PT-symmetric quantum field theory

PT-symmetric quantum theory began with an analysis of the strange-looking non-Hermitian Hamiltonian H = p(2) + x(2) (ix)(epsilon). This Hamiltonian is PT symmetric and the eigenvalues Hamiltonian are discrete, real, and positive when epsilon >= 0. In this talk we describe the properties of the corresponding quantum-field-theoretic Hamiltonian H = 1/2 (del phi)(2) + 1/2 phi(2) (i phi)(epsilon) in D-dimensional spacetime, where phi is a pseudoscalar field. We show how to calculate all of the Green's functions as series in powers of epsilon 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.