Re: Unitary matrix. - Rotation matrix
From: Roger Bagula (tftn_at_earthlink.net)
Date: 10/27/04
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Date: Wed, 27 Oct 2004 21:37:35 GMT
I dug up my Dirac six set up which I did
when messing around with C^2 minimal surfaces.
It is another (but complex matrix ) way to get an
six space that is four space based in 4by4 matrices
and this takes only four parameters!
It's based on the matrices from Michael Creutz's
"Quarks, Gluon and Lattices" that are determinant one
( Euclidean Diracs as opposed to Lorentzian Dirac's).
It's a six space such that:
x^2+y^2+z^2+t1^2+t2^2+t3^2=0
and the group is self congugate instead of complex congugate.
basically the good thging about it is that it is null Ricci like.
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Roger Bagula wrote:
> 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
>
>
>
-- 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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