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Transactions of the American Mathematical Society

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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
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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.

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  • [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

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Article copyright: © Copyright 1992 American Mathematical Society