NASA PIONEER ANOMALY & VACUUM TOPOLOGY
"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.
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.
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.
G/H = Pn-1
where P is the real projective space on n - 1 dimensions.
Definition of the physical Homotopy Groups PI
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.
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)
PIr(Sn-1) = 0 for r < n-1 UNSTABLE DEFECTS
PIn-1(Sn-1) = Z the group of all integers (winding numbers) STABLE DEFECTS
PIr(Pn-1) = PIr(Sn-1) for r > 1
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.