Utiyama's Phys Rev paper 1956 based on work he did in 1954 asks the "now familiar question what happens when a Lagrangian invariant with respect to a rigid Lie group G is required to be invariant with respect to the corresponding spacetime dependent group G(x)." p. 210

Examples:
1. Electromagnetism G = 1-parameter internal symmetry rigid U1, Noether's theorem in globally flat Minkowski spacetime with 10-parameter spacetime symmetry rigid Poincare group P10 = T4*O(1,3) where T4 is 4-parameter translation group whose Lie algebra generators are total energy-linear momentum 4-vector (1st rank tensor under P10) and O(1,3) is 6-parameter rigid Lorentz group of space-time rotations whose Lie algebra contains the rotational space-space total angular momentum and its mixed space-time "boosts" connecting observations of same events made by geodesic inertial observers in uniform relative motion. By definition "geodesic" means "zero acceleration" no "g-force". A curved spacetime geodesic in a real gravity field looks like an accelerated non-geodesic if you impose the wrong geometry, which is what Newton's theory is.
Localize rigid U1 to elastic U1(x) and the compensating gauge potential connection for parallel transport is the "vector potential" A 1-form with covariant exterior derivative, for c = h = 1

D = d + ieA/\

on the original source field Lagrangian Cartan form with charge e. This is "minimal coupling." /\ is antisymmetric exterior Cartan form multiplication

Note that

D = Dudx^u = ,udx^u + ieAudx^u/\

,udx^u = 1

Du = ,u + ieAu

If, for example, the source field is a momentum eigenstate |p)

|p),u = ipu|p)

+,- depends on signature choice

which is why we need i factor i^2 = -1 since in flat spacetime without gravity we do the Fourier analysis into 4-momentum eigenstates.

Therefore under e^i@(x) U1(x) on the source field, we get new terms

i@(x),uPsi + ieAuPsi - i@(x),uPsi

Where Au ---> Au' = Au - @(x),u

is the compensating gauge transformation induced by

Psi(x) -> e^i@(x)Psi(x)

Note if Psi(x) is a 2nd-quantized field operator then

Psi(x) -> Psi(x)' = e^-iQ@(x)Psi(x)e^IQ@(x) = e^i@(x)Psi(x)

where Q is the total conserved Noether charge of the source field Psi when @ is not a local function but is a global constant of the motion.

2. Yang-Mills G is non-Abelian with Lie algebra charge generators Qa where

[Qa,Qb ] = fab^cQc

f^a^bc = structure constants

B^a are the Yang-Mills gauge potential connections

D = d + QaB^a/\

3. General relativity 1916

Rigid T4 --> elastic T4(x) also non-rigorously called "General Coordinate Transformations" (GCT) and even "Diff(4)" though the latter is really a more general non-metrical idea. Compensating gauge potential is now the curved tetrads A^a where a = 0,1,2,3 are the still RIGID O(1,3) Lorentz group indices. "a" is like a Yang-Mills internal index.

e^a = I^a + A^a = Einstein-Cartan tetrad 1-forms.

Compare A^a to Yang-Mills B^a.

I^a are the flat Minkowski tetrads.

A^a is a connection in the "square root" spin 1 "substratum" of the GeoMetroDynamic (GMD) field where Einstein's

ds^2 = e^aea = I^aIa + I^aAa + A^a1a + A^aAa

The SPIN CONNECTION 1-form S^a^b = - S^b^a in 1916 GR is determined implicitly by constraint of vanishing torsion field 2-form T^a.

T^a = De^a = de^a +S^ac/\e^c = 0

Curvature 2-form is

R^a^b = DS^a^b

Lagrangian for pure gravity vacuum field (no dark ZPE energy) is

L = {abcd}R^a^b/\e^c/\e^d

4. Einstein-Cartan theory has T^a =/= 0 so that S^a^b has an additional piece T^a^b, i.e. independent dynamics of torsion field where now full P10 is locally gauged

D = d + (S^ac + T^ac)/\ ...

Choose L such that T^a^b propagates not just a contact interaction for a spinning Ricci source? Violates minimal coupling? On the other hand both dark energy and dark matter are spinning Ricci sources in exotic vacuum phases.

On Jun 2, 2007, at 2:46 PM, Jack Sarfatti wrote:

If you read, for example, Sean Carroll's recent text on general relativity you will see a passing reference to gauge theory where the Levi-Civita connection is compared to EM's Au. This is off the track. The real gauge potential is at the tetrad level. The Levi-Civita connection is secondary from the tetrads. This confusion is why quantum gravity theory does not work.

LO'R writes: "Meanwhile a completely independent approach to non-abelian gauge fields was being adopted by Utiyama in Japan. Utiyama's motivation was to find a structure that would be common to gravitation and the interactions of the electromagnetic type. The structure he found was the connection. Utiyama does not quote Weyl, and it is not clear whether he had read the 1929 paper ... He ... extended Weyl's gauge principle to general Lie groups ... he also included gravity ... not only for simple compact Lie groups (as in Yang-Mills internal SU(2), SU(3)) but for general Lie groups (e.g. non-compact spacetime 10-parameter Poincare group of four T4 translations and six space-time rotations Lorentz group O(1,3)) ... Utiyama was the only one to suggest that the broad analogy between gravitation and electromagnetism ... could be extended to include all the interactions, or conversely, that the Yang-Mills type theory might be extended to include gravitation. Utiyama wrote: ' ... my work on the general gauge theory ... first showed clearly that the theory of gravitation could fall into the framework of gauge theory ... my paper pointed out that fields carrying a fundamental force - either gravity or electromagnetism - must in fact be termed connections in mathematics, which are now called gauge potentials.'"

On Jun 2, 2007, at 12:25 PM, Jack Sarfatti wrote:

Clarification
On Jun 2, 2007, at 11:58 AM, Jack Sarfatti wrote:

Weyl's 1929 paper is not easy reading because of the archaic notation. However, a few glimpses

I think by "distant parallelism" Weyl simply means globally rigid 1905 special relativity in which there are global inertial frames, a single global tetrad that I call I^a in my theory that is renormalizable as a quantum field theory with Higgs-Goldstone ODLRO vacuum condensation.

However, there is also the "teleparallelism" assumed by Gennady Shipov. This means

1. Replace the symmetric connection of 1916 GR with it plus an added antisymmetric contortion GCT 3rd rank tensor.

2. The torsion gap is now not zero to second order

3. The larger disclination curvature has torsion gap dislocations as well. The total curvature is set to zero.

4. Shipov tries to interpret the torsion dislocations as matter fields in Tuv.

----- end of discussion of Shipov's theory above ---

In my original new theory: The complete Einstein-Cartan tetrad 1-form fields are the sum of the globally rigid Minkowski tetrads I^a and the intrinsically warped compensating renormalizable "spin 1" gauge field tetrads A^a.

These are 4-parameter GCT u-v index invariants that transform under the 6-parameter a-b index Lorentz group.

e^a = I^a + @A^a

@ = Lp^2/\zpf (dimensionless coupling of QGMD (QuantumGeoMetroDynamics) analogous to e^2/hc in QED (QuantumElectroDynamics)

e^a is a GCT (General Coordinate Transformation) invariant i.e.

e^a = e^audx^u is a u-scalar Cartan 1-form

e^a is a 4-vector under the 6-parameter Lorentz group. The globally rigid T4 translation group is replaced by its locally gauged GCT group. T4 is a subgroup of GCT.

Einstein's 1916 GR is simply, in this Cartan notation

ds^2 = e^aea

This bilinear invariant of two "spin 1" fields has spin 0, 1, & 2 quantum field fluctuations.

T^a = De^a = de^a + S^ac/\e^c = 0 vanishing torsion field 2-form

S^a^b = spin connection 1-form

R^a^b = DS^a^b

Einstein-Hilbert Lagrangian

L = {abcd}[R^a^b/\e^c/\e^d + /\zpfe^a/\e^b/\e^c/\e^d]

Weyl 2-Spinor covariant partial derivative

Du(Spinor) = (Spinor),u + S^a^b(Pauli Spin Matrix)abu(Spinor)

Weyl 1929

"I dispense with distant parallelism and stick with Einstein's classical theory of gravitation ... I prefer not to believe in distant-parallelism for a number of reasons. First my mathematical intuition objects to accepting such an artificial geometry; I find it difficult to accept the force that would keep the local tetrads at different points and in rotated positions in a rigid relationship. The loosening of the rigid relationship between the tetrads at different points converts the (U(1)) gauge factor e^iphase, which remains arbitrary with respect to (matter spinor) psi from a constant to an arbitrary function of spacetime. In other words, only through a loosening of the rigidity does the established gauge invariance become understandable. Secondly the freedom of rotating the tetrads independently at different points is, as we shall see, equivalent to the symmetry and (local) conservation of the (matter) energy-momentum tensor. ... Two components suffice if the requirement of left-right symmetry (parity) is dropped."

If the coordinates are hypercomplex matrices then it seems we get internal SU(2) & SU(3) in addition to the U(1) EM.

On May 30, 2007, at 5:09 PM, Jack Sarfatti wrote:

IT FROM QBIT

Part 1 (1929) Weyl starts with 4 projective coordinates x0, x1, x2, x3 on a 3D spacelike surface with coordinates x,y,z

x = x1/x0

y = x2/x0

z = x3/x0

The equation for the Einstein light cone unit sphere S2 is

x^2 + y^2 + z^2 = 1

equal to

x1^2 + x2^2 + x3^2 - x0^2 = 0

for null geodesic light rays in globally flat Minkowski spacetime of Einstein's 1905 special relativity.

The 2-component Weyl SPINOR comes from the equatorial stereographic projection i.e. project from SOUTH POLE of S2 to the z = 0 equatorial plane. In that plane define the complex number

w = x + iy = c(+)/c(-) =( rho)e^iphi

The UN-NORMALIZED spinor QBIT |q) is in Dirac bra-ket notation for the particular basis |+) & |-) implicitly defined as

|q) = c(+)|+) + c(-)|-)

Note when c(-) = 0

x + iy -> infinity

i.e. rho -> infinity is the |+) spinor eigenstate (base vector)

similarly rho = 0 is the |-) spinor eigenstate

Given the 2x2 Pauli spin matrices basis for a simple Clifford algebra

I, sigma(x), sigma(y), sigma(z)

First look at the LOCAL DIAGONAL matrix elements of the Pauli spin matrices with respect to the same QBIT spinor.

x0 = (q|I|q) = c*(+)c(+) + c*(-)c(-)

x1 = (q|sigma(x)|q) = c(+)*c(-) + c*(-)c(+)

x2 = (q|sigma(y)|q) = i[-c(+)*c(-) + c*(-)c(+)]

x3 = (q|sigma(z)|q) = i[-c(+)*c(+) - c*(-)c(-)]

*Now I do something new and original not in Weyl 1929.

These have an obvious Feynman-like diagram \/ for a single forward light cone in globally flat Minkowksi spacetime

The NONLOCAL OFF-DIAGONAL matrix elements between two different QBITs |q) & |q') "located" at different light cones gives a new kind of NONLOCAL PRE-GEOMETRY

x0(q,q') = (q|I|q') = c*(+)c(+)' + c*(-)c(-)'

x1(q,q') = (q|sigma(x)|q') = c(+)*c(-)' + c*(-)c(+)'

x2(q,q') = (q|sigma(y)|q') = i[-c(+)*c(-)' + c*(-)c(+)']

x3(q,q') = (q|sigma(z)|q') = i[-c(+)*c(+)' - c*(-)c(-)']

These have an obvious diagram \.../

with ... as the holonomic path independent unique globally flat geodesic connecting the two light cones.

The non-geodesics are zero point quantum vacuum fluctuations of spacetime itself.

In curved spacetime with gravity we have anholonomic path-dependence of course.

Note also that the off-diagonal matrix elements for 2 distinct qbits have the same formal syntax as the 4 Bell entangled pair states used in quantum teleportation protocols. Curious. Suggestive. Indeed!

The off-diagonal matrix elements are entangled vacuum correlation functions in second-quantized notation. For example

x1(q,q') = (q|sigma(x)|q') = (0|Psi(q)*sigma(x)Psi(q')|0)

x^1(q,q') is already a coherent long-range emergent vacuum condensate because the bilinear spinor form is bosonic. The classical space-time fabric geometrodynamic field is in fact a macroquantum vacuum condensate emergent artifact.


On May 26, 2007, at 6:53 PM, Jack Sarfatti wrote:

O'Raifeartaigh p. 110 writes of Weyl in 1929:
"It is remarkable that Weyl should even consider the possibility of time-reversal and parity-violation at this time. In fact Weyl not only considered these possibilities but ... made the statement: 'The problem of the proton and electron will be mixed with the symmetry properties of the quantum theory with respect to interchange of left and right, past and future, and positive and negative charge.' Thus ... he not only foreshadowed the later developments in P and T violation but foreshadowed the CPT theorem. All this was at a time when, as Yang put it, 'Nobody, absolutely nobody, was in any way suspecting that these symmetries were related in any manner. It was only in the 1950's that the deep connection between them was discovered. ... What had prompted Weyl in 1930 to write the above passage is a mystery to me.'

Yang's puzzlement is similar to Ed Teller's puzzlement over what prompted President Reagan to decide to do SDI. Teller, in his autobiography writes that he was out of that decision loop and was surprised. Ask Cap Weinberger Jr what really happened. Also http://sharonweinberger.com/?p=60 However, no precognition in the case of President Reagan's decision. ;-)

On May 26, 2007, at 3:46 PM, Jack Sarfatti wrote:

Define spinor

|s) = cos@|up> + sin@e^i&|down>

relative to a given basis

(TETRAD)^a s,s' = (s'|(Pauli)^a|s) inner product

s =/= s' possible.

However, it turns our that there is no Lorentz scalar with 2-component spinors, i.e. the spin 0 tetrad field is missing and is connected with the origin of inertia needing Dirac 4 component spinors.

On May 26, 2007, at 3:28 PM, Jack Sarfatti wrote:

Motivated by Part 1 of Hermann Weyl's 1929 seminal paper "Electron and Gravitation"

John A. Wheeler's "IT FROM BIT"

"BIT" = SPINOR = vector in basic rep of SL(2,C).

"IT" = TETRAD

TETRAD is bilinear in SPINOR

Einstein GEOMETRODYNAMIC FIELD is bilinear in TETRAD, therefore quartic in SPINOR

SPINOR QBIT is spin 1/2.

TETRAD = (SPINOR*|Pauli Spin Matrix|Spinor) Lorentz group 4-vector is an entangled EPR pair of spinors.

i.e. TETRAD is a 2 QBIT string

GEOMETRODYNAMIC FIELD is 4 QBIT string

Neglecting relative orbital angular momentum i.e. S-orbital

1/2 + 1/2 = 1 + 0

i.e.

2x2 = 3 + 1 Irrep dimensions

Therefore, spin 1 tetrads but is there also a spin 0 "scalar" tetrad that I missed before?

Note there are nonlocal tetrads if the two spinors in the inner product matrix element are at different space time events.

On May 22, 2007, at 10:59 PM, Jack Sarfatti wrote:

"I am as dissatisfied as you are with distant parallelism and your proposal to let the tetrads rotate independently at different space-time points is a true soluton." Pauli to Weyl (1929).

"IT FROM BIT" John Archibald Wheeler

e.g. Goldstone phases are macro-quantum BIT fields of physical information.

v(superfluid Helium 4) = (h/m)dTheta 1-form O(2) symmetry

A^a(warped tetrad) = M^a^a = dPhi^a/\Theta^a - Phi^a/\dTheta^a 1-form O(9) symmetry - gravitation

F^a = dA^a = -2dPhi^a/\dTheta^a 2-form

S^a^b = M^[a,b] - spin connection 1-form

A^a & S^a^b form the noncompact Poincare group Lie Algebra

On the conversion of Weyl's 1918 aborted scale factor on the uncharged metric IT field to a quantum phase factor on the electrically charged BIT pilot field:

"One can summarize Einstein's objection to Weyl's (1918) theory as the statement that, according to atomic spectroscopy, there is no Bohm-Aharonov effect for gravitation." p. 85.

Metricity in GR is like unitarity in QM, i.e. inner products are invariants of the evolution i.e. parallel transport whether in spacetime or Hilbert space. Nonmetricity, therefore, is like "collapse of the state" in von Neumann quantum measurement theory.

On May 22, 2007, at 7:42 PM, Jack Sarfatti wrote:

Weyl in 1918 made the mistake of applying the gauge principle to Einstein's 2nd rank symmetric metric tensor. What he actually found was a non-metricity vector field in which the magnitudes of vectors parallel transported along world lines depends on the world line. This would be a memory - a hysteresis non-integrability anholonomy not observed in the electromagnetic world where the spectral lines of atoms in the stars are recognizably the same once the gravity red shifts are subtracted out. Einstein pointed this out to Weyl back then. Slowly (Fock, Schrodinger, London ...) it was recognized that this path dependent electrodynamic (Bohm-Aharonov) non-integrability applied to the deBroglie pilot quantum waves not to the Einstein geometrodynamical field.

What Weyl really did in 1918 was to find the non-metricity piece of the connection beyond Einstein's 1916 metric connection. Later Cartan found the torsion field antisymmetric piece to the possible connection fields for parallel transport.

i.e.

Connection = Einstein metric connection + non-metricity vector field connection + antisymmetric (con) torsion

metric connection ~ disclination rotation defects in vectors around closed self-generated infinitesimal parallelograms

(con) torsion connection ~ dislocation defects in which the basic self-generated parallelograms has a gap to 2nd order of smallness (e.g. Penrose "The Road to Reality").

Weyl's 1918 non-metricity vector field means that the lengths of vectors are different around the closed loops i.e. disastrous multi-valuedness - a kind of Riemann surface fiber? In any case Weyl's 1918 vector field is not the EM 4-potential A 1-form but is some alien kind of geometrodynamic field whose flux field tensor 2-form F = dA is zero in our ordinary spacetime without topological defects giving non-vanishing deRham integrals of F through 2D surfaces even when A is closed. 1D string line defects involve surfaces with boundaries whose non-bounding loop integrals of A are quantized "winding numbers" (1st nontrivial homotopy) when the A -form derives from a single coherent Goldstone phase of two real Higgs fields. If there are three real Higgs fields with two independent Goldstone phases then A is not a closed 1-form the non-bounding 2D surface is closed without boundary and we have interior point "monopole defects" with quantized radial fluxes given by "wrapping numbers" (2nd non-trivial homotopy). What would be the physical effects of these non-metricity fluxes? Are they in the interior of the leptons and quarks as Bohm hidden variables that appear as points as the scattering probe magnification increases because of extreme micro-warping. The effective geometrodynamic coupling constant renormalization group flow is to larger values at the 1 fermi scale - it can then get smaller in both UV and IR directions with peaks between 10^-13 - 10^-16 cm.

L.O'R wrote (Ch 5):"(Hermann Weyl) had always been convinced that there was a close analogy between gravitation and electromagnetism ... in 1929 he was able to formulate the analogies ... by means of the tetrad formalism ... Weyl's formulation was complete and went beyond all previous ideas in proposing that electromagnetism be derived from the gauge principle ... that ... has turned out to be a powerful principle for deriving the nuclear interactions and to be the common principle underlying all the known fundamental interactions. ... the six sections of the paper ... two-component spinor theory in Minkowski space ... parity ("chirality" screw helical mirror reflection symmetry now known violated by quarks and leptons, but not known in 1929 of course) and time reversal invariance ... tetrads (AKA vierbeins) ... spinor theory in curved spaces ... Noether conservation laws ... spin connection ... invariant action ... (local) energy-momentum conservation laws from invariance with respect to both general coordinate transformations and Lorentz transformations of the tetrad (Noether's theorem) ... He then recast gravitational theory in the tetrad formalism with a view to exhibiting the analogies between it and electromagnetism. In the final section, he came to what he considered the most fundamental part of the paper, namely, the derivation of electromagnetic theory from the gauge principle."

All of fundamental theoretical physics today depends on only two battle-tested grand organizing ideas, and that includes the extra dimensions of string theory bye the bye, Witten need look no further IMHO.

I. The gauge principle

"AKA" here means "also known as" in a rough qualitative sense with minor differences of detail

II. "More is different" AKA "hidden symmetry" AKA "spontaneous breakdown of symmetry" AKA "ODLRO" AKS "macro-quantum coherence" AKA "Bose-Einstein condensation" AKA "macroscopic eigenvalues of correlation functions" AKA "collapse of phase space volume" AKA effective order parameters including nonlocal topological order of 2D Anyons in FQHE (Fractional Hall effect) as well as the more familiar local order of the Landau-Ginzburg phenomenology with the O(N) Mexican Hat effective low energy potentials for macroscopic phase transitions from quantum to misnamed "classical."

to be continued.

On May 20, 2007, at 1:43 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