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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 $\Psi$?

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Here's the symmetric energy-momentum tensor of the free Dirac field: \begin{equation}\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). \end{equation} 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.

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