Torsion Fields: Some toy model calculations
I. 1 + 1 string spacetime slice, neglecting Weyl conformal dilation but including energy and momentum translation generators i.e. A00 = Energy Generator, A11 = Linear Momentum Generator in sense of Noether's theorem relating symmetries to conservation laws for continuous groups. We mean appropriate matrix representations of the Lie algebra depending on what physical vector space they operate on, e.g. Cartan mobile tetrad local frames of reference for gravity and internal vector spaces for "charges" of U(1), SU(2) & SU(3).

The geometrodynamic gauge-covariant partial derivatives acting on Tuv(source) matter fields are

,u ---> ;u = ,u + (Connection)u^a^bAab

(use ' for base space indices)

;0' = ,0' + (Connection)0'^0^0A00 + (Connection)0'^1^1A11

+ (Connection)0'^1^0A10 + (Connection)0'^0^1A01

A01 = - A10 = space-time Lorentz boost

;0' = ,0' + (Connection)0'^0^0(Energy) + (Connection)0'^1^1(Linear Momentum)

+ [(Connection)0'^1^0 - (Connection)0'^0^1]A10

;0' = ,0' + (LC Connection)0'^0^0(Energy) + (LC Connection)0'^1^1(Linear Momentum)

+ (Tensor Torsion Field)0'^1^0(Lorentz Boost)

Note how the String Torsion Field is the Yang-Mills "Phase" of the Lorentz Boost.

The Equivalence Principle is encoded in the (LC Connection) terms.

Similarly for ;1

The tetrads tilt in infinitesimal displacements as:

ea(x + dx) = eb(x)(Connection at x)u^badx^u

e0(x + dx) = [(Connection at x)0'^00dx^0' + (Connection at x)1'^00dx^1']e0(x)

+ [(Connection at x)0'^10dx^0' + (Connection at x)1'^10dx^1']e1(x)

e1(x + dx) = [(Connection at x)0'^01dx^0' + (Connection at x)1'^01dx^1']e0(x)

+ [(Connection at x)0'^11dx^0' + (Connection at x)1'^11dx^1']e1(x)

II 2 + 1 slice for anyonic vacuum ODLRO fractional charge/quantum statistics "World Hologram Plate"

cc* + c*ce^i(Anyonic Phase) = 1

Anyonic Phase = 0 2D fermions

Anyonic Phase = pi 2D bosons

Control Anyonic Phase with perpendicular magnetic field on a 2D nano quantum well showing fractional quantum Hall effect.



Exotic matter effects are neither fermion nor boson.

What does that do to the spin-statistics connection?

Flux tubes pin to charged fermions. Effective charges are fractional like quarks, but only in 2D + 1 space-time "surfaces". In world holography there are no dynamically independent volume degrees of freedom for the geometrodynamic field, hence the Bekenstein bound for black hole thermodynamics and the amount of bits that can be packed into a Volume V ~ V^2/3/Lp^2 ~ (/\Lp^2)^-1 for our pocket universe on the cosmic landscape of parallel universes.

,u ---> ;u = ,u + (Connection)u^a^bAab

;0' = ,0' + (Connection)0'^a^bAab

;0' = ,0' + (Connection)0'^0^0A00 + (Connection)0'^1^1A11 + (Connection)0'^2^2A22

+ (Connection)0'^0^1A01 + (Connection)0'^1^0A10

+ (Connection)0'^0^2A02 + (Connection)0'^2^0A20

+ (Connection)0'^1^2A12 + (Connection)0'^2^1A21

;0' = ,0' + (Connection)0'^0^0(Energy) + (Connection)0'^1^1(Momentum1) +

(Connection)0'^2^2(Momentum2)

+ [(Connection)0'^0^1 - (Connection)0'^1^0]A01

+ [(Connection)0'^0^2 - (Connection)0'^2^0]A02

+ [(Connection)0'^1^2 - (Connection)0'^2^1]A12

;0' = ,0' + (Connection)0'^0^0(Energy) + (Connection)0'^1^1(Momentum1) +

(Connection)0'^2^2(Momentum2)

+ (TORSION)0'^0^1 (LORENTZ BOOST)01

+ (TORSION)0'^0^2 (LORENTZ BOOST)02

+ (TORSION)0'^1^2 (ROTATION)12

Similarly for ;1'

Note that in Einstein's 1916 theory, TORSION = 0, therefore the Lorentz boost and rotation terms vanish!

Homework, do this for 3D + 1.


On Aug 29, 2006, at 8:32 AM, Jack Sarfatti wrote:

Bottom Line
Gravity is "More is different" emergent from the Standard Model of quarks & leptons with gauge bosons. Standard quantum gravity theories are not even wrong, i.e. loop quantum gravity, string theory, canonical quantization are like trying to quantize elasticity theory.

This is the 11th dimension so to speak. 10 from the Poincare group.
From the principle fiber bundle

M(Base Space) = Cartan Mobile Tetrad Frame Fiber Bundle/(Poincare Group + Weyl Dilation)

dimM = 11?

This is intuitive - still thinking about it - non-rigorous.

That is, 4 translations of the tetrad local frame, 3 rotations, 3 boosts + 1 Weyl dilation. Each parameter "phase" "angle of rotation" is a "dimension" of M.

11 G-Orbit equivalence classes partition of the tetrad bundle.

G = Poincare Group + Weyl Dilation
(universal symmetry group for emergent spacetime physics)

Also an addition 4 special conformal transformations not accounted for (Tony Smith)

,u ---> ;u = ,u + (Connection)u^a^bAab

"Gauge covariant partial derivative" analog to internal symmetry Yang-Mills SU(2), SU(3) ...

{Aab} = Lie Algebra of G

The Cartan mobile tetrad frames tilt & stretch/contract relative to each other

ea(x + dx) = eb(x)(Connection at x)u^badx^u

(ea|eb) = (Minkowski metric)ab = nab

(eu|ev) = (Curvilinear Metric)uv = guv

e^a = eu^adx^u

eu = eu^a&a

Identity action = I = e^a&a = I' + B

F = dB ~ dTheta/\dPhi

Theta & Phi = Goldstone phases of vacuum ODLRO Higgs inflation field with 3 real components (in one toy model)

B ~ Theta/\dPhi - dTheta/\Phi

d^2 = 0

D = d + B/
DF = 0

D*F = *J

D*J = 0

Yang-Mills field equations for U(1)xSU(2)xSU(3) of standard model in false vacuum that is globally flat.

i.e. Gravity emergent from the Standard Model of quarks & leptons with gauge bosons.

On Aug 29, 2006, at 2:05 AM, Carlos Castro wrote:

Dear Jack :

Also in Weyl's geometry the non-metricity tensor Q is
zero as well.

The Weyl covariant derivative of the metric is zero.
Despite that the lengths of vectors change under
parallel transport (in Weyl spacetime )
the angle of two vectors remains the same ( conformal
property ) under paralell transport.

Best wishes

Carlos