Non-Relativistic Quantum Electrodynamics and the Coulomb Interaction

This review explores the foundations of non-relativistic quantum electrodynamics (QED) and its application to atoms and molecules. ​It follows the traditional route of placing classical electrodynamics in an Hamiltonian framework, followed by Dirac’s canonical quantisation algorithm. The properties of the resulting quantum Hamiltonian are reviewed from a non-perturbative perspective. It discusses the gauge invariance of the S-matrix, the Coulomb interaction, and the challenges posed by infinities in classical and quantum electrodynamics. The paper examines the mathematical frameworks used to address these issues, including the use of distributions and the Colombeau algebra. The review also highlights the limitations of the Coulomb Hamiltonian in explaining molecular structure and chemistry, emphasizing the need for additional theoretical modifications to bridge quantum mechanics and chemical phenomena. ​