Poincare sub-algebra and
gauge invariance
in nucleon structure
Xiang-Song Chen
Huazhong University of Science & Technology
陈相松 •华中科技大学•武汉
10 July 2012 @ KITPC-Beijing
Outline
I. Controversy in nucleon (spin) structure
II. Elliot Leader’s criteria of separating
momentum and angular momentum
III. Reconciling Poincare sub-algebra with
gauge invariance
IV. A practical thinking about nucleon
structure
V. A critical thinking about gauge invariance
A universally correct
statement for the
nucleon spin
Nucleon spin comes from
the polarization and orbital motion
of quarks and gluons
--- Chairman Mao
Controversy in nucleon spin structure
Leader [PRD 83:096012
Jaffe-Manohar [NPB337:509 (1990)]
1
1
J total   d 3 x     d 3 xx      d 3 xE  A   d 3 xx  E iAi
2
i
(2011)]
Ji [PRL78:610 (1997)], Chen-Wang [CTP27:212 (1997)]
J total   d 3 x 

1
1
   d 3 xx   D   d 3 xx  E  B
2
i

Chen-Lu-Sun-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]
1
1
i
J total   d 3 x     d 3 xx   Dpure   d 3 xE  Aphys   d 3 xx  E i Dpure Aphy
s
2
i
Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501 (2011)]
J total   d 3 x 
1
1
a
   d 3 xx   D   d 3 xE  Aphys   d 3 xx  ( E i D pure Api hys  Aphys
a)
2
i
Interacting theory:
Structure of Poincare generators
Without gauge symmetry, the issue is trivial:
Lagrangian: L  La  Lb  Lint
"Good" generators
"bad" generators
 P  Pa  Pb
 H = H a  H b  H int


 J  J a  J b
 K  K a  Kb  Kint
Spatial translation and rotation are kinematic
Time translation and Lorentz boost are dynamic
Interacting theory:
Poincare (sub)algebra
[ J i , J (ja ,b ) ]  i ijk J (ka ,b )
 i j
k
Kinematic transformation [ J , P( a ,b ) ]  i ijk P( a ,b )
[ P i , P j ]  0
( a ,b )

[ K i , J aj,b ]  i ijk K ak,b
Dynamic transformation  i j
[ K , Pa ,b ]  iH a ,b ij
[ K i , J j ]  i ijk K k
Only total J and P are covariant: 
i
j
[
K
,
P
]  iH  ij

Further criteria by Elliot Leader
Corollary
Examination of various decompositions
by Leader’s criteria
Leader [PRD 83:096012
Jaffe-Manohar [NPB337:509 (1990)]
1
1
J total   d 3 x     d 3 xx      d 3 xE  A   d 3 xx  E iAi
2
i
(2011)]
Ji [PRL78:610 (1997)], Chen-Wang [CTP27:212 (1997)]
J total   d 3 x 

1
1
   d 3 xx   D   d 3 xx  E  B
2
i

Chen-Lu-Sun-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]
1
1
i
J total   d 3 x     d 3 xx   Dpure   d 3 xE  Aphys   d 3 xx  E i Dpure Aphy
s
2
i
Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501 (2011)]
J total   d 3 x 
1
1
a
   d 3 xx   D   d 3 xE  Aphys   d 3 xx  ( E i D pure Api hys  Aphys
a)
2
i
Generators for the gauge-invariant
physical fields - translation
Generators for the gauge-invariant
physical fields - Rotation
The quark-gluon system
Generator for the gaugeinvariant quark field
Generator for the gaugeinvariant gluon field
Some detail in the proof
A practical thinking about
nucleon structure
 Hint from a forgotten practice: Why
photon is ignored for atomic spin?
 The fortune of choosing Coulomb gauge
 Quantitative differences
 Another example: momentum of a
moving atom and nucleon
Hint from a forgotten practice: Why
photon is ignored for atomic spin?
Do these solutions make sense?!
The atom as a whole
Close look at the photon contribution
The static terms!
Justification of neglecting photon field
A critical gap to be closed
The same story with Hamiltonian
The fortune of using Coulomb gauge
Gauge-invariant revision
– Angular Momentum
Gauge-invariant revision
-Momentum and Hamiltonian
The covariant scheme

spurious photon angular momentum
Gluon angular momentum in the nucleon:
Tree-level




3
J ' g   d x r  ( E  B)
0
One-gluon exchange has the same
property as one-photon exchange
Beyond the static approximation
Momentum of a moving atom
A stationary electromagnetic field carries
no momentum
Quark and gluon momentum in the nucleon
2ng

1

3



P

d
x

D

P


s
 q 
q
9
2 d
i

,
Q



2 
dQ  Pg  2  2ng
 Pq  d 3 xE  B



 9
Q :
2
Pg 
2ng
2ng  3n f
PN
1
PN (n f  5)
2
ng

1
ˆ
3


ˆ

P

d
x

D



P
pure
s
 q 
q
18
2 d
i



Q


2
 Pˆ  2  ng
dQ
 Pg  d 3 xE i Dpure Ai
g 



phys

 18
Q :
2
Pˆg 
ng / 2
ng / 2  3n f
PN
nf 
 P 
3  q
 
n f   Pg 
 
3 
nf 
  Pˆ 
3  q 
n f   Pˆg 
  
3 
1
PN (n f  5)
5
Weinberg’s approach: derivation of QED
with physical photons
S. Weinberg, Phys. Rev. 138 (1965) B988
S. Weinberg, Phys. Rev. 138 (1965) B988
S. Weinberg, Phys. Rev. 138 (1965) B988
The non-covariant propagator
of Physical photon
A delicate point:
the contact term and its effect
Cancelation of the contact term
Is gauge-invariance a “Compromise”,
or even “illusion”?
Conditions  Results
 First step in Physics：Complete Description
 Classic Physics: r and p（controllable）
 Quantum Mechanics：Wave Function
（Not completely controllable）
 Gauge Theory：Gauge potentials
(Completely uncontrollable）
Need for the physical variable:
Real emergence of a photon
A possible real difference
l=1
m=1
e
B
LY11
ikr
i
E    B  i A
k
E Flux
EB
J Flux
E  A  Ei x Ai

x EB

dP
dJ z
dJ z
(rad. gauge)
Dipole rad.
ikr
d
 1  cos  
d
 1  cos  
d
2
2
 2sin 2 
If we never need physical gluons ……
Then QCD is a true gauge theory, and the
only try gauge theory so far known
And all quark and gluon quantities are a
matter of definition
Do we sometimes need physical gluons?
Probably, in early universe
Then color gauge invariance may also be
an illusion!
What about SU(2)XU(1) and Higgs?
Derivation of QED with physical photons by
requiring Lorentz invariance
Derivation of non-Abelian gauge theory with
physical gluons by requiring Lorentz invariance???
Summary
I. Nucleon spin and momentum can be separated gauge
invariantly, with quark/gluon part acting as the
rotation and translation generators for the physical
quark/gluon field.
II. If adopting the naive free-form expression, Coulomb
gauge gives the simplest pictures for atomic and
nucleon structure.
III. Gauge symmetry might be an illusion. QED can be
derived from physical photons by requiring Lorentz
invariance of S matrix, but the situation for nonAbelian theory is more tricky and not yet proven.
Thank you!
谢谢!