In contrast to the imaginary unit, the unit vector [i'.sub.q] possesses three new degrees of freedom.
(a) Imaginary unit: in some simple physical problems, the three-dimensional unit vector [i.sub.q], of the complex-quaternion wave function [A.sub.([PSI])g], can be directly degraded into the imaginary unit, i.
In this case, the three-dimensional unit vector [i.sub.q], in the complex-quaternion wave function, [A.sub.([PSI])g], will be degraded into the imaginary unit i.
This new argument, that is, the color degree of freedom, enables the complex-quaternion wave function to be written as three conventional complex-number wave functions equivalently, while the unit vector, [i'.sub.q(k)], may be reduced into the imaginary unit, i, further.
Due to (5.4), to conclude the proof it is enough to show that there exists an imaginary unit g such that
Fix an arbitrary imaginary unit g and set h = [phi](g).
where u is an arbitrary imaginary unit, which is moreover orthogonal to g.
for some suitable choice of the imaginary unit [g.sub.x] and