  The particle-hole (PH) symmetry at half-filled Landau level requires the
relationship between the flux number N_phi and the particle number N on a
sphere to be exactly N_phi - 2(N-1) = 1. The wave functions of composite
fermions with 1/2 "orbital spin", which contributes to the shift "1" in the
N_phi and N relationship, are proposed, shown to be PH symmetric, and validated
with exact finite system results. It is shown the many-body composite electron
and composite hole wave functions at half-filling can be formed from the two
components of the same spinor wave function of a massless Dirac fermion at
zero-magnetic field. It is further shown that away from half-filling, the
many-body composite electron wave function at filling factor nu and its PH
conjugated composite hole wave function at 1-nu can be formed from the two
components of the very same spinor wave functions of a massless Dirac fermion
at non-zero magnetic field. This relationship leads to the proposal of a very
simple Dirac composite fermion effective field theory, where the two-component
Dirac fermion field is a particle-hole spinor field coupled to the same
emergent gauge field, with one field component describing the composite
electrons and the other describing the PH conjugated composite holes. As such,
the density of the Dirac spinor field is the density sum of the composite
electron and hole field components, and therefore is equal to the degeneracy of
the Lowest Landau level. On the other hand, the charge density coupled to the
external magnetic field is the density difference between the composite
electron and hole field components, and is therefore neutral at exactly
half-filling. It is shown that the proposed particle-hole spinor effective
field theory gives essentially the same electromagnetic responses as Son's
Dirac composite fermion theory does.
