  We discuss second quantization, discrete symmetry transformations and inner
products in free non-Hermitian scalar quantum field theories with PT symmetry,
focusing on a prototype model of two complex scalar fields with anti-Hermitian
mass mixing. Whereas the definition of the inner product is unique for theories
described by Hermitian Hamiltonians, its formulation is not unique for
non-Hermitian Hamiltonians. Energy eigenstates are not orthogonal with respect
to the conventional Dirac inner product, so we must consider additional
discrete transformations to define a positive-definite norm. We clarify the
relationship between canonical-conjugate operators and introduce the additional
discrete symmetry C', previously introduced for quantum-mechanical systems, and
show that the C'PT inner product does yield a positive-definite norm, and hence
is appropriate for defining the Fock space in non-Hermitian models with PT
symmetry in terms of energy eigenstates. We also discuss similarity
transformations between PT-symmetric non-Hermitian scalar quantum field
theories and Hermitian theories, showing that they would require modification
in the presence of interactions. As an illustration of our discussion, we
compare particle mixing in a Hermitian theory and in the corresponding
non-Hermitian model with PT symmetry, showing how the latter maintains
unitarity and exhibits mixing between scalar and pseudoscalar bosons.
