Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive.
Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive.
Non-Hermitian quantum-Hamiltonian-candidate combination H λ of a non-Hermitian unperturbed operator H = H 0 with an arbitrary "small" non-Hermitian perturbation λ W is given a mathematically consistent unitary-evolution interpretation. The formalism generalizes the conventional constructive Rayleigh-Schrödinger perturbation expansion technique. It is sufficiently general to take into account the well known formal ambiguity of reconstruction of the correct physical Hilbert space of states. The possibility of removal of the ambiguity via a complete, irreducible set of observables is also discussed.
To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R(t)) representing another observable. In the language of physics, the information carried by R(t)≠R†(t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ(t). From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ(t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator R(t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a(t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R(t) mimicking the birth of the Universe (“Big Bang”) at t=0.
Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space L, by a Hamiltonian h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h by its isospectral but non-Hermitian avatar H=Ω−1hΩ with Ω†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H, the amended Hamiltonian H=H† can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
In a way inspired by the brief 2002 note "The challenge of nonhermitian structures in physics" by Ramirez and Mielnik (with the text most easily available via arXiv: quant- ph/0211048 ) the situation in the theory is briefly summarized here as it looks twelve years later. Our text has three parts. In the first one we briefly mention the pre-history (dating back to the Freeman Dyson's proposal of the non-Hermitian-Hamiltonian method in 1956 and to its subsequent successful "interacting boson model" applications in nuclear physics) and, first of all, the amazing recent progress reached, in the stationary case, using, in essence, an inversion of the Dyson's approach. The impact on the latter idea upon abstract quantum physics is sampled, first of all, by the reference to papers by Bender et al. (who made the non-Hermitian model-building popular under the nickname of parity-times-time-reflection- symmetric alias PT-symmetric quantum mechanics) and by Mostafazadeh (who reinterpreted PT-symmetry as P-pseudo-Hermiticity). In the second part of our review the emphasis is shifted to the newest, non-stationary upgrade of the formalism which we proposed in the year 2009 and which is characterized by the simultaneous participation of a triplet of Hilbert spaces H in the representation of a single quantum system. In the third part of the review we finally emphasize that the majority of applications of our three-Hilbert-space (THS) recipe is still ahead of us because the enhancement of the flexibility is necessarily accompanied by an enhancement of the technical difficulties. An escape out of the technical trap is proposed to be sought in a restriction of attention to quantum models living in finite-dimensional Hilbert spaces H . As long as the use of such spaces is so typical for the quantum-control considerations, we conclude with conjecture that the THS formalism should start searching for implementations in the field of quantum control.
Nonlinearity of perturbations in 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.