  We discuss the implications of approximate particle-hole symmetry in a
half-filled Landau level in which a paired quantum Hall state forms. We note
that the Pfaffian state is not particle-hole symmetric. Therefore, in the limit
of vanishing Landau level mixing, in which particle-hole transformation is an
exact symmetry, the Pfaffian spontaneously breaks this symmetry. There is a
particle-hole conjugate state, which we call the anti-Pfaffian, which is
degenerate with the Pfaffian in this limit. We observe that strong Landau level
mixing should favor the Pfaffian, but it is an open problem which state is
favored for the moderate Landau level mixing which is present in experiments.
We discuss the bulk and edge physics of the anti-Pfaffian. We analyze a
simplified model in which transitions between analogs of the two states can be
studied in detail. Finally, we discuss experimental implications.
