Thursday, September 13, 2007

Meaning of the spin connection coefficients & Lie algebra structure constants

I cannot push the analogy of the GR spin-connection coefficients w^a^bc with the structure constants c^i^jk of the internal symmetry Yang-Mills Lie algebras too far, although there is a physical relation it is not direct.

In GR including torsion extension

S^a^b = w^a^bce^c = spin-connection 1-form

e^c = tetrad 1-form

a,b,c = 0,1,2,3

in all cases i.e.

Einstein 1916 localize 4-parameter T4 i,j,k = 0,1,2,3 gives GCT curvature no torsion

Utiyama 1956 localize 6-parameter SO(1,3) i,j,k = 4,5,6,7,8,9 gives a torsion induced curvature GCTs put in ad-hoc

Kibble 1961 localize 10-parameter P10 = T4xSO(1,3) gives curvature + torsion with GCTs built in

Therefore there is a mapping

w^a^bc = w^a^bc(c^i^jk)

where in general

a,b,c = 0,1,2,3

and for the Kibble 1961 theory

i,j,k = 0,1,2,3,4,5,6,7,8,9 i.e. Gennady Shipov's "oriented point"

where 4,5,6,7,8,9 is the connection to Calabi-Yau space of string theory's extra space dimensions with

e^i i.e. tetrads in 9D + 1 "oriented point" manifold

S^i^j = W^i^jke^k in Gennady Shipov's 10D manifold.

The W^i^jk should be the structure constants of the 10 generators of the Poincare group Lie algebra which determine the spin connection coefficients w^a^b,c in the 3D + 1 space-time where 3D volume-without-volume is the holographic image of something like the dark energy retro-causal future de Sitter horizon.

Something like that.

On Sep 13, 2007, at 5:13 PM, Paul Zielinski wrote:

Jack Sarfatti wrote:
PS Also Paul what you have been looking for is natural in this tetrad substratum. It is not natural on the metric tensor/Levi-Civita connection level because of bilinear tetrad cross terms.

Complete Einstein-Cartan tetrad is

e^a = I^a + B^a

Where B^a is the intrinsic curvature tetrad field - without committing to my specific hologram model where

B^a to N^-1/3A^a

N = (Closed Surrounding Horizon 2D Surface)/4Lp^2 Bekenstein's formula
Except that as I understand it your I-B tetrad split doesn't correspond to my decomposition at all, since your I component in isolation describes a flat intrinsic spacetime geometry,


while the B term when added in accounts for the presence of curvature;


whereas in my proposed L-C decomposition, there is no dependence whatever on the intrinsic geometry, flat or otherwise, in the "coordinate" component of the decomposition. So I think we may have our wires crossed here.

OK - I am just pointing out that the tetrad formalism does have a sharp clean split between flat and curved/torsioned that is non-perturbative and background-independent, but that disappears at the bilinear metric tensor level. Of course inertial g-forces are curvature independent, they only pertain to the center of mass of an extended test particle, the curvature is detected only in the relative coordinates of the extended test particle. In an N-particle system we can separate the CM coordinate from the relative coordinates. Simplest for idealized 2-body problem of course.

Granted the precise relationship between the two decompositions should be carefully investigated, which I plan to do.


Of course you are technically correct about the I-B cross terms that appear when you recover the LC connection from the tetrads.
No perturbation theory on background dependent Minkowski spacetime is implied here. That's a Red Herring.
That's "linearized GR", which I suppose corresponds nicely with the Kraitchnan-Deser-Feynman spin-2 model. I understand that your model does not rely on linearized GR, and your I + B decomposition has nothing to do with "perturbations" applied to a flat geometry. Your model is clearly non-perturbative.

My model is background-independent in Lee Smolin's sense.
Right -- although your I tetrad component in isolation describes a flat spacetime geometry, correct?


Nothing I say demands B^a << I^a as in perturbation theory.
I^a has all the inertial force effects of non-geodesic frames in Minkowski spacetime.

But only given a "coordinate" basis, as I understand the tetrad model.

Depends what is meant.

I^a = I^audx^u

I^au is a first-rank linear affine tensor in the u-index

so that I^a is an affine invariant.

In general when we have non-affine transformations, i.e. to accelerated frames in Minkowski spacetime i.e. no real gravity curvature - forget torsion for simplicity since it's not in 1916 GR anyway


I^au to I^au' = X^uu'I^au + X^uu'(d/dx^u")X^u"u

for linear affine frame transformations

X^uu'(d/dx^u")X^u"u = 0 by definition

Note that for GR

e^a = I^b + B^b is GCT invariant

i.e. GCT frame transformations include linear affine of 1905 SR + nonlinear (non-affine = localized T4) - of CMs of spatially extended detectors.

That is

B^au to B^au' = X^uu'B^au - X^uu'(d/dx^u")X^u"u

so that total Einstein-Cartan e^au is a GCT first rank tensor in u, but "Yang-Mills" spin 1 Bu is not - like the EM vector potential is not a tensor under U(1). This is precisely why Kibble's 1961 localized T4 = Einstein's 1916 GCTs. Very neat. Very pretty - showing natural Yang-Mills gauge structure is at tetrad level not where Ashtekar & LQG people put it at Levi-Civita composite level where it has obscure non-physical excess formal baggage leading to "non-renormalizability" puzzles. We need tetrads to couple spinor matter fields to gravity, so why this is not an obvious way to go for the Pundits astounds me.

I^a, B^a & e^a are affine 1st rank tensors under rigid Lorentz group - zero torsion field case.

I suppose we should play same game there, i.e. e^a still a torsion localized Lorentz group tensor & I^a & B^a not so, only their sum.

If e^a not a localized Lorentz group tensor then ds^2 = e^aea no longer invariant in curvature-torsion extension of 1916 GR - very dramatic new physics if that is the case! This is something to ponder.

Note that

T^a(Minkowski) = dI^a + w^abcI^b/\I^c = 0 zero torsion 2-form in Minkowski spacetime


R^a^b(Minkowski) = d(w^a^bcI^c) + w^ac'cI^c'/\w^b^cc"I^c" = 0 zero curvature 2-form in Minkowski spacetime

However, cross terms I^a with A^b occur in the general case mixing inertial with intrinsic effects.

I think this is only when you use a coordinate basis

I don't know what you mean. In 1905 SR - zero curvature, zero torsion

I^au to Kronecker-delta in all global inertial Minkowski geodesic frames related only by linear affine transformations. It is curvilinear in accelerated non-geodesic Minkowski frames from the X^uu'(d/dx^u")X^u"u "inertial g-forces".

-- otherwise your "I" has no discernible relationship to observer frames, but is simply an arbitrary orthonormal basis in a tangent space with no intrinsic relationship to spacetime coordinates.

I don't understand your sentence. Tetrad formalism is GCT invariant like using intrinsic vectors curlA, divA in vector calculus

i.e. e^a = e^audx^u is GCT scalar, i.e. local GCT frame invariant for all local observers independent of coordinate basis relative to the 4-parameter T4 group. However, it is a first rank tensor for the 6-parameter Lorentz group SO(1,3).

Note Utiyama 1956 localized only SO(1,3) he got torsion induced curvature and had to stick in Einstein's 1916 GCTs ad-hoc. Kibble 1961 localized complete P10 = T4xSO(1,3) and got GCTs from localized T4 absent in Utiyhama 1956. Kibble's theory larger than Einstein's 1916 curvature only theory. People got confused not understanding that Utiyama's torsion-induced curvature is not the same as Einstein's torsion-free curvature. This is trivial using tetrads as Cartan forms. In terms of crystalography, dislocation defects can induce disclination defects but not vice versa - or so it appears from

T^a = De^a torsion 2-form

R^a^b = DS^a^b

S^a^b = spin connection 1-form

with Yang-Mills mapping

S^a^b = w^a^bce^c

w^a^bc are the symmetry group G Lie algebra structure constants.

[Q^a,Q^b] = w^a^bcQ^c

for 1916 GR

G = T4

for Utiyama 1956

G = SO(1,3)

for Kibble 1961

G = P10 = T4 x SO(1,3).

Therefore, in general

w^a^bc = w^a^bc(T4) + w^a^bc(SO(1,3))


Einstein 1916 GR uses only w^a^bc(T4)

Utiyama 1956 uses only w^a^bc(SO(1,3))

Kibble 1961 uses the full w^a^bc = w^a^bc(T4) + w^a^bc(SO(1,3))

On Sep 13, 2007, at 4:21 PM, Jack Sarfatti wrote:

On Sep 13, 2007, at 12:12 PM, Paul Zielinski wrote:

Jack Sarfatti wrote:
The A^a q-number part is still emergent, it's just that it is the residual q-number random zero point part.
OK, but then how can you say that this part is equivalent to a quantized Yang-Mills field of the kind considered by t'Hooft, for purposes of renormalization?

Because it has a very similar formal structure to the internal symmetry Yang-Mills quantum field operators.

Positive frequency part creates a q-number A^a quantum out of the coherent c-number A^a condensate. Negative frequency part puts a quantum back into the c-number condensate etc. 2 independent polarizations if massless etc.

Let's just look at the intrinsic q-part, there is a natural "Yang-Mills" field 2-form

F^a = dA^a + w^ac'cA^c'/\A^c

With Lagrangian density 0-form ~ *[(1/4)*F^a/\Fa]

Note that A^a = A^a(condensate c-number) + A^a(q-number)

so that the bare Hamiltonian from the Lagrangian has quartic terms. Thus is same formal structure as in Yang-Mills.

Think of sound waves in a crystal. Sound, like gravity and torsion, is an emergent collective phenomenon out of the individual lattice atoms right? You can have "classical" "condensate" sound waves (many phonons in same momentum state - I mean narrow wave packet), but also you can detect "particle" like phonon quantum effects in the fluctuations - but the phonon itself is a collective object out of the atomic substratum.
And these phonon quantum effects can be treated as manifestations of an "emergent" quantized field?
Is that what you mean?

Yes. Sound is an emergent collective phenomenon. At low intensities you get quantum fluctuations - phonon analog to quantum optics effects Poisson noise, sub-Poisson et-al. Sound has both classical wavelike properties and quantized particle phonon properties for different kinds of experiments. I am saying that both intrinsic tetrad curvature ~ A^a and intrinsic torsion ~ w^a^bcA^c are both collective emergent both c-number and q-number like sound is. Sakharov basically had this idea in 1967 though not as detailed.

Note I suppress the possible model-dependent "hologram" N^-1/3 coupling factors and pure Minkowski I^a terms in the above rough heuristics,

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