# What's the field strength and energy density of a spinor field?

The field strength and energy density of a vector field $$A_\mu$$ can be described using the field strength tensor $$F_{\mu \nu}$$.

What is the field strength and energy density associated with a (Dirac) spinor field $$\Psi$$?

Here's the symmetric energy-momentum tensor of the free Dirac field: $$$$\tag{1} {T}_{\mu \nu}^{\textsf{D}} = i \, \frac{\hbar c}{4} \, \big( \, \bar{\Psi} \: \gamma_{\mu} \, (\, \partial_{\nu} \, \Psi \,) + \bar{\Psi} \: \gamma_{\nu} \, ( \, \partial_{\mu} \, \Psi \,) - (\, \partial_{\mu} \, \bar{\Psi} \,) \, \gamma_{\nu} \, \Psi - (\, \partial_{\nu} \, \bar{\Psi} \,) \, \gamma_{\mu} \, \Psi \, \big).$$$$ This guy could be found using the canonical energy-momentum (or the Noether current associated to translation in spacetime), but it would need to be symmetrized using the complicated Belefante-Rosenfeld procedure.