Quasi-classical Ground States. II. Standard Model of Non-relativistic QED

We consider a non-relativistic electron bound by an external potential and coupled to the quantized electromagnetic field in the standard model of non-relativistic QED. We compute the energy functional of product states of the form $u\otimes {\mathrm{\Psi }}_f$, where $u$ is a normalized state for the electron and ${\mathrm{\Psi }}_f$ is a coherent state in Fock space for the photon field. The minimization of this functional yields a Maxwell–Schr{ö}dinger system up to a trivial renormalization. We prove the existence of a ground state under general conditions on the external potential and the coupling. In particular, neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. Our results provide the convergence in the ultraviolet limit and the second-order asymptotic expansion in the coupling constant of the ground state energy of Maxwell–Schrödinger systems.