On Jun 21, 2007, at 10:19 AM, RKiehn2352@aol.com wrote:
"You might remember this old note that I called to the attention of JS:
Thermodynamics as a Finsler Space with Torsion
One of the most striking features of thermodynamics is the division into two sets of those variables that appear in thermodynamic constraints: there are intensities (like Pressure and Temperature, E and B) which appear in the thermodynamic functions as variables homogeneous of degree 0, and there are extensive quantities (like Volume, Energy and Entropy, D and H) which appear as variables homogeneous of degree 1. Classically, there is no way to distinguish (geometrically) these two different species. However, it has long been recognized in the calculus of variations of a singly parameterized function, L(x,V;t), with dx-Vdt=0 as a constraint, that homogeneous functions play a critical role. The variational integral is independent of the choice of parameterization, t, if and only if the function L(x,V,t) is homogeneous of degree one in the variables, V. This means that V, or anything proportional to V, such as U=b(x,V) V can be used in the constraints, dx-Vdt = 0 = dx-Uds, where the new parameterization function satisfies ds - bdt = 0. Then the variational procedure on the variational integrand L(x,V,t)dt or on the variational integrand L(x,U,s)ds will give the same minimal result.
When a vector field, V, is considered to be the same vector field, independent from any factor of scaling (or renormalization), then Chern calls the vector field, V, a projectivized vector field. A projective geometry is a geometry that is independent from scales, and can be modeled as the set of all rays through some point of perspective, named the origin."
(Jack wrote) Dirac quantum kets and bras are an example for orthodox QM. What are QBIT spaces with "curvature" and "torsion"?
(Kiehn wrote) "The mathematical vehicle of parametric independent variational calculus distinguishes functions which are homogeneous of degree 1. Such is the arena of thermodynamics, and therefor projective geometries share some of the properties of a thermodynamic system.
But there is more to the story, for in 1919 Finsler developed a thesis in which he studied a geometry which, if unconstrained, was not only non-euclidean, but also non-Riemannian. Special constraints would reduce Finsler geometries to the more common Riemannian geometries. The new feature permitted in Finsler spaces (a feature that is missing in Riemannian geometries because of quadratic, or quartic constraints) is the property of Torsion. Chern has shown how to interpret Finsler spaces in terms of the parametric independent calculus of variations, where the dual field, V, is projectivized.
Cartan recognized in 1922 that spaces with torsion existed, but he achieved this result from an entirely different tack than that used by Finsler. Cartan hypothecated that in the sense that a structural defect of closure in a basis frame implies the existence of curvature 2-forms, a structural defect of closure in the position of the origin could imply the existence of torsion 2-forms. "
(Jack Sarfatti wrote)
TORSION^a = D(TETRAD)^a = d(TETRAD)^a + (SPIN CONNECTION)^ac/\(TETRAD)^c
CURVATURE^a^b = D(SPIN CONNECTION)^a^b
(TETRAD)^a = (MINKOWSKI)^a + @(WARP)^a
(SPIN CONNECTION)^a^b has 2 independent pieces the one from localizing rigid 4-parameter T4, the other from localizing rigid 6-parameter O(1,3). The latter is Shipov's "oriented point" to Calabi-Yau space of superstring theory
Local T4(x) only gives disclination curvature defects in 4D world crystal lattice, local O(1,3)(x) gives BOTH disclination curvature and dislocation torsion gaps (failure of tiny parallelograms to close in 2nd order of smallness for parallel transport of vectors on them).
@ = (Lp^2/\zpf)^1/3 ~ 10^-41 ~ (Eddington number)^-1
Lp^2 = hG/c^3
h = Planck's QUANTUM of ACTION
G = scale-dependent Newtonian gravity coupling.
c = speed of real massless photon in classical vacuum without dark energy/matter
/\zpf at large scales is Einstein's cosmological constant /\(Einstein) = 1/R^2
R = radius of FUTURE dark energy deSitter horizon of our pocket Hubble bubble universe on the cosmic landscape of many worlds that reaches back in time to cosmic trigger the inflation -> big bang
Power of 1/3 in dimensionless renormalizable spin 1 vector coupling is from World Hologram
&R = @R = Lp^2/3R^1/3 ~ 1 fermi i.e. 1Gev (origin of nuclear physics scale - as above so below)
&R = quantum gravity fluctuation in the future de Sitter horizon retrocausal advanced signal cosmic trigger creating pocket universes in globally consistent Novikov loops in time.
/\zpf at small scales it is the quintessent field that we control for manufacture of warp drive and star gate time travel.
Note that dark energy density ~ (1/(LpR)^1/2)^4 ~ (10^-3 ev)4 ~ 10^-29 grams/cc
i.e. in more conventional units this is (c^4/G)/\zpf = (hc/Gh/c^3)/\zpf = hc/Lp^2R^2
(LpR)^1/2 ~ 10^-2 cm ~ geometric mean of smallest UV and largest length IR scales
in a IR-UV duality
(Kiehn) "As Brillouin (1959) states
'If one does not admit the symmetry of the (connection) coefficients, one obtains the twisted spaces of Cartan, spaces which scarcely have been used in physics to the present, but which seem to be called to an important role.'
To this day not much has been done with Cartan's spaces with torsion, except in the Kondo theory of dislocation defects in solids. However, Cartan's notions can be made precise and more general in terms of subspaces of a projectivized basis frame. Then not only the position of the origin (the point of perspective in projective geometry), but also its dual polar axis can have closure defects, leading to dislocations and disclinations respectively
NOWADAYS, my thinking has gone beyond geometrical concepts. Scaling and Renormalization should be replaced with homogenization and self-similarity, which are topological ideas. This means that Affine torsion (which is transitive) is not the key idea, and needs to be replaced by a concept that requires a "fixed" (or singular) point. The 3-form of topological torsion, A^dA, goes beyond affine torsion."
(Jack) Does that not that appear in 2D anyons? Chern something or other?
So T^a = De^a is "affine torsion" in your language?
Note in my theory
e^a = I^a + @A^a
So I can have a A^adAb.
Examples can be given for which affine torsion is not zero, but topological torsion is zero.
(Kiehn) "The vector arrray of Affine torsion 2-forms of a matrix basis frame of functions can be transformed away (but not by a diffeomorphism) if the components of affine torsion are integrable. The transforation is a matrix of integrating factors acting on the Basis Frame.
However, if the components of the vector of affine torsion 2-forms are not integrable, (A^dA not zero) then they cannot be transformed away.
Hence to focus attention on affine torsion and its possible dynamic effects IMO is useless.
The concept of emergence appears to require topological torsion, a key artifact of non-equilibirum thermodynamics.