Infrared divergence of a scalar quantum field model on a pseudo Riemann manifold

A scalar quantum field model defined on a pseudo Riemann manifold is considered. The model is unitarily transformed to the one with a variable mass. By means of a Feynman-Kac-type formula, it is shown that when the variable mass is short range, the Hamiltonian has no ground state. Moreover the infrared divergence of the expectation values of the number of bosons in the ground state is discussed.