Exact quantum mechanical description of the motion of spin-1/2 particles and of spin motion in a uniform magnetic field

The Dirac-Pauli equation is used to obtain the exact equation of spin motion for spin-1/2 particles with an anomalous magnetic moment in a constant and uniform magnetic field. Exact formulas are established for the angular velocity of the revolution of such particles along circular orbits and the rotation of the particle spin with respect to momentum. Finally, a quantum mechanical equation for the motion of the particles in a strong magnetic field is derived. Zh. éksp. Teor. Fiz. 114, 448–457 (August 1998)     

Electric and magnetic polarizabilities of pointlike spin-1/2 particles

The electric and magnetic polarizabilities of pointlike spin-1/2 particles with an anomalous magnetic moment (AMM) are calculated by the transformation of an initial Hamiltonian into the Foldy-Wouthuysen (FW) representation. Corresponding results for spin-1/2 and spin-1 particles are compared.     

Alternative approaches to the construction of the Poincare-invariant spin operators of the Dirac particles

At present a number of methods of constructing the Poincare-invariant spin operators for relativistic particles with half-integer spin in the one-particle theory are well known. The method of odd operator constructing, the Lorentz method of bilinear covariant form transformation, and the method with the Foldy–Wouthuysen representation belong to them. New approaches to the construction of spin operators are developed in the present work, namely, a method of separating space-like component directly from the spin matrices of bilinear covariant forms, including the method of multiplication of the covariant Hamiltonian of the Dirac equation by these matrices. By this means we succeeded in constructing the Poincare-invariant spin operators by simpler and mathematically faultless methods. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 10–15, November, 2008.     

Gravitational geometric phase in the presence of torsion

We investigate the relativistic and non-relativistic quantum dynamics of a neutral spin-1/2 particle subject to an external electromagnetic field in the presence of a cosmic dislocation. We analyze the explicit contribution of the torsion in the geometric phase acquired in the dynamics of this neutral spinorial particle. We discuss the influence of the torsion in the relativistic geometric phase. Using the Foldy–Wouthuysen approximation, the non-relativistic quantum dynamics is studied and the influence of the torsion on the Aharonov–Casher and He–McKellar–Wilkens effects are discussed. An erratum to this article can be found at     

Semiclassical dynamics of Dirac particles interacting with a static gravitational field

The semiclassical limit for Dirac particles interacting with a static gravitational field is investigated. A Foldy–Wouthuysen transformation which diagonalizes at the semiclassical order the Dirac equation for an arbitrary static spacetime metric is realized. In this representation the Hamiltonian provides for a coupling between spin and gravity through the torsion of the gravitational field. In the specific case of a symmetric gravitational field we retrieve the Hamiltonian previously found by other authors. But our formalism provides for another effect, namely, the spin hall effect, which was not predicted before in this context.     

Polarization of spin-1/2 particles in an axisymmetric magnetic field

The present paper shows that the nature of the polarization of charged spin-1/2 particles moving in a uniform magnetic field changes dramatically in a relatively weak transverse axisymmetric magnetic field. The direction along which the spin projection is quantized has a fixed orientation with respect to the axes of a cylindrical coordinate system and can form a substantial angle with the direction of the uniform magnetic field. The presence of spin quantization is proved both by the fact that the commutator of the Hamiltonian operator and the projection of the polarization operator in the direction of quantization is zero and by analyzing the Bargmann-Michel-Telegdi equation for this given case. Finally, the possibilities of detecting this effect and utilizing it are discussed. Zh. éksp. Teor. Fiz. 114, 1153–1161 (October 1998)     

Dirac Particle Polarization in Uniform Magnetic Field

For Dirac particles in a uniform magnetic field, the polarization operator projections onto three directions remain unchanged. If the orbital angular moment of particles is large with respect to 1, then the spin of moving particles retain the fixed orientation relative to the axes of the cylindrical coordinate system.     

Magnetic moment generation from non-minimal couplings in a scenario with Lorentz-symmetry violation

This paper deals with situations that illustrate how the violation of Lorentz symmetry in the gauge sector may contribute to magnetic moment generation of massive neutral particles with spin- and spin-1. The procedure we adopt here is based on Relativistic Quantum Mechanics. We work out the non-relativistic regime that follows from the wave equation corresponding to a certain particle coupled to an external electromagnetic field and a background that accounts for the Lorentz-symmetry violation, and we thereby read off the magnetic dipole moment operator for the particle under consideration. We keep track of the parameters that govern the non-minimal electromagnetic coupling and the breaking of Lorentz symmetry in the expressions we get for the magnetic moments in the different cases we contemplate. Our claim is that the tiny magnetic dipole moment of truly-elementary neutral particles might signal Lorentz-symmetry violation.     

Spin–rotation couplings: spinning test particles and Dirac field

The hypothesis of coupling between spin and rotation introduced long ago by Mashhoon is examined in the context of “1 + 3” and “3 + 1” space-time splitting techniques, either in special or in general relativity. Its content is discussed in terms of classical (Mathisson–Papapetrou–Dixon–Souriou model) as well as quantum physics (Foldy–Wouthuysen transformation for the Dirac field in an external field), reviewing and discussing all the relevant theoretical literature concerning the existence of such effect. Some original contributions are also included. Dedicated to Bahram Mashhoon for his 60th birthday.     

Black Holes: Interfacing the Classical and the Quantum

The central idea of this paper is that forming the black hole horizon is attended with the transition from the classical regime of evolution to the quantum one. We offer and justify the following criterion for discriminating between the classical and the quantum: creations and annihilations of particle-antiparticle pairs are impossible in the classical reality but possible in the quantum reality. In flat spacetime, we can switch from the classical picture of field propagation to the quantum picture by changing the overall sign of the spacetime signature. To describe a self-gravitating object at the final stage of its classical evolution, we propose to use the Foldy–Wouthuysen representation of the Dirac equation in curved spacetimes, and the Gozzi classical path integral. In both approaches, maintaining the dynamics in the classical regime is controlled by supersymmetry.     

Berry curvature in graphene: a new approach

In the present paper we have directly computed the Berry curvature terms relevant for graphene in the presence of an inhomogeneous lattice distortion. We have employed the generalized Foldy–Wouthuysen framework, developed by some of us. We show that a non-constant lattice distortion leads to a valley–orbit coupling which is responsible for a valley–Hall effect. This is similar to the valley–Hall effect induced by an electric field proposed in the literature and is the analogue of the spin–Hall effect in semiconductors. Our general expressions for Berry curvature, for the special case of homogeneous distortion, reduce to the previously obtained results. We also discuss the Berry phase in the quantization of cyclotron motion.     

In Appreciation

Leslie Foldy’s diminutive stature and modest demeanor gave little clue to the powerful intellect responsible for several significant advances in theoretical physics.Two were particularly important. His 1945 theory of the multiple scattering of waves laid out the fundamentals that most modern theories have followed (and sometimes rediscovered), while his work with Siegfried Wouthuysen on the nonrelativistic limit of the Dirac equation opened the way to a wealth of valuable insights. In this article we recall some of the milestones along Foldy’s path through a life in physics. Some of the anecdotes we report here were related to one of the authors (PLT) just before an event in 2000 celebrating Foldy’s 80th birthday, while others were told to us over the course of the nearly forty years during which we were colleagues. Still others were uncovered during the course of WJF’s research for his book, Physics at a Research University: Case Western Reserve 1830–1990 (Cleveland: Case Western Reserve University, 2006). Other details were provided by Foldy’s widow, Roma. Philip L. Taylor is the Perkins Professor of Physics and Professor of Macromolecular Science and Engineering at Case Western Reserve University. William J. Fickinger is Professor Emeritus of Physics at Case Western Reserve University.     

Dirac particle in Riemann–Cartan spacetimes

Dynamics of a Dirac particle in general Riemann–Cartan spacetimes is considered. The Hermitian Dirac Hamiltonian is derived and is transformed to the Foldy–Wouthuysen representation for an arbitrary spacetime geometry. The contribution of the torsion field to the Foldy–Wouthuysen Hamiltonian is found. The new bounds on Cartan’s spacetime torsion are obtained.     

Magnetic properties and spin waves of bilayer magnets in a uniform field

The two-layer square lattice quantum antiferromagnet with spins 12 shows a zero-field magnetic order-disorder transition at a critical ratio of the inter-plane to intra-plane couplings. Adding a uniform magnetic field tunes the system to canted antiferromagnetism and eventually to a fully polarized state; similar behavior occurs for ferromagnetic intra-plane coupling. Based on a bond operator spin representation, we propose an approximate ground state wavefunction which consistently covers all phases by means of a unitary transformation. The excitations can be efficiently described as independent bosons; in the antiferromagnetic phase these reduce to the well-known spin waves, whereas they describe gapped spin-1 excitations in the singlet phase. We compute the spectra of these excitations as well as the magnetizations throughout the whole phase diagram. Received 23 April 2001     

Pair Production and Solutions of Wave Equation for Arbitrary-Spin Particles

We investigate the theory of particles with arbitrary spin and magnetic moment in the Lorentz representation (0, s) (s, 0) in an external constant and uniform electromagnetic field. We obtain the density matrix of free particles in pure spin states. The differential probability of pair producing particles with arbitrary spin by an external constant and uniform electromagnetic field is found using the exact solutions. We calculate the imaginary and real parts of the Lagrangian in an electromagnetic field that takes into account the vacuum polarization.     

Quantum-Mechanical Description of the Electromagnetic Interaction of Relativistic Particles with Electric and Magnetic Dipole Moments

The Hamiltonian of relativistic particles with electric and magnetic dipole moments that interact with an electromagnetic field is determined in the Foldy-Wouthuysen representation. Transition to the semiclassical approximation is carried out. The quantum-mechanical and semiclassical equations of spin motion are derived. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 9–12, August, 2005.     

Pair production and vacuum polarization of vector particles with electric dipole moments and anomalous magnetic moments

The matrix 8-component Dirac-like form of the P-odd equations for boson fields of spin 1 and 0 are obtained and the symmetry group of the equations is derived. We found exact solutions of the field equation for vector particles with arbitrary electric and magnetic moments in external constant and uniform electromagnetic fields. The differential probability of pair production of vector particles with electric dipole moments and anomalous magnetic moments by an external constant and uniform electromagnetic field has been found using exact solutions. We have calculated the imaginary and real parts of the electromagnetic field Lagrangian that takes into account the vacuum polarization of vector particles. Received: 14 April 2001 / Revised version: 13 July 2001 / Published online: 19 September 2001     

Modelling of Lévy walk kinetics of charged particles in edge electrostatic turbulence in tokamaks

We model and discuss the possible types of motion that charged particles may undergo in a stationary and spatially periodic electrostatic potential and a homogeneous magnetic field. The model is considered to be the simplest approximation of more complex phenomena of plasma edge turbulence in tokamaks. Therein, low frequency turbulence appears in the plasma edge, resulting in a fluctuation of the electron density, and also in the generation of a turbulent electrostatic field. Typical parameters of this turbulent electrostatic field are an electrical potential amplitude of 10–100 V and wave numbers k≈10³ m^-1. In our model, we consider these regimes, together with a homogeneous magnetic field with a magnitude of 1 T. We investigate the dynamics of singly-ionized carbon ions – a typical plasma impurity – with kinetic energies on the order of 10 eV. Besides the obvious Larmor and drift motions, a motion of random-walk and of Lévy walk character appear therein. All of these types of motion can play an important role in the modelling of the anomalous diffusion of particles from the plasma edge turbulence region. The dynamics mentioned will cause an inevitable escape of energetic particles and thus of power loss from the thermonuclear reactor. Moreover, Lévy walk kinetics represents a very interesting kind of kinetics, currently of great interest, which was previously not so often discussed.     

Quantum corrections of the Dzyaloshinskii-Moriya interaction on the spin-$\frac{1}{2}$ AF-Heisenberg chain in an uniform magnetic field

We have investigated the ground state phase diagram of the 1D AF spin- Heisenberg model with the staggered Dzyaloshinskii-Moriya (DM) interaction in an external uniform magnetic field H. We have used the exact diagonalization technique. In the absence of the uniform magnetic field (H=0), we have shown that the DM interaction induces a staggered chiral phase. The staggered chiral phase remains stable even in the presence of the uniform magnetic field. We have identified that the ground state phase diagram consists of four Luttinger liquid, staggered chiral, spin-flop, and ferromagnetic phases.