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arXiv:1803.01949 · Physical Review A

Admissible perturbations and false instabilities in PT -symmetric quantum systems

Miloslav Znojil

Published Mar 16, 2018• 23 citations
arXivHTMLPDF

Abstract

One of the most characteristic mathematical features of the $\mathcal{PT}$-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states $\mathcal{H}=\mathcal{H}(H)$. Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but $\mathcal{PT}$-symmetric user-friendly family of $J$-parametric ``discrete anharmonic'' quantum Hamiltonians $H=H(\stackrel{P\vec}{\ensuremath{\lambda}})$ is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\in}{\mathcal{D}}^{(\mathrm{physical})}$. In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain ${\mathcal{D}}^{(\mathrm{physical})}$, even in the immediate vicinity of the phase-transition boundaries $\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}$.

DOI
10.1103/physreva.97.032114

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One of the most characteristic mathematical features of the PT\mathcal{PT}PT-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states H=H(H)\mathcal{H}=\mathcal{H}(H)H=H(H). Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but PT\mathcal{PT}PT-symmetric user-friendly family of JJJ-parametric ``discrete anharmonic'' quantum Hamiltonians H=H(\stackrel{P\vec}{\ensuremath{\lambda}}) is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, \stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\in}{\mathcal{D}}^{(\mathrm{physical})}. In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain D(physical){\mathcal{D}}^{(\mathrm{physical})}D(physical), even in the immediate vicinity of the phase-transition boundaries \ensuremath∂D(physical)\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}\ensuremath∂D(physical).
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the PTPTPT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.
Nonlinearity of perturbations in PT{\mathscr{P}}{\mathscr{T}}PT-symmetric quantum mechanics
In the so called crypto-Hermitian formulation of quantum theory (incorporating, in particular, the -symmetric quantum mechanics as its special case) the unitary evolution of a system is known to be described via an apparently redundant representation of the states in a triplet of Hilbert spaces. Two of them are unitarily equivalent while the auxiliary, zeroth one is unphysical but exceptionally user-friendly. The dynamical evolution equations are, naturally, solved in the friendliest space . The evaluation of experimental predictions then requires a Hermitization of the observables. This yields, as its byproduct, the correct physical Hilbert space . The formalism offers the conventional probabilistic interpretation due to the Dyson-proposed unitary equivalence between and a certain conventional but prohibitively complicated textbook space . The key merit of the innovation lies in its enhanced flexibility opening new ways towards nonlocal or complex-interaction unitary models. The price to pay is that the ad hoc inner product in the relevant physical Hilbert space is, by construction, Hamiltonian-dependent. We show that this implies that the Rayleigh-Schrödinger perturbation theory must be used with due care. The warning is supported by an elementary sample of quantum system near its exceptional-point phase transition. In contrast to a naive expectation, the model proves stable with respect to the admissible, self-consistently specified (i.e., physical) small perturbations.
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
Abstract In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric—the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss.
During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators $\mathrm{\ensuremath{\Lambda}}$ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space H(F){\mathcal{H}}^{(F)}H(F). The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition ${\mathcal{H}}^{(F)}\ensuremath{\rightarrow}{\mathcal{H}}^{(S)}$ to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators ${\mathrm{\ensuremath{\Lambda}}}_{0}$ and ${\mathrm{\ensuremath{\Lambda}}}_{1}$) in advance, such an upgrade need not exist in general.
Most quantum physicists believe that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under matrix transposition and complex conjugation) to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the non-Dirac-hermitian Hamiltonian H = p2+ix3 has a real positive discrete spectrum and generates unitary time evolution and defines a fully consistent and physical quantum theory. Evidently, Dirac Hermiticity is too restrictive. While H = p2+ix3 is not Dirac Hermitian, it is PT symmetric (invariant under combined space reflection P and time reversal T). Another PT-symmetric Hamiltonian whose energy levels are real, positive and discrete is H = p2−x4, which contains an upside-down potential. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics and quantum field theory are extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past two years some of these properties have been verified in laboratory experiments. Here, we first discuss PT-symmetric Hamiltonians at a simple intuitive level and explain why the energy levels of such Hamiltonians may be real, positive, and discrete. Second, we describe a recent experiment in which the PT phase transition was observed. Third, we briefly mention that PT-symmetric theories can be useful at a fundamental level. While the double-scaling limit of an O(N)-symmetric gφ4 quantum field theory appears to be inconsistent because the critical value of g is negative, this limit is in fact not inconsistent because the critical theory is PT symmetric.