Abstract
Given a manifoldM and a submanifoldC, together with a Lie groupG acting onM and leavingC invariant, it is shown how the algebra ofG-invariant functions onC can be described in terms of cohomology whenC is defined as the common zero level of an irreducible set of (G-covariant) constraints. The construction is independent of any additional structures such as, e.g., a symplectic structure onM, and therefore it provides a natural framework for a unified description of BRST cohomology both for Lagrangian and Hamiltonian systems. Finally, it is discussed how one can, in various typical situations, replace invariance under an infinite-dimensional gauge group by invariance under a suitable finite-dimensional Lie group; this is a necessary prerequisite for handling BRST cohomology for such systems within a completely finite-dimensional setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abraham, R., Marsden, J. E.: Foundations of mechanics, second edition, updated 1985 printing. Menlo Park: Benjamin Cummings 1978
Arms, J. M., Marsden, J. E., Moncrief, V.: Symmetry and bifurcations of momentum mappings. Commun. Math. Phys.78, 455–478 (1981)
Fradkin, E. S., Vilkovisky, G.: Quantization of relativistic systems with constraints Phys. Lett.55B, 224–226 (1975); Batalin, I. A., Vikovisky, G.: RelativisticS-matrix of dynamical systems with Boson and Fermion constraints. Phys. Lett.69B, 309–312 (1977)
Becchi, C., Rouet, A., Stora, R.: The Abelian Higgs Kibble Model, Unitarity of theS-Operator. Phys. Lett.52B, 344–354 (1974); Tyutin, I. V.: Report Fian39, (1975) (unpublished)
Bott, R., Tu, L. W.: Differential forms in algebraic topology. Berlin, Heidelberg, New York: Springer 1982
Dirac, P. A. M.: Lectures on quantum mechanics. Belfer Graduate School of Science Monograph Series, No.2, New York, 1964
Emmrich, C., Römer, H.: Orbifolds as configuration spaces of systems with gauge symmetries. Commun. Math. Phys.129, 69–94 (1990)
Faddeev, L. D., Popov, V. N.: Feynman diagrams for the Yang-Mills field. Phys. Lett.25B, 29–30 (1967)
Hanson, A., Regge, T., Teitelboim, C.: Constrained Hamiltonian systems. Roma: Academia Nazionale dei Lincei, 1976
Henneaux, M.: Hamiltonian form of the path integral for theories with a gauge freedom. Phys. Rep.126, No. 1, 1–66 (1985)
Henneaux, M., Teitelboim, C.: BRST cohomology in classical mechanics. Commun. Math. Phys.115, 213–230 (1988)
Fish, J., Henneaux, M., Stasheff, J., Teitelboim, C.: Existence, uniqueness and cohomology of the classical BRST charge with ghosts of ghosts. Commun. Math. Phys.120, 379–407 (1989)
Kostant, B., Sternberg, S.: Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Phys.176, 49–113 (1987)
Rund, H.: The Hamilton-Jacobi theory in the calculus of variations. Van Nostrand, 1966
Sundermeyer, K.: Constrained dynamics. Lecture Notes in Physics vol.169, Berlin, Heidelberg, New York: Springer 1982
Tate, J.: Homology of Noetherian rings and local rings. Illinois J. Math.1, 14–27 (1957)
Author information
Authors and Affiliations
Additional information
Communicated by N. Y. Reshetikhin
Rights and permissions
About this article
Cite this article
Forger, M., Kellendonk, J. Classical BRST cohomology and invariant functions on constraint manifolds. I. Commun.Math. Phys. 143, 235–251 (1992). https://doi.org/10.1007/BF02099008
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02099008