A Groenewold-Van Hove Theorem for $S^2$
HTML articles powered by AMS MathViewer
- by Mark J. Gotay, Hendrik Grundling and C. A. Hurst PDF
- Trans. Amer. Math. Soc. 348 (1996), 1579-1597 Request permission
Abstract:
We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold $S^2$ which is irreducible on the su(2) subalgebra generated by the components $\{S_1,S_2,S_3\}$ of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra $\mathcal {P}$ consisting of polynomials in $\{S_1,S_2,S_3\}$. Furthermore, we show that the maximal Poisson subalgebra of $\mathcal {P}$ containing $\{1,S_1,S_2,S_3\}$ that can be so quantized is just that generated by $\{1,S_1,S_2,S_3\}$.References
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
- Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
- Asim O. Barut and Ryszard Rączka, Theory of group representations and applications, PWN—Polish Scientific Publishers, Warsaw, 1977. MR 0495836
- Paul R. Chernoff, Mathematical obstructions to quantization, Hadronic J. 4 (1980/81), no. 3, 879–898. MR 613352
- Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
- Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
- Mark J. Gotay, Functorial geometric quantization and Van Hove’s theorem, Internat. J. Theoret. Phys. 19 (1980), no. 2, 139–161. MR 576457, DOI 10.1007/BF00669766
- Y. S. Kim and W. W. Zachary (eds.), The physics of phase space, Lecture Notes in Physics, vol. 278, Springer-Verlag, Berlin, 1987. Nonlinear dynamics and chaos, geometric quantization and Wigner function. MR 915780, DOI 10.1007/3-540-17894-5
- A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
- Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
- Helton, J.W. and Miller, R.L. [1994] NC Algebra: A Mathematica Package for Doing Non Commuting Algebra. v0.2 (Available from ncalg@@ucsd.edu, La Jolla).
- A. Joseph, Derivations of Lie brackets and canonical quantisation, Comm. Math. Phys. 17 (1970), 210–232. MR 293940
- Karasev, M. [1994] Private communication.
- Albert Messiah, Quantum mechanics. Vol. II, North-Holland Publishing Co., Amsterdam; Interscience Publishers (a division of John Wiley & Sons, Inc.), New York, 1962. Translated from the French by J. Potter. MR 0147125
- Marc A. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), no. 4, 531–562. MR 1002830
- G. M. Tuynman, Generalized Bergman kernels and geometric quantization, J. Math. Phys. 28 (1987), no. 3, 573–583. MR 877229, DOI 10.1063/1.527642
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- N. M. J. Woodhouse, Geometric quantization, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. Oxford Science Publications. MR 1183739
Additional Information
- Mark J. Gotay
- Affiliation: Department of Mathematics, University of Hawaii, 2565 The Mall, Honolulu, Hawaii 96822
- Email: gotay@math.hawaii.edu
- Hendrik Grundling
- Affiliation: Department of Pure Mathematics, University of New South Wales, P. O. Box 1, Kensington, NSW 2033 Australia
- Email: hendrik@solution.maths.unsw.edu.au
- C. A. Hurst
- Affiliation: Department of Physics and Mathematical Physics, University of Adelaide, G. P. O. Box 498, Adelaide, SA 5001 Australia
- Email: ahurst@physics.adelaide.edu.au
- Received by editor(s): March 23, 1995
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 1579-1597
- MSC (1991): Primary 81S99; Secondary 58F06
- DOI: https://doi.org/10.1090/S0002-9947-96-01559-0
- MathSciNet review: 1340175