  PT-symmetric quantum mechanics began with a study of the Hamiltonian
$H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian
Hamiltonian is that its eigenvalues are discrete, real, and positive when
$\varepsilon\geq0$. This paper examines the corresponding
quantum-field-theoretic Hamiltonian
$H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in
$D$-dimensional spacetime, where $\phi$ is a pseudoscalar field. It is shown
how to calculate the Green's functions as series in powers of $\varepsilon$
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 $\varepsilon$ are derived for $0\leq D<2$.
For $D\geq2$ 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.
