Friday, January 12, 2007

Unified Field Theory


1. Einstein-Cartan Curvature-Torsion Theory is a Local Gauge Theory (TWB Kibble, 1961).
The standard model of electro-weak-strong forces comes from locally gauging the internal symmetry groups U(1), SU(2) & SU(3) for "charged" quantum field actions. We also break parity by hand in the SU(2) weak force and spontaneously break symmetry in the SU(2) vacuum to get the masses of W-bosons and the quarks. We have somewhere between 18 - 23 fudge factors or epicycles but string theory and loop quantum gravity have nothing to offer here in terms of prediction and interpreting real data not only in HEP but also in the dark energy & dark matter measurements. So standard model is the worst model we have except for any other. General relativity with torsion added comes from locally gauging the 10-parameter Poincare space-time symmetry group for all non-gravity field actions. The equivalence principle is automatic because we do all actions of every quantum field in special relativity. The 16 tetrads Au^a = eu^a - I^u^a and 24 spin connections Su^a^b = - Su^b^a are the compensating gauge potentials for the curvature and torsion fields respectively. This is a pretty picture.

That is you start with a field, either classical or quantum. You construct its rigid Poincare group T4O(1,3) + rigid internal symmetry group (U(1)SU(2)SU(3)) invariant dynamical global action S. You now let the constant parameters of all of the groups be arbitrary functions of the 4 space-time coordinates. This breaks the symmetries, i.e. the field actions are no longer invariant. You restore this set of broken global symmetries with a set of connections for parallel transport of "tensors/spinors" relative to the group either in the base space-time and/or the internal fiber spaces as the case may be. These connections are also called "gauge potentials" A and they obey "gauge transformations" that in the case of T4 are Einstein's 1915 "General Coordinate Transformations" GCT's, i.e. infinitesimal local displacements for T4 -> T4(x).

2. Note when we only locally gauge 4-parameter translation group of displacements T4 generated by the total global 4-momentum Pu(phi) of the field phi(x) whose action we are dealing with. Since this all happens in globally flat Minkowski spacetime there is no problem defining conserved global Pu(phi). This simplicity breaks down in curved spacetime after we locally gauge T4 because bare invariance under space and time linear displacements makes no sense in curved space-time. That is, total energy, total linear and rotational momenta of all fields are not fundamental only their local current densities are. Note that it is impossible to measure global integrals over the entire history of the universe everywhere when. We can only do local measurements and we find conservation of fluxes through small closed bounding 2D surfaces only when the radii of curvature are large compared to the scale of the closed bounding surfaces. Note that not all closed surrounding 2D surfaces without boundary are themselves boundaries if the 3D interior has star gate wormholes.

3. Furthermore, if we only locally gauge T4 and keep O(1,3) globally rigid, then the Shipov 2nd-order torsion gap field of dislocation defects in the world crystal lattice (H. Kleinert) vanishes. That is, for the 4 tetrad 1-forms e^a = I^a + A^a, I^a are the globally flat tetrads, A^a are the intrinsically curved tetrads.

T^a = De^a = de^a + S(T4)^ac/\e^c = 0 (vanishing torsion field)

d^2 = 0

where S(T4)^a^b = - S(T4)^b^a are the 6 non-dynamical spin connection 1-forms with T4 curvature 2-form

R^a^b(T4) = DS(T4)^a^b = dS(T4)^a^b + S(T4)^ac/\S(T4)^c^b

with Einstein-Hilbert T4 action density

L(Einstein-Hilbert) = {abcf}R^a^b(T4)/\e^c/\e^f

Note that in a classical vacuum the Ricci scalar R vanishes and, therefore, the classical gravity vacuum action is zero, which is essentially why there is no non-zero local gravity field stress-energy tensor in that limit. That is, the functional derivative of the gravity vacuum action with respect to the tetrads vanishes.

4. Now locally gauge the 6-parameter Lorentz group O(1,3) of spacetime rotations giving naturally the 6 extra anholonomic dimensions of the Calabi-Yau space of superstring theory pictured metaphorically by Shipov in the first figure above. This gives the new dynamical spin connections S(1,3)^a^b such that we now have a non-zero torsion field

T^a = S(1,3)^ac/\e^c

The new curvature 2-form is

R^a^b(10) = D'(S(T4)+S(1,3))^a^b

= d(S(T4)+S(1,3))^a^b + (S(T4)+S(1,3))^ac/\(S(T4)+S(1,3))^c^b

Note the curvature-torsion cross-coupling terms in the second term on the RHS.

5. Shipov's "teleparallelism" is the "Clifford parallel" (e.g. Hopf fibration).

Shipov adhoc requires the constraint (Ansatz, conjecture)

R^a^b(10) = 0

Therefore the torsion spin connections S(1,3) act as sources of curvature.

Now Shipov has a physical interpretation of this as "inertial fields" and I am not sure if I buy that. Shipov hopes he can derive the standard model matter fields from S(1,3) alone but he has not proved that. On the other hand he has the extra 6 anholonomic dimensions similar to what string theory has.

6. However, my own approach is that the new torsion fields S(1,3) are simply added to the warp tetrads A(T4) as compensating gauge fields to the apriori spinor lepton/quark fields and their compensating gauge force bosons.

Jack Sarfatti
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein

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