Two-point Green's function in [formula omitted]-symmetric theories

Abstract

The Hamiltonian H= 1 2 p 2+ 1 2 m 2x 2+gx 2(ix) δ with δ, g⩾0 is non-Hermitian, but the energy levels are real and positive as a consequence of PT symmetry. The quantum mechanical theory described by H is treated as a one-dimensional Euclidean quantum field theory. The two-point Green's function for this theory is investigated using perturbative and numerical techniques. The Källen–Lehmann representation for the Green's function is constructed, and it is shown that by virtue of PT symmetry the Green's function is entirely real. While the wave-function renormalization constant Z cannot be interpreted as a conventional probability, it still obeys a normalization determined by the commutation relations of the field. This provides strong evidence that the eigenfunctions of the Hamiltonian are complete.