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

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