# Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

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?

Thanks.

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## marked as duplicate by Emilio Pisanty, Qmechanic♦Sep 25 '13 at 17:27

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.