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arXiv:2201.11894 · Chinese Physics Letters

Unitary Scattering Protected by Pseudo-Hermiticity

L. Jin

Published Feb 1, 2022• 12 citations
arXivHTMLPDF

Abstract

Hermitian systems possess unitary scattering. However, the Hermiticity is unnecessary for a unitary scattering although the scattering under the influence of non-Hermiticity is mostly non-unitary. Here we prove that the unitary scattering is protected by certain type of pseudo-Hermiticity and unaffected by the degree of non-Hermiticity. The energy conservation is violated in the scattering process and recovers after scattering. The subsystem of the pseudo-Hermitian scattering center including only the connection sites is Hermitian. These findings provide fundamental insights on the unitary scattering, pseudo-Hermiticity, and energy conservation, and are promising for light propagation, mesoscopic electron transport, and quantum interference in non-Hermitian systems.

DOI
10.1088/0256-307x/39/3/037302

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Hermitian systems possess unitary scattering. However, the Hermiticity is unnecessary for a unitary scattering although the scattering under the influence of non-Hermiticity is mostly non-unitary. Here we prove that the unitary scattering is protected by certain type of pseudo-Hermiticity and unaffected by the degree of non-Hermiticity. The energy conservation is violated in the scattering process and recovers after scattering. The subsystem of the pseudo-Hermitian scattering center including only the connection sites is Hermitian. These findings provide fundamental insights on the unitary scattering, pseudo-Hermiticity, and energy conservation, and are promising for light propagation, mesoscopic electron transport, and quantum interference in non-Hermitian systems.
Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows that real spectra and unitary evolution can still emerge through a suitably defined inner product. Motivated by this insight, we extend the pseudo-Hermitian framework to relativistic quantum field theory and construct a consistent formulation of scattering processes. A novel structural feature of this theory is the presence of distinct metric operators for the in and out sectors, connected through a nontrivial metric projector that guarantees global probability conservation under pseudo-unitary time evolution. We further develop a general symmetry formalism, showing that each symmetry generally corresponds to two pseudo-unitary operators associated with the in and out metrics, respectively. Within this framework, the scattering matrix admits a perturbative expansion through the Dyson series and remains Lorentz invariant and unitary, remarkably in complete agreement with the conventional Hermitian case. The fundamental CPT theorem is also shown to hold. Our results provide a rigorous foundation for interacting pseudo-Hermitian quantum field theories and open new directions for exploring their possible physical implications beyond the standard Hermitian paradigm.
We study the scattering problem for the non-Hermitian scattering center, which consists of two Hermitian clusters with anti-Hermitian couplings between them. Counterintuitively, it is shown that it acts as a Hermitian scattering center, that is, the SSS matrix is unitary, or the Dirac probability current is conserved, ∣r∣2+∣t∣2=1{|r|}^{2}+{|t|}^{2}=1∣r∣2+∣t∣2=1, when one of the two clusters is embedded in the waveguides. This conclusion can be applied to an arbitrary parity-symmetric real Hermitian graph with additional PT\mathcal{PT}PT-symmetric potentials, which is more feasible in the experiment. An exactly solvable model is presented to illustrate the theory. Bethe ansatz solution indicates that the transmission spectrum of such a cluster displays a peculiar feature arising from the non-Hermiticity of the scattering center.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.