From QED to the Schrödinger equation

Quantum electrodynamics (QED) is the fundamental theory of electromagnetic interactions. Thus, the Schrödinger equation, which yields the electron wavefunction in low energy electromagnetic interactions, must be a condition that can be derived from the fundamental QED theory. Due to the small value of the electromagnetic coupling constant, we can adequately describe low-energy interactions by classical potentials. Furthermore, the dominant (lowest order) term of the electron wavefunction is a superposition of plane waves. As a result, we show that to fulfill the total energy relations, the electron wavefunction must satisfy at each space-time point the Schrödinger, Pauli, and Dirac equations.