Parastatistics & Anyons in Thin Films I
Ex. 1) We need to be careful to distinguish the spontaneously broken symmetric vacuum manifold G/H for the space of local macro-quantum order parameters PSI(x) from the physical 3D spacelike slice of 4D spacetime.
For example, if for a given x in 4D spacetime, G/H is a complex plane fiber at each x, i.e. G/H ~ U(1) ~ S1 (circle in a plane), using the Bohm polar representation
PSI(x) = R(x)e^iS(x) = RePSI(x) + iImPSI(x)
Looking at the dual vector fields instead of the Cartan forms: define the ordinary partial derivatives in the fiber space i.e. ,R & ,S
Note that the Lie Bracket for the Cartesian partial derivatives vanishes, i.e.
[,RePSI,ImPSI] = 0
But the Lie Bracket for the polar partial derivatives does not because
,R = [RePSI,ImPSI + ImPSI,RePSI]/[RePSI^2 + ImPSI^2]^1/2
,S = [RePSI,ImPSI - ImPSI,RePSI]/[RePSI^2 + ImPSI^2]^1/2
Note the coordinate-dependent singularity at the zeros R = Higgs amplitude of the local vacuum ODLRO order parameter in these partial derivative vector fields.
In general for any G/H vacuum manifold
PSI = (PSI)ie^i = |Mean Higgs Amplitude|e^i(Mean Goldstone Phase)
B(x) = (hG/c^3)^1/2'd'(Mean Goldstone Phase(x))
The integral of the 1-form B(x) over a closed loop around the z-axis will be an integer if the order parameter is single-valued.
For example if G/H = SU(2) then i = 1,2,3
If G/H = SU(3) then i = 1,2,3, ... 8
The domain of i is the number of independent elements of the Lie algebra of the degenerate macro-quantum coherent vacuum manifold G/H, which is also a quotient group because the unbroken subgroup H of the broken symmetry group G is always a normal subgroup.
We always need to specify a "metric" in physical spacetime x to define a Hodge * dual.
3*'d'(Mean Goldstone Phase) is a closed non-exact 2-form (dual to closed non-exact 1-form B(x)) whose integral over the closed S2 wormhole mouth is quantized when the vacuum order parameter is single-valued.
Can we generalize this to multivalued order parameters to get anyon para-statistics? For example a macro-quantum spinor where one has to rotate 4pi to get back to same order parameter. I suspect that we can further generalize using the roots of unity to get fractions of the flux quantum like in the fractional quantum Hall effect.
In the non-rotating wormhole case we ignore the gravimagnetism of the Kerr metric in a toy model to simply show the idea. Further we restrict to a spacelike slice 3Geometry.
If 'd'(Mean Goldstone Phase) is a vortex string singularity along the z-axis from -infinity to 0, then we can associate a point defect on the string at the origin (x = y = 0) corresponding to one S2 mouth of the wormhole. The other mouth is for z from 0 to + infinity as shown below (in physical space not G/H space).
On Sep 27, 2005, at 9:27 PM, Jack Sarfatti wrote:
What is the relation of the shape functions of string and wormhole?
Easy in the toy model below, for the string defect
g -> x^2 + y^2 far from the string
for the wormhole
f -> (x^2 + y^2 + z^2)^1/2
f^2 = z^2 + g
for a "vortex string" on the z-axis for the Vacuum ODLRO field of the fabric of space-time.
whose 3D Hodge * dual is a wormhole (Einstein-Rosen Bridge)
On Sep 27, 2005, at 9:07 PM, Jack Sarfatti wrote:
The duality of strings (vortices) and wormholes, heralded as a triumph in "string theory" is almost trite in my vacuum ODLRO theory of emergent gravity in which extra dimensions plays no direct role.
Rule of thumb from the nonlinear non-unitary local Landau-Ginzburg type equations for topological defects when the Goldstone phase goes singular, the Higgs amplitude drops to zero. It need not be so extreme. When there is multiple connectivity the Goldstone phase peaks whilst the Higgs amplitude dips to compensate. This is certainly the case for the vortex line in a quantum liquid.
OK we have a coherent Goldstone phase field "Theta" 0-form with singularities so that the 1 form
B ~ 'd'(Theta) is multi-valued.
'd'(Theta) = (xdy - ydx)/g[(x^2 + y^2)]
is closed but not exact, i.e, it's closed loop integral need not vanish.
Far from the string (z-axis) where B peaks to a max, the Higgs amplitude |Psi| for
Psi = |Psi|e^iTheta dips. And if Psi is single valued we get a quantization (Bohr-Sommerfeld rule).
Asymptotically far from the z-axis (scale is the coherence length) g -> x^2 + y^2
We then introduce the multiply connected wormhole metric in 3D space that allows us to define a dual 2-form point defect
3*'d'(Theta) = (xdy/\dz + ydz/\dx + zdx/\dy)/f(r^3)
asymptotically f(r) -> (x^2 + y^2 + z^2)^1/2
df/dr = 0 determines size of wormhole portal with topology S^2
d^2/dr^2 > 0 for stability of the wormhole portal
f(r) = Star Gate Shape function needing a torsioned quintessent (variable) dark zero point energy field /\zpf(x,y,z) to achieve.
The non-vanishing "Flux without flux" integrals of closed, but non-exact, forms are always over multiply-connected manifolds and, therefore, do not vanish. If, in addition the order parameter is single-valued as it is in a "stationary" equilibrium state, then integrals are quantized. However one can always force non-integer fluxes through the holes in an "excited" non-equilibrium state that, if left to itself, will expel the excess flux out of the core of the defect in the generalized Meissner effect. The modulation of the vacuum phase for the fabric of space-time needed for the practical low-power shaping of warp drive and wormhole portals is such a pumped non-equilibrium process where
Phase of Vacuum Relative to Control ~ 2p(Forced Flux Through Hole)/(Quantum of Flux)
/\zpf ~ (impedance match)(Surface Density of Vacuum ODLRO)^1/2(Surface Density of Control ODLRO)^1/2cos(Relative Phase)
Guv(x) + /\zpf(x)guv(x) = 0
On Sep 27, 2005, at 5:57 PM, Jack Sarfatti wrote:
Some math intuitions (under construction)
Given a "multi-valued" 1-form
B = Lp'd'(Theta)
Lp^2 = hG/c^3
Theta = Mean Goldstone Phase corresponding to Vacuum Manifold G/H
Probably G = SU(2)hypercharge, H = U(1)
Restricting to the 3-geometry metric of a space-like slice of 4-D space-time to define the Hodge 3* operation
3*B is a 2-form.
Given a R1 1D string Goldstone phase singularity for the vacuum ODLRO field the along z-axis
B/Lp = (xdy - ydx)/(x^2 + y^2)
We have the 2D S2 dual multi-valued 2-form "Flux without flux"
3*B = 3*Lp'd'(Theta) = (xdy/\dz + ydz/\dx + zdx/\dy)/f(r)^3
with the wormhole space-metric with topology R1xS2
3g = dr^2 + f(r)^2(dtheta^2 + sin^2thetadphi^2)
f(r) is the wormhole shape function
f(r) -> r = (x^2 + y^2 + z^2)^1/2
<3|'d'3*B> = |<'&'3|3*B> = 4piN' (2D Brane Flux without flux quanta for 3*B 2-forms)
N', N = 0, +-1, +-2, ...
<'&'2|B> = <2|dB> = 2piN (1D Brane (AKA string) Flux without flux for B 1-forms)
Now extend this to a Kerr-Newman metric in which the interior of the wormhole between the two S2 2D wormhole mouths is filled with positive dark zero point energy density and negative pressure - sort of like George Chapline's "Dark Star" on micro-scale.
On Sep 27, 2005, at 9:17 AM, Jack Sarfatti wrote:
Memorandum for the Record
Subject: Advanced Space Flight Technology
No more Space Shuttles ever needed again.
I have been asked
"Ok Jack , I read it. Assuming its true, then pray tell how do we get some and bottle it as fuel?"
Forget fuel. There is no fuel! This is not a rocket ejecting mass. Also that is what is wrong with Hal's approach and Woodward's approach and even Gennady Shipov's torsion field drive. Even though they are PROPELLANTLESS, i.e. not ejecting mass, nevertheless they change mass M (by one alleged means or other, torsion for Shipov, Mach's principle for Woodward so there is a vdM/dt term just like in a rocket but without any ejected mass. This is intrinsically unstable and is a BOMB if they change e/m too much! Also it is a non-geodesic drive, i.e. they feel inertial g-forces! Mine is different. I have little LC oscillator loops embedded on nanoscale in thin very high Tc 2D layer anyon films "painted" on the fuselage. Forcing different non-integer continuously varying magnetic fluxes through the little Josephson "super-conducting loops" modifies the Goldstone phase of the Higgs vacuum phase that controls the curvature and the torsion of the local space-time geometrodynamic field. The trick is how to lock the phase of the vacuum to the magnetic fields (currents through the loops). I don't know enough solid state physics yet to do that, but obviously the alien ETs do because their ships fly.
Note that Maxwell's EM field equations in the curved space-time with Shipov's torsion field look like in invariant Cartan notation:
F = DA
DF = 0
D*F = *J
D = d + W/\ + S/
d = Cartan's exterior derivative
W = curvature spin-connection where
dB + W/\(1 + B) = 0
T = dS + W/\S + S/\(1 + B + S) = Russian Torsion Field 2-form
R = (d + W/\)W = Einstein's curvature 2-form from 1915 theory (not done this post-modern way of course)
R/\(1 + B) + /\zpf(1 + B)/\(1 + B)/\(1 + B) = 0
is 1915 Einstein vacuum equation with cosmological constant.
This all generalizes to include torsion to
R' = (d + W/\ + S/\)(W + S)
(d + W/\ + S/\)R' = 0
(d + W/\ + S/\)*R' = *J'
(d + W/\ + S/\)^2*R' = (d + W/\ + S/\)*J' = 0
this is key equation for local stress-energy current density conservation for practical low-power metric engineering of Weightless Warp Drive and Wormhole.
B is the curvature field gauge potential from locally gauging T4 -> Diff(4) of GCTs.
S is the torsion field gauge potential from locally gauging the Lorentz group O(1,3).
A is the electromagnetic gauge potential from locally gauging the internal U(1) group.
This local frame-invariant theory is easily extended to the standard model of quarks and leptons using
the SU(2) and SU(3) internal symmetry groups.
From the standard model SU(2)hypercharge spontaneous symmetry breakdown in pre -> post inflation of false to true vacuum I get (consistent with both standard particle model and standard cosmological model of mainstream physics)
B = (hG/c^3)^1/2'd'(Goldstone Phase of Higgs Field)
T = (hG/c^3)d'd'(Goldstone Phase of Higgs Field)
d^2 = 0
d'd' =/= 0
from PHASE SINGULARITIES
e.g. string vortex phase singularity along z-axis in 3D space
'd'(Phase) = (xdy - ydx)/(x^2 + y^2) 1-form
Note that the 1-form is infinite on the z-axis string.
<&2|'d'(Phase)> = <2|d'd'(Phase)> = 2piN
N = integer winding number
This is Bohm-Aharonov nonlocal "Flux without flux".
The Bohm hidden variable single electron (quark) is an extended microgeon with a strong dark zero point energy uniform core and a thin shell of charge in a "point defect" in the LOCAL vacuum order parameter ODLRO Higgs field manifold G/H.
On Sep 26, 2005, at 9:44 PM, Jack Sarfatti wrote:
Is the discovery of dark energy the key to warp drive?
Dark energy is positive zero point energy density with negative pressure. Einstein showed that the pressure is 3 times as powerful as the energy density in bending space-time. Therefore, dark energy anti-gravitates repelling everything and expanding space beyond its normal Hubble expansion. This is the accelerating universe in which ~ 73% of the large-scale stuff is this dark energy with negative pressure. In addition the Russians (Gennady Shipov in Moscow) are extending Einstein's theory to include torsion fields as well as curvature fields. In that case we can have quintessent fields in which the zero point energy density is not simply a cosmological constant, but is a variable local scalar field. We can now build a Weightless Warp Drive (WWD) like Alcubierre's by a kind of Josephson wave interference effect. We live in a Higgs Ocean vacuum field. The Higgs Ocean has vacuum waves. We use a local high Tc coherent wave like in the thin film anyon condensates that locks into the local phase of the vacuum wave. This is similar to a heterodyne radio receiver. By modulating the local vacuum wave's phase we can make both positive and negative zero point quantum pressures in different parts of the star ship's fuselage and, therefore, shape the weightless free float geodesic glide path of the ship. This does not take a lot of power. It's like Tao Chi. For more details see the new book Super Cosmos by Jack Sarfatti (2005) ISBN:1-4184-7663-3.