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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 $\Psi$?

jak
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  • The Dirac field do have an energy-momentum tensor, like the electromagnetic field. However, its mathematical expression is very different and pretty complicated. – Cham Oct 01 '19 at 13:27
  • There's a similar question here: https://physics.stackexchange.com/questions/86038/dirac-field-and-stress-energy-tensor-density?rq=1 and https://physics.stackexchange.com/questions/414556/energy-momentum-tensor-of-transformed-dirac-lagrangian?rq=1 and also https://physics.stackexchange.com/questions/268619/stress-energy-tensor-for-dirac-fields-and-its-dependence-on-connection?rq=1 and again https://physics.stackexchange.com/questions/459895/energy-momentum-tensor-of-the-dirac-field?noredirect=1&lq=1 – Cham Oct 01 '19 at 13:41

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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.

Cham
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