Unharmony within the Thematic Melodies of Twentieth Century Physics

1. Unharmony within the Thematic Melodies of Twentieth Century Physics X.S.Chen, X.F.Lu Dept. of Phys., Sichuan Univ. W.M.Sun, Fan Wang NJU and PMO Joint Center for particle nuclear physics and cosmology (J-CPNPC)

2. Outline • Introduction • Conflicts between Gauge invariance and Canonical Quantization • Quantum mechanics • QED • QCD VI. Summary

3. I.Introduction • Three Thematic Melodies of Twentieth Century Physics : Symmetry, Quantization, Phase (C.N.Yang) • The combination of symmetry and phase lead to Gauge Invariance Principle and gauge field theory (C.N.Yang) • There are conflicts between these three thematic melodies when one wants to apply them to study the internal structure.

4. Quantum Mechanics Even though the Schroedinger equation is gauge invariant, the matrix elements of the canonical momentum, orbital angular momentum, and Hamiltonian of a charged particle moving in an eletromagnetic field are gauge dependent, especially the orbital angular momentum and energy of the hydrogen atom are “not the measurable ones” !?

5. QED The canonical momentum and orbital angular momentum of electron are gauge dependent and so their physical meaning is obscure. The canonical photon spin and orbital angular momentum operators are also gauge dependent. Their physical meaning is obscure too.

6. Multipole radiation The multipole radiation theory is based on the decomposition of a polarized em wave into multipole radiation field with definite photon spin and orbital angular momentum coupled to a total angular momentum quantum number LM,

7. Multipole radiation measurement and analysis are the basis of atomic, molecular, nuclear and hadronic spectroscopy. If the orbital angular momentum of photon is gauge dependent and not measurable, then all determinations of the parity of these microscopic systems would be meaningless!

8. QCD • Because the parton (quark and gluon) momentum is “gauge dependent”, so the present analysis of parton distribution of nucleon uses the covariant derivative operator instead of the canonical momentum operator, the normal derivative operator as the quark momentum operator; uses the Poynting vector as the gluon momentum operator.

9. The quark spin contribution to nucleon spin has been measured, the further study is encumbered by the lack of gauge invariant quark orbital angular momentum, gluon spin and orbital angular momentum operators. The present gluon spin measurement is even under the condition that “they are measuring a not measurable quantity”.

10. II. Quantum Mechanics Gauge is an internal degree of freedom, no matter what gauge used, the canonical coordinate and momentum of a charged particle is and , the orbital angular momentum is , the Hamiltonian is

11. Gauge transformation The matrix elements transformed as even though the Schroedinger equation is gauge invariant.

12. New momentum operator inquantum mechanics Generalized momentum for a charged particle moving in em field: It is not gauge invariant, but satisfies the canonical momentum commutation relation. It is both gauge invariant and canonical momentum commutation relation satisfied.

13. We call physical momentum. It is neither the canonical momentum nor the mechanical momentum

14. Gauge transformation only affects the longitudinal part of the vector potential and time component it does not affect the transverse part, so is physical and which is used in Coulomb gauge. is unphysical, it is caused by gauge transformation.

15. Coulomb gauge: Hamiltonian of a nonrelativistic particle Gauge transformed one Hamiltonian of hydrogen atom

16. Follow the same recipe, we introduce a new Hamiltonian, which is gauge invariant, i.e., This means the hydrogen energy calculated in Coulomb gauge is gauge invariant and physical.

17. III.QED Different approach will obtain different energy-momentum tensor and four momentum, they are not unique: Noether theorem Gravitational theory (weinberg) It appears to be perfect and has been used in parton distribution analysis of nucleon, but do not satisfy the momentum algebra. Usually one supposes these two expressions are equivalent, because the integral is the same.

18. We are experienced in quantum mechanics, so we introduce They are both gauge invariant and momentum algebra satisfied. They return to the canonical expressions in Coulomb gauge.

19. We proved the renowned Poynting vector is not the correct momentum of em field It includes photon spin and orbital angular momentum

20. Electric dipole radiation field

22. Each term in this decomposition satisfies the canonical angular momentum algebra, so they are qualified to be called electron spin, orbital angular momentum, photon spin and orbital angular momentum operators. • However they are not gauge invariant except the electron spin. Therefore the physical meaning is obscure.

23. How to reconcile these two fundamental requirements, the gauge invariance and canonical angular momentum algebra? • One choice is to keep gauge invariance and give up canonical commutation relation.

25. However each term no longer satisfies the canonical angular momentum algebra except the electron spin, in this sense the second and third term is not the electron orbital and photon angular momentum operator. The physical meaning of these operators is obscure too. • One can not have gauge invariant photon spin and orbital angular momentum operator separately, the only gauge invariant one is the total angular momentum of photon. The photon spin and orbital angular momentum had been measured!

26. Dangerous suggestion It will ruin the multipole radiation analysis used from atom to hadron spectroscopy. Where the canonical spin and orbital angular momentum of photon have been used. Even the hydrogen energy is not an observable, neither the orbital angular momentum of electron nor the polarization (spin) of photon is observable either. It is totally unphysical!

28. Multipole radiation Multipole radiation analysis is based on the decomposition of em vector potential in Coulomb gauge. The results are physical and gauge invariant, i.e., gauge transformed to other gauges one will obtain the same results.

29. V. QCD • From QCD Lagrangian, one can get the total angular momentum by Noether theorem:

30. One can have the gauge invariant decomposition,

31. New decomposition

32. Esential task:to define properly the pure gauge field and physical one

33. VI. Summary • The renowned Poynting vector is not the right momentum operator of em field. • The gauge invariance and canonical quantization rule for momentum, spin and orbital angular momentum can be satisfied simultaneously. • The Coulomb gauge is physical, expressions in Coulomb gauge, even with vector potential, are gauge invariant, including the hydrogen atomic Hamiltonian and multipole radiation.

34. There is no proton spin crisisbut quark spin confusion

35. An Extended CQM with Sea Quark Components • To understand the nucleon spin structure quantitatively within CQM and to clarify the quark spin confusion further we developed a CQM with sea quark components,

41. Where does the nucleon get its Spin • As a QCD system the nucleon spin consists of the following four terms,

42. In the CQM, the gluon field is assumed to be frozen in the ground state and will not contribute to the nucleon spin. • The only other contribution is the quark orbital angular momentum . • One would wonder how can quark orbital angular momentum contribute for a pure S-wave configuration?

43. The quark orbital angular momentum operator can be expanded as,

44. The first term is the nonrelativistic quark orbital angular momentum operator used in CQM, which does not contribute to nucleon spin in a pure valence S-wave configuration. • The second term is again the relativistic correction, which takes back the relativistic spin reduction. • The third term is again the creation and annihilation contribution, which also takes back the missing spin.

45. It is most interesting to note that the relativistic correction and the creation and annihilation terms of the quark spin and the orbital angular momentum operator are exact the same but with opposite sign. Therefore if we add them together we will have where the , are the non-relativistic part of the quark spin and angular momentum operator.

46. The above relation tell us that the nucleon spin can be either solely attributed to the quark Pauli spin, as did in the last thirty years in CQM, and the nonrelativistic quark orbital angular momentum does not contribute to the nucleon spin; or • part of the nucleon spin is attributed to the relativistic quark spin, it is measured in DIS and better to call it axial charge to distinguish it from the Pauli spin which has been used in quantum mechanics over seventy years, part of the nucleon spin is attributed to the relativistic quark orbital angular momentum, it will provide the exact compensation missing in the relativistic “quark spin” no matter what quark model is used. • one must use the right combination otherwise will misunderstand the nucleon spin structure.

47. VI. Summary 1.The DIS measured quark spin is better to be called quark axial charge, it is not the quark spin calculated in CQM. 2.One can either attribute the nucleon spin solely to the quark Pauli spin, or partly attribute to the quark axial charge partly to the relativistic quark orbital angular momentum. The following relation should be kept in mind,

48. 3.We suggest to use the physical momentum, angular momentum, etc. in hadron physics as well as in atomic physics, which is both gauge invariant and canonical commutation relation satisfied, and had been measured in atomic physics with well established physical meaning.

49. Thanks