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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.
II.
Introduction
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 is
gauge invariant, the matrix elements of
the canonical momentum, orbital angular
momentum, and Hamiltonian of a charged
particle moving in an eletromagnetic field
are gauge dependent, especially
the orbital angular momentum and
energy of the hydrogen atom are
“not the measurable ones” !?
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.
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,
L
A   p eikr  2  L 1i L 2 L  1DMp
( , ,0)[ ALM (m)  ip ALM (e)]
ALM (e)  
L
L 1
 L1T LL 1M 
 L1T LL 1M
2L  1
2L  1
ALM (m)   L T LLM
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 r and p  i, the orbital
angular momentum is
 ,
L  r p  r
i
the Hamiltonian is
( p  e A) 2
H 
 e
2m
Gauge transformation
  '  eie ( x ) ,
A  A  A   ,
'
   '     t ,
The matrix elements transformed as
 | p |   | p |   | e | ,
 | L |   | L |   | er   | ,
 | H |   | H |   | et | ,
even though the Schroedinger equation is
gauge invariant.
New momentum operator in
quantum mechanics
Generalized momentum for a charged particle
moving in em field:
p  mr  q A  mr  q A  q A//
It is not gauge invariant, but satisfies the canonical
momentum commutation relation.
p  q A//  mr  q A
  A  0,
  A//  0
It is both gauge invariant and canonical momentum
commutation relation satisfied.
We call
D phy
1
 p  q A//    q A//
i
i
physical momentum.
It is neither the canonical momentum
1
p  mr  q A  
i
nor the mechanical momentum
1
p  q A  mr  D
i
Gauge transformation
 '  eiq ( x ) ,
A'  A   ( x),
only affects the longitudinal part of the vector potential
A//'  A//   ( x),
and time component
 '     t ( x),
it does not affect the transverse part,
A'  A ,
so A is physical and which is used in Coulomb gauge.
A // is unphysical, it is caused by gauge transformation.
Hamiltonian of hydrogen atom
Coulomb gauge:
c
//
c

A  0,
A  0,
A    0.
c
0
c
Hamiltonian of a nonrelativistic particle
c
2

(p  qA )
Hc 
 q c .
2m
Gauge transformed one
c
//
c

A//  A   ( x)   ( x), A  A ,    c   t ( x)
c
2

( p  q A)
( p  q  q A )
H
 q 
 q c  q t.
2m
2m
2
Follow the same recipe, we introduce a new Hamiltonian,
H phy
c
2

( p  q A//  q A )
 H  q t ( x) 
 q c
2m
   2  A//
which is gauge invariant, i.e.,
 | H phy |    c | H c |  c
This means the hydrogen energy calculated in
Coulomb gauge is gauge invariant and physical.
III.QED
Different approach will obtain different energy-momentum
tensor and four momentum, they are not unique:
Noether theorem

P   d 3 x{    E i Ai }
i
Gravitational theory (weinberg)
3
 D
P   d x{
  E  B}
i
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.
We are experienced in quantum mechanics, so we
introduce
D
P   d 3 x{ 
phy
i
  E iAi }
A  A//  A
D phy    ieA//
They are both gauge invariant and momentum
algebra satisfied. They return to the canonical
expressions in Coulomb gauge.
We proved the renowned Poynting vector is not the
correct momentum of em field
J    d xr  ( E  B)   d x E  A   d xr  E A 
3
3
3
photon spin and
orbital angular momentum
It includes
i
i
Electric dipole radiation field
i
B lm  a h (kr) LYlm ,......E lm  ik Alm    B lm
k
(1)
lm l

1
| a11 |2 3 1  cos 2 
sin 
Re[ E11 B11 ] 

[
nr 
n ]
2
2
(kr) 16
2
kr
2
2
1
|
a
|
3
1

cos

sin 
i
i
11
Re[ E11A11 ] 

[
nr 
n ]
2
2
(kr) 16
2
2kr
dP | a11 |2 3 1  cos 2 
dJ z


k
2
d
k
16
2
d
dJ z | a11 |2 3
2



sin

3
d
k
16





J QED Se  Le  S  L
• 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.




J QED  Se  L'e  J '
• 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.
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!





J QED  Se  Le ' ' S ' ' L ' '
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.
V. QCD
• From QCD Lagrangian, one can get the total
angular momentum by Noether theorem:
• One can have the gauge invariant decomposition,
New decomposition
''
q
''
g
J QCD  S q  L  S  L
Sq 
d
3
x
''
q

L   d 3 x  r 
S
L
''
g
''
g


d
3
a


2
D phy

i
xE  A
a
phy
3
a
a
d
xE
r


A
i
i phy

''
g
Esential task:to define properly the pure gauge
field A pure and physical one A phy
D phy    ig A pure
A pure  T A
a
a
pure
A  A pure  A phy
D phy  A pure    A pure  ig A pure  A pure  0
A phy  E  E  A phy  0
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.
There is no proton spin crisis
but 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 Lq .
• One would wonder how can quark orbital
angular momentum contribute for a pure
S-wave configuration?
• 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 qq creation and
annihilation contribution, which also takes back
the missing spin.
• It is most interesting to note that the relativistic
correction and the qq 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.
Thanks