×

Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. (English) Zbl 0642.17003

Authors’ abstract: “This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite dimensional Clifford algebras and their spin representations. We find that in the infinite dimensional case, the analog of the finite dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.”
Reviewer: A.Verona

MSC:

17B56 Cohomology of Lie (super)algebras
17B65 Infinite-dimensional Lie (super)algebras
15A66 Clifford algebras, spinors
81Q99 General mathematical topics and methods in quantum theory
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
15A90 Applications of matrix theory to physics (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Banks, T.; Peskin, M. E., Nucl. Phys. B, 263, 105 (1986)
[2] Batalin; Vilkovsky, G. A., Phys. Lett. B, 69, 309 (1977)
[3] Becchi, C.; Rouet, A.; Stora, R., Ann. Phys. (N.Y.), 98, 287 (1976)
[4] Bleuler, K., Helv. Phys. Acta, 23, 567 (1950)
[5] Bowick, M. J.; Gursey, F., Phys. Lett. B, 175, 182 (1986), Yale University, preprint
[6] M. J. Bowick and S. G. Rajeev; M. J. Bowick and S. G. Rajeev
[7] Corwin, L.; Ne’eman, Y.; Sternberg, S., Rev. Mod. Phys., 47, 573 (1975)
[8] Dirac, P. A.M., (Lectures on Quantum Mechanics (1964), Belfer Graduate School of Science, Yeshiva University: Belfer Graduate School of Science, Yeshiva University New York) · Zbl 0141.44603
[9] Feigin, B. L., Uspechi Mat. Nauk, 39, 195 (1984)
[10] Frenkel, I. B.; Garland, H.; Zuckerman, G. J., (Proc. Nat. Acad. Sci. (1986))
[11] Fradkin, E. S.; Vilkovisky, G. A., Phys. Lett. B, 55, 224 (1975)
[12] Greub, W., (Multilinear Algebra (1984), Springer-Verlag: Springer-Verlag Berlin/New York)
[13] Guillemin, V.; Sternberg, S., (Symplectic Techniques in Physics (1984), Cambridge Univ. Press: Cambridge Univ. Press London/New York) · Zbl 0576.58012
[14] Guillemin, V.; Sternberg, S., Invent. Math., 67, 515 (1982)
[15] Guillemin, V.; Sternberg, S., J. Funct. Anal., 47, 344 (1982)
[16] Guillemin, V.; Sternberg, S., (Proceedings Clausthal. Proceedings Clausthal, Lect. Notes in Math., Vol. 905 (1982), Springer: Springer New York/Berlin), 52
[17] Guillemin, V.; Sternberg, S., (Proceedings of the Summer Conference on Math. Phys.. Proceedings of the Summer Conference on Math. Phys., Trieste (1981), World Sci: World Sci Singapore) · Zbl 0469.58017
[18] Gupta, S. J., (Proc. Phys. Soc. London Sect. A, 63 (1950)), 681
[19] Kac, V., (Infinite Dimensional Lie Algebras (1985), Cambridge Univ. Press: Cambridge Univ. Press London/New York)
[20] Kac, V., (Sympossium on Topological and Geometric Methods in Field Theory. Sympossium on Topological and Geometric Methods in Field Theory, Helsinki, Finland (1986), World Scientific)
[21] Kac, V.; Peterson, D., (Proc. Nat. Acad. Sci. USA, 78 (1981)), 3308
[22] Kato, M.; Ogawa, K., Nucl. Phys. B, 212, 443 (1983)
[23] Kazhdan, D.; Kostant, B.; Sternberg, S., Comm. Pure Appl. Math., 31, 481 (1978)
[24] Kostant, B., (Diff. Geom. Meth. in Math. Phys. Bonn, 1975. Diff. Geom. Meth. in Math. Phys. Bonn, 1975, Lect. Notes in Math., Vol. 570 (1977), Springer: Springer New York/Berlin)
[25] Kostant, B., Ann. of Math., 74, 329 (1961)
[26] Kostant, B., Ann. of Math., 77, 72 (1963)
[27] Kumar, S., J. Diff. Geom., 20, 389 (1984)
[28] Lang, S., Algebra (1965), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0193.34701
[29] D. McMullan; D. McMullan
[30] Marsden, J.; Weinstein, A., Rep. Math. Phys., 5, 121 (1974)
[31] Ne’eman, Y.; Regge, T.; Thierry-Mieg, J., (Proceedings, 19th Int. Conf. High. Energy Phys.. Proceedings, 19th Int. Conf. High. Energy Phys., Tokyo, 1978 (1979), Phys. Soc. Japan: Phys. Soc. Japan Tokyo), 552
[32] Ne’eman, Y.; Thierry-Mieg, J., Ann. Phys. (N.Y.), 123, 247 (1979)
[33] Polyakov, A. M., Phys. Lett. B, 103, 207 (1981)
[34] Rawnsley, J.; Schmid, W.; Wolf, J. A., J. Funct. Anal., 51, 1 (1983)
[35] W. SiegelPhys. Lett. B151; W. SiegelPhys. Lett. B151
[36] Siegel, W.; Zwiebach, B., Nucl. Phys. B, 263, 105 (1986)
[37] M. Spiegelglass; M. Spiegelglass
[38] Thierry-Mieg, J., Nuovo Cimento A, 56, 396 (1980)
[39] I. V. Tyupin; I. V. Tyupin
[40] Witten, E., Nucl. Phys. B, 268, 253 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.