Poincare covariance, Canonical quantization and Gauge invariance

in the Nucleon internal structure

X.S.Chen, Dept. of Phys., Huazhong Univ. Sci. Tec.

X.F.Lu, Dept. of Phys., Sichuan Univ.

W.M.Sun, Dept. of Phys., Nanjing Univ.

Fan Wang, Dept. of Phys. Nanjing Univ.

T. Goldman, T-Division, LANL

C.W. Wong, Dept. of Phys., UCLA

fgwang@chenwang.nju.edu.cn

OutlineI. Introduction

II. The first proton spin crisis and quark spin confusion

III. The second proton spin crisis and the quark orbital angular momentum confusion

IV. A consistent decomposition of the momentum and angular momentum operators of a gauge field system

V. Poincare covariance

VI. Gauge invariant extension

VII. Summary

I. Introduction• It is still a popular idea that the polarized deep Inelastic scattering (DIS) measured quark spin invalidates the

constituent quark model (CQM), lead to the so-called proton spin crisis.

I will show that there is no proton spin crisis but has a misidentifying of the relativistic quark spin to the non-relativistic quark spin.

• There are different quark orbital angular momentumoperators used in the literature. This will cause further confusion in the nucleon spin structure study and might have already caused it, such as the second proton spin crisis.

• There is a new debate on what are the right quark, gluon spin and orbital angular momentum operators.

II.The first proton spin crisis and quark spin confusion

quark spin contribution to nucleon spin in

naïve non-relativistic quark model

consistent with nucleon magnetic moments.

4 1, , 0.

3 3u d s

/ 3 / 2p n

0, 0, 0.q GL G L

DIS measured quark spin

The DIS measured quark spin contributions are:

(E.Leader, A.V.Sidorov and D.B.Stamenov, PRD75,074027(2007); hep-ph/0612360)

(D.de Florian, R.Sassot, M.Statmann and W.Vogelsang, PRL101, 072001(2008); 0804.0422[hep-ph])

.

2 20.800 0.467 0.126 0.207( 1 , 0)Q GeV G 2 20.812 0.455 0.114 0.243( 1 , 0)Q GeV G

2 20.813 0.458 0.114 0.242( 10 , 0.084)Q GeV G

u d s

Proton spin crisis!?

It seems there are serious contradiction

between our understanding of nucleon spin structure and measurements:

The DIS measured total quark spin

contribution to nucleon spin is about 25%,

while we believe that the nucleon spin is

solely coming from quark spin;

Quark spin confusion

here a0= Δu+Δd+Δs which is not the quark spin contributions calculated in CQM. The CQM calculated one is the matrix element of the Pauli spin part only.

Quark axial vector current operator

The quark axial vector current operator can be

expanded instantaneously as

Contents of axial vector current operator

• Only the first term of the axial vector current operator, which is the Pauli spin, has been calculated in the non-relativistic quark models.

• The second term, the relativistic correction, has not been included in the non-relativistic quark model calculations. The relativistic quark model does include this correction and it reduces the quark spin contribution about 25%.

• The third term, creation and annihilation, has not been calculated in models with only valence quark configuration. Various meson cloud models have not calculated this term either.

qq

Extending the valence CQM to Fock states expansion quark model

• To understand the nucleon spin structure quantitatively within CQM and to clarify the quark spin confusion further we developed a CQM with sea quark components,

3 30 ( ) ( )N c q c q qq

(D.Qing, X.S.Chen and F.Wang,PRD58,114032(1998))

Model prediction of quark spin contribution to nucleon spin

Main spin reduction mechanism

III.The second proton spin crisis and quark orbital angular

momentum confusion

Quark orbital angular momentum confusion

• The quark “orbital angular momentum”

calculated in LQCD and measured in DVCS is not the

real orbital angular momentum used in quantum

mechanics. It does not satisfy the Angular Momentum

Algebra,

and the gluon contribution is ENTANGLED in it.

3 ( )q q qL d xx p gA

L L iL

Where does the nucleon get spin? Real quark orbital angular momentum

• As a QCD system the nucleon spin consists of the following four terms (in Coulomb gauge),

The Real quark orbital angular momentum operator

The real quark orbital angular momentum operator can be expanded instantaneously as

Quark orbital angular momentum will

compensate the quark spin reduction

• The first term is the non-relativistic quark orbital angular momentum operator used in CQM, which does not contribute to nucleon spin in the naïve CQM.

• The second term is again the relativistic correction, which will compensate the relativistic quark spin reduction.

• The third term is again the creation and annihilation contribution, which will compensate the quark spin reduction due to creation and annihilation.

qq

qq

Relativistic versus non-relativistic spin-orbital angular momentum sum

• It is most interesting to note that the relativistic correction and the creation and annihilation terms of the quark spin and the orbital angular momentum operators are exact the same but with opposite sign. Therefore if we add them together we will have

where the , are the non-relativistic quark spin and orbital angular momentum operator used in quantum mechanics.

qq

• The above relation tells us that the quark contribution to nucleon spin can be either attributed to the quark Pauli spin, as done in the last thirty years in CQM, and the non-relativistic quark orbital angular momentum which does not contribute to the nucleon spin in naïve CQM; or

• part of the quark contribution is attributed to the relativistic quark spin as measured in DIS, the other part is attributed to the relativistic quark orbital angular momentum which will provide

the exact compensation of the missing part in the relativistic “quark spin”

Prediction

IV.A consistent decomposition of the momentum and angular

momentum of a gauge system

Jaffe-Manohar decomposition:

R.L.Jaffe and A. Manohar,Nucl.Phys.B337,509(1990).

• Each term in this decomposition satisfies the canonical commutation relation of angular momentum operator, so they are qualified to be called quark spin, orbital angular momentum, gluon spin and orbital angular momentum operators.

• However they are not gauge invariant except the quark spin.

Gauge invariant decomposition

X.S.Chen and F.Wang, Commun.Theor.Phys. 27,212(1997).

X.Ji, Phys.Rev.Lett.,78,610(1997).

• However each term no longer satisfies the canonical commutation relation of angular momentum operator except the quark spin, in

this sense the second and third terms are not the real quark orbital and gluon angular momentum operators.

• One can not have gauge invariant gluon spin and orbital angular momentum operator separately, the only gauge invariant one is the total angular momentum of gluon.

• In QED this means there is no photon spin and orbital angular momentum! This contradicts the well established multipole radiation analysis.

Standard definition of momentum and orbital angular momentum

Momentum and angular momentum operators for charged particle moving

in electro-magnetic field

For a charged particle moving in em field,

the canonical momentum is,

• It is gauge dependent, so classically it is

Not measurable.

• In QM, we quantize it as no

matter what gauge is.

• It appears to be gauge invariant, but in fact

Not!

p mr qA

,pi

Under a gauge transformation

The matrix elements transform as

' ( ) ,iq xe ' ,A A A

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| | | | | | ,p p q

| | | | | | ,L L qr

| | | | | | ,tH H q

New momentum operator

.A A

We call

physical momentum.

It is neither the canonical momentum

nor the mechanical momentum

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Gauge invariance and canonical quantization both satisfied decomposition

• Gauge invariance is not sufficient to fix the decomposition of the angular momentum of a gauge system.

• Canonical quantization rule of the angular momentum operator must be respected. It is also an additional condition to fix the decomposition.

• Measurable one must be physical.

X.S.Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys.Rev.Lett. 100(2008) 232002.

arXiv:0806.3166; 0807.3083; 0812.4366[hep-ph]; 0909.0798[hep-ph]

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It provides the theoretical basis of the multipole radiation analysis

QCD

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,[ ] 0adj i ipure phys phys pure physD A A ig A A

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Consistent separation of nucleon momentum and angular momentum

3Standard construction of orbital angular momentum xL d x P

• Each term is gauge invariant and so in principle measurable.

• Each term satisfies angular momentum commutation relation and so can be compared to quark model ones.

• In Coulomb gauge it reduces to Jaffe-Manohar decomposition, the usual one used in QM and QFT.

• Jaffe-Manohar’s quark, gluon orbital angular momentum and gluon spin are gauge dependent. Ours are gauge invariant.

V.Poincare covariance

No full Poincare covariance for the individual operators

3-dimensional translation and rotationfor individual part can be retained

General Lorentz covariance for individual part are retained

QED

QCD

VI.Gauge Invariant extension

• Among the infinite possible gauge invariant extensions, only one might be used to construct the observable which is the physical part of the gauge potential.

• For QED, it is the Coulomb gauge fixing parts which are physical because there are only two transverse components or helicity components retained. They are observed in Compton scattering.

• For QCD a natural choice is to choose the two transverse or helicity components of the gluon field as physical parts which might be observed in gluon jet.

• The light cone gauge invariant extension is not physical one, because it includes unphysical pure gauge part and the residual gauge degree of freedom.

• Only in light cone gauge, the Jaffe-Bashinsky gluon spin operator can be related to gluon spin and so can the measured gluon helicity distribution be related to the matrix element of J-B operator, to the Collins-Soper-Manohar polarized gluon parton distribution function should be studied further.

• The kinematic momentum can never be reduced to canonical momentum in any gauge and no sulution of its eigen-value equation.

• It is not the right momentum operator, it has never been measured in quantum physics. It is measured in classic physics.

• DIS “measured” quark momentum is the light-cone momentum, not the kinematical momentum.

Agp

xAgp )(

Experimental Check• Energy-momentum (E-M) tensor is the

starting point of various gauge invariant decomposition and extension.

• The popular idea is that one has the freedom to choose the form of the E-M tensor because by adding a surface term the conservation law satisfied by the E-M tensor is unchanged.

• The gauge invariant symmetric E-M tensor is prefered in gauge field theory and the kinematical momentum is derived from the symmetric E-M tensor.

• In fact the classic em E-M tensor is measurable and there is already measurement.

T

Optical evidence

• For symmetric em E-M tensor, there should be no difference of the diffraction pattern for an orbital or spin polarized light beam if the total

• For asymmetric em E-M tensor, there should be difference of the diffraction pattern between orbital and spin polarized beams, because only for orbital polarized beam there is momentum density circular flow in the transverse plane.

A detailed analysis had been given in

arXiv:1211.4407[physics.class-ph]

1~~ zzz sorlJ

Spin is different from orbital angular momentum

• In 1920’s one already knew that the fundamental spin can not

be related to the orbital motion. So to use the symmetric E-M

tensor to express the angular momentum operator as

physically is misleading even though mathematically correct. It

also leads to the wrong idea that the Poynting vector is not only

the energy flow but also the momentum density flow of em field.

The Belinfante derivation of the symmetric E-M tensor and in

turn the derived momentum and angular momentum density

operators physically is misleading too.

Spin half electron field needs asymmetric energy-momentum tensor

• Symmetric E-M tensor for a spin half electron field will lead to contradiction between angular momentum and magnetic momentum measurement. Suppose we have a spin polarized electron beam moving along the z direction with momentum density flow ,

• For symmetric E-M tensor, the momentum density flow is the same as the energy density flow. For spin s=1/2 electron this will lead to a contradiction. The energy flow and momentum density flow should be different.

A detaild analysis had been given in arXiv:1211.2360[gr-qc].

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K

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j

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K 12)()( NdV

e

EdVjx

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To meet the requirement of a gauge invariant but asymmetric E-M tensor we derived the following one based on the

decomposition of gauge potential.Hope experimental colleagues to check

further which one is the correct one.

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T physpurecw ..

2

VI, Remarks about various proposals

• We appreciate E. Leader’s effort, especially on the full quantum version in the covariant gauge.

• E. Leader’s proposal: gauge non-invariant operators might have gauge invariant matrix elements for physical states.

We studied this approach in 1998.X.S. Chen and Fan Wang, Gauge invariance and hadron

structure, hep-ph/9802346.

At most, it might be true for gauge transformations within the covariant or the extended Lorentz gauge.

Hamiltonian of hydrogen atom

,0c

A

.2

)( 2c

c

c qm

AqpH

)(,),()(//// xAAxxAA tc

cc

Coulomb gauge:

Hamiltonian of a non-relativistic charged particle

Gauge transformed one

,0// c

A .00 ccA

.2

)(

2

)( 22

tc

c

qqm

Aqqpq

m

AqpH

c

c

tphy qm

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)()(

2//

cc

cphy HH ||||

2 A

A check

• We derived the Dirac equation and the Hamiltonian of electron in the presence of a massive proton from a em Lagrangian with electron and proton and found that indeed the time translation operator and the Hamiltonian are different, exactly as we obtained phenomenologically before.

W.M. Sun, X.S. Chen, X.F. Lu and F. Wang, arXiv:1002.3421[hep-ph]

• We appreciate Hatta’s effort .

But just say you use another gauge invariant one, it is not physical, because it includes unphysical gluon field.

• We also appreciate P.M. Zhang, D. Pak and Y.M. Cho’s effort. They claim their approach can avoid the Gribov ambiguity.

• We appreciate X. Ji’s criticism which pushed us very much to understand more in this topic.

VII. Summary

• There are different quark and gluon momentum and orbital angular momentum operators. Confusions disturbing or even misleading the nucleon spin structure studies.

• Quark spin missing can be understood within the CQM.

• It is quite possible that the real relativistic quark orbital angular momentum will compensate the missing quark spin.

• A LQCD calculation of the matrix elements of u,d quark real orbital angular momentum might illuminate the nucleon spin structure study.

• For a gauge system, the momentum and angular momentum operators of the individual part (quark~gluon, electron~photon), the existing ones are either gauge invariant or satisfy the canonical commutation relation only but not both.

• We suggest a decomposition which satisfies both the gauge invariance and canonical commutation relations. It might be useful and modify our picture of nucleon internal structure.

• It is not a special problem for quark and gluon angular momentum operators, but a fundamental problem of gauge field systems. Gauge potential can be separated into gauge invariant and pure gauge two parts. The pure gauge part should be subtracted from the operators of the individual momentum and angular momentum of quark and gluon to make them gauge invariant. The spin and orbital angular momentum of gluon or photon can be separated gauge invariantly. How to do these separation is still controversal.

X.S.Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys.Rev.Lett. 100,232002(2008), arXiv:0806.3166; 0807.3083; 0812.4336[hep-ph];0909.0798[hep-ph]

Thanks for the invitation.

Thanks for the discussion.