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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
Keywords:
heat kernel; worldline formalism
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