# Quantum Dirac fermions in half space and their interaction with electromagnetic field

## TL;DR

This paper investigates the polarization tensor of a Dirac fermion confined to a half space, modifies the regularization scheme for boundary conditions, and explores the resulting boundary-induced Chern-Simons action relevant to topological insulators.

## Contribution

It introduces a modified Pauli-Villars regularization with axial mass for boundary-confined Dirac fields and computes the renormalized polarization tensor in this setting.

## Key findings

- Derived the polarization tensor for Dirac fermions in half space.
- Identified the boundary-induced Chern-Simons action.
- Compared boundary effects to 2+1D Dirac fermion effective action.

## Abstract

We study the polarization tensor of a Dirac field in $(3+1)$ dimensions confined to a half space -- a problem motivated by applications to the condensed matter physics, and to Topological Insulators in particular. Although the Pauli-Villars regularization scheme has a number of advantages, like explicit gauge invariance and decoupling of heavy modes, it is not applicable on manifolds with boundaries. Here, we modify this scheme by giving an axial mass to the regulators and to the physical field. We compute the renormalized polarization tensor in coordinate representation. We discuss then the induced Chern-Simons type action on the boundary and compare it to the effective action of a $(2+1)$ dimensional Dirac fermion.