- #1

Dixanadu

- 254

- 2

So here's the deal. Consider the Lagrangian

[itex]\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi[/itex]

where [itex] \bar{\psi}=\psi^{\dagger}\gamma^{0} [/itex].

I need to find the Hamiltonian density from this, using

[itex]\mathcal{H}=\pi_{i}(\partial_{0}\psi_{i})-\mathcal{L}[/itex]

So I get the following:

[itex]\mathcal{H}=i\bar{\psi}\gamma^{i}\nabla\psi+\bar{\psi}\psi m[/itex]

But my teacher writes

[itex]\mathcal{H}=-i\bar{\psi}\gamma^{i}\nabla\psi+\bar{\psi}\psi m[/itex]

And I don't know how he gets that minus factor. The only part where I could be going wrong is when I expand [itex]\gamma^{\mu}\partial_{\mu}[/itex]...I'm using [itex]\gamma^{\mu}\partial_{\mu}=\gamma^{0}\partial_{0}-\gamma^{i}\partial_{i}[/itex]...because the metric signature is (+,---). But I guess this is wrong?