Volume 104B, number 1 PHYSICS LETTERS 13 August 1981 ON THE LAGRANGIAN FORMULATION OF A CHARGED SPINNING PARTICLE IN AN EXTERNAL ELECTROMAGNETIC FIELD G. COGNOLA, L. VANZO and S. ZERBINI Dipartimento di Fisica della Libera UniversitY, Trento, Italy and R. SOLDATI 1,2 Laboratoire de Physique Th~orique et Hautes Energies, Orsay, France Received 4 May 1981 We propose a completely consistent lagrangian description for a spinning test particle with anomalous magnetic moment in the presence of an external field; the Bargmann-Michel-Telegdi equations are derived in the limit of the weak field. In a recent paper [ 1 ] the motion of a spinning par- ticle in an external field has been discussed in the gen- eral framework of the manifold of reference frames [2]. The Dixon-Souriau equations [3,4] were derived following the well-known method based on the conser- vation laws. In this note we shall discuss a lagrangian approach to the motion of a charged spinning particle in an external electromagnetic field. The equations of motion which we shall derive, will give the well-known Bargmann-Michel-Telegdi (BMT) [5] equations in the limit of the weak and homogeneous external field. The problem of the lagrangian formulation has a very long history [6-8] and recently supersymmetric approaches have been proposed also [9] ; however, it should be noted that the anomalous magnetic moment cannot be described in this context, without breaking supersymmetry. Our starting point is the following lagrangian * 1 1 v L = - ~m(fc u + X Suv ) (~cu + XvSUV) - eAjcU + ~(~#A~ A- t}A~p A) -- ½kF•uS uv , (1) 1 Permanent address: Dipartimento di Fisica della Libera UniversitY, Trento, Italia. 2 NATO Research Fellow. ,1 The dot "." means derivation with respect to a scalar pa- rameter 0. Through the paper we use c = 1. where Su v = ~ A (~-" uv) A cB . (2) The dynamical variables are the position of the particle xu(O), a real vector ~0 A belonging to a finite-dimen- sional representation of the Lorentz-group, whose in- finitesimal generators are the quantities (Zuv~, a vec- tor ffA belonging to the conjugate representation, and the lagrangian multipliers Xu, which will play an impor tant role in the following. The constant m represents the mass of the particle and k is given by k = eg/2m, (3) where g represents the gyromagnetic factor. If we choose, for example, the four-vector repre- sentation of the Lorentz group, then q0 A and ffA be- come the components of the four-vectors bU and au, respectively, and Suv = a u b v - avb u . (4) In this case the kinetic term assumes the form ½ (au[~u -•ub u). These "internal" variables have been used by Grassberger [7]. We note that the equations of motion for all the relevant physical quantities do not depend on the par- ticular representation used to describe the "internal" degrees of freedom. 0 031 9163/81/0000 0000/$ 02.50 © North-Holland Publishing Company 67