# 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