Hamiltonians

latentcorpse

In equation 5.8 in this document

http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf

I am trying to derive this Hamiltonian. I find

[latex]H= \pi \dot{\psi} - L = i \psi^\dagger \dot{\psi} - \bar{\psi} ( i \gamma^\mu \partial_\mu - m ) \psi = i \bar{\psi} \gamma^0 \partial_0 \psi - i \bar{\psi} \gamma^\mu \partial_\mu \psi = m \bar{\psi} \psi = \bar{\psi} ( i \gamma^i \partial_i + m ) \psi[/latex]

so I get a minus sign the other way around because of the defn of the dot product of 4 vectors [latex]\gamma^\mu \partial_\mu = \gamma^0 \partial_0 - \gamma^i \partial_i[/latex]

So does anyone else think this is a typo? I'm sure it isn't since he uses it in the following pages!

Thanks.

dextercioby

That's wrong. It should be
[latex]\gamma^\mu \partial_\mu = \gamma^0 \partial_0 + \gamma^i \partial_i[/latex]
by the Einstein summation convention.

The space-time derivative is acting to the right.