Art of Spin Decomposition

Xiang-Song Chen 陈相松Huazhong Univ. of Sci. & Tech. 华中科技大学

Outline:

Spin sum rules and controversies

A most familiar example

What makes a good spin decomposition

Angular momentum of tensor gauge field

Critical thinking about gauge invariance

Related recent papers1) Art of spin decomposition Xiang-Song Chen, Wei-Min Sun, Fan Wang, T. Goldman, Phys. Rev. D 83, 071901(R) (2011).

2) Proper identification of the gluon spin Xiang-Song Chen, Wei-Min Sun, Fan Wang, T. Goldman, Phys. Lett. B 700, 21 (2011).

3) Physical decomposition of the gauge and gravitational fields Xiang-Song Chen, Ben-Chao Zhu, Phys. Rev. D 83, 084006 (2011).

4) Spin and orbital angular momentum of the tensor gauge field. Xiang-Song Chen, Ben-Chao Zhu, Niall Ó Murchadha, arXiv:1105.6300v1

5) Tensor gauge condition and tensor field decomposition Xiang-Song Chen, Ben-Chao Zhu, arXiv:1101.2809v4

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Earlier and recent Spin sum rules

A most familiar example: Solving the hydrogen atom

Does this 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

Momentum of a moving atom

A stationary electromagnetic field carries no momentum

Gauge-invariant revision

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

The tensor gauge field

Canonical expression of spin and OAM

Canonical expression of spin and OAM

Complete tensor gauge conditions

Vanishing of angular momentum for a stationary tensor gauge field

No spurious time-

dependence

The same property of momentum

Critical thinking about gauge invariance: Gauge theory as a “Compromising” Physical Theory

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

Gauge invariance: “The emperor’s new clothes”?

This clothes is made of

QCD vacuum. Intelligent

people see a splendid

structure. Stupid people see nothing!

Selling for one million dollars!

Wonderful! I buy it.

Great! We have

produced GDP of one million

dollars!

Demonstration with a key issue:Quantum Lorentz Invariance with

Tensor Gauge Coupling

Lorentz Invariance of Quantum Gravity

Two keywords in modern physics

• Quantization (量子化 )

• Symmetry (对称性 ) 1) Space-time symmetry

2) Gauge symmetry

Classical Lorentz Invariance

Four-vector gauge field

Quantum Lorentz Invariance

Clarification on Lorentz Covariance

1 1T. Goldman et al.: using (1,0) (0,1) instead of ( , )

2 2

Two approaches towards quantum Lorentz Invariance in gauge theory

• Covariant quantization with non-physical states

• Physical quantization with non-covariant propagator

Electrodynamics: vector gauge coupling

• Covariant quantization with non-physical states: Gupta-Bleuler

• Physical quantization with non-covariant propagator: Coulomb gauge

The non-covariant propagator of Physical photon

A delicate point: the contact term and its effect

Cancelation of the contact term

The fortune of Abelian gauge theory: Uniqueness of Coulomb gauge

Gravitation as tensor gauge coupling

The physical graviton propagator

Propagator of general gauge field

The contact term and its effect

Choice of tensor gauge conditions: Non-uniqueness

Hamiltonian in various gauges

Test for Quantum Lorentz Invariance!

Thank you!谢谢 !