  Some quantum field theories described by non-Hermitian Hamiltonians are
investigated. It is shown that for the case of a free fermion field theory with
a $\gamma_5$ mass term the Hamiltonian is $\cal PT$-symmetric. Depending on the
mass parameter this symmetry may be either broken or unbroken. When the $\cal
PT$ symmetry is unbroken, the spectrum of the quantum field theory is real. For
the $\cal PT$-symmetric version of the massive Thirring model in
two-dimensional space-time, which is dual to the $\cal PT$-symmetric scalar
Sine-Gordon model, an exact construction of the $\cal C$ operator is given. It
is shown that the $\cal PT$-symmetric massive Thirring and Sine-Gordon models
are equivalent to the conventional Hermitian massive Thirring and Sine-Gordon
models with appropriately shifted masses.
