We study non Hermitian quantum systems in noncommutative space as well as a \cal{PT}-symmetric deformation of this space. Specifically, a \mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative space, the iC(x_1+x_2) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not \mathcal{PT}-symmetric. A complex interacting anisotropic oscillator system has also been discussed.
With a view to eliminate an important misconception in some recent publications, we give a brief review of the notion of a pseudo-Hermitian operator, outline pseudo-Hermitian quantum mechanics, and discuss its basic difference with the indefinite-metric quantum mechanics. In particular, we show that the answer to the question posed in the title is a definite No.
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
We investigate quasi-hermitian quantum mechanics in phase space using standard deformation quantization methods: Groenewold star products and Wigner transforms. We focus on imaginary Liouville theory as a representative example where exact results are easily obtained. We emphasize spatially periodic solutions, compute various distribution functions and phase-space metrics, and explore the relationships between them.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the under…
The spectral singularity (SS) and coherent perfect absorption (CPA) have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian standard quantum mechanics (SQM) governed by Schrodinger equation. In the present work we explore these scattering features in the domain of non-Hermitian space fractional quantum mechanics (SFQM) governed by fractional Schrodinger equation which is characterized by Levy index $α$ ($ 1< α\leq 2$). We observe that non-Hermitian SFQM systems have more flexibility for SS and CPA and display some new features of …
We provide a careful analysis of the generating functional in the path integral formulation of pseudo-Hermitian and in particular PT-symmetric quantum mechanics and show how the metric operator enters the expression for the generating functional.
A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamil…
The He-McKellar-Wilkens(HMW) effect for spin one neutral particle in non-commutative quantum mechanics is studied. By solving the Kemmer like equations on non-commutative (NC) space and non-commutative phase space, we obtain topological He-McKellar-Wilkens phase on NC space and NC phase space respectively, where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.
We consider a model of non-commutative Quantum Mechanics given by two harmonic oscillators over a non-commutative two dimensional configuration space. We study possible ways of removing the non-commutativity based on the classical limit context known as anti-Wick quantization. We show that removal of non-commutativity from the configuration space and from the canonical operators are not commuting operations.
By using a generalized Bopp's shift formulation, instead of star product method, we investigate the Aharonov-Casher(AC) effect for a spin-1 neutral particle in non-commutative(NC) quantum mechanics. After solving the Kemmer equations both on a non-commutative space and a non-commutative phase space, we obtain the corrections to the topological phase of the AC effect for a spin-1 neutral particle both on a NC space and a NC phase space.
In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the …
We generalize the formulation of non-commutative quantum mechanics to three dimensional non-commutative space. Particular attention is paid to the identification of the quantum Hilbert space in which the physical states of the system are to be represented, the construction of the representation of the rotation group on this space, the deformation of the Leibnitz rule accompanying this representation and the implied necessity of deforming the co-product to restore the rotation symmetry automorphism. This also implies the breaking of rotational invariance on the level of the Schroedinger action …
We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau manifolds.
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=p^2/2m+ζ_-δ(x+a)+ζ_+δ(x-a), where ζ_\pm and a are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ_\pm where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator ηand the corresponding equivalent Hermitian H…
We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions. We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354]. In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former.
In this article, we define a non-commutative deformation of the "symplectic invariants" of an algebraic hyperelliptical plane curve. The necessary condition for our definition to make sense is a Bethe ansatz. The commutative limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus we define non-commutative deformations of some algebraic geometry quantities. In particular our non-commutative Bergmann kernel satisfies a Rauch variational formula. Those non-commutative invariants are inspired from the large N expansion of formal non-hermitian matrix models. Thus they are expe…
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 and elaborate on a number of rel…
Within the context of non-Hermitian quantum mechanics, we use the generators of eigenvectors of the Hamiltonian to construct a unitary inner product space. Such models have been of interest in recent years, for instance, in the context of ${\cal PT}$ symmetry, although our construction extends to the larger class of so-called pseudo-Hermitian Operators. We provide a detailed example to illustrate the concept and compare with known results.
By adding an imaginary interacting term proportional to ip_1p_2 to the Hamiltonian of a free anisotropic planar oscillator, we construct a new 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 positiv…
We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean ${\mathbb R}^4$ to symmetric two-tensors introduced in "Gravitational instantons from gauge theory," H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose $U(1)$ instantons on a noncommutative ${\mathbb C}^2$ as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric be…
We formulate quantum mechanics on SO(3) using a non-commutative dual space representation for the quantum states, inspired by recent work in quantum gravity. The new non-commutative variables have a clear connection to the corresponding classical variables, and our analysis confirms them as the natural phase space variables, both mathematically and physically. In particular, we derive the first order (Hamiltonian) path integral in terms of the non-commutative variables, as a formulation of the transition amplitudes alternative to that based on harmonic analysis. We find that the non-trivial ph…
Deformations of the canonical commutation relations lead to non-Hermitian momentum and position operators and therefore almost inevitably to non-Hermitian Hamiltonians. We demonstrate that such type of deformed quantum mechanical systems may be treated in a similar framework as quasi/pseudo and/or PT-symmetric systems, which have recently attracted much attention. For a newly proposed deformation of exponential type we compute the minimal uncertainty and minimal length, which are essential in almost all approaches to quantum gravity.
We develop relativistic wave equations in the framework of the new non-hermitian ${\cal PT}$ quantum mechanics. The familiar Hermitian Dirac equation emerges as an exact result of imposing the Dirac algebra, the criteria of ${\cal PT}$-symmetric quantum mechanics, and relativistic invariance. However, relaxing the constraint that in particular the mass matrix be Hermitian also allows for models that have no counterpart in conventional quantum mechanics. For example it is well-known that a quartet of Weyl spinors coupled by a Hermitian mass matrix reduces to two independent Dirac fermions; here…
Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commuta…
A non-Hermitian operator $H$ defined in a Hilbert space with inner product $\langle\cdot|\cdot\rangle$ may serve as the Hamiltonian for a unitary quantum system, if it is $η$-pseudo-Hermitian for a metric operator (positive-definite automorphism) $η$. The latter defines the inner product $\langle \cdot|η\cdot\rangle $ of the physical Hilbert space $\mathscr{H} _η$ of the system. For situations where some of the eigenstates of $H$ depend on time, $η$ becomes time-dependent. Therefore the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of …
A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian $ix^3$ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graph…
A general non-commutative quantum mechanical system in a central potential $V=V(r)$ in two dimensions is considered. The spectrum is bounded from below and for large values of the anticommutative parameter $θ$, we find an explicit expression for the eigenvalues. In fact, any quantum mechanical system with these characteristics is equivalent to a commutative one in such a way that the interaction $V(r)$ is replaced by $V = V ({\hat H}_{HO}, {\hat L}_z)$, where ${\hat H}_{HO}$ is the hamiltonian of the two-dimensional harmonic oscillator and ${\hat L}_z$ is z- component of the angular momentum. …
We examine the properties and consequences of pseudo-supersymmetry for quantum systems admitting a pseudo-Hermitian Hamiltonian. We explore the Witten index of pseudo-supersymmetry and show that every pair of diagonalizable (not necessarily Hermitian) Hamiltonians with discrete spectra and real or complex-conjugate pairs of eigenvalues are isospectral and have identical degeneracy structure except perhaps for the zero eigenvalue if and only if they are pseudo-supersymmetric partners. This implies that pseudo-supersymmetry is the basic framework for generating non-Hermitian PT-symmetric and non…
Some aspects of the "exotic" particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.
We show that when non-commutative quantum mechanics is formulated on the Hilbert space of Hilbert-Schmidt operators (referred to as quantum Hilbert space) acting on a classical configuration space, spectral triplets as introduced by Connes in the context of non-commutative geometry arise naturally. A distance function as defined by Connes can therefore also be introduced. We proceed to give a simple and general algorithm to compute this function. Using this we compute the distance between pure and mixed states on quantum Hilbert space and demonstrate a tantalizing link between statistics and g…
General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in physical systems with potentials involving spin and non-local interactions.
Harmonic oscillators with a centrifugal spike are analysed, via a non-Hermitian regularization, within a complexified SUSY quantum mechanics. The formalism enables us to construct the factorized creation and annihilation operators. We show how the real though, generically, non-equidistant spectrum complies with the current SUSY-type isospectrality and degeneracy in an unusual way.
Appearance of adiabatic geometric phase shift in the context of noncommutative quantum mechanics is studied using an exactly solvable model of 2D simple harmonic oscilator in Moyal plane, where momentum non-commutativity are also considered along with spatial noncommutativity. After finding a suitable Bopp shift, that bridges the noncommutative phase space operators with their effective commutative counterparts, having their dependence on the non-commutative parameters, we study the adiabatic evolution in the Heisenberg picture. An explicit expression for the geometric phase shift under adiaba…
This paper investigates the thermodynamics of a large class of non-Hermitian, $PT$-symmetric oscillators, whose energy spectrum is entirely real. The spectrum is estimated by second-order WKB approximation, which turns out to be very accurate even for small quantum numbers, and used to generate the quantum partition function. Graphs showing the thermal behavior of the entropy and the specific heat, at all regimes of temperature, are given. To obtain the corresponding classical partition function it turns out to be necessary in general to integrate over a complex "phase space". For the wrong-si…
Starting with the modified Dirac equations for free massive particles with the $γ_5$-extension of the physical mass $m\rightarrow m_1 + γ_5 m_2$, we consider equations of relativistic quantum mechanics in the presence of an external electromagnetic field. The new approach is developing on the basis of existing methods for study the unbroken $\cal PT$ symmetry of Non-Hermitian Hamiltonians. The paper shows that this modified model contains the definition of the mass parameter, which may use as the determination of the magnitude scaling of energy $M$. Obviously that the transition to the standar…
The ``exotic'' particle model with non-commuting position coordinates, associated with the two-parameter central extension of the planar Galilei group, can be used to derive the ground states of the Fractional Quantum Hall Effect. The relation to other NC models is discussed. Anomalous coupling is presented. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.
We investigate consequences of space non-commutativity in quantum mechanics of the hydrogen atom. We introduce rotationally invariant noncommutative space $\hat{\bf R}^3_0$ - an analog of the hydrogen atom ($H$-atom) configuration space ${\bf R}^3_0\,=\, {\bf R}^3\setminus \{0\}$. The space $\hat{\bf R}^3_0$ is generated by noncommutative coordinates realized as operators in an auxiliary (Fock) space ${\cal F}$. We introduce the Hilbert space $\hat{\cal{H}}$ of wave functions $\hatψ$ formed by properly weighted Hilbert-Schmidt operators in ${\cal F}$. Finally, we define an analog of the $H$-at…
We show that a possible violation of the Robertson-Schrödinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase-space (even if it violates the Robertson-Schrödinger uncertainty principle) is always a quantum state of an appropriate non-commutative extension of quantum mechanics. Conversely, all canonical non-commutative extensions of quantum mechanics display states that violate the Robertson-Schrödinger uncertainty principle.
One of the simplest example of non-commutative (NC) spaces is the NC plane. In this article we investigate the consequences of the non-commutativity to the quantum mechanics on a plane. We derive corrections to the standard (commutative) Hamiltonian spectrum for hydrogen-like atom and isotropic linear harmonic oscillator (LHO) and formulate the problem of the potential scattering on the NC plane. In the case of LHO we consider the noncommutativity of the momentum operators, too.
We study the Pauli equation on non-commutative plane. It is shown that the Supersymmetry algebra holds to all orders in the non-commutative parameter $θ$ in case the gyro-magnetic ratio $g$ is 2. Using Seiberg-Witten map, the first order in $θ$ correction to the spectrum is obtained in the case of uniform magnetic field. We find that the eigenstates in the non-commutative case are identical to the commutative case provided the magnetic field $B$ is everywhere replaced by $B(1+Bθ)$.
We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases. The "self-orthogonality" phenomenon is clarified in terms of a correct spectral decomposition and it is shown that "self-orthogonal" states never jeopardize resolution of identity and thereby quantum averages of observables. The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated. Some puzzles with zero-binorm bound states in continuous spectrum are…
The construction of exactly-solvable models has recently been advanced by considering integrable $T\bar{T}$ deformations and related Hamiltonian deformations in quantum mechanics. We introduce a broader class of non-Hermitian Hamiltonian deformations in a nonrelativistic setting, to account for the description of a large class of open quantum systems, which includes, e.g., arbitrary Markovian evolutions conditioned to the absence of quantum jumps. We relate the time evolution operator and the time-evolving density matrix in the undeformed and deformed theories in terms of integral transforms w…
One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if $H^\dagger = H$ is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite syste…
The quantum mechanical version of the four kinds of classical canonical transformations is investigated by using non-hermitian operator techniques. To help understand the usefulness of this appoach the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamiton-Jacobi theory is also employed to solve time dependent Schrödinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the correspondin…
Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into |psi_F>. Consider the following quantum brachistochrone problem: Subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least time tau? For Hermitian Hamiltonians tau has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, tau can be made arbitrarily small without violating the tim…
In this paper, imposing hermitian conjugate relations on the two-parameter deformed quantum group GL_{p,q}(2) is studied. This results in a non-commutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group U_{p,q}(2) with pq real via a non-commutative phase, the representation of the algebra is built by means of the action of the operators constituting the U_{p,q}(2) matrix on states.
We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S^2, must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2\times 2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the c…
The HMW effect in non-commutative quantum mechanics is studied. By solving the Dirac equations on non-commutative (NC) space and non-commutative phase space, we obtain topological HMW phase on NC space and NC phase space respectively, where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.
A model of an electron and a Dirac monopole interacting through an axially symmetric non-hermitian but \mathcal{PT}-symmetric potential is discussed in detail. The intriguing localization of the wave-packet as a result of the anomalous breaking of the scale symmetry is shown to provide a scale for the system. The symmetry algebra for the system, which is the conformal algebra SO(2,1), is discussed and is shown to belong to the enveloping algebra of the combined algebra, composed of the Virosoro algebra, \{L_n, n\in \mathbb{N}\} and an abelian algebra, \{P_n,n\in \mathbb{N}\}.
The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajectory from the conventional one. When the Berry curvature (alias noncommutative parameter) is a monopole in momentum space as found recently in some ferromagnetic semiconductors while observing the anomalous Hall effect, we get a transverse shift, similar to that in the optical Hall effect.
We consider non(anti)commutative (NAC) deformations of d=1 N=2 superspace. We find that, in the chiral base, the deformation preserves only a half of the original (linearly realized) supercharge algebra, as it usually happens in NAC field theories. We obtain in terms of a real supermultiplet a closed expression for a deformed Quantum Mechanics Lagrangian in which the original superpotential is smeared, similarly to what happens for the two dimensional deformed sigma model. Quite unexpectedly, we find that a second conserved charge can be constructed which leads to a nonlinear field realization…
In order to derive a large set of Hamiltonian dynamical systems, but with only first order Lagrangian, we resort to the formulation in terms of Lagrange-Souriau 2-form formalism. A wide class of systems derived in different phenomenological contexts are covered. The non-commutativity of the particle position coordinates are a natural consequence. Some explicit examples are considered.
We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasi-momentum space the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider net in contrast to the Hofstadter butterfly for unbiased motion.
Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table~1). Souriau's construction applied to the two-parameter central extension of the planar Galilei group leads to the ``exotic'' particle, which has non-commuting position coordinates. A Berry-phase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non-commutative parameter is momentum-dependent in this case, and can take the form of a monopole in momentum …
Quantum Mechanics in the non-commutative plane is shown to admit the ``exotic'' symmetry of the doubly-centrally-extended Galilei group. When coupled to a planar magnetic field whose strength is the inverse of the non-commutative parameter, the system becomes singular, and ``Faddeev-Jackiw'' reduction yields the ``Chern-Simons'' mechanics of Dunne, Jackiw, and Trugenberger. The reduced system moves according to the Hall law.
We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar $\bigstar$-product combines the usual phase space $\ast$ star and the noncommutative $\star$ star-product. Having dealt with subtleties of ordering present in this problem we show that the classical correspondent to the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-comm…
A consistent physical theory of quantum mechanics can be built 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 quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive defini…
