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arXiv · hep-th/9801054

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

Sergey N. Solodukhin
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Abstract

A quantum field described by the field operator $Δ_{a}=Δ+ aδ_Σ$ 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 $a_{ren}$.

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Sergey N. Solodukhin. 1998-01-27. Exact solution for a quantum field with $δ$-like interaction. https://doi.org/10.1016/s0550-3213(98)00789-5

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