Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains?
I heard many reasons, but I'm not sure which is true.
Some say it's just a matter of beauty, so Lagrangians are more beautiful because they don't break/separate the space-time variables (so space-time is a single variable, like in the Klein-Gordon Lagrangian and Hamiltonian).
Some say that Hamiltonians are not always Lorentz invariant.
Could someone explain in a little bit more details?
You have already mentioned the correct reason---the Lagrangian is manifestly Lorentz-invariant whereas the Hamiltonian is not. Since a relativistic field theory must be build of Lorentz-invariant quantities only the Lagrangian approach is good.
Compare for example the expressions for a free real scalar field $\phi$ $$ \mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi $$ This is Lorentz-invariant, because the Lorentz index $\mu$ is contracted in this way. The Hamiltonian for this theory is $$ H=\frac{1}{2}\dot\phi^2+\frac{1}{2}(\vec\nabla\phi)^2 $$ which is not manifestly Lorentz-invariant.
Is there any other reason?
