Derivation of the Schrödinger equation from QED

The Schrödinger equation relates the electron wavefunction and the electric potential, which are emergent physical quantities. At that emergent level, the Schrödinger equation is either postulated as a principle of quantum physics or obtained heuristically. However, the Schrödinger equation is a low energy condition we can derive from the foundations of QED. Due to the small value of the electromagnetic coupling constant, we show that, in low energy interactions, the electric potential accurately represents the contributions of the intermediate photon exchanges. Then, we see that the dominant term of the electron wavefunction is a superposition of plane (but not free) waves which, by fulfilling the total energy relations, satisfies the Schrödinger, Pauli, and Dirac equations. Furthermore, we show that what is considered the kinetic energy term of the Schrödinger equation does not represent the kinetic energy of the interacting electron. We analyze and clarify the dynamics of the Schrödinger equation.