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.
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.
One of the most characteristic mathematical features of the PT-symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states 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-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 D(physical), even in the immediate vicinity of the phase-transition boundaries $\ensuremath{\partial}{\mathcal{D}}^{(\mathrm{physical})}$.
In conventional Schrodinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, h→H=ΩhΩ=H† induces the change L→K of the Hilbert space of states, reflected by the loss of the Hermiticity of H=H†. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts K into the third, correct physical Hilbert space H, unitarily equivalent to L. The situation encountered, typically, in PT−symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.
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.
A physical theory of quantum mechanics can be based on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry ( PT symmetry).Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data.One would think that a non-Hermitian Hamiltonian would give a quantum theory that violates unitarity.However, if PT symmetry is not broken, it is possible to use a physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite.This construction is general and works for any PT -symmetric Hamiltonian.The dynamics is governed by unitary time evolution.This formulation does not conflict with the requirements of conventional quantum mechanics.There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.
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.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.