On pairs of difference operators satisfying [D, X] = Id

Journal of Mathematical Physics

Volumn 39, Issue 1, 1998, Pages 545-568

  Górski, Andrzej Z ^a,c   Szmigielski, Jacek ^b  

^a UNIVERSITY OF SOUTH CAROLINA   (United States)

^b UNIVERSITY OF SASKATCHEWAN   (Canada)

^c INSTITUTE OF NUCLEAR PHYSICS   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0347802106     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532322     Document Type: Article

References (41)

1. V. Ambarcumian and D. Ivanienko, Z. Phys. 64, 563 (1930).
2. The first questions about the world geometry were already posed by Riemann in the XIXth century: G. F. B. Riemann, in Über die Hypothesen, welche der Geometrie zugrunde liegen, edited by H. Weyl (Springer-Verlag, Berlin, 1919).
3. P. Gibbs, "The Small Scale Structure of Space-Time," preprint PEG-06-95 (1995), hep-th/9506171.
4. I. Klebanov and L. Suskind, Nucl. Phys. B 309, 175 (1988).
5. R. Penrose, in Quantum Theory and Beyond, edited by E. A. Bastin (Cambridge U. P. Cambridge, 1971).
6. G. Horowitz, talk at the Conference on Quantum Gravity, Durham (1994).
7. C. Rovelli and L. Smolin, Nucl. Phys. B 442, 593 (1995).
8. J. D. Bekenstein, Lett. Nuovo Cimento 11, 467 (1974).
9. V. F. Mukhanov, JETP Lett. 44, 63 (1986).
10. T. Jacobson, Phys. Rev. D 44, 1731 (1991).
11. J. D. Bekenstein and V. F. Mukhanov, Phys. Lett. B 360, 7 (1995).
12. P. O. Mazur, Acta Phys. Pol. B 27, 1849 (1996).
13. G. 't Hooft, Nucl. Phys. B 342, 471 (1990).
14. G. 't Hooft, Class. Quantum Grav. 10, 1023 (1993).
15. G. 't Hooft, Class. Quantum Grav. 10, 1653 (1993).
16. Y. Aharonov, H. Pendelton, and A. Petersen, Int. J. Theor. Phys. 2, 213 (1969).
17. K. Wilson, Phys. Rev. D 10, 2445 (1974).
18. T. Regge, Nuovo Cimento 19, 558 (1961).
19. L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Phys. Rev. Lett. 59, 521 (1987).
20. L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995).
21. J. Lukierski, A. Nowicki, and H. Ruegg, Phys. Lett. B 293, 344 (1992).
22. P. O. Mazur, Acta Phys. Pol. B 26, 1685 (1995).
23. P. O. Mazur, Phys. Rev. Lett. 57, 929 (1986); 59, 2380 (1987).
24. P. O. Mazur, Phys. Rev. Lett. 57, 929 (1986); 59, 2380 (1987).
25. M. R. Douglas, Phys. Lett. B 238, 176 (1990).
26. P. A. M. Dirac, Proc. R. Soc. London, Ser. A 109, 642,Sec.3 (1926).
27. There exists, however, a way of restoring the Leibniz rule by modyfying the product: Y. Bouguenaya and D. B. Fairlie, J. Phys. A 19, 1049 (1986) and R. S. Ward, Phys. Lett. A 165, 325 (1992). We are grateful to Prof. D. B. Fairlie for bringing these references to our attention.
28. There exists, however, a way of restoring the Leibniz rule by modyfying the product: Y. Bouguenaya and D. B. Fairlie, J. Phys. A 19, 1049 (1986) and R. S. Ward, Phys. Lett. A 165, 325 (1992). We are grateful to Prof. D. B. Fairlie for bringing these references to our attention.
29. Yu. Smirnov and A. Turbiner, Mod. Phys. Lett. A 10, 1795 (1995).
30. S. C. Coutinho, A Primer of Algebraic D-modules, LMSST 33 (Cambridge U.P., Cambridge, 1995).
31. It can be shown that for classical dynamical systems the optimal schemes are closer to continuous derivative with respect to the stability analysis and they preserve the Hamiltonian structure of the phase space, as well (A. Z. Górski and J. Szmigielski, in preparation).
32. E. A. B. Cole, Nuovo Cimento A 66, 645 (1970).
33. L. Frappat and A. Sciarrino, Phys. Lett. B 347, 28 (1995).
34. H. Widom, in Studies in Real and Complex Analysis, Vol. 3, edited by I. H. Hirschman, Jr., Prentice-Hall, Inc., Englewood Cliffs, NJ, 1965.
35. G. Dorfmeister and J. Dorfmeister, J. Funct. Anal. 57, 301 (1984).
36. D. Schmüdgen, J. Funct. Anal. 50, 8 (1983).
37. D. Schmüdgen, Publ. Res. Inst. Math. Sci., Kyoto Univ. 19, 601 (1983).
38. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1975), Vols. 1 and 2.
39. G-C. Rota, Finite Operator Calculus (Academic, New York, 1975).
40. A. Dimakis and F. Mueller-Hoissen, J. Phys. A 29, 6861 (1996).
41. A. Zygmunt, Trigonometric Series (Cambridge U. P., Cambridge, 1959).