28.2 Real Cliford algebras

28.2 Real Cliford algebras

Concept links · terms present in this machine draft; source roles are unverified: Pauli matrices

We can define real Cliford algebras $\mathrm{Cliff}(n,\mathbf{R})$ just as for the complex case, by taking only real linear combinations:

Definition (Real Cliford algebras)

The real Cliford algebra in $n$ variables is the algebra Clif(n, $R$) generated over the real numbers by $1,{\gamma }_{j}forj=1,2,\dots ,n$ satisfying the relations

$$[{\gamma }_j,{\gamma }_k{]}_{+}=2{\delta }_{jk}$$

For reasons that will be explained in the next chapter, it turns out that a more general definition is useful. We write the number of variables as $n=r+s,$ for $r,s$ non-negative integers, and now vary not just $r+s.$ but also $r-s.$ , the so-called “signature”:

Definition (Real Cliford algebras, arbitrary signature)

The real Cliford algebra in $n=r+s$ variables is the algebra $\mathrm{Cliff}(r,s,\mathbf{R})$ over the real numbers generated by $1,{\gamma }_{j}forj=1,2,...,n$ satisfying the relations

$$[{\gamma }_j,{\gamma }_k{]}_{+}=\pm 2{\delta }_{jk}1$$

where we choose $the+sign$ when $j=k=1,\dots ,r$ and the − sign when $j=k=$ $r+1,\dots ,n.$

In other words, as in the complex case diferent ${\gamma }_j$ anticommute, but only the first r of them satisfy ${\gamma }_j^2=1$ , with the other s of them satisfying ${\gamma }_j^2=-1$

Working out some of the low dimensional examples, one finds:

• Clif(0, 1, $R$). This has generators 1 and ${\gamma }_1$ , satisfying

$${\gamma }_1^2=-1$$

Taking real linear combinations of these two generators, the algebra one gets is just the algebra $C$ of complex numbers, with ${\gamma }_1$ playing the role of $i=\sqrt{-1}$

$\mathrm{Cliff}(0,2,\mathbf{R})$ . This has generators $1,{\gamma }_1,{\gamma }_2$ and a basis

$$1,{\gamma }_1,{\gamma }_2,{\gamma }_1{\gamma }_2$$

with

$${\gamma }_1^2=-1,{\gamma }_2^2=-1,({\gamma }_1{\gamma }_2{)}^2={\gamma }_1{\gamma }_2{\gamma }_1{\gamma }_2=-{\gamma }_1^2{\gamma }_2^2=-1$$

This four dimensional algebra over the real numbers can be identified with the algebra $H$ of quaternions by taking

$${\gamma }_1\leftrightarrow \mathbf{i},{\gamma }_2\leftrightarrow \mathbf{j},{\gamma }_1{\gamma }_2\leftrightarrow \mathbf{k}$$

• Clif(1, 1, $R$). This is the algebra $M(2,\mathbf{R})$ of real 2 by 2 matrices, with one possible identification as follows

$$1\leftrightarrow \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),{\gamma }_1\leftrightarrow \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_2\leftrightarrow \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),{\gamma }_1{\gamma }_2\leftrightarrow \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

Note that one can construct this using the $a_F,a_F^{†}$ for the complex case Clif(2, $C$) (see 28.1) as

$${\gamma }_1=a_F+a_F^{†},{\gamma }_2=a_F-a_F^{†}$$

since these are represented as real matrices.

$\mathrm{Cliff}(3,0,\mathbf{R})$ . This is the algebra $M(2,\mathbf{C})$ of complex 2 by 2 matrices, with one possible identification using Pauli matrices given by

$$1\leftrightarrow \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$ $$\begin{array}{c}{\gamma }_1\leftrightarrow {\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\gamma }_2\leftrightarrow {\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\gamma }_3\leftrightarrow {\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\\ {\gamma }_1{\gamma }_2\leftrightarrow i{\sigma }_3=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),{\gamma }_2{\gamma }_3\leftrightarrow i{\sigma }_1=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),{\gamma }_1{\gamma }_3\leftrightarrow -i{\sigma }_2=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\\ {\gamma }_1{\gamma }_2{\gamma }_3\leftrightarrow \left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)\end{array}$$

It turns out that $\mathrm{Cliff}(r,s,\mathbf{R})$ is always one or two copies of matrices of real, complex or quaternionic elements, of dimension a power of 2, but this requires a rather intricate algebraic argument that we will not enter into here. For the details of this and the resulting pattern of algebras one gets, see for instance [55]. One special case where the pattern is relatively simple is when one has $r=s$ . Then $n=2r$ is even dimensional and one finds

$$\mathrm{Cliff}(r,r,\mathbf{R})=M(2^r,\mathbf{R})$$

Previous section: 28.1 The Complex Weyl and Cliford algebras · Next section: 28.3 For further reading · 中文版本

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