Leading derivatives in the heat kernel and the nonlocal structure of the functional determinants

In [3,4,5] I applied the developed methods to study the nonlocal structure of the one-loop effective action $\Gamma_{(1)}=(1/2)\log {\rm Det} F =-(1/2)\zeta'(0)$ in quantum field theory determined by the functional determinant of a Laplace type operator F on a compact manifold without boundary. Here $\zeta(s)={\rm Tr}_{L^2}F^{-s}$ is the generalized zeta-function of the operator F. First, I proposed a new ansatz for the heat kernel in form of the inverse Mellin transform of a function b_q

$$U(t) =(4\pi t)^{-n/2}\Delta^{1/2}\exp\left(-{\sigma\over 2t}... ...}\int\limits_{c-i\infty}^{c+i\infty}dq\, t^q\,\Gamma (-q)\,b_q,$$

where c is a negative constant. The function b_q is shown to be an entire function of the complex variable q whose values at the positive integer points are equal to the HMDS-coefficients b_k=(-1)^kk!a_k, $(k=0,1,2,\dots)$. The heat equation is translated into a functional-differential equation for the function b_q. Using this ansatz I obtained very simple formulas for the zeta-function, Green function G=F^-1 and the functional determinant in terms of b_q. This reduced the problem of summation of asymptotic expansion to that of analytical continuation of the HMDS-coefficients. I calculated all HMDS-coefficients A_k in the approximation of rapidly varying background fields (when the covariant derivatives of the curvatures and the potential term are larger than their products of the same dimension) by picking up the leading higher derivative terms quadratic in curvatures R, ${\cal R}$ and Q and neglecting the terms of higher order in curvatures, and obtained manifestly covariant nonlocal expressions for the trace of the heat kernel ${\rm Tr}_{L^2}\exp(-tF)$, zeta-function, and the functional determinant.