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Two-point Green’s function in PT-symmetric theories. (English) Zbl 0998.81023
Summary: The Hamiltonian \(H=\frac 12p^2+\frac 12 m^2x^2+gx^2(ix)^{\delta}\) with \(\delta,g\geqslant 0\) 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.

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81T08 Constructive quantum field theory
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