  In all nontrivial cases renormalization, as it is usually formulated, is not
a change of integration variables in the functional integral, plus parameter
redefinitions, but a set of replacements, of actions and/or field variables and
parameters. Because of this, we cannot write simple identities relating bare
and renormalized generating functionals, or generating functionals before and
after nonlinear changes of field variables. In this paper we investigate this
issue and work out a general field-covariant approach to quantum field theory,
which allows us to treat all perturbative changes of field variables, including
the relation between bare and renormalized fields, as true changes of variables
in the functional integral, under which the functionals Z and W = ln Z behave
as scalars. We investigate the relation between composite fields and changes of
field variables, and show that, if J are the sources coupled to the elementary
fields, all changes of field variables can be expressed as J-dependent
redefinitions of the sources L coupled to the composite fields. We also work
out the relation between the renormalization of variable-changes and the
renormalization of composite fields. Using our transformation rules it is
possible to derive the renormalization of a theory in a new variable frame from
the renormalization in the old variable frame, without having to calculate it
anew. We define several approaches, useful for different purposes, in
particular a linear approach where all variable changes are described as linear
source redefinitions. We include a number of explicit examples.
