Communications in Mathematical Physics

, Volume 286, Issue 2, pp 459–494 | Cite as

A Gauge Model for Quantum Mechanics on a Stratified Space

  • J. HuebschmannEmail author
  • G. Rudolph
  • M. Schmidt
Open Access


In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure yields a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is SU(2), we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.


Hilbert Space Coherent State Poisson Structure Symplectic Structure Cotangent Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abraham, R., Marsden, J.E.: Foundations of mechanics. Reading, MA: Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1978Google Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions (abridged edition). Frankfurt am Main: Verlag Harri Deutsch, 1984Google Scholar
  3. 3.
    Aldrovandi R., Leal Ferreira P.: Quantum pendulum. Amer. J. Phys 48, 660–664 (1980)CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Arms, J.M., Cushman, R., Gotay, M.J.: A universal reduction procedure for Hamiltonian group actions. The geometry of Hamiltonian systems. In: Ratiu, T. (ed.), MSRI Publ 20 Berlin-Heidelberg, New York: Springer 1991, pp. 33–51Google Scholar
  5. 5.
    Arms J.M., Marsden J.E., Moncrief V.: Symmetry and bifurcation of moment mappings. Commun. Math. Phys. 78, 455–478 (1981)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Arms J.M., Marsden J.E., Moncrief V.: The structure of the space of solutions of Einstein’s equations. II. Several Killing fields and the Einstein-Yang-Mills equations. Ann. Phys. 144(1), 81–106 (1982)zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Arscott, F.M.: Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. London:Pergamon Press, 1964Google Scholar
  8. 8.
    Asorey M., Falceto F., López J.L., Luzón G.: Nodes, monopoles and confinement in (2 + 1)-dimensional gauge theories. Phys. Lett. B 349, 125–130 (1995)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Baker G.L., Blackburn J.A., Smith H.J.T.: The quantum pendulum: Small and large. Amer. J. Phys. 70, 525–531 (2002)CrossRefGoogle Scholar
  10. 10.
    Charzyński S., Kijowski J., Rudolph G., Schmidt M.: On the stratified classical configuration space of lattice QCD. J. Geom. Phys. 55, 137–178 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Charzyński S., Rudolph G., Schmidt M.: On the topological structure of the stratified classical configuration space of lattice QCD. J. Geom. Phys. 58, 1607–1623 (2008)zbMATHCrossRefADSGoogle Scholar
  12. 12.
    Condon E.U.: The physical pendulum in quantum mechanics. Phys. Rev. 31, 891–894 (1928)CrossRefADSGoogle Scholar
  13. 13.
    Cushman R.H., Bates L.M.: Global Aspects of Classical Integrable Systems. Birkhäuser, Basel-Boston (1997)zbMATHGoogle Scholar
  14. 14.
    Deser S., Jackiw R.: Classical and quantum scattering on a cone. Commun. Math. Phys. 118, 495–509 (1988)zbMATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Dimock J.: Canonical quantization of Yang-Mills on a circle. Rev. Math. Phys. 8, 85–102 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Driver B.K., Hall B.C.: Yang-Mills theory and the Segal-Bargmann transform. Commun. Math. Phys. 201, 249–290 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Emmrich C., Roemer H.: Orbifolds as configuration spaces of systems with gauge symmetries. Commun. Math. Phys. 129, 69–94 (1990)zbMATHCrossRefADSGoogle Scholar
  18. 18.
    Fischer E., Rudolph G., Schmidt M.: A lattice gauge model of singular Marsden-Weinstein reduction Part I. Kinematics. J. Geom. Phys. 57, 1193–1213 (2007)zbMATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Florentino C.A., Mourão J.M., Nunes J.: Coherent state transforms and vector bundles on elliptic curves. J. Funct. Anal. 204, 355–398 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Goresky M., MacPherson R.: Stratified Morse theory. Springer, Berlin-Heidelberg, New York (1988)zbMATHGoogle Scholar
  21. 21.
    Hall B.C.: The Segal-Bargmann “coherent state” transform for compact Lie groups. J. Funct. Anal. 122, 103–151 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hall B.C.: The inverse Segal-Bargmann transform for compact Lie groups. J. Funct. Anal. 143, 98–116 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hall B.C.: Phase space bounds for quantum mechanics on a compact Lie group. Commun. Math. Phys. 184, 233–250 (1997)zbMATHCrossRefADSGoogle Scholar
  24. 24.
    Hall B.C.: Coherent states and the quantization of 1+1-dimensional Yang-Mills theory. Rev. Math. Phys. 13, 1281–1306 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Hall B.C.: Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type. Commun. Math. Phys. 226, 233–268 (2002)zbMATHCrossRefADSGoogle Scholar
  26. 26.
    Hall B.C., Mitchell J.J.: The Segal-Bargmann transform for noncompact symmetric spaces of the complex type. J. Funct. Anal. 227, 338–371 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Helgason S.: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Academic Press, London-New York (1984)zbMATHGoogle Scholar
  28. 28.
    Hetrick J.E.: Canonical quantization of two-dimensional gauge fields. Int. J. Mod. Phys. A 9, 3153–3178 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Huebschmann J.: Poisson geometry of flat connections for SU(2)-bundles on surfaces. Math. Z. 221, 243–259 (1996)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Huebschmann J.: Symplectic and Poisson structures of certain moduli spaces. Duke Math. J. 80, 737–756 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Huebschmann, J.: Kähler spaces, nilpotent orbits, and singular reduction. Memoirs of the AMS 172 (814), Providence R.I.:Amer. Math. Soc., 2004Google Scholar
  32. 32.
    Huebschmann J.: Kähler quantization and reduction. J. reine. angew. Math. 591, 75–109 (2006)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Huebschmann J.: Stratified Kähler structures on adjoint quotients. Diff. Geom. Appl. 26, 704–731 (2008)CrossRefGoogle Scholar
  34. 34.
    Huebschmann, J.: Singular Poisson-Kähler geometry of certain adjoint quotients, In: Proceedings, The mathematical legacy of C. Ehresmann, Bedlewo, 2005, Banach Center Publications 76, 325–347 (2007)Google Scholar
  35. 35.
    Huebschmann J.: Kirillov’s character formula, the holomorphic Peter-Weyl theorem, and the Blattner-Kostant-Sternberg pairing. J. Geom. Phys. 58, 833–848 (2008)zbMATHCrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Jarvis P.D., Kijowski J., Rudolph G.: On the structure of the observable algebra of QCD on the lattice. J. Phys. A 38, 5359–5377 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Kay B.S., Studer U.M.: Boundary conditions for quantum mechanics on cones and fields around cosmic strings. Commun. Math. Phys. 139, 103–139 (1991)zbMATHCrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Kijowski J., Rudolph G.: On the Gauss law and global charge for quantum chromodynamics. J. Math. Phys. 43, 1796–1808 (2002)zbMATHCrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Kijowski J., Rudolph G.: Charge superselection sectors for qcd on the lattice. J. Math. Phys. 46, 032303 (2005)CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Kijowski J., Rudolph G., Śliwa C.: On the structure of the observable algebra for QED on the lattice. Lett. Math. Phys. 43, 99–308 (1998)CrossRefGoogle Scholar
  41. 41.
    Kijowski J., Rudolph G., C.: Charge superselection sectors for scalar QED on the lattice. Ann. Henri. Poincaré. 4, 1137–1167 (2003)CrossRefGoogle Scholar
  42. 42.
    Kijowski J., Rudolph G., Thielmann A.: Algebra of observables and charge superselection sectors for QED on the lattice. Commun. Math. Phys. 188, 535–564 (1997)zbMATHCrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Landsman, N.P.: Mathematical topics between classical and quantum mechanics. Berlin-Heidelberg, New York: Springer, 1998Google Scholar
  44. 44.
    Landsman N.P., Wren K.K.: Constrained quantization and θ-angles. Nucl. Phys. B. 502, 537–560 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Landsman, N.P., Wren, K.K.: Hall’s coherent states, the Cameron-Martin theorem, and the quantization of Yang-Mills theory on a circle., 1998
  46. 46.
    McLachlan N.W.: Theory and Application of Mathieu Functions. Dover Publications, New York (1964)zbMATHGoogle Scholar
  47. 47.
    Meixner J., Schaefke W.: Mathieusche Funktionen und Sphäroidfunktionen. Grundlehren Bd. 71. Berlin-Heidelberg, New York: Springer, 1954Google Scholar
  48. 48.
    Nelson E.: Analytic vectors. Ann. of Math. 70, 572–615 (1959)CrossRefMathSciNetGoogle Scholar
  49. 49.
    Pradhan T., Khare A.V.: Plane pendulum in quantum mechanics. Amer. J. Phys. 41, 59–66 (1973)CrossRefADSGoogle Scholar
  50. 50.
    Schwarz G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Śniatycki, J.: Geometric quantization and quantum mechanics. Applied Mathematical Sciences 30, Berlin-Heidelberg, New York: Springer, 1980Google Scholar
  52. 52.
    Stein, E.M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No 63. Princeton, NJ:Princeton University Press, 1970Google Scholar
  53. 53.
    Taylor J.: The Iwasawa decomposition and limiting behaviour of Brownian motion on symmetric spaces of non-compact type. Cont. Math. 73, 303–331 (1988)Google Scholar
  54. 54.
    Thiemann T.: Gauge field theory coherent states (GCS). I. General properties. Class. Quant. Grav. 18, 2025–2064 (2001)zbMATHCrossRefADSMathSciNetGoogle Scholar
  55. 55.
    Thiemann T., Winkler O.: Gauge field theory coherent states (GCS). II. Peakedness properties. Class. Quant. Grav. 18, 2561–2636 (2001)zbMATHCrossRefADSMathSciNetGoogle Scholar
  56. 56.
    Woodhouse N.M.J.: Geometric quantization. Clarendon Press, Oxford (1991)Google Scholar
  57. 57.
    Wren K.K.: Quantization of constrained systems with singularities using Rieffel induction. J. Geom. Phys. 24, 173–202 (1998)zbMATHCrossRefADSMathSciNetGoogle Scholar
  58. 58.
    Wren K.K.: Constrained quantisation and θ-angles II. Nucl. Phys. B 521, 471–502 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.USTL, UFR de Mathématiques, CNRS-UMR 8524Villeneuve d’Ascq CédexFrance
  2. 2.Institute for Theoretical PhysicsUniversity of LeipzigLeipzigGermany

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