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

Carl M. Bender·Stefan Boettcher·Peter N. Meisinger·Qinghai Wang
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Abstract

The Hamiltonian $H={1\over2} p^2+{1\over2}m^2x^2+gx^2(ix)^δ$ with $δ,g\geq0$ 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 Källen-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.

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Carl M. Bender, Stefan Boettcher, Peter N. Meisinger, Qinghai Wang. 2002-08-20. Two-Point Green's Function in PT-Symmetric Theories. https://doi.org/10.1016/s0375-9601(02)01196-9

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