Communications in Mathematical Physics

, Volume 151, Issue 3, pp 531–542 | Cite as

Equivalence of dirac and intrinsic quantization for non-free group actions

  • C. Emmrich


In this article we generalize some results on the equivalence of Dirac quantization and intrinsic quantization proven in [3]: We consider systems with first class constraints that may be considered as the vanishing of the momentum map to a lifted group action, but drop the assumption that the group action is free as well as the assumption that the group is compact. Using a generalized Weyl ordering prescription applicable to arbitrary cotangent bundles we derive necessary and sufficient conditions for the equivalence of the two approaches for different classes of functions analogous to those for the free case, although the proofs given in [3] must be considerably modified and refined due to the noncompactness of the orbits and the lack of sufficiently many invariant vector fields. The same strong obstruction as in the free case is found if one requires equivalence for all invariant functions, essentially only admitting trivial bundles.


Neural Network Statistical Physic Complex System Vector Field Nonlinear Dynamics 
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  1. 1.
    Besse, A.L.: Einstein manifold. Berlin-Heidelberg New York: Springer, 1987Google Scholar
  2. 2.
    Coquereaux, R., Jadczyk, A.: Riemannian Geometry, Fiber Bundles, Kaluza-Klein Theories and All That …. Singapore: World Scientific, 1988Google Scholar
  3. 3.
    Emmrich, C.: Equivalence of extrinsic and intrinsic quantization for observables not preserving the vertical polarization. Commun. Math. Phys. this issueGoogle Scholar
  4. 4.
    Gotay, M.J.: Constraints, reduction, and quantization.J. Math. Phys. 27, 2051–2066 (1986)CrossRefGoogle Scholar
  5. 5.
    Tuynman, G.M.: Reduction, quantization, and nonunimodular groups. J. Math. Phys.31, 83–90 (1990)CrossRefGoogle Scholar
  6. 6.
    Tuynman, G.M.: Quantization of first class constraints with structure functions. Lett. Math. Phys. 21, 205–213 (1991)CrossRefGoogle Scholar
  7. 7.
    Underhill, J.: Quantization on a manifold with connection. J. Math. Phys.19, 1932–1935 (1978)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • C. Emmrich
    • 1
  1. 1.Fakultät für Physik der Universität FreiburgFreiburgFRG

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