Clifford Algebras over Division Algebras

Section 6.8 Clifford Algebras over Division Algebras

We begin with a Weyl representation of \(\Cl(3,1)\text{.}\) Consider the gamma matrices

\begin{equation} \gamma_m = \left( \begin{array}{c|c} \zero \amp \sigma_m \\ \hline \tilde\sigma_m \amp \zero \end{array} \right)\tag{6.8.1} \end{equation}

where \(\zero\) denotes the zero matrix, the \(\sigma_m\) denote the Pauli matrices for \(m=x,y,z\text{,}\) and we have introduced the convention that \(\sigma_t=\one\text{.}\) We have also introduced the operation of trace reversal, denoted with a tilde and defined by

\begin{equation} \tilde{X} = X - \tr(X)\,\one\tag{6.8.2} \end{equation}

where \(X\) is a \(2\times2\) matrix. Since \(X\) satisfies its characteristic equation, namely

\begin{equation} X^2 - X\,\tr(X) + \det(X)\,\one = \zero\tag{6.8.3} \end{equation}

we see that

\begin{equation} X\tilde{X} = -\det(X)\,\one\tag{6.8.4} \end{equation}

so that \(\tilde{X}\) is closely related to the inverse of \(X\) (if it exists). It is easily checked that \(\{\gamma_t,\gamma_x,\gamma_y,\gamma_z\}\) generate \(\Cl(3,1)\text{.}\)

Since these gamma matrices are block off-diagonal, their products are block diagonal. But the elements of \(\Cl(3,1)\) of rank 2 generate the Lie algebra \(\so(3,1)\text{.}\) Thus, the spinors of \(\Cl(3,1)\text{,}\) represented here by 4-component complex column vectors, decompose into two irreducible representations corresponding to the “top” and “bottom” 2-component column vectors, which are acted on independently by \(\so(3,1)\text{.}\) Elements of these irreducible representations are called Weyl spinors; elements of the original, reducible representation are called Dirac spinors. ¹

Let's look at one of the \(2\times2\) blocks, which have the form

\begin{equation} X = t\,\sigma_t + x\,\sigma_x + y\,\sigma_y + z\,\sigma_z = \begin{pmatrix}t+z \amp x-iy \\ x+iy \amp t-z\end{pmatrix} .\tag{6.8.5} \end{equation}

Ignoring \(t\) and \(z\) for the moment, the remaining degrees of freedom in \(X\) clearly correspond to a complex number. We can thus imagine generalizing this construction so as to involve any of the division algebras \(\RR\text{,}\) \(\CC\text{,}\) \(\HH\text{,}\) or \(\OO\text{.}\) Notationally, we can rename \(\sigma_y=-is_y\) to \(\sigma_i\text{,}\) and then define \(\sigma_j=-js_y\text{,}\) etc.

The \(\kappa\) resulting gamma matrices (over \(\KK\)) clearly generate \(\Cl(\kappa)\text{,}\) where \(\kappa=\dim(\KK)\text{!}\) Furthermore, we can add \(t\) and \(z\) back in, resulting in \(\Cl(\kappa+1,1)\text{,}\) and hence also to the corresponding orthogonal algebras \(\so(\kappa+1,1)\text{.}\) Explicitly, the four division algebras \(\RR\text{,}\) \(\CC\text{,}\) \(\HH\text{,}\) \(\OO\) lead to the orthogonal groups in \(3\text{,}\) \(4\text{,}\) \(6\text{,}\) and \(10\) spacetime dimensions. Not coincidentally, these are precisely the spacetime dimensions that admit classical supersymmetry! ²

Returning to the \(t\) and \(z\) components of \(X\text{,}\) the key observation is that nothing changes in the above constructions if we replace \(\gamma_m\) with \(\gamma_t L\text{,}\) where \(L\in\CC'\) is the split complex unit (so \(L^2=+1\)). But now we can identify

\begin{equation} X = \begin{pmatrix}z+tL \amp 0 \\ 0 \amp -z+tL\end{pmatrix}\tag{6.8.6} \end{equation}

with a split complex number—and play the same game as before, generalizing to the split division algebras \(\CC'\text{,}\) \(\HH'\text{,}\) \(\OO'\text{.}\) The \(\kappa'\) resulting gamma matrices (over \(\KK'\)) again generate a Clifford algebra, with \(\kappa'=\dim(\KK')\text{.}\) However, the signature is now \((\frac{\kappa'}2,\frac{\kappa'}2)\text{.}\)

Furthermore, we can do both of these generalizations at once! That is, if we set

\begin{equation} X = \begin{pmatrix}A \amp \bar{a} \\ a \amp -\bar{A}\end{pmatrix}\tag{6.8.7} \end{equation}

with \(a\in\OO\) and \(A\in\OO'\text{,}\) then the 16 resulting gamma matrices generate \(\Cl(12,4)\text{!}\) This interpretation is supported by noting that

\begin{equation} -\det(X) = |a|^2 + |A|^2\tag{6.8.8} \end{equation}

which indeed has signature \((12,4)\text{.}\) We can in fact use (6.8.7) to define the gamma matrices by letting

\begin{equation} \sigma_a = \begin{pmatrix}0 \amp \bar{a} \\ a \amp 0\end{pmatrix} , \qquad \sigma_A = \begin{pmatrix}A \amp 0 \\ 0 \amp -\bar{A}\end{pmatrix}\tag{6.8.9} \end{equation}

with \(a\in\OO\) and \(A\in\OO'\text{,}\) then inserting these generalized Pauli matrices into (6.8.1).

We have carried out this construction using one division algebra and one split division algebra, but only the signature changes if both algebras are division algebras, or both are split division algebras. In summary, we have constructed a family of Clifford algebras “parameterized” by the choice of two (possibly split) division algebras.

This process of reducing a Weyl representation generated by block off-diagonal gamma matrices to a block-diagonal description of the underlying Lie algebra, and then looking at the blocks separately, is more formally the restriction of the original Clifford algebra to its even part, that is, to the subalgebra formed by elements of even rank.

The original proof of this fact used a complicated identity on gamma matrices that holds only in these four cases. Only later was it realized that this identity was equivalent to the composition property \(|pq|=|p||q|\) of the corresponding division algebra.