  In this paper DeWitt's formalism for field theories is presented; it provides
a framework in which the quantization of fields possessing infinite dimensional
invariance groups may be carried out in a manifestly covariant
(non-Hamiltonian) fashion, even in curved space-time. Another important virtue
of DeWitt's approach is that it emphasizes the common features of apparently
very different theories such as Yang-Mills theories and General Relativity;
moreover, it makes it possible to classify all gauge theories in three
categories characterized in a purely geometrical way, i.e., by the algebra
which the generators of the gauge group obey; the geometry of such theories is
the fundamental reason underlying the emergence of ghost fields in the
corresponding quantum theories, too. These "tricky extra particles", as Feynman
called them in 1964, contribute to a physical observable such as the
stress-energy tensor, which can be expressed in terms of Feynman's Green
function itself. Therefore, an entire section is devoted to the study of the
Green functions of the neutron scalar meson: in flat space-time, the choice of
a particular Green's function is the choice of an integration contour in the
"momentum" space; in curved space-time the momentum space is no longer
available, and the definition of the different Green functions requires a
careful discussion itself. After the necessary introduction of bitensors, world
function and parallel displacement tensor, an expansion for the Feynman
propagator in curved space-time is obtained. Most calculations are explicitly
shown.
