  We calculate the effective potentials for scalar, Dirac and Yang-Mills fields
in curved backgrounds using a new method for the determination of the heat
kernel involving a re-summation of the Schwinger-DeWitt series.
Self-interactions are treated both to one loop order as usual and slightly
beyond one-loop order by means of a mean-field approximation. The new approach
gives the familiar result for scalar fields, the Coleman-Weinberg potential
plus corrections such as the leading-log terms, but the actual calculation is
much faster. We furthermore show how to go systematically to higher loop order.
The Schwarzschild space-time is used to exemplify the procedure. Next we
consider phase transitions and we show that for a classical critical point to
be a critical point of the effective potential too, certain restrictions must
be imposed on as well its value as on the form of the classical potential and
the background geometry. We derive this extra condition for scalar fields with
arbitrary self couplings and comment on the case of fermions and gauge bosons
too. Critical points of the effective action which are not there classically
are also discussed. This has implications for inflation. The renormalised
energy-momentum tensor for a scalar field with arbitrary self-interaction and
non-minimal coupling to the gravitational background is also calculated to this
improved one-loop order as is the resulting conformal anomaly. Conditions for
the violation of energy conditions are described. Finally we discuss metric
fluctuations and a self-consistency condition for such fluctuations is written
down for spin 0,1/2,1 quantum fields. This is of importance for the study of
cosmic density fluctuations. All calculations are performed in the physically
relevant case of d=4 dimensions.
