Quasiclassical expansion in an external Yang-Mills field and the approximate calculation of functional determinants

We calculate the first several coefficients of a power-series expansion in the proper time t of the quantity Sp exp(-tK), where K is the differential operator for a gluon, ghost, or quark in an external Yang-Mills field. The expansion gives an estimate of the corresponding Gaussian integrals with respect to quantum fluctuations of the fields about a given external field, i.e., an approximate calculation of the functional determinants. In particular, for the field of an instanton, this rough method of calculating the determinants gives an accuracy of the order of a few percent.