HYPERCOMPLEX NUMBERS AND THE DESCRIPTION OF SPIN STATES

A family of hypercomplex numbers is introduced in which multiplication is commutative and members can have up to eight components. In partic ular, the eight basis elements {E} contain those for ordinary complex numbers, E* = E, as well as new elements where E** = -E; the operatio n being the generalization of complex conjugation. This family lends itself to the description of quantum mechanical spin states in that i t offers a simple treatment of time reversal, representations with the same conjugation properties as underlying operators, and explicit con tinuous-angle spherical harmonic functions Z(sm)(theta, phi) analogous to the Y-lm(theta, phi) for orbital angular momentum. The new element s are especially well suited for half-integral spin states, whereas co nventional complex numbers remain useful for integral spin states. (C) 1997 American Institute of Physics.