aa r X i v : . [ qu a n t - ph ] J u l Physical Decomposition of Photon Angular Momentum
Wei-Min Sun , ∗ School of Physics, Nanjing University, Nanjing 210093, China and Joint Center for Particle Nuclear Physics and Cosmology,Nanjing University and Purple Mountain Observatory, CAS, Nanjing 210093, China
We address the problem of angular momentum decomposition of a free photon. We propose anatural and physical scheme for separating the total angular momentum operator for a free photoninto an orbital part and a spin part with the hope that it solves this long-standing problem and itcould provide a natural basis for further study of the problem of angular momentum decompositionof a gauge field system.PACS numbers: 03.65.Ta, 14.70.Bh
The photon, as the quantum of the electromagneticfield, is a basic constituent particle of the StandardModel. Like any other microscopic particle, the photoncan possess a certain angular momentum. However, forthe photon, whether it is meaningful on physical groundto separate its total angular momentum into the contri-butions of an orbital part and a spin part is still a con-troversial issue. In standard textbooks of Quantum Elec-trodynamics (QED), for example, the book by Berestet-skii et al. [1], which represents the viewpoint of LandauSchool, it is explicitly stated that one could only talkabout the total angular momentum of a photon and theseparation of the photon angular momentum is of nophysical meaning. This viewpoint of Landau School isrecognized as true by the physics community for manyyears.In 2008, a paper by Chen, L¨u, Sun, Wang and Gold-man (hereafter referred to as Chen et al. ) [2] appearedin which the authors studied a problem related to this(and also more general than this), viz., that of separat-ing the total angular momentum of a gauge field system(both QED and QCD) into the contributions of orbitaland spin parts of the constituent matter particle (electronor quark) and gauge particle (photon or gluon) fields. Inthat paper, by introducing a decomposition of the gaugepotential into the so-called physical part and pure gaugepart, the authors obtained a separation of the total an-gular momentum operator of QED and QCD in whicheach individual part is gauge-invariant, and, as the au-thors claim, also satisfies the angular momentum algebra.Chen et al. ’s work evoked active responses of the commu-nity. After their work, some researchers further studiedthis approach. During these studies some different waysto perform the gauge potential decomposition were dis-covered and eventually people realized that there couldbe infinitely many different ways to perform such a de-composition. This implies that there are infinitely manydifferent ways to split the angular momentum of a QEDor QCD system in a gauge-invariant way! In this situ-ation, one needs to answer which separation scheme is ∗ [email protected] the physical one [3]. Besides, Chen et al. ’s assertion thatin their angular momentum decomposition scheme eachindividual part satisfies the angular momentum algebrahas also been questioned by some researchers. Waka-matsu pointed out that in the simplest case of a freeelectromagnetic field the photon spin operator as defined a la Chen et al. does not satisfy the angular momentumalgebra [4]. In fact, the very question of whether oneshould demand that in a physically meaningful angularmomentum decomposition scheme each part satisfy theangular momentum algebra is still controversial [5]. Uptill now, all these questions have no definite answers yet.Therefore, with regard to the problem of separation ofphoton angular momentum, we could say that a sepa-ration of the total angular momentum of a photon intoorbital and spin contributions is physically meaningful,but which separation scheme is the physically correct oneis still unclear. For a free photon, the physically correctseparation scheme should be unique. In this paper weshall address the problem of angular momentum decom-position of a free photon. Based on the physical char-acteristic of the motion of the photon, we shall proposesuch a physical separation scheme with the hope that itsolves this long-standing problem.In relativistic quantum mechanics of the photon, thestate of a free photon can be represented by its coordinaterepresentation wavefunction, which is just the vector po-tential of a free electromagnetic field ~A ( ~r, t ) in the trans-verse gauge ~ ∇ · ~A = 0. For a free photon, the Maxwellequation satisfied by the vector potential can be regardedas the ”Schr¨odinger eqution” of the photon: ∂ ~A∂t − △ ~A = 0 , ~ ∇ · ~A = 0 . (1)According to the standard definition of quantum theory,the total angular momentum operator is the generatorof spatial rotation of the system. Under a spatial rota-tion ~r ′ = R~r , the photon wavefunction transforms as avector wavefunction: ~A ′ ( ~r ) = R ~A ( R − ~r ). From this onecan deduce the angular momentum operator for a freephoton: ˆ ~J = ˆ ~L + ˆ ~S (2)with ˆ ~L = ˆ ~r × ˆ ~k being the usual orbital angular momen-tum operator and ˆ ~S = ( ˆ S x , ˆ S y , ˆ S z ) being the usual spinoperator for a spin-one particle:ˆ S x = − i i ˆ S y = i − i ˆ S z = − i i For a massive spin-one particle, ˆ ~L and ˆ ~S can be naturallyidentified as the orbital angular momentum operator andspin operator, respectively. However, for a photon, suchan identification fails because the actions of ˆ ~L and ˆ ~S donot preserve the transversality condition of the photonwavefunction [6] and hence they could not be regardedas operators acting on the physical state space of thephoton.Mathematically, a relativistic particle with definitemass and spin corresponds to a definite unitary represen-tation of the Poincare group. Here, we shall discuss theproblem of separation of photon angular momentum fromthe point of view of representation theory of the Poincaregroup. Since photon is a massless particle with spin one,according to representation theory of the Poincare group,a photon with definite momentum ~k has two differenthelicity states which we denote as | ~k, λ i with λ = ± ~J = ˆ ~L + ˆ ~S, (3)where the orbital angular momentum operator ˆ ~L and thespin operator ˆ ~S are both operators acting on the physicalstate space of the photon (here, for clarity of notation,we use the same notations as the above to denote theorbital angular momentum operator and spin operatorwe propose to define). Now, the spin operator ˆ ~S , whichdescribes the intrinsic angular momentum of the photon,should not depend on the specification of the origin ofthe coordinate system; hence it should commute with thegenerator of spatial translation, namely, the momentumoperator: [ ˆ S i , ˆ k j ] = 0.Now, we shall show that the three operators ˆ S x , ˆ S y ,ˆ S z , even though we call them spin operators, could notsatisfy the usual angular momentum algebra. The proofis as follows. Suppose one could define the three spinoperators ˆ S x , ˆ S y , ˆ S z in such a way that [ ˆ S i , ˆ S j ] = iε ijk ˆ S k is satisfied. Then, following the standard way of angular momentum theory, one could introduce the raising andlowering operators: ˆ S ± = ˆ S x ± i ˆ S y , which satisfy thefollowing set of commutation relations:[ ˆ S z , ˆ S + ] = ˆ S + (4)[ ˆ S z , ˆ S − ] = − ˆ S − (5)[ ˆ S + , ˆ S − ] = 2 ˆ S z . (6)The raising and lowering operators ˆ S ± also commute withthe momentum operator. Now, let us consider a photonstate with momentum ~k = (0 , , k ) along the z-directionand definite helicity λ . One then hasˆ J z | ~k, λ i = λ | ~k, λ i . (7)Now, according to the physical meaning of orbital angu-lar momentum, the component of orbital angular mo-mentum of the photon along its direction of motionshould vanish, thus ˆ L z | ~k, λ i = 0, and hence one hasˆ S z | ~k, λ i = λ | ~k, λ i . Next, let us first consider the stateˆ S + | ~k, λ = +1 i . We first note thatˆ ~k ˆ S + | ~k, λ = +1 i = ~k ˆ S + | ~k, λ = +1 i , (8)which shows that the state vector ˆ S + | ~k, λ = +1 i , if it isnonzero, also represents a one-photon state with momen-tum ~k . Then, from Eq. (4) one can deduceˆ S z ˆ S + | ~k, λ = +1 i = 2 ˆ S + | ~k, λ = +1 i . (9)Thus, the state vector ˆ S + | ~k, λ = +1 i should represent aone-photon state moving along the z-direction with he-licity two. However, in the physical state space of thephoton there is no such state, hence one necessarily hasˆ S + | ~k, λ = +1 i = 0. Then, let us consider the stateˆ S − | ~k, λ = +1 i . It can be easily seen that this state vec-tor also represents a one-photon state with momentum ~k . From Eq. (5) one can deduceˆ S z ˆ S − | ~k, λ = +1 i = 0 . (10)Now, we show that the state vector ˆ S − | ~k, λ = +1 i cannotbe zero. First, from Eq. (6) one can deduceˆ S + ˆ S − | ~k, λ = +1 i − ˆ S − ˆ S + | ~k, λ = +1 i = 2 | ~k, λ = +1 i . (11)One already has ˆ S + | ~k, λ = +1 i = 0, so, if one also hasˆ S − | ~k, λ = +1 i = 0, this will contradict Eq. (11). Thus,the state vector ˆ S − | ~k, λ = +1 i is nonzero and it repre-sents a one-photon state moving along the z-direction butwith helicity zero. But we all know that in the physicalstate space of the photon there is no such state! Clearly,the appearance of this contradictory conclusion is due toour presupposition that the three photon spin operatorsˆ S x , ˆ S y , ˆ S z satisfy the angular momentum algebra. There-fore, we conclude that in the physical state space of thephoton one cannot define a set of photon spin operatorssatisfying the angular momentum algebra.Then, how should one look upon the concept of photonspin? The usual definition of the spin of a particle as theangular momentum of this particle in its rest frame doesnot apply to the photon since a photon always moveswith the speed of light. For a photon there always existsa distinctive direction in space—the direction of its mo-mentum vector. In this case, the system does not havethe symmetry under the whole three-dimensional rota-tion group, but only the axial symmetry with respect tothat specific axis. Under the condition of axial symmetry,the conserved quantity is the helicity of the photon—thecomponent of the angular momentum along its directionof motion, which also coincides with the component ofthe spin angular momentum along the direction of mo-tion [1]. Thus, based on such a physical characteristicof the motion of the photon, we propose the followingphysical definition for the spin of a photon: a photonwith definite momentum ~k and helicity λ has a definitespin vector ~s = λ ~k | ~k | . Based on this definition, the actionof the photon spin operator (which we shall denote asˆ ~S phys ) on a photon state with definite momentum andhelicity can be expressed asˆ S phys,i | ~k, λ i = λ k i | ~k | | ~k, λ i , ( i = x, y, z ) . (12)The three components of the photon spin operator thusdefined commute with each other. This could be verifiedby direct calculation:[ ˆ S phys,i , ˆ S phys,j ] | ~k, λ i = k i k j | ~k | | ~k, λ i − k j k i | ~k | | ~k, λ i = 0 . (13)One can also verify that each component of ˆ ~S phys com-mutes with the momentum operator:[ ˆ S phys,i , ˆ k j ] | ~k, λ i = λ k i k j | ~k | | ~k, λ i − λ k j k i | ~k | | ~k, λ i = 0 , (14)which agrees with the usual concept of the spin angularmomentum of a particle.Now that we have given a physical definition for thephoton spin operator, we can give an explicit expressionfor this operator using the photon wavefunction as a con-crete realization of the photon state. The wavefunctionfor a photon with definite momentum ~k and helicity λ can be written as ~A ( λ ) ~k ( ~r ) = 1 √ ω ~e ( λ ) ~k e i~k · ~r , ω = | ~k | , (15)where ~e ( λ ) ~k denotes the polarization vector of the photonwith λ = +1 corresponding to the right-handed circu-lar polarization and λ = − ~e (+1) ~k = − √ ~e ξ + i~e η ) , ~e ( − ~k = 1 √ ~e ξ − i~e η ) , (16)where ( ~e ξ , ~e η , ~k | ~k | ) forms a right-handed orthogonaldreibein. Now, since one has ˆ ~S · ~k | ~k | ~e ( λ ) ~k = λ ~e ( λ ) ~k , oneobtains ˆ ~S · ~k | ~k | ~A ( λ ) ~k ( ~r ) = λ ~A ( λ ) ~k ( ~r ) (17)and ˆ ~J · ~k | ~k | ~A ( λ ) ~k ( ~r ) = ˆ ~L · ~k | ~k | ~A ( λ ) ~k ( ~r ) + ˆ ~S · ~k | ~k | ~A ( λ ) ~k ( ~r )= λ ~A ( λ ) ~k ( ~r ) . (18)Therefore, when one uses the coordinate representationwavefunction to represent a photon state, the photon spinoperator ˆ ~S phys has the following explicit expression:ˆ ~S phys = ˆ ~S · ˆ ~k | ˆ ~k | ˆ ~k | ˆ ~k | . (19)The orbital angular momentum operator of the photonis then naturally defined according to the separation ˆ ~J =ˆ ~L phys + ˆ ~S phys : ˆ ~L phys = ˆ ~L + ˆ ~S − ˆ ~S · ˆ ~k | ˆ ~k | ˆ ~k | ˆ ~k | . (20)It can be easily verified that ˆ ~L phys · ˆ ~k | ˆ ~k | = 0, which agreeswith the usual concept of orbital angular momentumof a particle. In his discussion on Einstein-Podolsky-Rosen-Bohm experiment with relativistic massive par-ticles, Czachor suggested that the orbital angular mo-mentum and spin operators defined via the relativisticcenter-of-mass operator should be the most physical def-inition of orbital angular momentum and spin for a rel-ativistic particle [7]. For a massless photon the orbitalangular momentum and spin operators suggested by Cza-chor reduce to the ones given in Eqs. (19) and (20). Wenote that the coordinate representation operators givenin Eqs. (19) and (20) were also obtained by Bliokh etal. [8] in the context of discussing angular momenta andspin-orbit interaction of nonparaxial light in free space.As discussed above, the three components of ˆ ~S phys commutes with each other: [ ˆ S phys,i , ˆ S phys,j ] = 0. Now,by construction both ˆ ~S phys and ˆ ~L phys are vector opera-tors, hence one has[ ˆ J i , ˆ S phys,j ] = iε ijk ˆ S phys,k [ ˆ J i , ˆ L phys,j ] = iε ijk ˆ L phys,k . (21)The above two relations together with [ ˆ S phys,i , ˆ S phys,j ] =0 imply[ ˆ L phys,i , ˆ S phys,j ] = iε ijk ˆ S phys,k , (22)[ ˆ L phys,i , ˆ L phys,j ] = iε ijk ( ˆ L phys,k − ˆ S phys,k ) . (23)Now, since the three operators ˆ S phys,i ( i = x, y, z ) com-mute with each other, they are not angular momentumoperators in the usual sense. Nevertheless, the compo-nent of ˆ ~S phys on the direction of motion of the photongenerates spin rotation of the polarization of the photon: e − iθ ˆ ~S phys · ~k | ~k | | ~k, λ i = e − iλθ | ~k, λ i . (24)The three operators ˆ L phys,i ( i = x, y, z ) also do not satisfythe angular momentum algebra. They even do not forma set of generators of some Lie algebra. Hence, they arealso not angular momentum operators in the usual sense.From Eq. (22) it is seen that different components ofˆ ~L phys and ˆ ~S phys do not commute. However, if one splitsthe component of the total angular momentum operatoralong an arbitrary direction ~n into the orbital part andspin part: ˆ ~J · ~n = ˆ ~L phys · ~n + ˆ ~S phys · ~n, (25)then it can be easily seen that the orbital part and the spin part commute:[ˆ ~L phys · ~n, ˆ ~S phys · ~n ] = 0 . (26)On the other hand, one has ˆ ~L phys · ~k | ~k | | ~k, λ i = 0, whichimplies e − iθ ˆ ~L phys · ~k | ~k | | ~k, λ i = | ~k, λ i . (27)Combining Eqs. (24) and (27), one has e − iθ ˆ ~J · ~k | ~k | | ~k, λ i = e − iθ ˆ ~L phys · ~k | ~k | e − iθ ˆ ~S phys · ~k | ~k | | ~k, λ i = e − iλθ | ~k, λ i , (28)which is in perfect accordance with the physical meaningof ˆ ~J · ~k | ~k | as the helicity operator of the photon.Thus, based on the physical characteristic of the mo-tion of the photon, we could obtain a natural and physicaldecomposition of the total angular momentum operatorfor a free photon into an orbital part and a spin part.We hope that this solves the long-standing problem ofangular momentum decomposition of a free photon andit could provide a natural basis for further study of theproblem of angular momentum decomposition of a gaugefield system. [1] V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii, Quan-tum Electrodynamics (Butterworth-Heinemann,1996), 2nded.[2] X.-S. Chen, X.-F. L¨u, W.-M. Sun, F. Wang and T. Gold-man, Phys. Rev. Lett. , 232002 (2008).[3] For a recent review of all these issues, see, E. Leader andC. Lorce, arXiv:1309.4235 [hep-ph].[4] For a discussion on this issue, see the recent review article:M. Wakamatsu, Int. J. Mod. Phys. A , 1430012 (2014);the discussion of this problem first appeared in S. J. VanEnk and G. Nienhuis, Europhys. Lett. , 497 (1994); J.Mod. Opt. , 963 (1994). [5] X. Ji, Phys. Rev. Lett. , 039101 (2010),arXiv:0810.4913 [hep-ph]; X.-S. Chen et al. ,arXiv:0812.4336 [hep-ph].[6] C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and Atoms: Introduction to Quantum Electrody-namics (Weinheim: Wiley-Vch, 2004) p.50.[7] See M. Czachor, Phys. Rev. A , 72 (1997) and referencestherein.[8] K.Y. Bliokh, M.A. Alonso, E.A. Ostrovskaya and A.Aiello, Phys. Rev. A82