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)



A Groenewold-Van Hove Theorem for $S^2$

Authors: Mark J. Gotay, Hendrik Grundling and C. A. Hurst
Journal: Trans. Amer. Math. Soc. 348 (1996), 1579-1597
MSC (1991): Primary 81S99; Secondary 58F06
MathSciNet review: 1340175
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $\cal P$ consisting of polynomials in $\{S_1,S_2,S_3\}$. Furthermore, we show that the maximal Poisson subalgebra of $\cal P$ 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 [Enhancements On Off] (What's this?)

  • [A-B-R] Axler, S., Bourdon P. and Ramey, W. [1992] Harmonic Function Theory. Grad. Texts in Math. 137 (Springer, New York). MR 93f:31001
  • [A-M] Abraham, R. and Marsden, J.E. [1978] Foundations of Mechanics. Second ed. (Benjamin-Cummings, Reading, MA). MR 81e:58025
  • [B-R] Barut, A.O. and Raczka, R. [1977] Theory of Group Representations and Applications. (Polish Scientific Publishers, Warsaw). MR 58:14480
  • [C] Chernoff, P. [1981] Mathematical obstructions to quantization. Hadronic J. 4, 879-898. MR 82i:81006
  • [D] Dixmier, J. [1977] Enveloping Algebras. (North-Holland, Amsterdam). MR 58:16803b
  • [F] Folland, G.B. [1989] Harmonic Analysis in Phase Space. Ann. Math. Studies 122 (Princeton Univ. Press, Princeton). MR 92k:22017
  • [Go1] Gotay, M.J. [1980] Functorial geometric quantization and Van Hove's theorem. Int. J. Theor. Phys. 19, 139-161. MR 81g:58016
  • [Go2] Gotay, M.J. [1987] Formal quantization of quadratic momentum observables. In: The Physics of Phase Space, Y.S. Kim and W.W. Zachary, Eds., Lect. Notes in Physics 278, 375-379. MR 88h:58003
  • [Gr] Groenewold, H.J. [1946] On the principles of elementary quantum mechanics. Physics 12, 405-460. MR 8:301a
  • [G-S] Guillemin, V. and Sternberg, S. [1984] Symplectic Techniques in Physics. (Cambridge Univ. Press, Cambridge). MR 86f:58054
  • [H-M] Helton, J.W. and Miller, R.L. [1994] NC Algebra: A Mathematica Package for Doing Non Commuting Algebra. v0.2 (Available from, La Jolla).
  • [J] Joseph, A. [1970] Derivations of Lie brackets and canonical quantization. Comm. Math. Phys. 17, 210-232. MR 45:3015
  • [K] Karasev, M. [1994] Private communication.
  • [M] Messiah, A. [1962] Quantum Mechanics II. (Wiley, New York). MR 26:4643
  • [R] Rieffel, M. A. [1989] Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531-562. MR 90e:46060
  • [T] Tuynman, G.T. [1987] Generalised Bergman kernels and geometric quantization. J. Math. Phys. 28, 573-583. MR 88g:58074
  • [VH1] Van Hove, L. [1951] Sur le problème des relations entre les transformations unitaires de la mécanique quantique et les transformations canoniques de la mécanique classique. Acad. Roy. Belgique Bull. Cl. Sci. (5) 37, 610-620. MR 13:519a
  • [VH2] Van Hove, L. [1951] Sur certaines représentations unitaires d'un groupe infini de transformations. Mem. Acad. Roy. Belgique Cl. Sci 26, 61-102. MR 15:198d
  • [vN] von Neumann, J. [1955] Mathematical Foundations of Quantum Mechanics. (Princeton. Univ. Press, Princeton). MR 16:654a
  • [W] Woodhouse, N.M.J. [1992] Geometric Quantization. Second ed. (Clarendon Press, Oxford). MR 94a:58082

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 81S99, 58F06

Retrieve articles in all journals with MSC (1991): 81S99, 58F06

Additional Information

Mark J. Gotay
Affiliation: Department of Mathematics, University of Hawaii, 2565 The Mall, Honolulu, Hawaii 96822

Hendrik Grundling
Affiliation: Department of Pure Mathematics, University of New South Wales, P. O. Box 1, Kensington, NSW 2033 Australia

C. A. Hurst
Affiliation: Department of Physics and Mathematical Physics, University of Adelaide, G. P. O. Box 498, Adelaide, SA 5001 Australia

Received by editor(s): March 23, 1995
Article copyright: © Copyright 1996 American Mathematical Society