Classifiying Clifford Algebras

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Section 6.6 Classifiying Clifford Algebras

Clifford algebras can be classified; it turns out that there are only five distinct classes (depending on how one counts). Although the classification itself is not obvious, it is easy to see that there are certain constraints.

For example, consider the unique (up to sign) element \(\omega\) of rank \(n=p+q\) in \(Cl(p,q)\) that squares to \(\pm\one\text{.}\) In an orthonormal basis (suitably ordered), we have

\begin{equation} \omega = \gamma_1 ... \gamma_n\tag{6.6.1} \end{equation}

from which we can compute

\begin{align} \omega^2 \amp= \left(\gamma_1 ... \gamma_n\right) \left(\gamma_1 ... \gamma_n\right)\notag\\ \amp= (-1)^{n(n-1)/2} \gamma_1^2 ... \gamma_n^2 = (-1)^{n(n-1)/2} (-1)^q \one .\tag{6.6.2} \end{align}

However, the exponent is

\begin{equation} \frac12\bigl((p+q)(p+q-1)\bigr)+q = 2pq+\frac12\bigl((p-q)(p-q-1)\bigr)\tag{6.6.3} \end{equation}

so that \(\omega^2\) depends only on \(p-q\text{.}\) Explicitly, \(\omega^2=\one\) if \(p-q\equiv0,1\pmod 4\text{,}\) and \(\omega^2=-\one\) if \(p-q\equiv2,3\pmod 4\text{.}\)

As this calculation suggests, Clifford algebras can be classified based on the value of \(p-q\text{.}\) Somewhat more surprisingly, it turns out that this classification depends only on the value of \(p-q\pmod8\text{,}\) a fact sometimes referred to as the spinorial chessboard.

This classification can be interpreted as finding the smallest matrices that can be used to represent \(\Cl(p,q)\text{.}\) We work here with real Clifford algebras, but the matrices can be real, complex, or even quaternionic—in each case thought of as a real vector space. The spinors of \(\SO(p,q)\) are then the column vectors on which these matrices act.

Complex Clifford algebras can be classified using a similar procedure. There are then two special cases. First, if the real and complex cases lead to matrices of the same size, then the (real) spinors are called Majorana spinors. Second, if the spinors are not an irreducible representation of \(\so(p,q)\) then they decompose into exactly two such representations, which are called Weyl spinors. ¹

In this case, the original, undecomposed spinors are referred to as Dirac spinors.