Scalar heat kernel with boundary in the worldline formalism.

*(English)*Zbl 1245.81109Summary: 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 \(\mathbb R_+\times\mathbb R^{D-1}\), based on an extension of the associated worldline path integral to the full \(\mathbb R^D\) 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, \(a_4\) and \(a_{9/2}\).