Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2002-08-20
Phys.Lett. A302 (2002) 286-290
Physics
High Energy Physics
High Energy Physics - Theory
Scientific paper
10.1016/S0375-9601(02)01196-9
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.
Bender Carl M.
Boettcher Stefan
Meisinger Peter N.
Wang Qing-hai
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