In deriving Dyson’s formula, it is assumed that the state ket vector evolves from an initial state to a final state according to Schrödinger's equation in the interaction picture. But isn’t the Schrödinger equation non-relativistic and thus can’t be assumed true when deriving scattering amplitudes in quantum field theory?
I know the Hamiltonian is consistent with relativity, but isn’t the Schrodinger equation only assumed to be true in non-relativistic quantum mechanics?
What is meant by the Schrödinger equation in this context is the general form $$i\hbar\frac{d}{dt}|\psi\rangle=H|\psi\rangle,$$ with a general expression for the Hamilton operator $H$. In a field theory, $H$ is obviously not Schrödinger's original expression, $-\frac{\hbar^{2}}{2m}\vec{\nabla}^{2}+V(\vec{r})$, but is instead a integral (but local) operator constructed from quantized field operators. However, with the correct relativistic $H$, the abstract form of the Schrödinger equation does still hold.
