"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.

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