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Section 6.2 Clifford Algebras

Given a vector space \(V\) with nondegenerate inner product \(g\text{,}\) the Clifford algebra \(\Cl(V)\) is the algebra obtained by introducing a bilinear, associative product on \(V\) such that

\begin{equation} X^2 = XX = g(X,X) = |X|^2\tag{6.2.1} \end{equation}

for all \(X\in V\text{.}\) Equivalently, by polarization,

\begin{equation} \{X,Y\} = XY+YX = 2g(X,Y)\tag{6.2.2} \end{equation}

for all \(X,Y\in V\text{,}\) where we have introduced the anticommutator using curly brackets. It is clear from this definition that the Clifford product is symmetric. If \(\{\gamma_m\}\) is an orthonormal basis of \(V\text{,}\) then

\begin{equation} \{\gamma_i,\gamma_j\} = \pm2\delta_{ij}\tag{6.2.3} \end{equation}

where the signs are determined by the signature of \(g\text{.}\) If \(V=\RR^{p,q}\text{,}\) that is, if \(g\) has signature \((p,q)\text{,}\) then \(\Cl(V)\) is usually written as \(\Cl(p,q)\text{.}\) Clifford algebras can be real or complex, depending on which scalars are allowed.

The Clifford algebra \(\Cl(p,q)\) can be represented using matrices in many ways. It is an interesting exercise to determine the minimum size of matrices \(\gamma_m\) for a given dimension \(n\) of \(V\) and signature \((p,q)\) of \(g\text{,}\) especially if one also demands that the matrices satisfy certain additional conditions. The gamma matrices used in physics are just a particular choice of such a basis for the underlying Clifford algebra, usually referring to \(\Cl(3,1)\text{.}\) Spinors can be thought of as elements of the vector space on which the Clifford algebra acts, that is, as the column vectors acted on by the gamma matrices.

The Clifford algebra \(\Cl(V)\) clearly contains a copy of \(V\text{;}\) these are the elements of rank 1. It also contains the product of any two elements of \(V\text{,}\) and so on. Some of these products will be multiples of the identity element (rank 0), while others will be independent (rank 2)—and others still will combine both rank 0 and rank 2 elements. This pattern continues for products of elements of higher rank.

Using these ideas, it is straightforward to verify that \(\Cl(V)\) is itself a vector space, and to give a basis for \(\Cl(V)\) in terms of an orthonormal basis \(\{\gamma_m\}\) of \(V\text{.}\) Each element of the basis of \(\Cl(V)\) must be the product of distinct basis elements of \(V\text{,}\) which we write as

\begin{equation} \gamma_{m_1...m_k} = \gamma_{m_1} ... \gamma_{m_k}\tag{6.2.4} \end{equation}

and which is clearly of rank \(k\) (so long as the indices \(m_i\) are distinct). We usually (but not always) assume that \(1\le m_1\lt m_2\lt...\lt m_k\le n\text{.}\) Thus, there are \({n\choose k}=\frac{n!}{k!(n-k)!}\) independent elements of rank \(k\text{,}\) and \(n^2\) total elements, in \(\Cl(V)\text{.}\)