Gauge Theory and Quantum Gravity

Einstein's 1916 theory of gravity, electromagnetism and the Yang-Mills
theory of the weak and strong sub-nuclear forces are all gauge theories,
yet in 4D there is a basic difference between gravity and the rest and
that is Einstein's local principle of equivalence. But what is the local
principle of equivalence other than the universality of localizing the
4-parameter translation subgroup T4 of the 10-parameter Poincare group
(1905 special relativity) for ALL non-gravity quantum fields?
EM-weak-strong gauge forces are from the internal U(1) SU(2) flavor and
SU(3) color groups with parity violation added so that the left and
right handed-chiral projections of the basic qubit spinor fields of
leptons and quarks see the nonabelian Yang-Mills fields very differently.

Note that Noether's theorem connecting global rigid symmetries to global
conserved "charges" including total energy and momentum Pu of a
non-gravity field for T4 does not survive localizing T4. This is the
basic reason why the purely conformal Weyl classical gravity total
energy is nonlocal and why in general one cannot define conserved total
energy, linear and angular momentum of any field globally in warped
spacetime. The local stress-energy current conservation law
Tuv(non-gravity field)^;v = 0 does not automatically define globally
conserved integrals over 3D spacelike slices of 4D spacetime in the
general case. One has approximately conserved global quantities only for
special asymptotically flat solutions of the generally covariant fields
with special Killing isometries in the metric field. In other words,
Noether's theorem connecting symmetries to conservation laws is
restricted to globally flat spacetimes in the absence of curvature and
torsion. Note, that the total dark energy of our accelerating expanding
universe is not conserved because its w = -1 Ricci energy density
(c^4/G)/\zpfg00 is "constant", i.e. in the IR cosmological equations


http://www-conf.slac.stanford.edu/ss...ew_Page_05.jpg

(c^4/G)/\zpfg00 --- (zero point quantum vacuum fluctuation energy density)

~ a(t)^0 = constant

where a(t) is the IR (long wave) FRW cosmological scale factor stretch
in the fabric of space.

Think of 3D space as a FLAT infinite elastic sheet that is at global
cosmic time t in the Hubble flow (lots of Killing isometries in this
particular solution) so that at every point on the sheet there is an
isotropic stretch a(t) in all directions from that point.

That is, let d(P1,P2) be the distance between galaxies 1 & 2 at cosmic
time t where t = 0 is the moment of inflation, then the distance between
those same galaxies at time t' will be d'(P1,P2) where

d'(P1,P2)/d(P1,P2) = a(t')/a(t)

Now back to the main point - what is the connection? In GR everyone was
thrown off track by thinking that the fundamental connection was at the
quadratic 2nd-rank tensor geometrodynamic level instead of the 1st rank
tensor vector tetrad field level!

The EM & Yang-Mills internal symmetry compensating gauge boson VECTOR
fields are all spin 1 quantum fields that are renormalizable.

Quantum gravity theorists have been thrown off course by noting that the
tensor metric fields are unrenormalizable like the old Fermi beta
radioactivity model.

However, the tetrad fields are spin 1 vector fields! They are the basic
gravity fields. The non-renormalizable tensor fields from unthinkingly
quantizing guv is asking the WRONG QUESTION of Nature. Quantize the spin
1 e^a tetrad Lorentz vector fields that are GCT invariants! Then it's a
piece of cake.

ds^2 = e^aea

Obviously you quantize e^a.

Also quite obviously

e^a = I^a + kA^a

where k is a dimensionless coupling constant like alpha = e^2/hc in QED.

A^a is the spin 1 Lorentz group VECTOR field, i.e. the fundamental
CONNECTION FIELD from localizing the 4-parameter translation subgroup of
P10 universally for ALL non-gravity field actions.

I^a is the trivial globally flat tetrad when k - 0. I^a's components
I^au are curvilinear for non-geodesic observers in globally flat
Minkowski spacetime - but so what?

From localizing the 10-parameter P10 we also get the spin connections
S^a^b = - S^b^a which enable GCT covariant derivatives of the Dirac,
Weyl & Majorana spinor fields as well as the tensor fields since we
easily get the Einstein Levi-Civita derived connection from products of
S^ab with the tetrad components e^au. See Rovelli's book Ch II for details.

Therefore there was a lot of wasted time in quantum gravity these past
50 years.

I have not even touched above on the emergence of the c-number vacuum
condensate parts of the above VECTOR tetrad 1-form fields e^a along with
their spin connection 1-forms S^a^b.

What are the Yang-Mills equations for the tetrad vector fields.

The Yang-Mills field tensor 2-form is obviously

F^a = dA^a + S^ac/\A^c

This is in analogy with

F = dA

in U(1) internal symmetry electromagnetism.

F^a is related to the torsion field i.e.

T^a = De^a = de^a + S^ac/\e^c = torsion field 2-form

where e^a = I^a + kA^a

The curvature field 2-form is

R^a^b = DS^a^b = dS^a^b + S^ac/\S^cb

Returning to the Yang-Mills spin 1 Lorentz group vector field equations

F^a = DA^a = dA^a + S^ac/\A^c

DF^a = 0

D*F^a = *J^a

*F^a = Hodge dual of F^a

*J^a is a current source density 3-form of the *F^a field.

D*J^a = 0

is local source current density

In general

[A^a, A^b] = K^a^bcA^c

Where K^a^bc are the structure constants of a Lie algebra.

What is the Lie group G here?

Is it T4? Is it O(1,3)? Is it P10?

On May 20, 2007, at 1:43 PM, Jack Sarfatti wrote:

On May 19, 2007, at 5:07 PM, Jack Sarfatti wrote:

Commentary 1 (Draft 2 expanded)

This is a very useful little book by Lochlainn O' Raifaeartaigh in
Dublin published by Princeton 1997. It has seminal papers by Weyl,
Klein, Fock, Schrodinger, London & Pauli in English from the original
German.

The two great battle-tested principles of basic theoretical physics are

1. The local gauge principle, i.e. "relativity" in the most general
sense of no action without reaction, no passive absolute Newtonian
arenas, Leibniz's relationism of Bohm's "dialogue" not monologue.

2. The spontaneous breaking (or "hiding" Sidney Coleman, Erice Lectures
1970's) of continuous symmetries in the ground state of real on mass
shell quanta and also in the vacuum of virtual zero point quanta. AKA
"More is different" (P.W. Anderson) "Emergent complexity." "ODLRO"
(Onsager & Oliver Penrose), "Goldstone theorem" "Macroquantum states",
"Glauber coherent & squeezed states" et-al. The idea of "hidden
symmetry" is that whilst the dynamical classical action in the Feynman
path integral alternative to second quantization is invariant under the
symmetry group G, the vacuum (ground state) is not. One familiar example
is a ferromagnet near absolute zero with a Galilean relativity 3-vector
order parameter "magnetization" in a coherent domain. This is three real
"Higgs fields" with two continuous 2pi periodic "Goldstone phases"
defining the S2 vacuum manifold of minima in the Landau-Ginzburg
effective quartic renormalizable interaction Higgs-type field
Lagrangian. Non-trivial 2nd order homotopy would give point "monopole"
defects not actually observed in real ferromagnetic phases. What is
observed are wall domain defects with a S0 vacuum manifold without any
continuous Goldstone phase at all corresponding to another broken
symmetry from three real Higgs fields to only one effective Higgs field
that vanishes on the domain wall. That is S3 broken to S1 - curious.

Emergence of "The Nine"

My original parsimonious theory explains the origin of gravity &
inertia, torsion (i.e. emergent "tetrad" local observer/detector frames
and "spinor" connections as macroquantum coherent "surface" world
hologram Goldstone phase modulations from an "M-Matrix" of non-closed
1-forms made from two Lorentz 4-vectors of eight 0-form continuous
periodic Goldstone phases Theta^b & Phi^a from nine post-inflation real
Higgs scalar vacuum ODLRO fields.

a,b = 0,1,2,3 are Lorentz group indices

M^a^b = Phi^a/\dTheta^b - dPhi^a/\Theta^b

dM^a^b = 2dPhi^a/\dTheta^b

A^a = M^a^a (diagonal elements of the Matrix)

Einstein-Cartan tetrad 1-forms are e^a = I^a + @A^a are spin 1
Yang-Mills type Lorentz group VECTOR renormalizable quantum translation
group T4 localized gauge fields.

@ = (Lp^2/\zpf)^1/3 dimensionless "world hologram" self-gravity coupling

I^a are the global Minkowski tetrad frames that we have when either

G - 0 gravity coupling switched off

h - 0 quantum action switched off

c - infinity, i.e. no causal retardation and/or advanced retro-causation

/\zpf - 0

i.e. no gravity when supersymmetry is perfect! /\zpf in the IR limit is
Lenny Susskind's cosmic landscape parameter ~ (area of future deSitter
horizon of pocket Hubble bubble universe in the "megaverse" of eternal
chaotic inflation.

Supersymmetry is square root of T4, i.e. anticommutator of
supersymmetries is T4.

"Spinor" connection 1-forms are (gets dynamical degrees of freedom when
total 10-parameter Poincare group P10 = T4*O(1,3) is locally gauged
(Utiyama, Kibble et-al 1961)

S^a^b = - S^b^a = M^[a,b]

Torsion field 2-forms are

T^a = de^a + S^ace^c

Curvature field 2-forms are

R^a^b = dS^a^b + S^acS^cb

Einstein-Hilbert (E-H) "classical" Lagrangian density is the 0-form

L(E-H) ~ {a,b,c,d}R^a^b/\e^c/\e^d

ds^2 = guvdx^udx^v = e^aea = I^aIa + @(I^aAa + A^aIa) + @^2A^aAa

Quantum noise corrections in second quantized formalism with
macro-quantum vacuum |0 condensate ODLRO is

A^a = 0|A^a|0 + &A^a

&A^a is the quantum excitation spin 1 tetrad vector field annihilation
operator

Note that R^a^b is quadratic in A^a and its gradients, therefore L(E-H)
has up to quartic terms in the tetrads consistent with renormalization.

Note that

F^a = dA^a = dM^a^a = dPhi^a/\dTheta^a =/= 0

We have "Maxwell" equations

dF^a = 0

d*F^a = *J^a (warped tetrad current density)

d*J^a = 0 (local conservation of warped tetrad current density)

The Einstein geometrodynamic field ds^2 = e^aea is quadratic in the
tetrads, therefore obviously made from entangled Einstein-Rosen-Podolsky
(1935) pairs of spin 1 tetrad quanta, hence we have quantum
geometrodynamic corrections of spin 2 tensor gravitions, spin 1 vector
gravi-photons and spin 0 scalar gravitons.

3x3 = 5 + 3 + 1

i.e.

1 + 1 = 2, 1, 0

for the irreducible spin/angular momentum representations of O(3)
subgroup of O(1,3).

This does not affect the macro-quantum c-number ODLRO sector of the
theory in most cases.

Dark energy and dark matter are both very simply virtual quanta inside
the total physical vacuum of positive and negative zero point
fluctuation energy density respectively.

Einstein showed that for an isotropic Ricci source in 3+1 spacetime, the
space-time bending power of the source is

(G/c^2)(energy density of source)(1 + 3w)

Lorentz invariance and general coordinate invariance imply

w = -1

for isotropic ZPF distributions of all quantum fields of all spins.

Bosons have positive ZPF energy density

Fermions have negative ZPF energy density.

Non-isotropic boundary conditions e.g. Casimir effect will shift the ZPF
w, but as long as w - 1/3 i.e. quintessent field region then

positive ZPF energy density - antigravity universal repulsion blue
shift (dark energy)

negative ZPF energy density - gravity universal attraction red shift
(clumping dark matter) - this is indistinguishable from w = 0
conventional CDM for distant observers looking at gravity lensing for
example.

Note when w - 1 that is "phantom energy" BIG RIP

I predict that the LHC will NOT find dark matter particles on mass shell
as a matter of basic principle. Looking for dark matter inside the LHC
is like looking for the motion of the Earth through the mechanical
Victorian aether with a Michelson-Morley interferometer. Null results
then and in the future I do declare. All of this is only based on 1 & 2
above. The book is mostly about the early history of 1 and how it
affected C.N. Yang as a graduate student. Robert Shaw's little-known
important independent work is also discussed.

"The role of geometry in physics has always been central. But ... it was
passive, providing only the stage on which physics took place ... the
theory of special relativity in 1905, when it became clear that space
and time were not independent ... But the most profound and astonishing
entry of geometry into physics came with Einstein's theory of
gravitation in 1916, which showed that gravitation was nothing but the
curvature of four-dimensional space. ... George Bernard Shaw ... wrote
'Asked to explain why planets did not move in straight lines and run
straight out of the universe, Einstein replied that they do not do so
because space is not rectilinear but curvilinear.' ... due almost
entirely to the genius of Einstein, geometry graduated from being the
stage on which the drama of physics took place to being a major player
in the drama.

There remained however the electromagnetic and nuclear forces, and the
geometrization of gravity raised the question as to whether these other
fundamental forces were 'true' forces operating in curved space of
gravitational theory or whether they also were part of the geometry.
This question has still not been fully answered. ... these forces and
gravitation have a common geometrical structure. This is ... the gauge
structure."

to be continued

Jack Sarfatti

"If we knew what it was we were doing, it would not be called research,
would it?"
- Albert Einstein
http://www.authorhouse.com/BookStore...x?bookid=23999
http://lifeboat.com/ex/bios.jack.sarfatti
http://qedcorp.com/APS/Dec122006.ppt
http://video.google.com/videoplay?do...ausation&hl=en
http://www.flickr.com/photos/lub/sets/72157594439814784

What do you mean by 4d ?

-y

On May 22, 3:41 am, Y wrote:

What do you mean by 4d ?

-y

"
"4-velocity and 4-acceleration"
http://farside.ph.utexas.edu/teachin...es/node15.html

Sue...