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Quantum Configuration Spaces of Extended Objects, Diffeomorphism Group Representations and Exotic Statistics

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Abstract

A fundamental approach to quantum mechanics is based on the unitary representations of the group of diffeomorphisms of physical space (and correspondingly, self-adjoint representations of a local current algebra). From these, various classes of quantum configuration spaces arise naturally, as well as the usual exchange statistics for point particles in spatial dimensions \( d\,\,\geq\, 3\), induced by representations of the symmetric group. For \( d\,\,=\, 3\), this approach led to an early prediction of intermediate or “anyon” statistics induced by unitary representations of the braid group. I review these ideas, and discuss briefly some analogous possibilities for infinite-dimensional configuration spaces, includinga nyonic statistics for extended objects in three-dimensional space.

Mathematics Subject Classification (2010). Primary 81R10; Secondary 81Q70.

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References

  1. H.S. Green, A Generalized Method of Field Quantization. Phys. Rev. 90 (1953), 270–273.

    Article  MATH  Google Scholar 

  2. A.M.L. Messiah and O.W. Greenberg, Symmetrization Postulate and its Experimental Foundation. Phys. Rev. 136 (1964), B248–B267.

    Article  MathSciNet  Google Scholar 

  3. Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory. Phys. Rev.115 (1959), 485–491.

    Article  MathSciNet  MATH  Google Scholar 

  4. M.G.G. Laidlaw and C.M. DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles. Phys. Rev. D 3 (1971), 1375–1378.

    Article  Google Scholar 

  5. J.M. Leinaas and J. Myrheim, On the Theory of Identical Particles. Nuovo Cimento 37B (1977), 1–23.

    Article  Google Scholar 

  6. G.A. Goldin, R. Menikoff and D.H. Sharp, Particle Statistics from Induced Representations of a Local Current Group. J. Math. Phys 21 (1980), 650–664.

    Article  MathSciNet  MATH  Google Scholar 

  7. G.A. Goldin, R.Menikoff and D.H. Sharp, Representations of a Local Current Algebra in Non-Simply Connected Space and the Aharonov-Bohm Effect. J. Math Phys. 22 (1981), 1664–1668.

    Article  MathSciNet  Google Scholar 

  8. F. Wilczek, Magnetic Flux, Angular Momentum, and Statistics. Phys. Rev. Lett. 48 (1982), 1144–1146.

    Article  MathSciNet  Google Scholar 

  9. F. Wilczek, Quantum Mechanics of Fractional Spin Particles. Phys. Rev. Lett. 49 (1982), 957–959.

    Article  MathSciNet  Google Scholar 

  10. G.A. Goldin and D.H. Sharp, Rotation Generators in Two-Dimensional Space and Particles Obeying Unusual Statistics. Phys. Rev. D 28 (1983), 830–832.

    Article  MathSciNet  Google Scholar 

  11. G.A. Goldin, R. Menikoff, and D.H. Sharp, Diffeomorphism Groups, Gauge Groups, and Quantum Theory. Phys. Rev. Lett. 51 (1983), 2246–2249.

    Article  MathSciNet  Google Scholar 

  12. G.A. Goldin and S. Majid, On the Fock Space for Nonrelativistic Anyon Fields and Braided Tensor Products. J. Math. Phys. 45 (2004), 3770–3787.

    Article  MathSciNet  MATH  Google Scholar 

  13. G.A. Goldin, R. Menikoff and D.H. Sharp, Comment on “General Theory for Quantum Statistics in Two Dimensions”. Phys. Rev. Lett. 54 (1985), 603.

    Article  MathSciNet  Google Scholar 

  14. C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, arXiv:0707.1889v2 [cond-mat.strel] (2008).

    Google Scholar 

  15. A.J. Niemi, Exotic Statistics of Leapfrogging Vortex Rings. Phys. Rev. Lett. 94 (2005), 124502–124505.

    Article  Google Scholar 

  16. G.A. Goldin and D.H. Sharp, The Diffeomorphism Group Approach to Anyons. In F. Wilczek (ed.), Fractional Statistics in Action (special issue), Int. J. Mod. Physics B 5 (1991), 2625–2640.

    Google Scholar 

  17. G.A. Goldin,The Diffeomorphism Group Approach to Nonlinear Quantum Systems. In M. Martellini and M. Rasetti (eds.), Topological and Quantum Group Methods in Field Theory and Condensed Matter Physics (special issue), Int. J. Mod. Physics B 6, 1905–1916.

    Google Scholar 

  18. G.A. Goldin, Lectures on Diffeomorphism Groups in Quantum Physics. In J. Govaerts et al. (eds.), Contemporary Problems in Mathematical Physics: Proceedings of the Third International Conference, Cotonou, Benin. World Scientific, 2004 (pp. 3–93).

    Google Scholar 

  19. R. Dashen and D.H. Sharp, Currents as Coordinates for Hadrons. Phys. Rev. 165 (1968), 1857–1866.

    Article  Google Scholar 

  20. G.A. Goldin, Non-Relativistic Current Algebras as Unitary Representations of Groups. J. Math. Phys. 12 (1971), 462–487.

    Article  MathSciNet  Google Scholar 

  21. G.A. Goldin and D.H. Sharp, Lie Algebras of Local Currents and their Representations. In V. Bargmann (ed.), Group Representations in Mathematics and Physics: Battelle-Seattle 1969 Rencontres. Lecture Notes in Physics 6, Springer, 1970 (pp. 300–310).

    Google Scholar 

  22. G.A. Goldin, J. Grodnik, R.T. Powers and D.H. Sharp, Nonrelativistic Current Algebra in the ‘N/V’ Limit. J. Math. Phys. 15 (1974), 88–100.

    Article  MathSciNet  Google Scholar 

  23. G.A. Goldin, Diffeomorphism Groups and Quantum Configurations. In S.T. Ali, G.G. Emch, A. Odzijewicz, M. Schlichenmaier and S.L. Woronowicz (eds.), Twenty Years of Bialowieza: A Mathematical Anthology, Aspects of Differential Geometric Methods in Physics. World Scientific, 2005 (pp. 1–22).

    Google Scholar 

  24. S. Albeverio, Y.G. Kondratiev and M. Röckner, Analysis and Geometry on Configuration Spaces. J. Funct. Anal. 154 (1998), 444–500.

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Albeverio, Y.G. Kondratiev and M. Röckner, Analysis and Geometry on Configuration Spaces: The Gibbsian Case. J. Funct. Anal. 157 (1998), 242–291.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Albeverio, Y.G. Kondratiev and M. Röckner, Diffeomorphism Groups and Current Algebras: Configuration Space Analysis in Quantum Theory. Rev. Math. Phys. 11 (1999), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  27. Y.G. Kondratiev and T. Kuna, Harmonic Analysis on Configuration Space I. General Theory. Infinite Dimensional Analysis 5 (2002), 201–233.

    MathSciNet  MATH  Google Scholar 

  28. R.S. Ismagilov, Representations of Infinite-Dimensional Groups. Translations of Mathematical Monographs 152, American Mathematical Society, 1996.

    Google Scholar 

  29. G.A. Goldin, U. Moschella and T. Sakuraba, Self-Similar Random Processes and Infinite-Dimensional Configuration Spaces. Institute of Physics, Physics of Atomic Nuclei 68 (2005), 1675–1684.

    Article  MathSciNet  Google Scholar 

  30. G.A. Goldin and R. Menikoff, Quantum-Mechanical Representations of the Group of Diffeomorphisms and Local Current Algebra Describing Tightly Bound Composite Particles. J. Math. Phys. 26 (1985), 1880–1884.

    Article  MathSciNet  MATH  Google Scholar 

  31. Y.G. Kondratiev, E.W. Lytvynov and G.F. Us, Analysis and Geometry on 𝑅+- marked Configuration Spaces. Meth. Funct. Anal. Topol 5 (1999), 29–64.

    MathSciNet  MATH  Google Scholar 

  32. S. Albeverio, Y.G. Kondratiev, E.W. Lytvynov and G.F. Us, Analysis and Geometry on Marked Configuration Spaces. In H. Heyer et al. (eds.), Infinite Dimensional Harmonic Analysis (Kyoto, September 20–24, 1999), Gräbner, 2000 (pp. 1–39); arXiv: math/0608344.

    Google Scholar 

  33. G.A. Goldin and D.H. Sharp, Diffeomorphism Groups, Anyon Fields, and q-Commutators Phys. Rev. Lett. 76, 1183–1187.

    Google Scholar 

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Authors and Affiliations

  1. Rutgers University, SERC BuildingR m. 239, Busch Campus 118 Frelinghuysen Road, Piscataway, NJ, 08854, USA

    Gerald A. Goldin

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  1. Gerald A. Goldin
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Correspondence to Gerald A. Goldin .

Editor information

Editors and Affiliations

  1. Departamento de Física, CINVESTAV, Mexico City, Distrito Federal, Mexico

    Piotr Kielanowski

  2. Dept. of Mathematics and Statistics, Concordia University, Montreal, Québec, Canada

    S. Twareque Ali

  3. Institute of Mathematics, University of Bialystok, Bialystok, Poland

    Anatol Odzijewicz

  4. Mathematics Research Unit, FSTC, University of Luxembourg, Luxembourg-Kirchberg, Luxembourg

    Martin Schlichenmaier

  5. School of Mathematics, University of Manchester, Manchester, United Kingdom

    Theodore Voronov

Additional information

Presented at the Felix Berezin Memorial Session, XXX Workshop on Geometric Methods in Physics, Białowieża, Poland, and dedicated also to my colleagues Bogdan Mielnik and Stanisław Woronowicz

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Goldin, G.A. (2013). Quantum Configuration Spaces of Extended Objects, Diffeomorphism Group Representations and Exotic Statistics. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_19

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