The complex multiplication naturally extends to a multiplication in four dimensions, which defines on the space $ \mathbb{R}^4 $ the structure of the algebra $ \mathbb{H} $ of Hamilton’s quaternions. This multiplication can be interpreted geometrically using the dot product and the cross product in space, which allows for the separation of a “scalar” component and a “vectorial” component, making $ \mathbb{H} $ an “algebraic space-time.”

1. The Algebraic Definition of Hamilton’s Quaternions

1.1. Recovering Complex Multiplication from $i^2 = -1$

When defining the algebraic structure of complex numbers, the structure of the set $\mathbb{R}$ is geometrically extended to that of the set $\mathbb{C}$. This extension consists of defining a “vector multiplication” on the plane $\mathbb{R}^2$, distributive over the usual addition of vectors coordinate by coordinate. The complex number $(0,1)$, denoted $i$, plays an essential role in the algebraic description of complex numbers, as any such number can be written in the form $(a,b) = a + ib$. The “rule” $i^2 = -1$ then allows us to recover complex multiplication from the usual properties of the product, in the form
$$ (a+ib)(c+id) = ac + i(ad + bc) + i^2(bd) = ac – bd + i(ad + bc). $$

1.2. Extending the Complex Structure to Higher Dimensions

The Irish mathematician and physicist William Rowan Hamilton sought to extend complex numbers to higher dimensions. While he desperately searched for an operation analogous to complex multiplication in three dimensions, the solution came to him in 1843 as an intuition during a walk with his wife. The key was to think in four dimensions to multiply higher-dimensional vectors!

The vectors of the space $\mathbb{R}^4$ are quadruples of real numbers $(t,x,y,z)$, and they are expressed as combinations of the basis vectors denoted $1 = (1,0,0,0)$, $i = (0,1,0,0)$, $j = (0,0,1,0)$, and $k = (0,0,0,1)$, in the form $(t,x,y,z) = t \cdot 1 + x \cdot i + y \cdot j + z \cdot k$. Hamilton’s intuition was to guess or invent the formulas
$$ i^2 = j^2 = k^2 = ijk = -1, $$
and to discover that a vector multiplication in dimension $4$ could be defined based on the multiplication rules of the basis vectors. These rules are: $i^2 = j^2 = k^2 = -1$, $ij = k = -ji$, $jk = i = -kj$, and $ki = j = -ik$. They are extended to vectors in $\mathbb{R}^4$ by writing
$$(t,x,y,z) \times (t’,x’,y’,z’) = (t \cdot 1 + x \cdot i + y \cdot j + z \cdot k) \times (t’ \cdot 1 + x’ \cdot i + y’ \cdot j + z’ \cdot k),$$
which expands to
$$(tt’ – xx’ – yy’ – zz’) \cdot 1 + (tx’ + xt’ + yz’ – zy’) \cdot i + (ty’ + yt’ + zx’ – xz’) \cdot j + (tz’ + zt’ + xy’ – yx’) \cdot k$$
using the usual properties of the product.

2. The Algebra $\mathbb{H}$ of Quaternions

2.1. A Division Algebra in Dimension 4

Thus, it is possible to define a multiplication in dimension $4$, following a similar idea to the one used to define complex multiplication in dimension $2$. This complex multiplication is recovered for vectors of the form $(t,x,0,0) = t \cdot 1 + x \cdot i$, which are identified with the vectors $(t,x)$ of the plane (complex numbers $t + xi$), and is extended here via the vectors $j$ and $k$. However, this multiplication in $4$ dimensions, which is distributive over vector addition, is no longer commutative: unlike complex multiplication, it is not true that if $q$ and $q’$ are two vectors in dimension $4$, we always have $q \times q’ = q’ \times q$. In other words, the order in which the operation is performed affects the result, as can already be seen with the basis vectors, since $i \times j = -j \times i$, $j \times k = -k \times j$, and $k \times i = -i \times k$.

The set $\mathbb{R}^4$, with its vector addition and this multiplication, is called the algebra of (Hamilton’s) quaternions and is denoted by $\mathbb{H}$. The vector $1 = (1,0,0,0)$ is the “one” of this multiplication, in the sense that if $q = (t,x,y,z)$ is a quaternion, we have $1 \times q = q \times 1 = q$, which can be easily verified by calculation. Furthermore, as in $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$, every nonzero element $q$ of $\mathbb{H}$ has a unique inverse $q^{-1}$, such that $q \times q^{-1} = q^{-1} \times q = 1$. We say that $\mathbb{H}$ is a noncommutative field, or more generally, a division algebra.

2.2 Geometric Interpretation: An Algebraic Space-Time

The previous description of multiplication in $\mathbb{H}$ is algebraic: it was defined in a purely “operational” manner. However, this structure holds profound geometric meanings. Indeed, Hamilton, who invented the concept of a “vector” to describe quaternions, distinguished one of the coordinates—the first one—which he called the “scalar part,” and the other three, which he grouped into a “vector part.”

Let us recall here the analytical definition of the natural dot product of two vectors $V = (x,y,z)$ and $V’ = (x’,y’,z’)$ in space: we have $V \cdot V’ = (xx’ + yy’ + zz’)$. Similarly, their cross product is $V \wedge V’ = (yz’ – zy’, zx’ – xz’, xy’ – yx’)$. Let us then describe any quaternion $q = (t,x,y,z)$ as a pair $(t,V)$, where $t \in \mathbb{R}$ and $V \in \mathbb{R}^3$ is a vector in Euclidean space.

Using the analytical definition of quaternionic multiplication and the dot and cross products in space, we can rewrite the product of two quaternions $q = (t,x,y,z)$ and $q’ = (t’,x’,y’,z’)$ in the form
$$ (t,V) \times (t’,V’) = (tt’ – V \cdot V’, tV’ + t’V + V \wedge V’), $$
if $V = (x,y,z)$ and $V’ = (x’,y’,z’)$.

In other words, quaternionic multiplication implicitly reveals the dot product and the natural cross product of a “spatial” component, which can be naturally separated from a “temporal” component. Thus, while the field $\mathbb{C}$ is an “algebraic plane,” the algebra $\mathbb{H}$ is, in a sense, an “algebraic space-time,” intrinsically mathematical.

3. Polar Representation of Hamilton’s Quaternions

3.1. The Unit Sphere in the Space $\mathbb{R}^4$

By considering complex numbers of modulus 1, that is, points on the unit circle, any nonzero complex number $z = x + iy$ can be represented in polar form, as $z = r(\cos t + i\sin t) = (r\cos t, r\sin t)$, where $r > 0$ is the modulus and $t \in \mathbb{R}$ is a determination of the argument of $z$. Similarly, a polar representation of nonzero quaternions can be given by considering quaternions of norm $1$.

The norm of a quaternion $q = (t,x,y,z) = t \cdot 1 + x \cdot i + y \cdot j + z \cdot k$ is generally understood as its arithmetic norm, that is, the positive real number $N(q) = t^2 + x^2 + y^2 + z^2$, while its Euclidean norm is $||q|| = \sqrt{t^2 + x^2 + y^2 + z^2} = \sqrt{N(q)}$. Either definition allows us to define the set of quaternions of norm $1$, which is the analogue in $\mathbb{R}^4$ of a sphere of radius $1$. This set is denoted $S^3$, a “sphere” of dimension $3$ as a geometric object (just as the sphere of radius $1$ in $\mathbb{R}^3$ is $S^2$, of dimension $2$, and the “sphere” of radius $1$ in $\mathbb{C} = \mathbb{R}^2$ is $S^1$, the unit circle, of dimension $1$).

3.2. Polar Representation of a Nonzero Quaternion

For the polar representation of quaternions, the sphere $S^3$ plays a role analogous to that of the unit circle in the polar representation of complex numbers. It can be shown that a vector in $S^3$ (that is, a quaternion of norm $1$) is always of the form $(\cos t, (\sin t)V)$ for a real number $t$ and a vector $V$ in the sphere $S^2$, this time in two dimensions, that is, a vector in $\mathbb{R}^3$.

Thus, if $q = (t,x,y,z)$ is a nonzero quaternion, dividing it by its Euclidean norm $||q||$ yields a quaternion of norm $1$, namely
$$ q’ = \frac{1}{||q||} \cdot q = \left(\frac{t}{||q||}, \frac{x}{||q||}, \frac{y}{||q||}, \frac{z}{||q||}\right), $$
which can then be written in the form $(\cos t, (\sin t)V)$. A polar representation of $q$ is therefore given by
$$ q = ||q|| \cdot (\cos t, (\sin t)V). $$