  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.
