A quaternion is composed of one real part and three imaginary parts and is typically written as; $$q = a + bi + cj + dk$$

Here, 𝑎, 𝑏, 𝑐, and 𝑑 are real numbers, and 𝑖, 𝑗 and 𝑘 are the fundamental quaternion units. The quaternion units satisfy the following multiplication rules; $$ i^{2} = j^{2} = k^{2} =ijk = -1 $$

An interesting possiblity is to describe a field using quaternions. For a scalar or vector field in three dimensions, each point of the field can be represented quite simply using quaternions. For example, a vector field 𝐹(𝑥,𝑦,𝑧) can be mapped to a quaternion field 𝑄(𝑥,𝑦,𝑧); $$ Q(x,y,z) = f_0(x,y,x) + f_1(x,y,z)i + f_2(x,y,z)j + f_3(x,y,z)k $$

Where \(f_0,f_1,f_2,f_3\) are scalar functions describing the field components.

Transformations of the three dimensional field can be performed on the field itself using quaternion algebra. For instance, to rotate the field, quaternion multiplication can be applied at each point. Quaternions offer a powerful way to handle three-dimensional fields and transformations, especially useful in physics.

Rotations in Quaternion Space

The beauty of quaternions lies in their ability to smoothly and consistently rotate objects without the complications that arise from other representations like Euler angles or rotation matrices. A rotation in three-dimensional space can be represented using a unit quaternion \(Q\), a quaternion with a magnitude or norm of 1, where: $$Q = e^{2\theta} (ai + bj + ck)$$ For a rotation by angle \(\theta\) around the unit vector \(n=(a,b,c)\) expanding via Euler’s formula:
$$Q = \cos^{2}\theta + (ai + bj + ck)\sin^{2}\theta$$ For a vector \(V\) (purely imaginary quaternion), the rotation is performed by: $$V' = QVQ^{-1}$$

Since unit quaternions satisfy \(Q^-1 = Q^\dagger\) the conjugate, this gives a correct 3D rotation.

Lorentz Boosts

A Lorentz boost is a type of transformation used in the theory of relativity, specifically in special relativity, to describe how the coordinates of an event change when observed from two different inertial frames of reference moving relative to each other at a constant velocity. Essentially, it's a way of transforming coordinates in spacetime to account for the relative motion between observers. A Lorentz boost along the x-axis (for simplicity) is given by the standard transformation: \[t' = \gamma (t - vx), \quad x' = \gamma (x - vt), \quad y' = y, \quad z' = z\] Where: $$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$ Is the Lorentz factor. It is notable that this transformation "mixes time and space", which suggests we need a quaternionic representation that includes both.

Boosting Quaternions

In quaternion algebra, boosts are represented similarly to rotations, but using hyperbolic functions instead of trigonometric ones.

Therefore a boost quaternion can be represented as:

\[Q = e^{2\phi}(iv_x + jv_y + kv_z)\] where: \(\theta\) is the rapidity, related to velocity by: \[\tanh \phi = \frac{v}{c}, \quad \text{or} \quad \gamma = \cosh \phi, \quad \gamma \frac{v}{c} = \sinh \phi\] \(v_x, \quad v_y, \quad v_z\) define the boost direction. For a boost along the x-axis, we define: \[Q = e^{2\phi}i = \cosh^{2}\phi + i\sinh^{2}\phi\] This acts on a quaternionic spacetime variable \(X = it + xi + yj + zk\) via a similarity transformation: \[X' = QXQ^{-1}\] Expanding, we obtain: \[( i \cosh^2 \phi + i i \sinh^2 \phi )(it + xi)(i \cosh^2 \phi - i i \sinh^2 \phi)\] Using \(i^2 = -1\) this simplifies to: \[X′=i(coshϕt−sinhϕx)+i(coshϕx−sinhϕt)+yj+zk.\] This interpretation perfectly matches the standard Lorentz transformation.