Consider the sixteen matrices (left{Gamma_{1}, Gamma_{2}, ldots, Gamma_{16} ight} equivleft{I, gamma^{mu},left.sigma^{mu u} ight|_{mu

Consider the sixteen matrices \(\left\{\Gamma_{1}, \Gamma_{2}, \ldots, \Gamma_{16}\right\} \equiv\left\{I, \gamma^{\mu},\left.\sigma^{\mu u}\right|_{\mu

(a) \(\gamma^{0}\left(\Gamma_{I}\right)^{\dagger} \gamma^{0}=\Gamma_{I}\) in any representation unitarily equivalent to the Dirac representation;

(b) \(\Gamma_{I}^{2}= \pm 1\);

(c) \(\Gamma_{I} \Gamma_{J}= \pm \Gamma_{J} \Gamma_{I}\);

(d) \(\Gamma_{J} \Gamma_{K}=a_{J K}{ }^{L} \Gamma_{L}\) for some \(L\) (no sum implied) with \(a_{J K}{ }^{L}= \pm 1, \pm i\);

(e) \(\Gamma_{I}^{-1} \Gamma_{J} \Gamma_{I}=-\Gamma_{J}\) for \(J eq 1\) for at least one \(\Gamma_{I}\);

(f) \(\operatorname{tr}\left(\Gamma_{J}\right)=0\) for \(J eq 1\);

(g) \(\operatorname{tr}\left(\Gamma_{I}^{-1} \Gamma_{J}\right)=\operatorname{tr}\left(\Gamma_{I} \Gamma_{J}^{-1}\right)=4 \delta_{I J}\);

(h) \(\sum_{I=1}^{16}\left(\Gamma_{I}^{-1}\right)_{i j}\left(\Gamma_{I}\right)_{k \ell}=\sum_{I=1}^{16}\left(\Gamma_{I}\right)_{i j}\left(\Gamma_{I}^{-1}\right)_{k \ell}=4 \delta_{i \ell} \delta_{k j}\);

(i) if \(\sum_{I=1}^{16} c_{I} \Gamma_{I}=0\), then we must have all \(c_{I}=0\) and hence the \(\Gamma_{J}\) are sixteen linearly independent matrices; and

(j) using the result of part (i), explain why any complex \(4 \times 4\) matrix can be written as \(A=\) \(\sum_{I=1}^{16} a_{I} \Gamma_{I}\) with \(a_{I} \equiv \frac{1}{4} \operatorname{tr}\left(\Gamma_{I}^{-1} A\right)\). So these sixteen matrices form a basis for the set of all \(4 \times 4\) complex matrices.