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{{wiki=Lagrangian_(field_theory)#Dirac_Lagrangian}}

$$
\mathcal{L} = \bar \psi ( i \hbar c {\partial}\!\!\!/ - mc^2) \psi
$$
where:
* ${\partial}\!\!\!$: <Feynman slash notation>
* $\bar \psi$: <Dirac adjoint>

Remember that $\psi$ is a 4-vetor, gamma matrices are 4x4 matrices, so the whole thing comes down to a <dot product> of two 4-vectors, with a modified $\psi$ by matrix multiplication/derivatives, and the result is a scalar, as expected for a <Lagrangian>.

Like any other <Lagrangian>, you can then recover the <Dirac equation>, which is the corresponding <equations of motion>, by applying the <Euler-Lagrange equation> to the Lagrangian.


Dirac Lagrangian

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Analogous to what the <Euler-Lagrange equation> is to <Lagrangian mechanics>, Hamilton's equations give the <equations of motion> from a given input <Hamiltonian>:
$$
\frac{\partial H}{\partial q_j} = -\dot{p}_j \\
\frac{\partial H}{\partial p_j} =  \dot{q}_j
$$
So once you have the Hamiltonian, you can write down this system of <partial differential equations> which can then be numerically solved.


Ciro Santilli @cirosantilli 34

 Incoming links: Equations of motion