PT -symmetric quantum field theory

PT -symmetric quantum theory began with an analysis of the strange-looking non-Hermitian Hamiltonian H = p 2 + x ( ix ) ε . This Hamiltonian is PT symmetric and the eigenvalues Hamiltonian are discrete, real, and positive when ε ≥ 0. In this talk we describe the properties of 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. We show how to calculate all of the Green’s functions as series in powers of ε directly from the Euclidean partition function. We derive exact finite expressions for the vacuum energy density, the renormalized mass, and the connected n -point Green’s functions for all n 0 ≤ D ≤ 2. For D ≥ 2 the one-point Green’s function and the renormalized mass become infinite, but perturbative renormalization can be performed. The beautiful spectral properties of PT -symmetric quantum mechanics appear to persist in PT -symmetric quantum field theory.