Composite system in noncommutative space and the equivalence principle

The motion of a composite system made of N particles is examined in a space with a canonical noncommutative algebra of coordinates. It is found that the coordinates of the center-of-mass position satisfy noncommutative algebra with effective parameter. Therefore, the upper bound of the parameter of noncommutativity is re-examined. We conclude that the weak equivalence principle is violated in the case of a non-uniform gravitational field and propose the condition for the recovery of this principle in noncommutative space. Furthermore, the same condition is derived from the independence of kinetic energy on the composition.     

Canonical transformations and exact invariants for time‐dependent Hamiltonian systems

An exact invariant is derived for n‐degree‐of‐freedom non‐relativistic Hamiltonian systems with general time‐dependent potentials. To work out the invariant, an infinitesimalcanonical transformation is performed in the framework of the extended phase‐space. We apply this approach to derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamiltonian systems, the invariant is obtained equivalently performing in the extended phase‐space a finitecanonical transformation of the initially time‐dependent Hamiltonian to a time‐independent one. It is furthermore shown that the invariant can be expressed as an integral of an energy balance equation. The invariant itself contains a time‐dependent auxiliary function ξ (t) that represents a solution of a linear third‐order differential equation, referred to as the auxiliary equation. The coefficients of the auxiliary equation depend in general on the explicitly known configuration space trajectory defined by the system's time evolution. This complexity of the auxiliary equation reflects the generally involved phase‐space symmetry associated with the conserved quantity of a time‐dependent non‐linear Hamiltonian system. Our results are applied to three examples of time‐dependent damped and undamped oscillators. The known invariants for time‐dependent and time‐independent harmonic oscillators are shown to follow directly from our generalized formulation.     

Dynamical realization of l-conformal Galilei algebra and oscillators

The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.     

Energy Separation in Oscillatory Hamiltonian Systems without any Non-resonance Condition

We consider multiscale Hamiltonian systems in which harmonic oscillators with several high frequencies are coupled to a slow system. It is shown that the oscillatory energy is nearly preserved over long times ${\varepsilon^{-N}}$ for arbitrary N > 1, where ${\varepsilon^{-1}}$ is the size of the smallest high frequency. The result is uniform in the frequencies and does not require non-resonance conditions.     

非对易空间的三模谐振子能谱

在非对易空间中,用不变本征算符方法(IEO),对非耦合、坐标耦合、动量耦合三种三模谐振子系统能谱进行求解,并将求解结果与一般对易空间的能谱进行比较分析.通过比较发现,当非对易参数为零时,所求能级差还原到了与普通空间相对应的一般量子系统哈密顿量能级差,验证了推导结果的正确性;同时讨论了耦合系数对非对易空间能谱的影响. 关键词： 不变本征算符 非对易空间 三模谐振子能谱 能级差     

Fate of the superconducting ground state on the Moyal plane

It is known that Berry curvature of the band structure of certain crystals can lead to effective noncommutativity between spatial coordinates. Using the techniques of twisted quantum field theory, we investigate the question of the formation of a paired state of twisted fermions in such a system. We find that to leading order in the noncommutativity parameter, the gap between the non-interacting ground state and the paired state is smaller compared to its commutative counterpart. This suggests that BCS type superconductivity, if present in such systems, is more fragile and easier to disrupt.     

Chaotic motion at the emergence of the time averaged energy decay

A system plus environment conservative model is used to characterize the nonlinear dynamics when the time averaged energy for the system particle starts to decay. The system particle dynamics is regular for low values of the N environment oscillators and becomes chaotic in the interval 13≤N≤15, where the system time averaged energy starts to decay. To characterize the nonlinear motion we estimate the Lyapunov exponent (LE), determine the power spectrum and the Kaplan-Yorke dimension. For much larger values of N the energy of the system particle is completely transferred to the environment and the corresponding LEs decrease. Numerical evidence shows the connection between the variations of the amplitude of the particles energy time oscillation with the time averaged energy decay and trapped trajectories.     

Influence of Noncommutativity on the Motion of Sun-Earth-Moon System and the Weak Equivalence Principle

Features of motion of macroscopic body in gravitational field in a space with noncommutativity of coordinates and noncommutativity of momenta are considered in general case when coordinates and momenta of different particles satisfy noncommutative algebra with different parameters of noncommutativity. Influence of noncommutativity on the motion of three-body Sun-Earth-Moon system is examined. We show that because of noncommutativity the free fall accelerations of the Moon and the Earth toward the Sun in the case when the Moon and the Earth are at the same distance to the source of gravity are not the same even if gravitational and inertial masses of the bodies are equal. Therefore, the Eotvos-parameter is not equal to zero and the weak equivalence principle is violated in noncommutative phase space. We estimate the corrections to the Eotvos-parameter caused by noncommutativity on the basis of Lunar laser ranging experiment results. We obtain that with high precision the ratio of parameter of momentum noncommutativity to mass is the same for different particles.     

Exact statements about many-particle systems with higher symmetries

A new method of calculating the energy spectrum of a system of A identical Fermi particles with translationally invariant interaction is developed under the assumption that there exists a high symmetry in the 3A-dimensional space of particle coordinates. For a special class of symmetries the many-body problem is split exactly into two sets of equations: one containing only totally symmetric combinations of the particle coordinates which are called “collective variables” and the other equation taking essentially into account the requirements of the Pauli principle and connected symmetry properties. In several cases it is possible to obtain the excitation spectra exactly showing qualitatively new features. They depend on “many-particle quantum numbers” varying independently of each other in an interval which sometimes depends on A. For special high symmetries the collective variables obey equations which are very similar to one-particle equations providing a new explanation of the “Independent-Particle Model” for arbitrary strength and form of the interaction potential. A manifold of unknown up to now excitation spectra of many-particle systems is obtained and discussed.     

Relativistic versus nonrelativistic solution of the N-fermion problem in a hyperradius-confining potential

We investigate the quantum system of N identical fermions in the relativistic limit. In this article the considered potential is a combination of Coulombic, linear confining and harmonic oscillator terms. By using Jacobi coordinates and introducing the hyperradius quantity we obtain the wave functions of the system as well as the corresponding energy eigenvalues. Assuming that all particles are confined within a hypersphere we calculate the corresponding x _bag . In particular we consider the case N = 3 which corresponds to baryonic systems. By using the experimental values of the charge radius of each baryon we calculate the potential coefficients. Within our treatment the results of the MIT bag model are recovered for N = 1. Finally we compare the results obtained by the Dirac equation with the corresponding results of the Schrödinger equation and we find that the energy spectra obtained by the former are much closer to experimental values.     

On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=ε_ij^k θ_k and a momentum noncommutativity matrix parameter β=ε_ij^k β_k, is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraints on this particular transformation, we firstly find that the product of the two parameters θ and β possesses a lower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on the physical system under study. This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Among the obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β, representing the same particle in presence of a magnetic field $\vec{B}=q^{-1}\vec{\beta}$. For the other examples, additional correction terms depending on β appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively feeled with opposite sign.     

The many-body problem with an energy-dependent confining potential

We consider systems of N bosons bound by two-body harmonic interactions, whose frequency depends on the total energy of the system. Such energy dependent confining interactions between the bosons yield remarkable properties of the many-body system. As the quantum numbers increase, the total energy cannot exceed a saturation energy, which is independent of the number of particles N. Moreover, the ground state energy increases with N. As a result, the density of states tends rapidly to infinity as N and/or the quantum numbers increase.     

Orthofermions versus Gutzwiller projection operator approach for the infinite U Hubbard model

A comparison of two well-known approaches for strongly correlated electron systems, namely, nested Bethe ansatz implemented through orthofermion algebra and Gutzwiller projection operator formalism, is made by calculating the energy spectrum of 1D infinite U Hubbard model for a finite system consisting of three particles on a four site anisotropic closed chain. It is shown that orthofermion algebra always leads to at least an eight hold degeneracy in the energy spectrum corresponding to all 2³ spin configurations, consistent with the nested Bethe ansatz solution leading to a N²-fold degeneracy of energy levels of an N electron system. Such a degeneracy is absent in the Gutzwiller projection operator approach. This finding shows the limitations of the Gutzwiller projection method and at the same time the relevance of orthofermion approach for the infinite U Hubbard model.     

The Schrödinger and Pauli-Dirac Oscillators in Noncommutative Phase Space

We investigate the non-relativistic Schrödinger and Pauli-Dirac oscillators in noncommutative phase space using the five-dimensional Galilean covariant framework. The Schrödinger oscillator presented the correct energy spectrum whose non isotropy is caused by the noncommutativity with an expected similarity between this system and the particle in a magnetic field. A general Hamiltonian for the 3-dimensional Galilean covariant Pauli-Dirac oscillator was obtained and it presents the usual terms that appears in commutative space, like Zeeman effect and spin-orbit terms. We find that the Hamiltonian also possesses terms involving the noncommutative parameters that are related to a type of magnetic moment and an electric dipole moment.     

非对易相空间三模坐标动量耦合谐振子的能谱

本文用不变本征算符方法研究非对易相空间中三模坐标动量耦合谐振子的能谱,分别得到了非耦合和坐标动量耦合两种情况下谐振子能谱的解析解,其中包括受非对易参数θ和φ影响的解λ0,1和λ1,1,以及不受非对易参数θ和φ影响的解λ0,2和λ1,2.然后,分析了两类耦合参数κ和η对三模坐标动量耦合谐振子能谱的影响.结果发现,耦合参数κ和η对λ1,1的影响是相同的,且当κ=η时,耦合系数κ和η对λ1,1是没有影响的.     

Quasi-exact solvability of Dirac-Pauli equation and generalized Dirac oscillators

In this paper we demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact solvability of the system based on the sl(2) symmetry are discussed separately in the spherical, cylindrical, and Cartesian coordinates. Some exactly solvable field configurations are also exhibited.     

The Quantum and Klein-Gordon Oscillators in a Non-commutative Complex Space and the Thermodynamic Functions

In this work we study the quantum and Klein-Gordon oscillators in a non-commutative complex space. We show that a particle described by such oscillators behaves similarly as an electron with spin in a commutative space in an external uniform magnetic field. Therefore the wave-function $\psi (z,\bar{z} )$ takes values in C ⁴, spin up, spin down, particle, antiparticle, a result which is obtained by the Dirac theory. We obtain the energy levels by exact solutions. We also derive the thermodynamic functions associated to the partition function, and show that the non-commutativity effects are manifested in energy at the high temperature limit.     

Non-traditional Approach to Quantum Computing

The N-qubit system characterized by an effective spin \(S = 2^{N - 1} - {1/2}\) is carried out in the representation of two coupled harmonic oscillators. It is shown that quantum computing results obtained with spinor algebra can be obtained also using the algebra of two coupled harmonic oscillators which is a convenient formalism, especially in the case of large number of qubits. In this formalism the non-abelian and abelian groups of the order of 16 related to one- and two-qubit systems were found. The structure of Cayley tables of those groups is different due to different commutation (anticommutation) relations for operators forming each group.     

Hydrogen atom in rotationally invariant noncommutative space

We consider the noncommutative algebra which is rotationally invariant. The hydrogen atom is studied in a rotationally invariant noncommutative space. We find the corrections to the energy levels of the hydrogen atom up to the second order in the parameter of noncommutativity. The upper bound of the parameter of noncommutativity is estimated on the basis of the experimental results for 1s–2s$1s\text{–}2s$ transition frequency.     

Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

We propose and solve exactly the Schrödinger equation of a bound quantum system consisting in four particles moving on a real line with both translationally invariant four particles interactions of Wolfes type [1] and additional non translationally invariant four-body potentials. We also generalize and solve exactly this problem in any D-dimensional space by providing full eigensolutions and the corresponding energy spectrum. We discuss the domain of the coupling constant where the irregular solutions becomes physically acceptable.