schoenfeld.one@gmail.com wrote:

Timothy Golden BandTechnology.com wrote:

Timo A. Nieminen wrote:

On Wed, 24 Oct 2006, Timothy Golden BandTechnology.com wrote:

In Hermann Weyl's book "The Theory of Groups and Quantum Mechanics"
chapter 3 section 12 toward the end of the first paragraph (pg. 160 on
my edition) he writes:

"The concept of an infinitesimal rotation will be familiar to the
reader from the kinematics of rigid bodies, as well as the fact that
these infinitesimal rotations in 3-dimensional space constitute a
3-dimensional linear family - in n-dimensional space an
[n(n-1)/2]-dimensional family."

I disagree with this and see that the rotations of an n-dimensional
space constitute an n-1 dimensional family.

You can write any rotation in n-d space in terms of a basis of
antisymmetric matrices. For 2D space, you only have one:

[ 0 -1; 1 0 ] (where ; means start a new row).

In 3D, [ 0 -1 0; 1 0 0; 0 0 0 ], [ 0 0 -1; 0 0 0; 1 0 0 ], and
[ 0 0 0; 0 0 -1; 0 1 0 ].

It basically comes down to "how many elements are there below the
diagonal of an n-by-n matrix?" - this is the dimensionality of rotations
in an n-d space. n^2 matrix elements in total, subtract n to remove the
diagonal which is all zero leaves n(n-1), and halve.

http://groups.google.com.au/group/sci.physics/msg/07fbb2e45180a599?hl=en&

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html

Thanks all for the clarification. For feedback purposes I do find
Nieminen's response the most legible. Perhaps I will need further
correction; here is my reflection of these comments:

The rotation Weyl speaks of is the most restrictive form; one degree of
freedom for the object under study. This means in a 2D problem rotation
about a point, in 3D about an axis, and in 4D about a plane, etc. The
"[n(n-1)/2]-dimensional family" refers to an n->n projection matrix
which fixes the position of the object. Ultimately such a matrix would
also contain a variable theta which would pertain to the rotational
angle accomodating rotations at a given fix. Though there are nxn
elements in the projection matrix the degree of freedom in such a
matrix is not simply nxn.

From the quote you gave, Weyl refers to the set of canonical rotations

which can be composed (i.e. multiplied) into an arbitrary rotation. In
3-space you have 3 canonical rotations one for (by convention) the XY,
YZ, ZX planes. Arbitrary N-space rotations are composed of minimum
N-choose-2 (or (n(n-1)/2) if you like) canonical rotations. That
derives from the generalization of Eulers rotation theorem which is
central to this entire approach. You can easily prove that theorem (see
previous post).

Now, by defining skew-symmetric matrices S(i,j)[i][j]=(-1)^(i+j) for
all i,j permutations you can construct the (ij)'th-plane canonical
rotation matrix as R(i,j)=exp(S(i,j)). Discard all reflections (i.e.
det(R(i,j))=-sin(x)^2-cos(x)^2=-1). Since rotation compositions do not
commute (past 2), apply canonical rotations arbitrarily but
consistently. Use the reverse order for inverse rotations.

[...]

I'm really interested in how this relates to angular momentum.
In making instances
J x , J y , J z
I have difficulty.
This is close to the planar representation you are steering us toward.
Does it relate?

There are systems of rotation that are constructable which do not
necessarily obey the strict rules that we have used thus far on this
thread. In that the complex numbers form a natural rotational system we
could extend this result downward to the real line and see that a
binary rotation is the only possibility via the arithmetic product.
This binary rotation can also be applied in higher dimensions yielding
the improper transform (determinant - 1). There are artifacts of this
in some work that I have done on polysigned numbers. Such systems are
flipping handedness and this sort of math is of interest when
considering spin and such qualities that are surprising to our ordinary
3D sense of space. Spacetime itself can be claimed to admit some
handedness if we attribute electromagnetic behavior to it rather than
building electromagnetic behavior on top of it. This is done somewhat
in the numerous tensor theories. Typically they will identify their
inherent support for Maxwell's equations up front.
All of this is just an aside I suppose unless you want to address some
of it. I am more interested in the attribution of a total energy of
| J x | | J x | + | J y | | J y | + | J z | | J z |

and verifying its validity. Some of the Lie stuff uses M and this form
would be equally discussable; It's the composition that I do not
understand.
If I were to attribute this energy in four components in three
dimensions would the result be equally valid?
The rotation considered is one dimensional right?
Like above on this thread?
How it comes to be broken into these components puzzles me.
I've always had a hard time with rotation yet also felt that it is deep
in its consequences.

-Tim