## Saturday, June 09, 2007

Utiyama's 1956 paper was work he actually finished in 1954 so he never got full credit. Details are in L O'Raifeartagh's Princeton book "The Dawning of Gauge Theory".
Utiyama in 1956 wrote a clear statement of the gauge-relativity organizing idea:

"Some systems of fields have been considered which are invariant on a certain group of transformations depending on n-parameters."

Stop there for a few key concrete examples:

1. Conservation of linear momentum and energy n = 4, i.e. RIGID translation group T4

2. Conservation of angular momentum, n = 3, i.e. RIGID 3D rotation group O(3)

4. Special relativity 1905 n = 10, i.e. RIGID Poincare group P(10) includes T4 & O(3) as subgroups.

5. Maxwell's electromagnetic field of 1865 is n = 1, i.e. localized U(1) S1 circle phase group. Local compensating spin 1 gauge potential is Au

6. Yang-Mills weak force is n = 3, i.e. localized SU(2) S2 double circle group. Local compensating connection gauge potentials are Bu^a, a = 1, 2, 3, u = 0,1,2,3

7. Yang-Mills strong force is n = 8, i.e. localized SU(3) S3 triple circle group. Local compensating gauge potentials are Cu^b, b = 1,2,3, ... 8, u = 0,1,2,3

Each S1 circle is a complex variable plane.

n = number of elements in the Lie algebra of conserved rigid "Noether" charges infinitesimally generating the continuous Lie group G of symmetry invariances of the field global actions.

8. General relativity 1916 (with disclination defect curvature fields but without torsion fields) is the localization of n = 4 RIGID T4 to what I non-rigorously call "GCT" i.e. Einstein's "General Coordinate Transformations"

x^u(P) -> x^u'(P) = x^u'(x^u(P))

at a fixed "local coincidence" (Einstein Hole Paradox 1917). Do not confused P with a bare manifold point p. P is a "gauge orbit" of a continuous infinity of manifold points p, i.e. P = {p} equivalence class. These GCTs are non-physical gauge transformations like

Au -> A'u = Au - Chi,u

in U(1) electromagnetism. The compensating gauge potential is NOT the UNRENORMALIZABLE spin 2 Levi-Civita connection {u,vw} but is the RENORMALIZABLE spin 1 warped tetrads Au^a where, a = 0,1,2,3 for the free-float zero g-force Local Inertial Frame (LIF) geodesic observers and u = 0,1,2,3 for the non-zero g-force Local Non-Inertial (LNIF) off-geodesic observers. Both LIF & LNIF at "same" P, i.e. separations small compared to local radii of curvature. {u,v,w} are bilinear in Au^a and its gradients. The antisymmetric 24 spin connection components S^a^bu = - S^b^au here are not independent fields, but are partially determined by the 16 tetrad components Au^a leaving the 8 vacuum ODLRO Goldstone phase gauge freedom that is in my 2006 archive paper.

9. Einstein-Cartan theory with torsion 4D world crystal dislocation gap fields in addition to curvature disclination fields has n = 10, i.e. localize rigid P10.