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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 912 (2016) 450–462

www.elsevier.com/locate/nuclphysb

On the Bargmann–Michel–Telegdi equations, and

spin–orbit coupling: A tribute to Raymond Stora

Christian Duval

Centre de Physique Théorique, Aix Marseille Université & Université de Toulon & CNRS UMR 7332, Case 907,

Luminy Campus, 13288 Marseille, Cedex 09, France

Received 17

March 2016; accepted 19 April 2016

Available

online 1 June 2016

Editor: Hubert

Saleur

Abstract

The

Bargmann–Michel–Telegdi equations describing the motions of a spinning, charged, relativistic par-

ticle

endowed with an anomalous magnetic moment in an electromagnetic ﬁeld, are reconsidered. They are

shown to duly stem from the linearization of the characteristic distribution of a presymplectic structure re-

ﬁning

the original one of Souriau. In this model, once specialized to the case of a static electric-like ﬁeld, the

angular momentum and energy given by the associated moment map now correctly restore the spin–orbit

coupling term. This is the state-of-the-art of unﬁnished joint work with Raymond Stora.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Forewords

Over a decade ago, I have been informed by Serge Lazzarini that Raymond Stora and Valen-

tine Telegdi were discussing, at CERN, an issue related to the expression of the spin–orbit

coupling term in Souriau’s classical presymplectic model of spinning particles leading to the

Bargmann–Michel–Telegdi (BMT) equations. Stora chose that model owing of the lack of a

proper Lagrangian formalism for these equations. A number of stimulating e

xchanges with him

then convinced me that the Souriau model (recalled in Section 3) should be somehow revisited so

as to yield again the “robust” BMT equations as well as the correct expression of the spin–orbit

E-mail address: duval@cpt.univ-mrs.fr.

URL: http://www

.cpt.univ-mrs.fr.

http://dx.doi.org/10.1016/j.nuclphysb.2016.04.031

The Author. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

C. Duval / Nuclear Physics B 912 (2016) 450–462 451

term, effectively model-dependent in the considered classical framework, and which was missing

in the above-mentioned model. In his carefully hand-written notes, Stora put forward an inge-

nious Ansatz, presented in Section 4, which proved quite useful to meet the above-mentioned

requirements. We discussed the merits and demerits of this new model which features tw

apriori independent, phenomenological parameters in the deﬁnition (4.1) of the presymplec-

tic (Lagrange) two-form σ

. This ﬂexibility ﬁnally enabled us to determine these adjustable

parameters, hence the sought after model that would guarantee the BMT equations (in the weak

ﬁeld limit) with gyromagnetic ratio g, and also provide the standard spin–orbit coefﬁcient, pro-

portional to g − 1, usually deduced from the Dirac–Pauli equation in the quantum mechanical

frame

work. This new construction is patently phenomenological, as is Souriau’s one yielding a

coefﬁcient g, instead of g −1. Now, the obtained ﬁxing (5.21) of the coefﬁcients of our model

should have preferably resulted from another, fully “predictive” model. This difﬁculty should be

imputed to the fundamental difference between the ph

ysics ruled by quantum mechanics and that

described by (semi-)classical models, even by those with a sound geometric basis. The arduous

quest of this conjectured predictive model has, since then, been placed on standby. After Ray-

mond’s passing, I found it fair and useful to make accessible to our community one of his yet

unanswered queries and to witness his great scientiﬁc insights.

2. The BMT equations

Consider a relativistic particle with mass m, spin s, electric charge q, and gyromagnetic ratio

g, under the inﬂuence of an external and constant electromagnetic ﬁeld F in Minkowski space–

time R

3,1

, with metric gof signature −3 +1.

Let P and S denote respecti

vely the linear momentum and spin vectors of our particle at

space–time location X. The celebrated Bargmann–Michel–Telegdi equations governing the clas-

sical motions of the particle with electric charge, q, and gyromagnetic ratio, g, read [1]

⎧

⎪

⎨

⎪

⎩

dτ

= P

dτ

=−qFP

dτ

=−q



FS +



1 −



PFS)



(2.1)

where τ is proportional to proper time.

These equations describe the motions of the particle in a weak electromagnetic ﬁeld. Let us

emphasize that the spin precession featured by the third equation in (2.1) is used by experimental-

ists to measure g − 2 with a very high accuracy; see, e.g., [5] for a survey of the ﬁeld-theoretical

computations and experimental data concerning the muon gyromagnetic ratio, g

. We recall that

the BMT equations have ﬁrst been deduced from a semi-classical approximation of the Dirac–

Pauli equation in [8].

The bar in the notation P = g(P, · ) denotes g-transposition, and P

= PP is used as a shorthand. The two-forms, F ,

are often traded as ﬁelds of skewsymmetric endomorphisms of space–time, F =−F , via F(P, S) ≡ PFS. Space–time

is oriented, and time-orientation is also assumed. We will put c =1 throughout the paper.

452 C. Duval / Nuclear Physics B 912 (2016) 450–462

3. Spinning particles in an electromagnetic ﬁeld: Souriau’s model

The system (2.1) was reconsidered by Jean-Marie Souriau (JMS) in the framework of presym-

plectic mechanics associated, in the relativistic framework, with the classiﬁcation of coadjoint

orbits of the (connected component of the) Poincaré group, SE(3, 1), i.e., of elementary relativis-

tic classical systems [9].

3.1. The free case

It has been shown that the BMT equations can be given by the kernel of a certain presymplec-

tic two-form on the nine-dimensional evolution space

V ={(X,I,J)∈(R

3,1

)



=−J

= 1, IJ = 0,I future-pointing} (3.1)

which is diffeomorphic to SE

(3, 1)/ SO(2).

Before dwelling on the coupling to an e

xternal electromagnetic ﬁeld, let us ﬁrst recall how to

describe the motions of a free relativistic particle endowed with mass, m, and spin, s. Start with

the closed two-form on V given by [9]

free

JMS



dI ∧dI − dJ ∧ dJ



−mdI ∧dX (3.2)

where

 = j(I,J) (3.3)

is the normalized spin tensor, i.e., the so(3, 1)-valued cross-product of I and J .

The un-

parametrized equations of free motion, namely δ(X, I, J) = (λI, 0, 0), with λ ∈ R, are given

by the null distribution of (3.2). As to the manifold V/ ker(σ

free

JMS

) of all the solutions of these

equations, it is actually symplectomorphic to the eight-dimensional SE

(3, 1)-coadjoint orbit

characterized by the invariants m and s.

3.2. Electromagnetic coupling

In order to switch on non-minimal electromagnetic coupling of the system to an otherwise

arbitrary external electromagnetic ﬁeld (EMF), F , Souriau has proposed to replace (3.2) by the

presymplectic (Lagrange) two-form [9,10]

JMS



dI ∧dI − dJ ∧ dJ



−d(MI)∧dX +

qdX ∧ FdX (3.4)

where the dressed mass may

be given by

M = m −

α (3.6)

and where

One has (I, J) =(I J − J I), where “” stands for the Hodge-star.

Another proposal [2], coming from a dequantization of the Dirac–Pauli equation, is

= m

−

gqs

α, (3.5)

and coincides with (3.6) up to a higher-order term in the ﬁeld strength.

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