What is the Hamiltonian density for a massive Dirac field?

1. May 12, 2015

Hey guys,

So here's the deal. Consider the Lagrangian

$\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi$

where $\bar{\psi}=\psi^{\dagger}\gamma^{0}$.

I need to find the Hamiltonian density from this, using

$\mathcal{H}=\pi_{i}(\partial_{0}\psi_{i})-\mathcal{L}$

So I get the following:

$\mathcal{H}=i\bar{\psi}\gamma^{i}\nabla\psi+\bar{\psi}\psi m$

But my teacher writes

$\mathcal{H}=-i\bar{\psi}\gamma^{i}\nabla\psi+\bar{\psi}\psi m$

And I dont know how he gets that minus factor. The only part where I could be going wrong is when I expand $\gamma^{\mu}\partial_{\mu}$....I'm using $\gamma^{\mu}\partial_{\mu}=\gamma^{0}\partial_{0}-\gamma^{i}\partial_{i}$...because the metric signature is (+,---). But I guess this is wrong?

2. May 17, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?