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Topic:
Gravity tetrad as the spin 1 gauge field
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Gravity tetrad as the spin 1 gauge field
Posted:
Sep 24, 2007 6:48 PM


The principle of local gauge invariance for spin 1 compensating vector fields that are renormalizable with the HiggsGoldstone hidden ODLRO symmetry (t'Hooft, 1973) is battletested experimentally (electroweak force & QCD) and is a direct consequence of Einstein locality for the Bohmian IT hidden variables when the effect of the BIT quantum (super) potential is small Q. The effect of the latter is, in essence, a tiny warpdrive bubble in which the, e.g. electron as a tiny T = 0 Kelvin nonradiating "black hole", can under the action of Q can literally turn around in time giving a literal meaning to the Feynman diagrams (electron positron pair creation & annihilation) as more than simply a heuristic of perturbation theory.
The EinsteinCartan "spin 1" tetrad 1form Lorentz group vector field decomposes as the local T4(x) minimal coupling to matter fields Psi
e^aPa = I^aPa + A^aPa
i.e. P > P + A acting on Psi
P = I^aPa
A =A^aPa
{Pa} is the Lie algebra of the universal rigid T4 symmetry group of all material field actions in 1905 Special Relativity (SR).
This is analogous in QED to the local U1(x) minimal coupling
P > P + A acting on Psi
Now what is I^a?
I^a are the 4 tetrad 1form fields I^a = I^audx^u that are AFFINE INVARIANTS under the restricted flat geodesic Global Inertial Frame (GIF) transformations in Minkowski spacetime under the 10parameter Poincare group. That is, only maps from one GIF to another GIF.
s^2(1905) = I^aIa AFFINE INVARIANT
As soon as one considers a transformation from a GIF to an accelerating off(flat) geodesic Global NonInertial Frame (GNIF) I^a is no longer invariant. Note these are constant accelerations the same everywherewhen (conformal boosts?)
In fact I^a > I^a + X^a
Therefore, we introduce a compensating gauge potential A^a such that
A^a > A^a  X^a
to keep the sum e^a = I^a + A^a INVARIANT under this wider NONAFFINE group of PHYSICAL FRAME transformations.
s^2("conformal"?) = (I^a + A^a)(I^a + A^a)
This is like in electromagnetism where the gauge transformation is
A > A  Gradf
This is the first BABY STEP to Einstein's 1916 General Relativity (GR).
There is a SPINCONNECTION 1form
S^a^b = w^a^bce^c = w^a^bc(I^c + A^c)
The curvature field 2form R^a^b must vanish in the above case
R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb = 0 everywherewhen
The torsion field 2form T^a must also vanish
T^a = De^a = de^a + S^ac/\e^c
Einstein's 1915 GR locally gauges the rigid 4parameter T4 translation subgroup of P10 to T4(x), i.e. locally inhomogeneous displacements where
I^a > I^a(x)
I^a(x) > I^a(x) + X^a(x)
A^a > A^a(x) > I^a(x)  X^a(x)
Now the HOMOGENEOUS global spacetime interval s connecting pairs of arbitrarily separated events is localized to the INHOMOGENEOUS local ds(x) connecting pairs of closely spaced events.
Where
R^a^b(x) =/= 0 is allowed but T^a(x) = 0 is enforced.
i.e. curvature but not torsion in 1916 GR.
Note that all states connected by a redundant gauge transformation (on same gauge orbit) are the same physical state.



