Effective Hamiltonian for non-minimally coupled scalar fields
  Performing a relativistic approximation as the generalization to a curved spacetime of the flat space Klein-Gordon equation, an effective Hamiltonian which includes non-minimial coupling between gravity and scalar field and also quartic self-interaction of scalar field term is obtained. 
Emine Mese, Nurettin Pirinccioglu, Irfan Acikgoz, and Figen Binbay