My question is can we get the Hamiltonain from Dirac Equation? We have the following for Dirac equation:
$$(i\gamma ^\mu \partial_\mu-m)\phi=0.$$
Then separating the time and space components:
$$(i\gamma ^0 \partial_0-i\gamma ^i \partial_i-m)\phi=0.$$
Then I try to write it in form $i\partial_0\phi=H_D\phi$ so if i take time component to one side and take the i components to the other side:
now is it right or should I multiply everything by $\gamma^0$? If I try to do that i get nothing like the dirac hamiltonian. any help is appreciated. Thank you.
PS. this question is from chapter 10 of Matthew Schwartz Quantum field theory and the Standard model Problem 10.1.