The Graviton Propagator with a Non-Conserved External Generating Source

A novel general expression is obtained for the graviton propagator from Lagrangian field theory by taking into account the necessary fact that in the functional differential approach of quantum field theory, in order to generate non-linearities in gravitation and interactions with matter, the external source $T_{\mu\nu}$, coupled to the gravitational field, should \textit{a priori} not be conserved $\partial^\mu T_{\mu\nu}\neq 0$, so variations with respect to its ten components may be varied \textit{independently}. The resulting propagator is the one which arises in the functional approach and does \textit{not} coincide with the corresponding time-ordered product of two fields and it includes so-called Schwinger terms. The quantization is carried out in a gauge corresponding to physical states with two polarization states to ensure positivity in quantum applications.