# Energy-momentum tensor of the Dirac field

I'm trying to compute the energy momentum tensor for the dirac field $$\mathcal{L}=\bar\psi(i\gamma_\mu\partial^\mu-m)\psi$$$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\partial^\nu\psi-\eta^{\mu\nu}\mathcal{L}$$ and I'm not clear on how to treat the term in $$\eta^{\mu\nu}\mathcal{L}$$: the first term gives $$i\bar\psi\gamma^\mu\partial^\nu\psi$$ which is the tensor given by Peskin-Schroeder but I don't get how to compute the term in $$\eta$$

• Related: physics.stackexchange.com/q/86038/2451 and links therein. – Qmechanic Feb 10 at 6:23
• I read it and I seem to understand that this answer is missing the mass term of the lagrangian but I still don't understand how to treat the term in $\eta^{\mu\nu}\mathcal{L}$ – Ringo_00 Feb 10 at 8:46
• Qmechanic my problem is basically how to multiply $\eta^{\mu\nu}$ with $\mathcal{L}$. – Ringo_00 Feb 10 at 8:55
• Why is it a problem to multiply $\eta^{\mu\nu}$ with $\mathcal{L}$? – Qmechanic Feb 10 at 8:58
• Because I can't contract the $\mu$ indices in $\eta$ with the one in $\gamma^\mu \partial_\mu$ since they are already contracted – Ringo_00 Feb 10 at 9:03