# Cayley-Klein parameters

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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