A. Bérard

Spin Hall effect and Berry phase of spinning particles

Abstract We consider the adiabatic evolution of the Dirac equation in order to compute its Berry curvature in momentum space. It is found that the position operator acquires an anomalous contribution due to the non-Abelian Berry gauge connection making the quantum mechanical algebra noncommutative. A generalization to any known spinning particles is possible by using the Bargmann–Wigner equation of motions. The noncommutativity of the coordinates is responsible for the topological spin transport of spinning particles similarly to the spin Hall effect in spintronic physics or the Magnus effect in optics. As an application we predict new dynamics for nonrelativistic particles in an electric field and for photons in a gravitational field.

monopole and Berry phase in momentum space in noncommutative quantum mechanics

To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter

Dirac monopole with Feynman brackets

\ensuremath{\theta}

Berry curvature in graphene: a new approach

to a field operator, which proves to be only momentum dependent. We find consequently that this field must be obligatorily a dual Dirac monopole in momentum space. Recent experiments in the context of the anomalous Hall effect provide evidence for a monopole in the crystal momentum space. We suggest a connection between the noncommutative field and the Berry curvature in momentum space which is at the origin of the anomalous Hall effect.

Magnetic monopole in the Feynman’s derivation of Maxwell equations

Abstract We introduce the magnetic angular momentum as a consequence of the structure of the sO(3) Lie algebra defined by the Feynman brackets. The Poincare momentum and Dirac magnetic monopole appear as a direct result of this framework.

Inverse problem and Bertrand’s theorem

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.

Lorentz-Covariant Hamiltonian Formalism

In 1992, Dyson published Feynman’s proof of the homogeneous Maxwell equations assuming only the Newton’s law of motion and the commutation relations between position and velocity for a nonrelativistic particle. Recently Tanimura gave a generalization of this proof in a relativistic context. Using the Hodge duality we extend his approach in order to derive the two groups of Maxwell equations with a magnetic monopole in flat and curved spaces.

Duality properties of Gorringe–Leach equations

Bertrand’s theorem is formulated as the solution of an inverse problem for classical one-dimensional motion. We show that the solutions of this problem, if suitably restricted, can be obtained by solving an elementary equation. This approach provides a compact and elegant proof of Bertrand’s theorem.

Electric and Magnetic Monopoles from a Lorentz- Covariant Hamiltonian

The dynamics of a classical system can be expressed by means of Poissonbrackets. In this paper we generalize the relation between the usual noncovariantHamiltonian and the Poisson brackets to a covariant Hamiltonian and new bracketsin the frame of Minkowski space. These brackets can be related to those usedby Feynman in his derivation Maxwells equations. The case of curved space isalso considered with the introduction of Christoffel symbols, covariant derivatives,and curvature tensors.

On Peres approach to Fradkin-Bacry-Ruegg-Souriau’s perihelion vector

In the category of motions preserving the angular momentum direction, Gorringe and Leach exhibited two classes of differential equations having elliptical orbits. After enlarging slightly these classes, we show that they are related by a duality correspondence of the Arnold–Vassiliev type. The specific associated conserved quantities (Laplace–Runge–Lenz vector and Fradkin–Jauch–Hill tensor) are then dual reflections of each other.