2002/08/20 by Carl M. Bender, Carl M. Bender, Stefan Boettcher +3 · 3 citations
Physics and Astronomy · #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Non-Hermitian Physics #Quantum and Classical Electrodynamics #hep-th
paper · pdf · doi:10.1016/s0375-9601(02)01196-9
published as Phys.Lett. A302 (2002) 286-290
arxiv created 2002/08/20 · openalex publication_date 2002/09/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The Hamiltonian H=1\over2 p2+1\over2m2x2+gx2(ix)δ with δ,g≥0 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.