Non Abelian Universe?
From: Russell E. Rierson (analog57_at_yahoo.com)
Date: 11/10/04
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Date: 10 Nov 2004 01:03:04 -0800
How can Stokes Theorem be used describe a succsession of
holographically encoded 2 dimensional surfaces, all causally
connected? The surfaces would all be closed geometric forms. An
example would be the 2 dimensional surface of a 3 dimensional sphere.
The the succsessive surfaces would be iterations from previous outer
sufaces, where the next surface is "inside" or projected inwardly
becoming an inner shell, from the larger previous iteration, and the
future iteration would be projected outwardly to the outer shells.
There would be no x-y plane though. How can it be done?
Photon paths are simply geodesics on the n-d space where arclength is
the proper time. Moreover, in view of Huygens' principle, the
elementary wavefronts are exemplified by the tangent
bundle[transverse-right angle] over the medium while the phase
surfaces are exemplified by the cotangent
bundle[longitudinal-parallel]. The relationship between these two
bundles is none other than the Legendre transformation. A Legendre
transform is where two differentiable functions, f nd g, have first
derivatives that are inverse functions of each other.
For vacuum spacetimes being asymptotically stationary
in past and future, such, that the paticles are always
created in pairs, where there will be a nonvanishing amplitude
for spontaneous particle creation - if and only if some
classical solution yields oscillations with purely
positive frequency in the asymptotic past consequently picks up
a nonvanishing negative frequency part in the asymptotic
future.
Ergo, there are two valid solutions for Maxwell's
equations, the Retarded Wave and the Advanced Wave.
So, basically, a linear dynamical system is simply a
collection of coupled harmonic oscillators whereby a linear field
in curved spacetime becomes essentially an infinite
collection of such harmonic oscillators. The empty vacuum of
space potentiates a dynamic energy feedback mechanism.
The mathematical apparatus of quantum mechanics
differs drastically from GR classical mechanics. A state in
quantum theory is represented by a vector in an infinite
dimenmsional Hilbert space rather than being represented by a point in
a finite dimensional manifold. An observable in quantum theory is
described by a self adjoint operator acting on the Hilbert space
instead of being represented as
a real valued function on the manifold.
There appears to be a slight discrepancy here...
Quantum mechanics and Einsteinian mechanics cannot disagree
with each other and still both be correct?
Differentially speaking, the definitive canonical formalism for the
cotangent bundle of a given configuration/phase space, exemplifies a
natural correspondence between the Hamiltonian vector fields that
govern the evolution of conservative ordinary differential equations
and the Hamiltonian functions which describe them. The natural arena
for the setting of tangent and cotangent bundles also provides a quite
natural setting for the Lagrangian description of the physical
dynamics and the Legendre transformation that connects the Lagrangian
and Hamiltonian points of view. In both cases, the use of vector
fields for the description of the dynamics is a most natural choice,
since, for ordinary differential equations, there is a single
distinguished variable, or parameter - time.
In contrast, for systems of partial differential equations who's
solutions depend on multiple variables, spatial as well as temporal,
one probably needs to recognize that a single vector field is not the
best point of view, because it would require collapsing all of the
spatial structure of a solution to a single point of phase space. This
will occur when a choice is made to consider the time coordinate
separately and describe the dynamics in terms of an
infinite-dimensional space of fields at a given instant in time.
Although this methodology has been very successful, availing itself to
the powerful organizing structure of the theory of evolution operators
from the point of view of functional analysis, its immediate affect is
a break of manifest generalized covariance.To maintain a covariant
description one can use a generalization of symplectic geometry known
as multisymplectic geometry.
Not to be jumping too far ahead, but consider the hypothetical
scenario - thought experiment, involving four hypothetical perfectly
equidistantly juxtaposed "co-moving" observers[A,B,C,D], with a flash
of light eminating from their equidistant center-point[P]:
A_________B
_____P
C_________D
The flash of light obeys what is known as Lorentz invariance, and will
reach all observers[A,B,C,D] simultaniously, since they are co-moving
and at rest with respect to each other.
If reality is totally discrete then the expanding circle of light that
reaches all observers simultaneously has a circumference that consists
of discrete Planck units, not a continuous circle described by
2*pi*radius?
Of course it could be that reality is described as being both
continuous AND discrete, or niether continuous nor discrete, but
something else entirely. The complementary formation of actualized and
nonactualized existence?
But if discrete it is[via actualization], then what is the value of
the constant that is approximately "pi" for an expanding circle of
light?
A closed path, or curve, C, in two dimensions, acquires a third
dimension when an enclosing overstretching elastic membrane - forms a
capping surface over it.
The capping surface and encompassed closed curve, shrinks relative to
...its "previous" outer self, as a noncommutative, nonlinear sequence
of iterative juxtapositioning via a vectorial, sequential arrow of
"symmetry breaking", generating the fourth "temporal" dimension.
Let the capping surface - membrane be a polyhedron of N faces, with
each N being defined as a "Planck length" area, on the membrane.
Conventional physics requires that the N faces become "infinitesimal"
and thus the membrane becomes a smooth continuous surface, which
probably does not correspond to what the surface of the physical
space-time membrane in the real universe, actually ...is. I postulate
a fractal geometry for the sub-microscopic N faces that generate the
observed symmetric macroscopic structure of space-time
It would really be a huge relief to discover that ours is a universe
that consists of simply connected regions at the Planck length scales,
even if, space-time is not a continuous differentiable manifold. That
is to say the interior of a sphere is simply connected but the
interior of a torus for example, is not. The fundamental "loops" of
string theory and Loop quantum gravity are probably[hopefully?] closed
curves. These loops consist in part, of compactified dimensions,
unfortunately forming tori or other possibly even unimaginable
configurations due to the extreme Planck scale dominance of Heisenberg
uncertainty.
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