The Gauge-Relativity Principle 1

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, 5:58 pm, Jack Sarfatti wrote:
4. Special relativity 1905 n = 10, i.e. RIGID Poincare group P(10)
includes T4 & O(3) as subgroups.

Newtonian Physics; Galilei group -- 11 parameters.

This raises an interesting question (and challenge) to you: define the
Galilean limit (that is, the limit as c - infinity) of Poincare (with
10) to Galilei (with 11).

The completely unexpected sucker-punchline is below item 9.

[8.] at a fixed "local coincidence" (Einstein Hole Paradox 1917).

Rovelli made a big deal about the hole paradox in his 2004 quantum
gravity book. He insisted on taking the tack that this leads
invariably to the notion that after you take out gravity, there's
nothing "in the background" left; not even spacetime. That is, after
factoring out "(4-dimensional) diffeomorphism invariance", you get
nothing at all.

Of course, this is fallacy. The analogous argument (a very close
analogy) is that after factoring out translation invariance from a
vector space, you get "nothing". Of course, you get something! It's
called an affine space.

By the same token once you remove diffeomorphism invariance, you still
have a structure left: the underlying manifold and its
differentiability structure and topology.

Rovelli would have you believe (as Loop quantum gravity seems to
insist) that even this is removed from the background and "quantized".
Non sequitur. What's actually described as having been quantized in
Rovelli's treatment (and in LQG in general) is NOT the whole thing --
not even the whole metric! Just the spin connection part (that part
that goes into Ashtekhar's formalism). Not even the tetrad field, just
the spin part.

This, of course, need to be, and is a central element of
Sardanashvily's (and Penrose's parallel) argument that you can't
quantize the inertial frame bundle since mutually accelerating vacuum
sectors do not have coherent superpositions (hence, the tetrad remains
fundamentally classical).

But even taking Rovelli on his own merits, he misses a major mark. The
"diffeomorphism invariance" that's actually made use of is only the
*three* dimensional invariance on spacelike layers in a foliation. The
causal structure is still left in the background! (Notwithstanding
Baez et al.'s striving to quantize this too). Nor can it be brought
out.

For, the very notion of signature (i.e. that the frame fields realize
SO(3,1) as opposed, say, to SE(3) or SO(4)) is built right into the
formalism, itself. This is a MAJOR part of the background!

The distinction between SO(3,1) and SO(4) (and even Newton-Cartan's
SE(3)) is not visible at the level of mere diffeomorphism invariance.
They all look the same at that level. So, right there, Rovelli's
programme fails right out of the starting gate. He's overreached.

The equivalence principle can't even be formulated at that level!
(Because it makes explicit reference to local SO(3,1) frames; vs. say
SE(3) or SO(4) frames).

This, of course, ties directly into Sardanashvily's formulation of the
equivalence principle and his linking the tetrad part of the field
with the symmetry breaking of GL(4) - SO(3,1) (and, more generally, I
might add, the symmetry breaking of the general affine group GA(4) -
Poincare').

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.

Since Poincare' has 10 degrees of freedom, while Galilei has 11; then
what's the Galilei limit of Einstein-Cartan gravity?

The spoler to the first question or "challenge" (which leads to a
resolution of the second)
Reference:
http://federation.g3z.com/Physics/in...eralizedWigner

You actually need an 11th degree of freedom to define the Galilei
limit. The 11th parameter (in Galilei) is the "central charge"
associated with mass. To make Poincare' suitable for taking the limit
to Galilei (that is: to implement the correspondence principle with
respect to non-relativistic physics!), one needs to split the energy
generator E into two parameters -- kinetic energy H and "relativistic
mass" M. E does not have a Galilean limit. The mass shell condition (E/
c)^2 - P^2 = (mc)^2 (where P is the momentum and m the rest mass)
needs to be generalized to P^2 - 2MH + (1/c)^2 H^2 = constant. An
additional invariant emerges: M - (1/c)^2 H = constant.

The rest mass exists only for those symmetry group orbits where the
invariant
M^2 - (1/c)^2 P^2 = (M - (1/c)^2 H)^2 - (1/c)^2 (P^2 - 2MH + (1/
c)^2 H^2)
is positive. One can transform these orbits to the rest state (P = 0),
at which point M - m. Thus, the invariant is
M^2 - (1/c)^2 P^2 = m^2, if M^2 (1/c)^2 P^2.

However, the generalized invariant allows for well-defined mass/energy/
momentum relations even in the absence of this condition. In
particular, in Galilean relativity (where (1/c)^2 = 0), one has the
sector M = 0, P != 0, where M^2 - (1/c)^2 P^2 = 0. (The "synchrons").
These are "action-at-a-distance" modes with the invariant
P^2 - 2MH + (1/c)^2 H^2 = p^2
giving you the momentum associated with the action-at-a-distance
momentum transfer.

The analogue exists in Poincare' relativity. Here, one can have
M^2 - (1/c)^2 P^2 = 0 -- "luxons" or light-like modes
M^2 - (1/c)^2 P^2 0 -- "tachyons"
Both have well-defined mass/energy/momentum relations; the latter of
the form
M = p/sqrt(v^2 - c^2)
P = p v/sqrt(v^2 - c^2)
H = U + p c^2/sqrt(v^2 - c^2)
where U is the internal energy (which is where the 11th parameter
ultimately goes to).

The "generalized" Einstein-Cartan theory has an 11th mode
corresponding to this split of total energy into kinetic energy and
relativistic mass. In effect, it gauges mass.

Reference:
The Wigner Classification for Galilei/Poincare/Euclid
http://federation.g3z.com/Physics/in...eralizedWigner