Exact solution for a quantum field with $δ$-like interaction

Sergey N. Solodukhin

doi:10.1016/S0550-3213(98)00789-5

paper

Exact solution for a quantum field with $δ$ -like interaction

arXiv:hep-th/9801054 · doi:10.1016/S0550-3213(98)00789-5

Abstract

A quantum field described by the field operator $Δ_{a} = Δ + a δ_{Σ}$ 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 $Δ$ . We give the general expressions for the resolvent and the heat kernel of the perturbed operator $Δ_{a}$ . 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 $Δ_{a}$ 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 $a_{r e n}$ .

18 pages, latex, no figures, new references added

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Exact solution for a quantum field with δ-like interaction

Abstract

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Exact solution for a quantum field with $δ$ -like interaction