In this work we present a general formalism to treat non-Hermitian and noncommutative Hamiltonians. This is done employing the phase-space formalism of quantum mechanics, which allows to write a set of robust maps connecting the Hamitonians and the associated Wigner functions to the different Hilbert space structures, namely, those describing the non-Hermitian and noncommutative, Hermitian and noncommutative, and Hermitian and commutative systems. A general recipe is provided to obtain the expected values of the more general Hamiltonian. Finally, we apply our method to the harmonic oscillator under linear amplification and discuss the implications of both non-Hermitian and noncommutative effects.
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 Ham…
We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian s…
When a non-hermitian hamiltonian has a certain symmetry, such as the PT pseudo-hermiticity, it is still possible that the hamiltonian has a real spectrum. In this note, by adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a PT pseudo-hermitian hamiltonian and obtain its real spectrum. In addition, we find that the reality of the energy spectrum remains when the model is extended to the canonical noncommutative space.
The author discusses a different kind of Hermitian quantum mechanics, called $J$-Hermitian quantum mechanics. He shows that $PT$-symmetric quantum mechanics is indeed $J$-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is $J$-Hermitian (or equivalently $PT$-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.
We investigate four different types of representations of deformed canonical variables leading to generalized versions of Heisenberg's uncertainty relations resulting from noncommutative spacetime structures. We demonstrate explicitly how the representations are related to each other and study three characteristically different solvable models on these spaces, the harmonic oscillator, the manifestly non-Hermitian Swanson model and an intrinsically noncommutative model with Poeschl-Teller type potential. We provide an analytical expression for the metric in terms of quantities specific to the g…
We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in $(2+1)$-space-time dimensions and the two-fold extension of the group of translations of $\mathbb R^4$. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build un…
We formulate a Born rule for families of quantum systems parametrized by a noncommutative space of control parameters. The resulting formalism may be viewed as a generalization of quantum mechanics where overlaps take values in a noncommutative algebra, rather than in the complex numbers.
This series of papers is devoted to an open-ended project aimed at the solution of Hilbert's sixth problem (concerning joint axiomatization of physics and probability theory) proposed to be constructed in the framework of an all-embracing mechanics. In this first paper, the bare skeleton of such a mechanics is constructed in the form of noncommutative Hamiltonian mechanics (NHM) which combines elements of noncommutative symplectic geometry and noncommutative probability in the framework of topological superalgebras; it includes, besides NHM basics, a treatment of Lie group actions in NHM and n…
We extend the formulation of pseudo-Hermitian quantum mechanics to eta-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator eta. In particular, we 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 eta and consequently its positive square root.
Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be $η$-pseudo-Hermitian with Hermitian, positive-definite and non-singular $η$ operators. We show that despite the fact that an $η$ operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is singular, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the hel…
We consider classical and quantum mechanics related to an additional noncommutativity, symmetric in position and momentum coordinates. We show that such mechanical system can be transformed to the corresponding one which allows employment of the usual formalism. In particular, we found explicit connections between quadratic Hamiltonians and Lagrangians, in their commutative and noncommutative regimes. In the quantum case we give general procedure how to compute Feynman's path integral in this noncommutative phase space with quadratic Lagrangians (Hamiltonians). This approach is applied to a ch…
In this paper we consider an electron moving on a two dimensional noncommutative plane immersed in a constant magnetic field under the influence of an anisotropic harmonic potential. We work out the gauge invariant energy spectra of this system using a consistent gauge prescription proposed in \cite{nctori,gauge-wigfunc}. We also show subsequently that straightforward generalization of the Landau problem in the noncommutative setting using minimal coupling prescription as is done naively on many occasions in the literature violates gauge invariance of the energy spectra.
We consider a problem of the consistent deformation of physical system introducing a new features, but preserving its fundamental properties. In particular, we study how to implement the noncommutativity of space-time without violation of the rotational symmetry in quantum mechanics or the Lorentz symmetry in f{i}eld theory. Since the canonical (Moyal) noncommutativity breaks the above symmetries one should work with more general case of coordinate-dependent noncommutative spaces, when the commutator between coordinates is a function of these coordinates. F{i}rst we describe in general lines h…
The spectral singularity have been extensively studied over the last one and half decade for different non-Hermitian potentials in non-Hermitian quantum mechanics. The nature of spectral singularities have not been studied for the case of quaternionic potential. In the present work we perform an analytical study for the scattering from a quaternionic point interaction represented by delta function. New features of spectral singularities are observed which are different than the case of complex (non-quaternionic) point interaction. Most notable difference is the occurrence of spectral singulari…
The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.
We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Hamiltonians, as well as Lagrangians, in their commutative and noncommutative regimes. The general procedure to compute Feynman's path integral on this noncommutative phase space with quadratic Lagrangians (Hamiltonians) is presented. Using this approa…
Classical nonlinear canonical (Poisson) maps have a distinguished role in quantum mechanics. They act unitarily on the quantum phase space and generate $\hbar$-independent quantum canonical maps. It is shown that such maps act in the noncommutative phase space as dictated by the classical covariance. A crucial observation made is that under the classical covariance the local quantum mechanical picture can become nonlocal in the Hilbert space. This nonlocal picture is made equivalent by the Weyl map to a noncommutative picture in the phase space formulation of the theory. The connection between…
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator $η_+$ and defining the annihilation and creation operators to be $η_+$-pseudo-Hermitian adjoint to each other. The operator $η_+$ represents the $η_+$-pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-$PT$-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator $η_+$ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanati…
We study some basic and interesting quantum mechanical systems in dynamical noncommutative spaces in which the space- space commutation relations are position dependent. It is observed that the fundamental objects in the dynamical noncommutative space introduced here are stringlike. We show that the Stark effect can be employed to determine whether the noncommutativity of space is dynamical or non-dynamical. It appears that unlike non-dynamical case there is a fundamental energy $\dfrac{τ\hbar^{2}}{m}$ in this dynamical space.}
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 present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governe…
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.
In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Dirac's proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit, which squares to zero. Thus, the crucial rôle in quantum theo…
We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.
We propose a model of dynamical noncommutative quantum mechanics in which the noncommutative strengths, describing the properties of the commutation relations of the coordinate and momenta, respectively, are arbitrary energy dependent functions. The Schrödinger equation in the energy dependent noncommutative algebra is derived for a two dimensional system for an arbitrary potential. The resulting equation reduces in the small energy limit to the standard quantum mechanical one, while for large energies the effects of the noncommutativity become important. We investigate in detail three cases, …
We utilize the close relation between the complex space $\textbf{C}^2$ and the real space $\textbf{R}^3$ to reformulate quantum mechanics in a manner which allows to, either or both, describe magnetic monopoles and quantize the underlying space, obtaining (noncommutative) quantum mechanics (with magnetic monopoles).
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.
The aim of this paper is to analyze the reconstructability of quantum mechanics from classical conditional probabilities representing measurement outcomes conditioned on measurement choices. We will investigate how the quantum mechanical representation of classical conditional probabilities is situated within the broader frame of noncommutative representations. To this goal, we adopt some parts of the quantum formalism and ask whether empirical data can constrain the rest of the representation to conform to quantum mechanics. We will show that as the set of empirical data grows conventional el…
We consider topological groupoids in finite and also in a compact settings. In the initial sections, we introduce definitions of typical observables and we studied them in the context of statistical mechanics and quantum mechanics. We exhibit explicit examples and one of them will be the so-called quantum ratchet. This is related to Schwinger's algebra of selective measurements. Here we consider $\mathcal{G}$-kernels, transverse functions, modular functions, and quasi-invariant measures for Haar systems. Later we present our main result which is a version of Tannaka's theorem for Hausdorff com…
In his recent paper arXiv:1312.7738, the author discussed $J$-Hermitian quantum mechanics and showed that $PT$-symmetric quantum mechanics is essentially $J$-Hermitian quantum mechanics. In this paper, the author discusses finite $J$-Hermitian quantum mechanics which is derived naturally from its continuum one and its relationship with finite $PT$-symmetric quantum mechanics.
We study the 1/N expansion in noncommutative quantum mechanics for the anharmonic and Coulombian potentials. The expansion for the anharmonic oscillator presented good convergence properties, but for the Coulombian potential, we found a divergent large N expansion when using the usual noncommutative generalization of the potential. We proposed a modified version of the noncommutative Coulombian potential which provides a well-behaved 1/N expansion.
We consider electrodynamics on a noncommutative spacetime using the enveloping algebra approach and perform a non-relativistic expansion of the effective action. We obtain the Hamiltonian for quantum mechanics formulated on a canonical noncommutative spacetime. An interesting new feature of quantum mechanics formulated on a noncommutative spacetime is an intrinsic electric dipole moment. We note however that noncommutative intrinsic dipole moments are not observable in present experiments searching for an EDM of leptons or nuclei such as the neutron since they are spin independent. These exper…
A demonstration is given that the simplest model of quantum mechanics formulated on a plane non-commutative geometry endowed with a Galilean symmetry group in which the position and linear momentum-variable commutators are first order in the dynamical variables (and thus constitute a true Lie algebra) is incompatible with the hypothesis of spacial isotropy.
We consider the probabilistic description of nonrelativistic, spinless one-particle classical mechanics, and immerse the particle in a deformed noncommutative phase space in which position coordinates do not commute among themselves and also with canonically conjugate momenta. With a postulated normalized distribution function in the quantum domain, the square of the Dirac delta density distribution in the classical case is properly realised in noncommutative phase space and it serves as the quantum condition. With only these inputs, we pull out the entire formalisms of noncommutative quantum …
One examines putative corrections to the Bell operator due to the noncommutativity in the phase-space. Starting from a Gaussian squeezed envelop whose time evolution is driven by commutative (standard quantum mechanics) and noncommutative dynamics respectively, one concludes that, although the time evolving covariance matrix in the noncommutative case is different from the standard case, the squeezing parameter dominates and there are no noticeable noncommutative corrections to the Bell operator. This indicates that, at least for squeezed states, the privileged states to test Bell correlations…
Quantum mechanics in noncommutative space modifies the standard result of the Aharonov-Bohm effect for electrons and other recent quantum effects. Here we obtain the phase in noncommutative space for the Spavieri effect, a generalization of Aharonov-bohm effect which involving a coherent superposition of particles with opposite charges moving along single open interferometric path. By means of the experimental considerations a limit $\sqrtθ\simeq\left( 0,13T\text{eV}\right) ^{-1}$ is achieved, improving by 10 orders of magnitude the derived by Chaichian et. al. for the Aharonov-Bohm effect. It…
It has earlier been argued that there should exist a formulation of quantum mechanics which does not refer to a background spacetime. In this paper we propose that, for a relativistic particle, such a formulation is provided by a noncommutative generalisation of the Hamilton-Jacobi equation. If a certain form for the metric in the noncommuting coordinate system is assumed, along with a correspondence rule for the commutation relations, it can be argued that this noncommutative Hamilton-Jacobi equation is equivalent to standard quantum mechanics.
In this work, we investigate the heat flow of two interacting quantum systems on the perspective of noncommutativity phase-space effects and show that by controlling the new constants introduced in the quantum theory, due to a deformed Heisenberg-Weyl algebra, the heat flow from the hot to the cold system may be enhanced, thus decreasing the time required to reach thermal equilibrium. We also give a brief discussion on the robustness of the second law of thermodynamics in the context of noncommutative quantum mechanics
We carry out an investigation into the possibility of developing a Bohmian interpretation based on the continuous motion of point particles for noncommutative quantum mechanics. The conditions for such an interpretation to be consistent are determined, and the implications of its adoption for noncommutativity are discussed. A Bohmian analysis of the noncommutative harmonic oscillator is carried out in detail. By studying the particle motion in the oscillator orbits, we show that small-scale physics can have influence at large scales, something similar to the IR-UV mixing.
We discuss various descriptions of a quantum particle on noncommutative space in a (possibly non-constant) magnetic field. We have tried to present the basic facts in a unified and synthetic manner, and to clarify the relationship between various approaches and results that are scattered in the literature.
Some fundamental aspects related with the construction of Robertson-Schrödinger like uncertainty principle inequalities are reported in order to provide an overall description of quantumness, separability and nonlocality of quantum systems in the noncommutative phase-space. Some consequences of the deformed noncommutative algebra are also considered in physical systems of interest.
The non-Hermitian Schrödinger equation is re-expressed generally in the form of Hamilton's canonical equation without any approximation. Its quantization called non-Hermitian quantum field theory is discussed. By virtue of the canonical equation, the theory of non-Hermitian quantum mechanics is totally reformulated, including the probability amplitudes of states, the expectations of operators, as well as the expressions of interaction terms. The conventional difficulties in non-Hermitian quantum mechanics are totally overcome by the reformulation. Specifically, the imaginary parts the non-Herm…
Starting with the first-order singular Lagrangian describing the dynamical system with 2nd-class constraints, the noncommutative quantum mechanics on a curved space is investigated by the constraint star-product quantization formalism of the projection operator method. Imposing the additional constraints to eliminate the reduntant degrees of freedom, it is shown that the resultant noncommutative quantum system on the curved space is represented with two kinds of the constrained quantum systems, which are equivalent with each other. Then, it is shown that the resultant Hamiltonians contain the …
The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kaehler manifold. The classical principles of Quantum Mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non linear geometrical context.
The breakdown of Ehrenfest's theorem imposes serious limitations on quaternionic quantum mechanics (QQM). In order to determine the conditions in which the theorem is valid, we examined the conservation of the probability density, the expectation value and the classical limit for a non-anti-hermitian formulation of QQM. The results also indicated that the non-anti-hermitian quaternionic theory is related to non-hermitian quantum mechanics, and thus the physical problems described with both of the theories should be related.
We study non-Hermitian quantum mechanics in the presence of a minimal length. In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty. The spectrum in both the cases are found to be real. It is also shown that the models are $η$ pseudo-Hermitian and the metric operator is found explicitly in both the cases.
Examples are given of non-Hermitian Hamiltonian operators which have a real spectrum. Some of the investigated operators are expressed in terms of the generators of the Weil-Heisenberg algebra. It is argued that the existence of an involutive operator $\hat J$ which renders the Hamiltonian $\hat J$-Hermitian leads to the unambiguous definition of an associated positive definite norm allowing for the standard probabilistic interpretation of quantum mechanics. Non-Hermitian extensions of the Poeschl-Teller Hamiltonian are also considered. Hermitian counterparts obtained by similarity transformat…
A condition to have a real spectrum for a non-Hermitian Hamiltonian is given. As special cases, it is shown that the condition is reduced to Hermiticity and PT symmetric conditions.
A Bose-Einstein condensation theory for any integer spin using noncommutative quantum mechanics methods is considered. The effective potential is derived as a multipolar expansion in the non-commutativity parameter ($θ$) and, at second order in $θ$, our procedure yields to the standard dipole-dipole interaction with $θ^2$ playing the role of the strength interaction parameter. The generalized Gross-Pitaevskii equation containing non-local dipolar contributions is found. For $^{52}$Cr isotopes $θ= C_{dd}/4π$ becomes $\sim 10^{-11}$ cm and, thus for this value of $θ$ one cannot distinguish inter…
Effects of noncommutativity are investigated in planar quantum mechanics in the coordinate representation. Generally these issues are addressed by converting to the momentum space. In the first part of the work we show noncommutative effects in a Gaussian wavepacket through the broadening of its width. We also rederive results on *-product of Gaussian wavepackets. In the second part, we construct a "Master" model for a noncommutative harmonic oscillator by embedding it in an extended space. Different gauge choices leading to different forms of noncommutativity, (such as between coordinates onl…
A series of recent papers ``Faster than Hermitian Quantum Mechanics'' and related articles made a point of the possibility of a non-Hermitian, but PT-symmetric, operator to play the role of a Hamiltonian. In particular, they show that with an appropriate choice of an inner product, the evolution generated by such an operator will conserve the norm and scalar product. Here we observe that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian, and the whole exercise is to a certain degree redundant.
The rationale for introducing non-hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.
The response of a test particle, both for the free case and under the harmonic oscillator potential, to circularly polarized gravitational waves is investigated in a noncommutative quantum mechanical setting. The system is quantized following the prescription in \cite{ncgw1}. Standard algebraic techniques are then employed to solve the Hamiltonian of the system. The solutions, in both cases, show signatures of the coordinate noncommutativity. In the harmonic oscillator case, this signature plays a key role in altering the resonance point and the oscillation frequency of the system.
In a recent paper we have suggested that a formulation of quantum mechanics should exist, which does not require the concept of time, and that the appropriate mathematical language for such a formulation is noncommutative differential geometry. In the present paper we discuss this formulation for the free point particle, by introducing a commutation relation for a set of noncommuting coordinates. The sought for background independent quantum mechanics is derived from this commutation relation for the coordinates. We propose that the basic equations are invariant under automorphisms which map o…
