# Why is this stress-energy tensor from Peskin and Schroeder not conserved?

In paragraph 19.5 Peskin and Schroeder discuss the difference between the canonical energy-momentum tensor $T^{\mu\nu}$ and the symmetric and gauge invariant energy-momentum tensor $\Theta^{\mu\nu}$. Ultimately they state that (I am taking the gauge fields zero here) for fermions the latter reads $$\Theta^{\mu\nu} = \frac{i}{2}\bar{\psi}\big(\gamma^{\mu}\partial^{\nu}+\gamma^{\nu}\partial^{\mu}\big)\psi - \eta^{\mu\nu}\bar{\psi}\big(i\gamma^{\rho}\partial_{\rho}-m\big)\psi. \tag{19.150}$$ Taking the divergence yields $\partial_{\mu}\Theta^{\mu\nu}\neq0$. The last term is zero because it is simply the Lagrangian, which vanishes when imposing the equations of motion. The first term ($\propto \gamma^{\mu}$) also vanishes by simply employing the equations of motion. The second term ($\propto \gamma^{\nu}$) however, does not vanish. Instead, one can show that the tensor $$\tilde{\Theta}^{\mu\nu} = \frac{i}{4}\Big[\bar{\psi}\gamma^{\mu}\partial^{\nu}\psi+\bar{\psi}\gamma^{\nu}\partial^{\mu}\psi - (\partial^\nu\bar{\psi})\gamma^{\mu}\psi -(\partial^{\mu}\bar{\psi})\gamma^{\nu}\psi \Big]- \eta^{\mu\nu}\bar{\psi}\big(i\gamma^{\rho}\partial_{\rho}-m\big)\psi,$$ is conserved by using the equations of motion.

Hence, my question is: did Peskin and Schroeder make a mistake or am I missing something here?

[I know that similar questions have been asked, for instance here and here, but they don't help me solve this problem]

• What are the "three terms" you are talking about (I only see two), and why would the Lagrangian vanish when imposing the equations of motion? $\delta L = 0$ on-shell, not $L=0$! – ACuriousMind Oct 31 '17 at 13:39
• @ACuriousMind The Lagrangian is first order so it vanishes upon enforcing the equations of motion. Or am I mistaken? – Funzies Oct 31 '17 at 13:52
• Ah, you're right. physics.stackexchange.com/q/348085/50583 might be relevant. – ACuriousMind Oct 31 '17 at 14:00