Cayley-Klein parameters

Special coordinates in the rotation group $ \mathop{\rm SO} ( 3) $ of three-dimensional space, the construction of which is based in the final analysis on the relationship between $ \mathop{\rm SO} ( 3) $ and the group $ \mathop{\rm SU} ( 2) $ of $ 2 \times 2 $ unitary matrices with determinant 1. There exists a mapping $ \phi : \mathop{\rm SU} ( 2) \rightarrow \mathop{\rm SO} ( 3) $ which is an epimorphism by virtue of its algebraic properties and a double covering by virtue of its topological properties. (Restricted to some neighbourhood of the identity matrix, $ \phi $ is an isomorphism; in other words, $ \mathop{\rm SO} ( 3) $ and $ \mathop{\rm SU} ( 2) $ are locally isomorphic.) Each matrix $ V \in \mathop{\rm SU} ( 2) $ may be written as

$$ \left \| \begin{array}{rr} \alpha &\beta \\ - \overline \beta &\overline \alpha \\ \end{array} \right \| , $$

where $ \alpha , \beta $ are complex numbers such that $ | \alpha | ^ {2} + | \beta | ^ {2} = 1 $. These are taken to be the Cayley–Klein parameters of $ A = \phi ( V) $. (The term is sometimes used for all four elements of the matrix $ V $.) As the actual construction of a mapping $ \phi $ with the above properties may be accomplished in various ways, different authors define the Cayley–Klein parameters in slightly different ways (see [2], [3]).

Since $ \phi $ is not a true isomorphism, but only a double covering, it is impossible to define the Cayley–Klein parameters as (continuous) coordinates on all of $ \mathop{\rm SO} ( 3) $; this can be done only locally. However, the Cayley–Klein parameters may nevertheless be used to study processes of rotation in which $ A $ is a continuous function of a real parameter $ t $ (and there is no necessity to restrict the domain of possible values of $ A $ in any way). Indeed, if some fixed value $ V ( t _ {0} ) \in \phi ^ {-1} ( A ( t _ {0} )) $ has been chosen at $ t = t _ {0} $, the corresponding values of $ V ( t) $ are uniquely defined by continuity for all $ t $. (The fact that the complete inverse $ \phi ^ {-1} $ is double-valued intrudes only in the observation that $ A ( t) = A ( s) $ not only when $ V ( t) = V ( s) $ but also when $ V ( t) = - V ( s) $.) Thus the Cayley–Klein parameters can be applied to investigate the motion of a rigid body with a fixed point (the configuration space of which is $ \mathop{\rm SO} ( 3) $). This approach was adopted in [1], but did not achieve popularity.

The group $ \mathop{\rm SU} ( 2) $ is isomorphic to the group of quaternions with norm 1 (cf. Quaternion); hence, by going over from $ V $ to the corresponding quaternion $ \rho + \lambda i + \mu j + \nu k $, one can replace the Cayley–Klein parameters by the Euler–Rodriguez parameters — four real numbers $ \rho , \lambda , \mu , \nu $ such that $ \rho ^ {2} + \lambda ^ {2} + \mu ^ {2} + \nu ^ {2} = 1 $. These stand in a simple relationship to the Cayley–Klein parameters (see [1], [3]) and possess the same "double-valuedness" property (for the history of the problem see [1]). It was essentially in this context that attention was first directed to two-valued representations of the rotation group (see Spinor).

References

[1]	F. Klein, A. Sommerfeld, "Ueber die Theorie des Kreises" , 1–2 , Teubner (1965)
[2]	H. Goldstein, "Classical mechanics" , Addison-Wesley (1953)
[3]	J.L. Synge, "Classical dynamics" , Handbuch der Physik , 3/1 , Springer (1960) pp. 1–225

References

[a1]	P. Du Val, "Homographies, quaternions and rotations" , Clarendon Press (1964)