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In curved spacetime, all equations only remain invariant when all
used matrices are
tensors and all used spacetime-derivatives are so-called
covariant derivatives.
Using covariant derivates, the order of
derivation matters! All
characteristics of covariant derivatives are given with the
so-called curvature(Riemann-Christoffel)-tensor.
This tensor has 20 degrees of
freedom. Using standard GR-theory, the degrees of freedom of the
curvature tensor reduces from 256 to 20
due to the Bianchi symmetry
relations, but the actual reason is the
non-commutativity of the covariant
derivatives! So, CAP
enforces all descriptions to use tensors and covariant derivatives,
also in SR flat space. The Euler-Lagrange
and Einstein-Hilbert
descriptions actually are point-descriptions. All described objects
are specified using a point in 4D
spacetime. As a result all equations of
motion follow from the
Ricci tensor, i.e. the non-zero
contracted curvature tensor, with the degrees of freedom halved (20
→ 10).
Mathematically the only way to solve this problem is
doubling the degrees of freedom
in the only possible non-reducible description of all symmetries of
our 4D universe. Any exact
description is a point-description! This is why all
Q(uantum)M(echanics)
descriptions assume elementary particles to be point-particles with so-called
intrinsic characteristics, like spin.
In April 1998 I concluded that all
elementary particles should be described as
extended particles using a point
description in the 2D-plane orthogonal to the observed direction of
motion given by the SR
worldline.
At infinitesimal level and whenever curvature can be neglected, the
solution can be described SR.
The extendedness is very small, i.e. always
non-observable. As a result a SR
description suffices in all cases. The used inertial frame is
chosen with origin at the average position of the harmonic
oscillating point in the 2D-plane giving the center of energy of the
extended particle. This origin specifies the position used as QM
point-particle on the SR worldline.
The solution shows constants of
motion following from symmetries. The time-like total energy
H=E(p)+U(r)
= hf = hω,
with p the particle's momentum observed from the inertial frame,
r the polar distance from
the origin, f the frequency of oscillation in the
2D-plane,
h
Planck's
constant, ω the
angular-frequency and h
Dirac's constant. The potential
energy must result in harmonic oscillation:
U(r)
= ˝kr2,
with k a force-constant to be determined. The space-like constant
just is the angular momentum, i.e. spin S =
hs in the
direction of motion given by the wordline in the direction of motion
of the particle (and inertial frame). Again
h is Dirac's
constant and s must be any
positive (half-)integer to describe any possible (fermion)boson.
I.e. the used intrinsic angular momentum of
QM comes to life when demanding
QM to comply to Einstein's
CAP.
The complete symmetry group of
SR is
the
Poincaré-group. Any transformation of the
CAP extended
Poincaré-group can be
represented with the sum of a symmetric transformation tensor and an
anti-symmetrical transformation tensor. All fundamental elementary
particles must follow from the most general symmetry group, i.e. the
CAP extended
Poincaré-group.
Elementary particles can be split into two
different kinds: 1.
Fermions, which are the sources of
all force fields and have half-integer spin.
2. Bosons, which
represent the force fields and have integer spin.
All members of the Poincaré-group are represented in a non-reducible
way as: spin2xspin˝
+ spin1xspin˝
The spin2 particle is the mass- and charge-less
graviton with source the multiplied
spin˝ mass.
The spin1 particle is the mass- and charge-less
photon with source the multiplied
spin˝ charge.
The average extendedness is proportional to the spin, so
CAP enforces
s > 0. Dimensional analysis now
allows: s є{2,
1˝, 1,
˝}
Both the
Euler-Lagrange and the
Einstein-Hilbert action mechanisms remain
valid as long as the particles are specified with the average
position on the SR worldline.
The
EM-field isn't given completely
with the Maxwell equations, which are given
SR with the
anti-symmetric EM-field tensor Fμν.
Only after imposing a
gauge-symmetry the
EM-field is specified
completely. All gauge-symmetry
is related to the anti-symmetrical
fields, because symmetric fields never yield any so-called
gauge-symmetry! As a result the total gauge-symmetry of our
3D-universe just is:
U(1)xSU(2)xSU(3).
The U(1)xSU(2) gauge bosons
describe the mixed (given by the Weinberg
angle) photon and chargeless and massive
Z-boson and the massive and
charged W± bosons.
The SU(3) gauge symmetry group
describes all charged massive spin3/2
quarks, which must combine into
hadrons, of which stable spin
baryons and interacting bosons
called gluons and
mesons appear in real life.
In standard QCD all
quarks are assumed to be
spin˝ with assumed so-called
isospin to end up with 4 degrees of freedom. However, the
fact that quarks are never observed on their own is not explained in
the standard model. To me, quarks must be intrinsic unstable,
i.e.
have spin3/2. The so-called
isospin is nonsense!
When solving
the equations of motion of extendedness in the
2D-plane orthogonal
to the direction of motion (worldline), one requires
B(oundary)C(onditions)
to solve the D(ifferential)E(quations).
Open BC
have one degree of freedom extra. It's the positive integer
giving the amount of rotations in the 2D-plane
before the motion repeats itself again. This degree of freedom must
be the quantum number of the particles
family. The higher this number, the more interaction with the
gravitational field, i.e. the higher the rest-mass. All
fermions must have open
BC, i.e. must have
non-zero rest-mass!
Closed
BC describe bosons that only interact with other
particles in the direction of motion, specified by the wordline. All
elementary bosons have only one
species. Only compound bosons have more
so-called families.
I observe our
universe as created from a singularity
with a fixed total energy which
resulted in our well-known Big Bang.
This constant total energy scattered in all directions. Just after
the Big Bang the energy density was the highest and decreased as
scattering took place. During this scattering the
entropy increased,
as it still does. Experiences show we only have
3 particle families
of fermions. All particle families came to life just after the start
of the Big Bang when the energy density was the highest. This is why
I assume that 3 particle families of
fermions are a fixed fact of
our universe.
In
2004 Grisha
Perelman showed that
knots are only possible in 3D-space,
i.e. 4D-spacetime. Fermions
always have mass, because they are
described using open
BC. As a result of this fact
it's always possible to create a knot
in the path of a fermion. I'm not saying that it'll ever happen, but
it's possible! Fermions are the sources of all
bosons, i.e. no
fermions implies
nothing at all. Based on this discovery of
Perelman I concluded that the only
possible universes have 3D-space,
i.e. 4D-spacetime.
Last change:
24-07-2009 20:32:00 |