Beschreibung:
<jats:p>A parametrization scheme for orthogonal and unitary groups is developed. This scheme is broadly similar to one developed earlier by Wigner, but differs from it in some essential details. It is shown that an arbitrary element g of the group SO(p, q), [SU(p, q)], 3 ≤ p &lt; ∞, 0 ≤ q &lt; ∞, p + q = N, can be written as a product of three factors, g = aBc, such that the extreme factors a and c belong to SO(p, q − 1) [SU(p, q − 1)] and the middle factor is an element of SO(p, q), [SU(p, q)] depending on at most two parameters and representing a ``rotation'' (real, complex, unitary or pseudo-unitary, as the case may be) in the (1 − N) plane. This parametrization includes the Euler parametrization R = R12(φ1) R13(θ)R12(φ2) of the group SO(3) as a special case, and is therefore a generalization of the Euler angle concept to more general orthogonal and unitary groups. The present factorization scheme differs from Wigner's, which is of the form g=[SO(p)⊗SO(q)]K[SO(p)⊗SO(q)] [and similarly for SU(p, q)], in that the extremal factors of this latter scheme are not the next natural subgroup of the original group. Consequently, the middle term K may depend on more than two parameters. Furthermore, the method of proof presented here is entirely different from Wigner's, and may serve as a useful alternative, if further generalization to real Lie groups that preserve some other types of bilinear forms is envisaged. By stepwise factorization, the element g may ultimately be expressed as a product of real, complex, unitary or pseudo-unitary ``rotations'' in the plane.</jats:p>