Exact solution for a quantum field with δ-like interaction

A quantum field described by the field operator ∆a = ∆+aδΣ involving a δ-like potential is considered. Mathematically, the treatment of the δ-potential is based on the theory of self-adjoint extension of the unperturbed operator ∆. We give the general expressions for the resolvent and the heat kernel of the perturbed operator ∆a. The main attention is payed to d = 2 δ-potential though d = 1 and d = 3 cases are considered in some detail. We calculate exactly the heat kernel, Green’s functions and the effective action for the operator ∆a 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. PACS number(s): 04.70.Dy, 04.62.+v e-mail: [email protected]