In the complex-quaternion space [H.sub.g], when the three-dimensional unit vector, [i.sub.q], can be degraded into the imaginary unit i, the complex-quaternion wave function will be degenerated into complex-number wave function, in the conventional quantum mechanics.
If the direction of unit vector, [i.sub.q], is not able to play a major role in the wave function [PSI]", this unit vector will be replaced by the imaginary unit, i.
Moreover, and of great relevance here, the coarser decomposition given by the parenthesis does not depend on the concrete choice of the imaginary units [e.sub.1], [e.sub.2], [e.sub.3].
Notice that there is not a canonical way to think of a quaternionic space as a complex one: there are as many complex structures, as the imaginary units are.
What is not canonical here, is just the choice of the imaginary units: i, j and k do not play any privileged role, every equally oriented orthonormal basis of [R.sup.3] doing exactly the same job.
Let p and q be imaginary units, and consider the equation gp = qg.
When X and Y are quaternionic spaces, the regularity of f has been introduced by adapting the Cauchy-Riemann equations to the increased number of imaginary units: see [6] or [23] for the one-dimensional case, and [8] for the general case.
Consider now two quaternionic spaces X and Y, and let g and h be the imaginary units which define their complex structures, respectively.
From the above quoted proof [LAMBDA] = P + Q where P and Q are imaginary quaternionic linear and quaternionic orthogonal, since u and v are real orthogonal imaginary units. ?