It's Utiyama not Umezawa

On Aug 7, 2011, at 1:20 PM, Paul Zielinski wrote:

There no is question that Kibble, building on the work of Utiyama, did this formally in his 1961 paper. There is no point in denying that. But there is a trick here, having something to do I suspect with the construction of the covariant derivative.

Show us. The covariant derivative is the choice of the connection field that you keep confounding with the gauge transformations because you have not gone through the algebra of the local gauging program with sufficient attention to detail.

The connection is never a tensor relative to the gauge group transformations.

The gauge covariant curl of the non-tensor connection is the physical tensor field.

A gauge transformation never morphs a zero tensor into a non-zero tensor. You are still confused about that.

Gauge transformations are single-valued 1-1 reversible non-singular redundant defining an equivalence class of solutions all with the same topology, homotopy, homology etc. They are merely Plato's shadows of the real Idea in The Cave.

What you are groping for, like the blinded Cyclops trashing about, is Hagen Kleinert's multi-valued singular topology-changing transformations.

Curvature changes are the creation and destruction of disclination topological defects in the 4D world crystal lattice of quantum gravity. Torsion changes are creation and destruction of dislocation topological defects in the same.

The two convert into each other as shown in detail by Hagen Kleinert.

So mass-energy-stress current densities are sources and sinks for these defects that are like Landau's "elementary excitations" and no doubt you can make an S-matrix theory of their collisions, mergers, splittings etc.

I mean quanta of curvature and torsion.

We've been here before. The criticisms of Kibble's use of passive coordinate displacements made by Hehl et al, are well known. Hehl in particular starts with active diffeomorphisms and ends up with precisely the same Einstein-Cartan type theory.

You are confused. The distinction you are looking for has nothing to do with "active" & "passive" single-valued diffeomorphisms, but with the topology-changing singular multivalued transformations of Hagen Kleinert - those only happened pumped by changes in mass-energy stress current densities.

Kibble's method works formally, but I suspect that Hehl's gauge theoretic argument is logically and mathematically sound whereas Kibble's is not -- although it may look OK on the surface.

Prove it.

On 8/7/2011 9:15 AM, Jack Sarfatti wrote:

If localizing the group T4 of coordinate displacements just yields a larger class of coordinate transformations, then why doesn't the same apply to the Poincare group?

It does. Umezawa localized Lorentz group got curvature declinations but put in GCT by hand. Kibble did whole Poincare got torsion + curvature.

Utiyama?

Yes, thanks