zbMATH — the first resource for mathematics

Scalar heat kernel with boundary in the worldline formalism. (English) Zbl 1245.81109
Summary: 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 \(a_{9/2}\).

81T15 Perturbative methods of renormalization applied to problems in quantum field theory
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58J90 Applications of PDEs on manifolds
Full Text: DOI arXiv