Effective Hamiltonian for non-minimally coupled scalar fields

In the post Newtonian limit, a non-relativistic Hamiltonian is derived for scalar fields with quartic self-interaction and non-minimal coupling to the curvature scalar of the background spacetime. These effects are found to contribute to the non-relativistic Hamiltonian by adding nonlinearities and by modifying the gravitational Darwin term. As we discuss briefly in the text, the impact of these novel structures can be sizable in dense media like neutron star core, and can have observable signatures in phase transitions, for example.