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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.

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