Discussion with a mathematician
"You might be heading in the same direction, but from an entirely different starting point -- but one that's got problems. For one, with the I's in the definition of the tetrad, you're still trying to have it both ways and inject a semblance of translation invariance in curved spacetimes. There is none."
No I don't think so. No global translational invariance. I have locally gauged the translation group and the compensating gauge potential is precisely A^a. The idea is this. The pre-inflation false vacuum is precisely global special relativity standard model. The trivial tetrads are global. I call them I^a. For globally flat unaccelerated geodesic observers, The components of I^a i.e. I^au = Kronecker delta. For non-geodesic accelerated observers in globally flat Minkowski spacetime, I^a components are general curvilinear functions but with zero Riemann-Christoffel curvature of course.
e^a = I^a + @A^a
Now this is not necessarily a perturbation theory. I am not assuming
@A^a << I^a
in general. However, obviously when @ -> 0 we return to globally flat Minkowski spacetime.
So I do not think this is a weak point of my theory.
I posit @ = hG*/\zpf/c^2 dimensionless like the fine structure constant in quantum electrodynamics.
"Rather, you should be looking at the more general notion of affine spaces and affine connections."
"This is where you begin to find a natural correspondence. When you start bringing in the M matrix and the various other constructs associated with it (your novel contributions), ultimately what you're doing or what you're going to end up doing is landing in
the same spot that Sardanashvily's already gotten to."
Well I have no understanding of Sardanashvily's work. Never read it. Note we both have "Sar" in our names. Why don't you be our Freeman Dyson and write a paper showing the correspondance?
"That is, you started from a somewhat problematic point of departure,"
What do you mean? I^a?
"took a turn on your path, and ended up landing right in the middle of a confluence with the gauge gravitation idea, which already has a perfectly sound starting point."
Well I have read Kibble's paper from 1961 on gravity as a gauge theory. I first read the Yang-Mills 1954 paper in the mid 60's and my PhD dissertation was influenced by the whole idea of local gauging. But I was operating in a vacuum back then where I was so not too much came of it, but I did predict the supersolid phase of helium in Physics Letters in 1969 before Tony Leggett. I also made a model of self-trapped laser filaments based on Landau-Ginzburg equation which helped Ray Chiao in his experiments in mid 60's - as he told Charles Townes, and I did write a paper with Marshall Stoneham on spontaneous broken symmetry in solid state physics in 1967 cited in American Institute of Physics "Resource Letter on Symmetry in Physics" in 1980 as a significant paper. So the two ideas of spontaneous symmetry breaking and local gauge invariance have been central to all my work from the 60's.
"Hence, the need to systematically compare notes. One of the major elements in Sardanashvily's treatment of mechanics (which comes straight out of the mathematical community) is the more general notion of a "connection". A connection is not just for gauge theory, but is a more general object that lives on a jet bundle."
You are thinking like a mathematician. Physics is very different. Why should a physicist be interested in a "jet bundle"? Mathematicians like to generalize, we physicists are primarily interested in phenomena and want to use as little excess formal baggage as possible - completely opposite to Max Tegmark's ideal of the "mathematical universe."
"Moreover, when the latter is related to an already-existing gauge-theoretic connection, then it has a decomposition into it plus a "soldering form". Ultimately, the tetrads come out of soldering forms
for affine connections. At least, that's my understanding of it."
Fine, but so what? I don't see any advantage there from a physics POV. If you could show one fine.
"Sardanashvily's Goldstone phases come about through the breaking of the general frame bundle's GL(4) symmetry group to the SL(2,C) group associated with fermions' local frames. That's enough to give you the spin coefficients -- but not the tetrad."
Sounds like Utiyama's paper that motivated Kibble in 1961. Well in that case my theory is much better since I get both. The tetrads from the diagonal elements of the M-matrix, the spin connections from the off-diagonal elements of the antisymmetrized part of the M-Matrix.
Hey! I just had a new idea from reading A.Zee's "Quantum Field Theory in a Nutshell"!
Matrix = Traceless Symmetric + Trace + Antisymmetric
So tetrads from the trace (I mean diagonals)
Spin connections from Antisymmetric off-diagonal part.
So what physical thing corresponds to the symmetric off diagonal part?
"The dimensions of GL(4) and SL(2,C) are 16 and 6. The symmetry breaking entails vacuum sectors associated with the quotient group GL(4)/SL(2,C) (10 dimensions, isomorphic to S_3 x R^7, in fact). This is where the tetrad lives. They're the Goldstone phases of the broken
symmetry brought about by the fermions' frames."
This sounds interesting if you can flesh it out in more detail. However, I remind you I have a very simple formula for the intrinsically warped pieces of the tetrads
A^a = Theta^a/\dPhi^a - dTheta^a/\Phi^a
where Theta^a & Phi^a are 8 zero-form Goldstone phases for 9 real scalar Higgs fields.
This is a SO(9) internal symmetry.
For example if only 2 real Higgs fields, they share only one Goldstone phase i.e. O(2) or U(1) - roughly speaking.
Vacuum manifold for Landau-Ginzburg potential minima is S1
3 real Higgs fields share 2 independent Goldstone phases i.e. O(3)
Vacuum manifold is S2
So for 9 real Higgs fields, the vacuum manifold is S8.
Suppose we have N real Higgs fields psi(i), i = 1 ... N
|psi|^2 = psi(1)^2 + psi(2)^2 + ... psi(N)^2
The vacuum manifold is the unit N-1 hypersphere
1 = [psi(1)^2 + psi(2)^2 + ... psi(N)^2]/|psi|^2
There are N-1 independent direction cosines of the independent Goldstone phases.
Theta^a & Phi^b are separately 4-vectors under the 6-parameter Lorentz group.
"Sardanashvily's papers and books has a large number of Hehl references, though I'm not entirely sure what the relation of the two is. He also seems to be caught up in the same general "clique" that I think may be centered on the 1979 Lecture Notes in Physics 107. It's probably out of there that the whole "covariant Hamiltonian" and
"polysymplectic" trends begun."
Well I don't see any physics here. Just a lot of math. I could be wrong, but it's your job to connect the above to physics.
"The more notable feature of this extra element, that should particularly interest you, is that it does not require the 3+1 decomposition of spacetime! That is, it's not only fully GR- compatible, but provides a natural starting point for anyone who wants to further study all matters related to achronal spacetimes, or spacetimes with causal anomalies (e.g. closed timelike curves). The
"covariant Hamiltonian" approach is general enough to accommodate this."
Sounds nice, but the tetrad and spin connection Cartan 1-forms are already independent of the 3+1 decomposition.
e^a = eu^adx^u
is a local scalar invariant under GCTs and so are the spin connections
S^a^b = - S^b^a = S^a^budx^u
ds^2 = e^aea = guvdx^udx^v
"Sardanashvily stays within the more rigid confines of globally hyperbolic spacetimes, however (in part, because there's already a well-known representation theorem that relates more general spacetimes to these). You've got a PhD in Physics. However, the subtleties that are brought out by the jet bundle formalism and all matters related requires a deeper probing into the Mathematical issues;"
No doubt, but the problem is we have too many mathematicians in physics with very little physics coming out of their efforts. I am interested in some very concrete physical issues
1. What is dark energy?
2. What is dark matter?
3. How do the silent-running "UFOs" work?
4. Pioneer anomaly?
5. Flat stellar rotation curves in galactic halos.
to name a few.
That is how do we make warp drive and traversable wormhole that we in fact see - though most mainstream physicists are in ignorance or denial of the UFO sightings and the strange "Skinwalker" phenomena at the Bigelow-Sherman Utah Ranch. USG Military Intelligence at high levels takes these anomalies very very seriously. I can tell you that.
"and this is probably where the greater focus may need to lie for a while."
I don't think so. I am making rapid progress.
"This is a language problem that's endemic to Math and Physics and it's serving as an obstacle to real progress in Physics."