-symmetric quantum field theory

Abstract

-symmetric quantum theory began with an analysis of the strange-looking non-Hermitian Hamiltonian H = p 2 + x (ix)ε. This Hamiltonian is 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 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 -symmetric quantum mechanics appear to persist in -symmetric quantum field theory.