Alexey Kalugin: Beyond Schwinger-DeWitt: higher-order minimal and nonminimal operators, off-diagonal expansions for operator functions, and Mellin-Barnes representations

We present an overview of generalizations of the heat kernel technique, a fundamental tool in QFT, gauge theory, and quantum gravity. As the classical Schwinger-DeWitt technique is restricted to Laplace-type operators, we propose extensions thereof to other operator types. First, we construct off-diagonal heat kernel expansions for minimal operators of arbitrary order. The resulting asymptotic expansion involves both positive and negative fractional powers of proper time and reduces to the standard Seeley-Gilkey expansion in the coincidence limit. Second, we present a manifestly covariant algorithm for constructing heat kernels for nonminimal causal operators. A special subtraction procedure guarantees finiteness and smoothness of resulting off-diagonal expansion, which is also consistent with the Seeley-Gilkey theory. Finally, we extend the Schwinger-DeWitt technique to integral kernels of arbitrary functions of Laplace-type operators. For these kernels, we obtain off-diagonal expansions over DeWitt coefficients with some functions of the Synge world function as coefficients, and, employing Mellin-Barnes representation, we analyze associated IR divergences.