| Topic: |
Science > Physics |
| User: |
"Jack Sarfatti" |
| Date: |
27 Oct 2005 02:23:50 PM |
| Object: |
Topology of the Vacuum 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.
<hedgehog.gif>
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|^2
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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