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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 \(a_{9/2}\).

MSC:
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
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