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

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

Introductions Dynamical Systems Newton Formalism Lagrange Formalism Hamilton Formalism Singular Lagrangian Systems Constrained Hamiltonian Systems Gauge Theories Quantization Of Gauge Systems Symmetries The Dirac Conjecture 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, 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) Symmetries in Constrained Hamiltonian Systems and Applications. Yong-Long Wang, De-Yu Zhao, Shandong People’s Publishing House, 2012 (in Chinese) ① Regular systems Canonical Systems

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

Introductions ③ Lagrange FormalismHamilton Formalism The higher-stage constraints The primary constraints

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

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 g 0, its value at time is Introductions ⑤ small arbitrary

Introductions ⑥

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

Introductions ⑧ The secondary constraints can be deduced by the consistency of the primary constraints as (1)The original Lagrangian equations of motion are inconsistent. (2)One kind of equations reduces as 0=0. (3)To determine the arbitrary function of the Lagrangian multiplier. (for second- class constraints) (4)Turn up new constraints.

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

Non-uniqueness of Lagrangian ① The prime 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, Classical and Quantum Constrained Systems and Their Symmetries. Zi-Ping Li, 1993 (in Chinese)

② Non-uniqueness of Lagrangian

③

④ A new annulation Terminate: No new constraint.

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

⑥ Non-uniqueness of Lagrangian i th -Stage S th -Stage No new constraints. End!

⑦ Non-uniqueness of Lagrangian (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. The Dirac Conjecture is valid. PRD32,405(1985); PRD42,2726(1990)

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)

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

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

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

Cawley’s Example ⑤

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)

“Counterexample” ①

“Counterexample” ②

Conclusions ① (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.

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