Srednicki — Quantum Field Theory · Problem 36.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · lagrangians-spinor-fields

Problem

a) Prove the Fierz identities

\begin{aligned} (\chi_1^\dagger\bar{\sigma}^\mu\chi_2)(\chi_3^\dagger\bar{\sigma}_\mu\chi_4) &= -2(\chi_1^\dagger\chi_3^\dagger)(\chi_2\chi_4) , \tag{36.58}\\ (\chi_1^\dagger\bar{\sigma}^\mu\chi_2)(\chi_3^\dagger\bar{\sigma}_\mu\chi_4) &= (\chi_1^\dagger\bar{\sigma}^\mu\chi_4)(\chi_3^\dagger\bar{\sigma}_\mu\chi_2) . \end{aligned}

b) Define the Dirac fields

\Psi_i \equiv \begin{pmatrix} \chi_i \\ \xi_i^\dagger \end{pmatrix}, \qquad \Psi_i^\mathrm{C} \equiv \begin{pmatrix} \xi_i \\ \chi_i^\dagger \end{pmatrix}. \tag{36.60}

Use eqs. (36.58) and (36.59) to prove the Dirac form of the Fierz identities,

\begin{aligned} (\overline{\Psi}_1\gamma^\mu P_\mathrm{L}\Psi_2)(\overline{\Psi}_3\gamma_\mu P_\mathrm{L}\Psi_4) &= -2(\overline{\Psi}_1 P_\mathrm{R}\Psi_3^\mathrm{C})(\overline{\Psi}_4^\mathrm{C} P_\mathrm{L}\Psi_2) , \tag{36.61}\\ (\overline{\Psi}_1\gamma^\mu P_\mathrm{L}\Psi_2)(\overline{\Psi}_3\gamma_\mu P_\mathrm{L}\Psi_4) &= (\overline{\Psi}_1\gamma^\mu P_\mathrm{L}\Psi_4)(\overline{\Psi}_3\gamma_\mu P_\mathrm{L}\Psi_2) . \end{aligned}

c) By writing both sides out in terms of Weyl fields, show that

\begin{aligned} \overline{\Psi}_1\gamma^\mu P_\mathrm{R}\Psi_2 &= -\overline{\Psi}_2^\mathrm{C}\gamma^\mu P_\mathrm{L}\Psi_1^\mathrm{C} , \tag{36.63}\\ \overline{\Psi}_1 P_\mathrm{L}\Psi_2 &= +\overline{\Psi}_2^\mathrm{C} P_\mathrm{L}\Psi_1^\mathrm{C} , \\ \overline{\Psi}_1 P_\mathrm{R}\Psi_2 &= +\overline{\Psi}_2^\mathrm{C} P_\mathrm{R}\Psi_1^\mathrm{C} . \end{aligned}

Combining eqs. (36.63--36.65) with eqs. (36.61--36.62) yields more useful forms of the Fierz identities.