I think there is a one to one correspondence between the solid angle
gauge theory and the EM gauge theory. Cooresponding to the EM field

Fμν = (∂Aμ / ∂xν ) - (∂Aν / ∂xμ )

There is a solid angle field like

Gμν,ηθ = (∂Pμν / ∂ξηθ ) - (∂Pηθ / ∂ξμν )

where ξηθ is the plane angle scale on ηθ-plane and Pμν is the
solid angle equivalent of the vector potential on μν-plane. For 3+1
Lorentz spacetime, there are 15 solid angle fields. For the more
natural 4+1 spacetime, there are 45 such fields. But more
correspondence, such as the spinors, need to be established.

Qchiang

Hi qchiang and Ken.
I am curious how you get the partials and things like psi into my fonts
here.
Anyhow, I love the solid angle hypothesis.
It has been a long held conviction of mine that rotation plays a deep
role in reality. It allows for a distance-wise dimensionless reality,
where rotation plays a deeper part. But how you get to distance is
beyond me. Perhaps my most recent thread about reciprocal reals will do
something for you.
It must be considered that angle can go on and on, yielding
relentlessly larger dimension without any gain in traditional distance
at the 3D level. But in allowing these greater dimensional angles a
larger distance can be quantified.
Perhaps I am only exposing my misunderstanding. None the less angle may
become distance.

-Tim

wrote:
I think there is a one to one correspondence between the solid angle
gauge theory and the EM gauge theory. Cooresponding to the EM field

Fμν = (∂Aμ / ∂xν ) - (∂Aν / ∂xμ )

There is a solid angle field like

Gμν,ηθ = (∂Pμν / ∂ξηθ ) - (∂Pηθ / ∂ξμν )

where ξηθ is the plane angle scale on ηθ-plane and Pμν is the
solid angle equivalent of the vector potential on μν-plane. For 3+1
Lorentz spacetime, there are 15 solid angle fields. For the more
natural 4+1 spacetime, there are 45 such fields. But more
correspondence, such as the spinors, need to be established.

Qchiang

Hi qchiang and Tim

wrote:
I think there is a one to one correspondence between the solid angle
gauge theory and the EM gauge theory. Cooresponding to the EM field

Fμν = (∂Aμ / ∂xν ) - (∂Aν / ∂xμ )

There is a solid angle field like

Gμν,ηθ = (∂Pμν / ∂ξηθ ) - (∂Pηθ / ∂ξμν )

where ξηθ is the plane angle scale on ηθ-plane and Pμν is the
solid angle equivalent of the vector potential on μν-plane.

That's starting to look like part of the
Riemann Chrisoffel Curvature tensor R_abcd.

For 3+1
Lorentz spacetime, there are 15 solid angle fields. For the more
natural 4+1 spacetime, there are 45 such fields. But more
correspondence, such as the spinors, need to be established.
Qchiang

I think I echo Tim's sentiments, I have a hard time
selling a spinning point on a continuum, but I accept
it as a postulate. If it goes somewhere and renders
clarity in the BIG PICTURE, then the postulate is
justified.
Ken

Hi Tim and Ken,

I just prepared my text in MS Word and pasted into the post.

Below is a portion of my initial post:

There is an inherent impossibility of conserving both plane angle and linear vector length under solid angle rotation. Such kind of rotation does not, and is not intended to, preserve vector lengths. Nor is it intended to be represented and visualized in "cartesian coordinates".

Therefore, distance cannot be visualized and measured either. However,
most people overlooked the fact that when one classifies particles on a
table, one is only concerned with plane angular momentum and solid
angular momentum (which is supposedly the iso-spin, strangeness, etc.)
but not with linear momentum. Hence there is no need to represent both
plane and solid angular momenta and linear momentum at the same time.
It is also exactly this fact which created the opportunity of parity
violation.

Qchiang

Hi Qchiang and Ken
wrote:
Hi Tim and Ken,
I just prepared my text in MS Word and pasted into the post.
Below is a portion of my initial post:

There is an inherent impossibility of conserving both plane angle and linear vector length under solid angle rotation. Such kind of rotation does not, and is not intended to, preserve vector lengths. Nor is it intended to be represented and visualized in "cartesian coordinates".

Therefore, distance cannot be visualized and measured either. However,
most people overlooked the fact that when one classifies particles on a
table, one is only concerned with plane angular momentum and solid
angular momentum (which is supposedly the iso-spin, strangeness, etc.)
but not with linear momentum. Hence there is no need to represent both
plane and solid angular momenta and linear momentum at the same time.
It is also exactly this fact which created the opportunity of parity
violation.

Qchiang
Hi Qchiang and Ken.
I appreciate the fundamental thought you are trying.
But I am not sure that I really understand what the difference between
solid angle and plane angle is. You state that:
Currently, only
symmetries under linear displacement (displacement of a 0-d point) and
plane angle rotation (displacement of a 1-d line) are recognized. I.e.
only linear and angular momenta are recognized. However, a little
sense of mathematics would dictate that solid angle rotation (or,
displacement of a 2-d surface) and solid angular momentum should
contribute equally to particle symmetries.
But as I see it plane angle rotation does already involve the entire 2d
surface, as opposed to just displacing a 1-d line. Regardless of the
dimension of the system, each point of rotation removes one degree of
freedom. an axis is comprised of two points and so drops the degree of
freedom by two, leaving one degree of freedom for a 3d system.
Would I be right in saying that you are considering a singular point of
rotation rather than an axis of rotation? This is the only way that I
see the dimensionality being raised by one. My argument is in terms of
geometry and may be too primitive for what you are doing. I'd like to
learn though.
I am trying to find a product space built from polysigned numbers,
which exhibit rotational behavior in any dimension. So my interest is
askance, but I'd like to learn the fundamentals as you see them.
-Tim

Hi Tim and Ken:

Sorry for your confusion. You are right, in a 3-d space, a plane angle
rotation does involve the whole 2d "flat" surface. By solid angle
I mean a cone. Its rotation is like the shrinking/expansion of the
cone, something like a satellite dish cone shrinks to the shape of a
telescope cone. It's a displacement of the whole 2-d surface of the
"cone" with, as you said, just "one singular point" (not the
whole axis) fixed.

Take a 3-d example, a plane angle rotation with z axis fixed will move
x-"axis" to the y-"axis" (the whole xz-plane move to the
yz-plane). But a solid angle rotation moves (shrinks/expands, can't
be visualized with Cartesian coordinates) the whole xz-"plane angle
scale" to the yz-"plane angle scale". The difference is between
"from axis to axis' and "from plane angle scale to plane angle
scale".

I realize it's not easy to recognize, because it can not be
visualized. But physics requires its existence. Its crucial role is
to establish a comparison between different plane angle scales in the
same way that magnetic fields establish comparison between plane angle
scales. Mathmeatically, there is no reason it cannot exist just like
plane angle rotation. It certainly can be defined mathematically and
cause symmetry without being visualized.

qchiang

wrote:
Hi Tim and Ken:

Sorry for your confusion. You are right, in a 3-d space, a plane angle
rotation does involve the whole 2d "flat" surface. By solid angle
I mean a cone. Its rotation is like the shrinking/expansion of the
cone, something like a satellite dish cone shrinks to the shape of a
telescope cone. It's a displacement of the whole 2-d surface of the
"cone" with, as you said, just "one singular point" (not the
whole axis) fixed.

Take a 3-d example, a plane angle rotation with z axis fixed will move
x-"axis" to the y-"axis" (the whole xz-plane move to the
yz-plane). But a solid angle rotation moves (shrinks/expands, can't
be visualized with Cartesian coordinates) the whole xz-"plane angle
scale" to the yz-"plane angle scale". The difference is between
"from axis to axis' and "from plane angle scale to plane angle
scale".

I realize it's not easy to recognize, because it can not be
visualized. But physics requires its existence. Its crucial role is
to establish a comparison between different plane angle scales in the
same way that magnetic fields establish comparison between plane angle
scales. Mathmeatically, there is no reason it cannot exist just like
plane angle rotation. It certainly can be defined mathematically and
cause symmetry without being visualized.

qchiang

A quick primitive comment.

I recall, most kids played with spinning tops,
after a while the axis about the rotation would
begin to rotate, I think that's called permutation.

Well that at least that allows the visualization
of the decay of a spinor, that begins to manifest
in more dimensions (3D instead of 2D), by playing
with a spinning top.

BTW qchiang, you sound oriental, is it polite to
ask your geographic location?

With respect
Ken S. Tucker

wrote:
Hi Tim and Ken:

Sorry for your confusion. You are right, in a 3-d space, a plane angle
rotation does involve the whole 2d "flat" surface. By solid angle
I mean a cone. Its rotation is like the shrinking/expansion of the
cone, something like a satellite dish cone shrinks to the shape of a
telescope cone. It's a displacement of the whole 2-d surface of the
"cone" with, as you said, just "one singular point" (not the
whole axis) fixed.

Take a 3-d example, a plane angle rotation with z axis fixed will move
x-"axis" to the y-"axis" (the whole xz-plane move to the
yz-plane). But a solid angle rotation moves (shrinks/expands, can't
be visualized with Cartesian coordinates) the whole xz-"plane angle
scale" to the yz-"plane angle scale". The difference is between
"from axis to axis' and "from plane angle scale to plane angle
scale".

I realize it's not easy to recognize, because it can not be
visualized. But physics requires its existence. Its crucial role is
to establish a comparison between different plane angle scales in the
same way that magnetic fields establish comparison between plane angle
scales. Mathmeatically, there is no reason it cannot exist just like
plane angle rotation. It certainly can be defined mathematically and
cause symmetry without being visualized.

qchiang
When linear momentum is eliminated from the picture these things can
get interesting.
But your selection of a cone as the basis is not obvious. The cone has
an axis. I can see that the apex of the cone is the 'center' of the
particle. But aren't you still spinning on an axis? As we look at the
cones of varying angle we are traveling along the surface of a sphere(
full 3D analogy) orthogonal to that axis. Endowing some quality to
this angle seems feasable. But the resultant would be symmetric about
the axis wouldn't it?
If I eliminate the axis the cone suggestion goes away. This then leaves
two angular degrees of freedom and an intuitively improper form of
rotation. But when linear momentum is eliminated the improper rotation
becomes quite an acceptable puzzle even if it does look like the inside
of a blender.
I guess I need a strict definition of solid angle in order to see the
distinction between it and the usage of an axis. Perhaps I am mixing
too much geometry into the particle.
-Tim

Hi Ken,

I lived in New York city for 20 years, but just moved to Vancouver this
summer. From your email address, it seems you are also in Canada.
Nice to be neighbors.

Qchiang

Hi Tim and Ken,

I am glad you dig into this subject. Let me give you some history of
my thinking, which would be hard to present in my initial post.

1. The linear scales of Lorentz spacetime can be defined in the
following way:

- Choose one light ray t in x direction. Let it travel from a to b.
Define the time it travels from a to b be 1 second and the distance ab
= 1 light-second. During this time, ray x travels from b to c, thus
define distance bc = 1 light-second. Define the time duration ray t
travels from b to c as the next second. Define the distance cd ray x
travels during the second second as cd = next light second, ... Thus
all the scales of x dimension and time dimension are defined.
- Run another ray y in y direction, mark the distance scale on y
dimension as the first light second, second light-second, third
light-second, ... during the first second, second second, third second,
....
- Do the same thing on z dimension. Thus the scales of time dimension
and all 3 spatial dimensions of this Lorentz spacetime are defined.
- Definition of simultaneity: Run two light rays from o in opposite
directions and let them be reflected back at two mirrors at p and q, if
the two reflected rays reach back at o at the same time, define the
time they reach p and q as simultaneous.
- Maxwell theory asserts any other light ray will be measured as of the
same speed as these 4 light rays.

However, if there are no magnetic fields running from one dimension to
another, or if the magnetic field line on xy-plane running from the 2nd
light-second on x-axis doesn not run through the 2nd light-second, but
through the 3rd light-second on the y-axis, the above definition would
still be meaningless. It is the whole Maxwell EM theory as an
integral, including light propagation and the electric and magnetic
fields running between different dimensioins, which guanrantees the
above definition of linear scales to make sense. From the fact that
all lights travel at the same speed in all directions, we can tell
there are guaranteed magnetic (and electric) fields running between
different dimensions. (Otherwise, lights would not travel at the same
speed in different dimensions).

Now, from the fact that plane angle scales on different planes are
equivalent to each other, we can also tell there are "guaranteed"
(solid angle) fields running from one plane (with its plane angle
scale) to another plane to ensure their equivalence. Otherwise, plane
angle scales cannot be equivalent to each other on different planes (as
manifested in numerous physics facts).

2 What we anticipate is a field running from one plane, with plane
angle scale, to another plane. Initially I thought it is something
like a cone shrinking from a plane (say xp-plane) to a needle, then
expand from the needle back to another plane (say zt-plane). But for
others which rotate from xy-plane to xz-plane, it becomes harder to
visualize from a Cartesian coordinates. It becomes more like a bent
cylinder coinnecting one plane angle scale to another plane angle
scale, while the axis and the fixed singular point become unnecessary.
To serve our purpose of comparing different plane angles, a cylinder is
essentially the same and topologically equivalent to a cone, even
though it is actually not a cone as visualized in a Cartesian
coordinates. That's why I continue to call it "solid angles".

3. Since particle classifications deal only with solid angular momentum
and plane angular momentum, without involvement of linear momentum, we
really don't need to bother the Cartesian coordinates. In fact, this
is the only way "my solid angle" can be defined. It is simply the
plane angle of a 6-d space, with each axis representing a plane angle
scale of the 3+1 Lorentz spacetime. (For the more natural 4+1
spacetime, it is a 10-d space representing the 10 palne angle scales).
We have to abandon Cartesian coordinates here, which is not wrong
because particle classifications don't bother linear momentum and, as
you can see, parity is also violated. On the other hand, when you
insist on a Cartesian view, such as in particle interactions (where
linear momentum is conserved), you will see conservation of solid
angular momentum (e.g. iso-spin, strangeness, etc.) is not observed.
So, there is an inherent incompatibility between the 6-d (or 10-d)
solid angle space and the Cartesian space.

Hope this calarifies a little bit more of this hard-to-convey concept.


Qchiang