Aha! Thanks. :-)
On Mar 8, 2007, at 3:06 AM, ROBERT BECKER (RB) wrote:
"The reason that Weinberg says in the excerpt you quote below on Torsion, 'Sorry, I still don't get it.' rather than just 'I don't get it.', is because Weinberg had already said "I don't get it" in response to my Letter published in the 4/06 Physics Today referenced by Hehl in his 3/07 Letter. One cannot conduct an extended tit-for-tat debate in the Letters section of Physics Today, so I could not respond further to Weinberg, but it looks like Hehl took up the mantle. Hehl cites some other authors on torsion; I had cited Vargas's papers and also Shipov. Vargas was left rather aghast to say the least when he saw the apparent obtuseness of Weinberg's response." - RB
It is amazing. The great professor is but a shadow of his former self it appears. Weinberg's 3 books on quantum field theory have all the basic ideas needed to understand torsion brilliantly presented. Old age I guess.
"Apart from any other consideration, Weinberg chose to overlook the potential utilitarian advantage to gravitation physics of what I said in my Letter regarding Teleparallelism (TP). To formally integrate objects such as energy-momentum densities that are used in conservation laws for gravitation and in the question of the definition and locality of energy-momentum in gravitation physics, one needs to "add together" objects belonging to tangent spaces of different, neighboring points. But this can not be done in a path independent manner, if at all, unless the affine curvature vanishes. The affine curvature, which is what is actually obtained from the Cartan moving frame method (or equivalently from the tetrads) in the types derivations you have provided many times in this forum, is not the same object in general as the familiar GR Riemannian curvature." - RB
The tetrad/spin connection substratum field equations are essentially spin 1 Yang-Mills type, but for the 10-parameter localized spacetime universal Poincare symmetry group for all non-gravity source field dynamical global actions. Therefore, according to t'Hooft, if you quantize them they are renormalizable in the local gauge invariant "square root" substratum of the spin 2 non-renormalizable composite geometrodynamical field of Einstein 1915.
"In particular, the vanishing affine curvature requires non-vanishing Torsion. This is the TP postulate, which Einstein tried to utilize in his reworkings of GR in the late 20s-early 30s. Since GR, as you correctly point out, has vanishing Torsion, it does not postulate TP. As Vargas has discussed in many Papers, this GR non-assumption, in turn, may have a profound effect on the issue of conservation laws and the definition of energy-momentum in GR and gravitation physics." - RB
Indeed, that may be the origin of the fact that 1915 GR with zero torsion (i.e. not all of the 10-parameter Poincare group is localized) has nonrenormalizable spin 2 nonlocality in the gravity vacuum stress-energy. I am not sure of that of course.
"Hehl brings out another important point: the relation of Torsion to translation. It is customary in physics to associate Torsion with spin one way or another. I believe both you and Shipov follow this interpretation. However, Vargas and others, like Pommaret, reject the association of Torsion spin or rotation because it is mathematically closely related to translation as Hehl highlights in his Letter. Vargas finds a geometrodynamical association of Torsion with the EM field, rather than spin." - RB
I pointed out this is a tricky point that Rovelli and Kibble clarify.
The GCT gauge freedom of 1915 GR comes from localizing only the 4-parameter translation subgroup T(4) of the 10-parameter Poincare group P[T(4),SO(1,3)]. Clearly the local GCT's
x^u = x^u(x^u') are localized infinitesimal translations a^u(x^u'), i.e. 4-parameter
x^u -> x^u + a^u(x^u')
In global 1905 special relativity the infinitesimal "elastic deformation" a^u(x^u') of Hagen Kleinert's 4D world crystal Planck lattice is a global constant over the entire infinity of Minkowski spacetime. This is global action at a distance that violates local objective light cone limited causality, hence localization is absolutely necessary to maintain orthodox causality. Global special relativity without gravity is a half-way house first-approximation that is not internally consistent from this POV.
On the other hand there is no question that dislocation defects in the Kleinert world crystal lattice (Burger's vectors et-al) form the torsion field gaps and that the disclination defects for parallel transport around closed loops of the lattice where
V^u(finish) - V^u(start) ~ SO(1,3)^uu'V^u'(start) ~ R^uvwlA^v^wV^l(start)
A^v^w = - A^w^v is the sectional area element of the small loop
form the curvature.
Therefore there are two dual POV
on the one hand
1915 GR with curvature only and zero torsion emerges from the local gauging of 4-parameter T(4)
on the other hand
1915 GR with curvature only and zero torsion is associated with a local Lorentz transformation LLT of the 4-vector around the closed loop (no torsion gap).
Of course when one does the actual Cartan form algebra there is no problem. The problem is only one of the informal language (Bohm) not of the mathematics.
Thus the 4 GCT invariant tetrad 1-forms are the LLT 4-vector components
e(4)^a = I^a + A(4)^a
I^a is the trivial globally flat 1905 SR tetrad
A(4)^a is the intrinsically warped "gauge connection potential" from localizing 4-parameter T(4).
Einstein's 1915 fundamental local GCT & LLT scalar invariant is
ds^2 = e^aea = (Minkowski metric)abe^ae^b = guvdx^udx^v
Einstein's 1915 constraint of zero torsion is the vanishing 2-form
T(4)^a = de(4)^a + S(4)^ac/\e(4)^c = 0
This with metricity, see Rovelli's explicit formula Ch 2 of his online "Quantum Gravity" gives the 6 zero torsion field effective spin-connnection 1-forms
S(4)^a^b = - S(4)^b^a
only from localizing T(4) to get the spin 1 Yang-Mills potential A(4)^a
A(4)^a spin 1 because it's a Lorentz group 4-vector in the a-index.
This warped localized tetrad resembles the EM 4-potential with "internal index" "a" i.e. "Yang-Mills".
The pure disclination curvature defects without torsion gaps then come from the curvature 2-form
R(4)^a^b = dS(4)^a^b + S(4)^ac/\S^cb
And no TP of course at this stage as you say.
Next step is to locally gauge SO(1,3) giving the additional independent torsion gap field spin connection 1-forms
The full connection is then S(4)^a^b + S(1,3)^a^b
T^a(1,3) = S(1,3)^ac/\e^c =/= 0
R(10)^a^b = d[S(4)^a^b + S(1,3)^a^b] + [S(4)^ac + S(1,3)^ac]/\[S(4)^c^b + S(1,3)^c^b]
Where the TP Ansatz is obviously in my transparent notation using only local objective GCT invariants (coordinate independent)
R(10)^a^b = 0
R(10)^a^b = R(4)^a^b + R(1,3)^a^b + S(4)^ac/\S(1,3)^cb + S(1,3)^ac/\S(4)^cb
Hence two "diagonal" curvature 2-forms. Utiyama in 1960 computed R(1,3)^a^b in effect without R(4)^a^b.
This settles the above informal language confusion you mention!
Note also the two T(4) x SO(1,3) cross-coupling terms.
*I am going to completely rewrite my emergent gravity archive paper with these new results of course.
Yes, thanks I know. :-)
BTW my theory for emergent gravity with torsion needs exactly 8 Higgs
bosons one for each Goldstone boson.
ï¿¼I have N = 8 to get Einstein's tetrads and spin connections emergent.
from my CIT talk under construction (outline sans math type equations)
M Theory for Idiots
Emergence of tetrads and spin connections from the spontaneous
breakdown of localized Poincare group symmetry in the post-inflation
Missing Organizing Idea
Witten and Green say they are missing the key idea denoted as â€œ?â€
No Rigid Symmetries
Locally gauging a nondynamical rigid symmetry group of the action
(either internal or spacetime) of a given source field introduces a
gauge force coupled to the source as the compensating connection
field needed to maintain gauge invariance for the now dynamical non-
rigid symmetry group.
Weinberg doesnâ€™t get it?
How is that possible?
â€œSorry I still donâ€™t get it. Is there any physical principle, such
as a principle of invariance that would require the Christoffel
symbol to be accompanied by some specific additional tensor? â€¦â€
Steven Weinberg to Fred Hehl on the role of the torsion field in
spacetime physics in March, 2007 Physics Today.
Spontaneous Symmetry Breaking
Spontaneously breaking a vacuum symmetry means macroscopic occupation
of a single-particle mode by a large number of virtual off-mass-shell
In a vacuum the ODLRO condensate is composed of virtual quanta in
contrast to the ground state of superfluid helium where the
condensate is made from real helium atoms on mass shell.
Local & Broken
The broken symmetry need not be the same as the locally gauged
symmetry, but can be. In gravity theory rigid translational symmetry
is spontaneously broken in order to have localized variable spacetime
curvature. The action still obeys the symmetry, only the vacuum does
not obey it.
Higgs Vacuum Manifold
N independent Goldstone phases require N+1 real Higgs scalar fields.
The degenerate vacua lie on the unit SN spherical hypersurface:
Higgs Vacuum Potential
The â€œMore is differentâ€ emergent macro-quantum coherent vacuum
order parameter has the effective potential
Dynamical L-G Eq.
Covariant dynamical Landau-Ginzburg equation for the macro-quantum
coherent hologram â€œVolume without volumeâ€ vacuum order parameter is
Local Gauge Invariance
Localizing the 10-parameter globally rigid Poincare group of 1905 SR
universally for all non-gravity source field actions gives the 4
Einstein-Cartan tetrads and the 6 spin connections as compensating
This is similar to localizing the internal symmetry groups in Yang-
Mills theory of the electroweak-strong forces.
M = My Mystery Matrix
Start with two Lorentz 4-vectors of 0-forms
Each Minkowski 4-vector has magnitude
I use Faradayâ€™s â€œlines of forceâ€ in a metaphorical way for the
geometrodynamic field compacted down from 9 + 1 to 3 + 1.
Two Goldstone phases have three single-valued real Higgs fields with
stable quantized (wrapping integers) point â€œmonopoleâ€ defects in
A single Goldstone phase has two real Higgs fields with stable
quantized (winding integers) line â€œvortex coreâ€ stringy defects.
Line Density Operator
The geometrodynamic line density operator is the non-closed 1-form
Area Density Operator
The geometrodynamic area density 2-form is closed. This is the world
hologram idea in action.
The deRahm integral of a p-form on a p-cycle (without boundary)
equals the p-formâ€™s exterior derivative on a p+1 co-form whose
boundary is the p-cycle.
A p-form is closed if its exterior derivative vanishes. When the
interior of the p-cycle is simply connected without â€œholesâ€ the
integral of the closed p-form vanishes.
Otherwise the integral is proportional to an integer analogous to
Bohr-Sommerfeld quantization in the old quantum theory and quantized
circulation in superfluid helium vortices.
The geometrodynamic volume density operator 3-form should be
proportional to the exterior derivative of the area density operator
2-form. However, the area operator is closed. Therefore, the volume
operator is locally zero.
Volume is a holographic projection of the area density operator
integrated on a nonbounding 2-cycle enclosing point defects in the
single-valued 3-real Higgs component field projection on 3+1 from 9
+ 1 spacetime.
The integral of the geometrodynamic area density operator 2-form,
over a non-bounding closed 2-surface (cycle) surrounding a multiply-
connected 3D interior with â€œholesâ€ where the 3 real Higgs fields
all vanish so that the 2 compacted Goldstone phases are undefined, is
Volume without volume
Although Stokeâ€™s theorem rigorously does not apply to closed non-
bounding 2D surfaces S2 , whose 3D interior M3 is multiply connected,
physical intuition demands the as-if holographic imaging
These 2 constraints leave 6 angular Kaluza-Klein Goldstone phases
whose radii determine coupling constants like the moduli of Calabi-
Gennady Shipov gets this structure in his version of torsion field
theory calling the extra 6 degrees of freedom an â€œoriented
pointâ€ (suggesting â€œbranesâ€)
The 8 Goldstone phases require 9 real Higgs scalar fields.
One degenerate vacuum manifold is S8. There are other topologies.
From the POV of nontrivial homotopy groups for stable topological
defects they fit most naturally in the 9D + 1 spacetime of
Shipovâ€™s Brane Worlds
The four GCT invariant 1-forms are
Analogy to Superfluid
The fabric of spacetime is an elastic 4D GCT covariant supersolid,
There is no curvature gravity field in my emergent â€œMore is
differentâ€ (P.W. Anderson) theory if
G = 0
h = 0
1/c = 0
In terms of GCT tensor scalar invariant products, the 4 tetrads and 6
antisymmetric spin connection 1-forms are
The GCT invariant globally flat tetrads and locally curved tetrads
The six GCT invariant connection 1-forms are
GCT Covariant D
The curved Cartan exterior derivative is
Torsion Field 2-Form
In 1915 GR this is zero
Take the covariant exterior derivative of the 1-form spin connection
to get the curvature 2-form
The GCT & LLT field Lagrangian density is
On Mar 7, 2007, at 8:48 PM, MPOGO@aol.com wrote:
> Particle X in rare decay could belong to a new physics model
> By Lisa Zyga
> A particle that may mediate the rare decay of a Sigma-plus hyperon
> appears to have close affiliations with a light Higgs boson found
> in one supersymmetric modelâ€”an interpretation suggesting
> unambiguous evidence for physics beyond the standard model (SM),
> scientists say.
> Xiao-Gang He of the National Taiwan University, Jusak Tandean of
> the University of La Verne, and German Valencia of Iowa State have
> investigated the so-called HyperCP result observed at Fermilab a
> little over two years ago. While the HyperCP Collaboration began as
> a search for CP symmetry violation, the rare decay of the Sigma-
> plus hyperon (made of a strange quark and two up quarks) opens the
> possibility for the existence of a new particle with unusual
> This unverified particle, which He and colleagues call particle X,
> would act as the intermediate state when the Sigma-plus decays to
> its final state, of a proton, muon-plus and muon-minus. With its
> extremely light mass (214.3 MeV), low energy, and smallest-ever
> branching ratio for a baryon decay, particle X would have less than
> a 1% chance of being accounted for within the SM.
> Although He and colleagues showed in an earlier paper that the
> HyperCP result may be explained by the SM if there is no new
> particle, the implications of a new particle are considerable. If
> scientists find that particle X is indeed a new particle belonging
> to a different model, the breakdown of the SM would open up new
> doors for future investigations in many areas, and possibly answer
> many questions unanswered by the SM.
> In their recent paper published in Physical Review Letters, He et
> al. turned their attention to a model called the â€œnext-to-minimal
> sypersymmetric standard modelâ€ (NMSSM) that contains seven Higgs
> bosons. The scientists showed that the lightest one is the main
> component of a particle called A01, and that A01 satisfies all the
> constraints of particle X.
> â€œIf the existence of a very light A01 is confirmed, other models
> such as the SM, which do not have any light Higgs bosons, will be
> ruled out,â€ Tandean told PhysOrg.com.
> Tandean also explained how, although the SM has withstood many
> experimental tests, there are some issues that the model doesnâ€™t
> â€œOne issue is the so-called hierarchy problem: why the electroweak
> scale (represented by the W and Z boson masses, of order 100 GeV)
> is so much smaller than the Planck scale (1019 GeV), at which
> gravity becomes important for elementary particle interactions,â€
> he said. â€œOne aspect of this issue is that quantum corrections to
> the Higgs mass make its value arbitrarily large, up to the Planck
> scale. This clearly contradicts the requirement that the Higgs be
> lighter than a few hundred GeV.â€
> However, as Tandean explained, supersymmetric models (where every
> SM particle has a corresponding superpartner) can provide a natural
> solution to this problem.
> â€The presence of the superpartners results in the cancellation of
> the large quantum corrections, leading to a Higgs mass at the
> desired level,â€ he said. â€œThe minimal version of such models is
> called the Minimal Supersymmetric Standard Model (MSSM). The MSSM
> is a very attractive model in many ways, but it does not address
> the question of why the electroweak scale is much smaller than the
> Planck scale to begin withâ€”this is the so-called mu problem.
> â€œInterestingly, the Next-to-Minimal Supersymmetric Standard Model
> (NMSSM) solves this problem by adding a set of two particles to the
> MSSM in such a way that the electroweak scale can be naturally
> small. The NMSSM has been extensively studied in the literature and
> has many other interesting features. It is therefore a well-
> motivated model.â€
> Among the constraints that the NMSSMâ€™s A01 can satisfy include
> explaining why X is very light: the mass of A01 can be as low as
> 100 MeV, and when the mass is 214.3 MeV, the decay into a muon-anti-
> muon dominates over other possible modes. Secondly, the
> interactions of A01 can produce the same rate found in the HyperCP
> Thirdly, A01 explains why previous experiments with kaons and B-
> mesons that thoroughly explored the same regions where X exists
> never saw X. For these reasons, kaon and B-meson decays impose
> severe constraints on the properties of X, specifically regarding
> two-quark couplings. This means that, even though A01 could explain
> the new particle over a wide range of parameters, there are only
> narrow ranges for which the kaon constraints are also satisfied.
> However, the scientists also suggest that revisiting these
> constraints might reveal some overlooked data.
> In the future, two new particle acceleratorsâ€”the Large Hadron
> Collider (LHC) and International Linear Collider (ILC)â€”might shed
> more light on the Higgs hypothesis. Supersymmetry may determine
> some parameters of particle X, where investigations of squark and
> chargino intermediate states in the NMSSM might provide more evidence.
> â€œThe LHC and ILC have the capability of finding the charginos
> predicted by supersymmetric theories, depending on the chargino
> masses,â€ Tandean said. â€œIn our study, we find that in order to
> explain the HyperCP results, the lighter chargino mass has to be
> around 100 GeV, which is within the range to be probed by the LHC
> and ILC. At the LHC and ILC, it is also possible to study the usual
> Higgs boson, h, in detail (or the one playing the role of h in
> NMSSM). If the A01 is the X particle, the process h --> XX can
> occur and may become the dominant decay mode if the h mass is
> relatively small (120 to 130 GeV). By studying the properties of h
> in detail, one may verify that X is the A01.â€
> Citation: He, Xiao-Gang, Tandean, Jusak, Valencia, G. â€œDoes the
> HyperCP Evidence for the Decay â€˜Sigma-plus to a proton, muon-plus
> and muon-minusâ€™ Indicate a Light Pseudoscalar Higgs Boson?â€
> Physical Review Letters 98, 081802 (2007).
> By Lisa Zyga, Copyright 2007 PhysOrg.com.
> All rights reserved. This material may not be published, broadcast,
> rewritten or redistributed in whole or part without the express
> written permission of PhysOrg.com.
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein