greater collective mind, and perhapsultimately to some yet more comprehensivemind in principle capable of going indefinitely beyond even the humanspecies as a whole. ...".

Jack Sarfatti's basic idea is:

Einstein's theory of General Relativity is the "two-way" physicsof 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 Geometrytelling it how to bend.

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

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

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

BEYOND the 4-dimensional geometry ofSpaceTime

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

In Bohm's original formulation, there is NO reciprocalBack-Reaction of the particle of Matter/Energy on the Quantum FieldPSI.

Sarfatii's Back-Reaction model INTRODUCES such areciprocal Back-Reaction of the particle of Matter/Energy on theQuantum Field PSI.

Sarfatti also notes that

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

and then Sarfatti asks the question:

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

IF A reciprocalBack-Reaction of the particle of Matter/Energy on the Quantum FieldPSI is allowed,

Bohm's Quantum Field becomes the physical field ofThought.

Just as Einstein identified gravitation as the direct back-actionof mass-energy on its spacetime geometry, Sarfatti has identifiedmind as the direct back-action of matter-geometry on its guidingquantum field.

The implications of this hypothesis are profound. One can nowcommunicate and directly influence distant events faster than thespeeding photon. The mind can effectively travel in time. This fitsreports not only by Shamans, but also by theUS Military-Intelligence Community on "remote-viewing". There arealso consequences for NASA's mission to develop "breakthroughpropellantless propulsion" to allow us to reach the stars andbeyond.

How do you build a concrete realization of JackSarfatti's application of Back-Reaction to Bohm's QuantumTheory?

First, you need a concrete description of the

MacroSpace

and whose structure is described in terms of

Symmetric SpaceGeometry

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

Implicate Order of theBohm formulation in terms of Lie Sphere Geometry

as describing trajectories "... as a kind of enfolded geometricstructure whose meaning can be seen all at once as a 'chain' ofsuccessively contacting spheres. ... The Lagrangian is thus, in ourapproach, a property of the implicate order which holds at any givenmoment. ... backward tracks in time are replaced by tracks in whichthe implication parameter is decreasing. "

In the Lie Sphere geometry of Lie and Klein, the Implicate Ordertrajectories are related to Huygens'Principle. As Bohm and Hiley say (at pages 355-356 of TheUndivided Universe): "... The process of enfoldment and unfoldmentwas already well known implicitly in the Huygens' construction. Wavesfrom each point unfold. But at the same time waves from many pointsare enfolding to give rise to a new wave front. ... The Huygens'construction is actually the basis of the Feynman graphs ... considerwaves which start at a point P and arrive at a point Q ... In thefirst 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 patheventually arrives at Q. The Huygens' construction implies that thewaves that arrive at Q from P are built up of contributions fromevery possible path. ... Originally Feynman wanted to regard thesepaths 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 amplitudeas implied by the Huygens' construction. .."

Each trajectory of the Implicate Order only describeslight-cone correlations using the Spacetime of OneWorld.

To write the Bohm Quantum Potential for the Many-Worlds Sum OverHistories of All Trajectories, you need the Super Implicate Order. AsBohm and Hiley say at pages 378-379: "... Thus far we have beenconsidering the implicate order mainly in relation to particletheories ... But when this field is quantized, a further kind ofimplicate order is introduced ... the Super Implicate Order. Thesuper implicate order is related to the implicate order as theimplicate 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 ofthe Many-Worlds,

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

In order to write the Bohm quantum potential in full detail, youneed to have a concrete geometrical model of the Super ImplicateOrder, analogous to the Minkowski geometry of Spacetime and the LieSphere geometry of trajectories correlated by lightconestructure.

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

Maybe I am not doing justice to David Deutsch, on whose Many-Worldsmodel my thinking is based, but I do not know of any description byhim of a geometry of the Super Implicate Order of the Many-Worldsbeyond his comment on page 285 of The Fabric of Reality (Allen Lane,Penguin, 1997) that "...unlike spacetime, the multiverse does notconsist of the mutually determining layers I have calledsuper-snapshots, which could serve as "moments" of the multiverse. Itis a complex, multi-dimensional jigsaw puzzle ... which neitherconsists 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, a27-dimensional geometry basedon the exceptional Jordan algebra of 3x3 Hermitian Octonionmatrices and the 27-complex-dimensionalsymmetric space E7 / (E6xU(1)), and

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

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

Another concrete candidate description forthe geometry of the Super Implicate Order is given by Saul-Paul Siragin his paper Consciousness: A Hyperspace View that is an appendix toThe Roots of Consciousness by Jeffrey Mishlove (Council Oak 1993). AsI understand it, he begins with the McKaycorrespondence between Finite Groups and Lie Groups, and then heuses the example of the 48-dim binary Octahedral Group OD and the133-dim exceptional Lie Group E7. The 48-dim Octahedral Finite GroupOD describes the physics of particles and forces in our spacetime,the Physical World, and the 133-dim Lie Group E7 describes thegeometry of the Super Implicate Order, the Mental World. The wholestructure of both things is then 48+133 = 181 dimensional, and iscalled the Spliced Bundle SB^181.

| Einstein SpaceTime |MacroSpace - E7 and E8 |

Einstein's Back-Reaction General Relativity is based onSpaceTime as the Geometric Object and Scalar Curvature of SpaceTimeas 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, scalarcurvature = R, and Ricci curvature tensor = R_mn ): "... It wasEinstein'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 thecovariant divergence ... the answer is identically zero. This meansthat the law of conservation of energy is a consequence simply of theform of the equation [and the Bianchi identities]. If we hadset the stress-energy tensor equal to the Ricci tensor alone, the lawof energy conservation would have been ... an additional requirement... we shall play with the equations for a while. First of all, wewill 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_gwith 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 avariational principle, its covariant divergence is necessarily zero... We have seen the connection from the other direction - that wecould deduce a variational principle provided that we started from adivergenceless tensor. ..."

Therefore, Einstein-Hilbert General Relativity is based on theCurvature of SpaceTime and the SpaceTime Metric. Further,Einstein-Cartan General Relativity is based on the Curvature ofSpaceTime, the SpaceTime Metric, and the SpaceTime SpinConnection.

Although Einstein-Cartan has Torsion and Spin phenomena that arenot present in purely Metric Einstein-Hilbert, present-dayexperiments cannot distinguish between Einstein-Cartan andEinstein-Hilbert.

In both Einstein-Cartan and Einstein-Hilbert, the basic geometricobject in the Lagrangian is the SpaceTime Curvature, which is roughlya measure of how much SpaceTime (Wick-rotated to its Euclideanversion) 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) theLorentz 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 usingnon-compact symmetric spaces like H4 instead of their compact dualspaces like S4. I find it easier to visualize compact spaces, so Isometimes wind up writing things like S4 for H4, which in my mind isreally OK to do if the underlying structure is complex (wheresignature does not matter). Such an underlying complex structure doesexist in my D4-D5-E6-E7-E8 VoDou Physicsmodel, where, for example, SpaceTimeis the Silov boundary of abounded 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 bederived by the MacDowell-Mansourimechanism.

For Sarfatti Back-Reaction withBohm Quantum Theory,

the Geometric Object should be the SuperImplicate Order, or

MacroSpace,

and the range of the Geometric Variable should be aGeneralized Curvature of the

27-complex-dimensionalSymmetric Space E7 /E6xU(1)

whose Algebra is described by E8 /E7xSU(2)

The Symmetric Space E7/ (E6 x U(1)), and its related BoundedComplex Domain and ShilovBoundary, seems to me to be "natural" for several points ofview:

Since the Lorentz Group Spin(4) corresponds to global rotations inflat SpaceTime, and since E6 is the global symmetry group of theD4-D5-E6 physics model, let E6 play the role of Einstein'sSpin(4).

Since SpaceTime is the Geometric arena for General RelativisticGravity, and since the MacroSpace of the Super Implicate Order ofBohm's Theory is the Geometric arena (at the Nearest Neighbor levelof Interconnectedness), let 27-dimensional MacroSpace Geometryplay the role of Einstein's SpaceTime S4 = Spin(5) / Spin(4)curvature.

MacroSpace is approximated by continuous 27-dimensional 3x3Hermitian Octonion Matrices and by the 27-complex dimensionalE7 / (E6 x U(1)) which is the133-78-1 = 54-real-dimensional set of complexified Octonionprojective planes (CxO)P2 that are in the octonionified Octonionprojective plane (OxO)P2.

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

Since the U(1) of E7 / (E6 xU(1)) just means that MacroSpace is a space of 27 complexdimensions 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 fromwhich the Einstein-Hilbert action can be derived by theMacDowell-Mansouri mechanism.

Therefore:

"... What I think Deutsch'smultiverse really is, is ... a stack of snapshots ... allconnected together by Threads ..."

To me, the Geometric MacroSpaceStructure shows how the Threads connect the snapshots.

The Threads can be thought of as Strings in a26-dimensional subspace of the 27-dimensional MacroSpace.

At short distances, there are two fundamental types ofThreads:

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

Null Link of a Massless Gauge Boson between a SourceParticle 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. Tospecify the location of the Loop in Physical SpaceTime requires 4more dimensions. To specify Internal Symmetry Space requires 4 moredimensions. To specify Fermion Particle and Antiparticle typerequires 8+8 = 16 more dimensions. The total dimensionality of StringTheory space is 26, with signature (1, 1+4+4+8+8) = (1,25).

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

V. I. Arnold (Remarks on the Stationary Phase Method and CoxeterNumbers, Russian Math. Surveys 28 no. 5 (1973) 19-48; and CatastropheTheory, 2nd ed., Springer-Verlag (1986)) has shown that SimpleSingularities are classified by the A-D-EClassification. Arnold also points out that the A-D-Eclassification appears in such apparently (but not really) diverseareas as critical points of functions, Lie algebras, categories oflinear spaces, caustics, wave fronts, regular polyhedra in3-dimensional space, and Coxeter crystallographic reflectiongroups.

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 superstringconformal field theory, being in 1-1 correspondence not only with themodular invariants of SU(2)k, but also with the special solutions ofsolutions of c=1 theory for two continuous classes and the threediscrete solutions. Kaku says that this is because of thecorrespondence between the simply laced groups and the finitesubgroups of SU(2).

The A-D-E Classificationalso classifies Quivers ofArrows. If the Multiverse Snapshots are thought of as Points,and short Threads from Point to Point are thought of as Arrows from aPoint at the Tail of the Arrow to a Point at the Head of the Arrow,then you can form Quivers of Arrows. In1972, Peter Gabriel represented a quiver (P, A, t, h) by representingthe Points as Complex vector spaces and the Arrows A as matrix mapsfrom the Complex vector space representing the Tail to the Complexvector space representing the Head, and proved Gabriel's theorem:

If aconnected Quiver has only finitely many non-isomorphic indecomposablerepresentations, its graph is a Coxeter-Dynkin diagram of one of theLie algebras An, Dn, E6, E7, or E8, and there is a 1-1 correspondencebetween the classes of isomorphic indecomposable representations andthe 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 Jordanalgebra J3(O).

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

Look at the traceless subalgebra J3(O)o that is26-dimensional.

The mathematicalstructure of 26-dimensional String Theory describes the Structureof the MacroSpace,

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

Since

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

Jack Sarfatti's Back-Reaction in the space ofpossible outcomes is motivated by analogy with Einstein's gravitationtheory, and

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

it seems that

26-dimensionalUnoriented Closed Bosonic String Theory gives a unitary model ofJack Sarfatti's Back-Reaction model of MacroSpacephenomena including QuantumConsciousness.

Resonant Connections inMacroSpace may be useful in making a ConsciousSelection of Fates among the Many Fates of the ManyWorlds.

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 LargeN limit of the AN Lie Algebras.

For a nice introductory discussion of the mathematics of BosonicClosed Strings, see Week126 and Week127 and other relevant works ofJohn Baez.

Here is how I got to E7, andthen on to E8:

E7 comes from E6:

I want a space that has local E6 symmetry, so I am looking for anirreducible symmetric space G / K such that K is E6 itself, or E6times 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-E8VoDou 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 aLagrangian for the Standard Model plus Gravity. There are at least 3parts:

Gauge Group, Fermions,and SpaceTime

The Gauge Group should have atleast 12 dimensions for the Standard Model U(1)xSU(2)xSU(3) plus 16dimensions for a U(2,2) group from which I can get Gravity by gaugingits 15-dim SU(2,2) conformal subgroup (this is well known, if notwidely known, being described for example in the textbook byMohapatra). 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 theCartesian product of U(1)xSU(2)xSU(3) and U(2,2). They come out ofSpin(8) in a complicated way that is NOT standard, but is related todimensional reduction of spacetime in the D4-D5-E6-E7-E8VoDou physics model.

The first-generation Fermionparticles should be represented by at least an 8-dim space, whosebasis I identify in the D4-D5-E6-E7-E8VoDou 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-handedantiparticle states come from the tree-level mass of the 7 fermionswith tree-level mass. (The neutrino is massless at tree-level. Asmall neutrino mass can come from non-tree-level effects.) The secondand third generations (but no further generations) come fromdimensional reduction of spacetime in the D4-D5-E6-E7-E8VoDou physics model.

4-dim SpaceTime is adimensionally reduced version of the 8-dim spacetime of the D4-D5-E6-E7-E8VoDou physics model that is also octonionic, and corresponds tothe 8-dim vector representation of Spin(8). The 4-dim spacetime isacted on by the U(2,2) that gives gauge gravity. The other 4dimensions form the internal symmetry space that is acted on by theStandard 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. Thereis in fact a 52-dim exceptional Lie group F4 and it does have suchstructure and it was the basis of my first attempt at a physicsmodel. However, my F4 model was deficient, because my calculations ofparticle masses and force strength constants required that spacetimehave an underlying complex structure (that is one reason that I likeWick rotations back and forth between Euclidean and Minkowskispacetime).

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

If I give the 8-dim "spacetime" complex structure, then I shouldalso give complex structure to the two 8-dim fermion representationspaces, because all three 8-dim spaces are isomorphic by the trialityautomorphism of the three 8-dim representations of Spin(8).

This does NOT require me to give complex structure to the GaugeGroups, 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-dimcomplex "spacetime" and a U(1) for the complex symmetry of the 32-dimfermion 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 theparts of the Lagrangian, including the needed complex structure, Isee E6 as the symmetry of the D4-D5-E6-E7-E8VoDou hysics model.

If you get a quantum theory by Many-Worlds path-integral-sum, yousee that each world or snapshot has E6symmetry, so that a Geometric Structure of all the worlds should havelocal E6 symmetry.

Therefore, the E7global Geometric MacroSpace Structure ought to be

MacroSpace according to Wheeler andThorne:

In Wheeler's picture, as described by Kip Thorne in his book BlackHoles and Time Warps (Norton 1994), MacroSpace (called superspace bythem) is a collection of 3-dim spatial spaces. As Thorne says on page476: "... Quantum Gravity then [at the Planck energy]radically changes the character of spacetime. It ruptures theunification of space and time into spacetime. It unglues space andtime from each other, and then destroys time as a concept anddestroys 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 spaceremains a coherent (although perhaps multiply connected) entity abovePlanck energy, while time is done away with. I think that the 3spatial dimensions would cease to exist as much as time would ceaseto exist, and in support of my view I note that, at blackhole horizons, time dimensions can become spatial, etc.

MacroSpace according toDeutsch:

David Deutsch's multiverse is madeup of snapshots that are each 3-dim spaces. Deutsch, like Wheelerand Thorne, uses 3-dim spaces as basic building blocks. Deutschsays on page 278 of Fabric of Reality (Allen Lane - Penguin 1997):"... there is no fundamental demarcation between snapshots of othertimes and snapshots of other universes. ... Other times are justspecial cases of other universes ... distinguished from otheruniverses ... only in that they are especially related to ours thelaws of physics ... the rest of the multiverse ... impinges on usvery weakly by comparison, through interference effects. ..." On page283, Deutsch says: "... in some regions of the multiverse, and insome places in space, the snapshots of some physical objects do fall,for a period, into chains, each of whose members determines all theothers to a good approximation. ... In those regions and places, themultiverse does indeed look as ... a collection of spacetimes, and ..[one] can distinguish approximately between different timesand different universes, and time is approximately a sequence ofmoments. But that approximation always breaks down if one examinesthe snapshots in more detail, or looks far forwards or backwards intime, 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 themultiverse ...", Deutsch does not define what "far afield" means. Hisagglomeration of snapshots does not have a mathematicallywell-defined overall geometric structure that allows him to define"far afield".

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

E7 / E6xS1MacroSpace:

The overall mathematical geometrical multiverse structure (which Icall a MacroSpace,from the macrosphere in Greg Egan'ssci-fi novel Diaspora) that I use is the 27-complex-dimensionalsymmetric space E7 / E6xS1 = E7 /E6xU(1). Each Point of the MacroSpace E7 / E6xU(1) is oneUniverse.

Each Point-Universe is NOT just a 3-dimensional spatial physicalspace, or a 4-dimensional physical SpaceTime, but is one PossibleConfiguration of elementary particle Fermions and Gauge Bosons on allthe Points of an entire SpaceTime.

Fundamentally, I regard the SpaceTime as a Planck-lengthHyperDiamond Lattice, but what I am describing here is a continuousmanifold approximation.

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

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

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

Just as a Standard Model gauge group acts as local symmetry groupindependently at each point of SpaceTime and the Standard Modelforces arise from the differences between the gauge group states, orphases, at different SpaceTime points, so the BohmQuantum Forces should arise from the differences between the E6states, or phases, at different multiverse MacroSpace points.

The 78-real-dimensions of E6, and therefore the E6 states orphases, have the following physical content in my D4-D5-E6-E7-E8VoDou 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 theirE6 state/phase difference, and all the Point-Universes, takentogether, have the overall geometric structure of the MacroSpaceE7 / E6xU(1).

27 = 8+8 8 3

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

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

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

Click here to see more detailedGeometric and Algebraic Structure of the MacroSpace ofMany-Worlds,

both of which are combined inE8 by the fibration E8 / E7 x SU(2)

and describable in terms of BosonicUnoriented Closed String Theory.

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