We compare three attempts that have been made to decompose the angular
momentum of the electromagnetic field into components of an "orbital" and
"spin" nature. All three expressions are different and it appears, on the basis
of classical electrodynamics, that there is no preferred way of decomposing the
angular momentum of the electromagnetic field into orbital and spin components,
even in an inertial frame.Comment: Some clarifications. 7 pages pdf. Earlier version published in
  International Journal of Optics:
  http://www.hindawi.com/journals/ijo/aip/728350

Stewart, A. M.

English

arXiv

We compare three attempts that have been made to decompose the angular momentum of the electromagnetic field into components of an “orbital” and “spin” nature. All three expressions are different, and there seems to be no reason to prefer one to another. It appears, on the basis of classical electrodynamics, that there is no unique way of decomposing the angular momentum of the electromagnetic field into orbital and spin components, even in a fixed inertial frame.</jats:p

A. M. Stewart

Crossref

Hindawi Publishing CorporationInternational Journal of OpticsVolume 2011, Article ID 728350, 4 pagesdoi:10.1155/2011/728350Research ArticleComparison of Different Forms for the “Spin” and “Orbital”Components of the Angular Momentum of LightA.M. StewartEmeritus Faculty, The Australian National University, Canberra, ACT 0200, AustraliaCorrespondence should be addressed to A. M. Stewart, parbuckle.stewart@gmail.comReceived 5 September 2010; Accepted 3 March 2011Academic Editor: Boris Y. ZeldovichCopyright © 2011 A. M. Stewart. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.We compare three attempts that have been made to decompose the angular momentum of the electromagnetic field intocomponents of an “orbital” and “spin” nature. All three expressions are diﬀerent, and there seems to be no reason to prefer one toanother. It appears, on the basis of classical electrodynamics, that there is no unique way of decomposing the angular momentumof the electromagnetic field into orbital and spin components, even in a fixed inertial frame.1. IntroductionThe total angular momentum J of the electromagnetic fieldis given (in SI units) [1] byJ(t) = ε0∫d3xx × [E(x, t)× B(x, t)]. (1)Henceforth, we will suppress the time coordinate t of thefields, all of which depend on time, and also the ε0 factor.There has been debate for a long time over whether thetotal angular moment J of the electromagnetic field can bedecomposed into an orbital part L and a spin part S so thatJ = L + S. (2)Some authors [2] argue that on the basis of the first principlesit is not possible to do this; others [3–5] show that forms canbe demonstrated that appear to be, at least algebraically, of aspin and orbital nature.By means of partial integration Ohanian [4] eﬀected adecomposition with J = L′ + S′ + J′b,S′ =∫d3xE(x)×A(x), (3)L′ =∑i∫d3xEi(x)(x×∇x)Ai(x), (4)where A is the vector potential {B(x) = ∇x × A(x)} and∇x is the gradient operator that operates on functions ofx. Ohanian assumed that the electric charge density ρ waszero and deemed (3) and (4) to be the spin and orbitalcomponents, respectively, of the electromagnetic field on thebasis that the integrand of (3) was not explicitly linear inthe x coordinate whereas the integrand of (4) was. When thecharge density is not zero, a bound term J′b, considered alsoto be of an orbital natureJ′b =∫d3xx × A(x)∇x · E(x) =∫d3xx× A(x)ρ(x), (5)is obtained on whose form all writers agree [6]. Althoughthe sum of (3) and (4) and (5) J = L′ + S′ + J′b is gaugeinvariant, the individual terms are not and so have nophysical interpretation until the gauge of the vector potentialis fixed completely.Cohen-Tannoudji et al. [3] used the Coulomb (ortransverse) gauge, defined by the gauge condition∇·At = 0,which givesS′′ =∫d3xE(x)× At(x), (6)L′′ =∑i∫d3xEi(x)(x ×∇x)Ait(x), (7)J′′b =∫d3xρ(x)x ×At(x). (8)2 International Journal of OpticsSince the Coulomb gauge is a completely fixed gauge andhas no remaining gauge arbitrariness, (6), (7), and (8) areindividually well defined with J = L′′ + S′′ + J′′b .Cohen-Tannoudji et al. [3] went further and expandedthe electric field E into transverse Et = −A˙t = −∂At/∂t andlongitudinal El = −∇φ parts, where φ is the scalar potentialwith E = Et + El, to getS′′′ =∫d3xEt(x)×At(x) = −∫d3xA˙t(x)× At(x), (9)L′′′ =∑i∫d3xEit(x)(x ×∇x)Ait(x)= −∑i∫d3xA˙it(x)(x ×∇x)Ait(x).(10)The bound component remains the same as (8). It willbe shown in Section 2 that the terms that involve the scalarpotential in (6) and (7) cancel so that J = L′′ +S′′ + J′′b in (6),(7), and (8) and J = L′′′+S′′′+J′′b in (9), (10), and (8) but S′′in (6) diﬀers from S′′′ in (9) and L′′ in (7) diﬀers from L′′′ in(10). The forms of (9) and (10) have also been used by vanEnk and Nienhuis [5]. The general explicit form for At , givenin (13), was not specified by these writers.On the other hand, Stewart [7] found a decompositionJ = L′′′′ + S′′′′ + J′′b from decomposing the electric field bythe Helmholtz theorem [8]:S′′′′ = 14π∫d3x∫d3 yB(x)× B˙(y)∣∣x− y∣∣ , (11)L′′′′ = 14π∫d3x∫d3 yB(x) · B˙(y) x× y∣∣x− y∣∣3 . (12)This decomposition uses the E and B fields throughout so noissues of gauge arbitrariness arise. Nor is there any complica-tion that arises from using the longitudinal and transversecomponents of the fields, as B is entirely transverse. Theexpression for J′′b was again given by (8) [6]. Equations (8),(11), and (12) have been used elsewhere [9] to show thatthe electromagnetic field makes zero contribution to theangular momentum of the physical electron described bythe Lagrangian of quantum electrodynamics and to resolvethe paradox concerning the angular momentum of a planeelectromagnetic wave [10]. Application has also been madeto paraxial rays [11] where it has been shown that the sameresults are obtained from (11) and (12) as from (9) and (10).The question addressed in this paper is how (11) and (12)are related to (9) and (10) when the relation below (13) [12,13] that expresses the vector potential of the Coulomb gaugeexplicitly in terms of the instantaneous magnetic field is used:At(x) = ∇x ×∫d3 yB(y)4π∣∣x− y∣∣= −∫d3 y4πB(y)×∇x 1∣∣x− y∣∣ .(13)In Section 2 we summarise the derivation [3] leading from(1) to (9) and (10). In Section 3 we obtain an expression forS′′′ of (9) and L′′′ of (10) using (13) and demonstrate thatL′′′ + S′′′ = L′′′′+S′′′′. Section 4 summarises the conclusionof the paper that there is, on the basis of classical physics,no unique decomposition of the angular momentum of theelectromagnetic field into spin and orbital components, evenin a fixed inertial frame.2. Derivation of the Standard Version ofthe AngularMomentumDecompositionTo obtain (3) and (4) from (1) we consider the componentof the cross productE× (∇x ×A)|i = εi jkεklmE j ∂Am∂xl. (14)By multiplying the Levi-Civita symbols we find [4]E× (∇x × A) = En∇xAn − (E · ∇x)A. (15)Next, consider the identity∂∂xm(x jEmAk)= δj,mEmAk + x j(Ak∂Em∂xm+ Em∂Ak∂xm)(16)and integrate this over xm from minus infinity to plusinfinity. The left-hand side vanishes because the integrand iszero at those points. Integrate over the other two componentsof x to produce a volume integral over all space of d3xthen sum over m. The result, when multiplied by εi jk andintegrated, substituted into the angular momentum obtainedfrom (15), and specialised to the Coulomb gauge, isJ =∫d3x{x× At(∇x · E) + E× At + En(x×∇x)Ant}.(17)This reproduces (3), (4), and (5) for general gauge and (6),(7), and (8) for the Coulomb gauge.The next step is to decompose the electric field into itslongitudinal −∇φ and transverse Et components. Since thedivergence of a transverse vector field is zero, the first termin (17) is unchanged from (8). The term associated with thepotential in the third term of (17) isΔJ3|i = −∑j,k,n∫d3x εi jk∂φ∂xnx j∂Ant∂xk. (18)Consider the identity∂∂xk(x jAnt∂φ∂xn)= δj,kAnt∂φ∂xn+ x j(∂Ant∂xk∂φ∂xn+ Ant∂2φ∂xk∂xn).(19)When (19) is put into (18), the term coming from the firstterm of the right-hand side of (19) vanishes because of theproduct of the Kronecker delta and the Levi Civita functions,and we getΔJ3|i =∑j,k,n∫d3x εi jkx jAnt∂2φ∂xk∂xn. (20)International Journal of Optics 3From the identity∂∂xn(x jAnt∂φ∂xk)= δj,nAnt∂φ∂xk+ x j(∂Ant∂xn∂φ∂xk+ Ant∂2φ∂xk∂xn)(21)and noting that∇ ·At = 0, we findΔJ3|i = −∑j,k,n∫d3x εi jkδ j,nAnt∂φ∂xk= −∑j,k∫d3x εi jkAjt∂φ∂xk.(22)The angular momentum of the second term of (17) arisingfrom the potential isΔJ2|i = −∑j,k∫d3x εi jk∂φ∂x jAkt . (23)Noting the order of the indices, (22) and (23) sum to zero;so we find that the contribution of the longitudinal electricfield to the total angular momentum is zero, so J = L′′′ +S′′′ + J′′b with the components L′′′ and S′′′ given by (9)and (10). We see already that the angular momentum ofthe electromagnetic field can be expressed in two diﬀerentways, as J = L′′′ + S′′′ + J′′b given by (6), (7), and (8) and asJ = L′′ + S′′ + J′′b given by (9), (10), and (8).3. Confirmation of the NewVersion ofthe AngularMomentumDecompositionIn this section of the paper we substitute the expression forthe vector potential of the Coulomb gauge (13) into (9) and(10) in order to find how (9) and (10) are related to (11) and(12). We will find that the sum of (9) and (10) is equal to thesum of (11) and (12). First we examine L′′′ of (10). This maybe expressed asL′′′ = −∫d3xx×∇x[A˙t(z) · At(x)]∣∣∣z=x= −∫d3xx×{A˙t(x)× B(x) +[A˙t(x) · ∇x]At(x)}.(24)Using (13), the first term of (24) comes toL′′′1 = −∫d3x∫d3 y4πx×{B(x)×[B˙(y)×∇x 1∣∣x − y∣∣]},(25)and, by multiplying out the triple vector product, thisbecomesL′′′1 = −∫d3x∫d3 y4πx×{B˙(y)[B(x) · ∇x 1∣∣x − y∣∣]−∇x 1∣∣x− y∣∣[B˙(y) · B(x)]}.(26)When the gradient in the second term of (26), L′′′12, isexpressed explicitly, the term becomesL′′′12 = 14π∫d3x∫d3 y B˙(y) · B(x) x× y∣∣x− y∣∣3 . (27)The first term in (26) is expressed in components asL′′′11|i = −∑j,k,m∫d3x∫d3 y4πεi jkx j B˙k(y)Bm(x)∂∂xm1∣∣x− y∣∣ .(28)By considering the identity∂∂xm[B˙k(y)x jBm(x)∣∣x− y∣∣]= B˙k(y){δj,mBm(x)∣∣x− y∣∣+x j[Bm(x)∂∂xm1∣∣x− y∣∣ +1∣∣x− y∣∣∂Bm(x)∂xm]},(29)we obtain, with∇ · B = 0,L′′′11|i =∑j,k,m∫d3x∫d3y4πεi jkB˙k(y)δjmBm(x)∣∣x − y∣∣ (30)orL′′′11 = 14π∫d3x∫d3 yB(x)× B˙(y)∣∣x− y∣∣ . (31)The second term L′′′2 of (24) is simplified by writing it incomponents and using the relation∂∂xm(x j A˙mt Akt)= δjmA˙mt Akt + x j(∂A˙mt∂xmAkt + A˙mt∂Akt∂xm)(32)with the gauge condition∇ · At = 0 to getL′′′2 =∫d3xA˙t(x)× At(x). (33)Accordingly, L′′′ = L′′′11 + L′′′12 + L′′′2 andL′′′ = 14π∫d3x∫d3 yB(x)× B˙(y)∣∣x − y∣∣+14π∫d3x∫d3yB˙(y) · B(x) x × y∣∣x − y∣∣3+∫d3xA˙t(x)× At(x).(34)Since S′′′ of (9) is given by the negative of the last term of(34), it follows that L′′′ + S′′′ = L′′′ + S′′′, confirming thevalidity of (11) and (12).4 International Journal of Optics4. ConclusionIn this paper we have compared three diﬀerent expressionsfor the angular momentum of the electromagnetic field; theyareJ =∫d3x{x × At ρ(x) + ε0E× At + ε0En(x ×∇x)Ant},(35)J =∫d3x{x × At ρ(x) + ε0Et × At + ε0Ent (x ×∇x)Ant},(36)J =∫d3xx × At(x)ρ(x) + ε04π∫d3x∫d3 yB(x)× B˙(y)∣∣x− y∣∣+ε04π∫d3x∫d3yB(x) · B˙(y) x× y∣∣x− y∣∣3 ,(37)where the vector potential of the Coulomb gauge At is givenby (13) and the ε0 factor has been restored. The expressionsall have the algebraic form of the sum of a spin and an orbitalcomponents, and each component of all three expressionsis gauge invariant. All three expressions are diﬀerent, and,on the basis of classical electrodynamics, there seems to beno reason to prefer one to another. The last form (37) [7]has the pedagogical advantage of being derived entirely interms of fields through the Helmholtz theorem; issues ofgauge arbitrariness do not have to be addressed. It appearsthat there is no unique way of decomposing the angularmomentum of the classical electromagnetic field into orbitaland spin components, even in an inertial frame.References[1] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons,New York, NY, USA, 3rd edition, 1999.[2] W. Heitler, The Quantum Theory of Radiation, ClarendonPress, Oxford, UK, 3rd edition, 1954.[3] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Gilbert, Photonsand Atoms, John Wiley & Sons, New York, NY, USA, 1989.[4] H. C. Ohanian, “What is spin?” American Journal of Physics,vol. 54, no. 6, pp. 500–505, 1986.[5] S. J. van Enk andG. Nienhuis, “Commutation rules and eigen-values of spin and orbital angular momentum of radiationfields,” Journal of Modern Optics, vol. 41, no. 5, pp. 963–977,1994.[6] A. M. Stewart, “Equivalence of two mathematical forms forthe bound angular momentum of the electromagnetic field,”Journal of Modern Optics, vol. 52, no. 18, pp. 2695–2698, 2005.[7] A. M. Stewart, “Angular momentum of light,” Journal ofModern Optics, vol. 52, no. 8, pp. 1145–1154, 2005.[8] A. M. Stewart, “Longitudinal and transversecomponents of avector field,” Classical Physics. In press, http://arxiv.org/abs/0801.0335v2.[9] A. M. Stewart, “Angular momentum of the physical electron,”Canadian Journal of Physics, vol. 87, no. 2, pp. 151–152, 2009.[10] A. M. Stewart, “Angular momentum of the electromagneticfield: the plane wave paradox resolved,” European Journal ofPhysics, vol. 26, no. 4, pp. 635–641, 2005.[11] A. M. Stewart, “Derivation of the paraxial form of the angularmomentum of the electromagnetic field from the generalform,” Journal of Modern Optics, vol. 53, no. 13, pp. 1947–1952, 2006.[12] A. M. Stewart, “Vector potential of the Coulomb gauge,”European Journal of Physics, vol. 24, no. 5, pp. 519–524, 2003.[13] A. M. Stewart, “Reply to Comments on ’Vector potential ofthe Coulomb gauge’,” European Journal of Physics, vol. 25, no.2, pp. L29–L30, 2004.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014High Energy PhysicsAdvances inThe Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014FluidsJournal of Atomic and  Molecular PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Advances in  Condensed Matter PhysicsOpticsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstronomyAdvances inInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014SuperconductivityHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Statistical MechanicsInternational Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014GravityJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AstrophysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Physics Research InternationalHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Solid State PhysicsJournal of Computational  Methods in PhysicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Soft MatterJournal ofHindawi Publishing Corporationhttp://www.hindawi.comAerodynamicsJournal ofVolume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014PhotonicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofBiophysicsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014ThermodynamicsJournal of

International Journal of Optics

Comparison of Different Forms for the “Spin” and “Orbital” Components of the Angular Momentum of Light

We compare three attempts that have been made to decompose the angular momentum of the electromagnetic field into components of an “orbital” and “spin” nature. All three expressions are different, and there seems to be no reason to prefer one to another. It appears, on the basis of classical electrodynamics, that there is no unique way of decomposing the angular momentum of the electromagnetic field into orbital and spin components, even in a fixed inertial frame

Comparison of different forms for the "spin" and "orbital" components of the angular momentum of light

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