Remarks on the formulation of quantum mechanics on noncommutative phase spaces

Journal of High Energy Physics

Volumn 2007, Issue 1, 2007, Pages

  Muthukumar, Balasundaram ^a  

^a SAHA INSTITUTE OF NUCLEAR PHYSICS   (India)

Author keywords

Non commutative geometry; Statistical methods

Indexed keywords

EID: 33947687767     PISSN: 10298479     EISSN: 10298479     Source Type: Journal    
DOI: 10.1088/1126-6708/2007/01/073     Document Type: Article

References (40)

1. N. Seiberg and E. Witten, String theory and noncommutative geometry 1999 J. High Energy Phys. JHEP09(1999)032 [hep-th/9908142]
2. M.R. Douglas and N.A. Nekrasov, Noncommutative field theory 2001 Rev. Mod. Phys. 73 977 [hep-th/0106048]
3. R.J. Szabo, Quantum field theory on noncommutative spaces 2003 Phys. Rep. 378 207 [hep-th/0109162]
4. J. Gomis, K. Landsteiner and E. Lopez, Non-relativistic non-commutative field theory and UV/IR mixing 2000 Phys. Rev. D 62 105006 [hep-th/0004115]
5. T. Mateos and A. Moreno, A note on unitarity of non-relativistic non-commutative theories 2001 Phys. Rev. D 64 047703 [hep-th/0104167]
6. M.A. Anacleto, M. Gomes, A.J. da Silva and D. Spehler, Noncommutative field theory: nonrelativistic fermionic field coupled to the Chern-Simons field in 2+1 dimensions 2005 Phys. Rev. D 71 107701 [hep-th/0411168]
7. B. Muthukumar, U(1) gauge invariant noncommutative Schroedinger theory and gravity 2005 Phys. Rev. D 71 105007 [hep-th/0412069]
8. P.A. Horvthy, L. Martina and P.C. Stichel, Galilean symmetry in noncommutative field theory 2003 Phys. Lett. B 564 149 [hep-th/0304215]
9. P.A. Horvthy, L. Martina and P.C. Stichel, Galilean noncommutative gauge theory: symmetries and vortices 2003 Nucl. Phys. B 673 301 [hep-th/0306228]
10. M. Chaichian, M.M. Sheikh-Jabbari and A. Tureanu, Hydrogen atom spectrum and the lamb shift in noncommutative QED 2001 Phys. Rev. Lett. 86 2716 [hep-th/0010175]
11. C. Duval and P.A. Horvathy, The 'Peierls substitution' and the exotic Galilei group 2000 Phys. Lett. B 479 284
12. V.P. Nair and A.P. Polychronakos, Quantum mechanics on the noncommutative plane and sphere 2001 Phys. Lett. B 505 267 [hep-th/0011172]
13. B. Muthukumar and P. Mitra, Non-commutative oscillators and the commutative limit 2002 Phys. Rev. D 66 027701 [hep-th/0204149]
14. S.-C. Jing, F. Zuo and T.-H. Heng, Deformation quantization of noncommutative quantum mechanics 2004 J. High Energy Phys. JHEP10(2004)049
15. G. Baker, Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space 1958 Phys. Rev. D 109 2198.
16. D. Fairlie, The formulation of quantum mechanics in terms of phase space functions 1964 Proc. Cambridge Phil. Soc. 60 581.
17. T. Curtright, D. Fairlie and C.K. Zachos, Features of time-independent Wigner functions 1998 Phys. Rev. D 58 025002 [hep-th/9711183]
18. N.C. Dias and J.N. Prata, Admissible states in quantum phase space 2004 Ann. Phys. (NY) 313 110.
19. N.C. Dias and J.N. Prata, Formal solutions of Stargenvalue equations 2004 Ann. Phys. (NY) 311 120.
20. B. Chakraborty, S. Gangopadhyay, A.G. Hazra and F.G. Scholtz, Twisted Galilean symmetry and the Pauli principle at low energies 2006 J. Phys. A: Math. Gen. 39 9557 [hep-th/0601121]
21. G. Landi, An introduction to noncommutative spaces and their geometry [hep-th/9701078]
22. J.C. Vrilly, An Introduction to noncommutative geometry [physics/9709045]
23. L. Castellani, Noncommutative geometry and physics: a review of selected recent results 2000 Class. Quantum Grav. 17 3377 [hep-th/0005210]
24. M.Khalkhali, Very basic noncommutative geometry [math.KT/0408416]
25. V.Ginzburg, Lectures on noncommutative geometry [math.AG/0506603].
26. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization: I deformation of symplectic structures 1978 Ann. Phys. (NY) 111 61.
27. R.Courant and D.Hilbert, Methods of mathematical physics Interscience Publishers, Inc., New York (1953) p.130
28. F. Riesz and B.Sz.-Nagy, Functional analysis Frederick Ungar Publishing Co., New York (1955)
29. J.E. Moyal, Quantum mechanics as a statistical theory 1949 Proc. Cambridge Phil. Soc. 45 99.
30. E.P. Wigner in, Perspectives in quantum theory ed. W. Yourgrau and A. Vander Merwe, Dover Publications Inc., New York, 1971
31. M. Hillery, R.F. O'Connell, M.O. Scully and E.P. Wigner, Distribution functions in physics: fundamentals 1984 Phys. Rep. 106 121
32. J.G. Kruger and A. Poffyn, Quantum mechanics in phase space. I. unicity of the Wigner distribution function 1976 Physica 85A 84.
33. F.J. Narcowich and R.F. O'Connell, Necessary and sufficient conditions for a phase-space function to be a Wigner distribution 1986 Phys. Rev. A 34 1.
34. A. Eftekharzadeh and B.L. Hu, The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation. 2005 Braz. J. Phys. 35 333
35. J.M. Romero, J.A. Santiago, J.D. Vergara, Newton's second law in a noncommutative space 2003 Phys. Lett. A 310 9.
36. G.D. Barbosa, On the meaning of the string inspired noncommutativity and its implications 2003 J. High Energy Phys. JHEP05(2003)024 [hep-th/0301106]
37. Y. Nambu, Generalized hamiltonian dynamics 1973 Phys. Rev. D 7 2405
38. G. Dito, M. Flato, D. Sternheimer, L. Takhtajan, Deformation quantization and Nambu mechanics 1997 Comm. Math. Phys. 183 1
39. H. Awata, M. Li, D. Minic and T. Yoneya, On the quantization of nambu brackets 2001 J. High Energy Phys. JHEP02(2001)013 [hep-th/9906248]
40. T. Curtright and C.K. Zachos, Classical and quantum nambu mechanics 2003 Phys. Rev. D 68 085001 [hep-th/0212267]