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Worldline formalism for a confined scalar field. (English) Zbl 1421.83107
Summary: The worldline formalism is a useful scheme in quantum field theory which has also become a powerful tool for numerical computations. The key ingredient in this formalism is the first quantization of an auxiliary point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field which is confined within some boundaries one needs to restrict the path integration domain of the auxiliary point-particle to a specific subset of worldlines enclosed by those boundaries. We show how to implement this restriction for the case of a scalar field confined to the $$D$$-dimensional ball under Dirichlet and Neumann boundary conditions, and compute the first few heat-kernel coefficients as a verification of our construction. We argue that this approach could admit different generalizations.
##### MSC:
 83E15 Kaluza-Klein and other higher-dimensional theories 83C75 Space-time singularities, cosmic censorship, etc. 81T10 Model quantum field theories 81S40 Path integrals in quantum mechanics 83C47 Methods of quantum field theory in general relativity and gravitational theory 35K08 Heat kernel
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