Sunday, November 20, 2005

Important discovery on macro-quantum geometrodynamics with multiple Goldstone phases.

It's well known that for constant magnitude of ODLRO VEV in V = G(disordered)/H(ordered) coset space:

Zero homotopy group non-trivial has 2D domain wall defects surrounding space of defect is S0 i.e. points -1, +1.
First homotopy group non-trivial has 1D string (vortices) with S1 loop surrounding space and G/H = S1 also.
Second homotopy group non-trivial has 0D point defect "monopole" (not necessarily magnetic, the geometrodynamic neutral monopole gives S/k = AREA/4Lp^2 + World Hologram + maybe NASA Pioneer Anomaly as dark energy hedgehog with wrapping number -1 defect in center of Sun?) with surrounding space = S2 and also G/H = S2.
Third homotopy group non-trivial has "texture" defect with G/H = S3.

Internal symmetries have no non-trivial tetrad valued in the Lie algebra of the locally gauged symmetry group.
Space-time symmetries DO and that is the DIFFERENCE from the EQUIVALENCE PRINCIPLE AKA EEP.

2 independent ideas here

1. Local gauging

2. Spontaneous vacuum symmetry breaking.

In my theory of emergent gravity i.e. my version of Andrei Sakharov's "metric elasticity"

I. I SPONTANEOUS BREAK INTERNAL SYMMETRY at Planck scale -> GUT vacuum phase transition

II. I locally gauge at least 10-parameter Poincare group - ultimately 16 parameter GL(4,R).

Note HOW TO GET GENNADY SHIPOV's 10 MANIFOLD (same as string theory) is to put the entire Lie algebra of the Poincare group into the TETRAD FIELD directly. This makes it a Kaluza-Klein theory. However, I DO NOT DO THIS - but I can later. I keep the tetrad field 4D from T4 and that's why I need the indirect quantities such as

B^I a 1-index 1-form curved tetrad from locally gauging T4

e^I = 1^I + B^I

W^IJ a 2-index 1-form CURVATURE spin-CONNECTION in

D = d + W^IJ/
And also

S^I a 1-index 1-form in the 4D tetrad from locally gauging O(1,3)

e'^I = e^I + S^I

Z^IJ a 2-index 1-form TORSION CONNECTION in

D' = D + Z^IJ/
Where we have the 2-forms

T^I = De = 0


T'I = D'e'

R'IJ = D'(W^IJ + Z^IJ)

With Bianchi identities (analog of dF = 0 of Maxwell internal U(1) EM field in flat space-time without gravity & torsion)

DR^IJ = 0

D'R'^IJ = 0

D'T'^I = 0

Source equations

D'*W^IK = *J'(T4)^IK

D'*Z^IK = *J(O(1,3)^IK

D'*T^I = *j(O(1,3)^I

Local covariant current density conservation

D'*J'(T4)^IK = 0

D'*J(O(1,3))^IK = 0

D'*j(O(1,3)^I = 0

OK, vortices have degenerate ODLRO macro-quantum manifold V = G/H of local order parameters with topology of the circle. That is we hold the Higgs amplitude fixed and we only have ONE independent Goldstone phase factor e^iTheta where the order parameter has 2 real scalar components.

Non-integrable Goldstone phase factors e^iQ^a(Theta)a, for Lie algebra [Q^a,Q^b] = f^a^bcQ^c give both the global and local properties of Yang-Mills internal gauge theories as well as gravity and torsion & non-metricity, dilaton gauge theories both classical and quantum via Feynman path integrals.

Point defects has V = S2 with 2 independent Goldstone phase factors e^iTheta and e^iPhi, i.e. polar and azimuthal angles in V = G/H space where the order parameter has 3 real scalar components.

The geometrodynamical fields.

I will only do T4 for simplicity.

CASE 1 V = G/H = S1 with ONE GOLDSTONE PHASE Theta that sweeps out a unit circle.

B^I(S1) = (LpP^I(Theta)/ih


e^I = 1^I + B^I

Einstein invariant is

ds^2 = B^I(Minkowski)IJB^J


Here we expect STRING DEFECTS!

In simpler completely invariant notation

B(S1) = Lpd(Theta)

Case 2 V = G/H = S2 with TWO INDEPENDENT GOLDSTONE PHASES Theta & Phi that sweep out a unit 2D spherical surface that is G/H topology.

So what do we do?

Simple! I was a DUMMY not to see this weeks ago! But I woke up this morning hundreds of miles north of SF on the beach with the idea crystal clear!

Define the 0-form (Theta)(Phi) these are local functions of Einstein's local coincidences "P" of course.

Our new geometrodynamic 1-form is obviously via product rule of Newton's "fluxions", i.e. "ghosts of departed quantities" (Bishop Berkeley)

B^I(S2) = (LpP^I(ThetaPhi)/ih

i.e. 2 -terms

B^I(S2) = (Lp(P^I(Theta)/ih)Phi + Theta(LpP^I(Phi)/ih = LENGTH OPERATOR

i.e. in completely invariant notation

B(S2) = Lp{d(Theta)Phi + Thetad(Phi)}

using d^2 = 0

We have the 2-form AREA OPERATOR (forget Loop Quantum Gravity & Spin Foams!)

Area = Lp^2dTheta/\dPhi

We also have

Closed surface integral of Area operator = Volume integral of d(Area) = Wrapping Number

when the closed surface surrounds the point defect since second homotopy group for S2 is Z

This is obviously essentially both Hawking-Bekenstein BITS and 't Hooft-Susskind hologram idea that is simply

VOLUME WITHOUT VOLUME as an example of Bohm-Aharonov's "Flux without flux" in which the gauge potentials have direct non-classical physical effects on phenomena.

We can continue this to case

G/H = S3 (textures) where now we have 3 independent Goldstone phases Theta, Phi & Chi with a VOLUME OPERATOR.

If we continue into higher dimensional hyperspace, I suppose V = G/H are what? Calabi-Yau spaces? I will check.

Here we need to extend the EEP tetrad idea missing in conventional internal symmetry Yang-Mills gauge force theories.

Basically we have S1xS1x... for possible V = G/H broken vacuum symmetries.

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