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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 $\eta$

Qmechanic
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Ringo_00
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  • Related: https://physics.stackexchange.com/q/86038/2451 and links therein. – Qmechanic Feb 10 '19 at 06:23
  • 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 '19 at 08:46
  • Qmechanic my problem is basically how to multiply $\eta^{\mu\nu}$ with $\mathcal{L}$. – Ringo_00 Feb 10 '19 at 08:55
  • Why is it a problem to multiply $\eta^{\mu\nu}$ with $\mathcal{L}$? – Qmechanic Feb 10 '19 at 08:58
  • 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 '19 at 09:03
  • Do you know that a symbol a for (summed-over) repeated index is immaterial, i.e. you can change it to another symbol to avoid notational clashes? – Qmechanic Feb 10 '19 at 09:08
  • Take note that the Dirac lagrangian is 0 for the on-shell Dirac field : $\mathscr{L} = 0$ if Dirac's equation is satisfied : $i , \gamma^a , \partial_a , \psi - m , \psi = 0$. Also, don't forget to add the $\bar{\psi}$ terms to your canonical energy-momentum tensor. – Cham Feb 10 '19 at 14:28

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