Two-dimensional zero-range potential and vacuum polarization of a scalar field

Abstract

We consider two problems related to the emergence and use of a two-dimensional delta-like potential: on a plane and in a three-dimensional space. We propose a nonperturbative calculation of the Green’s functions of a massless real scalar field \(\phi\). For an effectively two-dimensional zero-range potential, we compare two methods for calculating the Green’s functions: as a self-adjoint extension of the Laplacian on a plane with a punctured point and as a result of incomplete Fourier transforms with the direct action of the delta function. The second method involves the renormalization of the interaction coupling constant, as prescribed by the analogous quantum-mechanical problem. We present the regularized Green’s functions in three equivalent integral forms: as products of Bessel functions of the first kind, as products of modified Bessel functions, and as products of Kelvin functions. Using the Green’s function, we calculate the vacuum expectation value \(\langle \phi^2\rangle\) as well as the renormalized energy-momentum tensor operator. The two approaches yield equivalent results, which confirms the prescriptions imposed during the derivation in particular cases.