## Friday, September 30, 2005

Macro-Quantum Spinors

On Sep 30, 2005, at 2:00 PM, Jack Sarfatti wrote:

Super Cosmos is now at http://amazon.com
These rough notes below to appear (refined) in book Star Gate: Metric Engineering Warp Drive for Wormhole Space & Time Travel.

Lecture 4 Fractional Quantum Hall Effect for 2D Anyons
Note that the Hodge* as used in most elementary applications only works for static curved metrics and needs to be modified for rotating Kerr metrics with gravimagnetism H = (g01,g02,g03) as in frame-dragging in standard 1915 GR with zero torsion.
What about (anti)self-dual instantons in Euclidean metric? (Vacuum tunneling)

Given a 2-form, e.g. curvature, EM field, torsion, then relative to some metric define a *

The 2-form is self-dual if

*F = +F

it is anti self-dual if

*F = - F

*^2 = +1 for Euclidean signature ++++
*^2 = - 1 for Lorentzian signature - +++
where with causal light cones
F = F+ + F-
*F+- = +-iF+-

Self-interacting nonlinear Yang-Mills eqs in ++++ have (anti)self-dual "instanton" solutions.

Can we generalize this to the N roots of unity?

Closed and exact forms:

If dB = 0, B is a closed form.

If B = dA

B is an exact form.

All exact forms are closed, but not all closed forms are exact.

If we have a p-form A

B = 'd'A

('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) =/= 0

Where '&'(p+1) is a non-bounding p-cycle in a multiply-connected p+1 manifold.
For example, if p = 2, then &'3 is a closed 2D surface AKA wormhole mouth (portal) that is not a complete boundary of the interior 3D space. There is at least one other '&'3 closed surface some where-when in our universe or in a parallel universe next door in hyperspace. There can be many such '&'3 in a NONLOCAL multi-pronged network of Star Gate wormhole tunnels held open by negative pressure positive zero point dark energy density distributions.

The general theorem is

(&(p+1)|A) = Sum over all ('&'(p+1)|A) = Sum over all (p+1|'d'A) = (p+1|dA)

In the special case that dA = 0 then the final term on RHS = 0.

Of course there is no reason why dA = 0 is necessary. You can have dA =/= 0.

Restricting ourselves to a LOCAL wormhole mouth where

('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) =/= 0

Let p = 1

('&'2|A) = (2|'d'A)

Let

A = 'd'(Theta)

Theta = 0-form phase of a LOCAL macro-quantum order parameter valued in the complex plane.

Single-valuedness of the local order parameter around the closed loop ('&'2| that is not the complete boundary of the inner area (2| then gives us quantized "Flux without flux".

Consider winding around the closed non-bounding loop ('&'2| once. Imagine that the order parameter Goldstone phase changes by 2pi/n (n an integer) in order parameter space G/H for 1 winding of 2pi around the non-bounding closed loop in physical 3D space. In that case, the order parameter changes by e^i2pi/n in G/H vacuum manifold space for each 2pi circuit around the Theta Goldstone phase singularity (e.g. vortex core string) in 3D space.

In the simplest case n = 1, Then for N windings in physical 3D space we have the nonlocal Bohm-Aharonov effect

('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) = 2piN

of flux quanta through the closed loop in the "normal core" in a stationary state. We can actually pump non-integer flux through, but the state will not be stationary, it will shake off the excess flux in the form of radiating quasi-particles if not externally pumped in a Meissner effect.

Suppose n = 2, then for N windings

('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) = piN

This is a macro-quantum spinor condensate.

For n = 3

('&'(p+1)|A) = (p+1|'d'A) = (p+1|B) = 2piN/3

suggestive of the fractional Quantum Hall Effect for high Tc 2D films of anyon condensate with para-statistics of the normal fluctuations?

Lecture 3 Gravity energy is nonlocal and Yilmaz et-al are wrong.

The Bianchi identities generalize d^2 = 0 to D^2 = 0 when there are gauge field connections A, W etc. present.

For example:

D = d + A/\

for SU(2) & SU(3) internal symmetry non-Abelian Yang-Mills gauge force models of the parity-violating weak hypercharge and parity-conserving strong gluon forces respectively in the standard model. It is the weak hypercharge group SU(2) that is spontaneously broken to produce the small inertia of the leptons and quarks and to produce emergent gravity via my original formula

B = (hG/c^3)^1/2d(Goldstone Phase)

Here Goldstone Phase is the "mean" SU(2) hypercharge phase from using the Trace operation (details later on).

The large mass of the hadrons comes from the kinetic energy of the confined quarks in dark energy bags.

In 1915 General Relativity, locally gauging T4 -> Diff(4):

D = d + W/
Where the spin connection W obeys

T = De = d(1 + B) + W/\(1 + B) = 0

Therefore

dB + W/\(1 + B) = 0

B is the curvature tetrad field that corresponds to disclination topological defects in the Vacuum ODLRO Manifold G/H and in Hagen Kleinert's "World Crystal Lattice".

Gennady Shipov's torsion field is when the Lorentz group is locally gauged in addition to T4. This adds 6 new scalar fields that act like extra space dimensions, i.e. 10D space-time. Just as locally gauging T4 brings in the compensating 1-form curvature disclination connection B, provided SU(2) hypercharge spontaneously breaks in the inflation, so does locally gauging the Lorentz group O(1,3) bring in the compensating 1-form torsion dislocation connection S where the non-vanishing torsion 2-form T is now

T = dS + W/\S + S/\(1 + B + S)

Note the torsion-curvature coupling terms W/\S and S/\B.

The 1915 GR curvature 2-form is

R = DW = dW + W/\W

The Einstein-Hilbert dark energy vacuum action is the 4-form

R/\(1 + B)/\(1 + B)+ /\zpf(1 + B)/\(1 + B)/\(1 + B)/\(1 + B)

The 1915 Bianchi identity is, for /\zpf = 0

DR = 0

i.e.

(d + W/\)R = dR + W/\R = 0

The Einstein field equation is

D*R(Geometry) = *J(Matter)

D^2R = D*J(Matter) = 0

is the local conservation stress-energy current density.

Note

D*J(Matter) = d*J(matter) + W/\*J(Matter)

W/\*J(matter) = d*j(vacuum-matter)

*j(vacuum-matter) is a 3-form that is the pseudo-tensor of the (matter-vacuum coupling)

Nonlocality of total gravity energy is multiply-connected "Flux without flux", i.e.

(3|D*R) =/= 0

When *J(matter) = 0 everywhere-when

i.e.

*R = *"D"W

D^2 = 0

but

D"D" =/= 0

i.e. Flux without flux from multiple connectivity of 3-Manifold slices of space-time.

If no S, then /\zpf is really a constant, i.e. "Cosmological Constant" made small by vacuum ODLRO. When S =/= 0, then /\zpf is a local scalar field, indeed the one we need for metric engineering warp and wormhole with the Josephson homodyne detection method.

This is in complete analogy to Maxwell's U(1) equations

dF = 0

d*F = *J

d^2*F = d*J = 0

And the Yang-Mills SU(2) & SU(3) equations

DF = 0

D = d + A/
D*F = *J

D^2F = D*J = 0

The complete theory with torsion and U(1)xSU(2)xSU(3) is clearly the same template where now

D' = d + W/\ + S/\ + A/\ + C/
A is U(1)xSU(2)xSU(3) or whatever internal G symmetry group works.

W & S are from locally gauging the 10-parameter space-time symmetry Poincare group.

If we go to the 15 parameter conformal group we will get an additional conformal or Penrose twistor connection. If we go to Kaluza-Klein extra-dimensions and Grassmann fermion dimensions of supersymmetry (not observed) then everything generalizes in the same template. Now however

R' = D'(W + S + C + A ...)

D'R' = 0

D'*R' = *J'

As well as the Yang-Mills equations

F' = D'A

DF' = 0

D*F' = *j

etc.

Similarly for S, B & C?

On Sep 29, 2005, at 2:37 PM, Jack Sarfatti wrote:

Lecture 2 (Revised Draft #2)

Ex 4
Mechanical Model of a Phase Singularity

Simplest case.

Imagine a plane. Pick an origin O. Use polar coordinates, (r,theta) for arbitrary moving point P.

Pick a point 0' with fixed coordinates (a, chi).

Draw a circle of radius b < a centered at O' with coordinates (b,phi)

Let point P move around this circle whose center O' is displaced from origin O.

Obviously when a =/= 0 the total theta angle integral of the 1-form dtheta
swept out in one complete circuit round the circle is ZERO. Basically theta oscillates.

Note that the angle theta depends on the angles chi and phi.

Half of the movement is clockwise and then counter-clockwise for dtheta on successive semicircles as P winds around the circumference of the displaced circle. This is most easily seen intuitively all at once when O' is vertical compared to O (on y-axis ordinate).

Note what happens when you move the circle to different locations on the plane.

Draw tangents from O to the circle in different locations.

In contrast, when a = 0, or alternatively, b > a the total angle integral of dtheta is 2pi.

Homework Problem

Use trigonometry to make an algebraic proof.

-------------------

For 3D flat metric, the Hodge * is with the right-hand rule convention

*dx/\dy = dz
*dy/\dz = dx
*dz/\dx = dy

Left-hand rule is

*dy/\dx = dz
*dz/\dy = dx
*dx/\dz = dy

Parity transformation interchanges left and right hand rules in 3D.

(x,y,z) -> (-x,-y,-z)

SU(2)hypercharge breaks parity symmetry and it also may be the origin of inertia and gravity.

p-forms |p) = (p!)^-1Fuv ... dx^u/\dx^v ...

p-factors, p =< n = dim of manifold.

Important formula

|p)/\|q) = (-1)^(pq)|q)/\|p)

The exterior product /\ of forms is a parallelepiped in the co-tangent n-dim space of constant phase wave fronts in contrast to the tangent space of particle paths normal to the wave fronts.

For R^3

A = Axdx + Aydy + Azdz

F = dA = ( Az,y - Ay,z)dy/\dz + (Az,x - Ax,z)dz/\dx + (Ax,y - Ay,x)dx/\dy

2-form independent of metric

*F = *dA = ( Az,y - Ay,z)dx + (Az,x - Ax,z)dy + (Ax,y - Ay,x)dz

* dual 1-form in 3D manifold with a metric specified.

Note, if

A = df

F = dA = d^2f = 0

Therefore

( Az,y - Ay,z) = 0 etc

, is ordinary partial derivative

i.e. mixed second order partial derivatives of the 0-form f commute in that case.

However, in the case of a phase-singularity, there is some kind of region in the manifold where the mixed partials of the 0-form Goldstone phase of the local macro-quantum coherent vacuum order parameter Higgs field in our primary application to physics of this formalism do not commute. This is a topological defect in the vacuum manifold G/H, where I write

A = 'd'f

d'd' =/= d^2 = 0

due to multiply-connected manifolds

F = dA =/= 0

e.g. non-integrable anholonomic multi-valued gauge transformation of Hagen Kleinert

AKA

Flux without flux

see also the related idea of the nonlocal Bohm-Aharonov effect using Feynman amplitude Wilson loop operators.

In 3 space

d|0) is gradient of a function, i.e. scalar field

d|1) is curl of a vector field

d|2) is divergence of a vector field

B = Bxydx/\dy + Byzdy/\dz + Bzxdz/\dx

dB = (Bxy,z + Byz,x + Bzx,y)dx/\dy/\dz

Static 4D Metrics without gravimagnetism (non-rotating spacetimes) & without gravity waves (c = 1 convention) here

(curved metric) = g = -dt^2 + 3^g

The toy model wormhole is of this form.

We need positive dark zero point energy density with negative pressure to keep the wormhole open. There is no event horizon in this wormhole. It's not a black hole!

A metric allows the symmetric inner product { , }.

Classical energy density of the EM field in the absence of sources is

(1/2)[{E,E} + {B,B}]

The Lagrangian density is

(1/2)[{E,E} - {B,B}

E = (Ftx, Fty, Ftz) electric field

B = (Fyz, Fzx, Fxy) magnetic field

F = B + E/\dt

F & B are 2-forms

E is a 1-form

We need a classical EM stress-energy density tensor T to compute

T ~ &(Dynamical Action)/&(metric)

& is functional derivative of classical Lagrangian field theory (not particle mechanics).

w = (pressure/energy density)

Note, the above is classical without any quantum zero point fluctuations.

w = +1/3 for classical far-field radiation with only 2 transverse polarizations.

For example, the cosmic black body radiation has w = +1/3

http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_05_jpg.htm

It's wrong to use w = +1/3 for vacuum zero point energy that bends spacetime absolutely.

This is an error in SED used by HRP. The Casimir force is not important for metric engineering Weightless Warp Drive and Wormhole Time Travel to The Past (using old wormholes made at the beginning of our local universe and even connecting to the parallel universes of Super Cosmos).

Equivalence principle + local Lorentz invariance imply w = -1 for all kinds of zero point energy (isotropically distributed).

That is the Zero Point Energy Stress-Energy Current Density Tensor tuv(ZPF) diagonal is for Energy Density dE/dV & Pressure P

w(ZPF) = P/(dE/dV) = -1 from EEP & LI of GR

(dE/dV,-dE/dV,-dE/dV,dE/dV),

Therefore, since w = -1

Trace is -2(dE/dV) = (1 + 3w)(dE/dV)

Compare to EM radiation where

w = P/(dE/dV) = + 1/3

( dE/dV, +(1/3)dE/dV, +(1/3)dE/dV, +(1/3)dE/dV), i.e. Trace

= +2dE/dV = (1 + 3w)(dE/dV)

Note that dark energy is P < 0 & dE/dV > 0.

Dark Matter is P > 0 & dE/dV < 0

Assuming here above, of course, compact dark matter sources like the Galactic Halos when P > 0 and measurements by external observers.

If we stick in Casimir plates to break translational symmetry or somehow break the rotational symmetry (rotating superconducting disks that phase lock to the vacuum Goldstone phase?)

There is analogy here to homodyne detection of quantum information with continuous variables where the local oscillator in a beam splitter is like the vacuum ODLRO field. (Rev Mod Phys, p. 513, April 2005) Squeezed vacuum states in quantum optics is when one quadrature of the zero point virtual photons has less vacuum noise than does the other quadrature which has excess noise to compensate. http://www.cco.caltech.edu/~qoptics/squeeze.html

Then

( dE/dV, +a(dE/dV), + b(dE/dV), + c(dE/dV)), the trace is now (1 + a + b + c)dE/dV

1 + 3w = 1 + a + b + c

w = (a + b + c)/3

This is quintessence when w < -1/3 and it can perhaps be done with the Shipov torsion field. Phantom energy with the Big Rip tearing the fabric of the Universe apart is when w < -1.

On Sep 28, 2005, at 4:20 PM, Jack Sarfatti wrote:
Lecture 1 on Cartan Forms

I am using John Baez's Ch 4 of "Gauge Fields, Knots and Gravity" for the standard ideas.

All the local physical observables in classical gauge force field theories are examples of Eli Cartan's "differential forms", e.g., Au, Fuv, ju.

The integrals of forms over manifolds are premetrical until we define a Hodge * operation taking a p-form to a N-p form for N-dim manifolds.

The p-forms are very much like Bishop Berkeley’s “ghosts of departed quantities.” They “are neither finite, nor … infinitely small, nor yet nothing.”

The 1-forms are dual to tangent vector fields on the manifold. A vector field is like a bundle of particle paths in Bohm’s hidden variable picture of quantum theory. The 1-form (AKA “cotangent vector”) is like a stack of wave fronts (AKA “little hyperplanes”) of small extent as in Fig. 1 p. 45 (Baez) also in MTW’s “Gravitation.” “The bigger df is, the more tightly packed the hyperplanes are.” Given a Cartesian coordinate basis of tangent vector fields {,u} and a dual basis of 1-forms {dx^u}, then duality here is

dx^u,v = 1v^u = Kronecker delta NxN identity matrix.

Given a 1-form df and any vector field v, the directional real number df(v) “counts how many little hyperplanes in the stack df the vector v crosses.” Linearity is built in as a postulate. The Cartan forms are invariants of local coordinate LNIF transformations Diff(N). Diff(N) is what you get when you locally gauge the global ND translation group. In 1915 GR the Cartan forms are also invariant under the local LIF Lorentz transformations O(1,3). In general this would be O(N) pre-signature. That is TNxO(N).

/\ is the exterior product. Obviously we have a kind of quasi-algebra equivalent to dissecting an N-1 simplex or “brane” also giving partially ordered (non-Boolean?) lattices with the 0-form on bottom and the N-form on the top.

N = 1 (dx), i.e. 1

N = 2 (dx, dy, dx/\dy = -dy/\dx), i.e. 3

N = 3 (dx,dx,dz, dx/\dy, dx/\dz, dy/\dz, dx/\dy/\dz), i.e 7

N = 4 (dx,dy,dz,dt, dx/\dy, dx/\dz, dy/\dx, dt/\dx, dt/\dy,dt/\dz, dx/\dy/\dz, dt/\dx/\dy, dt/\dx/\dz, dt/\dy/\dz, dt/\dx/\dy/\dz), i.e. 15

If we include the 0-form we have 2, 4, 8, 16, i.e. 2^N elements in the quasi-algebra that suggests the Clifford Algebras. There are obviously N!/p!(N-p)! p-forms in N space. This is also like an information space of N c-Bit Shannon Boolean strings. Obviously there will be some kind of matrix representation. For example N = 2 should correspond to the 3 Paul 2x2 spin matrices with the unit matrix. Therefore, there is a connection to U(1)xSU(2) here. N = 3 should have something to do with the 8 SU(3) matrices, and N = 4 obviously connects with the Dirac algebra and possibly U(4) especially when we complexify each real number space-time dimension and even go beyond that to quaternions & octonians.

Classical gauge force theories include Maxwell's U(1) electromagnetic theory, Yang-Mills theories of the SU(2) weak and SU(3) strong forces of the leptons and quarks in the standard model and Einstein's theory of gravity (General Relativity, 1915 AKA GR) provided you do not work at the symmetric metric tensor level guv(x), but work at the "square root" 1-form tetrad "e" level. Note that Einstein's local equivalence principle is simply

(curved metric ) = e(flat metric)e

where e is the Einstein-Cartan 1-form tetrad field.

You can write

e = 1 + B

B = curvature tetrad field

Since the forms are local frame invariant this decomposition is objective.

Global Special Relativity 1905 AKA SR is when B = 0 everywhere-when.

Note that the (curved) metric has linear in B "elastic" terms and nonlinear quadratic in B "plastic" terms (H. Kleinert), i.e.,

(curved metric) = (flat metric) + 1(flat metric)B + B(flat metric)1 + B(flat metric)B

The B^2 terms show that the gravity field is self-interacting like the SU(2) & SU(3) gauge fields, but unlike the U(1) Maxwell EM field.

The Cartan exterior derivative operator d on forms generalizes the gradient, curl and divergence. Together with its dual boundary operator & on co-forms, there is a generalization of Stokes & Gauss's theorems to N-dimensional manifold integrations with multiple-connectivity (e.g. wormholes).

The p-dim form |p) is the thing integrated. The dual co-form (p| is the manifold on which the integral is done. I use a variation on the Dirac bra-ket notation.

The basic integration theorem, is like the adjoint operation in quantum theory, i.e.

(&(p+1)|p) = (p+1|d|p)

The two identities

d^2 = 0

&^2 = 0

are analogous to the antisymmetric Pauli exclusion principle in quantum field theory where

a^2 = 0

a*^2 = 0

a* creates a fermion, a destroys a fermion.

However, we use the notations ‘d’ and ‘&” partially introduced by John Baez on p. 130 of his book, where he writes:

Ex. 1:

‘dtheta’ = (xdy – ydx)/(x^2 + y^2)

for the polar angle “theta” where

dr = (xdy + ydx)/(x^2 + y^2)

x = rcos(theta)

y = rsin(theta)

The 1-form ‘dtheta’ above is closed, but not exact. In effect this means

d’d’ =/= 0

ONLY when the integral is over a non-bounding co-form (AKA non-bounding cycle).

Therefore, for this particular example, there is phase (theta) ambiguity at the origin r = 0. When the closed loop integral

(&’2|’d’theta’) = (2’|d’dtheta’) =/= 0

encircles the origin r = 0 it does not vanish. Note that if the closed loop integral of ‘dtheta’ does not encircle the branch point r = 0, it will vanish. In this sense, ‘dtheta’ is closed, but not exact and &2’ is not a true boundary because of the “hole” at r = 0. Note that the co-form (2’| is the area enclosed by the loop &’2 minus the “hole” at r = 0. If we extend this to cylindrical coordinates, then we have a vortex core string provided we have a local U(1) complex order parameter PSI(r,theta,z) such that

PSI(0,theta,z) = 0

PSI(r,theta,z) = PSI(r, theta + 2pi, z)

for equilibrium “stationary states” when the closed system relaxes expelling excess flux in the Meissner effect.

In that case,

(&’2|’d’theta’) = (2’|d’dtheta’) = 2piN

N = +-1, +-2 ….

N = winding number around the string vortex core on the z-axis.

To review, the rigorous theorem is

(&(p+1)|p) = (p+1|dp)

Where

(&(p+1)| is a true boundary, which means

(&^2(p+1)| = 0

When |p) is exact, that means

|p) = |d(p-1))

and

|dp) = |d^2(p-1)) = 0

However, when the topology of the co-form manifold is multiply connected we can have closed p-manifolds, AKA “non-bounding p-cycles”, (&’(p+1)| that are not true boundaries together with non-exact p-forms |d’(p-1)) such that

(&’(p+1)|d’(p-1) = (p+1’|dd’(p-1)) =/= 0

The non-bounding p-cycles are p-dim wormhole mouths or “Star Gate Portals” that are “Through The Looking Glass” Darkly as it were, down the Rabbit Hole in Hyperspace.

Ex. 2:

Consider the 3D space-like metric of a static spherically symmetric non-rotating uncharged wormhole Star Gate is

(3-metric) = dr^2 + f(r)^2(dtheta^2 + sin^2thetadphi^2)

Where f(r) is the wormhole shape function. Each wormhole mouth looks like a closed spherical surface of radius R where

R = f(r*)

df(r*)/dr = 0

d^2f(r*)/dr^2 > 0

This closed S2 surface is not a complete boundary (&3| enclosing a 3-space because it has a twin wormhole mouth somewhere-when else perhaps in a parallel universe next door in hyperspace. Therefore, all wormhole mouths for actual time travel to distant places in negligible proper time for the traveler are really (&’3| not (&3|. Furthermore, the curved tetrad field B = (hG/c^3)^1/2’d’(Goldstone Phase) is not exact, i.e. in a 1-D loop around the wormhole mouth S2 surface with the multiply-connected quasi Stoke’s theorem

(1’|B) = (&’3|dB) =/= 0

This is the curved tetrad flux through the closed loop that “cuts” the spherical wormhole mouth.

Ex. 3:

Given in cylindrical coordinates the vortex string along the z-axis

‘d’theta = (xdy – ydx)/(x^2 + y^2)

For any closed loop

(1’| = (&’2|

around the z-axis

(1’|’d’theta) = (2’|d’d’theta) =/= 0

i.e. Nonlocal Bohm-Aharonov “Flux without flux”

Given the above wormhole 3-metric define a Hodge * operation, with the non-exact 2-form

3*’d’theta = (xdy/\dz + ydz/\dx + zdx/\dy)/f(r)^3

Where now we have a multiply-connected quasi-divergence Gauss theorem

(3’|d3*’d’theta) = (&’3|3*’d’theta)=/= 0

when (&’3| is a wormhole portal. There is now a radial 3*’d’theta flux through the wormhole closed surface in addition to the ‘d’theta circulation around a closed loop that cuts the wormhole closed surface that is not a complete boundary.
Unified Field Theory, Warp Drive & Star Gate Time Travel

Super Cosmos is now at http://amazon.com
These rough notes below to appear (refined) in book Star Gate: Metric Engineering Warp Drive for Wormhole Space & Time Travel.

Lecture 3 Gravity energy is nonlocal and Yilmaz et-al are wrong.

The Bianchi identities generalize d^2 = 0 to D^2 = 0 when there are gauge field connections A, W etc. present.

For example:

D = d + A/\

for SU(2) & SU(3) internal symmetry non-Abelian Yang-Mills gauge force models of the parity-violating weak hypercharge and parity-conserving strong gluon forces respectively in the standard model. It is the weak hypercharge group SU(2) that is spontaneously broken to produce the small inertia of the leptons and quarks and to produce emergent gravity via my original formula

B = (hG/c^3)^1/2d(Goldstone Phase)

Here Goldstone Phase is the "mean" SU(2) hypercharge phase from using the Trace operation (details later on).

The large mass of the hadrons comes from the kinetic energy of the confined quarks in dark energy bags.

In 1915 General Relativity, locally gauging T4 -> Diff(4):

D = d + W/
Where the spin connection W obeys

T = De = d(1 + B) + W/\(1 + B) = 0

Therefore

dB + W/\(1 + B) = 0

B is the curvature tetrad field that corresponds to disclination topological defects in the Vacuum ODLRO Manifold G/H and in Hagen Kleinert's "World Crystal Lattice".

Gennady Shipov's torsion field is when the Lorentz group is locally gauged in addition to T4. This adds 6 new scalar fields that act like extra space dimensions, i.e. 10D space-time. Just as locally gauging T4 brings in the compensating 1-form curvature disclination connection B, provided SU(2) hypercharge spontaneously breaks in the inflation, so does locally gauging the Lorentz group O(1,3) bring in the compensating 1-form torsion dislocation connection S where the non-vanishing torsion 2-form T is now

T = dS + W/\S + S/\(1 + B + S)

Note the torsion-curvature coupling terms W/\S and S/\B.

The 1915 GR curvature 2-form is

R = DW = dW + W/\W

The Einstein-Hilbert dark energy vacuum action is the 4-form

R/\(1 + B)/\(1 + B)+ /\zpf(1 + B)/\(1 + B)/\(1 + B)/\(1 + B)

The 1915 Bianchi identity is, for /\zpf = 0

DR = 0

i.e.

(d + W/\)R = dR + W/\R = 0

The Einstein field equation is

D*R(Geometry) = *J(Matter)

D^2R = D*J(Matter) = 0

is the local conservation stress-energy current density.

Note

D*J(Matter) = d*J(matter) + W/\*J(Matter)

W/\*J(matter) = d*j(vacuum-matter)

*j(vacuum-matter) is a 3-form that is the pseudo-tensor of the (matter-vacuum coupling)

Nonlocality of total gravity energy is multiply-connected "Flux without flux", i.e.

(3|D*R) =/= 0

When *J(matter) = 0 everywhere-when

i.e.

*R = *"D"W

D^2 = 0

but

D"D" =/= 0

i.e. Flux without flux from multiple connectivity of 3-Manifold slices of space-time.

If no S, then /\zpf is really a constant, i.e. "Cosmological Constant" made small by vacuum ODLRO. When S =/= 0, then /\zpf is a local scalar field, indeed the one we need for metric engineering warp and wormhole with the Josephson homodyne detection method.

This is in complete analogy to Maxwell's U(1) equations

dF = 0

d*F = *J

d^2*F = d*J = 0

And the Yang-Mills SU(2) & SU(3) equations

DF = 0

D = d + A/
D*F = *J

D^2F = D*J = 0

The complete theory with torsion and U(1)xSU(2)xSU(3) is clearly the same template where now

D' = d + W/\ + S/\ + A/\ + C/
A is U(1)xSU(2)xSU(3) or whatever internal G symmetry group works.

W & S are from locally gauging the 10-parameter space-time symmetry Poincare group.

If we go to the 15 parameter conformal group we will get an additional conformal or Penrose twistor connection. If we go to Kaluza-Klein extra-dimensions and Grassmann fermion dimensions of supersymmetry (not observed) then everything generalizes in the same template. Now however

R' = D'(W + S + C + A ...)

D'R' = 0

D'*R' = *J'

As well as the Yang-Mills equations

F' = D'A

DF' = 0

D*F' = *j

etc.

Similarly for S, B & C?

On Sep 29, 2005, at 2:37 PM, Jack Sarfatti wrote:

Lecture 2 (Revised Draft #2)

Ex 4
Mechanical Model of a Phase Singularity

Simplest case.

Imagine a plane. Pick an origin O. Use polar coordinates, (r,theta) for arbitrary moving point P.

Pick a point 0' with fixed coordinates (a, chi).

Draw a circle of radius b < a centered at O' with coordinates (b,phi)

Let point P move around this circle whose center O' is displaced from origin O.

Obviously when a =/= 0 the total theta angle integral of the 1-form dtheta
swept out in one complete circuit round the circle is ZERO. Basically theta oscillates.

Note that the angle theta depends on the angles chi and phi.

Half of the movement is clockwise and then counter-clockwise for dtheta on successive semicircles as P winds around the circumference of the displaced circle. This is most easily seen intuitively all at once when O' is vertical compared to O (on y-axis ordinate).

Note what happens when you move the circle to different locations on the plane.

Draw tangents from O to the circle in different locations.

In contrast, when a = 0, or alternatively, b > a the total angle integral of dtheta is 2pi.

Homework Problem

Use trigonometry to make an algebraic proof.

-------------------

For 3D flat metric, the Hodge * is with the right-hand rule convention

*dx/\dy = dz
*dy/\dz = dx
*dz/\dx = dy

Left-hand rule is

*dy/\dx = dz
*dz/\dy = dx
*dx/\dz = dy

Parity transformation interchanges left and right hand rules in 3D.

(x,y,z) -> (-x,-y,-z)

SU(2)hypercharge breaks parity symmetry and it also may be the origin of inertia and gravity.

p-forms |p) = (p!)^-1Fuv ... dx^u/\dx^v ...

p-factors, p =< n = dim of manifold.

Important formula

|p)/\|q) = (-1)^(pq)|q)/\|p)

The exterior product /\ of forms is a parallelepiped in the co-tangent n-dim space of constant phase wave fronts in contrast to the tangent space of particle paths normal to the wave fronts.

For R^3

A = Axdx + Aydy + Azdz

F = dA = ( Az,y - Ay,z)dy/\dz + (Az,x - Ax,z)dz/\dx + (Ax,y - Ay,x)dx/\dy

2-form independent of metric

*F = *dA = ( Az,y - Ay,z)dx + (Az,x - Ax,z)dy + (Ax,y - Ay,x)dz

* dual 1-form in 3D manifold with a metric specified.

Note, if

A = df

F = dA = d^2f = 0

Therefore

( Az,y - Ay,z) = 0 etc

, is ordinary partial derivative

i.e. mixed second order partial derivatives of the 0-form f commute in that case.

However, in the case of a phase-singularity, there is some kind of region in the manifold where the mixed partials of the 0-form Goldstone phase of the local macro-quantum coherent vacuum order parameter Higgs field in our primary application to physics of this formalism do not commute. This is a topological defect in the vacuum manifold G/H, where I write

A = 'd'f

d'd' =/= d^2 = 0

due to multiply-connected manifolds

F = dA =/= 0

e.g. non-integrable anholonomic multi-valued gauge transformation of Hagen Kleinert

AKA

Flux without flux

see also the related idea of the nonlocal Bohm-Aharonov effect using Feynman amplitude Wilson loop operators.

In 3 space

d|0) is gradient of a function, i.e. scalar field

d|1) is curl of a vector field

d|2) is divergence of a vector field

B = Bxydx/\dy + Byzdy/\dz + Bzxdz/\dx

dB = (Bxy,z + Byz,x + Bzx,y)dx/\dy/\dz

Static 4D Metrics without gravimagnetism (non-rotating spacetimes) & without gravity waves (c = 1 convention) here

(curved metric) = g = -dt^2 + 3^g

The toy model wormhole is of this form.

We need positive dark zero point energy density with negative pressure to keep the wormhole open. There is no event horizon in this wormhole. It's not a black hole!

A metric allows the symmetric inner product { , }.

Classical energy density of the EM field in the absence of sources is

(1/2)[{E,E} + {B,B}]

The Lagrangian density is

(1/2)[{E,E} - {B,B}

E = (Ftx, Fty, Ftz) electric field

B = (Fyz, Fzx, Fxy) magnetic field

F = B + E/\dt

F & B are 2-forms

E is a 1-form

We need a classical EM stress-energy density tensor T to compute

T ~ &(Dynamical Action)/&(metric)

& is functional derivative of classical Lagrangian field theory (not particle mechanics).

w = (pressure/energy density)

Note, the above is classical without any quantum zero point fluctuations.

w = +1/3 for classical far-field radiation with only 2 transverse polarizations.

For example, the cosmic black body radiation has w = +1/3

http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_05_jpg.htm

It's wrong to use w = +1/3 for vacuum zero point energy that bends spacetime absolutely.

This is an error in SED used by HRP. The Casimir force is not important for metric engineering Weightless Warp Drive and Wormhole Time Travel to The Past (using old wormholes made at the beginning of our local universe and even connecting to the parallel universes of Super Cosmos).

Equivalence principle + local Lorentz invariance imply w = -1 for all kinds of zero point energy (isotropically distributed).

That is the Zero Point Energy Stress-Energy Current Density Tensor tuv(ZPF) diagonal is for Energy Density dE/dV & Pressure P

w(ZPF) = P/(dE/dV) = -1 from EEP & LI of GR

(dE/dV,-dE/dV,-dE/dV,dE/dV),

Therefore, since w = -1

Trace is -2(dE/dV) = (1 + 3w)(dE/dV)

Compare to EM radiation where

w = P/(dE/dV) = + 1/3

( dE/dV, +(1/3)dE/dV, +(1/3)dE/dV, +(1/3)dE/dV), i.e. Trace

= +2dE/dV = (1 + 3w)(dE/dV)

Note that dark energy is P < 0 & dE/dV > 0.

Dark Matter is P > 0 & dE/dV < 0

Assuming here above, of course, compact dark matter sources like the Galactic Halos when P > 0 and measurements by external observers.

If we stick in Casimir plates to break translational symmetry or somehow break the rotational symmetry (rotating superconducting disks that phase lock to the vacuum Goldstone phase?)

There is analogy here to homodyne detection of quantum information with continuous variables where the local oscillator in a beam splitter is like the vacuum ODLRO field. (Rev Mod Phys, p. 513, April 2005) Squeezed vacuum states in quantum optics is when one quadrature of the zero point virtual photons has less vacuum noise than does the other quadrature which has excess noise to compensate. http://www.cco.caltech.edu/~qoptics/squeeze.html

Then

( dE/dV, +a(dE/dV), + b(dE/dV), + c(dE/dV)), the trace is now (1 + a + b + c)dE/dV

1 + 3w = 1 + a + b + c

w = (a + b + c)/3

This is quintessence when w < -1/3 and it can perhaps be done with the Shipov torsion field. Phantom energy with the Big Rip tearing the fabric of the Universe apart is when w < -1.

On Sep 28, 2005, at 4:20 PM, Jack Sarfatti wrote:
Lecture 1 on Cartan Forms

I am using John Baez's Ch 4 of "Gauge Fields, Knots and Gravity" for the standard ideas.

All the local physical observables in classical gauge force field theories are examples of Eli Cartan's "differential forms", e.g., Au, Fuv, ju.

The integrals of forms over manifolds are premetrical until we define a Hodge * operation taking a p-form to a N-p form for N-dim manifolds.

The p-forms are very much like Bishop Berkeley’s “ghosts of departed quantities.” They “are neither finite, nor … infinitely small, nor yet nothing.”

The 1-forms are dual to tangent vector fields on the manifold. A vector field is like a bundle of particle paths in Bohm’s hidden variable picture of quantum theory. The 1-form (AKA “cotangent vector”) is like a stack of wave fronts (AKA “little hyperplanes”) of small extent as in Fig. 1 p. 45 (Baez) also in MTW’s “Gravitation.” “The bigger df is, the more tightly packed the hyperplanes are.” Given a Cartesian coordinate basis of tangent vector fields {,u} and a dual basis of 1-forms {dx^u}, then duality here is

dx^u,v = 1v^u = Kronecker delta NxN identity matrix.

Given a 1-form df and any vector field v, the directional real number df(v) “counts how many little hyperplanes in the stack df the vector v crosses.” Linearity is built in as a postulate. The Cartan forms are invariants of local coordinate LNIF transformations Diff(N). Diff(N) is what you get when you locally gauge the global ND translation group. In 1915 GR the Cartan forms are also invariant under the local LIF Lorentz transformations O(1,3). In general this would be O(N) pre-signature. That is TNxO(N).

/\ is the exterior product. Obviously we have a kind of quasi-algebra equivalent to dissecting an N-1 simplex or “brane” also giving partially ordered (non-Boolean?) lattices with the 0-form on bottom and the N-form on the top.

N = 1 (dx), i.e. 1

N = 2 (dx, dy, dx/\dy = -dy/\dx), i.e. 3

N = 3 (dx,dx,dz, dx/\dy, dx/\dz, dy/\dz, dx/\dy/\dz), i.e 7

N = 4 (dx,dy,dz,dt, dx/\dy, dx/\dz, dy/\dx, dt/\dx, dt/\dy,dt/\dz, dx/\dy/\dz, dt/\dx/\dy, dt/\dx/\dz, dt/\dy/\dz, dt/\dx/\dy/\dz), i.e. 15

If we include the 0-form we have 2, 4, 8, 16, i.e. 2^N elements in the quasi-algebra that suggests the Clifford Algebras. There are obviously N!/p!(N-p)! p-forms in N space. This is also like an information space of N c-Bit Shannon Boolean strings. Obviously there will be some kind of matrix representation. For example N = 2 should correspond to the 3 Paul 2x2 spin matrices with the unit matrix. Therefore, there is a connection to U(1)xSU(2) here. N = 3 should have something to do with the 8 SU(3) matrices, and N = 4 obviously connects with the Dirac algebra and possibly U(4) especially when we complexify each real number space-time dimension and even go beyond that to quaternions & octonians.

Classical gauge force theories include Maxwell's U(1) electromagnetic theory, Yang-Mills theories of the SU(2) weak and SU(3) strong forces of the leptons and quarks in the standard model and Einstein's theory of gravity (General Relativity, 1915 AKA GR) provided you do not work at the symmetric metric tensor level guv(x), but work at the "square root" 1-form tetrad "e" level. Note that Einstein's local equivalence principle is simply

(curved metric ) = e(flat metric)e

where e is the Einstein-Cartan 1-form tetrad field.

You can write

e = 1 + B

B = curvature tetrad field

Since the forms are local frame invariant this decomposition is objective.

Global Special Relativity 1905 AKA SR is when B = 0 everywhere-when.

Note that the (curved) metric has linear in B "elastic" terms and nonlinear quadratic in B "plastic" terms (H. Kleinert), i.e.,

(curved metric) = (flat metric) + 1(flat metric)B + B(flat metric)1 + B(flat metric)B

The B^2 terms show that the gravity field is self-interacting like the SU(2) & SU(3) gauge fields, but unlike the U(1) Maxwell EM field.

The Cartan exterior derivative operator d on forms generalizes the gradient, curl and divergence. Together with its dual boundary operator & on co-forms, there is a generalization of Stokes & Gauss's theorems to N-dimensional manifold integrations with multiple-connectivity (e.g. wormholes).

The p-dim form |p) is the thing integrated. The dual co-form (p| is the manifold on which the integral is done. I use a variation on the Dirac bra-ket notation.

The basic integration theorem, is like the adjoint operation in quantum theory, i.e.

(&(p+1)|p) = (p+1|d|p)

The two identities

d^2 = 0

&^2 = 0

are analogous to the antisymmetric Pauli exclusion principle in quantum field theory where

a^2 = 0

a*^2 = 0

a* creates a fermion, a destroys a fermion.

However, we use the notations ‘d’ and ‘&” partially introduced by John Baez on p. 130 of his book, where he writes:

Ex. 1:

‘dtheta’ = (xdy – ydx)/(x^2 + y^2)

for the polar angle “theta” where

dr = (xdy + ydx)/(x^2 + y^2)

x = rcos(theta)

y = rsin(theta)

The 1-form ‘dtheta’ above is closed, but not exact. In effect this means

d’d’ =/= 0

ONLY when the integral is over a non-bounding co-form (AKA non-bounding cycle).

Therefore, for this particular example, there is phase (theta) ambiguity at the origin r = 0. When the closed loop integral

(&’2|’d’theta’) = (2’|d’dtheta’) =/= 0

encircles the origin r = 0 it does not vanish. Note that if the closed loop integral of ‘dtheta’ does not encircle the branch point r = 0, it will vanish. In this sense, ‘dtheta’ is closed, but not exact and &2’ is not a true boundary because of the “hole” at r = 0. Note that the co-form (2’| is the area enclosed by the loop &’2 minus the “hole” at r = 0. If we extend this to cylindrical coordinates, then we have a vortex core string provided we have a local U(1) complex order parameter PSI(r,theta,z) such that

PSI(0,theta,z) = 0

PSI(r,theta,z) = PSI(r, theta + 2pi, z)

for equilibrium “stationary states” when the closed system relaxes expelling excess flux in the Meissner effect.

In that case,

(&’2|’d’theta’) = (2’|d’dtheta’) = 2piN

N = +-1, +-2 ….

N = winding number around the string vortex core on the z-axis.

To review, the rigorous theorem is

(&(p+1)|p) = (p+1|dp)

Where

(&(p+1)| is a true boundary, which means

(&^2(p+1)| = 0

When |p) is exact, that means

|p) = |d(p-1))

and

|dp) = |d^2(p-1)) = 0

However, when the topology of the co-form manifold is multiply connected we can have closed p-manifolds, AKA “non-bounding p-cycles”, (&’(p+1)| that are not true boundaries together with non-exact p-forms |d’(p-1)) such that

(&’(p+1)|d’(p-1) = (p+1’|dd’(p-1)) =/= 0

The non-bounding p-cycles are p-dim wormhole mouths or “Star Gate Portals” that are “Through The Looking Glass” Darkly as it were, down the Rabbit Hole in Hyperspace.

Ex. 2:

Consider the 3D space-like metric of a static spherically symmetric non-rotating uncharged wormhole Star Gate is

(3-metric) = dr^2 + f(r)^2(dtheta^2 + sin^2thetadphi^2)

Where f(r) is the wormhole shape function. Each wormhole mouth looks like a closed spherical surface of radius R where

R = f(r*)

df(r*)/dr = 0

d^2f(r*)/dr^2 > 0

This closed S2 surface is not a complete boundary (&3| enclosing a 3-space because it has a twin wormhole mouth somewhere-when else perhaps in a parallel universe next door in hyperspace. Therefore, all wormhole mouths for actual time travel to distant places in negligible proper time for the traveler are really (&’3| not (&3|. Furthermore, the curved tetrad field B = (hG/c^3)^1/2’d’(Goldstone Phase) is not exact, i.e. in a 1-D loop around the wormhole mouth S2 surface with the multiply-connected quasi Stoke’s theorem

(1’|B) = (&’3|dB) =/= 0

This is the curved tetrad flux through the closed loop that “cuts” the spherical wormhole mouth.

Ex. 3:

Given in cylindrical coordinates the vortex string along the z-axis

‘d’theta = (xdy – ydx)/(x^2 + y^2)

For any closed loop

(1’| = (&’2|

around the z-axis

(1’|’d’theta) = (2’|d’d’theta) =/= 0

i.e. Nonlocal Bohm-Aharonov “Flux without flux”

Given the above wormhole 3-metric define a Hodge * operation, with the non-exact 2-form

3*’d’theta = (xdy/\dz + ydz/\dx + zdx/\dy)/f(r)^3

Where now we have a multiply-connected quasi-divergence Gauss theorem

(3’|d3*’d’theta) = (&’3|3*’d’theta)=/= 0

when (&’3| is a wormhole portal. There is now a radial 3*’d’theta flux through the wormhole closed surface in addition to the ‘d’theta circulation around a closed loop that cuts the wormhole closed surface that is not a complete boundary.

## Thursday, September 29, 2005

Quintessence

Change

That is the ZPF stress-energy diagonal is for pressure P

(P,-P,-P,-P) i.e. Trace is -2P

If we stick in plates or somehow break the rotational symmetry (rotating superconducting disks that phase lock to the vacuum Goldstone phase?)

Then

(-P, +aP, + bP, + cP), the trace is now (1 - a - b - c)P

This is quintessence and it can perhaps be done with the Shipov torsion field.

to

That is the ZPF stress-energy diagonal is for pressure Energy Density dE/dV & Pressure P

w(ZPF) = P/(dE/dV) = -1 from EEP of GR

(dE/dV,-dE/dV,-dE/dV,dE/dV), i.e. Trace is -2(dE/dV) = (1 + 3w)(dE/dV)

Compare to EM radiation where

w = P/(dE/dV) = + 1/3

( dE/dV, +(1/3)dE/dV, +(1/3)dE/dV, +(1/3)dE/dV), i.e. Trace = +2dE/dV = (1 + 3w)(dE/dV)

Note that dark energy is P < 0 & dE/dV > 0.

If we stick in plates or somehow break the rotational symmetry (rotating superconducting disks that phase lock to the vacuum Goldstone phase?)

Then

( dE/dV, +a(dE/dV), + b(dE/dV), + c(dE/dV)), the trace is now (1 + a + b + c)dE/dV

1 + 3w = 1 + a + b + c

w = (a + b + c)/3

This is quintessence when w < -1/3 and it can perhaps be done with the Shipov torsion field.
No-cloning theorem limits

The no-cloning theorem is misleading. I mean, yes you cannot clone arbitrary quantum states perfectly, but you can clone arbitrary quantum states approximately with good enough overlap fidelity among the clones. This opens the door to presponse signal nonlocality. Gaussian continuous variable states in particular can be effectively cloned. It may mean an error in the Susskind-Hawking ideas about information recovery from evaporating black holes that depends upon naive use of the no-cloning theorem.
Stability of Matter in the Expanding Universe

More potential WMD danger in the Woodward, Puthoff & Shipov approach to change m if e/m ratios change in the process.

Woodward using Mach's Principle

Puthoff using PV (won't work fortunately)

Shipov using torsion (may work - dangerous)

Their schemes could increase e/m and increase the repulsive (J/mcr)^2 faster than the attractive - e^2/mc^2r in the case of the stability of the hydrogen atom studied in the paper by Price.

V = @hc/mr + e^2/mr + (J/mr)^2 + /\zpfr^2

see Price's eq (11) p.5 - note his r^2 term - interesting

r = a(t)R

Price's eq. (1)

http://www.arxiv.org/pdf/gr-qc/0508052

Internal dark energy core /\zpf may adjust to prevent the electrons and quarks as thin extended shells of charge from exploding. The atoms will still explode, which is bad enough. For hydrogen atom, m is reduced mass ~ electron mass with orbital angular momentum L

V/mc^2 = - e^2/mc^2r + (L/mcr)2 + /\r^2

Compare the Coulomb attraction between electron & proton with the centrifugal barrier

As m is decreased the repulsive centrifugal potential scales as 1/m^2 whilst the Coulomb attraction scales only as 1/m. Therefore the danger of instability grows if m is decreased to get NON-GEODESIC PROPELLANTLESS PROPULSION the way Woodward, Puthoff & Shipov all seem to want to do via different schemes.

Even in my GEODESIC WWD (Weightless Warp Drive) PROPELLANTLESS PROPULSION from tweaking the Goldstone Phase of the Higgs Field one must be careful not to disturb m on the scale of the atoms.

Begin forwarded message:

From: "newsletter@newscientist.com"
Date: September 29, 2005 10:25:12 AM PDT
To:
Subject: Expanding waistline? Don't blame the expanding universe
Reply-To: "newsletter@newscientist.com"

This week's top stories from the web's No.1 science and technology news service
29 September 2005
Welcome to the New Scientist newsletter, which this week reveals why we can't blame our expanding waistlines on the expansion of the universe, the vibrating clothes that help sports stars hone their skills, and the pill-sized camera that crawls through your intestine..Anil Ananthaswamy, Physical Sciences News Editor, Print Edition
Expanding Problem
Is your waistline spreading? Unfortunately you can no longer use the expansion of the universe as an excuse. While some things, such as clusters of galaxies, are known to stretch as the universe expands, physicists assumed that others, such as people, do not. But until now no one was sure why. It turns out that as long as the force - electromagnetic or gravitational - holding objects together exceeds a certain critical value, the expansion of the universe has no effect on them. This means that while you can't blame your bulging waistline on an expanding universe, at least we're not all about to be pulled apart...

http://www.newscientist.com/article.ns?id=mg18825194.800
Mechanical Model of Simple Phase Singularity

Simplest case.

Imagine a plane. Pick an origin O. Use polar coordinates, (r,theta) for arbitrary moving point P.

Pick a point 0' with fixed coordinates (a, chi).

Draw a circle of radius b < a centered at O' with coordinates (b,phi)

Let point P move around this circle whose center O' is displaced from origin O.

Obviously when a =/= 0 the total theta angle integral of the 1-form dtheta
swept out in one complete circuit round the circle is ZERO. Basically theta oscillates.

Note that the angle theta depends on the angles chi and phi.

Half of the movement is clockwise and then counter-clockwise for dtheta on successive semi-circles as P moves around the circumference of the displaced circle. This is most easily seen intuitively all at once when O' is vertical compared to O (on y-axis ordinate).

Note what happens when you move the circle to different locations on the plane.

Draw tangents from O to the circle in different locations.

In contrast, when a = 0, or alternatively, b > a the total angle integral of dtheta is 2pi.

Homework Problem

Use trigonometry to make an algebraic proof.

## Wednesday, September 28, 2005

Metric Engineering Star Gates

Lectures 2 & 1

Ex 4
Mechanical Model of a Phase Singularity

Simplest case.

Imagine a plane. Pick an origin O. Use polar coordinates, (r,theta) for arbitrary moving point P.

Pick a point 0' with fixed coordinates (a, chi).

Draw a circle of radius b < a centered at O' with coordinates (b,phi)

Let point P move around this circle whose center O' is displaced from origin O.

Obviously when a =/= 0 the total theta angle integral of the 1-form dtheta
swept out in one complete circuit round the circle is ZERO. Basically theta oscillates.

Note that the angle theta depends on the angles chi and phi.

Half of the movement is clockwise and then counter-clockwise for dtheta on successive semicircles as P winds around the circumference of the displaced circle. This is most easily seen intuitively all at once when O' is vertical compared to O (on y-axis ordinate).

Note what happens when you move the circle to different locations on the plane.

Draw tangents from O to the circle in different locations.

In contrast, when a = 0, or alternatively, b > a the total angle integral of dtheta is 2pi.

Homework Problem

Use trigonometry to make an algebraic proof.

-------------------

For 3D flat metric, the Hodge * is with the right-hand rule convention

*dx/\dy = dz
*dy/\dz = dx
*dz/\dx = dy

Left-hand rule is

*dy/\dx = dz
*dz/\dy = dx
*dx/\dz = dy

Parity transformation interchanges left and right hand rules in 3D.

(x,y,z) -> (-x,-y,-z)

SU(2)hypercharge breaks parity symmetry and it also may be the origin of inertia and gravity.

p-forms |p) = (p!)^-1Fuv ... dx^u/\dx^v ...

p-factors, p =< n = dim of manifold.

Important formula

|p)/\|q) = (-1)^(pq)|q)/\|p)

The exterior product /\ of forms is a parallelepiped in the co-tangent n-dim space of constant phase wave fronts in contrast to the tangent space of particle paths normal to the wave fronts.

For R^3

A = Axdx + Aydy + Azdz

F = dA = ( Az,y - Ay,z)dy/\dz + (Az,x - Ax,z)dz/\dx + (Ax,y - Ay,x)dx/\dy

2-form independent of metric

*F = *dA = ( Az,y - Ay,z)dx + (Az,x - Ax,z)dy + (Ax,y - Ay,x)dz

* dual 1-form in 3D manifold with a metric specified.

Note, if

A = df

F = dA = d^2f = 0

Therefore

( Az,y - Ay,z) = 0 etc

, is ordinary partial derivative

i.e. mixed second order partial derivatives of the 0-form f commute in that case.

However, in the case of a phase-singularity, there is some kind of region in the manifold where the mixed partials of the 0-form Goldstone phase of the local macro-quantum coherent vacuum order parameter Higgs field in our primary application to physics of this formalism do not commute. This is a topological defect in the vacuum manifold G/H, where I write

A = 'd'f

d'd' =/= d^2 = 0

due to multiply-connected manifolds

F = dA =/= 0

e.g. non-integrable anholonomic multi-valued gauge transformation of Hagen Kleinert

AKA

Flux without flux

see also the related idea of the nonlocal Bohm-Aharonov effect using Feynman amplitude Wilson loop operators.

In 3 space

d|0) is gradient of a function, i.e. scalar field

d|1) is curl of a vector field

d|2) is divergence of a vector field

B = Bxydx/\dy + Byzdy/\dz + Bzxdz/\dx

dB = (Bxy,z + Byz,x + Bzx,y)dx/\dy/\dz

Static 4D Metrics without gravimagnetism (non-rotating spacetimes) & without gravity waves (c = 1 convention) here

(curved metric) = g = -dt^2 + 3^g

The toy model wormhole is of this form.

We need positive dark zero point energy density with negative pressure to keep the wormhole open. There is no event horizon in this wormhole. It's not a black hole!

A metric allows the symmetric inner product { , }.

Classical energy density of the EM field in the absence of sources is

(1/2)[{E,E} + {B,B}]

The Lagrangian density is

(1/2)[{E,E} - {B,B}

E = (Ftx, Fty, Ftz) electric field

B = (Fyz, Fzx, Fxy) magnetic field

F = B + E/\dt

F & B are 2-forms

E is a 1-form

We need a classical EM stress-energy density tensor T to compute

T ~ &(Dynamical Action)/&(metric)

& is functional derivative of classical Lagrangian field theory (not particle mechanics).

w = (pressure/energy density)

Note, the above is classical without any quantum zero point fluctuations.

w = +1/3 for classical far-field radiation with only 2 transverse polarizations.

For example, the cosmic black body radiation has w = +1/3

http://www-conf.slac.stanford.edu/ssi/2005/lec_notes/Kolb1/kolb1new_Page_05_jpg.htm

It's wrong to use w = +1/3 for vacuum zero point energy that bends spacetime absolutely.

This is an error in SED used by HRP. The Casimir force is not important for metric engineering.

Equivalence principle + local Lorentz invariance imply w = -1 for all kinds of zero point energy (isotropically distributed).

That is the ZPF stress-energy diagonal is for pressure P

(P,-P,-P,-P) i.e. Trace is -2P

If we stick in plates or somehow break the rotational symmetry (rotating superconducting disks that phase lock to the vacuum Goldstone phase?)

Then

(-P, +aP, + bP, + cP), the trace is now (1 - a - b - c)P

This is quintessence and it can perhaps be done with the Shipov torsion field.

On Sep 28, 2005, at 4:20 PM, Jack Sarfatti wrote:

Lecture 1 on Cartan Forms

I am using John Baez's Ch 4 of "Gauge Fields, Knots and Gravity" for the standard ideas.

All the local physical observables in classical gauge force field theories are examples of Eli Cartan's "differential forms", e.g., Au, Fuv, ju.

The integrals of forms over manifolds are premetrical until we define a Hodge * operation taking a p-form to a N-p form for N-dim manifolds.

The p-forms are very much like Bishop Berkeley’s “ghosts of departed quantities.” They “are neither finite, nor … infinitely small, nor yet nothing.”

The 1-forms are dual to tangent vector fields on the manifold. A vector field is like a bundle of particle paths in Bohm’s hidden variable picture of quantum theory. The 1-form (AKA “cotangent vector”) is like a stack of wave fronts (AKA “little hyperplanes”) of small extent as in Fig. 1 p. 45 (Baez) also in MTW’s “Gravitation.” “The bigger df is, the more tightly packed the hyperplanes are.” Given a Cartesian coordinate basis of tangent vector fields {,u} and a dual basis of 1-forms {dx^u}, then duality here is

dx^u,v = 1v^u = Kronecker delta NxN identity matrix.

Given a 1-form df and any vector field v, the directional real number df(v) “counts how many little hyperplanes in the stack df the vector v crosses.” Linearity is built in as a postulate. The Cartan forms are invariants of local coordinate LNIF transformations Diff(N). Diff(N) is what you get when you locally gauge the global ND translation group. In 1915 GR the Cartan forms are also invariant under the local LIF Lorentz transformations O(1,3). In general this would be O(N) pre-signature. That is TNxO(N).

/\ is the exterior product. Obviously we have a kind of quasi-algebra equivalent to dissecting an N-1 simplex or “brane” also giving partially ordered (non-Boolean?) lattices with the 0-form on bottom and the N-form on the top.

N = 1 (dx), i.e. 1

N = 2 (dx, dy, dx/\dy = -dy/\dx), i.e. 3

N = 3 (dx,dx,dz, dx/\dy, dx/\dz, dy/\dz, dx/\dy/\dz), i.e 7

N = 4 (dx,dy,dz,dt, dx/\dy, dx/\dz, dy/\dx, dt/\dx, dt/\dy,dt/\dz, dx/\dy/\dz, dt/\dx/\dy, dt/\dx/\dz, dt/\dy/\dz, dt/\dx/\dy/\dz), i.e. 15

If we include the 0-form we have 2, 4, 8, 16, i.e. 2^N elements in the quasi-algebra that suggests the Clifford Algebras. There are obviously N!/p!(N-p)! p-forms in N space. This is also like an information space of N c-Bit Shannon Boolean strings. Obviously there will be some kind of matrix representation. For example N = 2 should correspond to the 3 Paul 2x2 spin matrices with the unit matrix. Therefore, there is a connection to U(1)xSU(2) here. N = 3 should have something to do with the 8 SU(3) matrices, and N = 4 obviously connects with the Dirac algebra and possibly U(4) especially when we complexify each real number space-time dimension and even go beyond that to quaternions & octonians.

Classical gauge force theories include Maxwell's U(1) electromagnetic theory, Yang-Mills theories of the SU(2) weak and SU(3) strong forces of the leptons and quarks in the standard model and Einstein's theory of gravity (General Relativity, 1915 AKA GR) provided you do not work at the symmetric metric tensor level guv(x), but work at the "square root" 1-form tetrad "e" level. Note that Einstein's local equivalence principle is simply

(curved metric ) = e(flat metric)e

where e is the Einstein-Cartan 1-form tetrad field.

You can write

e = 1 + B

B = curvature tetrad field

Since the forms are local frame invariant this decomposition is objective.

Global Special Relativity 1905 AKA SR is when B = 0 everywhere-when.

Note that the (curved) metric has linear in B "elastic" terms and nonlinear quadratic in B "plastic" terms (H. Kleinert), i.e.,

(curved metric) = (flat metric) + 1(flat metric)B + B(flat metric)1 + B(flat metric)B

The B^2 terms show that the gravity field is self-interacting like the SU(2) & SU(3) gauge fields, but unlike the U(1) Maxwell EM field.

The Cartan exterior derivative operator d on forms generalizes the gradient, curl and divergence. Together with its dual boundary operator & on co-forms, there is a generalization of Stokes & Gauss's theorems to N-dimensional manifold integrations with multiple-connectivity (e.g. wormholes).

The p-dim form |p) is the thing integrated. The dual co-form (p| is the manifold on which the integral is done. I use a variation on the Dirac bra-ket notation.

The basic integration theorem, is like the adjoint operation in quantum theory, i.e.

(&(p+1)|p) = (p+1|d|p)

The two identities

d^2 = 0

&^2 = 0

are analogous to the antisymmetric Pauli exclusion principle in quantum field theory where

a^2 = 0

a*^2 = 0

a* creates a fermion, a destroys a fermion.

However, we use the notations ‘d’ and ‘&” partially introduced by John Baez on p. 130 of his book, where he writes:

Ex. 1:

‘dtheta’ = (xdy – ydx)/(x^2 + y^2)

for the polar angle “theta” where

dr = (xdy + ydx)/(x^2 + y^2)

x = rcos(theta)

y = rsin(theta)

The 1-form ‘dtheta’ above is closed, but not exact. In effect this means

d’d’ =/= 0

ONLY when the integral is over a non-bounding co-form (AKA non-bounding cycle).

Therefore, for this particular example, there is phase (theta) ambiguity at the origin r = 0. When the closed loop integral

(&’2|’d’theta’) = (2’|d’dtheta’) =/= 0

encircles the origin r = 0 it does not vanish. Note that if the closed loop integral of ‘dtheta’ does not encircle the branch point r = 0, it will vanish. In this sense, ‘dtheta’ is closed, but not exact and &2’ is not a true boundary because of the “hole” at r = 0. Note that the co-form (2’| is the area enclosed by the loop &’2 minus the “hole” at r = 0. If we extend this to cylindrical coordinates, then we have a vortex core string provided we have a local U(1) complex order parameter PSI(r,theta,z) such that

PSI(0,theta,z) = 0

PSI(r,theta,z) = PSI(r, theta + 2pi, z)

for equilibrium “stationary states” when the closed system relaxes expelling excess flux in the Meissner effect.

In that case,

(&’2|’d’theta’) = (2’|d’dtheta’) = 2piN

N = +-1, +-2 ….

N = winding number around the string vortex core on the z-axis.

To review, the rigorous theorem is

(&(p+1)|p) = (p+1|dp)

Where

(&(p+1)| is a true boundary, which means

(&^2(p+1)| = 0

When |p) is exact, that means

|p) = |d(p-1))

and

|dp) = |d^2(p-1)) = 0

However, when the topology of the co-form manifold is multiply connected we can have closed p-manifolds, AKA “non-bounding p-cycles”, (&’(p+1)| that are not true boundaries together with non-exact p-forms |d’(p-1)) such that

(&’(p+1)|d’(p-1) = (p+1’|dd’(p-1)) =/= 0

The non-bounding p-cycles are p-dim wormhole mouths or “Star Gate Portals” that are “Through The Looking Glass” Darkly as it were, down the Rabbit Hole in Hyperspace.

Ex. 2:

Consider the 3D space-like metric of a static spherically symmetric non-rotating uncharged wormhole Star Gate is

(3-metric) = dr^2 + f(r)^2(dtheta^2 + sin^2thetadphi^2)

Where f(r) is the wormhole shape function. Each wormhole mouth looks like a closed spherical surface of radius R where

R = f(r*)

df(r*)/dr = 0

d^2f(r*)/dr^2 > 0

This closed S2 surface is not a complete boundary (&3| enclosing a 3-space because it has a twin wormhole mouth somewhere-when else perhaps in a parallel universe next door in hyperspace. Therefore, all wormhole mouths for actual time travel to distant places in negligible proper time for the traveler are really (&’3| not (&3|. Furthermore, the curved tetrad field B = (hG/c^3)^1/2’d’(Goldstone Phase) is not exact, i.e. in a 1-D loop around the wormhole mouth S2 surface with the multiply-connected quasi Stoke’s theorem

(1’|B) = (&’3|dB) =/= 0

This is the curved tetrad flux through the closed loop that “cuts” the spherical wormhole mouth.

Ex. 3:

Given in cylindrical coordinates the vortex string along the z-axis

‘d’theta = (xdy – ydx)/(x^2 + y^2)

For any closed loop

(1’| = (&’2|

around the z-axis

(1’|’d’theta) = (2’|d’d’theta) =/= 0

i.e. Nonlocal Bohm-Aharonov “Flux without flux”

Given the above wormhole 3-metric define a Hodge * operation, with the non-exact 2-form

3*’d’theta = (xdy/\dz + ydz/\dx + zdx/\dy)/f(r)^3

Where now we have a multiply-connected quasi-divergence Gauss theorem

(3’|d3*’d’theta) = (&’3|3*’d’theta)=/= 0

when (&’3| is a wormhole portal. There is now a radial 3*’d’theta flux through the wormhole closed surface in addition to the ‘d’theta circulation around a closed loop that cuts the wormhole closed surface that is not a complete boundary.
Discussion with Gennady Shipov in Moscow on Warp Drive

Thanks Gennady. You are like Schwinger I am like Feynman. I mean it's too hard for me to read through that archaic Schouten notation you use. The Cartan notation is like Feynman diagrams, one can see the point almost instantly without spending a lot of time with complicated notation.

Also your model

1. Does not explicitly have local vacuum order parameters out of which all the classical math emerges.

2. Although torsion fields can change the rest masses of elementary particles we really do not want to do that because changing e/m will explode the ship and possibly destroy not only the planet, but even the whole universe (e.g. Martin Rees, Ch 11 "Our Final Hour").

3. The key idea is to change the timelike geodesic glide path of the ship of mass M FROM the ship without changing M at all! M should cancel out of the problem. If not, it is a non-geodesic drive with g-force subject to the WMD instability above (see also Sir Martin Rees "Just Six Numbers").

I also tried to explain this to James Woodward, Hal Puthoff and Eric Davis, but they do not seem able to understand the danger of what they are trying to do. Fortunately, I don't think their physics is correct so we are not in any danger. However, your physics is essentially correct, but simply stops and does not go to "subspace" or the "substratum" out of which your equations EMERGE from:

B = (hG/c^3)^1/2'd'(Goldstone Phase)

= compensating gauge potential from T4 -> Diff(4) that also needs the Goldstone phase

i.e. BOTH local gauging of a space-time symmetry T4 and spontaneous breakdown of the internal symmetry SU(2) hypercharge is the mechanism for inflation and the low entropy of the early universe.

T = (d + W/\ + S/\)(1 + B + S) = dS + W/\S + S/\(1 + B + S) = (hG/c^3)d'd'(Goldstone Phase)

T = torsion field 2-form

S = torsion potential 1-form from locally gauging O(1,3) to get the extra 6 scalar fields (dimensions of "oriented point")

T =/= 0 when S =/= 0

On Sep 27, 2005, at 9:45 AM, Gennady Shipov wrote:

On Sep 27, 2005, at 11.39 AM, Jack Sarfatti wrote:

Jack Sarfatti

> 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.

Gennady Shipov
Yes Jack!

In the attachment I have presented the reasons about connection of torsion with a "dark matter".

Gennady.

Jack Sarfatti

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.
>
> On Sep 26, 2005, at 9:14 PM, Jack Sarfatti wrote:
>
> > The classical far-field transverse EM field stress-energy tensor is
> > traceless. Therefore, for a random SED EM field
> >
> > (Energy Density) + 3(Pressure) = 0
> >
> > w = +1/3
> >
> > QED
> >
> > This is indeed what Hal Puthoff assumed in his charge cluster paper.
> >
> > It is clearly wrong as shown below.
> >
> > The above stress-tensor is not a complete description. One must add
> >
> > (c^4/8piG)/\zpfguv(x)
> >
> > If there is no torsion /\zpf = constant.
> >
> > In QED /\zpf is infinite.
> >
> > It's a fudge to subtract. You can't subtract it in GR! If you put
> > in a quantum gravity cutoff then
> >
> > /\zpf = 1/Lp^2 = (c^3/hG) = 10^+66 cm^-2
> >
> > This is the cosmological constant problem.
> >
> > Note that traceguv = -2 for +--- signature
> >
> > This corresponds to w = -1
> >
> > i.e. energy density - 3(pressure) = -2(energy density)
> >
> > Therefore, you cannot consistently use the classical
> >
> > Tuv(EM) = -FuwF^wv - guvL
> >
> > in curved spacetime to get the effect of the virtual photons.
> >
> > We need to distinguish the zero point virtual photons that are on-
> > mass-shell in the purely QED Casimir force calculations from the
> > coherent states of off-mass-shell virtual photons that make up near
> > EM fields.
> >
> > PS Note that the BCS vacuum coherent state |0'> is
> >
> > |0'> = Product over modes k (uk + vkbk*)|0>
> >
> > where bk* creates a bound virtual electron-positron pair +k & - k
> > (may not be a singlet). Since this is neutral there is no problem
> > of charge conservation. We can extend this idea to virtual quark-
> > antiquark pairs.
> >
> > Note that zero point quantum gravity terms c^2/\zpfr^2 are like
> > "bag potentials" that confine quarks. With torsion fields /\zpf is
> > a local variable scalar Higgs field.
> >
> > On Sep 26, 2005, at 11:18 AM, Jack Sarfatti wrote:
> >
> >
> >> 1. Lorentz invariance and the equivalence principle imply
> >>
> >> w = -1
> >>
> >> for all zero point fluctuations not only for virtual photons.
> >>
> >> 2. Unlike QED you cannot simply stick in a plate(s) as in Casimir
> >> force and forget the absolute background of ZPE. This is precisely
> >> the "cosmological constant problem."
> >>
> >> i.e. zero point energy gravitates (and anti-gravitates depending)
> >> absolutely.
> >>
> >> The QED Casimir force calculation for virtual photons (excluding
> >> even vacuum polarization, which makes no essential difference
> >> here) is irrelevant to the "charge cluster", or, more
> >> fundamentally, the stability of a single electron as an extended
> >> micro-geon. More precisely the Casimir energy is merely one term
> >> of several in the effective potential for the micro-geon including
> >> gravity.
> >>
> >> i.e.
> >>
> >> V = @hc/mr + Q^2/mr + J^2/mr^2 + c^2/\zpfr^2
> >>
> >> The "gravity term" in this Newtonian approx is c^2/\zpfr^2 for a
> >> uniform core of positive ZPE of negative pressure. The Casmir term
> >> is @hc/mr that simply renormalizes the Coulomb barrier Q^2/mr.
> >> Everything Hal Puthoff talks about simply determines the pure
> >> number @ from QED.
> >>
> >> dV/dr =0
> >>
> >> d^2V/dr^2 > 0
> >>
> >> 3. The single electron Bohm hidden variable is not a point
> >> particle. Renormalization theory needs to be trashed as Feynman
> >> told me in his Cal Tech office in 1968. The electron (& quarks) is
> >> an extended shell of charge held in balance by its inner core of
> >> zero point dark energy where the vacuum coherence ODLRO field
> >> drops to zero exactly like in the core of a superfluid vortex
> >> except here we have a point defect rather than a string defect.
> >>
> >>
> >
> >
>
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

therefore

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.

e.g.

'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

Asymptotically

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

where

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

where

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)

Note that

d^2 = 0

but

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.

## Tuesday, September 27, 2005

PS on shapes of strings and wormholes

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

therefore

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)
String-Wormhole Duality in Gravity Emergent from Vacuum ODLRO

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.

e.g.

'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

Asymptotically

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

where

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

where

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)

Note that

d^2 = 0

but

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.