Exact solution for a quantum field with δ-like interaction: effective action and UV renormalization Original Research Article

A quantum field described by the field operator Δa = Δ + aδΣ involving a δ-like potential concentrated on a subspace Σ is considered. Mathematically, the treatment of the δ-potential is based on the theory of self-adjoint extension of the unperturbed operator Δ. We give the general expressions for the resolvent and the heat kernel of the perturbed operator Δa. The main attention is paid to d = 2 δ-potential though d = 1 and d = 3 cases are considered in some detail. We calculate exactly the heat kernel, Greenʹs functions and the effective action for the operator Δa in diverse dimensions and for various spaces Σ. The renormalization phenomenon for the coupling constant a of d = 2 and d = 3 δ-potentials is observed. We find the non-perturbative behavior of the effective action with respect to the renormalized coupling aren.