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 !