Two-Point Green's Function in PT-Symmetric Theories

TL;DR

This paper investigates the two-point Green's function in PT-symmetric quantum theories, demonstrating its reality and exploring properties like wave-function renormalization, thus supporting the completeness of eigenfunctions.

Contribution

It provides a detailed analysis of the Green's function in PT-symmetric theories, including perturbative and numerical methods, and establishes its reality and normalization properties.

Findings

Green's function is entirely real due to PT symmetry

Wave-function renormalization constant obeys a normalization condition

Eigenfunctions of the Hamiltonian are complete

Abstract

The Hamiltonian $H={1\over2} p^2+{1\over2}m^2x^2+gx^2(ix)^\delta$ with $\delta,g\geq0$ is non-Hermitian, but the energy levels are real and positive as a consequence of ${\cal PT}$ 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 K\"allen-Lehmann representation for the Green's function is constructed, and it is shown that by virtue of ${\cal PT}$ 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.