PT-symmetric quantum mechanics began with a study of the Hamiltonian H=p2+x2(ix)μ. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when μ≥0. This paper examines the corresponding quantum-field-theoretic Hamiltonian H=12( φ)2+12φ2(iφ)μ in D-dimensional spacetime, where φ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of μ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected n-point Green's functions, and the renormalized mass to order μ are derived for 0≤D<2. For D≥2 the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.