Gauge invariance, Canonical quantization and Lorentz covariance in the internal structure X.S.Chen, X.F.Lu Dept. of Phys., Sichuan Univ. W.M.Sun, Fan Wang NJU and PMO Joint Center for Particle Nuclear Physics and Cosmology (J-CPNPC) T.Goldman T.D., LANL, USA

Outline I.Introduction II.Conflicts between Gauge invariance and Canonical Quantization III.A new set of quark, gluon momentum, angular momentum, and spin operators III.0 A lemma:Decomposing the gauge field into pure gauge and physical parts III.1 Quantum mechanics III.2 QED III.3 QCD IV. Nucleon internal structure V. Summary

I. Introduction Fundamental principles of quantum physics: 1.Quantization rule: operators corresponding to observables satisfy definite quantization rule; 2.Gauge invariance: operators corresponding to observables must be gauge invariant; 3.Lorentz covariance: operators in quantum field theory must be Lorentz covariant.

How to apply these principles to the internal structure For nucleon, one has the quark, gluon momentum, orbital angular momentum and spin operators either satisfy the canonical quantization rule or gauge invariance but no one satisfies both. The atom internal structure has the same problem. No photon spin and orbital angular momentum operators which satisfy both requirements.

II. Conflicts between gauge invariance and canonical quantization

Quantum mechanics The classical canonical momentum of a charged particle moving in an electromagnetic field, an U(1) gauge field, is It is not gauge invariant! The gauge invariant one is, it does not satisfy the canonical momentum algebra. And so Feynman called it the velocity operator

Gauge is an internal degree of freedom, no matter what gauge is used, the canonical momentum of a charged particle is quantized as The orbital angular momentum is The Hamiltonian is

Under a gauge transformation, the matrix elements transformed as They are not gauge invariant, even though the Schroedinger equation is.

Relativistic quantum mechanics has the same problem The Dirac equation of a charged particle moving in electromagnetic field is gauge invariant. But the matrix elements of electron momentum, orbital angular momentum and Hamiltonian between physical states are not gauge invariant.

QED The canonical momentum and orbital angular momentum of electron are gauge dependent and so their physical meaning is obscure. The canonical photon spin and orbital angular momentum operators are also gauge dependent. Their physical meaning is obscure too. Even it has been claimed in some textbooks that it is impossible to have photon spin and orbital angular momentum operators. V.B. Berestetskii, A.M. Lifshitz and L.P. Pitaevskii, Quantum electrodynamics, Pergamon, Oxford, C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and atoms, Wiley, New York,1987.

Multipole radiation Multipole radiation measurement and analysis are the basis of atomic, molecular, nuclear and hadron spectroscopy. If the spin and orbital angular momentum of photon is gauge dependent and not measurable or even meaningless, then all determinations of the parity of these microscopic systems would be meaningless!

Multipole field The multipole radiation theory is based on the decomposition of an em field into multipole radiation field with definite photon spin and orbital angular momentum quantum numbers coupled to a total angular momentum quantum number LM,

QCD Because the canonical parton (quark and gluon) momentum is “gauge dependent”, so the present analysis of parton distribution of nucleon uses the covariant derivative operator instead of the canonical momentum operator ; uses the Poynting vector as the gluon momentum operator. They are not the proper momentum operators! Because they do not satisfy the canonical momentum algebra.

Because the canonical quark and gluon orbital angular momentum and gluon spin operators are not gauge invariant. The present nucleon spin structure analysis used the gauge invariant ones but do not satisfy angular momentum algebra. The present gluon spin measurement is even under the condition that “there is not a gluon spin can be measured”.

III. A New set of quark, gluon (electron, photon) momentum, orbital angular momentum and spin operators

III.0 Decomposing the gauge field into pure gauge and physical parts There were gauge field decompositions discussed before, mainly mathematical. Y.S.Duan and M.L.Ge, Sinica Sci. 11(1979)1072; L.Fadeev and A.J.Niemi, Nucl. Phys. B464(1999)90; B776(2007)38. We suggest a new decomposition based on the requirement: to separate the gauge field into pure gauge and physical parts. X.S. Chen, X.F.Lu, W.M.Sun, F.Wang and T.Goldman, Phys. Rev. Lett. 100(2008)

U(1) Abelian gauge field

The last expression shows that the is a local space-time function but determined nonlocally by the magnetic field in a whole region. The is measurable. The is also a measurable local space- time function.

One can also directly obtain first

Under a gauge transformation, The physical and pure gauge parts will be transformed as

SU(3) non-Abelian gauge field

The above equations can be rewritten as a perturbative solution in power of g through iteration can be obtained

Under a gauge transformation,

III.1 Quantum mechanics The classical canonical momentum of a charged particle moving in an electromagnetic field, an U(1) gauge field, is It satisfys the canonical momentum algebra but its matrix element is not gauge invariant!

New momentum operator The new momentum operator is, It satisfies the canonical momentum commutation relation and its matrix element is gauge invariant.

We call The physical momentum. It is neither the canonical momentum nor the mechanical momentum

Hamiltonian of hydrogen atom Coulomb gauge Gauge transformed one

Follow the same recipe, we introduce a new Hamiltonian, which is gauge invariant, i.e., This means the hydrogen energy calculated in Coulomb gauge is physical.

A rigorous derivation Start from a QED Lagrangian including electron, proton and em field, under the heavy proton approximation, one can derive a Dirac equation and a Hamiltonian for electron and proved that the time evolution operator is different from the Hamiltonian exactly as we obtained phenomenologically. The nonrelativistic approximation is the Schroedinger or Pauli equation.

III.2 QED Different approach will obtain different energy-momentum tensor and four momentum, they are not unique: Noether theorem They are not gauge invariant. Gravitational theory (Weinberg) or Belinfante tensor It appears to be perfect, but individual part does not satisfy the momentum algebra.

New momentum for QED system We are experienced in quantum mechanics, so we introduce They are both gauge invariant and momentum algebra satisfied. They return to the canonical expressions in Coulomb gauge.

The renowned Poynting vector is not the proper momentum of em field It includes photon spin and orbital angular momentum

Electric dipole radiation field

Each term in this decomposition satisfies the canonical angular momentum algebra, so they are qualified to be called electron spin, orbital angular momentum, photon spin and orbital angular momentum operators. However they are not gauge invariant except the electron spin. Therefore the physical meaning is obscure.

However each term no longer satisfies the canonical angular momentum algebra except the electron spin, in this sense the second and third term is not the electron orbital and photon angular momentum operator. The physical meaning of these operators is obscure too. One can not have gauge invariant photon spin and orbital angular momentum operator separately, the only gauge invariant one is the total angular momentum of photon. The photon spin and orbital angular momentum had been measured !

Dangerous suggestion It will ruin the multipole radiation analysis used from atom to hadron spectroscopy, where the canonical spin and orbital angular momentum of photon have been used. It is unphysical!

New spin decomposition for QED system

Multipole radiation Photon spin and orbital angular momentum are well defined now and they will take the canonical form in Coulomb gauge. Multipole radiation analysis is based on the decomposition of em vector potential in Coulomb gauge. The results are physical and these multipole field operators are in fact gauge invariant.

III.3 QCD three decompositions of momentum threethree

Three decompositions of angular momentum 1. From QCD Lagrangian, one can get the total angular momentum by Noether theorem:

2. One can have the gauge invariant decomposition,

3.New decomposition

IV. Nucleon internal structure it should be reexamined! The present parton distribution is not the real quark and gluon momentum distribution. In the asymptotic limit, the gluon only contributes ~1/5 nucleon momentum, not 1/2 ! arXiv: [hep-ph],Phys.Rev.Lett. 103, (2009) The nucleon spin structure should be reexamined based on the new decomposition and new operators. arXiv: [hep-ph], Phys.Rev.Lett. 100,232002(2008)

Consistent separation of nucleon momentum and spin

Quantitative example: Old quark/gluon momentum in the nucleon

Proper quark/gluon momentum in nucleon

One has to be careful when one compares experimental measured quark gluon momentum and angular momentum to the theoretical ones. The proton spin crisis is mainly due to misidentification of the measured quark axial charge to the nonrelativistic Pauli spin matrix elements. D. Qing, X.S. Chen and F. Wang, Phys. Rev. D58, (1998)

To clarify the confusion, first let me emphasize that the DIS measured one is the matrix element of the quark axial vector current operator in a nucleon state, Here a 0 = 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.

The axial vector current operator can be expanded as

The quark orbital angular momentum operator can be expanded as,

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 operator are exact the same but with opposite sign. Therefore if we add them together we will have where the, are the non-relativistic part of the quark spin and angular momentum operator.

The above relation tell us that the nucleon spin can be either solely attributed to the quark Pauli spin, as did in the last thirty years in CQM, and the nonrelativistic quark orbital angular momentum does not contribute to the nucleon spin; or part of the nucleon spin is attributed to the relativistic quark spin, it is measured in DIS and better to call it axial charge to distinguish it from the Pauli spin which has been used in quantum mechanics over seventy years, part of the nucleon spin is attributed to the relativistic quark orbital angular momentum, it will provide the exact compensation missing in the relativistic “quark spin” no matter what quark model is used. one must use the right combination otherwise will misunderstand the nucleon spin structure.

Conventional and new construction of parton distribution functions (PDFs)

The conventional gauge-invariant “quark” PDF The gauge link (Wilson line) restores gauge invariance, but also brings quark-gluon interaction, as also seen in the moment relation:

The new quark PDF With a second moment:

The conventional gluon PDF Relates to the Poynting vector:

The new gluon PDF Relates to the new gauge-invariant gluon momentum

Gauge-invariant polarized gluon PDF and gauge-invariant gluon spin

To measure the new quantities  The same experiments as to measure the conventional PDFs  New factorization formulae and extraction of the new PDFs needed  New quark and gluon orbital angular momentum can in principle be measured through generalized (off-forward) PDFs

VII. Summary: general The gauge field can be separated into pure gauge and physical parts. Physical part is measurable. The renowned Poynting vector is not the proper momentum operator of photon and gluon field. The canonical momentum, angular momentum operators of the Fermion part are not observables. The gauge invariant and canonical quantization rule both satisfied momentum, spin and orbital angular momentum operators of the individual part do exist. They had been measured in QM and QED.

special to nucleon internal structure The nucleon internal structure should be reanalyzed and our picture of it might be modified A new set of quark, gluon momentum, orbital angular momentum and spin operators for the study of nucleon internal structure is provided Gluon spin is indeed meaningful and measurable Gluons carry not much of the nucleon momentum, not ½ but 1/5

Prospect Computation of asymptotic partition of nucleon spin Reanalysis of the measurements of unpolarized quark and gluon PDFs New factorization formulas are needed Reanalysis and further measurements of polarized gluon distributions. A lattice QCD calculation of gluon spin contribution to nucleon spin.

For the quark (electron), gluon(photon) momentum and angular momentum operators the Lorentz covariance can be kept to what extent, the meaning of non Lorentz covariance. The possibility of the gauge non-invariant operator might have gauge invariant matrix element for special states should be studied further.

Thanks