Vacuum Polarization Effects of Pointlike Impurity

TL;DR

This paper formulates the effects of vacuum polarization caused by a pointlike impurity with zero-range potential in three-dimensional space, using self-adjoint extensions to analyze the scalar field's vacuum expectation values.

Contribution

It introduces a precise method to analyze vacuum polarization effects near pointlike sources using self-adjoint extensions in quantum field theory.

Findings

Computed renormalized vacuum expectation value of the field square.

Calculated renormalized vacuum energy-momentum tensor.

Discussed asymptotic behaviors of the effects.

Abstract

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 $\langle \phi^{ 2}(x)\rangle_{\rm ren}$ and the renormalized vacuum average of the scalar-field's energy-momentum tensor $\langle T_{\mu\nu}(x)\rangle_{\rm ren}$. Asymptotic cases are discussed in detail.