Scalar heat kernel with boundary in the worldline formalism

Abstract

The worldline formalism has in recent years emerged as a powerful tool for the computation of effective actions and heat kernels. However, implementing nontrivial boundary conditions in this formalism has turned out to be a difficult problem. Recently, such a generalization was developed for the case of a scalar field on the half-space ℝ<SUB>+</SUB> × ℝ<SUP>D-1</SUP>, based on an extension of the associated worldline path integral to the full ℝ<SUP>D</SUP> using image charges. We present here an improved version of this formalism which allows us to write down non-recursive master formulas for the n-point contribution to the heat kernel trace of a scalar field on the half-space with Dirichlet or Neumann boundary conditions. These master formulas are suitable to computerization. We demonstrate the efficiency of the formalism by a calculation of two new heat-kernel coefficients for the half-space, a<SUB>4</SUB> and a<SUB>9/2</SUB>.