  Building on previous work [N. Read, Phys. Rev. B 58, Z. Dong and T. Senthil,
16262 (1998); Phys. Rev. B 102, 205126 (2020)] on the system of bosons at
filling factor $\nu = 1$, we derive the Dirac composite fermion theory for a
half-filled Landau level from first principles and applying the Hartree-Fock
approach in a preferred representation. On the basis of the microscopic
formulation, in the long-wavelength limit, we propose a noncommutative
field-theoretical description, which in a commutative limit reproduces the
Son's theory, with additional terms that may be expected on physical grounds.
The microscopic representation of the problem is also used to discuss pairing
instabilities of composite fermions. We find that a presence of a particle-hole
symmetry breaking leads to a weak (BCS) coupling $p$-wave pairing in the lowest
Landau level, and strong coupling $p$-wave pairing in the second Landau level
that occurs in a band with nearly flat dispersion, a third power function of
momentum.
