The generalized Schwinger-Dewitt technique in gauge theories and quantum gravity

Abstract

The article contains a systematic presentation of the covariant diagrammatic technique for the effective action in gauge theories. The Schwinger-DeWitt technique is generalized and converted into a tool for the calculation of any covariant loop graphs in the effective action. In particular, this technique is extended to nonminimal but causal operators of any order. All calculations are carried out in a universal manner using only the form of the commutator of covariant derivatives. We show that one can even tabulate the universal loop graphs and present a table of a certain class of such graphs. The article can also serve as a reference book on the divergences of one-loop graphs. In particular, we present a complete algorithm for divergences in the determinant of the fourth-order operator. The calculational methods are developed for the recently proposed unique effective action in gauge theories. The unique counterterms in Einstein's theory are computed, and the question of asymptotic freedom in this theory is reexamined.