6.3 Quaternions and Clifford algebras

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Quaternions

In 1842 Hamilton showed that the next natural generalization to the complex numbers must occur in four dimensions. Let $\mathcal{Q}$ be the associative algebra over $\mathbb{R}$ generated by four elements $\{1,i,j,k\}$ satisfying

$$\begin{array}{l}{i}^2=j^2=k^2=-1,\\ ij=k,jk=i,ki=j,\\ 1^2=1,1i=i,1j=j,1k=k.\end{array}\text{\hspace{1em}(6.4)}$$

The element 1 may be regarded as being identical with the real number 1. From these relations and the associative law it follows that

$$ji=-k,kj=-i,ik=-j.$$

To prove the first identity use the defining relation $jk=i$ and the associative law,

$$ji=j(jk)=(jj)k=j^{2}k=(-1)k=-k.$$

The other identities follow in a similar way. The elements of this algebra are called quaternions; they form a non-commutative algebra since $ij-ji=2k\ne 0$

Exercise

Write out the structure constants of the quaternion algebra for the basis $E_1=1,E_2=i,E_3=j,E_4=k$.

Every quaternion can be written as

$$Q=q_{0}1+q_{1}i+q_{2}j+q_{3}k=q_0+\mathbf{q}$$

where $q_0$ is known as its scalar part and $\mathbf{q}=q_{1}i+q_{2}j+q_{3}k$ is its vector part. Define the conjugate quaternion $\overline{\mathbf{Q}}$ by

$$\overline{Q}=q_{0}1-q_{1}i-q_{2}j-q_{3}k=q_0-\mathbf{q}.$$

Pure quaternions are those of the form $Q=q_{1}i+q_{2}j+q_{3}k=\mathbf{q}$ , for which the scalar part vanishes. If $p$ and $q$ are pure quaternions then

$$\mathbf{pq}=-\mathbf{p}\cdot \mathbf{q}+\mathbf{p}\times \mathbf{q},\text{\hspace{1em}(6.5)}$$

a formula in which both the scalar product and cross product of ordinary 3-vectors make an appearance.

Exercise

Prove Eq. (6.5).

For full quaternions

$$\begin{array}{l}PQ=(p_0+\mathbf{p})(q_0+\mathbf{q})\\ =p_0{q}_0-\mathbf{p}\cdot \mathbf{q}+p_0\mathbf{q}+q_0\mathbf{p}+\mathbf{p}\times \mathbf{q}.\end{array}\text{\hspace{1em}(6.6)}$$

Curiously, the scalar part of $PQ$ is the four-dimensional Minkowskian scalar product of special relativity

$$\frac{1}{2}(PQ+\overline{PQ})=p_0{q}_0-p_1{q}_1-p_2{q}_2-p_3{q}_3=p_0{q}_0-\mathbf{p}\cdot \mathbf{q}.$$

To show that quaternions form a division algebra, define the magnitudet $|Q|$ of a quaternion $\mathbf{Q}$ by

$$|Q{|}^2=Q\overline{Q}=q_0^2+q_1^2+q_2^2+q_3^2.$$

The right-hand side is clearly a non-negative quantity that vanishes if and only if $Q=0$

Exercise

Show that $|\overline{Q}|=|Q|$

The inverse of any non-zero quaternion $Q$ is

$$Q^{-1}=\frac{\overline{Q}}{|Q{|}^2},$$

since

$$Q^{-1}Q=Q{Q}^{-1}=\frac{\overline{Q}Q}{|Q{|}^2}=1.$$

Hence, as claimed, quaternions form a division algebra.

Clifford algebras

Let $V$ be a real vector space with inner product $u\cdot v$, and $e_1,e_2,\dots ,e_n$ an orthonormal basis,

$$g_{ij}=e_i\cdot e_j=\{\begin{array}{ll}\pm 1& \text{if}i=j,\\ 0& \text{if}i\ne j.\end{array}$$

The Clifford algebra associated with this inner product space, denoted ${\mathcal{C}}_g$ , is defined as the associative algebra generated by 1, $e_1,e_2,\dots ,e_n$ with the product rules

$$e_i{e}_j+e_j{e}_i=2g_{ij}1,1e_i=e_{i}1=e_i.\text{\hspace{1em}(6.7)}$$

The case $n=1$ and $g_{11}=-1$ gives rise to the complex numbers on setting $i=e_1$ . The algebra of quaternions arises on setting $n=2$ and $g_{ij}=-{\delta }_{ij}$ , and making the identifications

$$i\equiv e_1,j\equiv e_2,k\equiv e_1{e}_2=-e_2{e}_1.$$

Evidently $k=ij=-ji$ , while other quaternionic identities in Eq. (6.4) are straightforward to show. For example,

$$ki=e_1{e}_2{e}_1=-e_1{e}_1{e}_2=e_2=j,\text{ etc. }$$

Thus Clifford algebras are a natural generalization of complex numbers and quaternions. They are not, however, division algebras – the only possible higher dimensional division algebra turns out to be non-associative and is known as an octonion.

The Clifford algebra ${\mathcal{C}}_g$ is spanned by successive products of higher orders ${\mathbf{e}}_i{\mathbf{e}}_j,{\mathbf{e}}_i{\mathbf{e}}_j{\mathbf{e}}_k$ , etc. However, since any pair $e_i{e}_j=-e_j{e}_i$ for $i\ne j$ , it is possible to keep commuting neighbouring elements of any product $e_{i_1}{e}_{i_2}\dots e_{i_r}$ until they are arranged in increasing order $i_1\le i_2\le \cdots \le i_r$ , with at most a change of sign occurring in the final expression. Furthermore, whenever an equal pair appear next to each other, $e_i{e}_i$ , they can be replaced by $g_{ii}=\pm 1$ , so there is no loss of generality in assuming $i_1<i_2<\cdots <i_r$ . The whole algebra is therefore spanned by

$$\begin{array}{c}1,\{e_i\mid i=1,\dots ,n\},\{e_i{e}_j\mid i<j\},\{e_i{e}_j{e}_k\mid i<j<k\},\dots \\ \{e_{i_1}{e}_{i_2}\dots e_{i_r}\mid i_1<i_2<\cdots <i_r\},\dots ,e_1{e}_2\dots e_n.\end{array}$$

Each basis element can be labelled $e_A$ where $A$ is any subset of the integers $\{1,2,\dots ,n\}$ the empty set corresponding to the unit scalar, $e_{\varnothing }\equiv 1$ . From Example 1.1 we have the dimension of ${\mathcal{C}}_g$ is $2^n$ . The definition of Clifford algebras given here depends on the choice of basis for $V$. It is possible to give a basis-independent definition but this involves the concept of a free algebra (see Problem 7.5 of the next chapter).

The most important application of Clifford algebras in physics is the relativistic theory of the spin $\frac{1}{2}$ particles. In 1928, Paul Dirac (1902–1984) sought a linear first-order equation for the electron,

$${\gamma }^{\mu }{\partial }_{\mu }\psi =-m_e\psi \text{ where }\mu =1,2,3,4,{\partial }_{\mu }\equiv \frac{\partial }{\partial x^{\mu }}.$$

In order that this equation imply the relativistic Klein–Gordon equation,

$$g^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }\psi =-m_e^2\psi$$

where

$$[g^{\mu \nu }]=[g_{\mu \nu }]=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right),$$

it is required that the coefficients ${\gamma }^{\mu }$ satisfy

$${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2g^{\mu \nu }.$$

The elements ${\gamma }_{\mu }$ defined by ‘lowering the index’, ${\gamma }_{\mu }=g_{\mu \rho }{\gamma }^{\rho }$ , must satisfy

$${\gamma }_{\mu }{\gamma }_{\nu }+{\gamma }_{\nu }{\gamma }_{\mu }=2g_{\mu \nu }$$

and can be used to generate a Clifford algebra with $n=4$ . Such a Clifford algebra has $2^4=16$ dimensions. If one attempts to represent this algebra by a set of matrices, the lowest possible order turns out to be $4\times 4$ matrices. The vectorial quantities $\psi$ on which these matrices act are known as spinors; they have at least four components, a fact related to the concept of relativistic spin. The greatest test for Dirac’s theory was the prediction of antiparticles known as positrons, shown to exist experimentally by Anderson in 1932.

Problems

Problem 6.5

Show the ‘anticommutation law’ of conjugation,

$$\overline{PQ}=\overline{Q}\overline{P}.$$

Hence prove

$$|PQ|=|P||Q|.$$

Problem 6.6

Show that the set of $2\times 2$ matrices of the form

$$\left(\begin{array}{cc}z& w\\ -\bar{w}& \bar{z}\end{array}\right),$$

where $z$ and $w$ are complex numbers, forms an algebra of dimension 4 over the real numbers.

(a) Show that this algebra is isomorphic to the algebra of quaternions by using the bijection

$$Q=a+bi+cj+dk⟷\left(\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right).$$

(b) Using this matrix representation prove the identities given in Problem 6.5.

Problem 6.7

Find a quaternion $\mathbf{Q}$ such that

$$Q^{-1}iQ=j,Q^{-1}jQ=k.$$

[Hint: Write the first equation as $iQ=Qj.]$ For this $\mathbf{Q}$ calculate $Q^{-1}kQ$

Problem 6.8

Let $e_A$ and $e_B$ where $A$, $B\subseteq \{1,2,\dots ,n\}$ be two basis elements of the Clifford algebra associated with the Euclidean inner product space having $g_{ij}={\delta }_{ij}$ . Show that $e_A{e}_B=\pm e_C$ where $C=A\cup B-A\cap B$ . Show that a plus sign appears in this rule if the number of pairs

$$\{(i_r,j_s)|i_r\in A,j_s\in B,i_r>j_s\}$$

is even, while a minus sign occurs if this number of pairs is odd.