Unharmony within the
Thematic Melodies of Twentieth Century Physics
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)
Outline
I. Introduction
II. Conflicts between Gauge invariance and Canonical Quantization
III. Quantum mechanics
IV. QED
V. QCD
VI. 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 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 TL
LT
L
LeA 1111 12
1
12)(
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
em
AepH
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 canonical
momentum commutation relation.
It is both gauge invariant and canonical momentum
commutation relation satisfied.
0// A
We call
physical momentum.
It is neither the canonical momentum
nor the mechanical momentum
////1
Aqi
Aqpi
D phy
i
Aqrmp1
Di
rmAqp1
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 qm
AqpH
)(,),()(//// xAAxxAA tc
cc
Coulomb gauge:
Hamiltonian of a nonrelativistic particle
Gauge transformed one
,0// cA .00 ccA
.2
)(
2
)( 22
tc
c
qqm
Aqqpq
m
AqpH
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 gauge invariant and physical.
c
c
tphy qm
AqAqpxqHH
2
)()(
2//
cc
cphy HH ||||
//2 A
III.QED
Different approach will obtain different energy-momentum
tensor 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 BEi
DxdP
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.
}{3 iiphy
AEi
DxdP
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 Bk
iAikEYLkrhaB ,......)()1(
]sin
2
cos1[
16
3
)(
||]Re[
2
1 2
2
211
1111
n
krn
kr
aBE r
]2
sin
2
cos1[
16
3
)(
||]Re[
2
1 2
2
211
1111
n
krn
kr
aAE rii
d
dJk
k
a
d
dP z
2
cos1
16
3|| 2
2
211
23
211 sin
16
3||
k
a
d
dJ z
• 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.
• 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!
Multipole radiation
Multipole radiation analysis is based on the
decomposition of em vector potential in
Coulomb gauge. The results are physical
and gauge invariant, i.e.,
gauge transformed to other gauges one will
obtain the same results.
New decomposition''''''
ggqqQCD LSLSJ
2
3xdS q
i
DrxdL
phyq
3''
a
phy
a
g AExdS 3''
phyai
aig ArxEdL 3
''
Esential task:to define properly the pure gauge field and physical one
purephy AigD a
purea
pure 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.
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 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.
• 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.
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.