  In a series of recent works based on foliation-based quantization in which
renormalizability has been achieved for the physical sector of the theory, we
have shown that the use of the standard graviton propagator interferes, due to
the presence of the trace mode, with the 4D covariance. A subtlety in the
background field method also requires careful handling. This status of the
matter motivated us to revisit an Einstein-scalar system in one of the sequels.
Continuing the endeavors, we revisit the one-loop renormalization of an
Einstein-Maxwell system in the present work. The systematic renormalization of
the cosmological and Newton's constants is carried out by applying the refined
background field method. One-loop beta function of the vector coupling constant
is explicitly computed and compared with the literature. The longstanding
problem of gauge choice-dependence of the effective action is addressed and the
manner in which the gauge-choice independence is restored in the present
framework is discussed. The formalism also sheds light on background
independent analysis. The renormalization involves a metric field redefinition
originally introduced by `t Hooft; with the field redefinition the theory
should be predictive.
