> Can these other equations be derived from the Schrödinger equation using a suitable Hamiltonian? No, it's the other way around: given a quantum field theory,

> Can these other equations be derived from the Schrödinger equation using a suitable Hamiltonian? No, it's the other way around: given a quantum field theory, under certain conditions (the main one I'm aware of is choosing a particular frame and taking the non-relativistic limit), you can derive a Schrodinger equation. > And do you mean that you need time to be in the Hamiltonian as an observable to satisfy Lorentz invariance? I mean that a fully relativistically correct treatment needs to treat time and space on the same footing. In quantum field theory that is usually done by making the quantum fields local operators, i.e., a quantum field is a mapping of points in spacetime to operators. In the simplest case, a free scalar field, these are the creation and annihilation operators of an infinite set of harmonic oscillators.