Download Review on Nucleon Spin Structure

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.
II.
Introduction
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
dr
pm
dt
 eA
It is not gauge invariant!
The gauge invariant one is p  eA , it does not satisfy
the canonical momentum algebra. And so Feynman
called it the velocity operator
( p  eA) / m
Gauge is an internal degree of freedom,
no matter what gauge is used, the canonical
momentum of a charged particle is quantized
as
p  i
The orbital angular momentum is

L  r p  r
i
The Hamiltonian is
( p  e A) 2
H 
 eA0
2m
Under a gauge transformation,
  '  eie ( x ) ,
A  A  A  ,
'
A0  A0'  A0   t ,
the matrix elements transformed as
 | p |   | p |   | e | ,
 | L |   | L |   | er  | ,
 | H |   | H |   | et | .
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, 1982.
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,
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
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 D instead
of the canonical momentum operator p ; uses
the Poynting vector E  B 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)232012.
U(1) Abelian gauge field



A  Apure  Aphys





Fpure   Apure   Apure  0
 Apure  0   Aphys   A
  Aphys  0
A phys ( x  )  0
Aphys
 ' A( x ') 3
B( x ')
  
d x'   
d 3x '
4 x  x '
4 x  x '
• The last expression shows that the Aphys ( x)
is a local space-time function but determined
nonlocally by the magnetic field B ( x ) in a
whole region.
• The Aphys ( x) is measurable.
0
0
i
i
i Aphys  i A  t ( A  Aphys )
0
x
Aphys
( x)   
dxi ( i A0   t Ai   t Aiphys )
1
  dx ( t (  '2 B)i  E i )

'i
'
0
( x) is also a measurable local space• The Aphys
time function.

One can also directly obtain Apure ( x)
first
Apure   ( x)
 ' A( x ') 3
 ( x)   
d x '  0 ( x)
4 x  x '
0
pure
A
 t ( x)
 20 ( x )  0
Apure    ( x)


Under a gauge transformation,
'


A  A    ( x)
The physical and pure gauge parts will be
transformed as
'
Aphys  Aphys
'
pure
A
 Apure   ( x)
0'
0
Aphys
 Aphys
,
0'
pure
A
A
0
pure
  ( x)
0
SU(3) non-Abelian gauge field

a


A  A T  Apure  Aphys
a



Fpure
   Apure   Apure
 ig[ Apure
, Apure ]  0
Dpure    igApure
Dpure  Apure   Apure  igApure  Apure  0
a
Dpure    ig[ Apure ,]
i
i
D

A



A

ig
[
A
,
A
a
pure
phys
phys
pure
phys ]  0
The above equations can be
rewritten as
  Aphys  ig[ Ai , Aiphys ]
 Aphys   A  ig ( A  Aphys )  ( A  Aphys )
0
0
i Aphys
 i A0  t ( Ai  Aiphys )  ig[ Ai  Aiphys , A0  Aphys
]
a perturbative solution in power of g
through iteration can be obtained
Under a gauge transformation,
  U ,
'
U e
 ig ( x )
i
 1
A  UA U  U  U
g
'

'
phys
A
'
pure
A
1
 UAphysU
1
i
1
 UApureU  U U
g
1
III.1 Quantum mechanics
The classical canonical momentum of a
charged particle moving in an electromagnetic
field, an U(1) gauge field, is
dr

pm
 eA 
dt
i
It satisfys the canonical momentum
algebra but its matrix element is
not gauge invariant!
New momentum operator
The new momentum operator is,
p pure  p  eApure

  eApure
i
It satisfies the canonical momentum
commutation relation and its matrix
element is gauge invariant.
We call
p phys  p  eA pure
Dpure
1
   eA pure 
i
i
The physical momentum.
It is neither the canonical momentum
1
p  mr  eA  
i
nor the mechanical momentum
1
p  eA  mr  D
i
Hamiltonian of hydrogen atom
Coulomb gauge
Aphys  A ,
Apure  0,
c
A 
0
c
1
H 
( p  eAc )2  e c
2m
c
Gauge transformed one
'
pure
A
 Apure    , A
'
phys
'
c



  t
 Aphys ,
1
H 
( p  eA' ) 2  e( c   t )
2m
'
Follow the same recipe, we introduce a new Hamiltonian,
H phy  H  et ( x) 
'
c
2
phys
( p  eApure  eA
)
2m
 e c
   2  A
which is gauge invariant, i.e.,
 ' | H phy |  '   c | H c |  c
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

i
i
P   d x{
  E A }
i

3
They are not gauge invariant.
Gravitational theory (Weinberg) or Belinfante tensor
D
P   d x{
  E  B}
i
3

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
D pure
P   d 3 x{ 
i
  E i Aiphys }
A  A pure  A phys
D pure    ieApure
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
J    d xx  ( E  B)   d x( x  E A
3
3
i
i
phys
 E  Aphys )
photon spin and
orbital angular momentum
It includes
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
Usual  Spin  decomposition





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.
Gauge  in var iant  spin  decomposition




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.
It is unphysical!
New spin decomposition for QED
system
J QED  Se  L  S  L
''
e
''
''

Se   d x
,
2

3
L   d x x 
''
e
3

Dpure
i

S''   d 3 xE  Aphys
L   d xE x  A
''
3
i
i
phys
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.
t
h
r
e
e
III.3 QCD
three decompositions of
momentum

P   d x{
  E i A i }
i

3
D
P   d x{
  E  B}
i

3
P   d x
3

Dpure
D pure    ig A pure
i
i
   d 3 xE i a Dpure Aphys
a
D pure    ig[ A pure , ]
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
''
q
''
g
J QCD  S q  L  S  L
Sq 
d
3
x

''
g


2
D pure
L   d x x 

i
''
q
S
''
g

3

3
d
 xE  A phy
L   d 3 xEi x  a Dpure Ai
''
g
phy
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:0904.0321[hep-ph],Phys.Rev.Lett. 103, 062001(2009)
• The nucleon spin structure should be
reexamined based on the new decomposition
and new operators.
arXiv:0806.3166[hep-ph], Phys.Rev.Lett. 100,232002(2008)
Consistent separation of nucleon
momentum and spin
Standard construction of orbital angular momentum L   d 3 x x  P
Quantitative example:
Old quark/gluon momentum in the nucleon

3
1
 Pq   d x i D
If: 
 Pg  d 3 xE  B


 2 ng


P
s
9
2 d  q 

Then Q

2 
dQ  Pg  2  2ng

 9
2 ng
2
Q   : Pg 
PN
2ng  3n f
nf 

3   Pq 
 
n f   Pg 
 
3 
1
PN (n f  5)
2
Proper quark/gluon momentum in
nucleon
 C
3
1
 Pq   d x i Dpure
if: 
 Pg C  d 3 xE ai Aai

phys

nf 
 ng

C
C






P
P
s
18
3
q
q
2 d

 C 
Then: Q


C
2 
n f   Pg 
dQ  Pg  2  ng
 

3 
 18
1
ng
1
2
C
2
Q   : Pg 
PN
PN (n f  5)
1
5
ng  3n f
2
• 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. D58,114032 (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 a0= 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 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.
Conventional and new construction of
parton distribution functions (PDFs)

The pure  gauge term Apure
can be used instead of
the full gauge field A to construct the gauge link
 Wilson line  to accomplish gauge invariance

 The physical term Aphys
can be used instead of
the field strength F  as the gauge  covariant
canonical variable
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
Its first moment gives the gauge-invariant local operator:
j
M g ij  F  i ij  Aphys
,
which is the + component of the gauge-invariant gluon spin
S g  E  Aphys
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