  We construct the adiabatic regularization method for a $U(1)$ gauge field in
a conformally flat spacetime by quantizing in the canonical formalism the gauge
fixed $U(1)$ theory with mass terms for the gauge fields and the ghost fields.
We show that the adiabatic expansion for the mode functions and the adiabatic
vacuum can be defined in a similar way using WKB-type solutions as the scalar
fields. As an application of the adiabatic method, we compute the trace of the
energy momentum tensor and reproduces the known result for the conformal
anomaly obtained by the other regularization methods. The availability of the
adiabatic expansion scheme for gauge field allows one to study the
renormalization of the de-Sitter space maximal superconformal Yang-Mills theory
using the adiabatic regularization method.
