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Physical Review A

Comment on “Spontaneous breaking of permutation symmetry in pseudo-Hermitian quantum mechanics”

Francisco Marcelo Fernández

Published Oct 10, 2017• 0 citations
Publisher

Abstract

Li and Miao [Phys. Rev. A 85, 042110 (2012)] proposed a non-Hermitian Hamiltonian that is neither Hermitian nor $PT$ symmetric but exhibits real eigenvalues for some values of the model parameters. In order to explain this fact, they resorted to $PT$-pseudo Hermiticity and to a so-called permutation symmetry. Here we show that the spectrum of this Hamiltonian can be easily analyzed in the usual way in terms of exact or broken antiunitary symmetries that appear to be more relevant than the permutation symmetry. In addition, we show why the authors' Hamiltonian and the well-known Pais-Uhlenbeck oscillator lead to the same fourth-order differential equation for the coordinates.

DOI
10.1103/physreva.96.046101

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Li and Miao [Phys. Rev. A 85, 042110 (2012)] proposed a non-Hermitian Hamiltonian that is neither Hermitian nor PTPTPT symmetric but exhibits real eigenvalues for some values of the model parameters. In order to explain this fact, they resorted to PTPTPT-pseudo Hermiticity and to a so-called permutation symmetry. Here we show that the spectrum of this Hamiltonian can be easily analyzed in the usual way in terms of exact or broken antiunitary symmetries that appear to be more relevant than the permutation symmetry. In addition, we show why the authors' Hamiltonian and the well-known Pais-Uhlenbeck oscillator lead to the same fourth-order differential equation for the coordinates.
By adding an imaginary interacting term proportional to ip1p2i{p}_{1}{p}_{2}ip1​p2​ to the Hamiltonian of a free anisotropic planar oscillator, we construct a model which is described by the PTPTPT-pseudo-Hermitian Hamiltonian with the permutation symmetry of two dimensions. We prove that our model is equivalent to the Pais-Uhlenbeck oscillator and thus establish a relationship between our PTPTPT-pseudo-Hermitian system and the fourth-order derivative oscillator model. We also point out the spontaneous breaking of permutation symmetry which plays a crucial role in giving a real spectrum free of interchange of positive and negative energy levels in our model. Moreover, we find that the permutation symmetry of two dimensions in our Hamiltonian corresponds to the identity (not in magnitude but in attribute) of two different frequencies in the Pais-Uhlenbeck oscillator, and reveal that the unequal-frequency condition imposed as a prerequisite upon the Pais-Uhlenbeck oscillator can reasonably be explained as the spontaneous breaking of this identity.
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.
We notice that non-linear oscillator Hamiltonian having the character of non-Hermitian (H≠H+)(H \ne H^{+})(H=H+), non-PT Symmetry ([H,PT]≠0)([H,PT] \ne 0)([H,PT]=0) and non-Pseudohermiticity (ηHη−1≠H+)(\eta H \eta^{-1}\ne H^{+})(ηHη−1=H+) can yield real eigenvalues.
PT symmetric quantum mechanics admits non-Hermitian Hamiltonians with real spectra but, as long as the evolution is pseudo-unitary, the probabilistic interpretation of wave functions is lost. We offer an effective Hermitization of the theory using the Feshbach-type elimination of ghosts. The energy-dependence of the resulting Hermitian effective Hamiltonians Heff is a nonlinearity which, in a strong-coupling regime, explains the spontaneous PT symmetry breaking and the simultaneous loss of the Hermiticity of Heff while the pseudo-unitary evolution law persists. PACS 03.65.Bz, 03.65.Ca 03.65.Fd The concept of extended, non-Hermitian quantum mechanics [1] with real spectra and PT symmetric Hamiltonians (P means parity and T denotes time reversal) grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [2] have discovered that a
It is generally believed that Parity-Time (PT)-symmetry breaking occurs when eigenvalues or both eigenvalues and eigenvectors coincide. However, we show that this well-accepted picture of PT-symmetry breaking is incorrect. Instead, we demonstrate that the physical mechanism of PT-symmetry breaking is the resonance between positive- and negative-action modes. It is proved that PT-symmetry breaking occurs when and only when this resonance condition is satisfied, and this mechanism applies to all known PT-symmetry breakings observed in different branches of physics. The result is achieved by proving a remarkable fact that in finite dimensions, a PT-symmetric Hamiltonian is necessarily pseudo-Hermitian, regardless whether it is diagonalizable or not.