Re: Test2 SPF

Hi Kwan and fellow SPFer's.

On Oct 4, 6:09 pm, qchiang <qchia...@xxxxxxxxx> wrote:
On 9月28日, 下午9时28分, "Ken S. Tucker" <dynam....@xxxxxxxxxxxx> wrote:
On Sep 25, 10:35 pm, qchiang <qchia...@xxxxxxxxx> wrote:
...
I can see quite naturally you are less than 100% convinced about the
plane angle space and solid angle rotation because it is a new concept
and not widely recognized just like any new concept.

The "new concept" you mention evidently relates to
geometry. Let me term that "Kwan's New Math".
This New Math needs to satisfy the mathematicians
(geometrists) who frequent the group, using as far as
possible, common terminology, to verify it's logical
soundness.
Where I stand, I'm unable to apply the New Math to
physics until I can understand it.

But I am totally
confident on it as I have spent a lot of time studying from various
angles and thinking deeply into the philosophy of physics. I'm
positive you will be gradually convinced,

I'm rather uncertain if we're working in 5D and whether
it's Euclidean, could you answer that please?

especially when physics
hasn't gone anywhere under conventional wisdom since the later part of
20th century.

Well there are quite a few experiments ongoing that
intend to test the new theories.
Regards
Ken S. Tucker

Hi Ken,

Thanks for your recommendations and sorry for the delayed reply, as I
had some trouble with posting in the correct ASCII formatting. Hope
this time it is ok.

Below is the new math which can be defined only in the PLANE ANGLE
SPACE, where each linear coordinate represents a plane angle scale,
while the plane angle IN THE PLANE ANGLE SPACE is actually the solid
angle.

For the Lorentz spacetime, there are x, y, z and t linear coordinates
and there are 6 plane angle scales for the 6 planes, xy-plane, yz-
plane, zx-plane, xt-plane, yt-plane and zt-plane. Correspondingly, on
the PLANE ANGLE SPACE, there are 6 LINEAR coordinates, each represents
a plane angle scale for the 6 planes, xy-plane, yz-plane, zx-plane, xt-
plane, yt-plane and zt-plane.

A PLANE angle rotation in the PLANE ANGLE SPACE, say from the axis of
xy-plane to the axis of yz-plane, is actually a SOLID angle rotation
from xy-plane to yz-plane as considered (conceptually) from the
viewpoint of the cartesian coordinates. (While it's termed as SOLID
angle, it's more like a curved cylinder running from xy-plane to yz-
plane. However, there is no direct translation of coordinates between
the cartesian coordinates and the PLANE ANGLE SPACE.)

We may define SOLID ANGLE w_xyyz in the following fashion: An
arbitrary plane angle arc alpha (ie, line segment in the plane angle
space) can be projected on the xy-plane and yz-plane through SOLID
ANGLE w_xyyz as

alpha_xy = alpha . sin w_xyyz
alpha_yz = alpha . cos w_xyyz

The absolute value of the arbitrary plane angle |alpha| remains a
constant

|alpha|^2 = (alpha_xy)^2 + (alpha_yz)^2 = (alpha . sin
w_xyyz )^2 + (alpha . cos w_xyyz )^2 = (alpha)^2 = constant

under any plane angle rotation in the plane angle space (or under the
solid angle rotation as considered conceptually from the viewpoint of
the cartesian coordinates). Likewise, for the 3+1 Lorentz spacetime,

|alpha|^2 = (alpha_xy)^2 + (alpha_yz)^2 + (alpha_zx)^2 +
(alpha_xt)^2 + (alpha_yt)^2 + (alpha_zt)^2 = constant

In this way, we can successfully decompose any plane angle arc into
components of the 6 planes, xy-plane, yz-plane, zx-plane, xt-plane, yt-
plane and zt-plane. And any solid angle rotation (or plane angle
rotation in the PLANE ANGLE SPACE) would not alter a plane angle arc.
Thus, a new symmetry has emerged. However, this is not a new
invention but actually a reflection from the Cisimir invariant.

Solid angle and the plane angle rotation is not visulizable from the
cartesian coordinates, and linear length cannot be conserved under
solid angle rotation.

Quote,
"We can either work in the PLANE ANGLE SPACE or
in the cartesian coordinates, but not both simultaneously."

This is
not a drawback though, as there are times we care only about the plane
angle and solid angle rotations, but not about the linear space, e.g.
in particle classifications.

The other major mathmatical aspect is that: for a 4+1 spacetime, a
point can be identified by identifying through the generalized polar
coordinates:

the 4-d surface in the 5-space according to the 5-d angles,
then the 3-d cubic surface in the 4-space according to the 4-d
angles,
then the 2-d surface in the 3-d space according to the solid (3-d)
angles,
then the 1-d surface (line) in the 2-d space according to plane (2-d)
polar angle,
then the 0-d point on the 1-d surface (line) according to the distance
(1-d angle).

Associated with this decomposition is that the quantum numbers to be
associated with a particle should be,

Ψ = ∑ E × D × C × B × A (6.1)

where:

is a wave function, exp[-iπ(p^0x^0 -p^1x^1 - p^2x^2 - p^3x^3 -
p^mx^m)], representing linear (1-dimensional) momentum, including
energy and mass. x^m is the extra dimension [1] and p^m = mc.
A spinor representing plane (2-dimensional) angular momentum.
A solid angle spinor representing solid (3-d) angular momentum. Solid
angle rotation runs from one plane (2-brane) to another (among the 10
planes) while preserving plane angular momentum. Symmetry of solid
angle rotation is suspected to be those of iso-spin, strangeness,
charm, etc. The interaction through solid angle rotation is believed
to be weak interaction.
A 4-d rotation spinor representing 4-d angular momentum. 4-d rotation
runs from one 3-plane (3-brane) to another (among the 10 3-planes)
while preserving solid angular momentum. This symmetry probably
generates K^L and K^S, the mixtures of K^0 and anti-K^0 mesons. The
interactions may be the CP-violation interactions.
A 5-d rotation spinor representing 5-d angular momentum. 5-d rotation
runs from one 4-d plane (4-brane) to another among the 5 4-d planes
while preserving 4-d angular momentum. Fields in 5-d rotations may be
causing the strong interactions. The symmetry of 4-d angular momentum
might be the color symmetry which exists but cannot be observed in
isolation.

I'm rather uncertain if we're working in 5D and whether
it's Euclidean, could you answer that please?

And then,

We are still in the 3+1 Lorentz space, but its PLANE ANGLE SPACE is
6D. In the case of 4+1 spacetime, the PLANE ANGLE SPACE is 10D. They
are all Euclidean.
Best regards,
Kwan Chiang (9/30/08)

As far as I know, a Euclidean space is characterized
by the Rieman tensor vanishing, ie.
http://en.wikipedia.org/wiki/Riemann_curvature_tensor
R_abcd=0.
Such that the metrics are transformable to constants,
specifically Cartesian CS's.
Naturally we can work with any metrics "g_uv" that
are compatible with R_abcd , "simultaneously", like
polar, cylindrical and on and on...
Does that reasoning always hold?
Regards
Ken S. Tucker
.