pseudo-Hermitian Quantum Mechanics Research Articles

On metric controlled operator-valued frames and their applications

The exploration of pseudo-Hermitian quantum mechanics has led to rapid progress in the methods of constructing inner products and determining the observables of the theory in quantum sciences. In this paper, we introduce metric controlled operator-valued frames and our motivation is to find the applications in the generalized setting of quantum mechanics. We give various characterizations of metric controlled operator-valued frames. We also explore various spectral properties of pseudo-Hermitian and pseudo-unitary operators. Finally, we give the method of representing infinte quantum channels in pseudo-Hermitian quantum mechanics using metric controlled operator-valued frames.

Enhancement of photon creation through the pseudo-Hermitian Dynamical Casimir Effect

We analyse here the pseudo-Hermitian Dynamical Casimir Effect, proposing a non-Hermitian version of the effective Law’s Hamiltonian used to describe the phenomenon. We verify that the average number of created photons can be substantially increased, a result which calls the attention to the possibility of engineering the time-dependent non-Hermitian Hamiltonian we have assumed. Given the well-known difficulty in detecting the Casimir photon production, the present result reinforces the importance of pseudo-Hermitian quantum mechanics as a new chapter of quantum theory and an important tool for the amplification of Hermitian processes such as the degree of squeezing of quantum states.

Closed operators in semi-Hilbertian spaces

Given a positive operator A on a Hilbert space, we introduce the notion of an A-closed linear operator as a natural extension of the usual notion of an A-bounded operator. We summarize a number of results and examples about this class of operators. We show that all A-bounded operators are A-closed and we prove that the class of A-closed operators is stable under perturbation by A-bounded operators. Moreover, we give sufficient conditions for the adjoint operator T* to be A-closed when T is A-closed. This study is motivated by recent developments of pseudo-Hermitian quantum mechanics.

Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

Hairy BTZ black hole and its analogue model in graphene

We obtain analytical solutions to Einstein–Maxwell-scalar gravity with negative cosmological constant in (2+1)-dimensions. The scalar field is non minimally coupled to gravity and electromagnetism by the metric ansatz. We first find analytical solutions to the quasinormal mode (QNM) spectrum of a rotating hairy Bañados–Teitelboim–Zanelli (BTZ) black hole by solving the associated Dirac equation within the framework of the pseudo-Hermitian quantum mechanics. We also develop an analogue model for this hairy BTZ black hole in graphene, by mapping its conformal counterpart onto the Beltrami pseudosphere. The model offers to make some predictions for the analogue model of the (2+1)-dimensional gravitational black hole in graphene.

Integrable minisuperspace models with Liouville field: energy density self-adjointness and semiclassical wave packets

The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum contexts of Wheeler–DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint, and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrödinger’s norm and a conserved Mostafazadeh’s norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.

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

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.

Deformed quons and bi-coherent states

We discuss how a q-mutation relation can be deformed replacing a pair of conjugate operators with two other and unrelated operators, as it is done in the construction of pseudo-fermions, pseudo-bosons and truncated pseudo-bosons. This deformation involves interesting mathematical problems and suggests possible applications to pseudo-hermitian quantum mechanics. We construct bi-coherent states associated to D -pseudo-quons, and we show that they share many of their properties with ordinary coherent states. In particular, we find conditions for these states to exist, to be eigenstates of suitable annihilation operators and to give rise to a resolution of the identity. Two examples are discussed in details, one connected to an unbounded similarity map, and the other to a bounded map.

Hamiltonians defined by biorthogonal sets

In some recent papers, studies on biorthogonal Riesz bases have found renewed motivation because of their connection with pseudo-Hermitian quantum mechanics, which deals with physical systems described by Hamiltonians that are not self-adjoint but may still have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed in some previous papers. However, in many physical models, one has to deal not with orthonormal bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of -quasi basis, and we show a series of conditions under which a definition of non-self-adjoint Hamiltonian with purely point real spectra is still possible.

Hilbert Space Inner Products for 𝓟 𝓣 $\mathcal {PT}$ -symmetric Su-Schrieffer-Heeger Models

A Su-Schrieffer-Heeger model with added PT-symmetric boundary term is studied in the framework of pseudo-hermitian quantum mechanics. For two special cases, a complete set of pseudometrics is constructed in closed form. When complemented with a condition of positivity, the pseudometrics determine all the physical inner products of the considered model.

Some invariant biorthogonal sets with an application to coherent states

We show how to construct, out of a certain basis invariant under the action of one or more unitary operators, a second biorthogonal set with similar properties. In particular, we discuss conditions for this new set to be also a basis of the Hilbert space, and we apply the procedure to coherent states. We conclude the paper considering a simple application of our construction to pseudo-Hermitian quantum mechanics.

Some remarks on quasi-Hermitian operators

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

Pseudo-Hermitian quantum mechanics with unbounded metric operators

I extend the formulation of pseudo-Hermitian quantum mechanics to η(+)-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η(+). In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η(+) and consequently √η(+).

Partial inner product spaces, metric operators and generalized hermiticity

Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (pip-space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties. Then we apply some of the previous results to operators on a particular pip-space, namely a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.

Spontaneous breaking of permutation symmetry in pseudo-Hermitian quantum mechanics

By adding an imaginary interacting term proportional to $i{p}_{1}{p}_{2}$ to the Hamiltonian of a free anisotropic planar oscillator, we construct a model which is described by the $PT$-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 $PT$-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.

Uniqueness and self-conjugacy of Dirac Hamiltonians in arbitrary gravitational fields

Proofs of two statements are provided in this paper. First, the authors prove that the formalism of the pseudo-Hermitian quantum mechanics allows describing the Dirac particles motion in arbitrary stationary gravitational fields. Second, it is proved that using the Parker weight operator and the subsequent transition to the \eta -representation gives the transformation of the Schroedinger equation for nonstationary metric, when the evolution operator becomes self-conjugate. The scalar products in the \eta -representation are flat, which makes possible the use of a standard apparatus for the Hermitian quantum mechanics. Based on the results of this paper the authors draw a conclusion about solution of the problem of uniqueness and self-conjugacy of Dirac Hamiltonians in arbitrary gravitational fields including those dependent on time. The general approach is illustrated by the example of Dirac Hamiltonians for several stationary metrics, as well as for the cosmologically flat and the open Friedmann models.

Solution of the problem of uniqueness and Hermiticity of Hamiltonians for Dirac particles in gravitational fields

The authors prove that the dynamics of spin 1/2 particles in stationary gravitational fields can be described using an approach, which builds upon the formalism of pseudo-Hermitian Hamiltonians. The proof consists in the analysis of three expressions for Hamiltonians, which are derived from the Dirac equation and describe the dynamics of spin 1/2 particles in the gravitational field of the Kerr solution. The Hamiltonians correspond to different choices of tetrad vectors and differ from each other. The differences between the Hamiltonians confirm the conclusion known from many studies that the Hamiltonians derived from the Dirac equation are non-unique. Application of standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the resulting Hamiltonian is the same as that of the Hamiltonians derived from the Dirac equation with any chosen system of tetrad vectors. For description of the dynamics of spin 1/2 particles in stationary gravitational fields can be used not only the formalism of pseudo-Hermitian Hamiltonians, but also an alternative approach, which employs the Parker scalar product. The authors show that the alternative approach is equivalent to the formalism of pseudo-Hermitian Hamiltonians.

Examples of pseudo-bosons in quantum mechanics

We discuss two physical examples of the so-called pseudo-bosons, recently introduced in connection with pseudo-hermitian quantum mechanics. In particular, we show that the so-called extended harmonic oscillator and the Swanson model satisfy all the assumptions of the pseudo-bosonic framework introduced by the author. We also prove that the biorthogonal bases they produce are not Riesz bases.

Mathematical aspects of intertwining operators: the role of Riesz bases

In this paper we continue our analysis of intertwining relations for both self-adjoint and not self-adjoint operators. In particular, in this last situation, we discuss the connection with pseudo-Hermitian quantum mechanics and the role of Riesz bases.

Application of pseudo-Hermitian quantum mechanics to a complex scattering potential with point interactions

We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p2/2m + ζ−δ(x + α) + ζ+δ(x − α), where ζ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h. η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ+) = −ℑ(ζ−). This in particular contains the -symmetric case: ζ+ = ζ*−. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of the -symmetry.