Saturday, June 09, 2007

Note that gauge-relativity theories are intrinsically quantum renormalizable spin 1 vector field theories with two signs of charges hence both attractive and repulsive classical forces. Gravity is bilinear in the spin 1 substratum tetrad "charges" hence is classically only "spin 2" attractive, with repulsion in the zero point quantum corrections, i.e. "dark energy."

Utiyama continued:

"Some systems of fields have been considered which are invariant on a certain group of transformations depending on n-parameters. A general rule is obtained for introducing a new field in a definite way with a definite type of interaction with the original fields by postulating the invariance of these systems under a wider group derived by replacing the parameters of the original group with a set of arbitrary functions. The transformation character of this new field under the wider group is determined from the invariance postulate. The possible types of equations of the new fields can also be derived, giving rise to a certain conservation law owing to the invariance. As examples, the electromagnetic, the gravitational and the Yang-Mills fields are reconsidered following this line of approach."

Paraphrasing Utiyama - the minimal coupling interaction illustrated for U(1) Maxwell electromagnetic field potential Au, use the gauge covariant partial derivative

DuQ = Q,u - ieAuQ

DuQ* = Q,u + ieAuQ

where Q is the charged source field (it's a spinor actually). Q is a complex number having the U(1) phase freedom. Here it's a spin 0 scalar field. With a spinor it will have 2 complex components if a massless Weyl spinor and 4 if a massive Dirac spinor.

The RIGID Noether theorem global conservation of total charge comes from the Q-field action being invariant under the local causality-violating rigid U(1) internal symmetry transformations

Q -> Q' = e^i@Q
Q* ->Q*' = e^-i@Q*

with the RIGID U(1) group parameter phase @ a constant for all events P over globally flat 4D Minkowski spacetime from Alpha to Omega everywhere-when - a dubious stretch of the imagination sans gravity sans torsion - too limited a model to be sure.

Now LOCALIZE i.e. destroy the RIGIDITY by letting @ -> @(x) a local arbitrary function, same basic idea as letting global rigid T4 go to GCT = T4(x) to get "spin 1" substratum gravity Au^a where spin 2 Levi-Civita {u^vw} ~ eu^ae^va,w such that

eu^a = Iu^a + (Lp^2/\zpf)^1/3Au^a = full Einstein-Cartan tetrad components (16).

(Lp^2/\zpf)^1/3 is the dimensionless coupling where in IR limit /\zpf -> /\ = Einstein's cosmological constant "landscape height" for each deSitter pocket parallel universe in the megaverse.

The factor 1/3 comes from the world-hologram Bekenstein -> Susskind postulate that the geometrodynamic field is 2D surface not 3D volume.

i.e. &R = (Lp^2/\zpf)^1/3R = Lp^2/3R^1/3

R = deSitter dark energy radius.

&R ~ 1 fermi in our pocket Hubble bubble parallel universe on the chaotically inflating cosmic landscape of the WAP.

Back from this future timeline in 2007 to Utiyama 1954-56 in the next episode.

On Jun 9, 2007, at 3:55 PM, Jack Sarfatti wrote:

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.

On Jun 9, 2007, at 2:43 PM, Jack Sarfatti wrote:

At this point in time I see no problem in theoretical physics in 2007, including both dark energy and dark matter, that cannot be solved satisfactorily in terms of the two great already battle-tested organizing ideas:

1. The gauge principle, i.e. Einstein's relativity principle expressed in its deepest form.

2. "More is different" (PW Anderson) emergence of new orders i.e. effective low/high energy theories including renormalization group flows to fixed points, ODLRO, "hidden symmetry", Goldstone-Higgs mechanism, spontaneous ground state symmetry breakdown leaving action invariant.

If anyone has a counter example - let's see it.

String theory and loop quantum gravity are over-mathematized clearly failed programs. The only surviving parts are already in 1 & 2 above.

On Jun 3, 2007, at 5:00 PM, Jack Sarfatti wrote:

PS To be more precise: Let Psi be the matter source fields forming a basis for an irreducible representation R of the rigid global Lie group G where the action field density L invariant under G. We need a new larger action density L' invariant under the larger locally gauged G(x) where the parameters of G are no longer constants but are arbitrary functions of the overlapping local coordinate charts. L(Psi, Psi,u) -> L(Psi, R(DuPsi), Au) where Au is the spin 1 compensating gauge field "connection".
GR is less direct than Yang-Mills because of the equivalence between local g-forces and stationary non-geodesic observers outside of event horizons. Not all GeoMetroDynamic fields are stationary of course - but the non-stationary corrections (gravity waves) are usually pretty small - definitely small Earthside. In GR the passive no gravity arena for Yang-Mills fields becomes at active player. A^a is the warped tetrad connection. It is spin 1 as a quantum field. The GMD affine connection is secondary essentially (,u is ordinary partial derivative):

GMD connection of Einstein-Cartan theory ~ e^ae^b,u = (I^a + @A^a)(I^b + @A^b),u

this includes possible torsion fields, but not a non-metricity Weyl conformal boost field in which the inner products are anholonomic (path dependent) - the non-metricity 4-potential has no relation to the EM potential which was Weyl's error of 1918. The curl of this conformal boost 4-potential vanishes in a simply connected way for phenomena observed so far.

Jack Sarfatti
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein

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