Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

How can I derive the Dirac equation from the Lagrangian density for the Dirac field?

share|improve this question
4  
Use the Euler-Lagarange equation for fields. –  ramanujan_dirac May 19 '13 at 8:24
5  
This standard derivation is done in every QFT text. If you are getting stuck can you please elaborate about what exactly is giving you trouble? –  Michael Brown May 19 '13 at 8:51
1  
Yeah, it's like a 2-line derivation –  user12345 May 19 '13 at 10:37

1 Answer 1

The Lagrangian density for a Dirac field is $$ \mathcal{L} = i\bar\psi\gamma^\mu\partial_\mu\psi -m \bar\psi\psi $$ The Euler-Lagrange equation reads $$ \frac{\partial\mathcal{L}}{\partial\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\psi)}\right] = 0 $$ We treat $\psi$ and $\bar\psi$ as independent dynamical variables. In fact, it is easier to consider the Euler-Lagrange for $\bar\psi$ $$ \frac{\partial\mathcal{L}}{\partial\bar\psi} - \frac{\partial}{\partial x^\mu}\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\bar\psi)}\right] = 0\\ \Rightarrow i\gamma^\mu\partial_\mu\psi -m\psi - \frac{\partial}{\partial x^\mu}[ 0] = 0\\ \Rightarrow i\gamma^\mu\partial_\mu\psi -m\psi=0 $$ The partial differentiation is trivial - remember that $\bar\psi$ and $\partial_\mu\bar\psi$ are treated as though independent. We recover the Dirac equation as expected. If we had instead chosen the Euler-Lagrange for $\psi$, we would have found the conjugate Dirac equation.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.