Quantum field theory is based on the Lagrangian and the path integral method explores, via the partition, all kinds of paths, like all paths in classical mechanìcs are explored, from which one is chosen, namely the one with the least action (which involves the Lagrangian). In QFT all paths actually contribute with varying weights.
Classical mechanics though can be done in a Hamiltonian way too. The Lagrangian represents $T-V$, while the Hamiltonian represents their sum, $T+V$. So both the potential energy as kinetic energy are involved. And indeed, in classical mechanics the Hamiltonian is used to determined position and momentum.
Can the Hamiltonian approach be used in QFT? Without trying all paths? Would this amount to cakculating all possible positions and associated particle momenta at all times, and the associated weights? If so, is there a reason why the Lagrangian is used and not the Hamiltonian? Why is the situation in classical quantum mechanics the opposite?
