The rotation group as an example of a continuous group

B. Schnizer

schnizer@itp.tu-graz.ac.at

2008-11-10

Rotations are given by the rotation tensor,  eq.(0.125):

	(Subscript[R, ij])((Subscript[e, k]), ω) =   (Subscript[e, i])(Subscript[e, j  ])+  ((Subscript[δ, ij])  -  (Subscript[e, i])(Subscript[e, j])) cos ω   -  (Subscript[ε, ijk])(Subscript[e, k]) sinω,      (Subscript[e, k])(Subscript[e, k ])= 1,  -π  ≤ ω  ≤ π.       (1)
	
ω gives the rotation angle; the unit vector (Subscript[e, k]) the rotation axis. These 4 constrained parameters are not
very well suited to give the group law of the rotation group. A rotation vector consisting of 3 new independent parameters is introduced in place of them by the following definitions: 

		(Overscript[ϖ, →])  = ((Subscript[ϖ, 1]), (Subscript[ϖ, 2]), (Subscript[ϖ, 3])) = (Overscript[e, →]) ϖ ,         ϖ  =  tan (ω/2),    -∞ ≤ ϖ   ≤ ∞.	           	(2)
	
The elements of the new representation of the rotation tensor are:

		(Subscript[R, ij])((Overscript[ϖ, →]) ) = (2/(1 + (ϖ^2)))[ ((1 - (ϖ^2))/2)(Subscript[δ, ij])  + (Subscript[ϖ, i]) (Subscript[ϖ, j])-  (Subscript[ε, ijk])(Subscript[ϖ, k])]				(3)
or		
		(Subscript[(Overscript[r, →]), 1]) = (2/(1 + (ϖ^2)))[ ((1 - (ϖ^2))/2)(Subscript[(Overscript[r, →]), 0]) + (Overscript[ϖ, →]) ((Overscript[ϖ, →])· (Subscript[(Overscript[r, →]), 0])) +   (Overscript[ϖ, →]) × (Subscript[(Overscript[r, →]), 0])]                                                     (4)

The group law is expressed in these parameters (Overscript[ϖ, →]) in the following way:

		(Subscript[R, ij])((Overscript[(Subscript[ϖ, 21]), →]) )   =   (Subscript[R, ik])((Overscript[(Subscript[ϖ, 2]), →]) ) (Subscript[R, kj])((Overscript[(Subscript[ϖ, 1]), →]) ) 						(5)

(Overscript[(Subscript[ϖ, 21]), →]) = (1/(1 - (Overscript[(Subscript[ϖ, 2]), →])· (Overscript[(Subscript[ϖ, 1]), →]) ))[(Overscript[(Subscript[ϖ, 2]) , →]) +(Overscript[(Subscript[ϖ, 1]), →])  +  (Overscript[(Subscript[ϖ, 2]), →]) × (Overscript[(Subscript[ϖ, 1]), →]) ]                                                                (6)

As is shown below the vector   (Overscript[(Subscript[ϖ, 2]) , →]) +(Overscript[(Subscript[ϖ, 1]), →])  +  (Overscript[(Subscript[ϖ, 2]), →]) × (Overscript[(Subscript[ϖ, 1]), →]) is an eigenvector of the rotation matrix 
belonging to the eigenvalue 1; it remains invariant under the joint rotation.
The factor is determined by selecting either (Overscript[(Subscript[ϖ, 1]), →])    and  (Overscript[(Subscript[ϖ, 2]) , →])  parallel or perpendicular to each other.
A more detailed justification is given at the end by a calculation.

A. Lindner: Drehimpulse in der Quantenmechanik. Teubner Studienbuecher Physik.1984.
§ 1.2, 3. Formel, Verknuepfung von  (Overscript[(Subscript[ϖ, 1]), →])und  (Overscript[(Subscript[ϖ, 2]), →])zu (Overscript[(Subscript[ϖ, 12]), →]).

The eigenvector of the joint rotation

om1 = {u1,v1,w1};
om2 = {u2,v2,w2};

o1q = om1.om1;
o2q = om2.om2;

r0 = {x0,y0,z0};

r1 = ((1 - o1q)/2 r0  + om1 om1.r0 + Cross[om1,r0])2/(1 + o1q);

r2 = FullSimplify[ ((1 - o2q)/2 r1  + om2 om2.r1 + Cross[om2,r1])2/(1 + o2q)];

At first it is shown that  o12 is the eigenvector of the tensor R(o12) belonging to the eigenvalue 1;
so it is the rotation axis of the joint rotation:

o12 = om1 + om2 + Cross[om2,om1]

s0 = Thread[r0 -> o12]

r2 /. s0 //FullSimplify

o12 == %

So om12 is proportional to o12.

The proportionality factor of the joint rotation

The proportionality factor of the joint rotation is denoted as ff:

om12 = ff o12

o12q = om12.om12;

rr2 = FullSimplify[ ((1 - o12q)/2 r0  + om12 om12.r0 + Cross[om12,r0])2/(1 + o12q)];

su = {ff->(1/(1-u1 u2-v1 v2-w1 w2))}

Table[r2[[k]] == rr2[[k]] /. su,{k,3}]//Simplify