Griffiths Introduction to Elementary Particles · Problem 7.33

Electrodynamics of Quarks and Hadrons — David J. Griffiths

Physics — QFT · levi-civita-tensor-identities

Problem

(a) Show that ϵμνλσϵμνλτ=−6δτσ\epsilon^{\mu\nu\lambda\sigma}\epsilon_{\mu\nu\lambda\tau} = -6\delta^\sigma_\tau. (Summation over μ\mu, ν\nu, λ\lambda implied.) (b) Show that ϵμνλσϵμνθτ=−2(δθλδτσ−δτλδθσ)\epsilon^{\mu\nu\lambda\sigma}\epsilon_{\mu\nu\theta\tau} = -2(\delta^\lambda_\theta\delta^\sigma_\tau - \delta^\lambda_\tau\delta^\sigma_\theta). (c) Find the analogous formula for ϵμνλσϵμϕθτ\epsilon^{\mu\nu\lambda\sigma}\epsilon_{\mu\phi\theta\tau}. (d) Find the analogous formula for ϵμνλσϵμϕθτ\epsilon^{\mu\nu\lambda\sigma}\epsilon_{\mu\phi\theta\tau}. [Here δνμ\delta^\mu_\nu is the Kronecker delta: 1 if μ=ν\mu = \nu, 0 otherwise. It could be written in terms of the mixed (co/contravariant) metric tensor: δνμ=gμν=gνμ\delta^\mu_\nu = g^\mu{}_\nu = g_\nu{}^\mu.]

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