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 papers 1)Art of spin decomposition Xiang-Song Chen, Wei-Min Sun, Fan Wang, T. Goldman, Phys. Rev. D 83, (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, (2011). 4) Spin and orbital angular momentum of the tensor gauge field. Xiang-Song Chen, Ben-Chao Zhu, Niall Ó Murchadha, arXiv: v1 5) Tensor gauge condition and tensor field decomposition Xiang-Song Chen, Ben-Chao Zhu, arXiv: v4

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 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

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 ）

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

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 !