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Quantum Conformal Gravity,Higgs, and SpinNetworks:

 


ConformalMacDowell-MansouriGravity

Conformal Groups are related to MoebiusTransformations and are used in the ConformalPhysics Theory. of IrvingEzra Segal.


The Conformal Group Spin(2,4) contains as a subgroup theanti-deSitter group Spin(2,3).

( Depending on convention, in some books andpapers those groups are written as Spin(4,2) and Spin(3,2), andsometimes the corresponding SO groups are described instead of theircovering Spin groups. )

M. Botta Cantcheff, in gr-qc/0010080,says: "... MacDowell and Mansouri proposed a gauge theory of gravitybased on the group SO(3,2) ... we focus our attention on theequations of motion ... of a Y-M's [Yang-Mills's] theory ...the real difference between GR [General Relativity withcosmological constant] and an YM-theory ... is ... a single[YM] constraint which has an extremely simple interpretation:the torsion-free condition. ... Torsion appears in a natural way inmodern formulations of the gravitational theories [ I. L.Shapiro, "Physical Aspects of the Space-Time Torsion", hep-th/0103093] ... by relaxing the constraint, we are naturally led to aparticularly elegant theory of gravity with torsion, whihc remarkablyenough turns out to be an ordinary ...[ SO(2,3) or SO(4,1)Yang-Mills ]... we observe that the cosmological constant must benon-vanishing. ...".


The D4-D5-E6 modelcoset spaces E6 /(D5 x U(1)) andD5 / (D4x U(1)) are Conformal Spaces. You can continue the chain toD4 / (D3x U(1)) where D3 is the 15-dimensional Conformal Group whose compactversion is Spin(6), and to D3 /(D2 x U(1)) where D2 is the6-dimensional Lorentz Group whose compact version is Spin(4).Electromagnetism, Gravity, and theZPF all have in common the symmetry of the 15-dimensional D3Conformal Group whose compact version is Spin(6), as can be seen bythe following structures with D3 Conformal Group symmetry:

Further, the 12-dimensional Standard Model Lie AlgebraU(1)xSU(2)xSU(3) may be related to the D3 Conformal Group Lie Algebrain the same way that the 12-dimensionalSchrodinger Lie Algebra is related to the D3 Conformal Group LieAlgebra.

 

Gravityin the D4-D5-E6 model beginswith the Clifford algebra Cl(0,6) = R(8) with spin group Spin(0,6) =SU(4) = compact conformal group of 4-dimensional spacetime. Thenon-compact conformal group Spin(4,2) = SU(2,2) has the same Cliffordalgebra R(8) = Cl(4,2), but, to simplify the discussion, the compactgroup Spin(0,6) will be used in this page.

The conformal group Spin(0,6) = SU(4) is 15-dimensional, with a10-dimensional subgroup Spin(0,5) = Sp(2) that is the de Sittergroup.

The 10 de Sitter infinitesimal generators correspond to gravitons.

The 15 infinitesimal generators of the conformal group are the 10Poincare group generators of the de Sitter group plus one scalegenerator and 4 conformal generators.

Mohapatra (in section 14.6 of Unification and Supersymmetry, 2ndedition, Springer-Verlag 1992) shows that if the scale and conformalgauge degrees of freedom are fixed, then a Lagrangian with theconformal group as gauge group gives the usual Hilbert action forgravity:

After the scale and conformal gauges have been fixed, theconformal Lagrangian becomes a de Sitter Lagrangian. Einstein-Hilbertgravity can be derived from the de Sitter Lagrangian, as was firstshown by MacDowell and Mansouri (Phys. Rev. Lett. 38 (1977)739). (Note that Frank Wilczek, in hep-th/9801184,says that the MacDowell-Mansouri "... approach to casting gravity asa gauge theory was initiated by MacDowell and Mansouri ... S.MacDowell and F. Mansouri, Phys. Rev. Lett. 38 739 (1977) ... , andindependently Chamseddine and West ... A. Chamseddine and P. WestNucl. Phys. B 129, 39 (1977); also quite relevant is A. Chamseddine,Ann. Phys. 113, 219 (1978). ...".


The Short-Range partof MacDowell-Mansouri Gravity is confined at the characteristicenergy level of about 309 MeV.

Above the 309 MeV energy level, and below the Higgs VacuumExpectation Value at v = 252 GeV, Gravity looks similar to the ColorForce.

Above the Higgs Vacuum Expectation Value at v = 252 GeV: the Higgsmechanism has not taken effect; SU(2) weak bosons and Diracfundamental fermions are massless; SU(2) gauge symmetry is unbroken;and the strength of Gravity is not so well defined in conventionalterms.


For further discussion of the MacDowell-Mansouri mechanism, seeFreund (chapter 21 of Introduction to Supersymmetry, Cambridge 1986),or Ne'eman and Regge (Riv. Nuovo Cim. v. 1, n. 5 (1978) 1, at pages25-28), or Nieto,Obregon, and Socorro, who have shown that the MacDowell-MansouriSpin(0,5) = Sp(2) de Sitter Lagrangian for gravity used in theD4-D5-E6 model plus

a Pontrjagin topological term

is equal to

the Lagrangian for gravity in terms of the Ashtekarvariables

plus

a cosmological constant term -which may vary duringExpansion of the InstantonUniverse,

Aldrovandi and Pereira, in gr-qc/9809061, show that de Sitter groups of the MacDowell-Mansouri Gravity mechanism can describe Special Relativity in SpaceTimes with varying Cosmological Constant. They use Inonu-Wigner contractions of de Sitter groups and spaces to show that in a weak cosmological constant limit the de Sitter groups are contracted to the Poincare group, and the de Sitter spaces are reduced to the Minkowski space, while in the strong cosmological-constant limit the de Sitter groups are contracted to another group which has the same abstract Lie algebra of the Poincare group, and the de Sitter spaces are reduced to a 4-dimensional cone-space of infinite scalar curvature, but vanishing Riemann and Ricci curvature tensors, in which the special conformal transformations act transitively and the equivalence between inertial frames is that of special relativity. If the fundamental spacetime symmetry of the laws of Physics is that given by the de Sitter instead of the Poincare group, the P-symmetry of the weak cosmological constant limit and the Q-symmetry of the strong cosmological constant limit can be considered as limiting cases of the fundamental symmetry. Minkowski and the cone-space can be considered as dual to each other, in the sense that their geometries are determined respectively by vanishing and infinite cosmological constants. The same can be said of their kinematical group of motions.
plus

an Euler topological term - which counts the number of handles ofa maniforld and for 4-dim spacetime is a 4-form that is proportionalto the square root of the determinant of the 4x4 matrix representingthe curvature 2-form (see sec. 11.4 of Nakahara, Geometry, Topology,and Physics, Adam Hilger 1990).

If you look at the chain of the 6-dim Lorentz Group Spin(4)included in the 10-dim de Sitter Spin(5) included in the 15-dimConformal Spin(6), you see that:

the 5 dimensions of Spin(6) outside Spin(5) are gauge-fixedConformal transformations and fixed Scale transformation; and

the 4 dimensions of Spin(5) outside Spin(4) (de Sitter/Poincaretranslations) combine to form the Metric of Einstein Gravity.

What about the 6 Lorentz Spin(4) dimensions?As Freund (chapter 21 of Introduction to Supersymmetry, Cambridge1986) says, they "... do not propagate, they are composite fields....".

They are the 3 spatial rotations and 3 Lorentz boosts of the 4vierbein basis vectors of 4-dim Spacetime, which basis vectors can berepresented as the 4 future links of a 4-dimensionalHyperDiamond Lattice.

Each of the 4 past links is the time-reversal negative of one ofthe 4 future links.

It is interesting to note that if you look at the permutationgroup of the 4 future links, S4 with 4! = 24 elements, you see thatit is the Weyl Group of the A3 = D3 LieAlgebra of the compact Conformal Group SU(4) = Spin(0,6), withnoncompact form SL(4) = Spin(2,4), which is where we started inconstructing the D4-D5-E6 model theoryof Gravity. It is also interesting that the 24 elements of S4 are in1-1 correspondence with the 24 root vectors of the D4 Lie Algebra ofSpin(0,8) used in constructing the entire D4-D5-E6model.

 

What does the Cosmologyof the D4-D5-E6-E7-E8 VoDou Physicsmodel say about the RelativeAbundance of Cosmological Constant, Primordial Black Holes, andOrdinary Matter?

In the D4-D5-E6-E7-E8 VoDou Physicsmodel, Gravity and the Cosmological Constant come from theMacDowell-Mansouri Mechanism and the 15-dimensional Spin(2,4) =SU(2,2) Conformal Group, which is madeup of:

According to gr-qc/9809061by R. Aldrovandi and J. G. Peireira:

"... By the process of Inonu&endash;Wigner group contraction with R -> oo ...[where R ]... the de Sitter pseudo-radius ... , both de Sitter groups ... with metric ... (-1,+1,+1,+1,-1) ...[or]... (-1,+1,+1,+1,+1) ... are reduced to the Poincare group P, and both de Sitter spacetimes are reduced to the Minkowski space M. As the de Sitter scalar curvature goes to zero in this limit, we can say that M is a spacetime gravitationally related to a vanishing cosmological constant.

On the other hand, in a similar fashion but taking the limit R -> 0, both de Sitter groups are contracted to the group Q, formed by a semi&endash;direct product between Lorentz and special conformal transformation groups, and both de Sitter spaces are reduced to the cone&endash;space N, which is a space with vanishing Riemann and Ricci curvature tensors. As the scalar curvature of the de Sitter space goes to infinity in this limit, we can say that N is a spacetime gravitationally related to an infinite cosmological constant.

If the fundamental spacetime symmetry of the laws of Physics is that given by the de Sitter instead of the Poincare group, the P-symmetry of the weak cosmological&endash;constant limit and the Q-symmetry of the strong cosmological&endash;constant limit can be considered as limiting cases of the fundamental symmetry.

Minkowski and the cone&endash;space can be considered as dual to each other, in the sense that their geometries are determined respectively by a vanishing and an infinite cosmological constants. The same can be said of their kinematical group of motions: P is associated to a vanishing cosmological constant and Q to an infinite cosmological constant.

The dual transformation connecting these two geometries is the spacetime inversion x^u -> x^u / sigma^2 . Under such a transformation, the Poincare group P is transformed into the group Q, and the Minkowski space M becomes the cone&endash;space N. The points at infinity of M are concentrated in the vertex of the cone&endash;space N, and those on the light&endash;cone of M becomes the infinity of N. It is interesting to notice that, despite presenting an infinite scalar curvature, the concepts of space isotropy and equivalence between inertial frames in the cone&endash;space N are those of special relativity. The difference lies in the concept of uniformity as it is the special conformal transformations, and not ordinary translations, which act transitively on N. ...

... in the light of the recent supernovae results ... favoring possibly quite large values for the cosmological constant, the above results may acquire a further relevance to Cosmology ...".

Since the Cosmological Constant comes from the 10 Rotation, Boost,and Special Conformal generators of the ConformalGroup Spin(2,4) = SU(2,2), the fractional part of our Universe ofthe Cosmological Constant should be about 10 / 15 = 67%.

Since Black Holes, including Dark Matter PrimordialBlack Holes, are curvature singularities in our 4-dimensionalphysical spacetime, and since Einstein-Hilbert curvature comes fromthe 4 Translations of the 15-dimensional ConformalGroup Spin(2,4) = SU(2,2) through the MacDowell-Mansouri Mechanism(in which the generators corresponding to the 3Rotations and 3 Boosts do not propagate), the fractional partof our Universe of Dark Matter PrimordialBlack Holes should be about 4 / 15 = 27%.

Since Ordinary Matter gets mass from the Higgs mechanism which isrelated to the 1 Scale Dilatation of the 15-dimensional ConformalGroup Spin(2,4) = SU(2,2), the fractional part of our universe ofOrdinary Matter should be about 1 / 15 = 6%.

Therefore, our Flat Expanding Universe should, according to thecosmology of the D4-D5-E6-E7-E8VoDou Physics model, have, roughly:

67% Cosmological Constant -

- related to GraviPhotons of SpecialConformal transformations and Akira/TetsuoEnergy.

27% Dark Matter -

- such as black holes, rangingin size from the stable Planckmass to Jupiter mass;possibly some gravitationalinteractions from other Worlds of the Many-Worlds; and/oreffective contributions from MOND.

6% Ordinary Matter -

- According to a New Scientist (22 March 2003pp. 41-42) article by Govert Schilling: "... Only around(1%) ... is ... in objects we can see ... stars, galaxies,and gas clouds ... Up to another ... (1%) .. may be...[in]... objects too faint for our telescopes too pickup, such as burned-out stars, small planets, or stars that failedto ignite ... The ...[ other 4% ] ...[maybe]... strung outlike cobwebsthroughout the cosmos ...".

In my opinion,

the WMAPobservations are quite consistent with the cosmologyof the D4-D5-E6-E7-E8 VoDou Physicsmodel.

 

 


WMAP  observation of theCosmic Background Radiation indicates that live in a FlatExpanding Universe with three types of stuff:

ordinary matter - 4% ;

cold dark matter (such as blackholes,ranging in size from the stable Planckmass to Jupiter mass, andpossibly some gravitationalinteractions from other Worlds of the Many-Worlds) - 23% ;and

a Cosmological Constant L(t) -73% .

What does the cosmologyof the D4-D5-E6-E7-E8 VoDou Physicsmodel say about those ratios?

 The ratio of the Cosmological Constant to Dark Matter plusOrdinary Matter is the ratio of the number of the 10 Spin(2,3) =Sp(2) generators

that involve time ( and thereforeevolution/expansion of the universe and therefore the CosmologicalConstant ), which is 3 boosts plus 4 translations( purely spacelike translations being a set ofmeasure zero, as most translations have some time component ),or 7

to the number of them

that are fixed in time, which is 3 rotations.

Therefore, our Flat Expanding Universe should have:

a Cosmological Constant - 70%

Dark Matter plus Ordinary Matter - 30%.

Since Ordinary Matter gets mass from the Higgs mechanism which isrelated to the 1 Scale Dilatation of the 15-dimensional ConformalGroup Spin(2,4) = SU(2,2), while Dark Matter Primordial BlackHoles are related to its 4 Special Conformal Transformations, theratio of Ordinary Matter to Dark Matter PrimordialBlack Holes should be 1 to 4, so that our Flat Expanding Universeshould, according to the cosmologyof the D4-D5-E6-E7-E8 VoDou Physicsmodel, have:

a Cosmological Constant - 70%

Dark Matter - 26%

Ordinary Matter - 6%.

In my opinion,

the WMAPobservations are quite consistent with the cosmologyof the D4-D5-E6-E7-E8 VoDou Physicsmodel.

 


Garcia-Compean,Obregon, and Ramirez have shown that the Pontrjagin term is aTHETA term, so that they can construct a dual theory associated withthe MacDowell-Mansouri theory.

The self-dual MacDowell-Mansouri theory corresponds to theAshetekar self-dual spin connection formalismand the approach of Nieto,Obregon, and Socorro.

When the anti-self-dual part (and the THETA term) are also takeninto account, it appears that it is possible that GravitationalMonopoles and/or Gravitational Solitons can be constructed.

Garcia-Compean,Obregon, and Ramirez use the group SO(3,2) in their paper. Theynote that, following the 1978 work of MacDowell and Mansouri, HeinzPagels (Phys. Rev. D 29 (1984) 1690) used the group O(5) to get aEuclidean formulation similar to the MacDowell-Mansouri theory.


A Pontrjagin topological term for 4-dim spacetime is proportionalto the trace of the square of the 4x4 matrix representing thecurvature 2-form (see sec. 11.4 of Nakahara, Geometry, Topology, andPhysics, Adam Hilger 1990). It may correspond to creation andexpansion of an InstantonUniverse.

Medina andNieto have shown that the Pontrjagin term may be related toChern-Simons theory for gauge group Spin(2,3), which in turn may berelated to conformal field theory in 1+1 dimensions. Smolinhas surveyed the Crane ladder of dimensions going from 3+1 dimensionsto 2+1 (Chern-Simons topological theory with massive gravitons) to1+1 (WZW conformal field theory). Smolin says that the ladder isrelated to the holographic hypothesis of 'tHooft and Susskindthat a quantum field theory on the interior of a blackhole is best described by a quantum field theory on itsboundary.

What is the physical reason for fixing the scale and conformaldegrees of freedom?

In the D4-D5-E6 model, the answer to that question comes from theanswer to another question:

Since all rest mass comes from the Higgs mechanism, and since restmass interacts through gravity, what is the relationship betweengravity and the Higgs mechanism?

 


Higgs andGravity and Torsion:

 

As remarked by Sardanashvily,Heisenberg and Ivanenko in the 1960s made the first atttempt toconnect gravity with a symmetry breaking mechanism by proposing thatthe graviton might be a Goldstone boson resulting from breakingLorentz symmetry in going from flat Minkowski spacetime to curvedspacetime.

Sardanashvily(see also gr-qc/9405013,gr-qc/9407032,and gr-qc/9411013)

proposes that gravity be represented by a gauge theory with groupGL(4), that GL(4) symmetry can be broken to either Lorentz SO(3,1)symmetry or SO(4) symmetry, and that the resulting Higgs fields canbe interpreted as either the gravitational field (for breaking toSO(3,1) or the Riemannian metric (for breaking to SO(4). Theidentification of a pseudo-Riemannian metric with a Higgs field wasmade by Trautman (Czechoslovac Journal of Physics, B29 (1979) 107),by Sardanashvily (Phys. Lett. 75A (1980) 257), and by Ivanenko andSardanashvily (Phys. Rep. 94 (1983) 1). In gr-qc/9711043,Sardanashvily has constructed a composite spinor bundle such that anyDirac spin structure is its subbundle and such that the compositespinor bundle admits general covariant transformations.Sardanashvily's Clifford algebra structure should allow treatment ofthe interaction between spinors andgravity to be represented by curvaturein a Clifford Manifold as done by WilliamPezzaglia in gr-qc/9710027 in his derivation of the PapapetrouEquations.

In the D4-D5-E6 model (using here the compact version) theconformal group Spin(0,6) = SU(4) is broken to the de Sitter groupSpin(0,5) = Sp(2) by fixing the 1 scale and 4 conformal gauge degreesof freedom.

The resulting Higgs field is interpreted in the D4-D5-E6 model asthe same Higgs field that gives mass to the SU(2) weak bosons and tothe Dirac fermions by the Higgs mechanism.

The Higgs mechanism requires "spontaneous symmetry breaking" of ascalar field potential whose minima are not zero, but which form a3-sphere SU(2). In particular, one real component of the complexHiggs scalar doublet is set to v / sqrt(2), where v is the modulus ofthe 3-sphere of minima, usually called the vacuum expectationvalue.

If the 3-sphere is taken to be the unit quaternions, then the"spontaneous symmetry breaking" requires choosing a (positive) realaxis for the quaternion space.

In the standard model, it is assumed that a random vacuumfluctuation breaks the SU(2) symmetry and in effect chooses a realaxis at random.

In the D4-D5-E6 model, the symmetry breaking from conformalSpin(0,6) to de Sitter Spin(0,5) by fixing the 1 scale and 4conformal gauge degrees of freedom is a symmetry breaking mechanismthat does not require perturbation by a random vacuumfluctuation.

Gauge-fixing the 1 scale degree of freedom fixes a length scale.It can be chosen to be the magnitude of the vacuum expectation value,or radius of the SU(2) 3-sphere.

Gauge-fixing the 4 conformal degrees of freedom fixes the(positive) real axis of the SU(2) 3-sphere consistently throughout4-dimensional spacetime.

Therefore, the D4-D5-E6 model Higgs field comes from the breakingof Spin(0,6) conformal symmetry to Spin(0,5) de Sitter gaugesymmetry, from which Einstein-Hilbert gravity (with Torsion)can be constructed by the MacDowell-Mansouri mechanism.

In Einstein-Cartantheory as a theory of defects in space-time, gr-qc/0306029, M. L.Ruggiero and A. Tartaglia say:

"... The Einstein-Cartan theory of gravitation and the classical theory of defects in an elastic medium are presented and compared. The former is an extension of general relativity and refers to four-dimensional space-time, while we introduce the latter as a description of the equilibrium state of a three-dimensional continuum. Despite these important differences, an analogy is built on their common geometrical foundations, and it is shown that a space-time with curvature and torsion can be considered as a state of a four-dimensional continuum containing defects. ... torsion refers to the non-symmetric part of the affine connection in a manifold, and in general relativity the torsion is assumed to be zero. ... From a physical point of view, torsion in ECT is generated by the spin. Hence, in ECT, both mass and spin, which are intrinsic and fundamental properties of matter, influence the structure of space-time. ...

... we want to introduce the concept of torsion by illustrating its use in the theory of defects, where curvature and torsion are used to describe the geometric properties of a material continuum. ... the Einstein-Cartan theory of gravitation and the theory of defects have similar fundamental equations, and we shall stress the analogies and differences in their underlying geometric structure. ...

... Trautman ... introduced a characteristic length to estimate the effects of torsion, the "Cartan" radius. To achieve the condition ...[that] spin effects ... be of the same order as mass effects ... or, alternatively, when the matter density is [about] 10^47 g cm^(-3) for electron-like matter and 10^54 g cm^(-3) for nucleon-like matter, ... we can imagine that a nucleon of mass m should be squeezed so that its radius coincides with the Cartan radius r_Cart ... For a nucleon we obtain r_Cart = 10^(-26) cm, which is very small when compared with macroscopical scales, but it is larger than the Planck length. Hence, torsion must be taken into account to achieve a quantum theory of gravity. ...

... formally, the Einstein-Cartan field equations describe the defect state of a three-dimensional continuum, at least when the defects are small so that we can use a linear approximation. The analogy is completed by the conservation equations, which, stated as geometric identities, give the correct conservation laws for dislocations and disclinations. ... Now we ask if we start from this three-dimensional correspondence in the linearized theory, is it possible to say something about the 3 + 1 space-time situation? ... When passing from 3 to 4 dimensions, there is an important difference in the geometric description of the medium. In three dimensions we can say that the effect of disclinations is to produce curvature. We used the Einstein tensor Gij to write the incompatibility equation, but we did not say explicitly that the curvature tensor and the Einstein tensor are equivalent. It is well known that in three dimensions they have the same number of independent components, which means that when the curvature tensor is zero, the Einstein tensor also is zero and vice versa. The presence of defects produces a nontrivial Einstein tensor, which also means that the curvature tensor is not zero. The same correspondence does not hold in four (or more) dimensions, because we can have curvature even if the Einstein tensoris zero. In particular, far from the sources, the curvature tensor could be nonzero. Indeed, this happens also in general relativity, because space-time is curved even far from the sources. If we extend the analogy to a four-dimensional context, we should expect that the effects of defects propagate through the manifold and are not purely local, as in three dimensions.

Kleinert ... adopted a linearized approach and showed that space-time with torsion and curvature can be generated from a flat space-time using "singular coordinate transformations," and is completely equivalent to a medium filled with dislocations and disclinations. In other words his singular coordinate transformations are the space-time equivalent of the plastic deformations which lead to incompatible states ... Hence, at least in this approximation, space-time can be thought of as a defect state, and defects are nothing but mass, mass current, and spin. The next important point is to try to go beyond the linear approximation.

We did our previous comparisons assuming small defects in order to use a linearized theory. As we have said, this assumption is fundamental for defining a density of disclinations. We must consider also that in real bodies there are physical constraints on the size of defects: additional or missing matter should not be such as to produce cracks in the structure. Hence, from a phenomenological point of view, it is often suffcient to use a linear theory, as it usually the case for the elastic theory of defects, where the linear Hooke's law is used.

However, curvature and torsion can always be introduced geometrically by the parallel transport procedure ... No approximation is contained in the equations governing curvature and torsion, so from the viewpoint of a geometric treatment no linearization is needed. ...

... four-dimensional equations which characterize the state of the medium ...[are]... a nonlinear generalization of the incompatibility equation, and ... the proportionality between torsion and the dislocation tensors. ... The correspondence between ...[them]... and ... the Einstein-Cartan field equations ... is obtained ...[for the proportionality between torsion and the dislocation tensors by]... using the definition of the Palatini tensor ...

... We can then say that Einstein-Cartan space-time can be considered as a defect state of a four-dimensional continuum, and the equations that describe the dynamical properties of this continuum correspond to the incompatibility equation and torsion source equation for space-time. This correspondence is an interesting analogy for the Einstein- Cartan theory. The meaning of the analogy becomes clear on physical grounds when we use three-dimensional equations in a linear theory of defects, where we have seen that the dislocation density is analogous to the moment stress tensor, and the total density of defects is analogous to the force stress tensor. ... In this analogy, the Poincare group, which is defined by the semidirect product P(10) = SO(1,3) x T(4), takes the place of the Euclidean group of R3. In this case we have six kinds of disclination-like deformations, and four kinds of dislocation-like deformations, which yield 10 different Riemann-Cartan spaces filled with topological defects. ... a Burgers vector B in T (4) and a Frank matrix G in SO(1,3) are defined by the parallel transport of a tetrad in the Riemann-Cartan space U4 around the line-like defect region. In this way, a space-time with curvature and torsion is thought of as a distorted medium filled with dislocations and disclinations ...".

 


Einstein-Hilbert gravity as aspin-2 field theory in flat spacetime:

The physical 4-dimensional SpaceTime of the D4-D5-E6-E7-E8VoDou Physics model is a 4-dimensional HyperDiamond latticeSpaceTime that is continuously approximated globally by RP1 x S3 andlocally by Minkowski SpaceTime, with Gravity coming from the15-dimensional Conformal Group Spin(2,4) by the MacDowell-Mansourimechanism. The curved SpaceTime of General Relativity is notconsidered fundamental, but is produced by by starting with a linearspin-2 field theory of massless gravitons in flat spacetime, and thenadding higher-order terms to get Einstein-Hilbert gravity(without a cosmological constant - toget a cosmological constant, use massive spin-2gravitons). The observed curved spacetime is thereforebased on an unobservable flat spacetime.

Richard Feynman's book Lectures on Gravitation (1962-63 lectures at Caltech), Addison-Wesley 1995, contains a section on Quantum Gravity by Brian Hatfield, who says: "... Feynman ... felt ... that ... the fact that a massless spin-2 field can be interpreted as a metric was simply a "coincidence" ... In order to produce a static force and not just scattering, the emission or absorption of a single graviton by either particle [of a pair of particles] must leave both particles in the same internal state ... Therefore the graviton must have integer spin. ... when the exchange particle carries odd integer spin, like charges repel and opposite charges attract ... when the exchanged particle carries even integer spin, the potential is universally attractive ... If we assume that the exchanged particle is spin 0, then we lose the coupling of gravity to the spin-1 photon ... the graviton is massless because gravity is a long ranged force and it is spin 2 in order to be able to couple the energy content of matter with universal attraction ... Hence, the gravitational field is represented by a rank 2 tensor field ... the antisymmetric part behaves like a couple of spin-1 fields ... and therefore should be dumped. This leaves a symmetric tensor field ... the higher spin possibilities are neglected ...".

Feynman's book also contains a Foreword by John Preskill and Kip S. Thorne in which they say: "... Feynman's investigations of quantum gravity eventually led him to a seminal discovery ... that a "ghost" field must be introduced into the covariantly quantized theory to maintain unitarity at one-loop order of perturbation theory .... it was eventually DeWitt ... and also Faddeev and Popov ... who worked out how to generalize the covariant quantization of Yang-Mills theory and gravitation to arbitrary loop order ...".

Frank Wilczek, in an article in Physics Today (August 2002, pages 10-11), says: "... There is a perfectly well-defined quantum theory of gravity that agrees accurately with all available experimental data ... A salient difference between how renormalization theory functions in the standard model and how it extends to include gravity is that, whereas in the standard model by itself we need only specify a finite number of parameters to fix all the integrals, after we include gravity we need an infinite number. But that's all right. ... The prescription is to put the coeficients of all nonminimal coupling terms to zero at some reference energy scale ... well below the Planck scale ... the consequences ... are ... far below the limits of observation ...".

Hatfield, in his Quantum Gravity section, also says: "... When a field is quantized, each mode of the field possesses a zero-point energy. Since the field is made up of an infinite number of modes, the vacuum energy of the quantum field is infinite. This infinity is quickly disposed of by normal ordering of the field operators. The justification for doing so is that we are just redefining the zero point of the energy scale which is arbitrary to begin with. However, since gravity couples to all energy, when we add gravity, we can no longer get away with this. ... such a vacuum energy density will appear in a gravity theory as a cosmological constant. ... this is a big problem. ... The hope was that some kind of hidden symmetry ... might render the pure gravity sector of theory finite. However, a computer calculation of two-loop corrections gave a divergent result dashing this hope. ... One way to get improved ultraviolet behavior is to have more symmetry built into a theory. ... One popular approach is called "supergravity" ... based on a symmetry between bosonic and fermionic fields called "supersymmetry." ... Unfortunately, when the dimension of space-time is 4 there still exist potential counterterms (starting at seven loops in the best case). ...". [Note that in the D4-D5-E6-E7-E8 Vodou Physics model, which includes both the standard model and gravity, not just gravity alone, there is a subtle supersymmetry that may give ultraviolet finiteness and solve the problem.]

Preskill and Thorne, in their Foreword, also say: "... the lectures ...[contain]... an unusual approach to the foundations of general relativity ... That appoach ... develops the theory of a massless spin-2 field (the graviton) coupled to the energy-momentum tensor of matter, and demonstrates that the effort to make the theory self-consistent leads inevitably to Einstein's general relativity. ... Feynman was not the very first to make such a claim. ... The field equation for a free massless spin-2 field was written down by Fierz and Pauli in 1939 .... Robert Kraichnan ... studied the problem of deriving general relativity as a consistent theory of a massless spin-2 field in flat space. He described his results in his unpublished 1946-47 Bachelor's thesis ... Kraichnan did not assume that gravity couples to the total energy-momentum tensor ... he derived this result as a consequence of the consistency of the field equatiions. .... Kraichnan continued to pursue this problem at the Institute for Advanced Study in 1949-50. ...he received some encouragement from Bryce DeWitt ... Einstein ... was appalled by an approach to gravitation that rejected Einstein's own hard-won geometrical insights. ... Kraichnan did not publish ... until 1955 ... in a 1954 paper ... Suraj N. Gupta ... proceeds as follows: We wish to construct a theory in which the "source" coupled to the massless spin-2 field h_uv is the energy-momentum tensor, including the energy-momentum of the spin-2 field itself. If the source is chosen to be the energy-momentum tensor 2T^uv of the free field theory (which is quadratic in h), then coupling this source to h_uv induces a cubic term in the Lagrangian. From this cubic term in the Lagrangian, a corresponding cubic term 3T^uv in the energy-momentum tensor can be inferred, which is then included in the source. This generates a quartic term 4T^uv , and so on. The iterative procedure generates an infinite series that can be summed to yield the full nonlinear Einstein equations. ... The first complete version of the argument (and an especially elegant one) was published by Deser in 1970 ...".

Stanley Deser's paper Gen. Rel. Grav. 1 (1970) 9-18 is described in Misner, Thorne, and Wheeler, Gravitation, Freeman 1973, pp. 424-425, where they say: ".... Deser summarizes the analysis at the end thus: "Consistency has therefore led us to universal coupling, which implies the equivalence principle. ... [The] initial flat "background" space is no longer observable." In other words, this approach to Einstein's field equations can be summarized as "curvature without curvature" or - equally well - as "flat spacetime without flat spacetime"! ...".

Preskill and Thorne, in their Foreword, also say: "... Weinberg showed ... From very reasonable assumptions about the analyticity properties of graviton-graviton scattering amplitudes ... that the theory of an intereacting massless spin-2 particle can be Lorentz invariant only if the particle couples to matter (including itself) with a universal strength; in other words, only if the strong principle of equivalence is satisfied. ... Once the principle of equivalence is established, one can proceed to the construction of Einstein's theory ...".

Steven Weinberg, in his book Gravitation and Cosmology (Wiley 1972), says in his Preface: "... I believe that the geometrical approach has driven a wedge between general relativity and the theory of elementary particles. ... the passage of time has taught us not to expect that the strong, weak, and electromagnetic interactions can be understood in geometrical terms, and too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics. In place of Riemannian geometry, I have based the discussion of general relativity on a principle derived from experiment: the Principle of Equivalence of Gravitation and Inertia. ... Riemannian geometry appears only as a mathematical tool for the exploitation of the Principle of Equivalence, and not as a fundamental basis for the theory of gravitation. ...".

If you were to start, not with locally Minkowski SpaceTime, butwith the curved SpaceTime of General Relativity, then you would seethat the Conformal transformations of Minkowski SpaceTime by the15-dimensional Conformal Group Spin(2,4) corresponds to the Conformaltransfomations of the curved SpaceTime by the infinite-dimensionalConformal subgroup of the group Diff(M4) of General Relativisticcoordinate transformations of the 4-dimensional SpaceTime M4 ofGeneral Relativity, which Conformal subgroup is defined as thoseGeneral Relativistic coordinate transformations that preserveconformal structure and which infinite-dimensional Conformal subgroupcan be called the Weyl Conformal Group. (See Ward and Wells, TwistorGeometry and Field Theory, Cambridge 1991, p. 261.)

To study the curvature of 4-dimensional curved metric SpaceTime(see Misner, Thorne, and Wheeler, Gravitation, Freeman 1973), startwith the 256 = 4^4 component tensor Rabcd, and then impose thesymmetries required by curvature Rabcd = R[ab][cd] =R[cd][ab] and R[abcd] = 0 and Ra[bcd]= 0 to see that the Riemann curvature tensor Rabcd has 20 independentcomponents. Then decompose Rabcd into the 10-component symmetricRicci tensor Rab and the 10-component conformal traceless Weyl tensorCabcd. Then the Einstein tensor Gabis given by Gab = Rab - (1/2)R, where R is the scalar curvature. TheRiemann tensor Rabcd obeys the Bianchi identities, and the Einsteintensor Gab is the only contraction that obeys contracted Bianchiidentities, which geometrically mean that the boundary of a boundaryis zero. To see the Riemann, Ricci, and conformal Weyl tensorswritten in terms of spinors or twistors, see Penrose and Rindler,Spinors and Space-Time, vols. 1 and 2, Cambridge 1986. Penrose (TheEmperor's New Mind, Oxford 1989) describes the Ricci tensor asmeasuring volume, while the Weyl tensor measures tidal distortion, sothat Ricci is a source like electric charges and currents, but Weylis like the electromagnetic field that carries waves ofradiation.

If you were to formulate the Conformal Gravity andHiggs structures of Sardinashvily by starting with the curvedmetric SpaceTime of General Relativity, you would replace the15-dimensional Conformal Group Spin(2,4) with theinfinite-dimensional Weyl Conformal Group, and you might getsomething very much like theProper Time Dynamics in General Relativity and Conformal UnifiedTheory of Gyngazov, Pawlowskiy, Pervushinz, and Smirichinski ingr-qc/9805083. In their model, as in the D4-D5-E6model, the Lagrangian has two sectors that are linked by theHiggs mechanism: a Gravitational Sector that acts to curve physicalSpaceTime with characteristic energythat is the energy of the PlanckMass; and a Standard Model Sector that acts on InternalSymmetry Space with characteristic energy that is the energy ofthe Proton Mass. As Gyngazov,Pawlowski, Pervushin, and Smirichinski say, "... Roughly speakingPlanck Mass is nothing but a multiplicity of the proton mass. ...",and as the D4-D5-E6 model says,the Planck Mass / Proton Massratio is on the order of 10^19.

The Conformal Unified Theory of Gyngazov,Pawlowski, Pervushin, and Smirichinski contains spinor particles,such as neutrinos. SpaceTime transformations of spinors producetorsion displacement in the 4th dimension if you go around a spatialloop that ends at the beginning spatial point. Gravity with torsioncomes from the Cartan point of view of varying the connection as anindependent variable as well as the metric, which is varied in theEinstein point of view. From the Cartan point of view, torsiondoes not propagate and its coupling to spin is of the magnitude ofthe Gravitational Constant G (Gockeler and Schucker, DifferentialGeometry, Gauge Theories, and Gravity, Cambridge 1987, p. 71, andFreund, Supersymmetry, Cambridge 1986, p. 104-105). Conformal Weylcurvature is discussed in relation to torsion and spin in the book ofBuchbinder, Odintsov, and Shapiro, Effective Action in QuantumGravity (IOP 1992).

 

Donoghue hasshown a way to formulate gravity as an effective field theory at lowenergies. He has also written a shortersurvey article.

 


 Geometry of Gravity

In their book Gravitation (W. H. Freeman 1973), Misner, Thorne,and Wheeler say (at pages 380, 378, 420-423):

"... Identify the stress-energy tensor (up to a factor of ... 8 piG / c^4 ...) with the moment of rotation ... the conservation ofmoment of rotation follows from ...

(1) The moment of rotation associated with any elementary3-cube is by definition a net value, obtainedby adding the six moments of rotation associated with the six facesof that cube.

(2) When one sums these net values for all 8 3-cubes ... which arethe boundary of the elementary 4-cube...[

]... one counts the contribution of a given 2-face twice, oncewith one sign and once with the opposite sign. In virtue of theprinciple that "the boundary of a boundary is zero," the conservationof moment of rotation [ and of stress-energy ] is thus anidentity. ...

"Mass-energy curves space" is the central principle ofgravitation. ...

"Space" means spacelike hypersurface. ...

the scalar curvature ... of the 3-geometry intrinsic to the [spacelike ] hypersurface ... is defined by ... measurements ofdistance made within the hypersurface ...

the "extrinsic curvature" of this 3-geometry relative to the4-geometry of the enveloping spacetime ... "how curved one cuts hisslice" ...

"curvature of space" must (1) be a single number ( a scalar ) that(2) depends on the inclination ... of the cut one makes throughspacetime ... in constructing the hypersurface ... but (3) must beunaffected by how one curves his cut. ...

All of Einstein's geometrodynamics is contained in this statement...

the "Moment of Rotation" associated with a unit element of3-volume ... in the hypersurface ... = ...

= Intrinsic Curvature Scalar + Extrinsic Curvature Scalar =

= 16 pi ( Local Density of Mass-Energy )

... [ which is ] valid for every spacelike slice throughspacetime at any arbitrary point ...

The factor 16 pi is appropriate for the ... system of units ...density ... in cm^(-2) given by

G / c^2 = 0.742 x 10^(-28) cm / g

multiplied by the density ... expressed in ... g / cm^3 ...".

 

In Lecture 11.2 of his Feynman Lectures on Gravitation(Addison-Wesley 1995), at page 154, Richard Feynman says:

 "... we may give an interpretation of the theory ofgravitation ... as follows: ... Consider a small three-dimensionalsphere ... [ in a ] three-space perpendicular to the timeaxis ... Its actual radius exceeds the radius calculated by Euclideangeometry ... by an amount proportional ... [ by the factor ]G / 3 c^2 ... to the amount of matter inside the sphere ... one fermiper 4 billion metric tons ... we require the same result to hold inany coordinate system regardless of its velocity. ...".

 

The D4-D5-E6-E7-E8 VoDou Physicsmodel has 5 ConformalGraviPhotons:

1 that scales/dilates spacetime, whose action is like justexpanding or contracting space at a single given point, and itsaction is suppressed because of Feynman's factor of G / 3 c^2 that isof the order of about 10^(-28); and

4 that do special conformal transformations, which have someglobal nonlocal effects

by which they can expand space atone point and produce a corresponding contraction of space at anotherpoint. Their effects are globally conservative in that there isno total net input or output of energy.

 


Geometric Algebra Gauge-Theory Gravity

 

In astro-ph/9707165,Lasenby, Doran, Dobrowski, and Challinor say:

"... Gauge-theory gravity,expressed in the language of Geometric Algebra [CliffordAlgebra],

allows very efficient numerical calculation of photon paths. ...We discuss ... applications of a gauge theory of gravity ... Thetheory employs [two] gauge fields in a flat Minkowskibackground spacetime to describe gravitational interactions. ... Thefirst of these, h(a), is a position-dependent linear function mappingthe vector argument a to vectors. The position dependence is usuallyleft implicit. Its gauge-theoretic purpose is to ensure covariance ofthe equations under arbitrary local displacements of the matterfields in the background spacetime. The second gauge field, W(a), isa position-dependent linear function which maps the vector a tobivectors. Its introduction ensures covariance of the equations underlocal rotations of vector and tensor fields, at a point, in thebackground spacetime. Once this gauging has been carried out, and asuitable Lagrangian for the matter fields and gauge fields has beenconstructed, we find that gravity has been introduced. ... the theoryis formally similar in its equations (hence local behaviour) to theEinstein-Cartan-Kibble-Sciama spin-torsiontheory, but it restricts the Lagrangian type and the torsion type(... torsion that is not trivector type leads to minimally coupledLagrangians giving non-minimally coupled equations for quantum fieldswith non-zero spin). ...

If we restrict attention to situations where the gravitatingmatter has no spin, then there are still differences between generalrelativity and the theory presented here. These differences arisewhen time reversal effects are important (e.g. horizons), whenquantum effects are important, and when topological issues areaddressed. ...

... within the framework of gauge-theory gravity, theKerr singularityis composed of a ring of matter, moving at the speed of light, whichsurrounds a disk of pure isotropic tension. ...

... As an interesting aside, we note that self-consistenthomogeneous cosmologies, based on a classical Dirac field, requirethat k = 0 (the universe is spatially flat). ... ".

 

In the paper Gravity, gauge theories and geometric algebra,downloadable from the webpage of The Geometric Algebra Research Group at Cavendish Laboratory,University of Cambridge, Lasenby, Doran, and Gull say: that"...fermionic matter would be able to detect the center of theuniverse if k=/= 0 [if the univese were not spatiallyflat] ...".

 

In the paper Effects of Spin-Torsionin Gauge Theory Gravity, downloadable from the webpage of The Geometric Algebra Research Group at Cavendish Laboratory,University of Cambridge, Doran, Lasenby, Challinor, and Gull say:that "... Within [Gauge-Theory Gravity],torsion is viewed as aphysical field derived from the gravitational gauge fields. Thisviewpoint has some conceptual advantages over that used indifferential geometry, where torsion is regarded as a property of anon-riemannian manifold. ... for a massive spinning point-particle,moving in a gravitational background with torsion ...the motion isnot generally geodesic, the spin vector is not Fermi-transported, andthe particle couples to the torsion. ... spinning point particlessee a preferred direction in space due to the spin of the matterfield. ... with spin there are extra physical fields presentwhich have observable consequences. ...".

 

In gr-qc/9910099,Chris Doran says: "...

A new form of the Kerrsolution

... is global and involves a time coordinate which represents thelocal proper time for free-falling observers on a set of simpletrajectories. ... The Kerr solution ... is global, making it suitablefor studying processes near the horizon. ... the time coordinatemeasured by a family of free-falling observers brings the Diracequation into Hamiltonian form ... This form of the equations alsopermits many techniques from quantum field theory to be carried overto a gravitational backgroundwith little modification. ... ".

 


 

Exotic Spacetime andGravity

J. Sladkowski, ingr-qc/9906037, Strongly Gravitating Empty Spaces, says:

"... We use various results concerning isometry groups of Riemannian and pseudo-Riemannian manifolds to prove that there are spaces on which differential structure can act as a source of gravitational force (Brans conjecture). ...

[If the] isometry group G of the solution acts nonproperly on R4 ... [then] G is locally isomorphic to SO(n,1) or SO(n,2 ). But the nonproper action of G on R4 means that there are points infinitely close together in R4 ... such that arbitrary large different isometries ... in G maps them into infinitely close points in R4 ... . There must exists quite strong gravity centers to force such convergence (even in empty spacetimes). ...

We see that ... Einstein gravity is quite nontrivial even in the absence of matter. ... Note that we have proven a weaker form of the Brans conjecture: there are four-manifolds (spacetimes) on which differential structures can act as a source of gravitational force just as ordinary matter does. ...".

Note that SO(n,1) and SO(n,2) correspond to Lorentz and Conformalgroups, respectively, and their geometry is related to LieSphere Geometry and SpaceTimeCorrelations.

 


Quantum Gravity

Renormalizable quantum theories are well known for the threeforces of the Standard Model (electromagnetism, the weak force, andthe color force), and they are part of the D4-D5-E6 model.

The standard model quantum path integral sum over histories breaksthe gauge group invariance of the Lagrangian, because the pathintegral must not overcount paths by including more than onerepresentative of each gauge-equivalence class of paths. Theremaining quantum symmetry is the symmetry of BRSTcohomology classes. Knowledge of the BRST symmetry tells youwhich ghosts must be used inquantum calculations, so the BRSTcohomology can be taken to be the basis for the quantumtheory.

A good description of BRST cohomology is in the paper of Garcia-Compean,Lopez-Romero, Rodriguez-Segura, and Socolovsky. As they state,the only force for which a renormalizable quantum theory is not wellknown is gravity.

They discuss two current approaches to quantum gravity:

string theory, which abandons point particles even at theclassical level; and

redefinition of classical general relativity in terms of newvariables, the Ashtekarvariables, and trying to use the new variables to construct aquantum theory of gravity.

Nieto, Obregon,and Socorro have shown that the MacDowell-Mansouri Spin(0,5) =Sp(2) de Sitter Lagrangian for gravity used in the D4-D5-E6model plus

a Pontrjagin topological term

is equal to

the Lagrangian for gravity in terms of the Ashtekarvariables plus

a cosmological constant term - which may vary during Expansionof the Instanton Universe, plus

an Euler topological term - which counts the number of handles ofa maniforld and for 4-dim spacetime is a 4-form that is proportionalto the square root of the determinant of the 4x4 matrix representingthe curvature 2-form (see sec. 11.4 of Nakahara, Geometry, Topology,and Physics, Adam Hilger 1990).

A Pontrjagin topological term for 4-dim spacetime is proportionalto the trace of the square of the 4x4 matrix representing thecurvature 2-form (see sec. 11.4 of Nakahara, Geometry, Topology, andPhysics, Adam Hilger 1990). It may correspond to creation andexpansion of an InstantonUniverse.

Medina andNieto have shown that the Pontrjagin term may be related toChern-Simons theory for gauge group Spin(2,3), which in turn may berelated to conformal field theory in 1+1 dimensions. Smolinhas surveyed the Crane ladder of dimensions going from 3+1 dimensionsto 2+1 (Chern-Simons topological theory with massive gravitons) to1+1 (WZW conformal field theory). Smolin says that the ladder isrelated to the holographic hypothesis of 'tHooft and Susskindthat a quantum field theory on the interior of a blackhole is best described by a quantum field theory on its boundary.By using the Crane ladder, you can study black holes with the methodsof topological quantum field theory, knot theory, and category theorythat are described by Baez in his series ThisWeek's Finds in Mathematical Physics as well as in his books andpapers.

Therefore, although the quantum gravity methods of string theorycannot be used in the D4-D5-E6 model because the D4-D5-E6 model usesfundamental point particles at the classical level, methods based onAshtekarvariables are available. Two such approaches are:

a topological approach based on loop groups and spin networks;and

an algebraic approach based on getting BRST transformations fromMaurer-Cartan horizontality conditions.

 


Spin Networks

Spin Networks are defined by Baezas "...graphs embedded in a manifold S (the space of spacetime) withedges labelled by representations of a Lie group G and with verticeslabelled by intertwining operators. Spin networks definegauge-invariant functions on the space A of connections of anyG-bundle over S...".

The intertwining operators are related to the fermion creation andannihilation operators in the 3x3 OctonionNilpotent Heisenberg Algebra Matrix Model.

The configuration space of gravity using the Ashtekarvariables is not the space of metrics, but the space ofconnections on an SL(2,C) bundle (with compact real form SU(2) =Spin(3)) over S.

The constraints of canonical quantum gravity are then polynomial,and the diffeomorphism constraint is invariance under diffeomorphismsof S. Since loops are invariant under diffeomorphisms of S, and sinceWilson loops can represent gauge theories, Rovelli and Smolinproposed a loop representation of quantum gravity.

John Baez,in week 55 of This Week's Finds in Mathematical Physics, discusseswork of Rovelli and Smolin and of Loll in which Rovelliand Smolin have shown that the loop representation of quantumgravity is equivalent to an SU(2) Spin Network representation and

Loll hascalculated that spatialvolume eigenstates of the Spin Network space of states arediscrete and nonzero, on the scale of the Planck length, for SpinNetworks with at least 4 edges at each vertex.

(Baez says that Rovelli and Smolin made a sign error that ledthem, in an earlier paper, to conclude that trivalent Spin Networkscould have nonzero spatial volume, and that Loll showed at a 1995Warsaw workshop that trivalent Spin Networks had zero spatial volume,but 4-valent Spin Networks had nonzero Planck-scale spatialvolume.)

This leads to the 4-dimensional HyperDiamondFeynman Checkerboard based on the 4-dimensional lattice structureof Michael Gibbs andDavidFinkelstein.

Gibbs views the 4-dimensional HyperDiamond Feynman Checkerboard asbeing a lattice in a 4-dimensional spacetime, while Finkelsteindeveloped the Quantum Graphs used in construction of the lattice as atheory of abstract Quantum Graphs not embedded in spacetime, but fromwhich spacetime should be derived.

The D4-D5-E6 model has the discrete structure of theFinkelstein-Gibbs HyperDiamond lattice, and has a gravity sectorbased on the conformal MacDowell-Mansouri mechanism, which has beenshown to be closely related to to the Ashtekar variable picture byNieto, Obregon, andSocorro.

Baez says thatSpin Networks "...were invented in the early 1970s by Penrose ...",and that while the Baez-Rovelli-Smolin-Loll Spin Networks "...involvegraphs embedded in a pre-existing manifold that represents space, his[Penrose's] spin networks were abstract graphs ... intendedas a purely combinatorial substitute for a spacetime manifold."

Since the Finkelstein-Gibbs 4-dimensional HyperDiamond lattice hasspatial structure of the 3-dimensional Diamond lattice, whose naturalSpin Network has 4-valent tetrahedral structure, Loll's resultssupport the Finkelstein-Gibbs HyperDiamond Spin Network and theD4-D5-E6 model. As MichaelGibbs says: a 3-valent Spin Network is spatially like 2-dimgraphite and a 4-valent Spin Network is spatially like 3-dimdiamond.

Barnett andCrane describe Relativistic Spin Networks in terms of thegeometry of the 2-dimensional faces of a 4-simplex and the groupSpin(4) = SU(2) x SU(2). Cranehas extended that model to An Octonionic Geometric State Model alsousesthe geometry of a Euclidean 4-simplex and the bivectors on itsfaces, but, instead of using the group Spin(4) = SU(2) x SU(2), Craneuses the group Spin(8) and its octonionic structure. Crane goes onto"... choose a triangulation of the base manifold, then choose aflat, affine lifting of each 4-simplex to the total space T of thebundle, identified with B4 x R8. This is equivalent to picking a atconnection on each 4-simplex. ...". As Crane says, "... Actually,this is only a partial analog of Kaluza-Klein theory, since thevariables correspond to a choice of a connection, but not of a metricon the base. ...". Crane then decomposes the 28-dimensional basis {Bij } of Spin(8) into 7 copies of a 4-dimensional Cartan subalgebrawhose basis is {B12, B34, B56, B78}, and for which the Spin(8)triality automorphism can be written as

F12 = (1/2) ( + B12 - B34 - B56 - B78)

F34 = (1/2) ( - B12 + B34 - B56 - B78)

F56 = (1/2) ( - B12 - B34 + B56 - B78)

F78 = (1/2) ( - B12 - B34 - B56 + B78)

The triality action Fij is the same on each of the 7 copies of the4-dim Cartan subalgebra; the { Fij } constitute a new basis for theSpin(8) Lie algebra with the same commutation relations as the { Bij}. In the { Fij } basis, the simple bivectors are characterized by|F12| = |F34| = |F56| = |F78|, which suggests a state sum formed bylabelling "... each face and each tetrahedron with an irreduciblerepresen tation of Uq(so(8)) whose highest weight satisfied |F12| =|F34| = |F56| = |F78|, join the representations into a 15Jq symbol[as done by Barnettand Crane for Spin(4)], multiply together the evaluations ofthe 15Jq symbols for all the 4-simplices in the triangulation,normalize with the product of powers of quantum dimensions ..., andsum over labellings. ...we would have to sum over a basis for thetensor operators at each trivalent vertex in our diagrams, since therepresentation category of so(8) does not have unique tensoroperators like the representation category of so(3). ..." Cranepredicts that his Octonionic Model has a nonzero cosmologicalconstant and is not chirally symmetric. He suggests that theexceptional Jordan algebra J3(O) and theLie algebra E6 might be useful infurther work on his model.

Crane'sstructure is similar to the structure of the 4-dimensionalHyperDiamond Feynman Checkerboard used in the D4-D5-E6model and in the models of Michael Gibbs and David Finkelstein,whose 4-link future lightcone is a 4-dimensional simplex, with the 4future ends of the links forming a 3-dimensional tetrahedron. It isshown here, with a stereo pair showing 3 dimensions and color coding(green = present, blue = future) for the 4th dimension.

The 4-link future lightcone leading from a vertex looks a lot likethe Quantum Pentacle of David Finkelstein and Ernesto Rodriguez (Int.J. Theoret. Phys. 23 (1984) 887), as well as a lot like an element ofthe Relativistic Spin Network of Barnettand Crane.

The future tetrahedron, not including the origin vertex, contains4 vertices and 4 bivector triangles. Those bivectors correspond tothe 4 translation bivectors of the 10-dimensional Spin(5) Liealgebra, which is based on the Cl(0,5) Cliffordalgebra with graded structure 1 5 10 10 5 1, so that there are: 1empty set, 5 vector vertices, 10 bivector edges, 10 triangles, 5tetrahedra, and 1 4-simplex. By Hodge duality, the 10 bivector edgescorrespond to the 10 triangles.

The future lightcone edges leading from the origin are edges onthe 6 bivector triangles that include the origin vertex. Thosebivectors correspond to the 6 bivectors of the Spin(4) subalgebra ofthe Spin(5) Lie algebra. The 6-dimensional Spin(4) subalgebra isreducible, isomorphic to SU(2) x SU(2), so that it reduces to 3rotations and 3 Lorentz boosts. That decomposition is most clearlyseen by looking at the dual to the 4-simplex illustrated above, inwhich its 5 vertices correspond to 5 tetrahedral 3-faces, and viceversa:

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

 

 


BRST Quantization is described HERE.


Chern-Simons Time

Nieto, Obregon,and Socorro have shown that Lagrangian action of the Ashtekarvariables is a Chern-Simons action if the Killing metric of thede Sitter group is used instead of the Levi-Civita tensor.

Smolin and Soohave shown that the Chern-Simons invariant of the Ashtekar-Senconnection is a natural candidate for the internal time coordinatefor classical and quantum cosmology, so that the D4-D5-E6-E7 modeluses Chern-Simons time.

 


Quantum Gravity plus Higgs

The D4-D5-E6-E7 model has

a quantum theory of the standard model forces (electromagnetism,the weak force, and the color force), and

a quantum theory of MacDowell-Mansouri-Ashtekar gravity.

The two quantum theories must be combined in order to calculatehow standard model particles and fields interact in the presence ofgravity.

Moritsch,Schweda, Sommer, Tataru, and Zerrouki have done this by usingMaurer-Cartan horizontality conditions to get BRST transformationsfor Yang-Mills gauge fields in the presence of gravity.

This gives a complete quantum structure for the D4-D5-E6-E7model.

 

What about the Aether in the D4-D5-E6-E7 model?

My personal opinion is that the Aether is compressible, but only at energies around the Vacuum Expectation Value of the Higgs field, around 250 GeV, which is corrresponds to the Superposition Separation of an entire single Tubulin in the Brain.

Since the unit Quaternions form the Lie Group Sp(1) = SU(2) = Spin(3) = S3, Maxwell's use of Quaternions in Electromagnetism anticipated the SU(2) Weak Force and the SU(2)xU(1) ElectroWeak unification, and Maxwell's consideration of a compressible general elastic Aether medium anticipated the Higgs mechanism and Torsion Physics.

In terms of the smallest charged Elementary Particle, the First-Generation Fermion Electron Compton Radius Vortex Particle, the Higgs VEV (about 250 GeV = 5 x 10^5 Me (Electron Masses)) gives the linear compressibility of the Aether,

Therefore, the Gravitational VEV should be given by the 4-volume compressibility of the Aether, so that the Gravitational VEV is about ( 5 x 10^5 )^4 Me = 6 x 10^22 Me = 3 x 10^22 MeV = 3 x 10^19 GeV.

Since the Gravitational VEV should correspond to a pair of Planck Mass Black Holes, the Planck Mass could be derived to be about 1.5 x 10^19 GeV.

 

 


 

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