Monday, September 24, 2007

The principle of local gauge invariance for spin 1 compensating vector fields that are renormalizable with the Higgs-Goldstone hidden ODLRO symmetry (t'Hooft, 1973) is battle-tested 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 warp-drive bubble in which the, e.g. electron as a tiny T = 0 Kelvin non-radiating "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 Einstein-Cartan "spin 1" tetrad 1-form 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 to 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 to P + A acting on Psi

Now what is I^a?

I^a are the 4 tetrad 1-form 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 10-parameter 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 Non-Inertial Frame (GNIF) I^a is no longer invariant. Note these are constant accelerations the same everywhere-when (conformal boosts?)

In fact I^a to I^a + X^a

Therefore, we introduce a compensating gauge potential A^a such that

A^a to A^a - X^a

to keep the sum e^a = I^a + A^a INVARIANT under this wider NON-AFFINE 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 to A - Gradf

This is the first BABY STEP to Einstein's 1916 General Relativity (GR).

There is a SPIN-CONNECTION 1-form

S^a^b = w^a^bce^c = w^a^bc(I^c + A^c)

The curvature field 2-form R^a^b must vanish in the above case

R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb = 0 everywhere-when

The torsion field 2-form T^a must also vanish

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

Einstein's 1915 GR locally gauges the rigid 4-parameter T4 translation subgroup of P10 to T4(x), i.e. locally inhomogeneous displacements where

I^a to I^a(x)

I^a(x) to I^a(x) + X^a(x)

A^a to 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.


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.

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