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

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

Outline

Nucleon spin comes from

the polarization and orbital motion

of quarks and gluons

--- Chairman Mao

A universally correct statement for the

nucleon spin

Controversy in nucleon spin structure

t3 3 3 3

3 3

otal

tota3

l

Jaffe-Manohar [NPB337:509 (1990)]

Ji [PRL78:610 (1997)], Chen-Wang [CTP27:

1 1

2

1

212 (1997)]

Chen

1

2

-Lu-S

i id x d x dx E A x E Ai

x D x E B

J x d x

d x d x d xi

J

3 3pure phys pure phy

3 3total s

t

un-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]

Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501

1 1D

2

(2011)]

i ix D E A x E Ai

d x x dJ

J

d x d x

3 3phys pure potal hys phys

3 31 1( D )

2i i a ax D E A x Ed x d x d x d

ix A A

Leader [PRD 83:096012 (2011)]

Interacting theory: Structure of Poincare generators

int

i t

n

n

i t

"bad" genera

Lagrangian:

Spatial translation and rotation are

"Good" generators

kine

tors

a b

a b

a b

a

a b

b

P P P

J

H = H H H

K K KJ KJ

L L L L

Time translation and Lorentz boost ar d

matic

e ynamic

Without gauge symmetry, the issue is trivial:

Interacting theory: Poincare (sub)algebra

( , ) ( , )

( , ) ( , )

( ,

,

)

,

, ,

[ , ]

Kinematic transformation [ , ]

[ , ] 0

[ , ]Dynamic transformat

Only total

ion [ , ]

and are

i j ka

i j ka b ijk

b ijk a bi j k

a b ijk

a bi

a b

ja b a b ij

i ja b

K J i K

K P

J

J J i J

J P i P

P

i

P

H

P

[ , ] covariant:

[ , ]

i j kijk

i jij

K J i K

K P iH

Further criteria by Elliot Leader

Corollary

Examination of various decompositions by Leader’s criteria

t3 3 3 3

3 3

otal

tota3

l

Jaffe-Manohar [NPB337:509 (1990)]

Ji [PRL78:610 (1997)], Chen-Wang [CTP27:

1 1

2

1

212 (1997)]

Chen

1

2

-Lu-S

i id x d x dx E A x E Ai

x D x E B

J x d x

d x d x d xi

J

3 3pure phys pure phy

3 3total s

t

un-Wang-Goldman [PRL100:232002 (2008); 103:062001 (2009)]

Wakamatsu [PRD81:114010(2010); 83:014012 (2011); 84:037501

1 1D

2

(2011)]

i ix D E A x E Ai

d x x dJ

J

d x d x

3 3phys pure potal hys phys

3 31 1( D )

2i i a ax D E A x Ed x d x d x d

ix A A

Leader [PRD 83:096012 (2011)]

Generators for the gauge-invariant physical fields - translation

Generators for the gauge-invariant physical fields - Rotation

The quark-gluon system

Generator for the gauge-invariant quark field

Generator for the gauge-invariant gluon field

Some detail in the proof

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

A practical thinking about nucleon structure

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

0

)( ' 3

BErxdJ g

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

2

3

32

1

,

2

9 322

9 3

q

q

g f

q qs

g gg f

P d x Di

P d xE B

n nP Pd

QP Pn ndQ

2 2:

1( 5)

22 3g

g Ng f

N f

nQ P P

nP n

n

phys

3pure

3pure

22

ˆ ˆ3

ˆ

1ˆ18

1

ˆ2

3

8

f

q qs

fg g

g

q

ig

gi

n

n

P d x Di

P

nP Pd

QndQ Pd xE A P

D

Quark and gluon momentum in the nucleon

2 ˆ: / 2 1

(5

52 3

)/g

gg N N f

f

P nPn

Qn

Pn

Weinberg’s approach: derivation of QED with physical photonsS. 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”?

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）

Conditions Results

Need for the physical variable:

Real emergence of a photon

A possible real difference

i iE A E x A

Dip

ole rad

.

11

ikreB LY

ikri

E B i Ak

(rad

. g

au

ge)

l=1

m=1

E B 21 cosdP

d

E

Flu

x

J Flu

x

x E B

21 coszdJd

22sinzdJd

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 non-Abelian gauge theory with physical gluons by requiring Lorentz

invariance???

Derivation of QED with physical photons by requiring Lorentz invariance

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 non-

Abelian theory is more tricky and not yet proven.

demonstrated.

Summary

Thank you!谢谢 !