Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Gauge invariant quantization on Riemannian manifolds


Authors: Zhang Ju Liu and Min Qian
Journal: Trans. Amer. Math. Soc. 331 (1992), 321-333
MSC: Primary 58G15; Secondary 58F05, 58F06
MathSciNet review: 1040266
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold $ M$, a family of differential operators is given, which acts on the space of smooth sections of a vector bundle on $ M$. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper.


References [Enhancements On Off] (What's this?)

  • [1] David Bleecker, Gauge theory and variational principles, Global Analysis Pure and Applied Series A, vol. 1, Addison-Wesley Publishing Co., Reading, Mass., 1981. MR 643361
  • [2] J. Czyż, On geometric quantization and its connections with the Maslov theory, Rep. Math. Phys. 15 (1979), no. 1, 57–97. MR 551131, 10.1016/0034-4877(79)90052-1
  • [3] C. DeWitt-Morette, K. D. Elworthy, B. L. Nelson, and G. S. Sammelman, A stochastic scheme for constructing solutions of the Schrödinger equations, Ann. Inst. H. Poincaré Sect. A (N.S.) 32 (1980), no. 4, 327–341. MR 594632
  • [4] K. D. Elworthy, Path integration on manifolds, Mathematical aspects of superspace (Hamburg, 1983) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 132, Reidel, Dordrecht, 1984, pp. 47–89. MR 773079
  • [5] Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
  • [6] André Lichnerowicz, Construction of twisted products for cotangent bundles of classical groups and Stiefel manifolds, Lett. Math. Phys. 2 (1977/78), no. 2, 133–143. MR 0488140
  • [7] Z. J. Liu, Semi-classical approximations on Riemannian manifolds, Acta Appl. Math. Sinica (to appear).
  • [8] Zhang Ju Liu, Quantum integrable systems constrained on the sphere, Lett. Math. Phys. 20 (1990), no. 2, 151–157. MR 1065243, 10.1007/BF00398280
  • [9] V. P. Maslov and M. P. Fedoriuk, Semi-classical approximation in quantum mechanics, Reidel, 1981.
  • [10] J. Underhill, Quantization on a manifold with connection, J. Math. Phys. 19 (1978), no. 9, 1932–1935. MR 0496146
  • [11] Alan Weinstein, Quasi-classical mechanics on spheres, Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973) Academic Press, London, 1974, pp. 25–32. MR 0385940
  • [12] Harold Widom, Szegő’s theorem and a complete symbolic calculus for pseudodifferential operators, Seminar on Singularities of Solutions of Linear Partial Differential Equations (Inst. Adv. Study, Princeton, N.J., 1977/78) Ann. of Math. Stud., vol. 91, Princeton Univ. Press, Princeton, N.J., 1979, pp. 261–283. MR 547022
  • [13] G. S. Agarwal and E. Wolf, Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators, Phys. Rev. D (3) 2 (1970), 2161–2186. MR 0395537
  • [14] Nicholas Woodhouse, Geometric quantization, The Clarendon Press, Oxford University Press, New York, 1980. Oxford Mathematical Monographs. MR 605306

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G15, 58F05, 58F06

Retrieve articles in all journals with MSC: 58G15, 58F05, 58F06


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1040266-0
Article copyright: © Copyright 1992 American Mathematical Society