Skip to main content
SpringerLink
Log in

On Hilbert-Schmidt Operator Formulation of Noncommutative Quantum Mechanics

  • Chapter
  • First Online:
Mathematical Structures and Applications

Abstract

This work gives value to the importance of Hilbert-Schmidt operators in the formulation of noncommutative quantum theory. A system of charged particle in a constant magnetic field is investigated in this framework.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S.T. Ali, F. Bagarello, Some physical appearances of vector coherent states and coherent states related to degenerate Hamiltonians. J. Math. Phys. 46, 053518 (2005)

    Article  MathSciNet  Google Scholar 

  2. S.T. Ali, F. Bagarello, G. Honnouvo, Modular structures on trace class operators and applications to Landau levels. J. Phys. A: Math. Theor. 43, 105202 (2010); S.T. Ali, An interesting modular structure associated to Landau levels. J. Phys: Conf. Ser. 237, 012001 (2010)

    Google Scholar 

  3. S.T. Ali, J.P. Antoine, J.P. Gazeau, Coherent States, Wavelets and their Generalizations. Theoretical and Mathematical Physics, 2nd edn. (Springer, New York, 2014)

    Book  Google Scholar 

  4. I. Aremua, M.N. Hounkonnou, Coherent states for the exotic Landau model and related properties (unpublished work)

    Google Scholar 

  5. I. Aremua, M.N. Hounkonnou, E. Baloïtcha, Coherent states for Landau levels: algebraic and thermodynamical properties. Rep. Math. Phys. 76(2), 247–269 (2015)

    Article  MathSciNet  Google Scholar 

  6. J. Ben Geloun, F.G. Scholtz, Coherent states in noncommutative quantum mechanics. J. Math. Phys. 50, 043505 (2009)

    Article  MathSciNet  Google Scholar 

  7. P. Bertozzini, Non-commutative geometries via modular theory, in RIMS International Conference on Noncommutative Geometry in Physics, Kyoto (2010)

    Google Scholar 

  8. O. Bratelli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. 1 (Springer, Berlin/Heidelberg, 2002); O. Bratelli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vol. 2 (Springer, Berlin/Heidelberg, 2002)

    Google Scholar 

  9. K.E. Cahill, R.J. Glauber, Density operators and quasiprobability distributions. Phys. Rev. 177, 1882 (1969)

    Article  Google Scholar 

  10. A. Connes, Groupe modulaire d’une algèbre de von Neumann. C. R. Acad. Sci. Paris Série A 274, 1923–1926 (1972); A. Connes, Caractérisation des algèbres de von Neumann comme espaces vectoriels ordonnés. Annales de l’Institut Fourier 24, 121–155 (1974); A. Connes, E. Stømer, Entropy for automorphisms of finite von Neumann algebras. Acta Math. 134, 289–306 (1975); A. Connes, M. Takesaki, The flow of weights on factors of type III. Tohoku Math. J. 29, 473–575 (1977); A. Connes, Von Neumann algebras, Proceedings of the International Congress of Mathematicians Helsinki, pp. 97–109 (1978); A. Connes, C. Rovelli, Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories. Class. Quantum Grav. 11, 2899–2917 (1994)

    Google Scholar 

  11. A. Connes, Noncommutative differential geometry. Inst. Hautes Etudes Sci. Publ. Math. 62, 257–360 (1985); A. Connes, Noncommutative Geometry (Academic Press, San Diego, CA, 1994); A. Connes, M. Douglas and A. Schwarz, JHEP 02003; [e-print hep-th/9711162] (1998).

    Google Scholar 

  12. J.P. Gazeau, Coherent States in Quantum Physics (Wiley, Berlin, 2009)

    Book  Google Scholar 

  13. M.O. Goerbig, P. Lederer, C.M. Smith, Competition between quantum-liquid and electron-solid phases in intermediate Landau levels. Phys. Rev. B 69, 115327 (2004)

    Article  Google Scholar 

  14. H. Grosse, P. Prešnajder, The construction on noncommutative manifolds using coherent states. Lett. Math. Phys. 28, 239 (1993)

    Article  MathSciNet  Google Scholar 

  15. M.N. Hounkonnou, I. Aremua, Landau Levels in a two-dimensional noncommutative space: matrix and quaternionic vector coherent states. J. Nonlinear Math. Phys. 19, 1250033 (2012)

    MathSciNet  MATH  Google Scholar 

  16. K. Husimi, Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn. 22, 264 (1940)

    MATH  Google Scholar 

  17. J.R. Klauder, B.S. Skagerstam, Coherent States, Applications in Physics and Mathematical Physics (World Scientific, Singapore, 1985)

    Book  Google Scholar 

  18. L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Non-relativistic Theory (Oxford, Pergamon, 1977)

    MATH  Google Scholar 

  19. F.J. Murray, J.v. Neumann, On rings of operators. Ann. Math. 37, 116–229 (1936)

    Article  MathSciNet  Google Scholar 

  20. A.M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986)

    Book  Google Scholar 

  21. E. Prugovečki, Quantum Mechanics in Hilbert Spaces, 2nd edn. (Academic Press, New York, 1981)

    MATH  Google Scholar 

  22. F.G. Scholtz, L. Gouba, A. Hafver, C.M. Rohwer, Formulation, interpretation, and applications of non-commutative quantum mechanics. J. Phys. A: Math. Theor. 42, 175303 (2009)

    Article  MathSciNet  Google Scholar 

  23. E. Schrödinger, Der stetige Übergang von der Mikro-zur Makromechanik. Naturwissenschaften 14, 664 (1926)

    Article  Google Scholar 

  24. S.J. Summers, Tomita-Takesaki Modular Theory (2005), arxiv: math-ph/0511034v1

    Google Scholar 

  25. M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras and Its Applications (Springer, New York, 1970); M. Takesaki, Theory of Operator Algebras. I. Encyclopaedia of Mathematical Sciences, vol. 124 (Springer, Berlin, 2002); Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5; M. Takesaki, Structure of Factors and Automorphism Groups. CBMS Regional Conference Series in Mathematics, vol. 51 (American Mathematical Society, Providence, 1983); M. Takesaki, Theory of Operator Algebras. II. Encyclopaedia of Mathematical Sciences, vol. 125 (Springer, Berlin, 2003); Operator Algebras and Non-commutative Geometry, vol. 6; M. Takesaki, Theory of Operator Algebras. III. Encyclopaedia of Mathematical Sciences, vol. 127 (Springer, Berlin, 2003); Operator Algebras and Non-commutative Geometry, vol. 8

    Google Scholar 

  26. M. Tomita, Standard forms of von Neumann algebras, in V-th Functional Analysis Symposium of the Mathematical Society of Japan, Sendai (1967)

    Google Scholar 

  27. J. von Neumann, On rings of operators III. Ann. Math. 41, 94–161 (1940)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by TWAS Research Grant RGA No.17-542 RG/MATHS/AF/AC_G -FR3240300147. The ICMPA-UNESCO Chair is in partnership with Daniel Iagolnitzer Foundation (DIF), France, supporting the development of mathematical physics in Africa.

Author information

Authors and Affiliations

  1. Département de Physique, Laboratoire de Physique des Matériaux et de Mécanique Appliquée, Université de Lomé, Faculté des Sciences, Lomé, Togo

    Isiaka Aremua & Komi Sodoga

  2. University of Abomey-Calavi, International Chair in Mathematical Physics and Applications (ICMPA), Cotonou, Benin

    Isiaka Aremua, Ezinvi Baloïtcha, Mahouton Norbert Hounkonnou & Komi Sodoga

Authors
  1. Isiaka Aremua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ezinvi Baloïtcha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mahouton Norbert Hounkonnou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Komi Sodoga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isiaka Aremua .

Editor information

Editors and Affiliations

  1. Mathematical Sciences Department, The University of Alabama in Huntsville, Hunstville, AL, USA

    Toka Diagana

  2. Department of Mathematics, Howard University, Washington, DC, USA

    Bourama Toni

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Aremua, I., Baloïtcha, E., Hounkonnou, M.N., Sodoga, K. (2018). On Hilbert-Schmidt Operator Formulation of Noncommutative Quantum Mechanics. In: Diagana, T., Toni, B. (eds) Mathematical Structures and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. Springer, Cham. https://doi.org/10.1007/978-3-319-97175-9_3

Download citation

Publish with us

Policies and ethics

Search

Navigation