  We discuss several aspects of a generalization of the Chern-Simons action
containing the pseudo-differential operator$\sqrt{-\Box}$, which we shall call
pseudo Chern-Simons (PCS). Firstly, we derive the PCS from the bosonization of
free massive Dirac particles in (2+1)D in the limit when $m^2\ll p^2$, where
$m$ is the fermion mass and $p$ is its momentum. In this regime, the whole
bosonized action also has a modified Maxwell term, involving the same
pseudo-differential operator. Furthermore, the large-mass $m^2\gg p^2$ regime
is also considered. We also investigate the main effects of the PCS term into
the Pseudo quantum electrodynamics (PQED), which describes the electromagnetic
interactions between charged particles in (2+1)D. We show that the massless
gauge field of PQED becomes massive in the presence of a PCS term, without the
need of a Higgs mechanism. In the nonrelativistic limit, we show that the
static potential has a repulsive term (given by the Coulomb potential) and an
attractive part (given by a sum of special functions), whose competition
generates bound states of particles with the same charge. Having in mind
two-dimensional materials, we also conclude that the presence of a PCS term
does not affect the renormalization either of the Fermi velocity and of the
band gap in a Dirac-like material.
