  Among various approaches in proving gauge independence, models containing an
explicit gauge dependence are convenient. The well-known example is the gauge
parameter in the covariant gauge fixing which is of course most suitable for
the perturbation theory but a negative metric prevents us from imaging a
dynamical picture. Noncovariant gauge such as the Coulomb gauge is on the
contrary used for many physical situations. Therefore it is desirable to
include both cases. More than ten years ago, Steinmann introduced a function
(distribution) which can play this role in his attempt on discussing quantum
electrodynamics (QED) in terms of the gauge invariant fields solely. The method
is, however, broken down in the covariant case: the invariant operators are
ill-defined because of 1/p^2 singularity in the Minkowski space. In this paper,
we apply his function to the path integral: utilizing the arbitrariness of the
function we first restrict it to be able to have a well- defined operator, and
then a Hamiltonian with which we can build up the (Euclidean) path integral
formula. Although the formula is far from covariant, a full covariant
expression is recovered by reviving the components which have been discarded
under the construction of the Hamiltonian. There is no pathological defects
contrary to the operator formalism. With the aid of the path integral formula,
the gauge independence of the free energy as well as the S-matrix is proved.
Moreover the reason is clarified why it is so simple and straightforward to
argue gauge transformations in the path integral. Discussions on the quark
confinement is also presented.
