# Angular-momentum of the Dirac spinor theory

The standard Dirac action $$S = \int d^4 x \bar \psi (i \gamma^\mu \partial_\mu - m) \psi$$ is invariant under Lorentz transformation.

In David Tong's lecture note, eq (4.96) lists that the corresponding Noether current is $$(J^\mu)^{\rho \sigma} = x^\rho T^{\mu \sigma} - x^\sigma T^{\mu \rho} - i \bar \psi\gamma^\mu S^{\rho \sigma} \psi \ ,$$ where $$T$$ is the conserved stress tensor and $$S^{ab} \propto \gamma^a\gamma^b - \gamma^b \gamma^a$$.

The first two terms clear conserve. However, I fail to show that the second term is conserved: \begin{align} \partial_\mu (\bar \psi i \gamma^\mu S^{\rho \sigma}\psi) = & \ (\partial_\mu\bar \psi i \gamma^\mu S^{\rho \sigma}\psi) + (\bar \psi i \gamma^\mu S^{\rho \sigma}\partial_\mu\psi)\\ = & \ - m (\bar \psi S^{\rho \sigma}\psi) + (\bar \psi i \gamma^\mu S^{\rho \sigma}\partial_\mu\psi)\ . \end{align} I'm not sure how to deal with the second term, since $$\gamma^\mu$$ sits away from $$\partial_\mu$$, and $$\gamma$$ does not simply commute with $$S^{\rho \sigma}$$.

I wonder if there is some magical trick I should do or if the current is incorrect?

• In the spinor theoy only the total angular momentum $J=L+S$ is conserved. May 20 at 8:46
• The first two terms are conserved only if $T^{\mu\nu} = T^{\nu\mu}$ which is not true here (assuming $T$ is the canonical stress tensor, not the Belinfante stress tensor). May 20 at 13:24
• @Prahar you are right, I mistaken the said $T$ with the symmetric one. May 21 at 7:19