I know that given the Hamiltonian of a theory, there can be many different associated Lagrangians, or even none at all, but why is that so? In classical mechanics the Hamiltonian and Lagrangian formulations are related by a Legendre transformation, what happens in QFT with this transformation to make it so ill-behaved?
I understand that Hamiltonians are not covariant, and if I understand correctly, a given Hamiltonian can be better understood as the time evolution operator of a given observer after fixing the Poincare symmetry. This, as we have to choose a explicit time coordinate, that determines the spatial coordinates and conjugate momenta that appear in the Legendre transformation (up to a rotation and translation, of course). Wouldn't that Legendre transformation then be unique? (As rotations and translations are still symmetries of our chosen frame of reference and thus return the same Lagrangian)
What's the reason we can have several different Lagrangians or maybe even none at all?
The only examples of no-Lagrangian QFTs that I have found in my studies are some CFTs , but I am not sure if those theories actually have a Hamiltonian at all, so I do not know if they can be a good working example.