PT-symmetric quantum field theory in D dimensions

PT-symmetric quantum mechanics began with a study of the Hamiltonian H = p2 + x2 ( ix ) ϵ. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when ϵ ≥ 0 . This paper examines the corresponding quantum-field-theoretic Hamiltonian H = 1/2 ( ∇ ϕ ) 2 + 1/2 ϕ 2 ( i ϕ ) ϵ 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 ϵ 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 ϵ 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.