Torsion Field Theory


On Oct 31, 2005, at 2:39 PM, ROBERT BECKER wrote:

"Jack, Gennady -

I am going to take a leap and jump in here. As I read the debate on this issue, there appears to be some distinct points of difference, but the overall approach may not be completely different.

I cannot intelligently comment on Gennady's complete statement that Riemannian geometry can not be a group manifold because it lacks the Cartan Structure Equations. But perhaps I can say a few things about parts of this statement."

There may be a problem of the Russian --> English translation here since, e.g. Rovelli's book Ch 2 clearly shows Cartan structure equations for 1915 GR without torsion. How to generalize correctly to torsion is not completely obvious. I need another go at that.

Only 2 key principles:
1. Locally gauge every continuous group, both internal & space-time.

2. Enforce equivalence principle by making weak force Higgs vacuum coherence mechanism also the source of emergent c-number gravity. Note that we have running coupling constants - weak force gets stronger at higher energy.

Side point:

Heisenberg's uncertainty principle is INCOMPLETE, i.e.

&r ~ h/&p + (G*/c^3)&p

G* is effective gravity parameter at scale &r.

Simplest Higgs mechanism in the standard model has TWO complex scalar fields for SU(2)weak

i.e. n = 4 real fields.

But only one complex scalar field develops the local order parameter VEV =/= 0. It is electrically neutral.

Remember U(1)hyperchargeSU(2)weak has 4 generators, combinations of which make ONE electric charge neutral massless photon and 3 massive Weak bosons. Not the Higgs bosons, which is another part of the theory.
From this POV I only have 2 real fields, n = 2, so in 3 space 1 + d' + r = 3. Stable defects d' = d - n = 1 string vortices rather than point defects. The two real VEV's give me 1 Goldstone Phase to make gravity with from

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

On the other hand, there are other possible models.

Of the 4 real fields, 3 feed mass to the W-bosons and the remaining massless one is the photon.

In this model N - K = 1 is dim of H, which in this case is U(1).
K = 3 = number of broken generators of G = U(1)xSU(2) has N = 4
and if n = 4, i.e., U(1) acts as multiple of identity on the two complex fields, rather than my earlier n = 6, then there is only 1 Higgs boson that should be ~ 250 Gev, which is the critical temperature for the onset of the VEV at least at this cosmological epoch. Presumably it was much larger when the universe was smaller consistent with the strength of the weak coupling decreasing as universe expands.




OK, Einstein-Cartan tetrad, showing the LIF tangent fiber indices explicitly

e^I = 1^I + B^I

B^I also from local gauging of T4 to Diff(4)

The 1 + B split is Diff(4) INVARIANT

Hence B = 0 really is a globally flat spacetime.

Zero torsion 2-form means

De^I = 0

where the SPIN CONNECTION is the 1-form W^IK

D = d + W^IK/
De^I = dB^I + W^IK/\B^K = 0

Therefore W determined from B, which comes from SINGULAR nonlocal Bohm-Aharonov "Flux-without-flux" 'd'(Goldstone Phase) d' = 1 vortex string singularities in simplest Higgs model that will not give a stable Hedghog where we need d' = 0.

The curvature 2-form is

R^IK = DW^IK = dW^IK + W^IJ/\W^JK

Bianchi identity is

DR^IK = 0

Einstein's field eq. is

D*RIK = *J(Matter)

Local conservation of matter current density is

D*J(Matter) = 0

That's 1915 GR in a NUTSHELL using SAME Cartan form eqs as in EM or Yang-Mills theory.
SAME FORMAL TEMPLATE.

Enter TORSION from local gauging of O(1,3) that is GLOBAL in 1915 GR.

This gives TORSION 1-form Potential S^IK where now

D' = D + S/
Therefore, the CURVATURE 2-form is now

R'^IK = D'W^IK = R ^IK+ S^IJ/\W^JK

Note the TORSION-CURVATURE COUPLING S/\W

The TORSION 2-form is

T'^IK = D'S^IK = dS^IK + W^IJ/\S^JK + S^IJ/\S^JK =/ = 0

The Bianchi identities are

D'R' = 0

D'T' = 0

The FIELD EQUATIONS are

D'*R' = *J(Translation)

D'*T' = *J(Rotation)

The LOCAL current conservation laws are

D'*J = 0 (both)

Everything is MOD Diff(4,) i.e. Einstein's "local coincidences" P = {Diff(4) Orbits of manifold points} as explained by Rovelli Ch 2.

One can add Yang-Mills fields from internal symmetry groups trivially into

D" = D' + A^IK/
where A^IK = A^IKaT^a

T^a are the charges of G which spontaneously breaks to H etc.


"A formalism is not a geometry. The Cartan Equations of Structure arising from the what is often called the Cartan Moving Frame Method, which I think is synomous with Gennady's Oriented Point Method, provides a formalism upon which one can impose additional constraints and assumptions."

Yes, Gennady uses a 10Dim manifold from the local gauging of O(1,3), which will be internal parameters in my notation

i.e. Bu^I ~ BuP^I

The 4 P^I form Lie algebra of T4 for 4D translations

Su^IK ~ SuL^IK

L^IK are the 6 O(1,3) Lie algebra generators for 4D rotations.

There are conjugate phases @^I & @^IK, I =/= K, for each, i.e. a 10 Dim manifold of "dynamical phases" (not Goldstone phases of VEV's of course)

"From what I can see, Jack is using the Cartan Structure Equations in his derivation. However, I do not think Gennady's objection is to Jack's use of the Cartan Structure Equations specifically. What Gennady appears to be saying is that the Curvature Form which appears in the Equations of Structure is not the same kind of mathematical object as the Curvature that shows up in the Riemannian Differential Geometry of GR. Gennady is correct on this so far as I know."

Yes, but there is a TETRAD MAPPING connecting them.

R^IK = Ruv^IKdx^udx^v

Ruv^wl = Ruv^IKeI^wel^K

The base space u,v indices ARE HIDDEN in the Cartan formalism, that is AUTOMATICALLY DIFF(4) SCALAR LOCAL INVARIANT in sense of P "local coincidences" of course.


"What shows up in the Equations of Structure is what Vargas calls the "Total Curvature", of which the classical Riemannian Curvature is only a piece."


That's exactly my

R' = D'W = R + S/\W

The difference is the direct torsion-curvature coupling.

If you want TELEPARALLELISM, then I suppose that means

R' = 0

T' =/= 0

"By making the assumption that Torsion vanishes, one obtains Riemannian GR and a Riemannian Curvature."

Yes, that's simply S = 0 in my notation.

"(One can readily use the Cartan formalism to recover Riemannian GR, it just does not contain the full richness of the mathematics that can be obtained from the Cartan formalism if one does not make the vanishing Torsion assumption.)

Therefore, if the vanishing Torsion assumption is used, then the more restricted version of the Curvature Form shows up in the Equation of Structure and the full theory is not obtained. Jack in the first part of his derivation does appear to be using a vanishing Torsion assumption, which forces the Curvature in Jack's equations to not be the full Total Curvature of the Cartan formalism. So, in this sense, Gennady is correct that there is a discrepancy."

Not really. I simply did that at first for PEDAGOGY to show how to get Einstein's 1915 theory as THE FIRST BABY STEP. After a few FUMBLES I think I got the TORSION extension of GR above (correcting some previous errors).

"However, about 2/3 to 3/4 of the way down in Jack's sequence of equations, there appears a T', a "modified" Torsion, which is not assumed to vanish. So, with this final set of equations, Jack seems to be satisfying Gennady's concern over the apparent use of a vanishing Torsion in the Cartan formalism."

Yes. But also I have the CONNECTION to Higgs mechanism for origin of inertia of leptons and quarks and GRAVITY so that SPIRIT of equivalence principle is obeyed GRAVITY & INERTIA EMERGE TOGETHER! Still lots of mopping up details of course. But basic idea I think is sound. Everything FROM COHERENCE including ZPF COHERENCE i.e. ZPF, like ALL EPR ENTANGLEMENTS is LOCALLY RANDOM but NONLOCALLY COHERENT or PHASE-LOCKED. In contrast Higgs field CONDENSATE is BOTH LOCALLY AND NONLOCALLY COHERENT. The origin of GRAVITY and INERTIA TOGETHER is from the LOCAL COHERENCE. The DARK ENERGY/MATTER is from the LOCAL w = -1 INCOHERENCE of the NONLOCALLY EPR CORRELATED ZPF in the T = 0 Kelvin "Vacuum".

"A few other comments:

Assuming vanishing Torsion yields GR. Assuming the vanishing of the Total Curvature (in which case, the Torsion does not vanish) is called Teleparallelism (TP) and is one primary leg of Vargas Theory."

Yes, that's easy in my notation, that's

R' = R + S/\W = 0

T' = dS' + W/\S + S/\S =/= 0

But why R' = 0 is not physically compelling for me. I prefer

D'R' = 0

D'*R = *J (Translation)

D'T' = 0

D'*T' = *J(Rotation)

D'*J = 0 etc.

"Another difference is that Vargas Theory does not seek to introduce gauge fields into the frame forms (i.e. Jack's e) or into theory at all. Finally, no assumption of a spin-connection is made in Vargas Theory and the resulting "geometric" Maxwell-Einstein Equations resulting from the Torsion Equation of Structure do not have a rotating source. Torsion in Vargas Theory is not related to rotation, but rather, in a certain matehmatically detailed sense, to EM. The geometric Maxwell Equations essentially represent a geometrization of the EM (at that level of theory).

Take care,

Robert E. Becker"

I think my theory makes more physical sense than Vargas theory BECAUSE

I get Gravity and Dark Energy/Matter and Inertia ALL EMERGING TOGETHER in a way consistent with standard model of leptons & quarks. My theory is very close to observation and is FALSIFIABLE.

In this way my theory is better not only than Vargas theory, it is better than STRING THEORY and better than LOOP QUANTUM GRAVITY.

I make some strong predictions of principle:

1. Dark Matter Omega ~ 0.23 is VIRTUAL NONLOCALLY COHERENT ZPF that is LOCALLY RANDOM with positive pressure for compact sources.

2. Hence DARK MATTER DETECTORS will BE SILENT with RIGHT STUFF same a Michelson-Morley not showing Ether Drift - barring some bizarre claims to the contrary of course. No exotic on-mass-shell quanta will explain Galactic Halo etc.

3. Quantum foam does not exist and high energy cosmic photons will not show it!

4. Universal slope of Regge trajectories for hadronic resonances is STRONG SHORT RANGE ZPF induced gravity.

Basically Kerr solution J/h ~ GM^2/hc (I published that in 1973)

i.e. anticipated string-blackhole correspondance in 1973.





Gennady Shipov wrote:

----- Original Message -----
From: Jack Sarfatti
To: Gennady Shipov
Cc: ROBERT BECKER ; Paul Zielinski ; Burinskii Ya. ; James Woodward ; Andrey Sidorov ; hammond Richard ; Hal Puthoff ; Vladimir Poponin ; Bill Page; Marc G. Millis ; Creon Levit ; Kay zum Felde ; Evgeniya Chizhikova ; Gubarev Eugeni ; john dering ; Eric Davis ; Richard Amoroso ; Ken Shoulders ;RKiehn2352@aol.com ; David M Mcmahon ; Mark Pesses ; Tony Smith ; S-P Sirag ; michael ibison
Sent: Monday, October 31, 2005 3:13 AM
Subject: Local Curvature & Torsion fields as Nonlocal Bohm-Aharonov Topological Singularities


On Oct 30, 2005, at 6:25 AM, Gennady Shipov wrote:




Jack Sarfatti


Witten et-al WRONG that GR is not a LOCAL GAUGE THEORY! (when properly
framed of course using P physical coincidence not x = manifold point).
Gennady Shipov
I agree with Witten et-al .
Jack Sarfatti

Is that what you mean? What exactly do you agree with?

Gennady Shipov
Riemann geometry which has been used by Einstein, has no Cartan structural equationsà and cannot be submitted as a group manifold. It is only a shadow of the "true" geometry.


Jack Sarfatti

I do not understand what you just wrote. Rovelli has the structural equations in Ch 2 of his online "Quantum Gravity" book. Indeed I have written them down now several times.
http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html
has free pdf early version of book.

Rovelli's eqs are essentiall

Given W^IJ = spin-connection 1-form

I,J in tangent space fiber indices raised and lowered with constant flat Minkowski metric nIJ in accord with EEP.

Exterior covariant derivative is

D = d + W/
The curvature 2-form is

R^IJ = dW^IJ + W^IK/\W^KJ

Bianchi identities

DR = 0

Einstein's field eqs.

D*R = *J(Matter)

Local conservation of stress-energy current densities

D*J(Matter) = 0

Einstein-Hilbert Field Lagrangian Density 0-form

L = {IJKL}R^I^J/\e^K/\e^L

{IJKL} is completely antisymmetric tensor

e^I = e^Iudx^u

ds^2 = guvdx^udx^v = eu^InIJev^Jdx^udx^v

e = e^I&I

where &I are basis vector fields in fiber dual to dx^I

So above is standard with zero torsion 2-form assumed, i.e.

T^I = De^I = de^I + W^IJ/\e^J = 0

In my theory, I have additional structure:

e = 1 + B

B = (hG/c^3)^1/2'd'(Goldstone Phase of Charge Neutral Higgs Field Component)

B also comes from local gauging of T4 to Diff(4)

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

W determined from B completely.

If we also locally gauge O(1,3)

we get in addition to W(Curvature) an S(Torsion)

D' = d + W^IJ/\ + S^IJ/\ = D + S/
We now have to make some choices

R' = D'W = R + S/\W

T' = D'S = dS + 'S/\S' + W/\S

Note T' has 2 indices, T only had 1.

'S/\S' = S^IKS^KJ etc.

Bianchi identities

D'R' = 0

D'T' = 0

Field equations

D'*R' = *J(Translation Sources)

D'*T' = *J(Rotating Sources)

Local conservation of current densities

D*J(Translation) = 0

D*J(Rotation) = 0
Gennady Shipov
Jack! Rovelli used that has been made by Cartan in 1926. Nothing new. A source of a mistake that Riemann geometry (1854) is based on a point manifold, and Cartan geometry on the orinted point manifold. These are the different approaches giving different results. In Cartan approach there is no Riemann curvature without Ricci torsion.


Jack Sarfatti

OK these are MY NEW CURVATURE-TORSION FIELD EQs (replacing my earlier ones that may not have been quite correct) from LOCAL GAUGING full 10-parameter Poincare space-time symmetry group with spontaneous broken WEAK FORCE internal symmetry supplying the charge neutral Goldstone phase that generates BOTH CURVATURE and TORSION as Bohm-Aharonov "Flux-without-flux" nonlocal SINGULAR GAUGE TRANSFORMATIONS on the charge-neutral Goldstone phase.

B(Curvature) = (hG/c^3)^1/2'd'(Neutral Goldstone Phase of SU(2)weak Higgs Vacuum ODLRO)

'd' means SINGULAR gauge transformation i.e. off the singular topological defect 'd'(Phase) is closed i.e.

d'd'(Phase) = 0 like what looks like an exact 1-form with a 0-form Phase.

Nevertheless when the closed loop 1-cycles surround a topological defect Phase Singularity the NONLOCAL DeRham integral of the 1-form around this singular NON-BOUNDING 1-cycle LOOP not only does not vanish, but it is quantized!

The SINGULAR INTEGRAL of the B-1form about the topological defect is a MEMBER of the HOMOTOPY GROUP ~ Z

i.e. PI1[S1] = Z in this case.

In general PI(surrounding cycle)[Order Parameter Fiber Space] is the physically relevant Homotopy Group.

In simple terms this is Bohr-Sommerfeld quantization from single-valuedness of the "More is different" emergent LOCAL zero thermodynamical entropy vacuum order parameter from the weak force spontaneous broken symmetry giving both gravity and inertia together in ONE UNIFIED MECHANISM consistent with the EQUIVALENCE PRINCIPLE.

"Flux-without-Flux is DEFINING the "circulation integral of the p-form" as equal to the flux integral of its exterior derivative over the surrounded p+1 manifold including its topological defect.

This is essentially the NONLOCAL BOHM-AHARONOV ACTION-AT-A-DISTANCE of the topological defect in which the Goldstone Phase is undefined at the ZEROs of its complementary Higgs Amplitude

i.e. Order Parameter Component = (Higgs Amplitude)e^i(Goldstone Phase) c-number field

However, we still have Number-Phase complementarity for the small vibration quanta either real or virtual.

There are two approaches to the geometry - Riemannian and Cartanian. The Riemannian approach is wrong from the beginning. Only a method of orthogonal reper used by Cartan gives the correct approach, and it means, that torsion of space is different from zero. If torsion is equal to zero we have Minkovski space (Einstein).

Jack Sarfatti



I have proved that GR is a local gauge theory
R = DW

D = d + W/
DR = 0

D*R = *J

D*J = 0

R = Curvature 2-form

above is only for 1915 GR limit of zero torsion 2-form

T = De = D(1 + B) = 0

R is like EM field F in

F = dA

dF = 0

d*F = *j

dj* = 0

Where B(P), W(P), R(P)

P is local coincidence not a bare manifold point x

Technically P = {x} = Coset Orbit of All x in manifold mod Diff(4).


Jack Sarfatti

It's straight-forward to generalize to Shipov's torsion field theory
using the Cartan forms that are local frame invariant automatically.
Gennady Shipov

Using of Cartan forms cannot give generalization of the physical theory.

Formalism of external forms it only the mathematical tool.

Besides my theory is finished with experiment and, even up to technology, but the theory of strings is phenomenological model which after some time willl die.

Yang-Mills Gauge Theory with Higgs Mechanism
Locally gauging the global rigid internal symmetry Lie group G makes a big qualitative difference. Note, there is NO gravity as yet in this theory. It is done on globally flat Minkowski space-time, yet rest masses of all the fundamental particles, i.e. leptons, unconfined quarks, W-bosons emerge from the vacuum coherent ODLRO local order parameter in the U(1)xSU(2) electro-weak force sector. There is NO need for the Haisch-Puthoff transverse virtual photon acceleration drag in the semi-classical SED picture, and no need for James Woodward's "Mach's principle" model. The rest mass of the Higgs particle(s) themselves is irrelevant to the formulae for the rest masses of the leptons and quarks. To say that the high rest mass(s) of the Higgs shows that standard model is wrong is not correct. However, there is a contradiction with the GR equivalence principle of the inseparabilty of gravity and inertia that I think I have corrected by arguing that the curvature of space-time is emergent from a charge neutral Goldstone phase of the electro-weak force.

To review, we have the false vacuum disordered continuous internal symmetry Lie group G, with Lie algebra T^a in a nxn matrix representation. The range of a is not necessarily the same as n, except in the "adjoint representation" where

(T^a)bc = -if^abc the structure constants of the Lie algebra of G

The unbroken ordered vacuum phase has an invariant subgroup H of G such that the H transformations leave the order parameters invariant. This implies that the Lie sub-algebra of H annihilates the order parameter, that are eigenvectors of the sub-algebra charges with zero eigenvalue. That is, the order parameters are "charge neutral" relative to the residual internal symmetry group H of the more ordered COHERENT vacuum phase. The generators of G not in H are called the "broken generators". We use the Mexican Hat quartic potential for the order parameters because the associated quantum operator field theory for small vibrations in Higgs amplitudes and Goldstone phases are then renormalizable. The mass matrix of the global gauge theory is not of direct physical interest. Now locally gauge G, this introduces compensating Cartan 1-form Yang-Mills potentials (in most general non-Abelian G)

A = Audx^u = Au^aT^adx^u

where T^a is the nxn matrix representation of the elements of the G Lie algebra.

Define the internal symmetry gauge covariant partial derivatives in flat Minkowski space-time as

Du = ,u - ieAu^aTa

for a real Lie group representation of n REAL SCALAR FIELDS T^a replaced by iQ^a, where Q^a are real antisymmetric nxn matrices. Let D(g) be a nxn matrix representation of g in G. Then

The scalar field c- number local order parameters transform as

(0|phi(x)i|0) --> D(g)ij(0|phi(x)j|0)

i = 1 ... n

Note n = 2 is a U(1) complex scalar field here. The phij are real functions of space-time in global Minkowski metric.

The internal symmetry Yang-Mills gauge potentials transform INHOMOGENEOUSLY (like Levi-Civita connection under Diff(4)) as

Au --> D(g)AuD(g)^-1 + ie^-1D(g)^-1D(g),u

For the internal symmetry gauge covariant partial derivatives

Du(0|phi|0) --> D(g)Du(0|phi|0)

And for the Yang-Mills field tensors under G (internal curvature)

Fuv --> D(g)FuvD(g)^-1

Fuv^a = A^av,u - A^au,v + ef^abcAu^bAv^c

Fuv = Fuv^aTa

The Lagrangian Density is

L = (1/2)DuphiiD^u(phi^i) - V(phii) - (1/4)F^auvF^auv

Make small vibrations about the VEVs (0|phi(x)i|0)

When we quantize these small vibrations into quantum harmonic oscillators, the theory will be renormalizable.

phi(q-number)j = (0|phi(x)j0) (c-number local order parameter) + phi'(x) q-number operator (boson CR)

The resulting Lagrangian density for the q-number excited states above the vacuum is

L' = (1/2)(phi'(x)^j)^,u(phi'(x)j),u - (1/2)mij^2phi'(x)^iphi'(x)^j - (1/4)(Fauv(x)F^a^u^v(x)) + (1/2)Mab^2A^au(x)A^b^u(x) + Lint

Note there will be THREE kinds of INDEPENDENT MASS MATRICES

I. mij for the HIGGS BOSONS of the UNBROKEN GENERATORS of ordered H.

II. Mab for the MASSIVE GAUGE BOSONS OF THE BROKEN GENERATORS OF G ---> H

III. Yukawa interactions for the Leptons & Quarks with

Fermion(lepto-quark) MASS MATRIX

mij = (Gamma)j(0|phi|0)j

*Eric Davis and James Woodward have confounded I with III in their Red Herring objections.

DEFINE A GAUGE CONSTRAINT for the BROKEN GENERATORS o unorderedf G i.e. T^a NOT in the ordered subgroup H in which the BROKEN VEVS vanish. These broken generators WOULD have had gapless Goldstone phase modes if there were no local gauge fields.

The sum over ij in the mij^2 mass matrix for the scalar fields does not include the broken generators, but only goes over the H invariant subgroup generators in the selected gauge constraint.

mij^2 (over H only) = Functional Second Derivative of V relative to phi(x)i & phi(x)j at (0|phi(x)|0) vacuum in G/H.

*These mij describe the REST MASSES of the HIGGS BOSONS on mass-shell, the number of them depends on the dimension of H.

The gauge force boson MASS matrix is

Mab^2 = e^2(T^aT^b)ij(0|phi(x)^i|0)(0|phi(x)^j)

WHERE THESE T^a are ONLY FOR THE BROKEN GENERATORS OUTSIDE OF H!

That is, the MEISSNER EFFECT of MASSIVE GAUGE BOSONS ONLY APPLIES TO THE BROKEN GENERATORS OF THE UNORDERED LIE GROUP G. For example, W+,W-,W0 is from SU(3)weak that is completely broken. Only U(1)hypercharge is unbroken with a massless gauge boson.

"We see that the vector fields associated with the broken generators acquire non-zero masses, while the gauge fields of the unbroken subgroup H remain massless. The WOULD-BE Goldstone bosons have DISAPPEARED, and the corresponding degrees of freedom have been absorbed as additional spin states of the massive vector fields.

Fermions (leptons & quarks) can be EASILY INCORPORATED .. by adding appropriate kinetic and interaction terms to the Lagrangian ...

psi*Gammaipsiphi^i

psi are the fermion leptons & quarks with their OWN independent COUPLINGS Gammai

* That is, the rest masses of the leptons & quarks are INDEPENDENT of the Higgs boson rest masses!

Summary

N = number of generators for UNORDERED FALSE VACUUM internal symmetry Lie group G

N - K = number of UNBROKEN GENERATORS of ORDERED INVARIANT SUBGROUP H of G

i.e. K = number of BROKEN GENERATORS

n = Dimension of REAL REPRESENTATION of G.

e.g. n = 2 is a U(1) single complex scalar field over Minkowski space-time.

There are then

n - K MASSIVE HIGGS BOSONS that are now being looked for.

K massive gauge bosons ---> K = 3 in standard mode,l i.e. only W+, W- & W0

Massless gauge bosons N - K

Note, if we start from

G = U(1)hyperchargeSU(2)weak

N = 4

K = 3

N - K = 1

U(1)xSU(2) is n = 6 real scalar fields 2 for U(1) and 4 for SU(2)

This means 3 massive Higgs bosons in this simplest version of standard model.

Curvature and Torsion as Bohm-Aharonov Effects from Topological Defect Phase Singularities


On Oct 30, 2005, at 12:13 PM, Jack Sarfatti wrote:


On Oct 30, 2005, at 6:25 AM, Gennady Shipov wrote:




Jack Sarfatti


Witten et-al WRONG that GR is not a LOCAL GAUGE THEORY! (when properly
framed of course using P physical coincidence not x = manifold point).
Gennady Shipov
I agree with Witten et-al .
Jack Sarfatti

Is that what you mean? What exactly do you agree with?

Gennady Shipov
Riemann geometry which has been used by Einstein, has no Cartan structural equationsà and cannot be submitted as a group manifold. It is only a shadow of the "true" geometry.


I do not understand what you just wrote. Rovelli has the structural equations in Ch 2 of his online "Quantum Gravity" book. Indeed I have written them down now several times.
http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html
has free pdf early version of book.

Rovelli's eqs are essentiall

Given W^IJ = spin-connection 1-form

I,J in tangent space fiber indices raised and lowered with constant flat Minkowski metric nIJ in accord with EEP.

Exterior covariant derivative is

D = d + W/
The curvature 2-form is

R^IJ = dW^IJ + W^IK/\W^KJ

Bianchi identities

DR = 0

Einstein's field eqs.

D*R = *J(Matter)

Local conservation of stress-energy current densities

D*J(Matter) = 0

Einstein-Hilbert Field Lagrangian Density 0-form

L = {IJKL}R^I^J/\e^K/\e^L

{IJKL} is completely antisymmetric tensor

e^I = e^Iudx^u

ds^2 = guvdx^udx^v = eu^InIJev^Jdx^udx^v

e = e^I&I

where &I are basis vector fields in fiber dual to dx^I

So above is standard with zero torsion 2-form assumed, i.e.

T^I = De^I = de^I + W^IJ/\e^J = 0

In my theory, I have additional structure:

e = 1 + B

B = (hG/c^3)^1/2'd'(Goldstone Phase of Charge Neutral Higgs Field Component)

B also comes from local gauging of T4 to Diff(4)

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

W determined from B completely.

If we also locally gauge O(1,3)

we get in addition to W(Curvature) an S(Torsion)

D' = d + W^IJ/\ + S^IJ/\ = D + S/
We now have to make some choices

R' = D'W = R + S/\W

T' = D'S = dS + 'S/\S' + W/\S

Note T' has 2 indices, T only had 1.

'S/\S' = S^IKS^KJ etc.

Bianchi identities

D'R' = 0

D'T' = 0

Field equations

D'*R' = *J(Translation Sources)

D'*T' = *J(Rotating Sources)

Local conservation of current densities

D*J(Translation) = 0

D*J(Rotation) = 0

OK these are MY NEW CURVATURE-TORSION FIELD EQs (replacing my earlier ones that may not have been quite correct) from LOCAL GAUGING full 10-parameter Poincare space-time symmetry group with spontaneous broken WEAK FORCE internal symmetry supplying the charge neutral Goldstone phase that generates BOTH CURVATURE and TORSION as Bohm-Aharonov "Flux-without-flux" nonlocal SINGULAR GAUGE TRANSFORMATIONS on the charge-neutral Goldstone phase.

B(Curvature) = (hG/c^3)^1/2'd'(Neutral Goldstone Phase of SU(2)weak Higgs Vacuum ODLRO)

'd' means SINGULAR gauge transformation i.e. off the singular topological defect 'd'(Phase) is closed i.e.

d'd'(Phase) = 0 like what looks like an exact 1-form with a 0-form Phase.

Nevertheless when the closed loop 1-cycles surround a topological defect Phase Singularity the NONLOCAL DeRham integral of the 1-form around this singular NON-BOUNDING 1-cycle LOOP not only does not vanish, but it is quantized!

The SINGULAR INTEGRAL of the B-1form about the topological defect is a MEMBER of the HOMOTOPY GROUP ~ Z

i.e. PI1[S1] = Z in this case.

In general PI(surrounding cycle)[Order Parameter Fiber Space] is the physically relevant Homotopy Group.

In simple terms this is Bohr-Sommerfeld quantization from single-valuedness of the "More is different" emergent LOCAL zero thermodynamical entropy vacuum order parameter from the weak force spontaneous broken symmetry giving both gravity and inertia together in ONE UNIFIED MECHANISM consistent with the EQUIVALENCE PRINCIPLE.

"Flux-without-Flux is DEFINING the "circulation integral of the p-form" as equal to the flux integral of its exterior derivative over the surrounded p+1 manifold including its topological defect.

This is essentially the NONLOCAL BOHM-AHARONOV ACTION-AT-A-DISTANCE of the topological defect in which the Goldstone Phase is undefined at the ZEROs of its complementary Higgs Amplitude

i.e. Order Parameter Component = (Higgs Amplitude)e^i(Goldstone Phase) c-number field

However, we still have Number-Phase complementarity for the small vibration quanta either real or virtual.

There are two approaches to the geometry - Riemannian and Cartanian. The Riemannian approach is wrong from the beginning. Only a method of orthogonal reper used by Cartan gives the correct approach, and it means, that torsion of space is different from zero. If torsion is equal to zero we have Minkovski space (Einstein).

Jack Sarfatti



I have proved that GR is a local gauge theory
R = DW

D = d + W/
DR = 0

D*R = *J

D*J = 0

R = Curvature 2-form

above is only for 1915 GR limit of zero torsion 2-form

T = De = D(1 + B) = 0

R is like EM field F in

F = dA

dF = 0

d*F = *j

dj* = 0

Where B(P), W(P), R(P)

P is local coincidence not a bare manifold point x

Technically P = {x} = Coset Orbit of All x in manifold mod Diff(4).


Jack Sarfatti

It's straight-forward to generalize to Shipov's torsion field theory
using the Cartan forms that are local frame invariant automatically.
Gennady Shipov



Using of Cartan forms cannot give generalization of the physical theory.

Formalism of external forms it only the mathematical tool.

Besides my theory is finished with experiment and, even up to technology, but the theory of strings is phenomenological model which after some time willl die.

Vacuum Instability for any Lie Group of Internal Symmetries

n components of macro-quantum ODLRO order parameter are each local complex functions of Einstein's "local coincidences" P (in general). However, for now I only use Minkowski space-time without gravity and we can be sloppy and use "x" instead of "P".

We use Mexican Hat quartic potential because no other seems to permit renormalizable quantum field theories sans gravity of the SSB scheme.

Using the previous notation of n scalar fields for the order parameter in fiber space V over physical base space M of dimension d, with a defect of dimension d' surrounded by space S of dim r in d.

1 + r + d' = d

Topological stability means homotopy group PIr(V) more then identity group generally implies

d' = d - n

1. SImplest global U(1) spin 0 complex scalar field SSB no gauge field.

n = 2 real scalar fields = 1 complex field. In spacelike slice of Minkowski space-time d = 3, therefore the only stable defects are d' = 3 - 2 = 1 line vortex Goldstone phase defects where the Higgs amplitude vanishes surrounded by non-bounding closed loops giving quantized Bohm-Aharonov "Flux without flux" in stable "stationary states" (Bohr-Sommerfeld).

Small vibrations (real quanta on-mass-shell) of the local zero entropy order parameter Goldstone phase along the minimum circle of the Mexican Hat Potential are massless - no energy gap at infinite wavelength. In contrast the small vibrations of the Higgs amplitude up the slope of the potential have a mass gap m ~ Higgs amplitude itself, i.e.

Let (0|phi|0) be the order parameter (VEV = Vacuum Expectation Value), the Mexican Hat Potential of the dynamical complex scalar field phi is

V(phi) = (k/4)(|phi|^2 - |(0|phi|0)|^2)^2

m(Higgs) = k^1/2|(0|phi|0)| real and positive

2. Same spin 0 n = 2 field above with a massless Abelian local gauge field. That is we locally gauged the global n= 2 U(1) above with minimal coupling.

Now, m(Higgs) = k^1/2|(0|phi|0)|

same as before, but now no massless Goldstone phase quantum. It is absorbed into the gauge force field giving it a mass

m(gauge boson) = 2^1/2e|(0|phi|0)

Similarly if we put in a MASSLESS Dirac spinor with a Yukawa coupling @ to the scalar field

m(fermion) ~ @^1/2(0|phi|0)

so that failing to find Higgs boson has NOTHING to do with the origin of inertia of the fermion leptons & quarks in the standard model that uses the more complicated Yang-Mills theory, but this basic idea remains the same.

The objections of James Woodward & Eric Davis about this have no foundation in fact in terms of the actual mainstream theory for the origin of inertia of the basic particles. Hadrons are trapped bags of light quarks and most of the hadronic mass is trapped kinetic energy (e.g. F. Wilczek).

Yang-Mills Nonabelian Gauge Theories beyond n = 2.

Let there be n-complex scalar fields, i.e. n here is not same as in the above notation for real fields. One can see that stable topological defects may require extra space dimensions. For now we will not worry about the stability of the topological defects.

Let G be a Lie group of RIGID global internal symmetry transformations on the n complex scalar field components. "Scalar" under the Poincare space-time group. Let g be an element of G. The nxn matrix representation of G is D(g).

phij --> phi'j = Dj^j'(g)phij'

g = e^-i@aL^a

@a are "phases" (not Goldstone phases as yet) conjugate to the Lie algebra generators L^a where the commutators are

[L^a,L^b] = f^a^bcL^c

fab^c are the "structure constants"

(range of a = N, which need not be same as n)

The N-dim rep is the "adjoint rep" in which

(L^a)bc = -if^abc

Tr[L^aL^b] = (1)^a^b (Identity NxN matrix)

More generally for any n that need not = N but can of course:

T^a = Dnxn(L^a)

a = 1 ... N

The Vacuum Manifold V = G/H.

SSB of G to normal subgroup H.

H is a normal AKA "invariant" subgroup because

H = {g< G|D(g)(0|phi|0) = (0|phi|0)}

Therefore since

g = e^-i@aT^a ~ 1 - i@aT^a + powers of T^a

T^a(0|phi|0) = 0

note this is a nxn matrix eigenvalue equation with ZERO EIGENVALUE with the n-component order parameter as the singular eigenvector.

ZERO EIGENVALUES always signal INSTABILITY IN THE FALSE VACUUM.

Do not confuse the above with the mass matrix.

Invariance of the ACTION under the full G implies

V,iTij^aphi^j = 0

V,i is functional derivative of V with respect to phii.

Again do small vibrations about the equilibrium order parameter minima to get real quanta excited states.

There is a squared mass matrix

m^2ij = V,i,j > 0 at the minima on the landscape

where V,i = 0

m^2ijT^a^j^k(0|phi|0)k = 0

Since T^a(0|phi|0) =/= 0 for the BROKEN GENERATORS of G not in H. They are linearly independent and therefore have ZERO MASSES.

That is, the BROKEN GENERATORS of G (non-zero eigenvalues in previous sense) have MASSLESS Goldstone phase excitations on mass shell outside the vacuum. The unbroken generators of H CAN have massive Goldstone phase excitations. All of this BEFORE putting in any gauge fields mind you!

The degenerate Vacuum Manifold of order parameters is the coset space

V = G/H

Next step Yang-Mills gauge fields.

to be continued.

On Oct 29, 2005, at 12:34 PM, Jack Sarfatti wrote:

PS Cartan form generalization of GR to torsion.

B the curved tetrad 1-form comes from local gauging of T4 to Diff(4)

And, in addition, from Spontaneous Broken Vacuum ODLRO of standard model Higgs in U(1)xSU(2)

B = (hG/c^3)^1/2'd'(Effective Goldstone Phase from Higgs Field in Standard Model)

Therefore gravity & inertia together in accord with equivalence principle
i.e. origin of inertia of leptons, quarks, W bosons & Higgs itself seamlessly integrated with emergence of gravity without gravitons, without quantum foam, but with classical gravity waves (LIGO, LISA)

To get torsion 2-form locally gauge O(1,3) in addition to T4

This gives the torsion tetrad 1-form that is also equal to

T' = (hG/c^3)d'd'(Goldstone Phase) = Torsion 2-form

in sense of Bohm-Aharonov "Flux-without-flux", i.e. T = 0 locally on the surrounding non-bounding 1-cycle, but not nonlocally in the sense to the total 2-form integral over the interior of the non-bounding 1-cycle including the Goldstone Phase singularity. That is, the Goldstone Phase Singularities have Bohm-Aharonov NONLOCAL "Flux-without flux" actions-at-a-distance.

Furthermore, in the global deRham integral sense:

T' = dS + W/\S + S/\(1 + B + S) = (hG/c^3)d'd'(Goldstone Phase) =/= 0 when S =/= 0

Note that S/\S means SK^I/\S^KJ

Both S & W are 1-forms with 2 tangent space indices.

D' = d + W/\ + S/
The extended torsion field equations I postulate are:

R' = D'(W + S) = dW + W/\W + S/\W = R + S/\W + S/\S

D'R' = 0 "Bianchi identity"

D'*R' = *J(4D Translational Sources)

D*J(4D Translational Sources) = 0

D'T' = 0

D'*T' = *J(4D Rotating Sources)

D*J(4D Rotating Sources)

ANSATZ: Hypothesis
All local gauge theories (using Einstein's "local coincidences" P MOD Diff(4)) obey the SAME UNIVERSAL
Cartan-form TEMPLATE

A = potential 1-form

F = DA field 2-form

DF = 0 Bianchi identity

D*F = *J Source Equation

D*J = 0 Local covariant conservation of Source Current Densities

Lagrangian Density ~ F/\F

In the case of 1915 GR we start with B and from there get spin connection W where

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

Determines W from B and

D = d + W/\ = Diff(4) covariant exterior derivative

in 1915 GR


I have proved that GR is a local gauge theory
R = DW

D = d + W/
DR = 0

D*R = *J

D*J = 0

R = Curvature 2-form

above is only for 1915 GR limit of zero torsion 2-form

T = De = D(1 + B) = 0

R is like EM field F in

F = dA

dF = 0

d*F = *j

dj* = 0

Where B(P), W(P), R(P)

P is local coincidence not a bare manifold point x

Technically P = {x} = Coset Orbit of All x in manifold mod Diff(4).

Weinberg-Witten theorem refuted
http://infeld.harvard.edu/sidneyfest/11-witten.mov

George Chapline first mentioned this apparent fly in my ointment for emergent gravity.
Now that I watched Ed Witten talk about it above, I see why the theorem, while technically true, is physically irrelevant because it provides a correct answer to the WRONG Question.

"The Question is: What is The Question?" J.A. Wheeler

Weinberg & Witten proved that you cannot make a massless spin 2 graviton from a local gauge theory in 4D Minkowski space-time. Well that's fine, because my theory for the emergence of curved 4D space-time from local massless gauge theory in unstable 4D Minkowski space-time does not have any gravitons or quantum foam as a matter of principle. Therefore, whilst the theorem, I'm sure is mathematically correct, it is physically irrelevant IMHO.

Strangley, DeWitt, Witten, Weinberg et-al do not seem to know about the Einstein Hole Problem. No wonder Lubos Motl panned Rovelli's online "Quantum Gravity" book, because it shows that, contrary to the folklore superstition given by Witten at the Sidney Coleman Fest at Harvard that you cannot have local gauge invariant observables in general relativity, in fact you can when you use Einstein's idea of "local coincidences" P rather than bare local coordinate points x subject to the Diff(4) group x -> x' =/= x. Indeed all the Diff(4) orbits of any given x in the manifold are like gauge transformations for the internal groups and form an equivalence class {x} for a single P = {x}. That is, look at Fig. 2.4 p. 49 to see the idea.
"The diffeomorphism moves the nonflat region as well as the intersection point of the two particles a & b from the point x = B to the point x' = A. The LOCAL OBJECTIVE EVENT P on which to PIN local Diff(4) observables is Einstein's "local coincidence" a SCATTERING of 2 particles a and b. True the old pre-GR idea of the objectivity of points in Minkowski space-time does not work in GR as it kind of worked in QED. DeWitt, Weinberg, Witten et-al have seemed to overstate the problem making it a lot harder than it need be.

Given a LOCAL Vacuum ODLRO parameter PSI, I do NOT mean PSI(x) in Witten's sense. I mean PSI(P), where P = {x} = Manifold/Diff(4) the COSET equivalence class of all bare manifold points connected by Diff(4) local coordinate transformations. .Diff(4) is the local gauging of T4 in Minkowski space-time.

P is a COINCIDENCE COLLISION of particles a & b as explained by Rovelli in his 2.2. This is basically what Einstein finally understood after much struggle to be sure.

Now, the curved part of the Einstein-Cartan tetrad field is

B(P) = (hG/c^3)^1/2'd'argPSI(P)

where 'd' means there are Goldstone phase = argPSI(P) singularities of Bohm-Aharonov "Flux-without flux" in which, including the singularities on the interior of the non-bounding 1-cycle:

DeRham Integral of the 1-form 'd'(argPSI(P)) on a non-bounding 1-cycle surrounding the singularity does not vanish and is indeed, in stationary Bohr-Sommerfeld sense, quantized from the single-valuedness of PSI(P). That is, using a QUASI-Stoke's theorem, it's AS IF

Line integral of B over non-bounding loop = Surface integral of "Curl B" (in 4D).

That is, in heuristic notation

d'd'argPSI =/= 0 GLOBALLY even though d'd'argPSI = 0 locally on the nonbounding loop. The interior surrounding argPSI singularity makes the outer loop not a boundary.

This is quantized Flux without flux.

Einstein's local metric field is the symmetric tetradic bi-linear form of the EEP

ds^2(P) = guv(P)dx^udx^v = (1^I + B(P)^I)nIJ(1^J + B(P)^J)

B(P) = B^I&I

&I dual to dx^I

I,J in tangent space, nIJ is constant Minkowski metric.

There BE NO GRAVITONS HERE. I have NO MASSLESS SPIN 2 QUANTA.

Therefore, I LEAPFROG over Weinberg-Witten theorem. It is irrelevant. We do not RE-QUANTIZE an emergent BOTTOM -> UP c-number ODLRO field whose Goldstone phase gives Einstein's 1915 GR!

What happens at the singularities where the Higgs Amplitude |PSI(P)| = 0 is NO QUANTUM GRAVITY FOAM, but MASSLESS FLAT QUANTUM FIELD THEORY.

But is it in 2+1 D on virtual holographic boundary of curved 3+1 interior as Witten mentioned? Could be. I have been thinking massless pre-inflation false vacuum was 3+1 but maybe it is 2+1?

Breakdown of G/H formulation of Spontaneous Broken Symmetry?
"There is an interesting group action of S1 (thought of as the group of complex numbers of absolute value 1) on S3 (thought of as a subset of C2): λ·(z1,z2) = (λz1,λz2). The orbit space of this action is naturally homeomorphic to the two-sphere S2. The resulting map from the 3-sphere to the 2-sphere is known as the Hopf bundle. It is the generator of the homotopy group π3(S2). ... Thus, S3 as a Lie group is isomorphic to SU(2)." http://en.wikipedia.org/wiki/3-sphere

However S2 is not a Lie group. So we cannot think of S2 as a G/H quotient group where H is a normal subgroup of G.
"Group structure

When considered as the set of unit quaternions, S3 inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, S3 takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, S3 can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3. When thought of as a Lie group S3 is often denoted Sp(1) or U(1, H).

... The set of unit quaternions is then given by matrices of the above form with unit determinant. It turns out that this group is precisely the special unitary group SU(2). Thus, S3 as a Lie group is isomorphic to SU(2).

It turns out that the only spheres which admit a Lie group structure are S1, thought of as the set of unit complex numbers, and S3, the set of unit quaternions. One might think that S7, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give S7 one important property: parallelizability. It turns out that the only spheres which are parallelizable are S1, S3, and S7."http://en.wikipedia.org/wiki/3-sphere

Therefore, it appears that we cannot think of the n = 3 order parameter with V = S2 as a quotient group G/H. The common intuitive idea that the group G is broken down to invariant subgroup H with vacuum manifold V = quotient group G/H does not seem to work for order parameters n = 3 whose manifold is S2 because S2 does not correspond to any group the way S3 and S1 do.


3.Fundamental Homotopic Group PI1(V) for STRINGS (1-Branes)
"Each vortex can be set in correspondence to an element of the fundamental group PI1(S1), which is called the fundamental group of the circle. The latter is isomorphic to the group of integers Z ... each vortex can be characterized by N .. the number of circulation quanta of the superfluid velocity vs around the core of the vortex." Volovik & Mineev

Each Circulation Quantum = Vorticity Flux-without-flux Quantum

for single-valued local order parameters.

via QUASI AS IF "Stoke's theorem" for non-bounding p-cycles without boundary surrounding singular p + 1 interior manifolds. The singularities mean multiple connectivity of the surrounded interior p + 1 manifolds.

"Each linear singularity of the degeneracy parameters" [Goldstone Phase(s)] can be set in correspondence with an element of the fundamental group PI1(V). A NONSINGULAR CONFIGURATION of the degeneracy parameter [Goldstone Phase] field CORRESPONDS TO THE UNIT ELEMENT of this group, and COALESCENCE OF THE SINGULARITIES CORRESPONDS TO MULTIPLICATION OF ELEMENTS OF THIS GROUP." Volovik ...

Unstable singularities of dimension d' < d in physical base space of dimension d that are surrounded by closed (without boundary) hypersurfaces AKA "cycles" of dim r, are homotopic to zero. That is, if the single-valued order parameter has dimension n in the fiber space V, then if there is no TORSION GAP, a cycle of dimension r in base space MAPS, via the Goldstone Phase(s), into an image cycle of dimension n-1 in fiber space. If the image cycle is homotopic to zero, i.e. can be continuously deformed to a single point, then the base space singularity is unstable. If the image cycle is stable, then it cannot be continuously deformed into a point, but has an integer winding number from the single-valuedness of the order parameter. This is Bohr-Sommerfeld quantization. The winding number defines a coset space of non-overlapping homotopic equivalence classes. All mappings from the surrounding cycle of the singularity in base to the image cycle in fiber space with the same set of winding integers can be continuously deformed into each other. They are homotopically equivalent. Each winding number is an element of the DISCRETE homotopy group PIr(V).

r = 1 is the fundamental homotopy group PI1(A,V) where A is an arbitrary fixed point in V from which the cycles in the fiber space of Goldstone phases start and finish.
This fundamental group can be non-Abelian (e.g. cholesteric liquid crystal). Usually we can drop the A notation.

For superfluid helium 4 with only ONE Goldstone phase

PI1(S1) = Z


If the fundamental homotopy group is abelian with a set of generators of order pi for a set of winding numbers {Ni}, fusion of the singularities means that the Ni add up mod pi. For example if p = 2, then 1 + 1 = 0 like particle-antiparticle annihilation.

Energy barriers can separate different defects each with same set {Ni} of winding numbers in different regions of physical base space. That is, the formal homotopy equivalence is not physically complete.

Imagine collisions of vortices, splitting of vortices, fusion of them. Think "strings" for "vortices".

e.g. a split where 2 flux quanta -> 1 + 1

there may be an energy barrier Fc(coherence length)^2 for this reaction, need to tunnel through it.

S1 needs to be continued into whole complex plane because in these reactions (Higgs) changes in intermediate states along the reaction pathway Fc(coherence length)^3 (Volovik ...)

Conservation laws of string 1-Brane reactions are

Sum over j of Nj^i(mod p^i) = N^i fixed invariant

for all singularities inside a closed region.

Homotopic Group PI2(V) and Point Defects for 2-BRANES (e.g. isotropic ferromagnet, physical vacuum?) dimV = n = 3, the degeneracy parameters lie on the spherical surface S2 in fiber space (not physical base space). The two Goldstone Phase angles are in fiber space. There is a map from a direction in 3D physical space (with polar and azimuthal angles) to the two Goldstone phases. The ferromagnet is a bit misleading because the magnetization there is the order parameter and so the two Goldstone phases in that special case are identical with the polar and azimuthal angles in physical base space. But that case is not general enough. It's a degenerate case.

Now we have d' = 0, r' = 2, n = 3 for the stable point defect in physical space.

V = S2

In point defect collisions we may need to go off S2 into the entire 3D FIBER SPACE.

There are no stable string singularities for the n = 3 order parameter with two independent Goldstone phases in the ground state manifold fiber space, i.e. PI1(S2) = 0, i.e. identity group (ZERO FLUX-WITHOUT-FLUX QUANTA through the closed 1-loops.

There can be metastable string defects maintained by potential energy barriers.

The 2 order parameter Goldstone Phases are not defined at the isolated point defect because their corresponding Higgs Amplitudes are zero at that isolated point in physical base space.

Imagine a physical closed surface s2 of dim r = 2 surrounding the point defect of dim d' = 0 in physical space of dim d = 3. There is a map from points of s2 to points of the fiber space S2 = V of dim n = 3.

Here r = n - 1 = 2

Hedgehog "a field of the type m(r) = r^, a singular point at the origin (in physical space) where r^, theta^, phi^ are the UNIT VECTORS (local triad) of a spherical coordinate system in ordinary space), then the sphere s2 surrounding the singular point is mapped by the function m(r) on the entire sphere S2" not homotopic to the identity. The image point on S2 cannot be continuously deformed on S2 to a point on S2. You can only do so by GOING OFF S2 into the entire 3D fiber space, which can happen in brane reactions that "change topology".

Note that PI2(S1) = 0 i.e. no topologically stable point hedgehogs in superfluid helium with n = 2.

But PI2(S2) = Z where n = 3. Here the integer N is the WRAPPING NUMBER of times the order parameter (2 Goldstone Phases THETA & PHI) ranges over the entire S2 when a point in physical space moves over all of s2 ONCE.

That is, the physical polar and azimuthal angles theta & phi vary over

0 < theta < pi, 0 < phi < 2pi inducing

deltaTHETA = N2pi

deltaPHI = N2pi

if only one N.

We obviously can also have {Ntheta =/= Nphi}.

However, when Ntheta = Nphi this is the "degree of the mapping".

N = (1/4pi)Surface Integral over theta & phi in physical space of Gaussian curvature of the surface for which m(r) is the normal.

In the special case of the Hedgehog m(r) is the special function +-r^ (radial unit vector in the local spherical polar triad. This corresponds to N = + 1 & -1 respectively.

If and only if PI1(V) = 0, then there is a 1-1 correspondence between the point singularities of all N with the elements of PI2(V) (which is always Abelian). If PI1(V) is non-trivial then the classification of the point defects is more complicated.


The NASA Pioneer Anomaly shows a Hedgehog vacuum point defect needing TWO Goldstone Phases, i.e. a 2-component "internal" Vacuum Higgs Condensate SPINOR - on a scale of tens of AUs. Does every star contain such a defect at its center?

I had written

This NASA Pioneer Anomaly must correspond to a 2-component macro-quantum "SPINOR" c-number vacuum order parameter PSIi, i = 1,2 each PSIi is a complex function of space-time.

The effective Landau-Ginzburg potential then must be of the form

V = a|PSI1 + PSI2|^2 + b|PSI1 + PSI2|^4

We can pull out the absolute phase of say PSI1 to get

V = a||PSI1| + e^iphi|PSI2||^2 + b||PSI1| + e^iphi|PSI2||^4

= a[|PSI1|^2 + |PSI2|^2 + 2|PSI1||PSI2|cosphi]

+ b[|PSI1|^2 + |PSI2|^2 + 2|PSI1||PSI2|cosphi]^2

i.e. the ABSOLUTE PHASE will be as in the U(1) Mexican Hat Potential Picture and there will be an internal phase degree of freedom in the VEV.

We have only begun to scratch the surface of the physical vacuum structure here.

On the other hand, if we think of the spinor in terms of Dirac's Bra-Ket then there may not be any relative interference between the two components. In that case we would have two decoupled degrees of freedom i.e.

The effective Landau-Ginzburg potential would then be of the form

V = a|PSI1|^2 + a'|PSI2|^2 + b|PSI1|^4 + b'|PSI2|^4

We no longer have Feynman's micro-quantum rules on adding amplitudes for indistinguishable alternative histories in this new macro-quantum domain.


On Oct 27, 2005, at 12:20 PM, Jack Sarfatti wrote:

2. Superfluid String Vortices

Was Descartes correct after all? ;-)
"Still, in spite of its crudeness and its inherent defects, the theory of vortices marks a fresh era in astronomy, for it was an attempt to explain the phenomena of the whole universe by the same mechanical laws which experiment shews to be true on the earth."
http://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html
String theory is now the fashion, but at the very tiniest level of course.

Superfluid helium II has a single component local complex macro-quantum order parameter.

PSI = (Higgs Amplitude)e^i(Goldstone Phase)

Therefore

dimV = n = 2

V has the topology of S1 the unit circle on the plane (fiber) - locus of points in the Goldstone phase fiber for arbitrary fixed non-zero value of Higgs amplitude.

Stable defects obey

d' = d - n = 3 - 2 = 1

Therefore the stable topological defects in this Galilean system are lines or string defects in the physical base space of the order parameter fiber bundle.

The surrounding hypersurface has dim r

1 + d' + r = d

1 + 1 + r = 3

r = 1

i.e. surround the line defect with a closed 1D loop.

This loop is a NON-BOUNDING CYCLE because it encloses a singularity in the physical space where the Goldstone Phase is undefined because the Higgs Amplitude is ZERO on the the singular line in 3D physical base space.

The homotopy group PI1(S1) = Z

i.e. integer winding numbers from single-valuedness of PSI in a single non-bounding loop in physical base space that corresponds to N windings in V fiber space if the vortex has circulation Nh/m.

Flux without flux

Including the singularity we use a PSEUDO-Stoke's theorem as a DEFINITION of an EFFECTIVE VORTICITY FLUX

The non-vanishing loop integral of the superfluid velocity

vs = (h/2pim)'Grad'(Goldstone Phase)

is defined to be the surface integral of curlvs on the interior to this non-bounding loop.

*Of course, the rigorous Stoke's theorem only works for a bounding loop and this loop does not bound. But physicists have different standards of rigor. Since the non-bounding loop is far from the vortex core and since we do not directly measure inside the vortex core in these experiments, it's AS IF there were a vorticity inside the loop in the core where the Goldstone Phase is ill-defined. This is a kind of NONLOCAL Bohm-Aharonov effect since the LOCAL curl of vs on the loop far outside the vortex core is zero, but the interior surface integral of the curl is not zero because we include the singularity. This is like integrating around a pole in the theory of complex functions of a single complex variable.

That is, if theta is the angle of rotation in around the single 1D loop in 3D base space, then the Goldstone phase is THETA = Ntheta for a vortex singularity with N quanta of circulation.

PSI(N) = (Higgs)e^iNtheta

for that vortex string singularity.

(Higgs) = 0 on the string singularity

The scale over which Higgs spontaneously rises from zero to its asymptotic constant value is the vortex core size, AKA "coherence length". There is ZPF and normal fluid inside the core whose relative amounts depend on temperature T and pressure P. This is not the Goldstone phase coherence length, which is effectively infinite, i.e. over entire pot of superfluid that is one giant quantum system with coherent ZPF that is locally random, but globally non-locally Einstein-Podolsky-Rosen (EPR) correlated. This is distinct from the condensate density that is not locally random at all.

Superfluid Density = Condensate Density + Coherent ZPF Density

Total Density = Superfluid Density + Normal Fluid Density

The Coherent ZPF Density is virtual inside the ground state (at T = 0).

The Normal Fluid Density are classically thermally excited quasi-particles and possibly collective modes outside the ground state. The normal fluid density is zero at absolute zero. The locally random, but nonlocally EPR phase-locked ZPF density dominates the locally non-random smooth condensate at T = 0 in HeII.

To be more precise at T = 0 degrees Kelvin:

Superfluid Density = |Higgs Amplitude|^2 + Virtual ZPF Density

At finite T:

Total Density = Superfluid Density + Real Normal fluid Density

For a pot of liquid HeII below the critical lambda temperature |Higgs Amplitude| is fixed (uniform and stationary) at C(T,P) minimizing the condensation thermodynamic Landau-Ginzburg semi-phenomenological Free Energy Density Fc(|Higgs Amplitude|).

The uniform stationary Goldstone Phase Theta is the degeneracy parameter on

V = G/H = S1.

In NON-EQUILIBRIUM both Higgs Amplitude and Goldstone Phase (they live in the fiber space) are inhomogeneous and dynamic in the physical base space of the fiber bundle. There is then an additional gradient Free Energy Density Fgrad that depends on gradients in space and time of both the Higgs and the Goldstone macro-quantum degrees of freedom.

The best studied case for HeII is the IR (Infra-Red) steady weakly inhomogeneous one where the Higgs and Goldstone fields vary slowly relative to the vortex core "coherence length". In this regime, we can do time-independent perturbation theory since

Fgrad << Fc

In effect, |Higgs| ~ uniform homogeneous and the main variation is in the Goldstone Phase field.

Fgrad ~ (1/2)(Superfluid Density)vs^2

vs = (h/2pim)Grad(Goldstone Phase)

*This gives "phase rigidity". Unlike the micro-quantum Bohm potential, which is fragile to warm environmental decoherence, the macro-quantum Bohm potential for the local order parameter is robust and permits signal nonlocality in violation of the no-cloning theorem of micro-quantum information theory. The Born probability interpretation does not work for the local giant quantum order parameters. See the papers by Antony Valentini. It is not easy to "collapse" a giant order parameter like it is for a pigmy micro-quantum wave function.

Inside the core Fgrad ~ Fc and Higgs -> 0. Note at T = 0 there is zero normal fluid, but Higgs --> 0 leaving only the ZPF inside the core. In the curved vacuum case

tuv(ZPF) = (c^4/8piG*)/\zpfguv

Where G* is the effective ZPF induced gravity from the Sakharov effect.

Let L be the effective short wave UV cutoff, therefore

L^2 = hG*/c^3

That is

tuv(ZPF) = (hc/L^2)/\zpfguv

The ZPF vacuum density is then

(hc/L^2)/\zpfg00.

Similarly in the superfluid, the vortex core coherence length is the effective short wave cutoff for smooth modulations of the Goldstone phase. This is like the lattice spacing for sound waves in a crystal lattice.

Therefore at T = 0 only:

(hc/(Vortex Core Size)^2)/\ ~ F - Fc

F = total free energy density of the liquid

For distances far from the vortex core Higgs ~ constant, and the single Goldstone Phase maps the points of the fluid onto the S1 circle fiber space. Each point in the stationary fluid has a S1 circle fiber and the value of the Goldstone Phase at that point in the fluid base space is a single point on the S1 circle fiber.

Stability of the vortices. Physically, the unstable vortices can be eliminated by a continuous deformation of the Goldstone Phase field. The closed loop l in the physical base space, is mapped into a closed loop L in the S1 fiber space (ASSUMING NO TORSION!). If the vortex is UNSTABLE, then the closed loop L is the NO-LOOP, i.e. L = 0 is a fixed point on the circle fiber S1 in the following sense. The image point on S1 begins to move away from the initial point on S1 in a clockwise sense, but then returns to it in a counter-clockwise movement in a complete single circuit in the physical base space. These reversals in fiber space can happen more than once of course in the single circuit in base space. In contrast, on the other hand, if the image point of the mapping Goldstone Phase (x) -> S1 goes around the circle fiber in a STEADY WAY IN A FIXED CIRCULATION SENSE NEVER REVERSING an integer number of 2pi circuits for a single circuit in base space around the singular vortex string, then the vortex is stable. Obviously the unstable vortex has ZERO FLUX-WITHOUT-FLUX quanta through the closed loop's interior singular family of surfaces in physical base space. This is the physical meaning of the homotopy group formula:

PI1(S1) = Z


On Oct 26, 2005, at 5:38 PM, Jack Sarfatti wrote:


1."Spontaneous broken symmetry" AKA "More is different" AKA Bottom -> Up "Emergent Order" beyond reductionism.

Homogeneous equilibrium special case: The equilibrium state for homogeneous control parameters (e.g. external EM fields, temperature, pressure ...) are degenerate with respect to some subset of control parameters. There is an entire "Equilibrium State Manifold" of non-equivalent states for different values of the subset of control parameters with the same thermodynamic potential.

Example: Superfluid helium in homogeneous thermal equilibrium at absolute temperature T. The control parameter is the Goldstone Phase "Theta" whose manifold is the unit circle S1 on a plane. The square of the Higgs amplitude is the superfluid density. That is,

Local Macro-Quantum Zero Entropy U(1) Order Parameter in S1 manifold is

PSI = |Higgs Amplitude(x)|e^i(Goldstone Phase)

Superfluid Number Density Per Unit Volume is |Higgs Amplitude|^2

Coherent Superfluid Density + Incoherent Normal Fluid Density = Constant

as temperature, pressure, external fields, rotation vary.

Normal Fluid Density = 0 at Absolute Zero Temperature.

Superfluid Density = 0 at Lambda Point critical temperature.

ODLRO Condensate Density =/= Superfluid Density

But they are linearly proportional.

Macro-Quantum ODLRO Condensate Density + Micro-Quantum Zero Point Jiggle Density = Phenomenological Superfluid (or Supersolid) Density

Finite Temperature adds additional "density matrix" classical jiggle.

Robert Becker has a good intuitive description here. At Absolute Zero where total classical entropy vanishes, locally the Zero Point Jiggle is completely random, but the random jiggle is phase-locked over the entire sample. That is, perfect Einstein-Rosen-Podolsky nonlocal correlation of the local random jiggle in space and time.

In the case of the virtual processes inside the physical vacuum, the Quantum Zero Point Jiggle Density is either Dark Energy or Dark Matter depending if the Zero Point Pressure is negative or positive respectively. Lorentz invariance + Equivalence Principle imply

w = Pressure/Energy Density = -1

for all locally random, but globally coherent, Einstein-Podolsky-Rosen nonlocally correlated micro-quantum zero point jiggle motion inside the vacuum (or degenerate ground state for on-mass-shell excited states outside the vacuum).

Anti-gravitating Dark Energy has POSITIVE zero point jiggle energy density with equal and opposite NEGATIVE PRESSURE. Gravitating Dark Matter is the exact opposite.

Inhomogeneous States: Now the degeneracy parameters (e.g. Phase and amplitude of the local order parameter) depend on space and time.

TOPOLOGICAL OBSTRUCTIONS OR DEFECTS AKA SINGULARITIES
At isolated points, on lines, or on surfaces (walls) one may find, depending on the topology of the manifold of degenerate vacuum/ground states of the effective emergent dynamical fields, REGIONS WHERE THE DEGENERACY PARAMETER IS NOT DEFINED.

Example, the U(1) Goldstone Phase of Superfluid Helium is not defined at the stringy vortex cores where the Superfluid Density (square of Higgs Amplitude) VANISHES.

Note, in the case of the actual physical vacuum of our universe, the core of the defect will contain the pre-inflation false vacuum phase without gravity or inertia.

Enter "Goldstone Coherent Phase Rigidity", e.g. "Space-Time Stiffness" AKA "String Tension" --> "Brane Tension" i.e. effective energy barrier against environmental decoherence of the emergent macro-quantum coherent order (e.g. conscious human mind field): "this singular point, or line" [or domain wall] "cannot be eliminated without destroying at the same time the ordered state in a large volume ..."
G.E. Volovik, V.P. Mineev "Investigations of singularities in superfluid He3 in liquid crystals by the homotopic topology methods" Sovietsky JETP 1977 reprinted in "Topological Quantum Numbers in Nonrelativistic Physics" David J. Thouless (World Scientific, 1998)

In the U(1) S1 order parameter of superfluid helium HeII, the quantized circulation vortex is a singular line in which the ground state degeneracy parameter in the Mexican Hat Potential of the emergent macro-quantum Landau-Ginzburg eq. replacement of the micro-quantum Schrodinger eq,, i.e. the now inhomogeneous Goldstone Phase Theta(x) changes by 2Npi after circling this vortex line in physical 3D space an integer "winding number" N full circuits in either right-hand or left-hand sense, i.e. + & - integers. The Goldstone phase Theta(x) is undefined on the singular vortex line itself which is a continuous locus of zeros, or branch cut, of the Higgs field amplitude.

You need to destroy the superfluid coherence in a large volume of helium to eliminate the vortex. This gives the vortex a robust stability.

Note that tornadoes and even hurricanes also have metastable vortices, but they are not macro-quantum.

These seemingly local topological defects in physical 3D base space have nonlocal global properties in the associated fiber space of degenerate ground/vacuum states.

We are interested in STABLE topological defects in the order parameter fiber space that induces singular subregions in physical 3D base space where the degeneracy parameters distinguishing different points of the fiber are undefined. Therefore, the defect in the fiber of order parameters corresponds is a FUZZINESS or FOG that maps to a singular region of the base space where the Higgs intensity of the coherent order vanishes.

The Higgs amplitude and Goldstone phase of the local order parameter (a single multi-dimensional point in the fiber) are canonically conjugate (complementary) quadratures in the "phase space" of the emergent macro-quantum coherent order.

A precise zero in the Higgs amplitude wipes out all discrimination in the conjugate Goldstone phase just like knowing exactly and precisely WHERE an electron is wipes out all knowledge of the speed of the electron.

The Glauber coherent states of large numbers of bosons condensed into the same single-boson quantum state with squeezing of the conjugate quadrature zero point noise fluctuations is the obvious mathematics for these local macro-quantum order parameters with zero thermodynamic entropy.

Problem
What results from squeezing the Higgs amplitude quadrature of the local order parameter? What results from squeezing the complementary Goldstone phase?

The homotopy groups classify the topological defects. In particular they identify the stable topological defects. Each stable topological defect is in 1-1 correspondence with one element of the relevant homotopy group that is NOT the identity. Any topological defect that is associated with the identity is not stable.

Recall from an earlier message.

Later this will generalize to fractal non-integer dimensions I would suppose.

Physical 3D space (or 4D space-time depending on the problem - or ND boson hyperspace) has dimension d.

The singularity inside physical space of dimension d has dimension d' < d.

The singularity is "surrounded" by a subspace of dimension r inside physical space.

Therefore,

1 + d' + r = d

all of the above inside of physical space, i.e. base space of the fiber bundle.

Next we go to the spontaneous broken symmetry V fiber space of degenerate ground/vacuum/equilbrium etc. states depending on which problem we are doing. This is a very general scheme.

dimV = n = dim of "Vacuum Manifold" fiber in the key problem of interest here.

Theorem: STABLE TOPOLOGICAL DEFECTS obey

d' = d - n

Example: Superfluid Helium 4 AKA HeII. V = S1 = U(1) i.e. n = 2. Think of unit circle S1 in the 2D plane. The broken internal symmetry group of the Goldstone phase here is U(1).

Note, the Higgs amplitude is factored out in the definition of the V AKA Vacuum Manifold. Basically only the Goldstone Phases matter in the Homotopy. However, singularities are ZEROS of the Higgs amplitude where the conjugate Goldstone phases are undefined. Sn-1 are the unit spheres embedded in n-dim.

If in another case G/H = S0, these are the two points +1 & -1 on a line where n = 1

If V = S2, then n = 3 e.g. ferromagnet

V = Sn-1

dimV = n

is a class of possible topological defects.

Another is

V = Pn-1

where P is the real projective space on n - 1 dimensions.

Definition of the physical Homotopy Groups PI

PIr(V)

This is a MAP of a point in the surrounding subspace of dim r to the degenerate manifold of V of dim n.

Given 2 such maps, if one can be continuously deformed into the other then they are equivalent. Each homotopy group element corresponds to an infinite equivalence class of maps that can be continuously deformed into each other. That is the homotopy group is itself a quotient group of non-overlapping cosets mod the just given equivalence relation.

Theorem

if V = Sn-1 the unit sphere boundary of n-dim FIBER sub-space that splits it into two pieces if the FIBER n-space is simply-connected

Then the MAPS from SURROUNDING SUBSPACE of PHYSICAL BASE SPACE to the FIBER SPACE G/H of DEGENERATE VACUA of the SPONTANEOUS BROKEN SYMMETRY, where at least in some instances G --> H(normal subgroup of G)

are

PIr(Sn-1) = 0 for r < n-1 UNSTABLE DEFECTS

PIn-1(Sn-1) = Z the group of all integers (winding numbers) STABLE DEFECTS

Also

PIr(Pn-1) = PIr(Sn-1) for r > 1

And

PI1(Pn-1) = Z2 i.e. integers mod 2 STABLE

Ref: "Principles of a Classification of Defects in Ordered Media"
G. Tolouse, M. Kleman, 1976 reprinted in Thouless op-cit.


The NASA Pioneer Anomaly looks like a hedgehog topological defect in the physical vacuum i.e.



In the NASA Pioneer data, the arrows point inward to Sun at center in physical space of dim d = 3. The arrows in physical space are of EQUAL LENGTH between the 2 concentric spherical boundaries. The first spherical boundary is at the orbit of Jupiter ~ 20 AU from the Sun. This can only happen if the vacuum order parameter has dim n = 3 for a point defect of dim d' = 0 in the center of the Sun. The vacuum manifold G/H has the topology of S2 which, contingently in this case, also is the same topology as the surrounding regions isolating the point defect.

Each arrow has length a_g = -cH(t) = 1 nanometer per sec^2

H(t) = a(t)^-1da(t)/dt

a(t) is the cosmological scale parameter of expanding space.

Obviously then, n = 3 and d' = 0 and r = 2.

The only stable defect will be at PI2(S2) = Z

V = S2

The S2 unit sphere has 2 Goldstone phases. Recall that S1 has only 1 Goldstone phase.

What about d' + 1 + r = d

i.e. 0 + 1 + 2 = 3

So that here d = 3 physical space with a point defect, but the order parameter FIBER space is 3D.

d' = d - n for stability is obeyed

i.e. 0 = 3 - 3.

Remember that the Goldstone phases live in the fiber space V = G/H of dim n not in physical space of dim d. In this special case however n = d because d' = 0.

This NASA Pioneer Anomaly must correspond to a 2-component macro-quantum "SPINOR" c-number vacuum order parameter PSIi, i = 1,2 each PSIi is a complex function of space-time.

The effective Landau-Ginzburg potential then must be of the form

V = a|PSI1 + PSI2|^2 + b|PSI1 + PSI2|^4

We can pull out the absolute phase of say PSI1 to get

V = a||PSI1| + e^iphi|PSI2||^2 + b||PSI1| + e^iphi|PSI2||^4

= a[|PSI1|^2 + |PSI2|^2 + 2|PSI1||PSI2|cosphi]

+ b[|PSI1|^2 + |PSI2|^2 + 2|PSI1||PSI2|cosphi]^2

i.e. the ABSOLUTE PHASE will be as in the U(1) Mexican Hat Potential Picture and there will be an internal phase degree of freedom in the VEV.

We have only begun to scratch the surface of the physical vacuum structure here.

Vacuum Topology 2 Flux without flux
2. Superfluid String Vortices

Was Descartes correct after all? ;-)
"Still, in spite of its crudeness and its inherent defects, the theory of vortices marks a fresh era in astronomy, for it was an attempt to explain the phenomena of the whole universe by the same mechanical laws which experiment shews to be true on the earth."
http://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html
String theory is now the fashion, but at the very tiniest level of course.

Superfluid helium II has a single component local complex macro-quantum order parameter.

PSI = (Higgs Amplitude)e^i(Goldstone Phase)

Therefore

dimV = dim(G/H) = n = 2

V = G/H has the topology of S1 the unit circle on the plane (fiber) - locus of points in the Goldstone phase fiber for arbitrary fixed non-zero value of Higgs amplitude.

Stable defects obey

d' = d - n = 3 - 2 = 1

Therefore the stable topological defects in this Galilean system are lines or string defects in the physical base space of the order parameter fiber bundle.

The surrounding hypersurface has dim r

1 + d' + r = d

1 + 1 + r = 3

r = 1

i.e. surround the line defect with a closed 1D loop.

This loop is a NON-BOUNDING CYCLE because it encloses a singularity in the physical space where the Goldstone Phase is undefined because the Higgs Amplitude is ZERO on the the singular line in 3D physical base space.

The homotopy group PI1(S1) = Z

i.e. integer winding numbers from single-valuedness of PSI in a single non-bounding loop in physical base space that corresponds to N windings in V fiber space if the vortex has circulation Nh/m.

Flux without flux

Including the singularity we use a PSEUDO-Stoke's theorem as a DEFINITION of an EFFECTIVE VORTICITY FLUX

The non-vanishing loop integral of the superfluid velocity

vs = (h/2pim)'Grad'(Goldstone Phase)

is defined to be the surface integral of curlvs on the interior to this non-bounding loop.

*Of course, the rigorous Stoke's theorem only works for a bounding loop and this loop does not bound. But physicists have different standards of rigor. Since the non-bounding loop is far from the vortex core and since we do not directly measure inside the vortex core in these experiments, it's AS IF there were a vorticity inside the loop in the core where the Goldstone Phase is ill-defined. This is a kind of NONLOCAL Bohm-Aharonov effect since the LOCAL curl of vs on the loop far outside the vortex core is zero, but the interior surface integral of the curl is not zero because we include the singularity. This is like integrating around a pole in the theory of complex functions of a single complex variable.

That is, if theta is the angle of rotation in around the single 1D loop in 3D base space, then the Goldstone phase is THETA = Ntheta for a vortex singularity with N quanta of circulation.

PSI(N) = (Higgs)e^iNtheta

for that vortex string singularity.

(Higgs) = 0 on the string singularity

The scale over which Higgs spontaneously rises from zero to its asymptotic constant value is the vortex core size, AKA "coherence length". There is ZPF and normal fluid inside the core whose relative amounts depend on temperature T and pressure P. This is not the Goldstone phase coherence length, which is effectively infinite, i.e. over entire pot of superfluid that is one giant quantum system with coherent ZPF that is locally random, but globally non-locally Einstein-Podolsky-Rosen (EPR) correlated. This is distinct from the condensate density that is not locally random at all.

Superfluid Density = Condensate Density + Coherent ZPF Density

Total Density = Superfluid Density + Normal Fluid Density

The Coherent ZPF Density is virtual inside the ground state (at T = 0).

The Normal Fluid Density are classically thermally excited quasi-particles and possibly collective modes outside the ground state. The normal fluid density is zero at absolute zero. The locally random, but nonlocally EPR phase-locked ZPF density dominates the locally non-random smooth condensate at T = 0 in HeII.

To be more precise at T = 0 degrees Kelvin:

Superfluid Density = |Higgs Amplitude|^2 + Virtual ZPF Density

At finite T:

Total Density = Superfluid Density + Real Normal fluid Density

For a pot of liquid HeII below the critical lambda temperature |Higgs Amplitude| is fixed (uniform and stationary) at C(T,P) minimizing the condensation thermodynamic Landau-Ginzburg semi-phenomenological Free Energy Density Fc(|Higgs Amplitude|).

The uniform stationary Goldstone Phase Theta is the degeneracy parameter on

V = G/H = S1.

In NON-EQUILIBRIUM both Higgs Amplitude and Goldstone Phase (they live in the fiber space) are inhomogeneous and dynamic in the physical base space of the fiber bundle. There is then an additional gradient Free Energy Density Fgrad that depends on gradients in space and time of both the Higgs and the Goldstone macro-quantum degrees of freedom.

The best studied case for HeII is the IR (Infra-Red) steady weakly inhomogeneous one where the Higgs and Goldstone fields vary slowly relative to the vortex core "coherence length". In this regime, we can do time-independent perturbation theory since

Fgrad << Fc

In effect, |Higgs| ~ uniform homogeneous and the main variation is in the Goldstone Phase field.

Fgrad ~ (1/2)(Superfluid Density)vs^2

vs = (h/2pim)Grad(Goldstone Phase)

*This gives "phase rigidity". Unlike the micro-quantum Bohm potential, which is fragile to warm environmental decoherence, the macro-quantum Bohm potential for the local order parameter is robust and permits signal nonlocality in violation of the no-cloning theorem of micro-quantum information theory. The Born probability interpretation does not work for the local giant quantum order parameters. See the papers by Antony Valentini. It is not easy to "collapse" a giant order parameter like it is for a pigmy micro-quantum wave function.

Inside the core Fgrad ~ Fc and Higgs -> 0. Note at T = 0 there is zero normal fluid, but Higgs --> 0 leaving only the ZPF inside the core. In the curved vacuum case

tuv(ZPF) = (c^4/8piG*)/\zpfguv

Where G* is the effective ZPF induced gravity from the Sakharov effect.

Let L be the effective short wave UV cutoff, therefore

L^2 = hG*/c^3

That is

tuv(ZPF) = (hc/L^2)/\zpfguv

The ZPF vacuum density is then

(hc/L^2)/\zpfg00.

Similarly in the superfluid, the vortex core coherence length is the effective short wave cutoff for smooth modulations of the Goldstone phase. This is like the lattice spacing for sound waves in a crystal lattice.

Therefore at T = 0 only:

(hc/(Vortex Core Size)^2)/\ ~ F - Fc

F = total free energy density of the liquid

For distances far from the vortex core Higgs ~ constant, and the single Goldstone Phase maps the points of the fluid onto the S1 circle fiber space. Each point in the stationary fluid has a S1 circle fiber and the value of the Goldstone Phase at that point in the fluid base space is a single point on the S1 circle fiber.

Stability of the vortices. Physically, the unstable vortices can be eliminated by a continuous deformation of the Goldstone Phase field. The closed loop l in the physical base space, is mapped into a closed loop L in the S1 fiber space (ASSUMING NO TORSION!). If the vortex is UNSTABLE, then the closed loop L is the NO-LOOP, i.e. L = 0 is a fixed point on the circle fiber S1 in the following sense. The image point on S1 begins to move away from the initial point on S1 in a clockwise sense, but then returns to it in a counter-clockwise movement in a complete single circuit in the physical base space. These reversals in fiber space can happen more than once of course in the single circuit in base space. In contrast, on the other hand, if the image point of the mapping Goldstone Phase (x) -> S1 goes around the circle fiber in a STEADY WAY IN A FIXED CIRCULATION SENSE NEVER REVERSING an integer number of 2pi circuits for a single circuit in base space around the singular vortex string, then the vortex is stable. Obviously the unstable vortex has ZERO FLUX-WITHOUT-FLUX quanta through the closed loop's interior singular family of surfaces in physical base space. This is the physical meaning of the homotopy group formula:

PI1(S1) = Z


On Oct 26, 2005, at 5:38 PM, Jack Sarfatti wrote:

1."Spontaneous broken symmetry" AKA "More is different" AKA Bottom -> Up "Emergent Order" beyond reductionism.

Homogeneous equilibrium special case: The equilibrium state for homogeneous control parameters (e.g. external EM fields, temperature, pressure ...) are degenerate with respect to some subset of control parameters. There is an entire "Equilibrium State Manifold" of non-equivalent states for different values of the subset of control parameters with the same thermodynamic potential.

Example: Superfluid helium in homogeneous thermal equilibrium at absolute temperature T. The control parameter is the Goldstone Phase "Theta" whose manifold is the unit circle S1 on a plane. The square of the Higgs amplitude is the superfluid density. That is,

Local Macro-Quantum Zero Entropy U(1) Order Parameter in S1 manifold is

PSI = |Higgs Amplitude(x)|e^i(Goldstone Phase)

Superfluid Number Density Per Unit Volume is |Higgs Amplitude|^2

Coherent Superfluid Density + Incoherent Normal Fluid Density = Constant

as temperature, pressure, external fields, rotation vary.

Normal Fluid Density = 0 at Absolute Zero Temperature.

Superfluid Density = 0 at Lambda Point critical temperature.

ODLRO Condensate Density =/= Superfluid Density

But they are linearly proportional.

Macro-Quantum ODLRO Condensate Density + Micro-Quantum Zero Point Jiggle Density = Phenomenological Superfluid (or Supersolid) Density

Finite Temperature adds additional "density matrix" classical jiggle.

Robert Becker has a good intuitive description here. At Absolute Zero where total classical entropy vanishes, locally the Zero Point Jiggle is completely random, but the random jiggle is phase-locked over the entire sample. That is, perfect Einstein-Rosen-Podolsky nonlocal correlation of the local random jiggle in space and time.

In the case of the virtual processes inside the physical vacuum, the Quantum Zero Point Jiggle Density is either Dark Energy or Dark Matter depending if the Zero Point Pressure is negative or positive respectively. Lorentz invariance + Equivalence Principle imply

w = Pressure/Energy Density = -1

for all locally random, but globally coherent, Einstein-Podolsky-Rosen nonlocally correlated micro-quantum zero point jiggle motion inside the vacuum (or degenerate ground state for on-mass-shell excited states outside the vacuum).

Anti-gravitating Dark Energy has POSITIVE zero point jiggle energy density with equal and opposite NEGATIVE PRESSURE. Gravitating Dark Matter is the exact opposite.

Inhomogeneous States: Now the degeneracy parameters (e.g. Phase and amplitude of the local order parameter) depend on space and time.

TOPOLOGICAL OBSTRUCTIONS OR DEFECTS AKA SINGULARITIES
At isolated points, on lines, or on surfaces (walls) one may find, depending on the topology of the manifold of degenerate vacuum/ground states of the effective emergent dynamical fields, REGIONS WHERE THE DEGENERACY PARAMETER IS NOT DEFINED.

Example, the U(1) Goldstone Phase of Superfluid Helium is not defined at the stringy vortex cores where the Superfluid Density (square of Higgs Amplitude) VANISHES.

Note, in the case of the actual physical vacuum of our universe, the core of the defect will contain the pre-inflation false vacuum phase without gravity or inertia.

Enter "Goldstone Coherent Phase Rigidity", e.g. "Space-Time Stiffness" AKA "String Tension" --> "Brane Tension" i.e. effective energy barrier against environmental decoherence of the emergent macro-quantum coherent order (e.g. conscious human mind field): "this singular point, or line" [or domain wall] "cannot be eliminated without destroying at the same time the ordered state in a large volume ..."
G.E. Volovik, V.P. Mineev "Investigations of singularities in superfluid He3 in liquid crystals by the homotopic topology methods" Sovietsky JETP 1977 reprinted in "Topological Quantum Numbers in Nonrelativistic Physics" David J. Thouless (World Scientific, 1998)

In the U(1) S1 order parameter of superfluid helium HeII, the quantized circulation vortex is a singular line in which the ground state degeneracy parameter in the Mexican Hat Potential of the emergent macro-quantum Landau-Ginzburg eq. replacement of the micro-quantum Schrodinger eq,, i.e. the now inhomogeneous Goldstone Phase Theta(x) changes by 2Npi after circling this vortex line in physical 3D space an integer "winding number" N full circuits in either right-hand or left-hand sense, i.e. + & - integers. The Goldstone phase Theta(x) is undefined on the singular vortex line itself which is a continuous locus of zeros, or branch cut, of the Higgs field amplitude.

You need to destroy the superfluid coherence in a large volume of helium to eliminate the vortex. This gives the vortex a robust stability.

Note that tornadoes and even hurricanes also have metastable vortices, but they are not macro-quantum.

These seemingly local topological defects in physical 3D base space have nonlocal global properties in the associated fiber space of degenerate ground/vacuum states.

We are interested in STABLE topological defects in the order parameter fiber space that induces singular subregions in physical 3D base space where the degeneracy parameters distinguishing different points of the fiber are undefined. Therefore, the defect in the fiber of order parameters corresponds is a FUZZINESS or FOG that maps to a singular region of the base space where the Higgs intensity of the coherent order vanishes.

The Higgs amplitude and Goldstone phase of the local order parameter (a single multi-dimensional point in the fiber) are canonically conjugate (complementary) quadratures in the "phase space" of the emergent macro-quantum coherent order.

A precise zero in the Higgs amplitude wipes out all discrimination in the conjugate Goldstone phase just like knowing exactly and precisely WHERE an electron is wipes out all knowledge of the speed of the electron.

The Glauber coherent states of large numbers of bosons condensed into the same single-boson quantum state with squeezing of the conjugate quadrature zero point noise fluctuations is the obvious mathematics for these local macro-quantum order parameters with zero thermodynamic entropy.

Problem
What results from squeezing the Higgs amplitude quadrature of the local order parameter? What results from squeezing the complementary Goldstone phase?

The homotopy groups classify the topological defects. In particular they identify the stable topological defects. Each stable topological defect is in 1-1 correspondence with one element of the relevant homotopy group that is NOT the identity. Any topological defect that is associated with the identity is not stable.

Recall from an earlier message.

Later this will generalize to fractal non-integer dimensions I would suppose.

Physical 3D space (or 4D space-time depending on the problem - or ND boson hyperspace) has dimension d.

The singularity inside physical space of dimension d has dimension d' < d.

The singularity is "surrounded" by a subspace of dimension r inside physical space.

Therefore,

1 + d' + r = d

all of the above inside of physical space, i.e. base space of the fiber bundle.

Next we go to the spontaneous broken symmetry G/H fiber space of degenerate ground/vacuum/equilbrium etc. states depending on which problem we are doing. This is a very general scheme.

dim(G/H) = n = dim of "Vacuum Manifold" fiber in the key problem of interest here.

Theorem: STABLE TOPOLOGICAL DEFECTS obey

d' = d - n

Example: Superfluid Helium 4 AKA HeII. G/H = S1 = U(1) i.e. n = 2. Think of unit circle S1 in the 2D plane. The broken internal symmetry group of the Goldstone phase here is U(1).

Note, the Higgs amplitude is factored out in the definition of the V = G/H Vacuum Manifold. Basically only the Goldstone Phases matter in the Homotopy. However, singularities are ZEROS of the Higgs amplitude where the conjugate Goldstone phases are undefined. Sn-1 are the unit spheres embedded in n-dim.

If in another case G/H = S0, these are the two points +1 & -1 on a line where n = 1

If G/H = S2, then n = 3 e.g. ferromagnet

G/H = Sn-1

dimG/H = n

is a class of possible topological defects.

Another is

G/H = Pn-1

where P is the real projective space on n - 1 dimensions.

Definition of the physical Homotopy Groups PI


PIr(G/H)

This is a MAP of a point in the surrounding subspace of dim r to the degenerate manifold of G/H of dim n.

Given 2 such maps, if one can be continuously deformed into the other then they are equivalent. Each homotopy group element corresponds to an infinite equivalence class of maps that can be continuously deformed into each other. That is the homotopy group is itself a quotient group of non-overlapping cosets mod the just given equivalence relation.

Theorem

if V = G/H = Sn-1 the unit sphere boundary of n-dim FIBER sub-space that splits it into two pieces if the FIBER n-space is simply-connected

Then the MAPS from SURROUNDING SUBSPACE of PHYSICAL BASE SPACE to the FIBER SPACE G/H of DEGENERATE VACUA of the SPONTANEOUS BROKEN SYMMETRY G --> H(normal subgroup of G)

are

PIr(Sn-1) = 0 for r < n-1 UNSTABLE DEFECTS

PIn-1(Sn-1) = Z the group of all integers (winding numbers) STABLE DEFECTS

Also

PIr(Pn-1) = PIr(Sn-1) for r > 1

And

PI1(Pn-1) = Z2 i.e. integers mod 2 STABLE

Ref: "Principles of a Classification of Defects in Ordered Media"
G. Tolouse, M. Kleman, 1976 reprinted in Thouless op-cit.


The NASA Pioneer Anomaly looks like a hedgehog topological defect in the physical vacuum i.e.



In the NASA Pioneer data, the arrows point inward to Sun at center in physical space of dim d = 3. The arrows in physical space are of EQUAL LENGTH between the 2 concentric spherical boundaries. The first spherical boundary is at the orbit of Jupiter ~ 20 AU from the Sun. This can only happen if the vacuum order parameter has dim n = 3 for a point defect of dim d' = 0 in the center of the Sun. The vacuum manifold G/H has the topology of S2 which, contingently in this case, also is the same topology as the surrounding regions isolating the point defect.

Each arrow has length a_g = -cH(t) = 1 nanometer per sec^2

H(t) = a(t)^-1da(t)/dt

a(t) is the cosmological scale parameter of expanding space.

Obviously then, n = 3 and d' = 0 and r = 2.

The only stable defect will be at PI2(S2) = Z

G/H = S2

The S2 unit sphere has 2 Goldstone phases. Recall that S1 has only 1 Goldstone phase.

What about d' + 1 + r = d

i.e. 0 + 1 + 2 = 3

So that here d = 3 physical space with a point defect, but the order parameter FIBER space is 3D.

d' = d - n for stability is obeyed

i.e. 0 = 3 - 3.

Remember that the Goldstone phases live in the fiber space V = G/H of dim n not in physical space of dim d. In this special case however n = d because d' = 0.

This NASA Pioneer Anomaly must correspond to a 2-component macro-quantum "SPINOR" c-number vacuum order parameter PSIi, i = 1,2 each PSIi is a complex function of space-time.

The effective Landau-Ginzburg potential then must be of the form

V = a|PSI1 + PSI2|^2 + b|PSI1 + PSI2|^4

We can pull out the absolute phase of say PSI1 to get

V = a||PSI1| + e^iphi|PSI2||^2 + b||PSI1| + e^iphi|PSI2||^4

= a[|PSI1|^2 + |PSI2|^2 + 2|PSI1||PSI2|cosphi]

+ b[|PSI1|^2 + |PSI2|^2 + 2|PSI1||PSI2|cosphi]^2

i.e. the ABSOLUTE PHASE will be as in the U(1) Mexican Hat Potential Picture and there will be an internal phase degree of freedom in the VEV.

We have only begun to scratch the surface of the physical vacuum structure here.