Sunday, November 18, 2007

In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear group GL(n, C).
The simplest case, SU(1), is a trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of absolute value 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is { + I, − I}.

Three real Goldstone phases correspond ot the 3-sphere of internal SU(2). Therefore, my eight Goldstone phases THETA^a and PHI^b forming two Lorentz group SO(1,3) 4-vectors project to two internal SU(2) groups when their respective magnitudes are real. These subspaces correspond to stable GMD string quantized vortex filaments in ordinary 3D space with an attached internal SU(2). That is, the "lines" of the 3D "spin lattice" naturally carry SU(2) q-numbers in my model! The two spaces with two Goldstone phase magnitudes taken together give the GMD point defect monopoles at the ends of open string vortices.

The residual 6 internal Goldstone phases after 2 magnitudes are fixed for the 3D spin lattice require one extra real Higgs scalar. Therefore, there is actually the 10D + 1 manifold of Ed Witten's M-Theory naturally in my model along with its 9D + 1 sub-manifold.

On Nov 18, 2007, at 12:03 AM, Jack Sarfatti wrote:

Now if you think the logical flow of ideas below is hard to follow, for comparison look at a paper by almost any string theorist and tell me if what they say is easier for you to follow? ;-)

On Nov 17, 2007, at 11:56 PM, Jack Sarfatti wrote:

On Nov 17, 2007, at 11:15 PM, Jack Sarfatti wrote:

Note the EM vector potential A(em) is compared to the tetrad e^I, i.e. (2.101) & (2.105). Really A(em) should be compared with A^I(gravity) where the torsion field vanishes.

e^I(gravity) = I^a(1905SR) + @A^I(gravity)

where from P.W. Anderson's "More is different" emergent complexity I posit

A^I(gravity) = M^I^I = (dTheta)^I(Phi)^I - (Theta)^I(dPhi)^I


F^I(gravity) = DA^I ~ 2(dTheta)^I/\(dPhi)^I + w^IJ/\A^J

w^I^J = M^[I,J]

M^I^J = (dTheta)^I(Phi)^J - (Theta)^I(dPhi)^J

DF^I = 0

D*F^I = *J^I

D*J^I = 0

The 8 Goldstone phases form 2 Lorentz group 4-vectors THETA^I and PHI^J. Their 2 magnitudes THETA & PHI are the 2 effective Goldstone phases of the world hologram for 3D + 1 spacetime. This requires 3 real Higgs fields living in our 3D spacelike world. The possible stable topological defects are domain walls S0 vacuum manifold, line vortices S1 vacuum manifold, and point monopole defects S2 vacuum manifold. The 3D "spin network" is a "Abrikosov" lattice of point monopoles connected by line vortices. The finite cores of these soldered defects is the residual random zero point energy density from the pre-inflation false vacuum just like in superfluid helium. The residual 6 Goldstone phases lead to the internal symmetry Calabi-Yau space. E8 obviously applies to my 8 post-inflation Goldstone phases. Since the magnitudes THETA & PHI can be imaginary as well as zero and real, the 3D Goldstone phase space is non-compact. Brane worlds are simply the stable topological defects in the space of the 9 real Higgs fields that form the 8 Goldstone phases.

Horizon area is quantized and the number of area quanta N is the wrapping number of the non-trivial 2nd homotopy group using the S2 vacuum manifold of all 3 Higgs fields for the "visible universe" - the top of the iceberg as it were. Each real Higgs field is essentially a "space dimension" in order to have stable topological defects where subsets of the real Higgs fields vanish leaving subsets of Goldstone phases undefined i.e. Berry phase singularities.

The area element is ~ dTHETA/\dPHI with N quanta, N = # Bekenstein bits

The volume element is d(area element) = 0, i.e. volume without volume (Wheelerism)

The world hologram imaging equation is Jack Ng's equation

&L = (Lp^2L)^1/3 = N^1/6Lp

L = scale of the dominating horizon, which in our universe is the Hubble scale of the retro-causal future de Sitter horizon with dark energy (DE) density

t00(DE) = hc/NLp^4 ~ 10^-29 gm/cc

N ~ 10^122

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