  We propose a particle-hole symmetric theory of the Fermi-liquid ground state
of a half-filled Landau level. This theory should be applicable for a Dirac
fermion in the magnetic field at charge neutrality, as well as for the
$\nu=\frac12$ quantum Hall ground state of nonrelativistic fermions in the
limit of negligible inter-Landau-level mixing. We argue that when particle-hole
symmetry is exact, the composite fermion is a massless Dirac fermion,
characterized by a Berry phase of $\pi$ around the Fermi circle. We write down
a tentative effective field theory of such a fermion and discuss the discrete
symmetries, in particular, $\mathcal C\mathcal P$. The Dirac composite fermions
interact through a gauge, but non-Chern-Simons, interaction. The particle-hole
conjugate pair of Jain-sequence states at filling factors $\frac n{2n+1}$ and
$\frac{n+1}{2n+1}$, which in the conventional composite fermion picture
corresponds to integer quantum Hall states with different filling factors, $n$
and $n+1$, is now mapped to the same half-integer filling factor $n+\frac12$ of
the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are
interpreted as $d$-wave Bardeen-Cooper-Schrieffer paired states of the Dirac
fermion with orbital angular momentum of opposite signs, while $s$-wave pairing
would give rise to a particle-hole symmetric non-Abelian gapped phase. When
particle-hole symmetry is not exact, the Dirac fermion has a $\mathcal
C\mathcal P$-breaking mass. The conventional fermionic Chern-Simons theory is
shown to emerge in the nonrelativistic limit of the massive theory.
