  Gravity is perturbatively renormalizable for the physical states which can be
conveniently defined via foliation-based quantization. In recent sequels,
one-loop analysis was explicitly carried out for Einstein-scalar and
Einstein-Maxwell systems. Various germane issues and all-loop renormalizability
have been addressed. In the present work we make further progress by carrying
out several additional tasks. Firstly, we present an alternative 4D covariant
derivation of the physical state condition by examining gauge
choice-independence of a scattering amplitude. To this end, a careful dichotomy
between the ordinary, and large gauge symmetries is required and appropriate
gauge-fixing of the ordinary symmetry must be performed. Secondly, vacuum
energy is analyzed in a finite-temperature setup. A variant optimal
perturbation theory is implemented to two-loop. The renormalized mass
determined by the optimal perturbation theory turns out to be on the order of
the temperature, allowing one to avoid the cosmological constant problem. The
third task that we take up is examination of the possibility of asymptotic
freedom in finite-temperature quantum electrodynamics. In spite of the debates
in the literature, the idea remains reasonable.
