Abstract
One may introduce at least three different Lie algebras in any Lagrangian field theory: (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin–Vilkovisky antibracket, which has attracted considerable interest recently; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number −1 and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number −1, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from −1, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin, and Vilkovisky.
REFERENCES
- 1. J. Zinn-Justin, “Renormalisation of gauge theories,” in Trends in Elementary Particle Theory, Lecture Notes in Physics No. 37 (Springer-Verlag, Berlin, 1975); Google Scholar
B. W. Lee, “Gauge theories,” in Methods in Field Theory, Les Houches Lectures 1975, edited by R. Balian and J. Zinn-Justin (North-Holland, New York, 1976); Google Scholar
H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D12, 467, 482 (1975); Google Scholar
J. A. Dixon, Nucl. Phys. B99, 420 (1975). Google ScholarCrossref - 2. I. A. Batalin and G. A. Vilkovisky, Phys. Lett. 102B, 27 (1981); Google ScholarCrossref
I. A. Batalin and G. A. Vilkovisky, Phys. Rev. D 28, 2567 (1983); , Google ScholarCrossref
I. A. Batalin and G. A. Vilkovisky, Phys. Rev. D 30, 508 (1984). , Google ScholarCrossref - 3. M. Bochicchio, Phys. Lett. B 193, 31 (1987); Google ScholarCrossref
C. B. Thorn, Phys. Rep. 175, 1 (1989); , Google ScholarCrossref
B. Zwiebach, Nucl. Phys. B390, 33 (1993); , Google ScholarCrossref
E. Witten, Phys. Rev. D 46, 5467 (1992); , Google ScholarCrossref
E. Verlinde, Nucl. Phys. B381, (1992); , Google Scholar
H. Hata and B. Zwiebach, Ann. Phys. 229, 177 (1994). Google ScholarCrossref - 4. L. A. Dickey, Soliton Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics (World Scientific, Singapore, 1991), Vol. 12. Google Scholar
- 5. E. S. Fradkin and G. A. Vilkovisky, Phys. Lett. B 55, 224 (1975); Google ScholarCrossref
I. A. Batalin and G. A. Vilkovisky, Phys. Lett. B 69, 309 (1977); , Google ScholarCrossref
E. S. Fradkin and T. E. Fradkina, Phys. Lett. B 72, 343 (1977). , Google ScholarCrossref - 6. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, 1992). Google Scholar
- 7. P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics (Springer-Verlag, New York, 1986), Vol. 107. Google Scholar
- 8. G. Barnich, F. Brandt, and M. Henneaux, “Local BRST cohomology in the antifield formalism: I. General theorems”, Commun. Math. Phys. 174, 57 (1995). Google ScholarCrossref
- 9. M. Dubois-Violette, M. Henneaux, M. Talon, and C. M. Viallet, Phys. Lett. B 267, 81 (1991). Google ScholarCrossref
- 10. G. Barnich, M. Henneaux, and C. Schomblond, Phys. Rev. D 44, 939 (1991). Google ScholarCrossref
- 11. J. Fisch and M. Henneaux, Phys. Lett. B 226, 80 (1989); Google ScholarCrossref
W. Siegel, Int. J. Mod. Phys. A 4, 3951 (1989). , Google ScholarCrossref - 12. P. Mc Cloud, Class. Quantum Grav. 11, 567 (1994). Google ScholarCrossref
- 13. D. J. Saunders, The Geometry of Jet Bundles, London Mathematical Society Lecture Note Series 142 (Cambridge University Press, Cambridge, 1989). Google Scholar
- 14. I. M. Anderson, The Variational Bicomplex (Academic Press, Boston, 1994); Google Scholar
Contemp. Math. 132, 51 (1992). Google ScholarCrossref - 15. Th. De Donder, Théorie Invariantive du Calcul des Variations (Gauthier-Villars, Paris, 1935). Google Scholar
- 16. E. Witten, Mod. Phys. Lett. A 5, 487 (1990). Google ScholarCrossref
- 17. G. Barnich, Mod. Phys. Lett. A 9, 665 (1994). Google ScholarCrossref
- © 1996 American Institute of Physics.
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
