  In a remarkable development Bender and coworkers have shown that it is
possible to formulate quantum mechanics consistently even if the Hamiltonian
and other observables are not Hermitian. Their formulation, dubbed PT quantum
mechanics, replaces hermiticity by another set of requirements, notably that
the Hamiltonian should be invariant under the discrete symmetry PT, where P
denotes parity and T denotes time reversal. All prior work has focused on the
case that time reversal is even (T^2 = 1). We generalize the formalism to the
case of odd time reversal (T^2 = -1). We discover an analogue of Kramer's
theorem for PT quantum mechanics, present a prototypical example of a PT
quantum system with odd time reversal, and discuss potential applications of
the formalism. Odd time reversal symmetry applies to fermionic systems
including quarks and leptons and a plethora of models in nuclear, atomic and
condensed matter physics. PT quantum mechanics makes it possible to enlarge the
set of possible Hamiltonians that physicists could deploy to describe
fundamental physics beyond the standard model or for the effective description
of condensed matter phenomena.
