Unharmonywithin the

Thematic Melodies of Twentieth Century Physics

X.S.Chen, X.F.LuDept. 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

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

• There are conflicts between these three thematic melodies

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, especiallythe 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 multipoleradiation 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

parityof 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 isgauge 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

AqpiD 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

,0��cA

.2

)( 2 cc

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 qmAqAqpxqHH � ������� �

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 partondistribution 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 AEiDxdP ��

��� AAA //

//ieAD phy ��

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

It includes photon spin andorbital 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 rii �

�����

��

���

� 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 analysisused 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

3 xdS q

� �� � �� iDrxdL phyq 3

''

� ��aphy

ag AExdS 3''

� �� phyaiaig 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

quantization 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.

Thanks

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