Tuesday, November 01, 2005

Classical Unified Field Theory

On Nov 1, 2005, at 1:42 PM, Jack Sarfatti wrote:


Please answer to me the following questions:

1. What group is located in your theory?

If the Poincare's group, is located:

2. What quantities are the structural functions of the group of translations and to what equations they satisfy?

3. What quantities are structural functions of the group of rotations and to what equations they satisfy?"

I assume you mean "localized" by "located".

u,v ... indices raised and lowered with guv etc.

I,J ... indices raised and lowered with nIJ (Minkowski)

Equivalence Principle (EEP) is

guv = (Iu^I + Bu^I)nIJ(1^Jv + B^Jv)

ds^2(P) = guv(P)dx^udx^v

P = {x} mod Diff(4)

Step 1 Localize T4 the translation subgroup of Poincare Group

The curved tetrad B is the compensating gauge potential from T4

B ~ dx^u&IBuP^I(Goldstone Phase)

Bu^I = eu^I - Iu^I

|u^I = Kronecker delta MAP connecting tangent space frames to base space frames in ALIGNMENT when B = 0.

The globally flat Minkowski space-time is degenerate, i.e. tangent space and base space MERGE, which is why we can have global frames. Curvature splits off local tangent fibers from the curved base space.

P^I are the 4 Lie algebra generators of T4, their conjugate phases are 4 dimensions of the base manifold.

Given the above preliminary structure, define the Cartan 1-form SPIN-CONNECTION W^IK implicitly from

e = e^I&I same for 1 & B

&I are base tangent vectors

&Idx^J = 1I^J duality of wave Cartan form and particle Cartan co-form.

i.e. dx's are wave front isophase surfaces

& are particle directional propagation vectors

De^I = d(1^I + B^I) + W^IK/\(I^K + B^K) = dB^I + W^IK/\(I^K + B^K) = 0 ZERO TORSION CONSTRAINT

Therefore, W^IK determined completely from B^I.

Map W^IK to Levi-Civita connection

W^IK = dx^uWu^IK

This is LOCAL DIFF(4) scalar INVARIANT (nice feature of this Cartan notation. This is the INTRINSIC GEOMETRY that is Diff(4) INVARIANT and LOCAL in Einstein's sense of P = {x} = Coset Space of Manifold/Diff(4).

It seems that Witten, Baez et-al do not know this? Hard to believe. Hence their odd statements about non-existence of LOCAL Diff(4) gauge invariant observables in Quantum Gravity?

Levi-Civita = {u^vw} = Wu^IKeI^ve^Kw

1915 GR Cartan Structure eqs are the same FORM as U(1) EM & SU(2)SU(3)Yang-Mills, but now for T4:

R^IK = DW^IK = curvature 2-form

R^IK = dx^udx^vRuv^IK

DR^IK = 0 Bianchi identities

D*R^IK = *J^IK(T4 sources) ---> Guv = kTuv(Matter)

D*J^IK(T4) = 0 local conservation of matter current densities

Step 2 BEYOND 1915 GR locally gauge O(1,3)

SIK ~ 6 generators of O(1,3). Their conjugate phases are 6 extra-dimensions of the "oriented" base manifold.

SIK ~ dx^uSuIK(Goldstone Phase)

D' = D + S^IJ/\. ...

Extended Curvature 2-form is

R'^IK = D'W^IK = R^IK + S^IJ/\W^JK

Note the torsion-curvature coupling term S^IJ/\W^JK

The field equations are

D'R'^IK = 0 Bianchi

D'*R^IK = *J'^IK(T4) Einstein Marble - Wood

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

Similarly the TORSION 2-form is

T'^IK = D'S^IK = dS^IK + W^IJ/\S^JK + S^IJ/\S^JK

with field equations

D'T'^IK = 0 Bianchi

D'*T^IK = *J(O(1,3)) Einstein

D'*J(O(1,3)) = 0 Current conservation

This is at least classical UNIFIED FIELD THEORY FORMALLY speaking.

Same basic template for ALL fields for ALL continuous symmetry groups internal & space-time.

On Nov 1, 2005, at 3:15 AM, Gennady Shipov wrote:

----- Original Message -----
From: Jack Sarfatti

Subject: Re: Local Curvature & Torsion fields as Nonlocal Bohm-Aharonov Topological Singularities

On Oct 31, 2005, at 2:39 PM, ROBERT BECKER wrote:

"Jack, Gennady -
I am going to take a leap and jump in here. As I read the debate on this issue, there appears to be some distinct points of difference, but the overall approach may not be completely different.
I cannot intelligently comment on Gennady's complete statement that Riemannian geometry can not be a group manifold because it lacks the Cartan Structure Equations. But perhaps I can say a few things about parts of this statement."
Jack Sarfatti

There may be a problem of the Russian --> English translation here since, e.g. Rovelli's book Ch 2 clearly shows Cartan structure equations for 1915 GR without torsion. How to generalize correctly to torsion is not completely obvious. I need another go at that.

Only 2 key principles:
1. Locally gauge every continuous group, both internal & space-time.

2. Enforce equivalence principle by making weak force Higgs vacuum coherence mechanism also the source of emergent c-number gravity. Note that we have running coupling constants - weak force gets stronger at higher energy.

Side point:

Heisenberg's uncertainty principle is INCOMPLETE, i.e.

&r ~ h/&p + (G*/c^3)&p

G* is effective gravity parameter at scale &r.

Simplest Higgs mechanism in the standard model has TWO complex scalar fields for SU(2)weak

i.e. n = 4 real fields.

But only one complex scalar field develops the local order parameter VEV =/= 0. It is electrically neutral.

Remember U(1)hyperchargeSU(2)weak has 4 generators, combinations of which make ONE electric charge neutral massless photon and 3 massive Weak bosons. Not the Higgs bosons, which is another part of the theory.
From this POV I only have 2 real fields, n = 2, so in 3 space 1 + d' + r = 3. Stable defects d' = d - n = 1 string vortices rather than point defects. The two real VEV's give me 1 Goldstone Phase to make gravity with from

B = (hG/c^3)^1/2'd' (Goldstone Phase)

On the other hand, there are other possible models.

Of the 4 real fields, 3 feed mass to the W-bosons and the remaining massless one is the photon.

In this model N - K = 1 is dim of H, which in this case is U(1).
K = 3 = number of broken generators of G = U(1)xSU(2) has N = 4
and if n = 4, i.e., U(1) acts as multiple of identity on the two complex fields, rather than my earlier n = 6, then there is only 1 Higgs boson that should be ~ 250 Gev, which is the critical temperature for the onset of the VEV at least at this cosmological epoch. Presumably it was much larger when the universe was smaller consistent with the strength of the weak coupling decreasing as universe expands.


OK,? Einstein-Cartan tetrad, showing the LIF tangent fiber indices explicitly

e^I = 1^I + B^I

B^I also from local gauging of T4 to Diff(4)

The 1 + B split is Diff(4) INVARIANT

Hence B = 0 really is a globally flat spacetime.

Zero torsion 2-form means

De^I = 0

where the SPIN CONNECTION is the 1-form W^IK

D = d + W^IK/
De^I = dB^I + W^IK/\B^K = 0

Therefore W determined from B, which comes from SINGULAR nonlocal Bohm-Aharonov "Flux-without-flux" 'd'(Goldstone Phase) d' = 1 vortex string singularities in simplest Higgs model that will not give a stable Hedghog where we need d' = 0.

The curvature 2-form is

R^IK = DW^IK = dW^IK + W^IJ/\W^JK

Bianchi identity is

DR^IK = 0

Einstein's field eq. is

D*RIK = *J(Matter)

Local conservation of matter current density is

D*J(Matter) = 0

That's 1915 GR in a NUTSHELL using SAME Cartan form eqs as in EM or Yang-Mills theory.

Enter TORSION from local gauging of O(1,3) that is GLOBAL in 1915 GR.

This gives TORSION 1-form Potential S^IK where now

D' = D + S/
Therefore, the CURVATURE 2-form is now

R'^IK = D'W^IK = R ^IK+ S^IJ/\W^JK


The TORSION 2-form is?

T'^IK = D'S^IK = dS^IK + W^IJ/\S^JK + S^IJ/\S^JK? =/ = 0

The Bianchi identities are

D'R' = 0

D'T' = 0


D'*R' = *J(Translation)

D'*T' = *J(Rotation)

The LOCAL current conservation laws are

D'*J = 0 (both)

Everything is MOD Diff(4,) i.e. Einstein's "local coincidences" P = {Diff(4) Orbits of manifold points} as explained by Rovelli Ch 2.?

One can add Yang-Mills fields from internal symmetry groups trivially into

D" = D' + A^IK/
where A^IK = A^IKaT^a

T^a are the charges of G which spontaneously breaks to H etc.

"A formalism is not a geometry. The Cartan Equations of Structure arising from the what is often called the Cartan Moving Frame Method, which I think is synomous with Gennady's Oriented Point Method, provides a formalism upon which one can impose additional constraints and assumptions."

Yes, Gennady uses a 10Dim manifold from the local gauging of O(1,3), which will be internal parameters in my notation

i.e. Bu^I ~? BuP^I

The 4 P^I form Lie algebra of T4 for 4D translations

Su^IK ~ SuL^IK

L^IK are the 6 O(1,3) Lie algebra generators for 4D rotations.

There are conjugate phases @^I & @^IK, I =/= K, for each, i.e. a 10 Dim manifold of "dynamical phases" (not Goldstone phases of VEV's of course)
"From what I can see, Jack?is?using the Cartan Structure Equations in his derivation. However, I do not think Gennady's objection is to Jack's use of the Cartan Structure Equations specifically. What Gennady appears to be saying is that the Curvature Form which appears in the Equations of Structure is?not?the same kind of mathematical object as the Curvature that shows up in the Riemannian Differential Geometry of GR. Gennady is correct on this so far as I know."

Yes, but there is a TETRAD MAPPING connecting them.

R^IK = Ruv^IKdx^udx^v

Ruv^wl = Ruv^IKeI^wel^K

The base space u,v indices ARE HIDDEN in the Cartan formalism, that is AUTOMATICALLY DIFF(4) SCALAR LOCAL INVARIANT in sense of P "local coincidences" of course.

"What shows up in the Equations of Structure is what Vargas calls the "Total Curvature", of which the classical Riemannian Curvature is only a piece."

That's exactly my

R' = D'W = R + S/\W

The difference is the direct torsion-curvature coupling.

If you want TELEPARALLELISM, then I suppose that means

R' = 0

T' =/= 0

"By making the assumption that Torsion vanishes, one obtains Riemannian GR and a Riemannian Curvature."

Yes, that's simply S = 0 in my notation.

"(One can readily use the Cartan formalism to recover Riemannian GR, it just does not contain the full richness of the mathematics that can be obtained from the Cartan formalism if one does not make the vanishing Torsion assumption.)
Therefore, if the vanishing Torsion assumption is used, then the more restricted version of the Curvature Form shows up in the Equation of Structure and the full theory is not obtained. Jack in the first part of his derivation?does?appear to be using a vanishing Torsion assumption, which forces the Curvature in Jack's equations to not be the full Total Curvature of the Cartan formalism. So, in this sense, Gennady is correct that there is a discrepancy."

Not really. I simply did that at first for PEDAGOGY to show how to get Einstein's 1915 theory as THE FIRST BABY STEP. After a few FUMBLES I think I got the TORSION extension of GR above (correcting some previous errors).
"However, about 2/3 to 3/4 of the way down in Jack's sequence of equations, there appears a T', a "modified" Torsion, which is?not assumed to vanish.?So,?with this final set of equations, Jack seems to be satisfying Gennady's concern over the apparent use of a vanishing Torsion in the Cartan formalism."

Yes. But also I have the CONNECTION to Higgs mechanism for origin of inertia of leptons and quarks and GRAVITY so that SPIRIT of equivalence principle is obeyed GRAVITY & INERTIA EMERGE TOGETHER! Still lots of mopping up details of course. But basic idea I think is sound. Everything FROM COHERENCE including ZPF COHERENCE i.e. ZPF, like ALL EPR ENTANGLEMENTS is LOCALLY RANDOM but NONLOCALLY COHERENT or PHASE-LOCKED. In contrast Higgs field CONDENSATE is BOTH LOCALLY AND NONLOCALLY COHERENT. The origin of GRAVITY and INERTIA TOGETHER is from the LOCAL COHERENCE. The DARK ENERGY/MATTER is from the LOCAL w = -1 INCOHERENCE of the NONLOCALLY EPR CORRELATED ZPF in the T = 0 Kelvin "Vacuum".
"A few other comments:
Assuming vanishing Torsion?yields GR. Assuming the?vanishing of the Total Curvature (in which case, the Torsion does not vanish) is called Teleparallelism (TP) and is one?primary leg of Vargas Theory."

Yes, that's easy in my notation, that's

R' = R + S/\W = 0

T' = dS' + W/\S + S/\S =/= 0

But why R' = 0 is not physically compelling for me. I prefer

D'R' = 0

D'*R = *J (Translation)

D'T' = 0

D'*T' = *J(Rotation)

D'*J = 0 etc.

"Another difference is that Vargas Theory does not seek to introduce gauge fields into the frame forms (i.e. Jack's e) or into theory at all. Finally, no assumption of a spin-connection is made in Vargas Theory and the resulting "geometric" Maxwell-Einstein Equations resulting from the Torsion Equation of Structure do not have a rotating source. Torsion in Vargas Theory is not related to rotation, but rather, in a certain matehmatically detailed sense, to EM. The geometric Maxwell Equations essentially represent a geometrization of the EM (at that level of theory).
Take care,
Robert E. Becker"

I think my theory makes more physical sense than Vargas theory BECAUSE

I get Gravity and Dark Energy/Matter and Inertia ALL EMERGING TOGETHER in a way consistent with standard model of leptons & quarks. My theory is very close to observation and is FALSIFIABLE.?

In this way my theory is better not only than Vargas theory, it is better than STRING THEORY and better than LOOP QUANTUM GRAVITY.

I make some strong predictions of principle:

1. Dark Matter Omega ~ 0.23 is VIRTUAL NONLOCALLY COHERENT ZPF that is LOCALLY RANDOM with positive pressure for compact sources.

2. Hence DARK MATTER DETECTORS will BE SILENT with RIGHT STUFF same a Michelson-Morley not showing Ether Drift - barring some bizarre claims to the contrary of course. No exotic on-mass-shell quanta will explain Galactic Halo etc.

3. Quantum foam does not exist and high energy cosmic photons will not show it!

4. Universal slope of Regge trajectories for hadronic resonances is STRONG SHORT RANGE ZPF induced gravity.

Basically Kerr solution J/h ~ GM^2/hc? (I published that in 1973)

i.e. anticipated string-blackhole correspondance in 1973.

No comments: