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

Abstract

The Hamiltonian $H=\frac{1}{2}{p}^2+\frac{1}{2}{m}^2{x}^2+g{x}^2(ix{)}^{\delta }$ with $\delta ,g\ge 0$ is non-Hermitian, but the energy levels are real and positive as a consequence of $\mathcal{P}\mathcal{T}$ 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 $\mathcal{P}\mathcal{T}$ 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.

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