Complex quaternion functions

The exp, sqrt, and log complex quaternion functions are treated and then applied to obtain the general Lorentz transformation and obtain a simple technique to represent it as a boost followed by a rotation or vice versa. What's treated is how to compute these functions, when they are defined, and what their multiplicities are.

The quaternions were discovered by William Rowan Hamilton in 1843.^[1] He had long searched for an algebra that was to three dimensions what the complex numbers are to two dimensions. He sought to multiply and divide these sought after numbers for many years before, in a flash of insight, the solution came to him. The problem was that there are no such numbers in three dimensions, only in four dimensions.

A quaternion Q can be written as $${\displaystyle {\textbf {Q}}=a\,+b\,{\textbf {I}}+c\,{\textbf {J}}\,+d\,{\textbf {K}}}$$ where $${\displaystyle {\textbf {I}}\;{\textbf {I}}={\textbf {J}}\;{\textbf {J}}={\textbf {K}}\;{\textbf {K}}={\textbf {I}}\;{\textbf {J}}\;{\textbf {K}}=-1}$$ From these, using associativity, it follows that $${\displaystyle {\textbf {I}}\;{\textbf {J}}=-{\textbf {J}}\;{\textbf {I}}={\textbf {K}}\quad {\textbf {J}}\;{\textbf {K}}=-{\textbf {K}}\;{\textbf {J}}={\textbf {I}}\quad {\textbf {K}}\;{\textbf {I}}=-{\textbf {I}}\;{\textbf {K}}={\textbf {J}}\quad }$$

This was the first non-commutative algebra. In hindsight, non-commutativity is to be expected since rotations about the origin in three dimensions do not in general commute but in two dimensions they do. For Hamilton a,b, c, and d were all real. Define the norm as

$${\displaystyle {\textbf {N}}({\textbf {Q}})=a^{2}+b^{2}+c^{2}+d^{2}}$$

It is easily verified that

$${\displaystyle {\textbf {Q}}^{-1}={\frac {a-b\,{\textbf {I}}-c\,{\textbf {J}}-d\,{\textbf {K}}}{{\textbf {N}}({\textbf {Q}})}}}$$

Since N(Q) is always positive for a non-zero real quaternion, the inverse always exists. This makes the algebra a division algebra. Also, the norm of a product is the product of the norms. This makes the algebra a composition algebra.^[2]

The real quaternions can be used to do spatial rotations,^[3] but not to do Lorentz transformations with a boost. But if a, b, c, and d are allowed to be complex, they can.^[4][5] This is what motivated the study of functions of a complex quaternion or biquaternion, such as how they are to be computed, when they are defined, and what their multiplicities are.

Since there are non-zero complex quaternions with zero norm, the inverse does not always exist. So they are not a division algebra. But they almost are. And there are zero divisors, as evidenced by (1 + i I) (1 - i I) = 0. In a way, the need for complex quaternions is not surprising, since in special relativity the Minkowski invariant, which is the norm of a 4-vector, can be any real number, including zero (null rays).

A 4-vector ${\displaystyle (c\,t,x,y,z)}$ is represented by the complex quaternion ${\displaystyle c\,t+i\,x\,{\textbf {I}}+i\,y\,{\textbf {J}}+i\,z\,{\textbf {K}}}$, which is called a Minkowski quaternion. Its scalar time-like component is real and its spatial vector component is pure imaginary. This is the convention used by P. A. M. Dirac,^[6] which gives the metric ${\displaystyle \eta _{00}>0}$. Choosing the time-like component imaginary and the spatial vector component real has also been done and gives the metric ${\displaystyle \eta _{00}<0}$.

The basis quaternions I, J, and K can be represented in terms of the Pauli spin matrices as ${\displaystyle -i\,\sigma _{1}}$, ${\displaystyle -i\,\sigma _{2}}$, and ${\displaystyle -i\,\sigma _{3}}$, respectively,^[7] as one possibility. . These have the same multiplication table. The Pauli spin matrices are used in particle physics for Lorentz transformations of 2-spinors and can do Lorentz transformations of a 4-vector by representing it as a 2x2 matrix, which is obtained from its equivalent Minkowski quaternion by replacing I, J, and K by their Pauli spin matrix representations and adding. The scalar term ${\displaystyle c\,t}$ is replaced by multiplying it by the 2x2 identity matrix. Working with complex quaternions is simpler and more transparent and intuitive than working with matrices. But there is a one-to-one correspondence between the two viewpoints.

The exponential function with a complex quaternion argument is used to generate finite Lorentz transformations and the square root function with a complex quaternion argument is used to express a Lorentz transformation either as a pure boost followed by a spatial rotation or vice versa.

Functions with a complex quaternion argument are treated first, followed by a discussion of their application to Lorentz transformations.