Energy-momentum tensor of a fermion

Question

I'm wondering how does one write the energy momentum tensor of a fermion (like an electron)? I've seen the formula for deriving it through the action. What I'm specifically asking is how would each of the 16 different components of the $T_{\mu\nu}$ of the fermion look like and what is the physics meaning for each of those components?

Answer 1

The energy-momentum tensor for a spinor is given by $T^{\mu\nu}=i\bar\psi\gamma^\mu\partial^\nu\psi$. Its components have the usual meaning, e.g., $T_{00}$ is energy density, etc.

To obtain it, just follow the usual definition: it is the Noether current of spacetime translations, given the spinor Lagrangian ${\cal L}=i\bar\psi\gamma^\mu\partial_\mu\psi$.

More details can be found in every basic field theory textbook (Peskin & Schroeder...)