Saturday, September 02, 2006

Kibble's 1961 paper on gravity as a local gauge theory

Sean Carroll's text book lightly touches on this at the end. Sunny Auyang makes a short cryptic remark in her book on the philosophy of quantum field theory that is intriguing but too incomplete.

My original unique approach to gravity as an emergent collective phenomenon from the inflation process itself has the tetrads (AKA "vierbeins") as the macro-quantum emergent 4D covariant supersolid field in analogy with the 3D Galilean superfluid velocity field. The tetrad field is renormalizable spin 1 as a quantum field. Einstein's geometrodynamic field is quadratic in the tetrad field, therefore any residual zero point micro-quanta outside of the Bose-Einstein vacuum ODLRO condensate forming the random anti-gravitating dark energy are Einstein-Podolsky-Rosen entangled spin 2 triplet pair states of the spin 1 tetrad quanta.

T.W.B. Kibble's 1961 paper "Lorentz Invariance and the Gravitational Field" JMP 2, March-April 1961 was a marked improvement over Utiyama's partial solution of the problem that locally gauged only the 6-parameter homogeneous Lorentz group (AKA Poincare group) to get the spin connection 1-form w^ab = w^abudx^u for the parallel transport of orientations of the tetrad 1-forms e^a = e^audx^u, a = 0,1,2,3 AKA Cartan mobile frames. Utiyama had to stick in the curved metric ad-hoc - not very satisfactory. Kibble locally gauged the entire 10-parameter inhomogeneous Lorentz group. This was prior to the elegant math of fiber bundles in physics where the compensating local gauge potential comes from the principle bundle and the source fields come from an associated bundle. Gauge theories use internal symmetry groups G for action dynamics with the Poincare group as a rigid non-dynamical background enforcing globally flat spacetime without any gravity at all. The equivalence principle forces the Poincare group to be dynamical and this introduces an added layer of complexity, ambiguity and confusion when trying to cast gravity as a local gauge theory. One must use Dirac's idea of the "substratum" in which the tetrad fields are well-behaved spin 1 vector fields when quantized rather than the unrenormalizable spin 2 tensor fields. It is curious that Kibble, or Penrose later, did not locally gauge the 15-parameter massless conformal group that is the basis of twistor theory. Locally gauging the 4-parameter translation subgroup T4 of the 10-parameter Poincare group gives the Einstein-Cartan tetrads e^a as the compensating field. However, because of the equivalence principle, these tetrads are also in the associated bundle as source fields like the spinor electron field in U(1) QED. That is, the equivalence principle has a feature like Godel's self-reference. In a sense this is true of all non-Abelian gauge theories that are self-interacting forming "geons" or "solitons" or "glue balls" (QCD), i.e. the gauge field carries the source charge. In the case of gravity the source charge is stress-energy density. Although the spin 2 geometrodynamic field does not have a local stress-energy tensor, one cannot jump to that conclusion for the spin 1 tetrad field in the substratum. Locally gauging the 6-parameter homogeneous group O(1,3) gives a dynamically independent spin connection. Note, that in Einstein's 1916 theory, the spin connection is not dynamically independent. The tetrads are dynamically independent and forcing the constraint of zero torsion gaps to second order in closed loops of parallel transport means that the spin connection components are determined by the tetrad components. This is not so in the general case treated by Kibble in 1961.

"The extended transformations for which the 10 parameters become arbitrary functions of position may be interpreted as general coordinate transformations and rotations of the vierbein system."

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