Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher-derivative quantum gravity

The main results are: 1. A manifestly covariant technique for the calculation of De Witt coefficients is elaborated; 2. The coefficients $a_3$ and $a_4$ are calculated; 3. Covariant methods for the study of the nonlocal structure of the effective action are developed. The terms of first and second order in background fields in De Witt coefficients are calculated. The summation of these terms is carried out and nonlocal covariant expression for the Green function, the heat kernel and the effective action are obtained. It is shown that in the conform-invariant case the Green function is finite. A finite effective action in the conform-invariant case of massless scalar field in two-dimensional space is obtained; 4. The off-shell one-loop divergences of the effective action in arbitrary covariant gauge as well as those of the `unique' effective action in higher-derivative quantum gravity are calculated; The ultraviolet asymptotics of coupling constants are found. It is shown that in the `physical' region of coupling constants (no tachyons!) the conformal sector has `zero-charge' behavior contrary to previous authors. This means that the theory goes beyond the limits of weak conformal coupling at higher energies. In other words, the condition of conformal stability of flat background is incompatible with the asymptotic freedom in the conformal sector. There is a stable non-flat ground state but only in the case of positive definite Euclidean action. In this case the theory is asymptotically free both in tensor and conformal sectors. The off-shell one-loop effective action in arbitrary covariant gauge and the `unique' effective action on De Sitter background are calculated.