Saturday, October 15, 2005

Super Cosmos

Super Cosmos is now at
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)


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


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