# Hamilton's equations for Dirac Hamiltonian [duplicate]

The Dirac Lagrangian $$\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_\mu \psi - m \bar{\psi}\psi$$ gives a Hamiltonian $$\mathcal{H}(\Pi,\bar{\Pi},\psi,\bar{\psi})=\Pi \dot{\psi}-\mathcal{L}=-\bar{\psi}\gamma^{i}\partial_i \psi + m \bar{\psi}\psi,$$ $$\text{where}\quad \Pi=i\bar{\psi}\gamma^0.$$ So the Hamiltonian is actually independent of $$\Pi,\bar{\Pi}$$. Suppose I wanted to start from this Hamiltonian and tried to find the equations of motion via Hamilton's equations $$\dot{\psi} = \partial\mathcal{H}/\partial \Pi$$ and $$\dot{\Pi} = -\partial\mathcal{H}/\partial \psi$$, plus equivalent ones for conjugate field.

Clearly something is wrong as the first of Hamilton's equations gives $$\dot{\psi}=0$$, while the second gives $$\dot{\Pi}=i \partial_i\bar{\psi}\gamma^i +m\bar{\psi}$$ and I don't see the Dirac equation popping out...

• in the transformation from lagrangian to hamiltonian you change variables $(\psi,\dot{\psi})\rightarrow(\psi,\Pi)$. I still see $\dot{\psi}$ in your hamiltonian instead of $\Pi$. Perhaps changing the variables would solve the issue? Jun 3, 2019 at 19:30
• I don't think there's a $\dot{\psi}$ in my Hamiltonian. $\partial_i$ is a spatial derivative, which sstays as is in the Hamiltonian formulation. Jun 3, 2019 at 19:37
• Possible duplicates: physics.stackexchange.com/q/43502/2451 and links therein. Jun 3, 2019 at 19:54