Gauge invariance of the S-matrix for atoms, molecules and electromagnetic radiation

The Hamiltonian for non-relativistic quantum electrodynamics can be expressed in terms of the usual dynamical variables for the field and charged particles and as a functional of a c-number vector field p(x; x') which is essentially arbitrary. This corresponds to the freedom to use the vector potential of he field in an arbitrary gauge since every p(x; x') determines a gauge condition for the potential. We study the S-matrix for two different choices of p(x; x') and calculate the difference. This is done directly in low-orders of perturbation theory and suggests a conjecture for the general formula for the difference, valid in any order of perturbation theory; we verify the validity of this conjecture by elementary means. On the energy shell the S-matrices are identical, i.e. gauge-invariant.