  In this paper we generalize the construction of generally covariant quantum
theories given in the work of Brunetti, Fredenhagen and Verch to encompass the
conformal covariant case. After introducing the abstract framework, we discuss
the massless conformally coupled Klein Gordon field theory, showing that its
quantization corresponds to a functor between two certain categories. At the
abstract level, the ordinary fields, could be thought as natural
transformations in the sense of category theory. We show that, the Wick
monomials without derivatives (Wick powers), can be interpreted as fields in
this generalized sense, provided a non trivial choice of the renormalization
constants is given. A careful analysis shows that the transformation law of
Wick powers is characterized by a weight, and it turns out that the sum of
fields with different weights breaks the conformal covariance. At this point
there is a difference between the previously given picture due to the presence
of a bigger group of covariance. It is furthermore shown that the construction
does not depend upon the scale mu appearing in the Hadamard parametrix, used to
regularize the fields. Finally, we briefly discuss some further examples of
more involved fields.
