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We show here that an extension of the Hamiltonian theory developed by us overthe years furnishes a composite fermion (CF) description of the $\nu=\frac{1}{2}$ state that is particle-hole (PH) symmetric, has a charge densitythat obeys the magnetic translation algebra of the lowest Landau level (LLL),and exhibits cherished ideas from highly successful wave functions, such as aneutral quasi-particle with a certain dipole moment related to its momentum. Wealso a provide an extension away from $\nu=\frac{1}{2}$ which has the featuresfrom $\nu=\frac{1}{2}$ and implements the the PH transformation on the LLL asan anti-unitary operator ${\cal T}$ with ${\cal T}^2=-1$. This extension of ourpast work was inspired by Son, who showed that the CF may be viewed as a Diracfermion on which the particle-hole transformation of LLL electrons is realizedas time-reversal, and Wang and Senthil who provided a very attractiveinterpretation of the CF as the bound state of a semion and anti-semion ofcharge $\pm {e\over 2}$. Along the way we also found a representation with allthe features listed above except that now ${\cal T}^2=+1$. We suspect itcorresponds to an emergent charge-conjugation symmetry of the $\nu =1$ bosonproblem analyzed by Read.

The $\nu={1\over2}$ Landau level: Half-full or half-empty?