Abstract
This is the first part of a two-part paper dedicated to the definition of BRST quantization in the framework of geometric quantization. After recognizing prequantization as a manifestation of the Poisson module structure of the sections of the prequantum line bundle, we define BRST prequantization and show that it is the homological analog of the symplectic reduction of prequantum data. We define a prequantum BRST cohomology theory and interpret it in terms of geometric objects. We then show that all Poisson structures correspond under homological reduction. This allows to prove, in the BRST context, that prequantization and reduction commute.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Guillemin, V., Sternberg, S.: Invent. Math.67, 515 (1982)
Duval, C., Elhadad, J., Tuynman, G.M.: CPT Marseille Preprint CPT-89/P.2248
Figueroa-O'Farrill, J.M., Henneaux, M., Kimura, T.: In preparation
Huebschmann, J., Poisson cohomology and quantization, Heidelberg Preprint 1989
Guillemin, V., Sternberg, S.: Sympletic techniques in physics. Cambridge: Cambridge University Press, 1984
Marsden, J., Weinstein, A.: Rep. Math. Phys.5, 121 (1974)
Batalin, I.A., Vilkovisky, G.A.: Phys. Lett. B69, 309 (1977); Henneaux, M.: Phys. Rep.126, 1 (1985); McMullan, D.: J. Math. Phys.28, 428 (1987); Browning, A.D., McMullan, D.: J. Math. Phys.28, 438 (1987); Kostant, B., Sternberg, S.: Ann. Phys.176, 49 (1987); Stasheff, J.: Bull. Am. Math. Soc.19, 287 (1988); Dubois-Violette, M.: Ann. Inst. Fourier37, (4),45 (1987); Henneaux, M., Teitelboim, C.: Commun. Math. Phys.115, 213 (1988)
Figueroa-O'Farrill, J.M., Kimura, T.: Stony Brokk Preprint ITP-SB-88-81 (rev.)
Lang, S.: Algebra. Reading, MA: Addison-Wesley 1984
Figueroa-O'Farrill, J.M.: The topological characterization of classical BRST cohomology. Commun. Math. Phys.127, 181–186 (1990)
Kostant, B., Sternberg, S.: Ann. Phys.176, 49 (1987)
Henneaux, M.: Phys. Rep.126, 1 (1985)
Stasheff, J.: Bull. Am. Math. Soc.19, 287 (1988)
Van Hove, L.: Acad. Roy. Belgique Bull. Sci.37 (5), 610 (1951); Abraham, R., Marsden, J.: Foundations of mechanics. New York, NY: Benjamin 1978
Kostant, B.: Quantization and unitary representations. I. Prequantization. In: Lecture Notes in Math. Vol.170 pp. 87:208. Berlin, Heidelberg, New York: Springer 1970
Souriau, J.-M.: Structure des systèmes dynamiques. Paris: Dunod 1970
Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology. I. New York, NY: Academic Press 1972
Woodhouse, N.: Geometric quantization. Oxford: Oxford University Press 1980; Hurt, N.: Geometric quantization in action. Amsterdam: D. Reidel 1983
Stasheff, J.: Homological Reduction of Constrained Poisson Algebras, North Carolina Preprint, 1989
Śniatycki, J., Weinstein, A.: Lett. Math. Phys.7, 155 (1983)
Arms, J., Marsden, J., Montcrief, V.: Commun. Math. Phys.78, 455 (1981)
Fisch, J., Henneaux, M., Stasheff, J., Teitelboim, C.: Commun. Math. Phys.120, 379 (1989)
Author information
Authors and Affiliations
Additional information
Communicated by L. Alvarez-Gaumé
Rights and permissions
About this article
Cite this article
Figueroa-O'Farrill, J.M., Kimura, T. Geometric BRST quantization, I: Prequantization. Commun.Math. Phys. 136, 209–229 (1991). https://doi.org/10.1007/BF02100022
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02100022