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The Heat Kernel Expansion on a Cone and Quantum Fields Near Cosmic Strings

An asymptotic expansion of the trace of the heat kernel on a cone where the heat coefficients have a delta function behavior at the apex is obtained. It is used to derive the renormalized effective action and total energy of a self-interacting quantum scalar field on the cosmic string space-time. Analogy is pointed out with quantum theory with boundaries. The surface infinities in the effective action are shown to appear and are removed by renormalization of the string tension. Besides, the total renormalized energy turns out to be finite due to cancelation of the known non-integrable divergence in the energy density of the field with a counterterm in the bare string tension.