Unharmony within the

Thematic Melodies of Twentieth Century Physics

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

W.M.Sun, Fan WangNJU and PMO

Joint Center for particle nuclear physics and cosmology

(J-CPNPC)

Outline

I. IntroductionII. Conflicts between Gauge invariance

and Canonical Quantization III. Quantum mechanicsIV. QEDV. QCDVI. Summary

I.Introduction

• Three Thematic Melodies of Twentieth Century Physics :

Symmetry, Quantization, Phase (C.N.Yang)• The combination of symmetry and phase l

ead to Gauge Invariance Principle and gauge field theory (C.N.Yang)

• There are conflicts between these three thematic melodies when one wants to

apply them to study the internal structure.

Quantum Mechanics

Even though the Schroedinger equation isgauge invariant, the matrix elements of the canonical momentum, orbital angularmomentum, and Hamiltonian of a charged particle moving in an eletromagnetic fieldare gauge dependent, especially

the orbital angular momentum and energy of the hydrogen atom are

“not the measurable ones” !?

QEDThe canonical momentum and orbitalangular 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.

Multipole radiation

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

LLMLLM TmA )(

)]()()[0,,(122 1 eAipmADLieA LMLMLMp

LL

rikp

MLLLMLLLLM TLLT

LLeA 1111 12

112

)(

Multipole radiation measurement and analysis are the basis of atomic, molecular, nuclear and hadronic spectroscopy. If the orbital angular momentum of photon is gauge dependent and not measurable, then all determinations of the

parity of these microscopic systems would be

meaningless!

QCD

• Because the 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, the normal derivative operator as the quark momentum operator; uses the Poynting vector as the gluon momentum operator.

• The quark spin contribution to nucleon spin has been measured, the further study is encumbered by the lack of gauge invariant quark orbital angular momentum, gluon spin and orbital angular momentum operators. The present gluon spin measurement is even under the condition that

“they are measuring a not measurable quantity”.

II. Quantum Mechanics

Gauge is an internal degree of freedom, no matter what gauge used, the canonical coordinate and momentum of a charged particle is and , the orbital angular momentum is

, the Hamiltonian is

r ip

irprL

emAepH

2)( 2

Gauge transformation

The matrix elements transformed as

even though the Schroedinger equation is gauge invariant.

,)(' xiee,' AAA ,' t

,|||||| epp

,|||||| reLL

,|||||| teHH

New momentum operator inquantum mechanics

//AqAqrmAqrmp

AqrmAqp //

,0 A

Generalized momentum for a charged particle moving in em field:

It is not gauge invariant, but satisfies the canonicalmomentum commutation relation.

It is both gauge invariant and canonical momentumcommutation relation satisfied.

0// A

We call

physical momentum.It is neither the canonical momentum

nor the mechanical momentum

////1 Aqi

Aqpi

D phy

i

Aqrmp 1

Di

rmAqp 1

Gauge transformation

only affects the longitudinal part of the vector potential

and time component

it does not affect the transverse part,

so is physical and which is used in Coulomb gauge. is unphysical, it is caused by gauge transformation.

),(//'// xAA

,)(' xiqe ),(' xAA

,' AA

),(' xt

A//A

Hamiltonian of hydrogen atom

,0cA

.2

)( 2c

c

c qmAqpH

)(,),()(//// xAAxxAA tccc

Coulomb gauge:

Hamiltonian of a nonrelativistic particle

Gauge transformed one

,0// cA .00 ccA

.2

)(2

)( 22

tc

c

qqm

AqqpqmAqpH

Follow the same recipe, we introduce a new Hamiltonian,

which is gauge invariant, i.e.,

This means the hydrogen energy calculated inCoulomb gauge is gauge invariant and physical.

c

c

tphy qm

AqAqpxqHH

2)()(2

//

cc

cphy HH ||||

//2 A

III.QEDDifferent approach will obtain different energy-momentumtensor and four momentum, they are not unique:Noether theorem

Gravitational theory (weinberg)

It appears to be perfect and has been used in parton distribution analysis of nucleon, but do not satisfy the momentum algebra.Usually one supposes these two expressions are equivalent, because the integral is the same.

}{3 ii AEi

xdP

}{3 BEiDxdP

We are experienced in quantum mechanics, so we introduce

They are both gauge invariant and momentum algebra satisfied. They return to the canonicalexpressions in Coulomb gauge.

}{3 iiphy AE

iDxdP

AAA //

//ieAD phy

We proved the renowned Poynting vector is not the correct momentum of em field

It includes photon spin and

orbital angular momentum

ii AErxdAExdBErxdJ 333 )(

Electric dipole radiation field

lmlmlmlmllmlm BkiAikEYLkrhaB ,......)()1(

]sin2cos1[

163

)(||]Re[

21 2

2

211

1111

n

krn

kraBE r

]2sin

2cos1[

163

)(||]Re[

21 2

2

211

1111

n

krn

kraAE r

ii

ddJk

ka

ddP z

2cos1

163|| 2

2

211

23

211 sin

163||

k

addJ z

LSLSJ eeQED

• 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.

• How to reconcile these two fundamental requirements, the gauge invariance and canonical angular momentum algebra?

• One choice is to keep gauge invariance and give up canonical commutation relation.

'' JLSJ eeQED

• 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 suggestionIt 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.Even the hydrogen energy is not an observable, neither the orbital angular momentum of electron nor the polarization (spin) of photon is observable either.It is totally unphysical!

'''''' LSLSJ eeQED

Multipole radiation

Multipole radiation analysis is based on thedecomposition of em vector potential in Coulomb gauge. The results are physical and gauge invariant, i.e.,gauge transformed to other gauges one willobtain the same results.

V. QCD

• From QCD Lagrangian, one can get the total angular momentum by Noether theorem:

• One can have the gauge invariant decomposition,

New decomposition''''''ggqqQCD LSLSJ

2

3xdS q

i

DrxdL phyq

3''

aphy

ag AExdS 3''

phyai

aig ArxEdL 3''

Esential task:to define properly the pure gauge field and physical one

purephy AigD apure

apure ATA

pureA phyA

phypure AAA

0 purepurepurepurephy AAigAAD

0 phyphy AEEA

VI. Summary• The renowned Poynting vector is not the right

momentum operator of em field.• The gauge invariance and canonical quantiza

tion rule for momentum, spin and orbital angular momentum can be satisfied simultaneously.

• The Coulomb gauge is physical, expressions in Coulomb gauge, even with vector potential, are gauge invariant, including the hydrogen atomic Hamiltonian and multipole radiation.

There is no proton spin crisisbut quark spin confusion

An Extended CQM with Sea Quark Components

• 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,

Where does the nucleon get its Spin

• As a QCD system the nucleon spin consists of the following four terms,

• In the CQM, the gluon field is assumed to be frozen in the ground state and will not contribute to the nucleon spin.

• The only other contribution is the quark orbital angular momentum .

• One would wonder how can quark orbital angular momentum contribute for a pure S-wave configuration?

qL

• The quark orbital angular momentum operator can be expanded as,

• The first term is the nonrelativistic quark orbital angular momentum operator used in CQM, which does not contribute to nucleon spin in a pure valence S-wave configuration.

• The second term is again the relativistic correction, which takes back the relativistic spin reduction.

• The third term is again the creation and annihilation contribution, which also takes back the missing spin.

qq

• 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.

qq

• 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.

VI. Summary

1.The DIS measured quark spin is better to be called quark axial charge, it is not the quark spin calculated in CQM.

2.One can either attribute the nucleon spin solely to the quark Pauli spin, or partly

attribute to the quark axial charge partly to the relativistic quark orbital angular momentum. The following relation should be kept in mind,

3.We suggest to use the physical momentum, angular momentum, etc.

in hadron physics as well as in atomic physics, which is both gauge invariant and canonical commutation relation satisfied, and had been measured in atomic physics with well established physical meaning.

Thanks