Gauge invariant quantization on Riemannian manifolds

Liu, Zhang Ju; Qian, Min

doi:10.1090/S0002-9947-1992-1040266-0

Gauge invariant quantization on Riemannian manifolds
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by Zhang Ju Liu and Min Qian

Trans. Amer. Math. Soc. 331 (1992), 321-333

DOI: https://doi.org/10.1090/S0002-9947-1992-1040266-0

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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.

References

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
J. Czyż, On geometric quantization and its connections with the Maslov theory, Rep. Math. Phys. 15 (1979), no. 1, 57–97. MR 551131, DOI 10.1016/0034-4877(79)90052-1
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
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
Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
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 488140, DOI 10.1007/BF00398579

Semi-classical approximations on Riemannian manifolds

Zhang Ju Liu, Quantum integrable systems constrained on the sphere, Lett. Math. Phys. 20 (1990), no. 2, 151–157. MR 1065243, DOI 10.1007/BF00398280

Semi-classical approximation in quantum mechanics

J. Underhill, Quantization on a manifold with connection, J. Math. Phys. 19 (1978), no. 9, 1932–1935. MR 496146, DOI 10.1063/1.523913
Alan Weinstein, Quasi-classical mechanics on spheres, Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica & Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Rome, 1973) Academic Press, London, 1974, pp. 25–32. MR 0385940
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
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 395537, DOI 10.1103/PhysRevD.2.2161
Nicholas Woodhouse, Geometric quantization, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1980. MR 605306