Quaternions

Prerequisites: Imaginary Numbers.

• Intuitive
• Concrete
• Abstract

Intuitive

Multiplying by a complex number of length one rotates a point in the plane. This makes complex numbers a convenient language for rotations in two dimensions. William Rowan Hamilton spent years looking for a similar number system for rotations in three dimensions. A system with three components did not work. In 1843 he found that it works with four components: one ordinary real part and three imaginary units, called i, j and k, each of which squares to minus one.

The price is that the order of multiplication matters. In the quaternions i times j is k, but j times i is minus k. This matches a familiar fact about rotations in space: turning a book first about one axis and then about another gives a different result from doing the two turns in the opposite order.

Quaternions of length one describe rotations in three dimensions. Each rotation is described by exactly two of them, a quaternion and its negative. This "two to one" relation is the same one that appears for the spin of the electron, where a rotation by a full turn changes the sign of the state and only two full turns bring it back. In practice, quaternions are used in computer graphics, robotics and spacecraft attitude control, because composing two rotations is a single multiplication and they avoid the coordinate singularities of Euler angles.

• see Curious quaternions | plus.maths.org
• and the excellent video on Quaternions by 3Blue1Brown
• The Strange Numbers That Birthed Modern Algebra by Charlie Wood in Quanta Magazine.

Concrete

A quaternion is an expression

$$q = a + b\,i + c\,j + d\,k, \qquad a,b,c,d\in\mathbb{R},$$

with the multiplication rules found by Hamilton,

$$i^2 = j^2 = k^2 = ijk = -1 .$$

From these one derives $ij = k = -ji$, $jk = i = -kj$ and $ki = j = -ik$. Multiplication is associative and distributive, but not commutative. We call $a$ the real part and $\vec v = (b,c,d)$, written $v = bi+cj+dk$, the pure (imaginary) part. For two pure quaternions $u$ and $v$ the product contains both familiar vector products,

$$uv = -\,\vec u\cdot\vec v + \vec u\times\vec v ,$$

where on the right the vector $\vec u\times\vec v$ is read as a pure quaternion.

The conjugate is $\bar q = a - bi - cj - dk$, and $q\bar q = \bar q q = a^2+b^2+c^2+d^2 = |q|^2$. Every $q\neq 0$ has the inverse $q^{-1} = \bar q/|q|^2$, and the norm is multiplicative, $|pq| = |p|\,|q|$. Note $\overline{pq} = \bar q\,\bar p$.

Rotations. A unit quaternion can be written as $q = \cos(\theta/2) + \sin(\theta/2)\, n$ with $n$ a pure unit quaternion. For a pure quaternion $v$ the map

$$v \mapsto q\, v\, \bar q$$

gives again a pure quaternion, and it is the rotation of the vector $\vec v$ by the angle $\theta$ about the axis $\vec n$ (counterclockwise when seen from the tip of $\vec n$, i.e. by the right-hand rule). As a check, take $\theta = \pi/2$ about the $z$-axis, $q = (1+k)/\sqrt 2$, and $v = i$:

$$q\,i\,\bar q = \tfrac12 (1+k)\, i\, (1-k) = \tfrac12 (i+j)(1-k) = \tfrac12(i - ik + j - jk) = \tfrac12(i + j + j - i) = j .$$

The $x$-axis is rotated into the $y$-axis, as expected. Doing first the rotation $q_1$ and then $q_2$ corresponds to the product $q_2 q_1$. Since $q$ and $-q$ give the same map, the unit quaternions cover the rotation group twice.

Matrices. The assignment

$$1\mapsto \mathbb{1},\quad i\mapsto -i\sigma_x,\quad j\mapsto -i\sigma_y,\quad k\mapsto -i\sigma_z$$

with the Pauli matrices $\sigma_{x,y,z}$ respects the multiplication rules (on the right, $i$ is the complex unit). It maps the unit quaternions one to one onto the group SU(2), which is why the spin-1/2 rotation operators $e^{-i\theta\,\vec n\cdot\vec\sigma/2}$ have the same structure as $q = \cos(\theta/2)+\sin(\theta/2)n$.

Abstract

The quaternions $\mathbb{H}$ form a four-dimensional associative algebra over $\mathbb{R}$ in which every non-zero element has an inverse, i.e. a division algebra. Its centre is $\mathbb{R}$. By Frobenius' theorem the finite-dimensional associative division algebras over $\mathbb{R}$ are, up to isomorphism, exactly $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$. Dropping associativity, Hurwitz's theorem states that the normed division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and the octonions $\mathbb{O}$. $\mathbb{H}$ is also the real Clifford algebra generated by two anticommuting elements squaring to $-1$ (with $e_1 = i$, $e_2 = j$, $e_1e_2 = k$), and $\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C}\cong M_2(\mathbb{C})$.

The norm $|q|^2 = q\bar q$ is the Euclidean norm on $\mathbb{H}\cong\mathbb{R}^4$, so the unit quaternions form the three-sphere $S^3$. This is a Lie group,

$$Sp(1) = \{q\in\mathbb{H} : |q| = 1\} \cong SU(2) \cong \mathrm{Spin}(3),$$

see Sp(1), with Lie algebra the pure quaternions $\mathrm{Im}\,\mathbb{H}$ and bracket $[u,v] = uv-vu = 2\,\vec u\times\vec v$. Two standard double covers follow from the algebra:

• $Sp(1)\to SO(3)$, $q\mapsto (v\mapsto qv\bar q)$ on $\mathrm{Im}\,\mathbb{H}\cong\mathbb{R}^3$, with kernel $\{\pm1\}$.
• $Sp(1)\times Sp(1)\to SO(4)$, $(p,q)\mapsto(x\mapsto p\,x\,\bar q)$ on $\mathbb{H}\cong\mathbb{R}^4$, with kernel $\{(1,1),(-1,-1)\}$. Hence $\mathrm{Spin}(4)\cong SU(2)\times SU(2)$, which underlies the splitting of four-dimensional (Euclidean) spinors into two chiralities and of two-forms on $\mathbb{R}^4$ into self-dual and anti-self-dual parts.

Matrices with quaternionic entries give the compact symplectic groups $Sp(n) = \{A\in M_n(\mathbb{H}) : A^\dagger A = \mathbb{1}\}$, where $\dagger$ is transpose combined with quaternionic conjugation. Quaternionic structure also appears inside complex quantum mechanics. A complex representation $V$ is called quaternionic (or pseudoreal) if it carries an antilinear map $J$ that commutes with the group action and satisfies $J^2 = -1$. The spin-1/2 representation of $SU(2)$ is of this type. Time reversal for a system of half-integer total spin is an antiunitary operator with $T^2 = -1$; it turns the Hilbert space into a quaternionic vector space, and this is the origin of Kramers degeneracy. Together with the real and complex cases this gives Dyson's "threefold way" classification of irreducible representations.

Leads to: Sp(1).