Science > Physics > General Relativity, Quantum Field Theory & Fiber Bundles
| Topic: |
Science > Physics |
| User: |
"Jack Sarfatti" |
| Date: |
17 Nov 2003 06:25:53 PM |
| Object: |
General Relativity, Quantum Field Theory & Fiber Bundles |
Commentary 4
Synopsis of where we are at so far in the emergent evolution of our
understanding of how the mathematics of fiber bundles with a natural
idea of "hyperspace" and "Super Cosmos" (Linde's "chaotic inflation") is
interpreted as the physics of classical relativity, local quantum field
theory with the objective of using it also in the macro-quantum theory
of emergent Einstein gravity with exotic vacuum dark energy/matter for
"metric engineering" and possibly also in micro-quantum delocalized
string theory.
We have taken a top -> down approach for the "principal bundle". Start
with a large higher dimensional hyperspace H. Do not assume any metric
in it to begin with. Assume a CONTINUOUS Lie symmetry group G
equivalence relation ~ that partitions H into disjoint G-orbits that are
equivalence classes of points X of H where X' ~ X mod G. Each distinct
point of the base space M is a projection from a single G-orbit where M
= H mod G or H/G. The G-orbit is an internal hidden structure of the
base space event M that can include extra compactified boson dimensions
and also the fermi dimensions of supersymmetry. How Planck's h and
Heisenberg's uncertainty fit in is not apparent yet; The construction
so far seems classical. h seems to demand fractals that are continuous
but not differentiable like the classical manifolds are.
Hyperspace H is locally a product of a the "beyond space-time" fiber and
a small neighborhood of the base space.
Around each point x of base space M there is a coordinate patch C(x) and
a fiber f(x) and a special diffeomorphism Trivial (x) that maps H at x
into the product C(x)f(x). If the hyperspace is globally not oriented
like a one-sided Mobius strip or a Klein Bottle then Trivial(x) locally
unwraps the global twists. A transition function is
Trivial(x)Trivial(x')^-1 in the overlap of the local coordinate patch
neighborhoods around x and x' with different G-orbits (I think?)
6. There is a purely "vertical" inverse bottom -> up "emergent"
projection P^-1 from base space C(x) to fiber f(x).
P^-1 is a rule for associating each point fo in the fiber f(x) with a
group element g < G of the principal bundle for the gauge forces i.e.
electroweak + strong NOT gravity yet.
7. P^-1 does not establish a "horizontal" connection for identifying
points on different fibers f(x) and f'(x') in different regions of the
base space with the same continuous symmetry group element g in the
global group G.
The global Cartesian product space is "like a broad staircase with
vertical handrails." In contrast the fiber space is like "a set of
identical escalators moving up and down independently." S. Y. Auyang
"How is Quantum Field Theory Possible?"
p. 217, Oxford, 1995.
8. The local gauge force potential interaction dynamics allows parallel
transport of fiber information along continuous paths in the base space
of "control parameters", which in special applications can be the
space-time manifold, but generally it can be other kinds of spaces.
9. The ALL-IMPORTANT "section": A section is an inverse projection C(x)
-> P^-1[C(x)] mapping a neighborhood of base space back into a region of
hyperspace H. The section creates a local coordinate patch in the
hyperspace from the local coordinate patch in the base space by
arbitrarily CALIBRATING a single point in the vertical fiber fo(x) above
each x in C(x) as the identity e of G. If a single section works
globally for the whole hyperspace then the bundle is trivial like a
two-sided orientable cylinder not like a one-sided non-orientable Mobius
strip that resembles a spinor needing a 4pi rotation to return to its
original "normal vector."
The idea of connection is implicit in the idea of the section.
10. The special section called the "principal connection" maps the
tangent spaces of the base space to the tangent spaces of the
hyperspace. Let Tx(M) be a tangent space of M at point x. Let TX(H) be
the tangent space of hyperspace at hyper-point X. Then the principal
connection is
P^-1[Tx(M)] = TX(H)
X = (x,fo)
TX(H) = TX(H)horizontal + TX(H)vertical
TX(H)horizontal =Tx(M).
Note that Einstein's smooth c-number gravity is essentially from the
tangent bundle {M, Tx(M)} with an additional
"metric" or alternatively a "tetrad" spanning both x < M and Tx(M) that
embodies Einstein's Equivalence Principle (EEP). The symmetry group G
acts like the identity in Tx(M) and should not be confused with Diff(4)
in Einstein's gravity theory. The principal connection splits any path
in hyperspace into a horizontal path in base space and a vertical path
in the extended fiber region of hyperspace. Presumably we can extend
this from paths to world sheets for strings rather than points?
11. Given some principal connection |~ and a "worldline" in base space
M. The worldline can be "horizontally lifted" into the extra space
dimensions of the Calabi-Yau spaces (anticipating the string
generalization yet to come) such that all tangent vectors of the hyper
world line are horizontal. This is PARALLEL TRANSPORT IN HYPERSPACE as
distinct from parallel transport of world tensors along worldlines in
Einstein's gravity theory in the special tangent bundle [M, Tx(M)].
12. The horizontal lift of a M world line into hyperspace is UNIQUE and
this allows us to associate different points fo(x) and fo'(x') in
different NON-OVERLAPPING regions of hyperspace with disjoint patches
C(x) & C'(x') with the same g < G relative to that specific M worldline
connecting the two points.
13. EEP (Einstein's Equivalence Principle) of GR is an approximate
statement that: i. far from a space-time singularity and ii. at a scale
larger than
Lp^2 = hG(Newton)/c^3
One can "freely float/fall" feeling no weight (i.e. no g-force) along a
slower-than-light "time-like geodesic" in a non-rotating LIF (Local
Inertial Frame) with comfortably small stretch-squeeze torture rack
local curvature tidal force inhomogeneities in the g-force.
13. Thus GR is a specialized kind of fiber bundle not the same as the
fiber bundles in local quantum field theory. Indeed, I claim that the
former is emergent from a false vacuum instability in the latter.
On Monday, November 17, 2003, at 01:33 PM, Jack Sarfatti wrote:
Commentary 3
On Monday, November 17, 2003, at 02:05 AM, Arkadiusz Jadczyk wrote:
On 16 Nov 2003 at 15:30, Jack Sarfatti wrote:
The hyperspace H consists of "fibers" f(x) that are
either copies of or representations of the symmetry
group G.
Jack, this is not quite correct. They are "homogenous spaces" on
which the group operate transitively. Example, for the group SU(2),
you can take as the fibre a copy of SU(2) itself (3-dimensional), or
you can take sphere S^2, on which SU(2) operate (2-dimensional).
Notice that S^2 is not a "representation" of SU(2). It is a quotient
SU(2)/SO(2).
Early Kaluza-Klein theories were operating with group Manifolds.
Souriau, and later Witten, suggested more "realistic" theories where
fibers could be of lesser dimensions. Thie rigorous mathematics and
examples of this latter approach have been developed in the
monograph:
"Riemannian Geometry, Fibre Bundles, Kaluza-Klein Theories and All
That..." (World Scientific Lecture Notes in Physics, Vol 16)
by Robert Coquereaux, Arkadiusz Jadczyk :-)
ark
Thanks Ark. :-)
Principal fiber bundles for the local gauge forces:
1. A transformation g of the symmetry group G acts on the ordered pair X
= (x, fo) in "hyperspace" H with output gX.
Question: Can gx = x' =/=x i.e. can one move the base point in this
operation or must G always be the identity in the base space? That is,
we always need, in addition to G a "connection" and a "path" in order to
change location in the "horizontal" base space and the "vertical" fiber
space that is "beyond space-time." G certainly moves fo up and down the
vertical fiber for every element g =/= identity. Does it also move x ->
x' = gx =/= x horizontally along the base manifold without a connection
field and a path specified? Clearly the answer must be NO. See below.
"The modern understanding of gauge invariance, as a symmetry under
transformations of
quantum-mechanical wave functions, was reached by Weyl himself and also
by London very
shortly after the new quantum mechanics was first proposed. In this
understanding of
abelian gauge invariance, and in its nonabelian generalization [2], the
space-time aspect is
lost. The gauge transformations act only on internal variables. This
formulation has had
great practical success. Still, it is not entirely satisfactory to have
two closely related, yet
definitely distinct, fundamental principles, and several physicists have
proposed ways to
unite them.
One line of thought, beginning with Kaluza [3] and Klein [4], seeks to
submerge gauge
symmetry into general covariance. Its leading idea is that gauge
symmetry arises as a reflec-
tion in the four familiar macroscopic space-time dimensions of general
covariance in a larger
number of dimensions, several of which are postulated to be small,
presumably for dynam-
ical reasons.
Here we should take the opportunity to emphasize a point that is somewhat
confused by the historically standard usages, but which it is vital to
have clear for what
follows. When physicists refer to general covariance, they usually mean
the form-invariance
of physical laws under coordinate transformations following the usual
laws of tensor calculus,
i
cluding the transformation of a given, preferred metric tensor.
Without a metric tensor,
one cannot form an action principle in the normal way, nor in particular
formulate the ac-
cepted fundamental laws of physics, viz. general relativity and the
Standard Model. From
a purely mathematical point of view one might consider doing without the
metric tensor;
in that case general covariance becomes essentially the same concept as
topological invari-
ance. The existence of a metric tensor reduces the genuine symmetry to a
much smaller one,
in which space-times are required not merely to be topologically the
same, but congruent
(isometric), in order to be considered equivalent. In the Kaluza-Klein
construction, for this
reason, the gauge symmetries arise only from isometries of the
compactified dimensions.
Another line of thought proceeds in the opposite direction, seeking to
realize general
covariance – in the metric sense – as a gauge symmetry. "
arXiv:hep-th/9801184 v4 23 Apr 1998
IASSNS-HEP-97/142
Riemann-Einstein Structure from Volume and Gauge Symmetry
Frank Wilczek
Thanks to Tony Smith for alerting me to this relevant paper by Wilczek.
BTW Wilczek shows that Gennady Shipov's "torsion theory" is closely
related to
Roger Penrose's "spinors in curved spacetime" with the anti-symmetric
"spin connection" as the locally induced compensating torsion field.
It all comes from locally gauging the O(3,1) subgroup of the Conformal
Group
as I said previously based on Utiyama's and Kibble's papers from the
mid-1960's.
Whether or not Akimov's claims from Moscow that "torsion waves" from
O(1,3) of
sufficient intensity to have "psychotronic weapon"s bio-toxic effects
can easily be generated when,
in contrast, gravity waves from T4 are so hard to find is another issue
not considered here.
The gravity wave T4 coupling parameter is essentially Ed Witten's alpha'
= (superstring tension)^-1.
What is the corresponding O(1,3) spin connection coupling parameter?
Akimov's claims hang
on the answer to that question. Is it easier to make propagating torsion
dislocation topological string defects
than to make propagating curvature disclination topological string
defects in the MACRO-QUANTUM
Vacuum Coherence Field's "Goldstone Phase"? That's what Akimov's claims
come down to in terms of
my new theoretical paradigm for the emergence of Einstein's Gravity and
the Unified Exotic Vacuum Field of
w = -1 "Dark Energy/Matter."
2. The action of the symmetry group G on the total hyperspace H induces
an equivalence relation ~ .
That is, if X' = gX, g < G, then X' ~ X.
3. ~ partitions hyperspace H into disjoint non-overlapping equivalence
classes called "G-orbits"
G(X) = {gX, for all g < G}
Remember that in this principal bundle fo is also a g < G.
All G-orbits have identical structure and are diffeomorphic to G.
4. This disjoint partition of hyperspace H gives the quotient space H/G
that is the base space M with points x.
Every point x of the base space M is really an equivalence class or
"G-orbit" of a continuous infinity of points of a larger dimensional
Hermetic or occult hidden hyperspace implicate inside it. Worlds within
worlds. Wheels within wheels. Shades of Bohm's "Implicate Order"?
5. The Projection Map P is simply P:G-orbit -> x.
This means that each individual G-Orbit is really associated with a
single vertical fiber at a single horizontal base space event. The
G-orbit is the vertical fiber beyond, in the usual physics applications,
a localized spacetime event x, although we can have delocalized base
spaces of twistors whose intersections are points. We can also perhaps
have base spaces of finite strings both open and closed and even base
spaces of higher dimensional "brane worlds"?
On Sunday, November 16, 2003, at 09:05 PM, Jack Sarfatti wrote:
Commentary 2
Given coordinate patch C(x) in the base space M in a neighborhood of
point x and fiber f(x)
form the local Cartesian product C(x)f(x) with ordered pair X = (x,fo).
Take the union C(x)f(x)\/C(x')f(x')\/... of all such local products.
There are redundant ordered pairs X because the coordinate patches C(x)
and C(x') as sets overlap
with non-vanishing intersection C(x)/\C(x')=/= Empty Set.
Identify the redundant multiple images of the same actual point of the
base space M using
the symmetry group G as an equivalence relation. That is, two ordered
pairs X and X' are
identified or equivalent if x = x' < C(x)/\C(x') and if fo' = gfo where
g < G to form disjoint
equivalence classes {f(x)} that are the distinct points of the fiber in
hyperspace H.
This is all local at a fixed base point x like in an internal gauge
force symmetry.
g is also called a "transition function."
The hyperspace H is the factor space of the union
C(x)f(x)\/C(x')f(x')\/ ... mod G.
The projection map P:(x,{fo}) -> x
When M is the curved space-time of Einstein's gravity theory in addition
to the G equivalence
in the extra "space" dimensions of the fiber, x'(E) = Diff(4)x(E) at
fixed event E
to make disjoint equivalence classes {x(E)} mod Diff4(E).
One can imagine a hybrid where the fiber is a discrete space of strings
of c-bits.
One can also imagine a fiber of strings of qubits.
1 qubit is a parallel infinity of c-bits.
i.e.
|qubit> = |1 c-bit><1c-bit|qubit> + |0 c-bit><0 c-bit|qubit>
Where there is a continuous infinity of different c-bit bases
or orthonormal frames each corresponding, for example,
the the angular orientation of an inhomogeneous field
magnet in a Stern-Gerlach filter for spin qubits
in the DARPA "spintronics" project or like the billion billion
Single Electron Transistors inside the human brain at the
sub-microtubular protein dimer hydrophobic cage level forming
the hardware interface with external world whose software is our stream
of inner consciousness.
Each possible orientation is a primitive "parallel quantum universe."
The quantum computer computes in all possible
orientations simultaneously like a continuous
infinity of classical Turing machines in a
distributed network working on the same problem
- or so the folklore goes.
to be continued.
On Sunday, November 16, 2003, at 03:30 PM, Jack Sarfatti wrote:
Commentary 1
The "fiber bundle" as an idea has 4 parts.
1. A "structure" symmetry group G.
2. The "total" hyperspace H or, in some applications Wheeler's "BIT."
3. The "projection map" P.
4. The "base space" M or, in some applications. Wheeler's "IT."
The hyperspace H consists of "fibers" f(x) that are
either copies of or representations of the symmetry
group G.
The projection map P "collapses" a fiber f(x) in the hyperspace H to
a point x in the base space M.
All of these objects are continuum differential manifolds
depending on the continuum of real numbers which its
associated issues of Cantor's infinity of infinities of
Cabalistic Aleph's in an ascending "Jacob's Ladder".
This is not a discrete combinatoric mathematics although
such a skeletal structure is associated with it as in
Herman Weyl's "Theory of Groups and Quantum Mechanics"
and as in Saul-Paul Sirag's presentation of V.I. Arnold's
A-D-E "mathematics of everything."
The base space is covered by an "atlas" of "local coordinate patches"
with all important "overlap" "transition functions" sewing the
patches together like a quilt.
M is "space-time" in "local micro-quantum field" theory of point
particles and also in delocalized string theory.
The extra-dimensions of hyperspace form
the Calabi-Yau space of vibrations of the
superstring beyond space-time.
The "connection" on the total "hyperspace" H is the "potential"
of a "local gauge force."
Examples of "connections" is the 4 potential Au(x) in
Maxwell's electromagnetism with G as U(1).
There are similar connections for the Yang-Mills weak force
with G = SU(2) and the strong force with G = SU(3).
Classical general relativity, as distinct from local micro-quantum
field theory, has the torsion-free symmetric three-index non-tensor
Levi-Civita connection with G as the Diff(4) group.
The latter comes from locally gauging the 4 parameter translation
subgroup (generated by the 4-momentum Pu of globally flat special
relativity ) of the 15 parameter conformal group of Roger Penrose's
"massless twistors."
Bottom -> Up: Given base space M and symmetry group G construct the
hyperspace H as a quilt patchwork.
Top -> Down: Given hyperspace H and symmetry group G construct the
base space M as the non-overlapping partition of hyperspace into G-orbits
called the "quotient space" of H mod G in the "principal bundle."
Micro-quantum source renormalizable local fields of spin 1/2
lepto-quarks are associated "vector bundles".
Micro-quantum force renormalizable local fields of spin 1 gauge force
bosons (electro-weak and strong) are from the principal bundle.
There is no renormalizable quantum gravity in this precise sense.
This is because "classical" Einstein gravity is a "More is different"
(P.W. Anderson) emergent collective effect as in Andrei Sakharov's
"metric elasticity" of an instability in the globally flat "false
vacuum" of the interacting lepto-quark source/electroweak-strong force.
Einstein's gravity + unified exotic vacuum "dark energy/matter" with
Andrei Linde's "chaotic inflationary cosmology" are the result of the
continual phase transitions from globally flat false high entropy
micro-quantum vacua to locally curved macro-quantum low entropy
metastable vacua.
to be continued:
.
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