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Canonical Quantization of Scalar Fields — Mark Srednicki
Physics — QFT · left-right-handed-spinor-fields
Problem Show that the Levi-Civita symbol obeys
ε μ ν ρ σ ε α β γ σ = − δ μ α μ δ ν β ν δ ρ γ ρ − δ μ β μ δ ν γ ν δ ρ α ρ − δ μ γ μ δ ν α ν δ ρ β ρ + δ μ β μ δ ν α ν δ ρ γ ρ + δ μ α μ δ ν γ ν δ ρ β ρ + δ μ γ μ δ ν β ν δ ρ α ρ , ε μ ν ρ σ ε α β ρ σ = − 2 ( δ μ α μ δ ν β ν − δ μ β μ δ ν α ν ) , ε μ ν ρ σ ε α ν ρ σ = − 6 δ μ α μ . (34.44) \begin{aligned}
\varepsilon^{\mu\nu\rho\sigma}\varepsilon_{\alpha\beta\gamma\sigma} &= -\delta^\mu_{\phantom{\mu}\alpha}\delta^\nu_{\phantom{\nu}\beta}\delta^\rho_{\phantom{\rho}\gamma} - \delta^\mu_{\phantom{\mu}\beta}\delta^\nu_{\phantom{\nu}\gamma}\delta^\rho_{\phantom{\rho}\alpha} - \delta^\mu_{\phantom{\mu}\gamma}\delta^\nu_{\phantom{\nu}\alpha}\delta^\rho_{\phantom{\rho}\beta} \nonumber\\
&\quad + \delta^\mu_{\phantom{\mu}\beta}\delta^\nu_{\phantom{\nu}\alpha}\delta^\rho_{\phantom{\rho}\gamma} + \delta^\mu_{\phantom{\mu}\alpha}\delta^\nu_{\phantom{\nu}\gamma}\delta^\rho_{\phantom{\rho}\beta} + \delta^\mu_{\phantom{\mu}\gamma}\delta^\nu_{\phantom{\nu}\beta}\delta^\rho_{\phantom{\rho}\alpha} , \tag{34.44}\\
\varepsilon^{\mu\nu\rho\sigma}\varepsilon_{\alpha\beta\rho\sigma} &= -2(\delta^\mu_{\phantom{\mu}\alpha}\delta^\nu_{\phantom{\nu}\beta} - \delta^\mu_{\phantom{\mu}\beta}\delta^\nu_{\phantom{\nu}\alpha}) , \\
\varepsilon^{\mu\nu\rho\sigma}\varepsilon_{\alpha\nu\rho\sigma} &= -6\,\delta^\mu_{\phantom{\mu}\alpha} .
\end{aligned} ε μν ρ σ ε α β γ σ ε μν ρ σ ε α β ρ σ ε μν ρ σ ε α ν ρ σ = − δ μ α μ δ ν β ν δ ρ γ ρ − δ μ β μ δ ν γ ν δ ρ α ρ − δ μ γ μ δ ν α ν δ ρ β ρ + δ μ β μ δ ν α ν δ ρ γ ρ + δ μ α μ δ ν γ ν δ ρ β ρ + δ μ γ μ δ ν β ν δ ρ α ρ , = − 2 ( δ μ α μ δ ν β ν − δ μ β μ δ ν α ν ) , = − 6 δ μ α μ . ( 34.44 ) Browse the problem library