Spinless Duffin-Kemmer-Petiau Oscillator in a Galilean Non-commutative Phase Space

We examine Galilei-invariant linear wave equations in a non-commutative phase space. Specifically, we establish and solve the Galilean covariant Duffin-Kemmer-Petiau equation for spin-0 fields in a harmonic oscillator potential. We obtain these wave equations with a Galilean covariant approach, based on a (4+1)-dimensional manifold with light-cone coordinates followed by a reduction to a (3+1)-dimensional spacetime. We find the exact wave functions and their energy levels, and we examine the effects of non-commutativity.     

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.     

Galilean-Covariant Clifford Algebras in the Phase-Space Representation

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.     

Weak-field approximation of effective gravitational theory with local Galilean invariance

We examine the weak-field approximation of locally Galilean invariant gravitational theories with general covariance in a (4+1)-dimensional Galilean framework. The additional degrees of freedom allow us to obtain Poisson, diffusion, and Schrödinger equations for the fluctuation field. An advantage of this approach over the usual (3+1)-dimensional General Relativity is that it allows us to choose an ansatz for the fluctuation field that can accommodate the field equations of the Lagrangian approach to MOdified Newtonian Dynamics (MOND) known as AQUAdratic Lagrangian (AQUAL). We investigate a wave solution for the Schrödinger equations.     

Nonrelativistic Wave Equations with Gauge Fields

We illustrate a metric formulation of Galilean invariance by constructing wave equations with gauge fields. It consists of expressing nonrelativistic equations in a covariant form, but with a five-dimensional Riemannian manifold. First we use the tensorial expressions of electromagnetism to obtain the two Galilean limits of electromagnetism found previously by Le Bellac and Lévy-Leblond. Then we examine the nonrelativistic version of the linear Dirac wave equation. With an Abelian gauge field we find, in a weak field approximation, the Pauli equation as well as the spin—orbit interaction and a part reminiscent of the Darwin term. We also propose a generalized model involving the interaction of the Dirac field with a non-Abelian gauge field; the SU(2) Hamiltonian is given as an example.     

Logics generated by causality structures. covariant representations of the Galilean logic

We consider the causal structure of space-time in the logic approach. The general form of covariant representations of the Galilean logic, which correspond to a localizable Galilean system, is found.     

Generally Covariant Schrödinger Equation in Newton–Cartan Space–Time. Part I

The covariant Schrödinger equation is obtained with the use of standard geometrical objects of the Galilean space–time. It's symmetry and covariance are investigated. Gauge freedom is eliminated by invariance condition. Family of the plane wave solutions in any coordinate system is found. Connection with previous investigations is discussed.     

Non-commutative Lorentz symmetry and the origin of the Seiberg–Witten map

We show that the non-commutative Yang–Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The non-commutative Yang–Mills action is invariant under combined conformal transformations of the Yang–Mills field and of the non-commutativity parameter . The Seiberg–Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action. Received: 6 November 2001 / Published online: 5 April 2002     

Exact solutions of （3＋1）-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations

The symmetries and the exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space-time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.     

Relativistic spin-zero bosons in a Som–Raychaudhuri space–time

We investigate the Duffin–Kemmer–Petiau equation for spin-zero bosons in a (\(3+1\))-dimensional Som–Raychaudhuri space–time. We establish the covariant Duffin–Kemmer–Petiau equation in this curved space–time for the so-called oscillator and we include interaction with a scalar potential. We determine eigenfunctions and the corresponding eigenvalues for the oscillator with the Cornell potential. We investigate the effect of the space–time’s parameters, oscillator’s frequency and the Cornell potential’s parameters on the wave functions.     

Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method.     

The twistor geometry of the covariantly quantized Brink-Schwarz superparticle

We perform the Lorentz-covariant BFV-BRST quantization of the Brink-Schwarz (BS) superparticle by reducing it to a system whose constraints are all covariant, first class and independent (CFI).This procedure requires the presence of certain auxiliary pure gauge variables which are then recognized as (super-)twistors. In a particular covariant gauge, the system reduces to a purely (super-)twistor one.We present several extensions of this 4-dimensional construction to higher dimensions.     

Non-relativistic holography and singular black hole

We provide a framework for non-relativistic holography so that a covariant action principle ensuring the Galilean symmetry for dual conformal field theory is given. This framework is based on the Bargmann lift of the Newton–Cartan gravity to the one-dimensional higher Einstein gravity, or reversely, the null-like Kaluza–Klein reduction. We reproduce the previous zero temperature results, and our framework provides a natural explanation about why the holography is co-dimension 2. We then construct the black hole solution dual to the thermal CFT, and find the horizon is curvature singular. However, we are able to derive the sensible thermodynamics for the dual non-relativistic CFT with correct thermodynamical relations. Besides, our construction admits a null Killing vector in the bulk such that the Galilean symmetry is preserved under the holographic RG flow. Finally, we evaluate the viscosity and find it zero if we neglect the back reaction of the singular horizon, otherwise, it could be non-zero.     

Generally Covariant Schrödinger Equation in Newton–Cartan Space–Time. Part II

Schrödinger equation in Newton–Cartan space–time can be obtained from Einstein equivalence principle, that is firstly one should obtain generally covariant Schrödinger's equation in Galilean space–time (using generally covariant Hamilton–Jacobi formalism and Schrödinger's Ansatz, as was previously shown) and get Schrödinger's equation in Newton–Cartan space–time with the help of equivalence principle. The equation possesses a gauge freedom f connected with phase transformation of the wave function. But absolute elements of the space–time possess a symmetry group and they depend on f. So, a natural problem arises: to find the gauge f in which absolute elements become invariant with respect to the group. In the paper the gauge f is found with the help of space–time properties, that is transformation rule of the wave function is obtained from the space–time structure (in Newton–Cartan space–time). The special form of f is found in the case when the space–time as a whole possesses a symmetry.     

Wilson Loops and Topological Phases in Closed String Theory

Using covariant phase space formulations for the natural topological invariants associated with the world-surface in closed string theory, we find that certain Wilson loops defined on the world-surface and that preserve topological invariance, correspond to wave functionals for the vacuum state with zero energy. The differences and similarities with the 2-dimensional QED proposed by Schwinger early are discussed. PACS Numbers : 81T30, 81T45     

非自治物质畸形波的传播操控

研究了(1+1)维的变系数Gross-Pitaevskii方程, 获得了该方程的精确畸形波解. 基于该精确畸形波解, 深入研究了非自治物质畸形波在随时间指数变化的相互作用下的传播动力学行为, 发现非自治畸形波除具有“来无影、去无踪”的不可预测特性外, 也可实现完全激发、抑制激发以及维持激发等操控. 研究表明, 畸形波操控的关键是对累积时间的最大值T_max 与峰值位置T₀ (或T_I,T_II)值大小关系的调节. 当T_max > T₀ (或T_I,T_II)时畸形波被快速地完全激发, 热原子团中的原子增加到凝聚体中. 当T_max = T₀ (或T_I,T_II) 时畸形波激发到最大振幅, 可以维持相当长的时间而不消失, 热原子团中的原子增加到凝聚体中. 当T_max < T₀ (或T_I,T_II)时畸形波没有充足的时间来激发而被抑制甚至消失, 凝聚体中的原子减少. 这些结果在理论和实际应用上具有启迪意义.     

On Hopf algebroid structure of κ-deformed Heisenberg algebra

Physics of Atomic Nuclei - The (4 + 4)-dimensional κ-deformed quantum phase space as well as its (10 + 10)-dimensional covariant extension by the Lorentz sector can be described as Heisenberg...     

Null-plane dynamics for few-quark systems

A Lorentz-invariant formulation of the 3-dimensional harmonic oscillator is proposed for serving as the confining part of the kernel of a Bethe-Salpeter equation for aqq system, and likewise for aqqq system. Such a kernel is amenable to an effective 3-dimensional reduction in the null-plane approximation (NPA), so as to maintain explicit NPA-covariance for the (reduced) BS equations as well as the corresponding wave functions. This rectifies the mathematical limitation of an earlier BS formulation (by Mitra and coworkers) forqq andqqq systems with non-covariant h. o. kernels which was restricted to the instantaneous approximation and was therefore valid only for slowly moving hadrons. The present (covariant) formulation agrees exactly with the earlier (non-covariant) treatment for hadrons at rest, thus preserving all the physical results on mass spectra, etc., obtained with the latter, but is now formally applicable to processes involving hadrons in motion.