2010/02/15 by H. F. Jones
Mathematics · Physics and Astronomy · #Hamiltonian (control theory) #Hermitian matrix #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum field theory #Quartic function #Sign (mathematics) #hep-th
paper · pdf · doi:10.1007/s10773-010-0541-9
arxiv created 2010/02/15 · openalex publication_date 2010/10/22 · arxiv updated 2015/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
It has been shown that the Schwinger-Dyson equations for non-Hermitian theories implicitly include the Hilbert-space metric. Approximate Green functions for such theories may thus be obtained, without having to evaluate the metric explicitly, by truncation of the equations. Such a calculation has recently been carried out for various PT-symmetric theories, in both quantum mechanics and quantum field theory, including the wrong-sign quartic oscillator. For this particular theory the metric is known in closed form, making possible an independent check of these approximate results. We do so by numerically evaluating the ground-state wave-function for the equivalent Hermitian Hamiltonian and using this wave-function, in conjunction with the metric operator, to calculate the one- and two-point Green functions. We find that the Green functions evaluated by lowest-order truncation of the Schwinger-Dyson equations are already accurate at the (6-8)% level. This provides a strong justification for the method and a motivation for its extension to higher order and to higher dimensions, where the calculation of the metric is extremely difficult.