PS
Consider

F = T + R = (I/\ + A/\ + S/\)(I + A + S)

OK so now it really looks like Yang-Mills F with A & S appearing symmetrically!

Note that

(I/\ + A/\ + S/\)I = 0

therefore we have the 2-form field from the ten compensating gauge potentials of Poincare group

F = T + R = d(A + S) + (A + S)/\(A + S)

F is peculiar with mixed Lorentz group indices

T^a + S^a^b = dA^a + dS^a^b + A^a/\A^b + S^ac/\A^c + S^ac/\S^c^b

which splits into

T^a = dA^a + S^ac/\A^c torsion field

R^a^b = dS^a^b + A^a/\A^b + S^ac/\S^c^b curvature field


On Oct 1, 2007, at 3:11 PM, Jack Sarfatti wrote:

v2 (expanded & corrected too many "/\" in R formulae)
On Sep 29, 2007, at 6:22 PM, Jack Sarfatti wrote:

Note also, the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields.

Look at the exterior covariant derivative 1-form

D = (d + S)/\

note that d is usually written without the /\ that is tacitly understood since

d(p-form) = (p + 1)-form

dual to boundary operator

&(p + 1) dim manifold = p-dim manifold so that Stokes theorem generalizes to

Integral of d(p-form) on p + 1 dim manifold = Integral of (p-form) on the boundary &(p + 1)manifold

for example when p = 0 in 3D space i.e. fundamental definite integral formula of calculus

d(0-form) = gradient of a function

p = 1 e.g. loop integral of vector (1-form) field is interior flux (2-form) integral

d(1-form) - curl of a vector (2-form)

p = 2 Gauss's divergence theorem - end of short digression

d(2-form) = divergence of a 1-form vector field in 3D space.

can generalize to Minkowski spacetime of 1905 SR

Stoke's theorem is key. A star gate traversable wormhole time machine to past (or equivalently in this regard a weightless zero g-force geodesic glider warp drive bubble in sense of Alcubierre's toy model metric) held open by universally antigravitating positive zero point dark energy density with equal and opposite negative pressure (w = -1 with GR source factor 1 + 3w) depends on multiply-connected spacetime corresponding to topological defects in the vacuum ODLRO field whose phase modulation determines the local curved tetrad fields and the local torsion field spin connections from localizing the 10-parameter Poincare symmetry group of special relativistic quantum field theory. The latter is background-dependent the former is not.

Multiple connectivity means, in our specific problem of metric engineering practical warp and wormhole, closed 2 surfaces (portals or wormhole mouths) without boundary that are not boundaries of the 3D space wormhole tunnel.


http://www.kahl.net/astro/graphics/wormhole.gif

the 2D mouth is a circle (1 space dimension removed in picture at a fixed "time"

the 3D space Dr. Who walks through is the 2D tube in the picture - you are flatlander bug constrained to surface in the picture.

"dark energy" = "exotic matter"

Since the vacuum ODLRO field is single-valued that means that the geometrodynamic field area density flux through a nonbounding closed 2D surface is quantized and this explains the Hawking-Bekenstein formula corresponding to point gravity monopole defects in the fabric of dynamical 3D space.

Entropy/kB = Horizon Area/4 Quantum of Area Flux = N

i.e. Newton's Planck area hG/c^3 ~ 10^-66 cm^2

with the world hologram formulae

Size of wavelet Quantum Foam Bubble ~ N^1/6(Quantum of Area)^1/2

i.e. &L ~ N^1/6Lp = (Lp^2L)^1/3

L ~ N^1/2Lp (hologram)

Horizon area ~ (10^28 cm)^2 in our pocket universe on the Cosmic Landscape with &L ~ 10^-13 cm and observed dark energy density ~ hc/NLp^4 ~ 10^-29 gm/cc


Back to main point

d in a sense is e

i.e. in a flat spacetime in a geodesic GIF coordinate basis

d(0-form) is "4-gradient" d/dx^a on a 0-form function since ea^u = Kronecker delta

ea = ea^u(d/dx^u)

e^a = e^audx^u

That is we can think of d/\ as e/\

or

D/\ = (e + S)/\

e = I + A


i.e.

D/\ = I/\ + A/\ + S/\

I is when we have globally flat Minkowski spacetime and a Global Inertial Frame (GIF)

A & S are the compensating geometrodynamic field gauge potentials that first appear in a Global Non-Inertial Frame

in globally flat Minkowski spacetime where the curvature R^a^b = 0 and the torsion T^a = 0 but in 1916 GR R^a^b =/= 0 while still T^a = 0.


where

I/\ is in flat Minkowski spacetime of 1905 SR

A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.

Note that A induces S that is not independent when the torsion T field vanishes globally.

In a GNIF

R = D/\S = 0


i.e.

R = (I/\ + A/\ + S/\)S = 0

and also

T = De = 0

i.e.

(I/\ + A/\ + S/\)(I + A) = 0

when rigid T4 is localized to T4(x) then we have possibility that

R = (I/\ + A/\ + S/\)S =/= 0

This notation makes the Yang-Mills field structure more apparent.

Consider

F = T + R = (I/\ + A/\ + S/\)(I + A + S)

OK so now it really looks like Yang-Mills F with A & S appearing symmetrically!


In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89

S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged.

That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli (2.89). Hence GR is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field that may give Salam's strong short-range f-gravity in addition to zero mass geometrodynamic field quanta. Indeed the composite quanta are spin 0, spin 1 and spin 2 from pairs of the fundamental spin 1 geometrodynamic quanta i.e. zero point fluctuations of the uncondensed part of the post-inflation vacuum ODLRO field. Think of the geometrodynamic field random "dark energy" quanta like the zero point motions of helium 4 atoms in the T = 0 ground state that is only 10% coherent condensate even though the effective superfluid density is 100%.





typo corrected draft 2
On Sep 29, 2007, at 5:37 PM, Jack Sarfatti wrote:

The key equation is Rovelli's (2.89) for only the torsion-free curvature-only spin connection in terms of the tetrads. It has quadratic and quartic parts. The quartic part can be put into the desired form but the quadratic part cannot. Also both parts depend on gradients in the tetrad component fields. It may be that only the torsion part of the spin connection can be put into the Yang-Mills covariant derivative form. I have not yet confirmed that. However, this is really a side issue, as in general we need to treat the 6 spin connection 1-forms S^a^b and the 4 tetrad 1-forms e^a as independent Yang-Mills type compensating local gauge field potentials in which we define the exterior covariant derivative as

D = d + S/\

Suppressing indices for simplicity. This is analogous to a Yang-Mills theory where the curvature two form field is

R = DS

i.e. curvature field 2-form = exterior covariant derivative of the spin connection Yang-Mills potential with itself, i.e. in 1916 GR

R = dS + S/\S

This is completely analogous to the Yang-Mills theory where

F = DA

= dA + A/\A

DF = 0

D*F = J*

DJ* = 0

In 1916 GR

DR = 0

D*R = *J

must translate in ordinary tensor notation to

Guv = kTuv

D*J = 0

corresponds to

Tuv^;v = 0 i.e. local energy-momentum stress current densities conserved - all bets off on global integrals over spacelike surfaces.

All of the above is for zero torsion fields

T = De = 0

This is an auxiliary equation not found in the internal Yang-Mills theories. The theory is more complex of course when T =/= 0 i.e. locally gauging the full 10-parameter Poincare spacetime symmetry group. One must be careful on how to make the analogy of GR with Yang-Mills theories. The analogy is perfect in Utiyama 1956 where there is only S and no e in the sense of the compensating field A where e = I + A because T4 is not locally gauged there. GCTs are put in adhoc - not pretty.

On Sep 28, 2007, at 4:25 PM, Jack Sarfatti wrote:

In trying to make gravity tetrad GR into a formal analog of Yang-Mills I have posited

S^ac = w^acc'e^c'

e^c' are the Einstein tetrad 1-forms

S^ac are the spin-connection 1-forms (involving gradients of the tetrads in 2.88)

Rovelli has (2.88) for example. Now I had thought I had seen the equivalent of S^ac = w^acc'e^c' in Rovelli's book, but now I cannot find it.

Using it, the torsion field 2-form is

T^a = de^a + S^ac/\e^c

= de^a + w^acc'e^c'/\e^c

which is like the Yang-Mills field 2-form

F^a = dA^a + w^acc'A^a/\A^c'

It is not clear that S^ac = w^acc'e^c' is consistent with (2.88)

Note also, the effect of the equivalence principle makes a difference in comparing gravity fields to Yang-Mills fields.

Look at the exterior covariant derivative 1-form

D = (d + S)/\

note that d is usually written without the /\ that is tacitly understood since

d(p-form) = (p + 1)-form

dual to boundary operator

&(p + 1) dim manifold = p-dim manifold so that Stokes theorem generalizes to

Integral of d(p-form) on p + 1 dim manifold = Integral of (p-form) on the boundary &(p + 1)manifold

for example when p = 0 in 3D space i.e. fundamental definite integral formula of calculus

d(0-form) = gradient of a function

p = 1 e.g. loop integral of vector (1-form) field is interior flux (2-form) integral

d(1-form) - curl of a vector (2-form)

p = 2 Gauss's divergence theorem - end of short digression

d(2-form) = divergence of a 1-form vector field in 3D space.

can generalize to Minkowski spacetime of 1905 SR

Back to main point

d in a sense is e

i.e. in a flat spacetime in a geodesic GIF coordinate basis

d(0-form) is "4-gradient" d/dx^a on a 0-form function since ea^u = Kronecker delta

ea = ea^u(d/dx^u)

e^a = e^audx^u

That is we can think of d/\ as e/\

or

D/\ = (e + S)/\

e = I + A


i.e.

D/\ = I/\ + A/\ + S/\

where

I/\ is in flat Minkowski spacetime of 1905 SR

A comes from using a GNIF and finally from localizing T4 to LIFs & LNIFs.

Note that A induces S that is not independent when the torsion T field vanishes globally.

In a GNIF

R = D/\S = 0

i.e.

R = (I/\ + A/\ + S/\)/\S = 0

and also

T = De = 0

i.e.

(I/\ + A/\ + S/\)(I + A) = 0

when rigid T4 is localized to T4(x) then we have possibility that

R = (I/\ + A/\ + S/\)/\S =/= 0

This notation makes the Yang-Mills field structure more apparent.

In 1916 GR S = S(A) is redundant as shown in Rovelli's eq. 2.89

S has and independent part when there is a torsion field i.e. full Poincare group is locally gauged.

That is 1916 GR is really a theory of the spin 1 Yang-Mills curvature tetrad field A with a redundant S as given in Rovelli (2.89). Hence GR is renormalizable in t'Hooft's sense including vacuum ODLRO Higgs field that may give Salam's strong short-range f-gravity in addition to zero mass geometrodynamic field quanta. Indeed the composite quanta are spin 0, spin 1 and spin 2 from pairs of the fundamental spin 1 geometrodynamic quanta i.e. zero point fluctuations of the uncondensed part of the post-inflation vacuum ODLRO field. Think of the geometrodynamic field random "dark energy" quanta like the zero point motions of helium 4 atoms in the T = 0 ground state that is only 10% coherent condensate even though the effective superfluid density is 100%.

On Sep 11, 2007, at 5:54 PM, Creon Levit wrote:

Jack, this is very important, right? 

Very.

It might help me (and a lot of us struggling to understand) if you could write very sort answers to the following questions:

1) If gravity = tetrads = spin-1 = renormalizable, then do you have renormalizable quantum gravity formulation? 

I believe so. But I am not expert in the details of t'Hooft's 1972 paper for which he got the Nobel. That my formulation falls within his paper seems self-evident to me, but I could be wrong here.

Or is it old news because you have essentially classical gravity, and it is only when you have (nonclassical) gravitational quanta that normalization becomes problematic for gravity theories.

I do not understand your sentence. The steps I use are:

1. Local gauge principle

2. 10-parameter Poincare group P10 of 1905 special relativity

3. A^a is the compensating "curvature" "Yang-Mills" spin 1 vector classical field from locally gauging ONLY the 4-parameter subgroup T4 of P10. The gauge transformations are precisely Einstein's local coordinate transformations of 1916. This was shown by Kibble clearly in 1961! t'Hooft's argument is when you quantize the classical A^a field. In t'Hooft "a" is an internal index. In my theory "a" is the tetrad Minkowski space index. The Lie algebras are not the same, t'Hooft's Lie groups are compact, P10 is not, but the formal structure seems to me to be essentially the same as far as physics is concerned.

From Rovelli I think

S^a^b = w^a^bce^a which gives instantly the FORMAL Yang-Mills structure for gravity!

S^a^b is the "spin connection" whose independent torsion field dynamics comes from locally gauging the 6-parameter Lorentz subgroup of P10 that Einstein did NOT do in 1916. This is the new torsion field physics of dark energy IMHO.

ds^2 = guvdx^udx^v = e^aea found in Rovelli's "Quantum Gravity"

My original WORLD HOLOGRAM Ansatz - completely original to me, and perhaps wrong, is

e^a = I^a + (1/N)^1/3A^a

I^a is the global Minkowski space-time tetrad 1-form

A^a is the intrinsically CURVED tetrad 1-form.

A^a is obviously a spin 1 vector field under the Lorentz group

A^a' = L^a'aA^a

L^a'a' is a Lorentz transformation

A^a = A^audx^u

A^au' = X^u'uA^au

X^u'u = Einstein local coordinate transformation (aka GCT) = T4 local gauge transformation.

A^a is a GCT INVARIANT!

N = Bekenstein's BITs

i.e. N = Surrounding 2D Area/4Lp^2

Lp^2 = hG/c^3 ~ 10^-66 cm^2 = quantum of area

G = Newton's gravity constant

Surrounding Area here has no boundary, but is itself not a boundary because it encloses N point defects in the macro-quantum coherent vacuum ODLRO order parameter with three real Higgs spin 0 scalar fields that is the "fabric of 4D spacetime". These point defects are somewhat like the simple poles (Residue)(z - zi)^-1 of a complex function w = f(z) of complex variable z.

Note also A^a = M^a^a

S^a^b = M^[a,b]

M^a^b = dTheta^a/\Phi^b - Theta^a/\dPhi^b

Theta^a & Phi^b are 8 zero-form Goldstone phase Lorentz 4-vectors whose two magnitudes Theta & Phi (3 real Higgs fields) make the point monopoles of the geometrodynamic field (AKA fabric of 4D spacetime).

This is the WORLD HOLOGRAM of "Volume without volume" that there is a 1-1 correspondence between an area quantum and its hologram image "volume quantum".

2) What are the [gravitational] quanta in your theory?

Obviously there are 3 kinds, spin 0 gravi-scalar, spin 1 gravi-vector and spin 2 gravi-tensor AKA "graviton".

This is completely elementary because Einstein's metric field guv is a quadratic form in A^a i.e.

ds^2 = guvdx^udx^v = I^aIa + N^-1/3(I^aAa + A^aIa) + N^-2/3A^aAa

N^-1/3 term gives only spin 1 quanta

N^-2/3 terms gives ALL THREE! spin 0, spin 1, spin 2 from elementary quantum field theory when

A^a classical field is replaced by A^a(classical ODLRO) + A^a(quantum fluctuation) as in Gorkov's theory of BCS superconductor for example.

Is there any graviton-like entity that emerges?  Is there any need for one?

Explained above. Spin 0, spin 1 & spin 2 gravitons.

3) What happens in your theory at the Planck scale, where the uncertainty principle "demands" the existence of complementary huge (or should we say "massive")  momentum and energy fluctuations?

Planck scale is trivial. It's when the post-inflation field order parameter vanishes like at the point monopole defect. No big deal. Same as in Yang-Mills quantum field theory. I have reduced gravity to a Yang-Mills theory. When the order parameter vanishes curvature and torsion also vanish - we are back to special relativity quantum field theory! This IS the pre-inflation false vacuum! The core of the point monopole is a small remnant of the pre-inflation false vacuum. This is really simple and elegant conceptually. The point monopoles are the equilibrium points of Hagen Kleinert's world crystal lattice. There disclination defects are curvature and their dislocation defects are torsion. Torsion induces curvature, but not vice versa. This is obvious from the local gauge theory of P10 BTW.

4) Is it not these fluctuations which drive the nonlinearities in GR to dominate and thus make the standard formulations analytically intractable and/or unrenormalizable?  Is this the basic quantum gravity conundrum, or is that something else?

I have no idea what you mean. Again:

1. Lorentz vector classical field theories WHEN QUANTIZED are renormalizable in general says t'Hooft 1972

2. The Einstein-Cartan tetrad theory of curvature and torsion is clearly formally a vector classical field theory.

That's all folks.

5) Does your theory have a way around this conundrum?  How so?

There is no conundrum here only a badly formulated question.

"The Question is: What is The Question?" (J.A. Wheeler) :-)

Jack Sarfatti wrote: OK I need a copy of the current STAIF Log 1 paper that the referees have. :-)
What's good about tetrads is that they are spin 1 vector field like Yang-Mills they are renormalizable because of t'Hooft's work - was it in 1972? Formally my A^a is same structure as t'Hooft's Yang-Mills - that he used compact internal SU(2) & SU(3) and I use non-compact Poincare group is not an essential difference in regard to renormalizablity I suspect.

On Sep 11, 2007, at 5:11 PM, Andrew Beckwith wrote:

Jack,
 
That is a start.
 
Now I want you to make a short, one paragraph description of what you just said.
A PARAGRAPH. Not a long one.
 
Start with the quantum fluctuations in a spin 1 vector field. Then jump from there
to the tetrad field theory.
 
...

 
If your theory has such a compact description of it, you need to include
this in the paper. That in itself is a MAJOR result. And it should definitely
be in your STAIF Log 1 paper.
 
I cannot stress how important this is.
 
Andy


Jack Sarfatti wrote:
i gave you my definition which is more than adequate for this paper's
purposes
quantum foam is trivial in my tetrad A^a field theory - same as quantum
fluctuations in any spin 1 vector field

quantum foam = virtual A^a quanta in QGMD

just like virtual photons are virtual A quanta in QED

my theory is much simpler than the previous theories

quantum foam = zero point fluctuations in the A^a field where

e^a = I^a + (1/N)^1/3A^a   already in paper

ds^2 = guvdx^udx^v = e^aea = I^aIa + (1/N)^1/3(I^aAa + A^aIa) + (1/N)^2/3A^aAa   already in paper

no big deal on "quantum foam"