  We review the parity anomaly of the massless Dirac fermion in $2+1$
dimensions from the Hamiltonian, as opposed to the path integral, point of
view. We have two main goals for this note. First, we hope to make the parity
anomaly more accessible to condensed matter physicists, who generally prefer to
work within the Hamiltonian formalism. The parity anomaly plays an important
role in modern condensed matter physics, as the massless Dirac fermion is the
surface theory of the time-reversal invariant topological insulator (TI) in
$3+1$ dimensions. Our second goal is to clarify the relation between the
time-reversal symmetry of the massless Dirac fermion and the fractional charge
of $\pm\frac{1}{2}$ (in units of $e$) which appears on the surface of the TI
when a magnetic monopole is present in the bulk. To accomplish these goals we
study the Dirac fermion in the Hamiltonian formalism using two different
regularization schemes. One scheme is consistent with the time-reversal
symmetry of the massless Dirac fermion, but leads to the aforementioned
fractional charge. The second scheme does not lead to any fractionalization,
but it does break time-reversal symmetry. For both regularization schemes we
also compute the effective action $S_{\text{eff}}[A]$ which encodes the
response of the Dirac fermion to a background electromagnetic field $A$. We
find that the two effective actions differ by a Chern-Simons counterterm with
fractional level equal to $\frac{1}{2}$, as is expected from path integral
treatments of the parity anomaly. Finally, we propose the study of a bosonic
analogue of the parity anomaly as a topic for future work.
