Pauli Quaternions
Quaternions are elements of a four dimensional vector space. The basis elements of the quaternion space are denoted as s1,s2,s3,s4. They obey the following multiplication rules:
(1)
Note these rules are the same as the Pauli spin matrices (see review article by Rastall)
(2)
They differ from the classical quaterion basis of W.H Hamilton by the factor
i Note that this basis set does not include a multiplicative identity. This important quaternion can be defined in
terms of the basis as I=-is4.
One can also replace s4 with I as a basis element and
perhaps that is the best way.
There is also a relation between the Pauli quaternions and Clifford Algebra of three dimensions
A general quaternion is a linear combination of these basis elements. We denote quaternions here as column vectors containing the expansion coefficients. The column vector
(3)
denotes the quaternion q1s1+q2s2+q3s3+q4s4`
Basic Quaternion Functions
Quaternion conjugate and Hermitian conjugate:
(4)
Norm, magnitude:
(5)
A quaternion q is unimodular if Norm(q)=1. Note that the Norm is complex in general. The unimodularity condition imposes two constraints on the components of the quaternion q. A unimodular quaternion has 8-2=6 independent components.
Inverse of q and the Unitizing operator
(6)
Unitizing operator U(Q) constructs a muimodular quatneinion in the same direction as Q. (Throughout these pages the square root symbol denotes the branch of the square root having positive real part)
The vector part of q:
(7)
The argument of q:
(8)
Geometrically the argument is the angle from the Identity quaternion to the vector part of q. (cos-1q is the branch between 0 and p for real q.)
Square root of a quaternion:
(9)
The exponential and logarithm of a quaternion:
(10) 
(11)
These functions have properties one would expect them to have: For instance.
(12)
However note that 
Quaternions and Rotations
Quaternions describe rotations both in three and four dimensional
space. The Pauli quaternions defined here describe Lorentz transformations in
four dimensional space-time. That’s
because the Lorentz transformation is mathematically analogous to a rotation.
Mathematically a rotation is a linear transformation that leaves the norm
invariant. These are called orthogonal
transformations.
It follows from the rule:
(13)
that multiplication by a unimodular quaternion effects an orthogonal transformation. Conversely any orthogonal transformation can be effected thru multiplication by unimodular quaternions. The most general orthogonal transformation on quaternion Q is effected by two unimodular quaternions QR and QL
(14)
For relativity applications we restrict the orthogonal transformations to those that transform one Hermitian quaternion to another Hermitian quaternion: These are the Lorentz transformations. They are of the form:
(15)
Where QH and Q¢H are Hermitian and QL is a unimodular quaternion. It can be shown that for a given pair of Hermitian quaternions QH and Q¢H, the unimodular quaternion connecting them is
(16)