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 $varepsilongeq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=frac{1}{2}( ablaphi)^2+frac{1}{2}phi^2(iphi)^varepsilon$ in $D$-dimensional spacetime, where $phi$ is a pseudoscalar field. It is shown how to calculate the Greens 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 Greens functions, and the renormalized mass to order $varepsilon$ are derived for $0leq D<2$. For $Dgeq2$ the one-point Greens 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.
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