gauge invariance and canonical quantization applied in the study of internal structure of gauge...

Gauge invariance and canonical quantization applied in the study ofinternal structure of gauge field systems

Fan Wanga, Xiang-Song Chenb, Xiao-Fu Lub, Wei-Ming Suna and T. Goldmanc

aDepartment of Physics, Nanjing University and J-CPNPC (Joint Center for ParticleNuclear Physics and Cosmology, Nanjing University and Purple Mountain Observatory,Chinese Academy of Sciences) Nanjing, 210093, P.R. China

bDepartment of Physics, Sichuan University, Chendu, 610064, P.R. China

cTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

It is unavoidable to deal with the quark and gluon momentum and angular momen-tum contributions to the nucleon momentum and spin in the study of nucleon internalstructure. However, we never have the quark and gluon momentum, orbital angularmomentum and gluon spin operators which satisfy both the gauge invariance and thecanonical momentum and angular momentum commutation relations. The conflicts be-tween the gauge invariance and canonical quantization requirement of these operators arediscussed. A new set of quark and gluon momentum, orbital angular momentum andspin operators, which satisfy both the gauge invariance and canonical momentum andangular momentum commutation relations, are proposed. The key point to achieve sucha proper decomposition is to separate the gauge field into the pure gauge and the gaugecovariant parts. The same conflicts also exist in QED and quantum mechanics and havebeen solved in the same manner. The impacts of this new decomposition to the nucleoninternal structure are discussed.

1. INTRODUCTION

In quantum physics, any observable is expressed as a Hermitian operator in Hilbertspace. The fundamental operators, such as momentum, orbital angular momentum, spin,satisfy the canonical momentum and angular momentum commutation relations. Thesecommutation relations or Lie algebras define the properties of these operators.Gauge invariance has been recognized as the first principle through the development of

the standard model. In classical gauge field theory, gauge invariance principle requires anyobservable must be expressed in terms of gauge invariant variable. In quantum gauge fieldtheory, in general one only requires the matrix elements of an operator in between physicalstates to be gauge invariant. However one usually requires the operators themselves tobe gauge invariant. This is called the strong gauge invariance in [1]. We will restrict ourdiscussion in strong gauge invariance in this paper and leave the other possibility to thefuture study [1,2].

Nuclear Physics A 844 (2010) 85c–94c

0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/nuclphysa

doi:10.1016/j.nuclphysa.2010.05.019

In the study of nucleon (atom) internal structure, it is unavoidable to study the quark,gluon (electron, photon) momentum, orbital angular momentum and spin contributions tothe nucleon (atom) momentum and spin. However, we never have the quark, gluon (elec-tron, photon) momentum, orbital angular momentum and spin operators which satisfyboth the gauge invariance and canonical momentum and angular momentum commutationrelations except the quark (electron) spin. Even it has been claimed in some textbooksthat one can not define separately the photon spin and orbital angular momentum opera-tors [3] and almost a proper gluon spin operator search has been given up in the nucleonspin structure study for the last ten years. This situation has left puzzles in quantummechanics, quantum electrodynamics (QED) and quantum chromodynamics (QCD). Forexample, the expectation value of the Hamiltonian of hydrogen atom is gauge dependentunder a time dependent gauge transformation [4], the meaning of the multipole radia-tion analysis from atom to hadron spectroscopy would be obscure if the photon spin andorbital angular momentum operators were not well defined especially the parity of thesemicroscopic states determined from the multipole radiation analysis would be obscure,there will be no way to compare the measured gluon spin contribution to nucleon spinwith the theoretically calculated one if one has not a proper gluon spin operator, etc.In section II the conflict between gauge invariance and canonical quantization of the

usual quark gluon (electron photon) momentum, orbital angular momentum and spinoperators are discussed from the simple quantum mechanics of a charged particle movingin an electromagnetic field to those of quark and gluon in QCD. In the third section anew set of momentum, orbital angular momentum and spin operators, which satisfy boththe gauge invariance and canonical momentum and angular momentum commutationrelations, are given. The key point to achieve this is to separate the gauge field into puregauge and gauge covariant (invariant) parts. The possible impacts of these modificationto the nucleon internal structure will be discussed in section IV. the last section is asummary and a prospect of further studies.

2. GAUGE INVARIANCE AND CANONICAL QUANTIZATION OF MO-MENTUMAND ANGULAR MOMENTUM OPERATORS OF FERMIONAND GAUGE FIELD PARTS

The conflict between gauge invariance and canonical quantization of the momentum andorbital angular momentum operators of a charged particle moving in the electromagneticfield, a U(1) Abelian gauge field, has existed in quantum mechanics since the establishmentof gauge invariance principle. Starting from the Lagrangian of a non-relativistic chargedparticle with mass m, velocity �v and charge e moving in an electromagnetic field Aμ =(A0, �A),

L(m,�v, e, Aμ) =1

2m(m�v)2 − e(A0 − �v · �A), (1)

one obtains the canonical momentum �p, the orbital angular momentum �L and the Hamil-tonian H

�p = m�v + e �A, �L = �r × �p, H =1

2m(�p− e �A)2 + eA0. (2)

F. Wang et al. / Nuclear Physics A 844 (2010) 85c–94c86c

All of these three classical dynamical variables are gauge dependent. In the coordinate

representation, the momentum operator �p is quantized as �p =�∇i, no matter what kind

gauge is fixed on even though the classical canonical momentum operator is gauge de-pendent. The orbital angular momentum and Hamiltonian operators are quantized by

replacing �p by�∇i. These quantized momentum and angular momentum operators satisfy

the canonical commutation relation or the Lie algebra. In general, the [pi, H] �= 0, whichis different from the Poincare algebra of the total momentum Pi, (i = 1, 2, 3) and total Hof the whole system where [Pi, H] = 0.However, after a gauge transformation, ψ′ = e−ieω(x)ψ, the matrix elements of the above

operators transform as,

〈ψ′|�p|ψ′〉 = 〈ψ|�p|ψ〉 − e〈ψ|�∇ω(x)|ψ〉,〈ψ′|�L|ψ′〉 = 〈ψ|�L|ψ〉 − e〈ψ|�r × �∇ω(x)|ψ〉,〈ψ′|H ′|ψ′〉 = 〈ψ|H|ψ〉+ e〈ψ|∂tω(x)|ψ〉.

(3)

It is obvious that the matrix elements of these three operators are all gauge dependent.Therefore they are not measurable and so these operators do not correspond to observ-ables. This problem has been left in quantum mechanics since the gauge principle wasproposed.The relativistic version of the quantum mechanics has the same problem. The gauge

dependence of the expectation value of the Hamiltonian of a charged particle moving inelectromagnetic field under a time dependent gauge transformation had been discussedby T. Goldman [4].This conflict had been carried over to QED. Starting from a QED Lagrangian,

L = ψ[iγμ(∂μ + ieAμ)−m]ψ − 1

4FμνF

μν , Fμν = ∂μAν − ∂νAμ. (4)

By means of the Noether theorem one obtains the momentum and angular momentumoperators as follows:

�P = �Pe + �Pph =∫d3xψ† �∇

iψ +

∫d3xEi�∇Ai, (5)

�J = �Se+�Le+�Sph+�Lph =∫d3xψ† �Σ

2ψ+

∫d3x�x×ψ† �∇

iψ+

∫d3x�E× �A+

∫d3x�x×Ei�∇Ai.(6)

Here Σj = i2εjklγ

kγl. These electron and photon momentums, orbital angular momentumsand spin, after quantization, satisfy momentum and angular momentum Lie algebra.However, they are not gauge invariant except the electron spin.The multipole radiation analysis is the basis of atomic, molecular, nuclear and hadron

spectroscopy. The multipole field is based on the decomposition of the electromagneticfield into field with definite orbital angular momentum and spin quantum numbers. If thephoton spin and orbital angular momentum operators were gauge dependent, then thephysical meaning of the multipole field would be obscure especially the parity of thesemicroscopic states determined by the measurement of the orbital angular momentumquantum number of the multipole radiation field would be obscure.

F. Wang et al. / Nuclear Physics A 844 (2010) 85c–94c 87c

QCD has the same problem as QED. The quark gluon momentum, orbital angularmomentum and spin operators derived from QCD Lagrangian by Noether theorem havethe same form as those of electron and photon if one omits the color indices. They satisfythe momentum and angular momentum Lie algebra but they are not gauge invariantexcept the quark spin.Because of the lack of gauge invariant quark, gluon momentum operators, the present

operator product expansion (OPE) used the following two operators as quark gluon mo-mentum operators,

�P = �Pq + �Pg =∫

d3xψ† �Diψ +

∫d3x�E × �B, �D = �∇− ig �A. (7)

Both the quark and gluon “momentum” operators �Pq and �Pg defined in Eq. (7) are gauge

invariant but neither the quark “momentum” �Pq nor the gluon “momentum” �Pg satisfiesthe momentum algebra, for example,

[Dl, Dm] = −ig(∂lAm − ∂mAl)− ig2CabcAalA

bmT

c, (8)

where Cabc is the SU(3) group structure constant. The �Pg does not satisfy the momentumalgebra either in the interacting quark-gluon field, i.e., QCD case. Therefore neither the�Pq nor the �Pg used in the OPE is the proper momentum operator.The gluon spin contribution is under intensive study, PHENIX, STAR, COMPASS,

HERMES, and others are measuring the gluon spin contribution to nucleon spin. How-ever there is no gluon spin operator which satisfies both gauge invariance and angularmomentum algebra. There is also no quark, gluon orbital angular momentum opera-tor which satisfies the gauge invariance and orbital angular momentum algebra. Thesesituations hindered the study of the nucleon spin structure.

3. A NEW SET OF MOMENTUM, ORBITAL ANGULAR MOMENTUMAND SPIN OPERATORS FOR THE FERMION AND GAUGE FIELDPARTS

3.1. Decomposing the gauge field Aμ into pure gauge part and gauge invariant(covariant) part

Let us start from the simpler QED case. It is well known that to use gauge potentialAμ to describe the electromagnetic field the Aμ is not unique, i.e., there is gauge freedom.Under a gauge transformation,

A′μ = Aμ + ∂μω(x), (9)

one obtains a new gauge potential A′μ from Aμ. Aμ and A′μ describe the same electro-magnetic field,

Fμν = ∂μAν − ∂νAμ = ∂μA′ν − ∂νA

′μ. (10)

Such a gauge freedom is necessary because the gauge potential Aμ plays two role ingauge field theory: the first is to provide a pure gauge field Apure to compensate theinduced field due to the phase change in a local gauge transformation of the Fermion field


ψ′(x) = e−ieω(x)ψ(x) which must be varied with the arbitrary changed phase parameterω(x); the second is to provide a physical field Aphys for the physical interaction betweenFermion field and gauge field which should be gauge invariant under gauge transformation.The pure gauge potential Apure should not contribute to electromagnetic field, F μν

pure =∂μAν

pure − ∂νAμpure = 0. This equation can not fix the Apure. One has to find additional

condition to fix it. The spatial part of the above equation is ∇× �Apure = 0, which means�Apure does not contribute to magnetic field. This equation can be expressed in anotherform,

∇× �Aphys = ∇× �A. (11)

A natural choice of the additional condition in QED case is

∇ · �Aphys = 0, (12)

which is the transverse wave condition and we know that this part is the physical one fromthe Coulomb gauge quantization. Combining these two conditions, Eq. (11,12), under

the natural boundary condition, �Aphys(|x| → ∞) = 0, for any given set of gauge field �A,one can decompose it uniquely as follows,

�Aphys(x) = �∇× 1

4π

∫d3x′ �∇′ × �A(x′)

|�x− �x′| =1

4π

∫d3x′ �B(x′)

|�x− vec(x′)| ,�Apure = �A− �Aphy.(13)

We have to emphasize that for fixed �A(x), the integration can be done and the obtained�Aphys(x) is a local function of space-time x. It is easy to prove that these two partstransform as follows in a gauge transformation Eq. (9),

�A′phy =

�Aphy, �A′pure =

�Apure − �∇ω(x). (14)

The time component A0 can be decomposed in the same manner. From the conditionF i0pure = 0, one obtains

∂iA0phys = ∂iA

0 + ∂t(Ai − Ai

phys), A0phys =

∫ x

−∞dxi(∂iA

0 + ∂tAi − ∂tA

iphys). (15)

To decompose the gauge potential Aμ = AaμT

a for the gluon field is more complicatedthan QED case. We first define the pure gauge potential Aμ

pure (hereafter we omit thecolor indices if not necessary) by the same condition, i.e., it does not contribute to colorelectromagnetic field,

F μνpure = ∂μAν

pure − ∂νAμpure + ig[Aμ

pure, Aνpure] = 0. (16)

In order to make this defining condition looks similar to Eq.(11), we introduce a notation,

�Dpure = �∇− ig �Apure, �Dpure × �Apure = �∇× �Apure − ig �Apure × �Apure = 0. (17)

The additional condition is even more complicated, i.e., one does not have a natural choiceas Eq. (12) in QED. We make the following choice [5],

�Dpure = �∇− ig[ �Apure, ], �Dpure · �Aphys = �∇ · �Aphys − ig[Aipure, A

iphys] = 0. (18)


The summation over the vector component i has been assumed in the above equation andfollowing ones. Please note that in the above adjoint representation of the new covariantderivative operator �D, the bracket [Ai

pure, Aiphys] is not the quantum bracket but a color

SU c(3) group one,

[Aipure, A

iphys] = iCabcA

ibpureA

icphysT

a. (19)

These equations can be rewritten as follows,

�∇ · �Aphys = ig[Ai, Aiphys],

�∇× �Aphys = �∇× �A− ig( �A− �Aphys)× ( �A− �Aphys), (20)

∂iA0phys = ∂iA

0 + ∂t(Ai − Ai

phys)− ig[Ai − Aiphys, A

0 − A0phys].

These equations can be solved perturbatively: in the zeroth order, i.e., assuming g = 0,these equations are the same as those of QED, one can obtain the zeroth order solu-tion; then taking into account the nonlinear coupling through iteration one obtains aperturbative solution as a power expansion in g.Under a gauge transformation,

ψ′ = Uψ, A′μ = UAμU

† − i

gU∂μU

†, (21)

where U = e−igω. The �Apure and �Aphys will be transformed as

�A′phys = U �AphysU

†, �A′pure = U �ApureU

† − i

gU∂μU

†. (22)

3.2. Quantum mechanicsWe have mentioned in the introduction part that even in quantum mechanics, there are

already puzzles related to the fundamental operators, the matrix elements of canonicalmomentum, orbital angular momentum and Hamiltonian of a charged particle moving inan electromagnetic field are all not gauge invariant. In order to get rid of these puzzles,gauge invariant operators have been introduced,

�P = �p− e �A, �L = �x× �P . (23)

It is easy to check that the matrix elements of these operators are gauge invariant. How-ever, the gauge invariant “momentum” �P does not satisfy the canonical momentum Liealgebra, so they are not the real momentum. The gauge invariant “orbital angular mo-mentum” �L does not satisfy the angular momentum Lie algebra either.Based on the gauge field decomposition proposed in above section, we introduce a new

set of momentum and orbital angular momentum operators which satisfy both gaugeinvariance and the corresponding commutation relations,

�ppure = �p− e �Apure, �Lpure = �x× �ppure. (24)

The long standing puzzle, the gauge non-invariance of the expectation value of theHamiltonian [4] can be solved in the same manner. For the non-relativistic quantummechanics, the new Hamiltonian is

H =(�p− e �A)2

2m+ eA0 − e∂t

1

∇2∇ · �A. (25)


The last term is a pure gauge term, it cancels the unphysical energy appeared in eA0

induced by the pure gauge term and then guarantees the expectation value of this Hamil-tonian gauge invariant. It is a direct extension of Eq. (24) to the fourth momentumcomponent.The Dirac Hamiltonian has the same unphysical energy part and has to be canceled in

the same manner as that for the Schrodinger Hamiltonian. Here we have done a check:starting from a QED Lagrangian with both electron and proton, under the infinite protonmass approximation, we derived the Dirac equation of electron and the gauge invariantHamiltonian of the electron part and verified the difference between the Dirac Hamiltonianobtained from the Dirac equation and the gauge invariant one from the energy-momentumtensor.Our study shows that the canonical momentum, orbital angular momentum and the

Hamiltonian used in quantum mechanics are not observables, one must subtract the puregauge part, the unphysical one, from these operators as we did in Eq. (24,25) to obtainthe observable ones.

3.3. QEDWe have explained that the momentum and angular momentum operators of the Fermion

and gauge field part, Eq. (5) and (6), derived from the QED Lagrangian by means ofNoether theorem are not gauge invariant except the Fermion spin. One can obtain agauge invariant decomposition by adding a surface term,

�P = �Pe + �Pph =∫d3xψ† �D

iψ +

∫d3x�E × �B, (26)

�J = �Se + �Le + �Jph =∫d3xψ† �Σ

2ψ +

∫d3x�x× ψ† �D

iψ +

∫d3x�x× ( �E × �B). (27)

There are two problems with this decomposition: (1), �Le and �Jph do not satisfy the angularmomentum commutation relation even though in free electromagnic field the photon totalangular momentum �Jph does; (2), there is no separate photon spin and orbital angularmomentum operators and this feature will ruin the multipole radiation analysis as wediscussed in the second section.Based on the decomposition of the gauge field potential into pure gauge and the physical

parts, Eq. (13), we obtain the following decomposition,

�P = �Pe + �Pph =∫d3xψ† �Dpure

iψ +

∫d3xEi �DpureA

iphys. (28)

�J = �Se+�Le+�Sph+�Lph =∫d3xψ† �Σ

2ψ+

∫d3x�x×ψ† �Dpure

iψ+

∫d3x�E× �Aphys+

∫d3x�x×Ei �DpureA

iphys.

(29)

Here the operator �Dpure and �Dpure are the same as giving in Eq. (17) and (18) but withg replaced by e. Because of the Abelian property of the U(1) gauge field, the adjoint

representation of the operator �D is simplified to be a simple �∇. It is not hard to checkthat each operator in the above decomposition, Eq. (28) and (29) is gauge invariant andsatisfies the momentum, angular momentum commutation relation.The photon spin and orbital angular momentum operators are well defined as shown in

Eq. (29). The multipole radiation analysis is theoretically sound now as it should be.


3.4. QCDOne can copy results for QED, the Eqs. (28,29), to QCD to obtain the quark, gluon

momentum, orbital angular momentum and spin operators which satisfy both the gaugeinvariance and canonical momentum and angular momentum commutation relations.A decomposition of the form of Eq. (26,27) has been used in the nucleon spin structure

study for the last ten years [6]. Each operator in those decomposition is gauge invariantand so corresponding to observable. However, because they do not satisfy the momentum,angular momentum Lie algebra so the measured ones are not the quark, gluon momen-tum and orbital angular momentum and can not be compared to those used in hadronspectroscopy.The gluon spin operator had been searched for more than ten years in the nucleon spin

structure study and no satisfied one was obtained. Now one can calculate the matrixelement of the gluon spin operator �Sg =

∫d3x�E × �Aphys between the polarized nucleon

state |N(p, s)〉 to obtain the gluon spin contribution to nucleon spin and compare it withthe measured ones.

4. REEXAMINATION OF THE NUCLEON INTERNAL STRUCTURE

For the past years, nucleon internal structure has been studied based on operators givenin Eq. (26,27) which are gauge invariant but violate canonical momentum, angular mo-mentum Lie algebra. This led to a distorted picture of the nucleon internal structure. Forexample, that the quark and gluon carry half of the nucleon momentum in the asymptoticlimit has been a deeply rooted picture of nucleon internal momentum structure. Usingthe new quark, gluon momentum operator, Eq. (28), we recalculated their scale evolutionand obtained the new result [8],

�PRg =

12ng

12ng + 3nf

�Ptotal. (30)

For typical gluon number ng = 8 and quark flavor number nf = 5, the above equation

gives �pRg � 15�Ptotal. This is distinctly different from the renowned results �PR

g � 12�Ptotal.

The asymptotic nucleon spin structure [7] is obtained based on the decomposition Eq.(6), a QED analog of QCD angular momentum decomposition. The authors had pointedout that the quark and gluon orbital angular momentum operators are not gauge invariant.As we have mentioned in the beginning, in the present gauge field theory an observablemust be expressed in terms of a gauge invariant operator. The gauge dependent operatorsused in this analysis [7] are not measurable ones. Therefore this asymptotic limit ofnucleon spin content should be reexamined.The present polarized and unpolarized parton distribution functions are related to the

“momentum”, “angular momentum” operators given in Eq. (26,27), they do not satisfythe canonical momentum and angular momentum commutation relations. A new set ofparton distribution functions are proposed in [8] and a new factorization theorem relatingthe polarized and unpolarized lepton-nucleon deep inelastic scattering cross section withthese new parton distribution functions are called for.


5. SUMMARY AND PROSPECT

Since the establishment of gauge invariance principle, we enjoy that the total momen-tum, angular momentum and the Lorentz boosting operators of a gauge system satisfyboth the gauge invariance and Poincare algebra, however we never have the separate mo-mentum, orbital angular momentum operators of the Fermion (electron in QED, quarkin QCD) and boson (photon in QED, gluon in QCD) part which satisfy both the gaugeinvariance and the canonical momentum, angular momentum Lie algebra. We have theelectron, quark spin operator but we never have the photon, gluon spin operator whichsatisfy both the gauge invariance and spin Lie algebra. Even it had been claimed in sometextbooks that it is impossible to have a well defined photon spin [3]. The nucleon spinstructure study needs the gluon spin operator, but after about ten years effort in searchingfor a gluon spin operator since the so-called proton spin crisis such an effort has almostbeen given up for the last ten years. In this report we proposed a new set of quark (elec-tron), gluon (photon) momentum, orbital angular momentum and spin operators whichsatisfy both the gauge invariance and the canonical momentum, angular momentum Liealgebra.To achieve this a key point is to separate the gauge field into pure gauge and physical

parts: the former is unphysical and can be gauged away as in Coulomb gauge, it is usedto compensate the induced unphysical gauge field due to the local gauge transformationof the Fermion field to keep the gauge invariance; the physical part is responsible for thephysical coupling between Fermion and boson field. It is physical and should be gaugeinvariant (covariant). We provide a method to do this separation both for the AbelianU(1) and the non-Abelian SU(3) gauge field.Our proposed momentum operators for the Fermion part are different from the canonical

ones, the latter ones are not gauge invariant and so do not represent observables becausethey include the unphysical pure gauge field contribution. The new ones subtract theunphysical pure gauge field contribution and so they are gauge invariant and representthe observables.We achieved to obtain a gauge invariant orbital angular momentum and spin operators

of the photon and gluon by means of the physical part of the gauge field, Eq. (29), whichprovides the theoretical basis of the widely used multipole radiation analysis, the photonspin and orbital angular momentum used in quantum computation and communicationstudy, the gluon spin contribution in the nucleon spin structure study.The Poincare algebra can not be fully maintained for the momentum, angular mo-

mentum and Lorentz boosting operators of the individual Fermion and boson part ofan interacting gauge field system. What is the meaning of these observables if they arenot Lorentz covariant? We have shown that the momentum and angular momentum al-gebra can be maintained simultaneously with the gauge invariance. How much part ofthe Poincare algebra can be maintained for the operators of the interacting Fermion andboson separately, especially the Lorentz covariance can be maintained to what extent areleft for further study.The new asymptotic limit of quark and gluon parton momentums of a nucleon have

been obtained, the immediate problem is the asymptotic limit of the quark and gluonorbital angular momentums and spins.


The gluon spin contribution to the nucleon spin is under measurement in different labs.A lattice QCD calculation with the gauge invariant gluon spin operator is called for.To obtain the new PDFs the factorization theorem with respect to the new PDF formula

should be examined.In summary, the nucleon internal structure is better to be reexamined based on the

new quark, gluon momentum, orbital angular momentum, spin operators and partondistribution functions and our picture of the nucleon internal structure might be modified.

REFERENCES

1. F. Strocchi and A. S. Wightman, J. Math. Phys. 15 (1974) 2198.2. X. S. Chen and F. Wang, hep-ph/9802346.3. V. B. Berestetskii, E. M. Lifshitz and L. P. Pitaevskii, Quantum Electrodynamics.

(Oxford, Pergamon) 1982; C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg,Photons and Atoms. (New York, Wiley) 1987.

4. T. Goldman, Phys. Rev. D15 (1977) 1063.5. X. S. Chen, X. F. Lu, W. M. Sun, F. Wang and T. Goldman, Phys. Rev. Lett. 100

(2008) 232002.6. X. Ji, Phys. Rev. Lett. 78 (1997) 610; X. S. Chen, and F. Wang, Commun. Theor.

Phys. 27 (1997) 121.7. X. Ji, J. Tang and P. Hoodbhoy, Phys. Rev. Lett. 76 (1996) 740.8. X. S. Chen, X. F. Lu, W. M. Sun, F. Wang and T. Goldman, Phys. Rev. Lett. 103

(2009) 062001.


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