# A field-theoretical approach to non-relativistic QED

## TL;DR

This paper presents a field-theoretical derivation of the non-relativistic QED Hamiltonian in Coulomb gauge, emphasizing the correct use of Euler-Lagrange equations and canonical formalism over classical approaches.

## Contribution

It introduces a systematic field-theoretical derivation of the non-relativistic QED Hamiltonian, clarifying foundational formalism issues.

## Key findings

- Derivation aligns with traditional methods using minimal coupling.
- Highlights the importance of proper Euler-Lagrange application.
- Clarifies the limitations of classical starting points for many-body quantum theory.

## Abstract

We give here a field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This approach illustrates however the proper use of the Euler-Lagrange equations and the canonical formalism that fail if one tries to build up the many-body quantum theory starting from the classical theory of point-like particles interacting with electromagnetic fields.