Re: Unitary matrix. - Rotation matrix

From: Roger Bagula (tftn_at_earthlink.net)
Date: 10/26/04 

Date: Tue, 26 Oct 2004 20:45:10 GMT

Dear JEMebius,
I'm just an old quantum chemist
and using quaternoins (upward) or octonions ( downward) seems
the really hard way to me.
And SO(4) type real 4d electromagnetic field has energy density
like ( M=SO(4) group sum: F(u,v)=a*M, a= scale)
T(em)=(1/4)*F(u,v)*F(u,v)^(-1)
I don't see any singularities since this gives a nice diagonal matrix.
A better approach is possibly F(u,v)=a*g(u,v)*M for a Lorentz Minkowski
geometry,
but it still gives a nice diagonal matrix.
The representation of SO(3) is such that that (x,y,z) in a sphere surface
In
Mso3={{0,x,-z},{-x,0,y},{z,-y,0}}
give
Mso3^2
 such that the new coordinates are a projective plane ( Steiner Roman
surface)
which is basically a tetrahedral torus type.
I think these may be the "singularities" you are seeking
to avoid which have their root in the spherical vibrations ( Legendre,
etc.).
These are only higher energy Hilbert space states.
Can avoid Hilbert spaces in Quantum mechanics by using higher symmetry
derivations?
It is possible to define an Dirac like intermediate between the
quaternoins and octonions
but is isn't a "nice" set of equations in my experience.
I can dig up those matrices for you if you like,
since I had to prove to myself a Clifford algebra could be made for a
Dirac like
symmetry.
I'm sorry I got carried away by this answer, ha, ha...

If we all knew everything we could all stop struggling to understand.

For the rotation matrices I gave are good for this kind of rotations.
Here are the checked matrices ( all unitary) and their product is unitary
as the original question asked for.
(I did may some typing mistakes)
m1 = {{Cos[a], Sin[a], 0, 0}, {-Sin[a], Cos[a], 0, 0}, {0, 0, 1, 0}, {0,
0, 0, 1}}

m2 = {{Cos[b], 0, -Sin[b], 0}, {0, 1, 0, 0}, {Sin[b], 0, Cos[b], 0}, {0,
0, 0, 1}}

m3 = {{1, 0, 0, 1}, {0, Cos[c], Sin[c], 0}, {0, -Sin[c], Cos[c], 0}, {0,
0, 0, 1}}

m4 = {{Cos[d], 0, 0, -Sin[d]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sin[d], 0,
0, Cos[d]}}

m5 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, Cos[e], -Sin[e]}, {0, 0,
Sin[e], Cos[e]}}

m6 = {{1, 0, 0, 0}, {0, Cos[f], 0, Sin[f]}, {0, 0, 1, 0}, {0, -Sin[f],
0, Cos[f]}}

Simplify[Det[m1]]

Simplify[Det[m2]]

Simplify[Det[m3]]

Simplify[Det[m4]]

Simplify[Det[m5]]

Simplify[Det[m6]]

M = m1 . m2 . m3 . m4 . m5 . m6

M={{Cos[a] Cos[b] Cos[d] + Cos[a] Cos[b] Sin[d], Cos[f] (Cos[c] Sin[a] +
Cos[a] Sin[b] Sin[c]) - (Cos[e] (Cos[a] Cos[b] Cos[d] - Cos[a] Cos[b]
Sin[d]) - (-Cos[a] Cos[c] Sin[b] + Sin[a] Sin[c]) Sin[e]) Sin[f],
 
   Cos[e] (-Cos[a] Cos[c] Sin[b] + Sin[a] Sin[c]) + (Cos[a] Cos[b]
Cos[d] - Cos[a] Cos[b] Sin[d]) Sin[e],
 
   Cos[f] (Cos[e] (Cos[a] Cos[b] Cos[d] - Cos[a] Cos[b] Sin[d]) -
(-Cos[a] Cos[c] Sin[b] + Sin[a] Sin[c]) Sin[e]) + (Cos[c] Sin[a] +
Cos[a] Sin[b] Sin[c]) Sin[f]},
 
  {-Cos[b] Cos[d] Sin[a] - Cos[b] Sin[a] Sin[d], Cos[f] (Cos[a] Cos[c] -
Sin[a] Sin[b] Sin[c]) - (Cos[e] (-Cos[b] Cos[d] Sin[a] + Cos[b] Sin[a]
Sin[d]) - (Cos[c] Sin[a] Sin[b] + Cos[a] Sin[c]) Sin[e]) Sin[f],
 
   Cos[e] (Cos[c] Sin[a] Sin[b] + Cos[a] Sin[c]) + (-Cos[b] Cos[d]
Sin[a] + Cos[b] Sin[a] Sin[d]) Sin[e],
 
   Cos[f] (Cos[e] (-Cos[b] Cos[d] Sin[a] + Cos[b] Sin[a] Sin[d]) -
(Cos[c] Sin[a] Sin[b] + Cos[a] Sin[c]) Sin[e]) + (Cos[a] Cos[c] - Sin[a]
Sin[b] Sin[c]) Sin[f]},
 
  {Cos[d] Sin[b] + Sin[b] Sin[d], -Cos[b] Cos[f] Sin[c] - (Cos[e]
(Cos[d] Sin[b] - Sin[b] Sin[d]) - Cos[b] Cos[c] Sin[e]) Sin[f], Cos[b]
Cos[c] Cos[e] + (Cos[d] Sin[b] - Sin[b] Sin[d]) Sin[e],
 
   Cos[f] (Cos[e] (Cos[d] Sin[b] - Sin[b] Sin[d]) - Cos[b] Cos[c]
Sin[e]) - Cos[b] Sin[c] Sin[f]}, {Sin[d], -Cos[d] Cos[e] Sin[f], Cos[d]
Sin[e], Cos[d] Cos[e] Cos[f]}}
  
Simplify[Det[M]]

JEMebius wrote:

> Dear mr Bagula, my thanks for your interest in my SO(4) work.
>
> You propose to perform rotations in each of the 6 coordinate planes;
> this is a most natural approach to obtain more general rotations than
> just a rotation in a single plane. Actually this was Euler's approach
> in studying N-dimensional rotations; however he failed to give a
> complete proof that one can obtain == all == N-dimensional rotations
> about a fixed point in this way.
> In the 4D case Arthur Cayley and Van Elfrinkhof observed that one
> obtains 4D rotations by means of quaternions, but both of them failed
> to prove, like Euler, that == all == 4D rotations can be obtained by
> quaternions. This gap is filled by Bouman's 1932 article and in an
> entirely different manner by my paper.
>
> In rotations in 3D space (SO(3)) the three Euler angles are
> well-known. Several different conventions are in common use as regards
> the order of rotations in the successive coordinate planes. All of
> them have 3 independent parameters. A rather deep theorem says that
> any representation of SO(3) by means of 3 independent parameters has
> singularities somewhere. It is the same for representations of SO(4)
> by means of 6 independent parameters.
>
> I checked that with the three angles of yaw, pitch and roll of
> aircraft one can reach each attitude in 3D space in precisely one way,
> except the vertical nose-up and nose-down attitudes, which can reached
> in infinitely many ways.
> This is because in vertical flight the roll and pitch axes coincide
> and one cannot any longer distinguish roll and pitch angles from each
> other.
> I never checked how this works for the 6 rotation angles in your letter.
>
> In conclusion: indeed quaternions in SO(4) seem not necessary, but
> turn out to be the only practical way to handle 4D rotations without
> stumbling upon singularities sooner or later.
>
> ===== reply to =====
>
>
>>
Respectfully, Roger L. Bagula

tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn

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