Levy-Leblond Equations Research Articles

Generalized Dirac and Klein-Gordon equations for spinor wavefunctions

A novel method is developed to derive the original Dirac equation and demonstrate that it is the only Poincare invariant dynamical equa- tion for 4-component spinor wavefunctions. New Poincare invariant generalized Dirac and Klein-Gordon equations are also derived. In the non-relativistic limit the generalized Dirac equation gives the general- ized Levy-Leblond equation and the generalized Pauli-Schrodinger equa- tion. The main difference between the original and generalized Dirac equations is that the former and latter are obtained with zero and non- zero phase functions, respectively. Otherwise, both equations describe free elementary particles with spin 1/2, which have all other physical properties the same except their masses. The fact that the generalized Dirac equation describes elementary particles with larger masses is used to suggest that non-zero phase functions may account for the existence of three families of elementary particles in the Standard Model. This suggestion significantly differs from those previously made to account for the three families of particle physics.

Hidden symmetries of Eisenhart-Duval lift metrics and the Dirac equation with flux

The Eisenhart-Duval lift allows embedding non-relativistic theories into a Lorentzian geometrical setting. In this paper we study the lift from the point of view of the Dirac equation and its hidden symmetries. We show that dimensional reduction of the Dirac equation for the Eisenhart-Duval metric in general gives rise to the non-relativistic Levy-Leblond equation in lower dimension. We study in detail in which specific cases the lower dimensional limit is given by the Dirac equation, with scalar and vector flux, and the relation between lift, reduction and the hidden symmetries of the Dirac equation. While there is a precise correspondence in the case of the lower dimensional massive Dirac equation with no flux, we find that for generic fluxes it is not possible to lift or reduce all solutions and hidden symmetries. As a by-product of this analysis we construct new Lorentzian metrics with special tensors by lifting Killing-Yano and Closed Conformal Killing-Yano tensors and describe the general Conformal Killing-Yano tensor of the Eisenhart-Duval lift metrics in terms of lower dimensional forms. Lastly, we show how dimensionally reducing the higher dimensional operators of the massless Dirac equation that are associated to shared hidden symmetries it is possible to recover hidden symmetry operators for the Dirac equation with flux.

Supersymmetry and the physical-phase-space formulation of spinning particles

We present an action for the relativistic spin- 1 2 particle, with non-linearly realised Lorentz invariance, in terms of physical phase-space variables only. It is the relativistic counterpart of a standard form of the non-relativistic spin- 1 2 particle action. On quantisation of the non-relativistic particle, galilean supersymmetry is crucial to the recovery of the Levy-Leblond equation as the square root of the Schrödinger equation. The Dirac equation is not obtained in a similar way from quantisation of the relativistic particle because supersymmetry implies positive energy. Instead, the pair of Schrödinger equations for positive and negative energy yield the Foldy-Wouthuysen transform of the Dirac equation.

Perturbation theory of relativistic corrections

After a discussion of the problems associated with the non-relativistic limit of the Dirac equation and of the expansion of the exact eigenvalues and eigenfunctions of the H atom in powers ofc−2 the traditional approaches for a perturbation theory of relativistic effects are critically reviewed. Then a direct perturbation theory is presented, that is characterized by a change of the metric in 4-component spinor space such that the Levy-Leblond equation appears as the straightforward non-relativistic limit of the Dirac equation. The various orders in perturbation theory of the energy and the wave function are derived first in a direct way, then in a resolvent formalism. The formulas are very compact and easily generalizeable to arbitrary order. All integrals that arise to any order exist, and no controlled cancellation of divergent terms (as in other approaches) is necessary. In the same philosophy an iterative approach towards the solution of the Dirac equation is derived, in which the solution of the Schrodinger equation is the first iteration step.

Pseudoclassical model of a particle with arbitrary spin.

A pseudoclassical model for a free spinning particle with arbitrary spin is presented. It is a generalization of a previously studied model for spin one-half. Quantization of the Poincar\'e-invariant version of the model yields the Bargmann-Wigner equations while a tower of Levy-Leblond equations emerge from the Galilean-invariant version. Classicalization yields an invariant world line and the expected multipole structure in both cases.

Pseudoclassical description for a nonrelativistic spinning particle. I. The Levy-Leblond equation.

A pseudoclassical model for a spinning nonrelativistic particle is presented. The model contains two first-class constraints which after quantization give rise to the Levy-Leblond equation for a spin-1/2 particle.

Galilean relativistic wave equations

The locally operating realisations of the Galilei group are studied. Levy-Leblond's equation for spin-one-half Galilei relativistic massive systems is rederived by making use of group theoretical methods and a generalisation to spin one is proposed. Finally, the relation to the corresponding Poincare relativistic equations is analysed.