1. 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

2. Related recent papers 1)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

3. Earlier and recent Spin sum rules

4. A most familiar example: Solving the hydrogen atom Does this make sense???

5. The atom as a whole

6. Close look at the photon contribution The static terms!

7. Justification of neglecting photon field

8. A critical gap to be closed

9. The same story with Hamiltonian

10. The fortune of using Coulomb gauge

11. Momentum of a moving atom A stationary electromagnetic field carries no momentum

12. Gauge-invariant revision

13. The covariant scheme spurious photon angular momentum

14. Gluon angular momentum in the nucleon: Tree-level One-gluon exchange has the same property as one-photon exchange

15. Beyond the static approximation

16. The tensor gauge field

17. Canonical expression of spin and OAM

19. Complete tensor gauge conditions

20. Vanishing of angular momentum for a stationary tensor gauge field No spurious time- dependence

21. The same property of momentum

22. 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 ）

23. 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!

24. Demonstration with a key issue: Quantum Lorentz Invariance with Tensor Gauge Coupling Lorentz Invariance of Quantum Gravity

25. Two keywords in modern physics Quantization ( 量子化 ) Symmetry ( 对称性 ) 1) Space-time symmetry 2) Gauge symmetry

26. Classical Lorentz Invariance Four-vector gauge field

27. Quantum Lorentz Invariance

28. Clarification on Lorentz Covariance

29. Two approaches towards quantum Lorentz Invariance in gauge theory Covariant quantization with non-physical states Physical quantization with non-covariant propagator

30. Electrodynamics: vector gauge coupling Covariant quantization with non- physical states: Gupta-Bleuler Physical quantization with non- covariant propagator: Coulomb gauge

31. The non-covariant propagator of Physical photon

32. A delicate point: the contact term and its effect

33. Cancelation of the contact term

34. The fortune of Abelian gauge theory: Uniqueness of Coulomb gauge

35. Gravitation as tensor gauge coupling

36. The physical graviton propagator

37. Propagator of general gauge field

38. The contact term and its effect

39. Choice of tensor gauge conditions: Non-uniqueness

40. Hamiltonian in various gauges Test for Quantum Lorentz Invariance!

41. Thank you! 谢谢 !