Gravity as a Yang-Mills Theory

On Sep 2, 2006, at 10:31 PM, Jack Sarfatti wrote:

on p. 222 of "How is Quantum Field Theory Possible?" Oxford,

"Gravity, like the other three fundamental interactions, can be
represented by the connection on a principle fiber bundle, the bundle of
orthonormal frames. In general relativity, there is no phase factor,
which is a quantum characteristic. What corresponds to the phase and
what the gravitational potential couples to can be interpreted as the
orientations and deployments of our measuring equipments."

Therefore the "tetrads" are an abstraction, an idealization, of
"measuring equipments". The tetrad mobile frames are independent of the
local coordinate patches. "Gravitational potential" below does NOT mean
Newton's potential V(Newton) in g00 = 1 + V(Newton)/c^2, but means the
non-tensor connection gradients of V(Newton).

"The tangent bundle underlies the Lagrangian formulation of
classical-dynamical systems. However, in these applications, it does not
have a local symmetry group. In general relativity, the tangent space Tx
over each point x in M, is equipped with orthonormal reference frames
called tetrads {ei(x)}. ... The tetrads are nonholonomic or
non-coordinate bases to distinguish them from the coordinates defined in
the coordinate patches. ... The orientations of the tetrads correspond
to the phases in the quantum fields and like the phases they vary with
x. Physically, a specific tetrad represents a particular experimental
configuration ... Conceptually our experimental arrangements are
infinitesimal. The collection of all tetrads at all points in M
constitutes a principal fiber bundle ... The gravitational potential is
represented by a connection on the bundle of orthonormal frames. ... To
ensure that the local Poincare transformations yield equivalent
measurements, the gravitational potential is introduced. The orientation
of a tetrad axis can be expressed by its components with respect to a
coordinate {x^u} at x; ei(x) = ei^u(x)(d/dx^u). IN A GRAVITATIONAL
FIELD, THE ACTUAL ROTATIONS OF THE TETRADS ARE DERIVED BY REPLACING THE
PARTIAL DERIVATIVE BY THE COVARIANT DERIVATION

d/dx^u - D/dx^u = d/dx^u + (Connection)u^i^jQij

where Qij is a generator of the Lorentz group. (Connection)u^i^j is the
coefficient of the affine connection. It represents the gravitational
potential that reconciles the orientations of the tetrads. GOING FROM x
to x + dx, WE FIND THE TETRADS HAVE ROTATED BY ei(Connection)^i^judx^u."

The problem is the Qij in the Yang-Mills formula

D/dx^u = d/dx^u + (Connection)u^i^jQij

Qij is not there, for example in the world tensor formula

DA^v/dx^u = dA^v/dx^u + (Connection)^vulA^l

(Connection)^vul = (Connection)^vijeu^iel^j

Kibble does have a Yang-Mills formulae like

D/dx^u = d/dx^u + (Connection)u^i^jQij

i.e., (4.7) - more on that anon - the key is Kibble's Section 6 to get
from the Yang-Mills formula cited by Auyang to the more familiar world
tensor formula. That is the relation of the substratum spin 1 Yang-Mills
formalism for gravity to the Einstein's earlier spin 2 world tensor
formalism is subtle and not direct.

of course that

&e^a(x + dx) = e^b(x)(Connection)bu^adx^u

is not special to tetrads but is generally true.

Thus, for example,

&A^v = (Connection)^vuwA^udx^w (67)

Einstein "The Meaning of Relativity"

BTW
Just read Einstein's own words in "The Meaning of Relativity". When he
speaks of "gravitational field" in context of "equivalence principle" he
means pulling g's. He does not mean the tidal curvature field at all.
For example p. 63. P... objection to Einstein's equivalence principle
based on the presence of curvature is a Red Herring completely beside
the point, that all locally coincident frames, whether geodesic LIF
with zero internal g-force (weight) or non-geodesic LNIF with non-zero
internal g-force (weight) are equally good local descriptions of
proximate events.

"Let now K be an inertial system ... also K', uniformly accelerating
with respect to K. Relatively to K' all the masses have equal and
parallel accelerations; with respect to K' they behave just as if a
gravitational field was present."

Note that Einstein here by "gravitational field" is NOT talking about
tidal curvature geodesic deviation, but about local pulling of g's or
"g-force" in common parlance.

"There is nothing to prevent our conceiving this gravitational field as
real, that is, the conception that K' is 'at rest' and a gravitational
field is present we may consider as equivalent to the statement that
only K is an 'allowable' system of coordinates and no gravitational
field is present. THE ASSUMPTION OF THE COMPLETE PHYSICAL EQUIVALENCE OF
THE SYSTEMS OF COORDINATES K AND K', WE CALL THE 'PRINCIPLE OF
EQUIVALENCE'."

Note that the gravitational field here is 100% inertial force. Presence
or absence of curvature is irrelevant for this local statement. If we
contingently stipulate a static hovering shell LNIF observer then we do
enforce a relationship of the g-force to the local curvature, but that
is not an intrinsic objective property of the fabric of curved
spacetime. That is merely a matter of subjective desire not of objective
necessity. Zielinski's error is not distinguishing a non-objective
stipulative contingency from a non-negotiable objective necessity.


On Sep 2, 2006, at 9:30 PM, Jack Sarfatti wrote:

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."