Vacuum polarization effects of pointlike impurity

We develop precise formulation for the effects of vacuum polarization near a pointlike source with a zero-range (\(\delta\)-like) potential in three spatial dimensions. There are different ways of introducing \(\delta\)-interaction in the framework of quantum theory. We discuss the approach based on the concept of self-adjoint extensions of densely defined symmetric operators. Within this approach, we consider the real massive scalar field in three-dimensional Euclidean space with a single extracted point. Appropriate boundary conditions, imposed at this point, enable one to consider all self-adjoint extensions of \(- \Delta\) as operators which can describe a pointlike source with a zero-range potential. In this framework, we compute the renormalized vacuum expectation value of the field square \(\langle \phi ^{{\hspace{1.0pt}}2}(x)\rangle _{\textrm{ren}}\) and the renormalized vacuum average of the scalar-field’s energy-momentum tensor \(\langle T_{\mu \nu }(x)\rangle _{\textrm{ren}}\). Asymptotic cases are discussed in detail.