Dirac equation in low dimensions: The factorization method

We present a general approach to solve the (1+1)$(1+1)$ and (2+1)$(2+1)$-dimensional Dirac equations in the presence of static scalar, pseudoscalar and gauge potentials, for the case in which the potentials have the same functional form and thus the factorization method can be applied. We show that the presence of electric potentials in the Dirac equation leads to two Klein–Gordon equations including an energy-dependent potential. We then generalize the factorization method for the case of energy-dependent Hamiltonians. Additionally, the shape invariance is generalized for a specific class of energy-dependent Hamiltonians. We also present a condition for the absence of the Klein paradox (stability of the Dirac sea), showing how Dirac particles in low dimensions can be confined for a wide family of potentials.     

Noncommutative Dirac oscillator in an external magnetic field

We show that (2+1)$(2+1)$-dimensional noncommutative Dirac oscillator in an external magnetic field is mapped onto the same but with reduced angular frequency in absence of magnetic field. We construct the relativistic Landau levels by solving corresponding Dirac equation in (2+1)$(2+1)$-dimensional noncommutative phase space. All the Landau levels become independent of noncommutative parameter for a critical value of the magnetic field. Several other interesting features along with the relevance of such models in the study of atomic transitions in a radiation field have been discussed.     

Pseudo-Hermitian generalized Dirac oscillators

We study generalized Dirac oscillators with complex interactions in (1+1)$(1+1)$ dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are η$\eta$-pseudo-Hermitian with respect to certain metric operators η$\eta$. Exact solutions for the generalized Dirac oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with an anti-Jaynes–Cummings-type model and by spin flipping they can also be identified with a Jaynes–Cummings-type model.     

Quantum criticality and Yang–Mills gauge theory

We present a family of nonrelativistic Yang–Mills gauge theories in D+1$D+1$ dimensions whose free-field limit exhibits quantum critical behavior with gapless excitations and dynamical critical exponent z=2$z=2$. The ground state wavefunction is intimately related to the partition function of relativistic Yang–Mills in D   dimensions. The gauge couplings exhibit logarithmic scaling and asymptotic freedom in the upper critical spacetime dimension, equal to 4+1$4+1$. The theories can be deformed in the infrared by a relevant operator that restores Poincaré invariance as an accidental symmetry. In the large-N limit, our nonrelativistic gauge theories can be expected to have weakly curved gravity duals.     

Supersymmetric quantum mechanics of magnetic monopoles: A case study

We study, in detail, the supersymmetric quantum mechanics of charge-(1,1)$(1,1)$ monopoles in N=2$N=2$ supersymmetric Yang–Mills–Higgs theory with gauge group SU(3)$\mathit{SU}(3)$ spontaneously broken to U(1)×U(1)$U(1)\times U(1)$. We use the moduli space approximation of the quantised dynamics, which can be expressed in two equivalent formalisms: either one describes quantum states by Dirac spinors on the moduli space, in which case the Hamiltonian is the square of the Dirac operator, or one works with anti-holomorphic forms on the moduli space, in which case the Hamiltonian is the Laplacian acting on forms. We review the derivation of both formalisms, explicitly exhibit their equivalence and derive general expressions for the supercharges as differential operators in both formalisms. We propose a general expression for the total angular momentum operator as a differential operator, and check its commutation relations with the supercharges. Using the known metric structure of the moduli space of charge-(1,1)$(1,1)$ monopoles we show that there are no quantum bound states of such monopoles in the moduli space approximation. We exhibit scattering states and compute corresponding differential cross sections.     

Hidden symmetry in the presence of fluxes

We derive the most general first-order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogeneous form ω   which is a solution to a coupled system of first-order partial differential equations which we call the generalized conformal Killing–Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U(1)$U(1)$ field, the system of generalized conformal Killing–Yano equations decouples into the homogeneous conformal Killing–Yano equations with torsion introduced in D. Kubiznak et al. (2009) [8] and the symmetry operator is essentially the one derived in T. Houri et al. (2010) [9]. We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.     

Anisotropic harmonic oscillator,non-commutative Landau problem and exotic Newton–Hooke symmetry

We investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with non-commutative coordinates. It includes the exotic Newton–Hooke particle and the non-commutative Landau problem as special, isotropic and maximally anisotropic, cases. The system is described by the same (2+1$2+1$)-dimensional exotic Newton–Hooke symmetry as in the isotropic case, and develops three different phases depending on the values of the two central charges. The special cases of the exotic Newton–Hooke particle and non-commutative Landau problem are shown to be characterized by additional, so(3)$\mathit{so}(3)$ or so(2,1)$\mathit{so}(2,1)$ Lie symmetry, which reflects their peculiar spectral properties.     

Mass mixing,the fourth generation,and the kinematic Higgs mechanism

We describe how to construct explicit chiral fermion mass terms using Dirac–Kähler (DK) spinors. Classical massive DK spinors are shown to be equivalent to four generations of Dirac spinors with equal mass coupled to a background U(2,2)$U(2,2)$ gauge field. Quantization breaks U(2,2)$U(2,2)$ to U(2)×U(2)$U(2)\times U(2)$, lifts mass spectrum degeneracy, and generates a non-trivial CKM mixing.     

An xp model on AdS2 spacetime

In this paper we formulate the xp model on the AdS₂ spacetime. We find that the spectrum of the Hamiltonian has positive and negative eigenvalues, whose absolute values are given by a harmonic oscillator spectrum, which in turn coincides with that of a massive Dirac fermion in AdS₂. We extend this result to generic xp models which are shown to be equivalent to a massive Dirac fermion on spacetimes whose metric depend of the xp   Hamiltonian. Finally, we construct the generators of the isometry group SO(2,1)$\mathit{SO}(2,1)$ of the AdS₂ spacetime, and discuss the relation with conformal quantum mechanics.     

Supersymmetric Chern–Simons vortex systems and extended supersymmetric quantum mechanics algebras

We study N=2$N=2$ supersymmetric Chern–Simons Higgs models in (2+1)$(2+1)$-dimensions and the existence of extended underlying supersymmetric quantum mechanics algebras. Our findings indicate that the fermionic zero modes quantum system in conjunction with the system of zero modes corresponding to bosonic fluctuations, are related to an N=4$N=4$ extended 1-dimensional supersymmetric algebra with central charge, a result closely connected to the N=2$N=2$ spacetime supersymmetry of the total system. We also add soft supersymmetric terms to the fermionic sector in order to examine how this affects the index of the corresponding Dirac operator, with the latter characterizing the degeneracy of the solitonic solutions. In addition, we analyze the impact of the underlying supersymmetric quantum algebras to the zero mode bosonic fluctuations. This is relevant to the quantum theory of self-dual vortices and particularly for the symmetries of the metric of the space of vortices solutions and also for the non-zero mode states of bosonic fluctuations.     

Stable compact motions of a particle driven by a central force in six-dimensional spacetime

We give a counterexample to the well-known Ehrenfest’s assertion that the existence of stable electromagnetic bound systems is impossible in spaces of more than three dimensions. If we require that the Maxwellian laws of electromagnetism be preserved for any even spacetime dimension, and that the dynamics as a whole be consistent, then the laws of mechanics must be amended by the addition of terms with higher derivatives. We consider a nonrelativistic particle with an acceleration-dependent Lagrangian which moves in an attractive 1/r³$1/r^3$ potential in five-dimensional space. There are compactly supported motions whose projections on the SO(5)$\left(5\right)$-reduced Hamiltonian system are Poisson equilibrium points. The nonlinearly stable equilibria correspond to physically stable motions over the direct product of two three-spheres in configuration space. The Energy-Casimir method turns out to be not appropriate for checking the stability. The studied system is shown to be stable through an analysis of numerical solutions to the equations of motion for small perturbations on the reduced phase space. This implies that falling to the center is prevented.     

Electric dipole moments in pseudo-Dirac gauginos

The SUSY CP problem is one of serious problems in construction of realistic supersymmetric standard models. We consider the problem in a framework in which adjoint chiral multiplets are introduced and gauginos have Dirac mass terms induced by a U(1)$\mathrm{U}(1)$ gauge interaction in the hidden sector. This is realized in hidden sector models without singlet chiral multiplets, which are favored from a recent study of the Polonyi problem. We find that the dominant contributions to electron and neutron electric dipole moments (EDMs) in the model come from phases in the supersymmetric adjoint mass terms. When the supersymmetric adjoint masses are suppressed by a factor of ∼100 compared with the Dirac ones, the electron and neutron EDMs are suppressed below the experimental bound even if the SUSY particle masses are around 1 TeV. Thus, this model works as a framework to solve the SUSY CP problem.     

Relativistic quantum dynamics of a neutral particle in external electric fields: An approach on effects of spin

The planar quantum dynamics of a spin-1/2 neutral particle interacting with electrical fields is considered. A set of first order differential equations is obtained directly from the planar Dirac equation with nonminimum coupling. New solutions of this system, in particular, for the Aharonov–Casher effect, are found and discussed in detail. Pauli equation is also obtained by studying the motion of the particle when it describes a circular path of constant radius. We also analyze the planar dynamics in the full space, including the r=0$r=0$ region. The self-adjoint extension method is used to obtain the energy levels and wave functions of the particle for two particular values for the self-adjoint extension parameter. The energy levels obtained are analogous to the Landau levels and explicitly depend on the spin projection parameter.     

Yangian symmetries and integrability of the Dirac equation with spin symmetry

We show that a Yangian symmetry, namely, Y(su(2))$Y\left(su\left(2\right)\right)$, exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2))$Y\left(su\left(2\right)\right)$ explicitly and get the energy spectrum of this model from the representation theory for Y(su(2))$Y\left(su\left(2\right)\right)$. We also show that this model is integrable, from RTT relations.     

Relativistic particles and the geometry of 4-D null curves

We study actions in (d+1)$(d+1)$-dimensions associated with null curves, mainly when d=3$d=3$, whose Lagrangian is a linear function on the curvature of the particle path, showing that null helices are always possible trajectories of the particles. We find Killing vector fields along critical curves of the action which correspond to the linear and the angular momenta of the particle. They provide two constants of the motion which can be interpreted in terms of the mass and the spin of the system. Moreover, we are able to integrate both the Euler–Lagrange equations and the Cartan equations in cylindrical coordinates around a certain plane.     

Trapping neutral fermions with a PT-symmetric trigonometric potential in the presence of position-dependent mass

The relativistic problem of neutral fermions subject to PT-symmetric trigonometric potential (∼iαtanαx)$(\sim \mathrm{i}\alpha \tan \alpha x)$ in 1+1$1+1$ dimensions is investigated. By using the basic concepts of the supersymmetric quantum mechanics formalism and the functional analysis method, we solve exactly the position-dependent effective mass Dirac equation with the vector coupling scheme and obtain the bound state solutions in closed form. The behavior of the energy spectra is discussed in detail.     

Dirac equation in noncommutative space for hydrogen atom

We consider the energy levels of a hydrogen-like atom in the framework of θ  -modified, due to space noncommutativity, Dirac equation with Coulomb field. It is shown that on the noncommutative (NC) space the degeneracy of the levels 2S_1/2$2S_{1/2}$, 2P_1/2$2P_{1/2}$ and 2P_3/2$2P_{3/2}$ is lifted completely, such that new transition channels are allowed.