The Dirac Conjecture and the Non-uniqueness of Lagrangian

1. The First Sino-Americas Workshop and School on the Bound-State Problem in Continuum QCD Oct. 22-26, 2013, USTC, Hefei The Dirac Conjecture and the Non-uniqueness of Lagrangian Wang Yong-Long Department of Physics, School of Science, Linyi University Department of Physics, Nanjing University

2. Outline Introductions Non-uniqueness of Lagrangian Cawley’s Example “Counterexample” Conclusions . arXiv:1306.3580

3. Introductions Dynamical Systems ① Hamilton Formalism Newton Formalism Lagrange Formalism Regular systems Canonical Systems Singular Lagrangian Systems Constrained Hamiltonian Systems Gauge Theories Classical and Quantum Constrained Systems and Their Symmetries. Zi-Ping Li, Press of Beijing University of Technology, 1993 (in Chinese) Constrained Hamiltonian Systems and Their Symmetries. Zi-Ping Li. Press of Beijing University of Technology, 1999 (in Chinese) Symmetries in Constrained Canonical Systems, Li Zi-Ping, Science Press, 2002 Quantum Symmetries in Constrained Systems. Zi-Ping Li, Ai-Min Li. Press of Beijing University of Technology, 2011 (in Chinese) Quantization Of Gauge Systems Symmetries Quantization of Gauge Systems, edited by M. Henneaux, C. Teitelboim, Princeton University, 1991 Gauge Fields Introduction to Quantum Theory, edited by L. D. Faddeev and A. A. Slavnov, The Benjamin, 1980. Symmetries in Constrained Hamiltonian Systems and Applications. Yong-Long Wang, De-Yu Zhao, Shandong People’s Publishing House, 2012 (in Chinese) The Dirac Conjecture

4. Introductions Quantization of Gauge Systems ② Canonical Quantization Path Integral Quantization BRST Batalin- Fradkin- Vilkovsky Faddeev- Popov BRST Batalin- Fradkin- Vilkovsky Dirac’s Formalism Faddeev- Jackiw’s Formalism Faddeev- Senjanovic The Dirac Conjecture

5. Introductions ③ Lagrange Formalism Hamilton Formalism The primary constraints The higher-stage constraints

6. Introductions ④ According to the consistency of the , we can obtain the Lagrange multipliers with respect to primary second-class constraints

7. small arbitrary Introductions ⑤ In terms of the total Hamiltonian, for a general dynamical variable g depending only on the q’s and the p’s, with initial value g0, its value at time is

8. Introductions ⑥

9. It is arbitrary Introductions ⑦ P. C. Dirac, Can. J. Math. 2, 147(1950); Lectures on Quantum Mechanics

10. Introductions ⑧ The secondary constraints can be deduced by the consistency of the primary constraints as • The original Lagrangian equations of motion are inconsistent. • One kind of equations reduces as 0=0. • To determine the arbitrary function of the Lagrangian multiplier. (for second-class constraints) • Turn up new constraints.

11. generators generators ? left by Dirac Introductions ⑨ Dirac conjecture: All first-class constraints are generators of gauge transformations, not only primary first-class ones.

12. Non-uniqueness of Lagrangian ① Theprime Hamiltonian consists of the canonical Hamiltonian and all primary second-class constraints, the number can be determined by the rank of the matrix . denotes all first-class primary constraints. Classical Mechanics, H. Goldstein, 1980. Classical and Quantum Constrained Systems and Their Symmetries. Zi-Ping Li, 1993 (in Chinese)

13. Non-uniqueness of Lagrangian ②

14. Non-uniqueness of Lagrangian ③

15. Non-uniqueness of Lagrangian ④ A new annulation Terminate: No new constraint.

16. 0th-stage 1st-stage Non-uniqueness of Lagrangian ⑤

17. Non-uniqueness of Lagrangian ⑥ ith-Stage Sth-Stage No new constraints. End!

18. Non-uniqueness of Lagrangian ⑦ The Dirac Conjecture is valid. (2) The total time derivatives of constraints to Lagrangian may turn up new constraints. In terms of the stage total Hamiltonian, the consistencies of constraints can generate all constraints implied in the constrained system. PRD32,405(1985); PRD42,2726(1990)

19. Cawley’s Example ① R. Cawley, PRL, 42,413(1979); PRD21, 2988(1980) L. Lusanna, Phys. Rep. 185,1(1990); Riv. Nuovo Cimento 14(3), 1(1991)

20. Cawley’s Example ② L. Lusanna, Riv. Nuovo Cimento 14(3), 1-75(1991)

21. Cawley’s Example ③ 0th-stage 1th-stage

22. Cawley’s Example ④ 2th-stage 3th-stage

23. Cawley’s Example ⑤

24. Cawley’s Example ⑥ In the Cawley example, we must consider the secondary constraints. A. A. Deriglazov, J. Phys. A40, 11083(2007); J. Math. Phys. 50,012907(2009)

25. “Counterexample” ①

26. “Counterexample” ②

27. Conclusions ① • The Dirac conjecture is valid to a system with singular Lagrangian. • (2) The extended Hamiltonian shows symmetries more obviously than the total Hamiltonian in a constrained system.

28. Thanks!