Quaternions in polar coordinate form

doug  <sweetser@world.com>

Any quaternion can be written in polar coordinate form, which involves a scalar magnitude and angle, and a 3-vector I (which in some cases can be the more familiar i).

\(q\  = \ 
      \(\( || q || \ exp[θ\_q\ \(I\_q\)\&⇀]\)\  = \ 
        \(q\^*\) q 
          \((Cos[θ\_q]\  + \ 
              \(I\_q\)\&⇀\ Sin[θ\_q])\)\)\)

\(q\  = \ 
      \(\( || q || \ exp[θ\ I\&⇀]\)\  = \ 
        \(q\^*\) q 
          \((Cos[θ]\  + \ I\&⇀\ Sin[θ])\)\)\)

This representation can be useful due to the properties of the exponential function, cosines and sines.

The absolute value of a quaternion is the square root of the norm, which is the transpose of a quaternion multiplied by itself.

abs[nq_] :=  (Sqrt[(Transpose[nq].nq)])[[1, 1]]

\(\( | q | \) = \@\(\(q\^*\)\ q\)\)

The angle is the arccosine of the ratio of the first component of a quaternion divided by the norm.

\(θq[polarq_]\  := 
	
      Module[{firstterm}, 
		
        firstterm\  = \ \ \((polarq + \ Transpose[polarq])\)[\([1, 1]\)]\/2; 
        
		ArcCos[firstterm\/abs[polarq]\ ]\ ]\)

\(θ\  = \ ArcCos \((\(q\  + \ \(q\^*\)\)\/\(2 | q | \))\)\)

The vector component is generated by normalizing the pure quaternion (the final three terms) to the norm of the pure quaternion.

\(iq[polarq_]\  := 
	
      Module[{pureq}, 
		pureq\  = \ \(polarq - \ Transpose[polarq]\)\/2; 
        
		pureq\/abs[pureq]]\)

\(I = \(q\  - \ \(q\^*\)\)\/\(2 | q\  - \ \(q\^*\) | \)\)

I^2 equals -1 just like i^2.  Let (0, V) = (q - q*)/2.

Simplify[(iq[q[t, x, y, z]] . iq[q[t, x, y, z]]).{1, 0, 0, 0}]{-1, 0, 0, 0}

(I^2)  =  
      (((0,  V) (0,  V))/(|(0,  V) || (0,  V) | )) = 
        (((-V.V,  VxV))/(((V^2), 0))) = -1

It should be possible to do Fourier analysis with quaternions, and to form a Dirac delta function (or distribution).  That is a project for the future.  Those tools are necessary for solving problems in quantum mechanics.