Absence of ground state for the Nelson model on static space–times

We consider the Nelson model on some static space–times and investigate the problem of absence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass m(x) tends to 0 at spatial infinity. Using path space techniques, we show that if m(x)⩽C|x|−μ at infinity for some C>0 and μ>1 then the Nelson Hamiltonian has no ground state.