Eigenfunction Diffeomorphism

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Posted by xjgraviton on November 11, 2002 at 01:54:19:

Hello

Perturbative methods probably do not hold the key to a
"consistent and comprehensive" theory. Since there is
Godel's incompleteness theorem, a theory of reality
will always be incomplete. But can we use the concept
of "identity" to logically formulate the Eigenfunction
diffeomorphism?

A diffeomorphism is basically a map between manifolds
with the term "manifold" a topological space which is
locally Euclidean. An infinitely differentiable
bijection also with a differentiable inverse.

An "inverse expansion" would be material and radiative
contraction. A type of mathematical "inverse" of spatial expansion.

I figure that both expansion AND inverse expansion ARE
logically valid explanations!

Remember the "T-Duality" of string theory? A type of
isomorphism.

R<------>[1/R]

Really T-Duality says:

R<------->[(L_st)^2]/R

The physics for a circle of radius R is the same for a
circle of radius 1/R .

So a type of diagram for the "self organizing"(self
creating), self containing universe, would be:

{<-{->{U}<-}->}

Infinity and continuity would be explained as quantum
wavefunction "potential". Discrete finite particles
would be explained as "actualization".

How can the "relativistic effects" be described by
quantum wavefunctions, when the wavefunctions describe
the position and momentum of particles in a backround OF
space?

In quantizing spacetime geometry, we won't get
wavefunctions based on a background space. The space of wavefunctions can be thought of the space of square-integrable wavefunctions over classical configuration space. In ordinary quantum mechanics, configuration space is space itself (i.e.,to describe the configuration of a particle, location in space is specified). In general relativity, there is a more general kind of configuration space: taken to be the space of 3-metrics
("superspace", not to be confused with supersymmetric space) in the geometrodynamics formulation,or the space of connections (of an appropriate gauge group)in the Ashtekar/loop formulation. So the wavefunctions will be functions over these abstract spaces, not space itself-- the wavefunction _defines_ "space itself".

The *process* is the "function of functions".

With flat sheets, foliations of space, or equatorial planes, the light cone cross section corresponding to a circle would be a "rotated" light cone near a massive object. Using abstract generalizations of course!

The two light cones form a relationship, describing degrees of
rotation and circular-elliptic cross sections.

It should be possible to derive a set of equations from these
rotational perspective effects!

Non-Euclidean geometry has great explanatory power, yet there must be a type of "dynamics" involved, possibly related to a type of configuration space or varying density gradients. Static geometry cannot be the whole, complete explanation.

Mathematician Roger Penrose demonstrated how many of the properties of three dimensional space can be created out of networks of spinors, the simplest possible "quantum mechanical objects". These spinors are used to define the two possible values of an electron's spin.

He then generalized the spinor into a mathematical quantity called a twistor. The mathematics of complex numbers is used, which makes twistors hard to visualize. Geometrically, the notion of a point becomes more complicated, and secondary, defined by a conjunction of many individual twistors. A daunting approach mathematically.

Theoretically speaking, does the "absolute spacetime metric"
exist?...Yes. Relativity does not prove that spacetime geometry is in all respects relative. "All-everything is relative" cannot be true. The overall spacetime structure must be stable and symmetric. The stability and symmetry are ultimately related to the existence of an absolute metric. Absolute relativity.

Time cannot merely be added on to a theory via an assumption as in the ADM formalism, whith Lorentzian manifolds, diffeomorhic to R x S with manifold S reresenting "space" and "t" an element of R representing
time...

phi: M---> R x S

We need not invoke the "lumeniferous aether". The absolute metric must be a type of "meta-relation".

The laws of physics would be distributed over space-time. Thus the equivalence principle results from the law: conservation of
momentum-energy. This could also be interpreted as a type of Lenz' law for all mass-energy interacting with "space-time". The laws of physics are then a subset of mathematics.

Dr. Georg Cantor proved that a one dimensional line of length "s" has the same number of points as the 2-D plane with sides "s". In fact, the number of points on the line would be the same as 3-D, 4-D, ... n-D and higher dimensional space. Very interesting.

The gravitational field, described by the metric of spacetime g_uv , is generated by the stress-energy tensor T^uv of matter. Various field equations relating g_uv to T^uv have been proposed. The most succsessful have been the Einstein field equations which are of course, the foundation of general relativity.

G_uv == R_uv - 1/2 g_uv R = 8pi T_uv

where R_uv and R are the Ricci tensor and scalar curvature derived from the metric g_uv , and G_uv is the Einstein tensor. The equations are non-linear, since the left hand side is not a linear function of the metric.

When the gravitational field is weak, the geometry of spacetime is nearly flat and the equation is: g_uv = n_uv + h_uv

where all h_uv are << 1.

This linearized theory is very interesting.

But really, what is needed is a quantum theory of gravity-spacetime.

Successive sheets of space, sequential hypersurfaces parameterized by time, where time is not added on as an assumption.

Since the geometry of the universe can be explained as a type of
language with grammatical structure, and the structure of that language is relational, space has no existence outside the relationships between things in the universe.

As Lee Smolin says: "There is nothing outside the universe". A closed circuit.

I am reading the book "A Journey Into Gravity and Spacetime" by John Archibald Wheeler.

OK, "A directed or 'oriented' line - a one dimensional 'manifold' - has
for its boundary the starting point and the terminal point, both zero dimensional." A closed circuit.

The starting point is "negative".

The end point is "positive".

-A + A = 0.

-A------0------A

The starting point is also the ending point.

--------->
<---------

"The zero dimensional boundary of a one dimensional boundary of a two dimensional region is zero".

A true statement.

"The one dimensional boundary of the two dimensional boundary of a three dimensional region is zero".

A true statement.

Space is three dimensional. A cube or tetrahedron is three dimensional.

Also symmetrical