  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.
