Standard Model and Beyond · Problem 2.11

This identity reveals the deep structure of Dirac algebra by decomposing a triple product of gamma matrices into symmetric and antisymmetric pieces. The key is recognizing that any product of three gammas can be separated into a symmetric part (involving the metric $g^{\mu\nu}$) and an antisymmetric part (involving the Levi-Civita tensor and $\gamma^5$). Start by using the fundamental anticommutation relation $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}$ repeatedly to rearrange terms, then collect what survives into these two irreducible structures—this mirrors how any rank-3 tensor decomposes into its symmetric and antisymmetric components.

This identity reveals the deep structure of Dirac algebra by decomposing a triple product of gamma matrices into symmetric and antisymmetric pieces. The key is recognizing that any product of three gammas can be separated into a symmetric part (involving the metric $g^{\mu\nu}$) and an antisymmetric part (involving the Levi-Civita tensor and $\gamma^5$). Start by using the fundamental anticommutation relation $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}$ repeatedly to rearrange terms, then collect what survives into these two irreducible structures—this mirrors how any rank-3 tensor decomposes into its symmetric and antisymmetric components.