(a) Define (left{V_{1}, V_{2}, ldots, V_{16} ight} equiv frac{1}{2}left{I, gamma^{0}, i gamma^{j}, i sigma^{0 j},left.sigma^{j k} ight|_{j...

(a) Define \(\left\{V_{1}, V_{2}, \ldots, V_{16}\right\} \equiv \frac{1}{2}\left\{I, \gamma^{0}, i \gamma^{j}, i \sigma^{0 j},\left.\sigma^{j k}\right|_{j

(b) Using the \(\gamma\)-matrix anticommutation relations show that (i) \(V_{J}^{2}=\frac{1}{4} I\); (ii) \(V_{J} V_{K}=\) \(\pm V_{K} V_{J}\); and (iii) \(V_{K} V_{J} V_{K}=-\frac{1}{4} V_{J}\) for every \(J eq 1\) for at least one \(K\) ( \(K=16\) for \(J=2, \ldots, 5\) and \(12, \ldots, 15 ; K\) is any of \(2, \ldots, 5\) for \(J=16\); and \(K\) is at least one of \(2, \ldots, 5\) for \(J=6, \ldots, 11)\).

(c) Show for every \(V_{J}\) and \(V_{K}\) that \(V_{J} V_{K}=\frac{1}{2} c_{J K L} V_{L}\) where \(c_{J K L}= \pm 1, \pm i\) for only one \(V_{L}\) and where no sum over \(L\) is implied. Note that for fixed \(J\) each \(K\) leads to a different \(L\) and for fixed \(K\) each \(J\) leads to a different \(L\). This is referred to as the rearrangement lemma.

(d) By considering \(\operatorname{tr}\left(V_{K} V_{J} V_{K}\right)\) explain why \(\operatorname{tr} V_{J}=0\) for every \(J eq 1\).

(e) Explain why trace identities lead to "orthonormality", \(\operatorname{tr}\left(V_{J}^{\dagger} V_{K}\right)=\operatorname{tr}\left(V_{J} V_{K}\right)=\delta_{J K}\).

(f) Show that \(c_{J K L}=2 \operatorname{tr}\left(V_{J} V_{K} V_{L}\right)\).

(g) Define the 16 vectors \(\left(X_{J}\right)_{r} \equiv\left(V_{J}\right)_{i j}\) with \(r \equiv j+4(i-1)=1, \ldots, 16\). Hence show that \(\operatorname{tr}\left(V_{J}^{\dagger} V_{K}\right)=\sum_{r}\left(X_{J}\right)_{r}^{*}\left(X_{K}\right)_{r}=X_{J}^{\dagger} X_{K}=\delta_{J K}\) so that the \(X_{J}\) are 16 orthonormal sixteen component complex vectors and hence are an orthonormal basis of \(\mathbb{C}^{16}\). Show that the completeness relation for this orthonormal basis \(\sum_{J}\left(X_{J}\right)_{r}\left(X_{J}^{*}\right)_{s}=\delta_{r s}\) can be written as \(\sum_{J}\left(V_{J}\right)_{i j}\left(V_{J}^{*}\right)_{k \ell}=\sum_{J}\left(V_{J}\right)_{i j}\left(V_{J}^{\dagger}\right)_{\ell k}=\sum_{J}\left(V_{J}\right)_{i j}\left(V_{J}\right)_{\ell k}=\delta_{i k} \delta_{j \ell}\).

(h) Any \(4 \times 4\) complex matrix can be regarded as a vector in \(\mathbb{C}^{16}\). Completeness of the \(X_{J}\) then means completeness of the matrices \(V_{J}\) and so for any matrix \(A\) we can write \(A \equiv\) \(\sum_{J} c_{J} V_{J}\). Show that \(c_{J}=\operatorname{tr}\left(V_{K} A\right)\).

(i) Explain why a matrix that commutes with all four \(\gamma\)-matrices must commute with all matrices and so must be a multiple of the identity. This is sometimes referred to Schur's lemma.

(j) Show \(\left(V_{J}\right)_{i j}\left(V_{K}\right)_{k \ell}=\sum_{m, n}\left(V_{J}\right)_{i m}\left(V_{K}\right)_{k n} \delta_{m j} \delta_{n \ell}=\sum_{L}\left(V_{J} V_{L}\right)_{i \ell}\left(V_{K} V_{L}\right)_{k j}\) using completeness. Then show using the rearrangement lemma that \(\left(V_{J}\right)_{i j}\left(V_{K}\right)_{k \ell}=\) \((1 / 4) \sum_{L} c_{J L M} c_{K L N}\left(V_{M}\right)_{i \ell}\left(V_{N}\right)_{k j}\). Explain why we can write for some coefficients \(C_{J K M N} \in \mathbb{C}\),\[\begin{equation*}\left(V_{J}\right)_{i j}\left(V_{K}\right)_{k \ell} \equiv \sum_{M, N} C_{J K MN}\left(V_{M}\right)_{i\ell}\left(V_{N}\right)_{k j}, \tag{6.5.84}\end{equation*}

which is referred to as the general form of the Fierz identity. Explain why \(C_{J K M N}=0\) if \(J=K\) unless \(M=N\) and vice versa.

(k) Using this result and orthonormality show that \(C_{J K M N}=\operatorname{tr}\left(V_{M} V_{J} V_{N} V_{K}\right)\).

(1) Rewrite the 16 basis matrices as \(2 V_{J} \rightarrow \Gamma_{\mathcal{A}}^{a}\), where \(a\) labels the appropriate combination of spacetime indices and where \(\mathcal{A}=S, V, T, A, P\) denote scalar, vector, tensor, axial vector and pseudoscalar, respectively. Specifically, we define \(\Gamma_{S}^{1} \equiv I, \Gamma_{V}^{1}, \ldots, \Gamma_{V}^{4} \equiv \gamma^{\mu}\), \(\Gamma_{T}^{1}, \ldots,\left.\Gamma_{T}^{6} \equiv \sigma^{\mu u}\right|_{\mu