Dynamics of gauge systems and Dirac’s conjecture

M. E. V. Costa; H. O. Girotti; T. J. M. Simes

doi:10.1103/PhysRevD.32.405

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Dynamics of gauge systems and Dirac’s conjecture

M. E. V. Costa, H. O. Girotti, and T. J. M. Simes

Instituto de Física, Universidade Federal do Rio Grande do Sul, 90000 Porto Alegre, RS, Brasil

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Phys. Rev. D 32, 405 – Published 15 July, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.405

Abstract

For systems possessing only first-class constraints, we rigorously prove that the secondary constraints act as independent generators of gauge transformations (Dirac’s conjecture). The proof essentially consists in demonstrating that the total and the extended Hamiltonian generate the same time evolution for the canonical realization, f(q,p), of the gauge-invariant quantities. We also discuss the alternative realization of these quantities.

References (23)

G. R. Allcock, Philos. Trans. R. Soc. London A279, 487 (1974).
R. Cawley, Phys. Rev. Lett. 42, 413 (1979); Phys. Rev. D 21, 2988 (1980).
A. Frenkel, Phys. Rev. D 21, 2986 (1980).
N. Mukunda, Phys. Scr. 21, 783 (1980).
L. Castellani, Ann. Phys. (N.Y.) 143, 357 (1982).
R. Sugano, Prog. Theor. Phys. 68, 1377 (1982).
R. Sugano and H. Kamo, Prog. Theor. Phys. 67, 1966 (1982).
R. Sugano and T. Kimura, Prog. Theor. Phys. 69, 252 (1983); ibid. 69, 1256 (1983).
R. Sugano and T. Kimura, J. Phys. A 16, 4417 (1983).
N. H. Christ, A. H. Guth and E. J. Weinberg, Nucl. Phys. B114, 61 (1976).
N. H. Christ and T. D. Lee, Phys. Rev. D 22, 939 (1980).
M. E. V. Costa and H. O. Girotti, Phys. Rev. D 24, 3323 (1981).
E. S. Fradkin and G. A. Vilkovisky, Report No. CERN TH. 1232, 1977 (unpublished).
L. Faddeev, Teor. Mat. Fiz. 1, 3 (1969) [Theor. Math. Phys. 1, 1 (1970)].
E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (Wiley, New York, 1974).
N. Mukunda, Ann. Phys. (N.Y.) 99, 408 (1976).
P. Senjanovic, Ann. Phys. (N.Y.) 100, 227 (1976).
J. L. Anderson and P. G. Bergmann, Phys. Rev. 83, 1018 (1951).
We notice that the SFCC's arising through the algorithm of Castellani (Ref. 6) are, in the general case, linear combinations of the PFCC's and SFCC's detected via Dirac's algorithm (Ref. 1).
R. Di Stefano, Phys. Rev. D 27, 1752 (1983).
Our metric convention is $g_{00}$ = - $g_{11}$ = - $g_{22}$ = - $g_{33}$ = + 1,~ $g_{μ ν}$ = 0 if μ != nu.
Omitted end note.
H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959), p. 227.

[c1] G. R. Allcock, Philos. Trans. R. Soc. London A279, 487 (1974).

[c2] R. Cawley, Phys. Rev. Lett. 42, 413 (1979); Phys. Rev. D 21, 2988 (1980).

[c3] A. Frenkel, Phys. Rev. D 21, 2986 (1980).

[c4] N. Mukunda, Phys. Scr. 21, 783 (1980).

[c5] L. Castellani, Ann. Phys. (N.Y.) 143, 357 (1982).

[c6] R. Sugano, Prog. Theor. Phys. 68, 1377 (1982).

[c7] R. Sugano and H. Kamo, Prog. Theor. Phys. 67, 1966 (1982).

[c8] R. Sugano and T. Kimura, Prog. Theor. Phys. 69, 252 (1983); ibid. 69, 1256 (1983).

[c9] R. Sugano and T. Kimura, J. Phys. A 16, 4417 (1983).

[c10] N. H. Christ, A. H. Guth and E. J. Weinberg, Nucl. Phys. B114, 61 (1976).

[c11] N. H. Christ and T. D. Lee, Phys. Rev. D 22, 939 (1980).

[c12] M. E. V. Costa and H. O. Girotti, Phys. Rev. D 24, 3323 (1981).

[c13] E. S. Fradkin and G. A. Vilkovisky, Report No. CERN TH. 1232, 1977 (unpublished).

[c14] L. Faddeev, Teor. Mat. Fiz. 1, 3 (1969) [Theor. Math. Phys. 1, 1 (1970)].

[c15] E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (Wiley, New York, 1974).

[c16] N. Mukunda, Ann. Phys. (N.Y.) 99, 408 (1976).

[c17] P. Senjanovic, Ann. Phys. (N.Y.) 100, 227 (1976).

[c18] J. L. Anderson and P. G. Bergmann, Phys. Rev. 83, 1018 (1951).

[c19] We notice that the SFCC's arising through the algorithm of Castellani (Ref. 6) are, in the general case, linear combinations of the PFCC's and SFCC's detected via Dirac's algorithm (Ref. 1).

[c20] R. Di Stefano, Phys. Rev. D 27, 1752 (1983).

[c21] Our metric convention is $g_{00}$ = - $g_{11}$ = - $g_{22}$ = - $g_{33}$ = + 1,~ $g_{μ ν}$ = 0 if μ != nu.

[c22] Omitted end note.

[c23] H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959), p. 227.

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Dynamics of gauge systems and Dirac’s conjecture

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