Even before the Dirac string, you have the field outside an infinite
thin solenoid carrying a current.

Even though B = CurlA = 0 outside the solenoid, A is not = 0 and the
closed loop integral of A on a loop outside the solenoid must equal the
enclosed magnetic flux.

That this is real, is seen in the fringe shift of electron patterns in
Bohm-Aharonov experiments.

In Cartan form language

A is a 1-form (automatically Diff(4) invariant in curved space-time)

A = Audx^u

B is part of a 2-form (F includes the electric field as well).

F = Fuvdx^u/\dx^v

Nonrigorously

F = dA

d is the exterior topological derivative of Cartan

To make F a Diff(4) local frame invariant (both LIF & LNIF) replace d by
D = d + C/\ the exterior covariant derivative relative to a given 1-form
connection C (more on that elsewhere)

OK

if A = d@

@ is a 0-form scalar phase

Therefore, outside the solenoid

F = d^2@ = 0

everywhere except for a kind of Dirac string PHASE SINGULARITY or BRANCH
CUT where @ jumps by N2pi, where N is the number of Flux-without-flux
quanta.

This is true if there is some kind of order parameter

Psi(x) = |Psi(x)|e^i@(x)

that must be single-valued when x is taken around a closed loop L in
space-time.

For the solenoid let's stay inside a space-like slice for now to keep it
simple.

What we have is generalized Stoke's theorem

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

p| is a p-form

& is the boundary operator dual to the exterior derivative operator d

Now in the case of the long thin solenoid above, the Loop L outside the
solenoid is not a boundary! That is, it is a non-bounding cycle with a
non-trivial cohomology group

H1 = (All Loops)/(Bounding Loops)

(Bounding Loops) is a normal subgroup of (All Loops)

This is dual to the homology group

H^1 = (All Closed Forms)/(Exact Forms)

Essentially a closed form A defined as dA = 0 that is not exact is an
exact form

A = d@ + a PHASE SINGULARITY

If you take integral over the finite annulus with closed loops L & L'
you will get zero, because L + L' is a boundary of the enclosed finite
annulus.

But the inner loop L' integral is equal to the enclosed flux since L' is
by itself a bounding cycle of its interior where the flux really "is".

For all practical purposes however, you can throw away the real flux
inside L' and replace the imaginary string by a REAL string where PHASE
jumps! You can also require that Psi = 0 somewhere inside L' and this
makes a multiply-connected manifold for the order parameters Psi!

i.e. imaginary string PHASE SINGULARITY (JUMP) + REAL FLUX WITH FLUX is
EQUIVALENT TO

REAL string PHASE SINGULARITY (JUMP) + IMAGINARY FLUX WITHOUT FLUX

The latter picture is Wheeler-Feynman (Hoyle-Narlikar) replacement of
ALL Gauge Force Fields by STRINGS of ORDER PARAMETER PHASE JUMPS - in
effect the force fields are redundant non-dynamical degrees of freedom
- Schwinger "source" theory?



On Aug 3, 2005, at 9:34 AM, Jack Sarfatti wrote:


Note that if one does not like "VEV" (Vacuum Expectation Value) use
"ODLRO parameter", since the ODLRO parameter in thermal equilibrium
closed systems, for example, does not vanish up until some critical Tc
and Hc.

e.g. kTc ~ binding energy of pair in ODLRO condensate.

When T 0 there is some random "normal fluid".

Here the "normal fluid" is the residual random micro-quantum zpf dark
energy.

However

The Free Lunch Ansatz is

Total Global Energy over Hubble Bubble = Total Energy of Gravity
(effective B = Lpd(GoldstoneargODLRO) field) + Total Energy of |Higgs
ODLRO| + Total Energy of "Matter" + Total Zero Point Energy (summed over
all fields) = 0

For example the D^uDu(ODLRO) gradient term in the Diff(4)
Landau-Ginzburg Euler-Lagrange equation always appears with a
compensating B = Lpd(argODLRO) contribution.

That is, in the Action inhomogeneous terms like

(D^uODLRO*)(DuODLRO)

are compensated by terms in B and its derivatives giving Einstein's
Ricci scalar R for example. The important physics of the B-fields are in
the "Dirac string" GOLDSTONE PHASE SINGULARITIES inside the VACUUM
MANIFOLD G/H(unbroken) or LANDSCAPE of possible ODLRO configurations -
not all of which need be homogeneous in space-time with constant
homogeneous Goldstone phase.

On Aug 3, 2005, at 8:33 AM, Jack Sarfatti wrote:


George Chapline objects to my new equation

e(x) = 1 + Lpd(Goldstone-Higgs VEV Phase)

for the Einstein-Cartan tetrad 1-form e(x) where Einstein's

ds^2 = |e(x)|^2 = eI(x)e^I(x)

e^I(x) = eu^I(x)dx^u

d is the exterior derivative on the 0-form Goldstone-Higgs VEV phase

saying that the vacuum has constant phase. However the VEV does not
vanish above this homogeneous unstable GLOBALLY FLAT FALSE VACUUM
WITHOUT GRAVITY (pre-inflation).

Note that

e(x) = 1 + Lpd(Goldstone-Higgs VEV Phase)

B = Lpd(Goldstone-Higgs VEV Phase)

is in analogy with Bohm's

v(IT) = (h/m)Grad(Phase of QUANTUM BIT PILOT WAVE)

where quantum of circulation h/m is replaced by the quantum of
distortion Lp.

Of course, George's example is an oversimplification. For example, take
a Type II superconductor with a lattice of magnetic flux vortices as the
temperature approaches absolute zero. The vortices do not go away and
the phase is definitely not identically zero.

Note - inside the region where |VEV| =/= 0 you can replace the magnetic
flux by "Dirac strings" of argVEV phase singularities (e.g. Abrikosov's
Nobel Lecture Rev Mod Phys for a picture and the math). I call this
"Flux without flux" in accord with Wheeler-Feynman eliminating EM field
completely as independent dynamical fields (see also Hoyle-Narlikar).

Note that

d(VEV Phase) creates gravity with total NEGATIVE ENERGY to COMPENSATE
the any positive energy from it in the total action S(Gravity, VEV),
i.e. Universe is a Free Lunch.
This happens even though local gravity energy density stress tensor is
zero when /\zpf = 0. Here is where we need NONLOCALITY OF THE TOTAL
NEGATIVE GRAVITY ENERGY that compensates any TOTAL POSITIVE NON-GRAVITY
ENERGY.

Note that if one does not like "VEV" (Vacuum Expectation Value) use
"ODLRO parameter", since the ODLRO parameter in thermal equilibrium
closed systems, for example, does not vanish up to Tc and Hc.
Furthermore, our local universe, if we are stuck on a BRANE for example
in extra dimensions is an OPEN NON-EQUILIBRIUM SYSTEM since at least
GRAVITY LEAKS OFF the 3D BRANE!