106
G. Vinod
5 Conclusion
Though the interest in quantum groups and q-deformation started as early as 1989,
they still remain hot areas of research in mathematics and physics. Since two major
theories of the last century, namely, relativity and quantum mechanics involved some
type of deformation, we expect this new mathematical structure will also lead to some
new physics. The idea of non-commuting space fascinates mathematicians as well
as physicists. It is reasonable to assume that a quantum theory of gravity will be
associated with some type of deformation of the space-time structure. This opens
up possibilities of q-deformed theories of gravitation [23, 24]. Besides giving rise
to new paradigms in statistical mechanics [25, 26] and quantum optics [27, 28],
q-deformed oscillators find phenomenological applications in areas like molecular
physics [23, 24], condensed matter physics [29, 30] and nuclear physics [31, 32]. A
large collection of publications in these areas are available in the literature and for
brevity, I cite only a few. The study of q-deformed quantum mechanics will enhance
our knowledge of the standard quantum mechanics.
The purpose of this article is to introduce the reader to one of the fast developing
fields in theoretical physics and to mention Prof. K Babu Joseph’s contribution in this
field. His contribution is mainly in the application of the ideas of quantum groups
and q-deformations to physical problems. It includes the application of q-oscillator
to quantum optics [33], formulation of q-deformed quantum mechanics [22] and
phenomenological applications in condensed matter physics [34].
References
1. P. Kulish, Y. Reshetikhin, J. Sov. Math. 23, 2435 (1983)
2. E.K. Sklyanin, Func. Anal. Appl. 16, 263 (1982)
3. Y. Reshetikhin, L.A. Takhtajan, L.D. Faddeev, Leningrad Math. J. 1, 193 (1990)
4. V.G. Drinfeld, Proceedings of the International Congress of Mathematicians (Berkeley, 1986)
5. M. Jimbo, Lett. Math. Phys. 10(63), 247 (1986)
6. V.F.R. Jones, Bull. Am. Mathemat. Soc. 12, 103 (1985)
7. Y.I. Mannin, Ann. Inst. Fourier 37, 191 (1987)
8. L.C. Biedenharn, Int. J. Theoret. Phys. 32, 1789 (1990)
9. S. Majid, Int. J. Modern Phys. A 5, 1 (1990)
10. V. Chari, D. Takur, Current Sci. 59, 1297 (1990)
11. T. Tjin, Int. J. Modern Phys. A 8, 231 (1992)
12. E. Calegini, R. Giachetti, E. Sorace, M. Tarlini, J. Math. Phys. 31, 11 (1990)
13. M. Chaichian, R.G. Felipe, C. Montonen, J. Phys. A Math. Gen. 26, 4017 (1993)
14. E. Heine, J. Math. 34, 285 (1846)
15. H. Exton, q-Hypergeometric Functions and Applications (Ellis Horwood, Chichester, 1983)
16. L.C. Biedenharn, J. Phys. A Math. Gen. 22, L873 (1989)
17. M.J. Macfarlane, J. Phys. A Math. Gen. 22, 4581 (1989)
18. M. Chaichan, P. Kulish, Phys. Lett. 234B, 72 (1990)
19. O.W. Greenberg, Phys. Rev. Lett. 64, 705 (1990)
20. S. Chaturvedi, A.K. Kapoor, R. Sandhya, V. Srinivasan, R. Simon, Phys. Rev. A 43, 4555
(1991)
G. Vinod
5 Conclusion
Though the interest in quantum groups and q-deformation started as early as 1989,
they still remain hot areas of research in mathematics and physics. Since two major
theories of the last century, namely, relativity and quantum mechanics involved some
type of deformation, we expect this new mathematical structure will also lead to some
new physics. The idea of non-commuting space fascinates mathematicians as well
as physicists. It is reasonable to assume that a quantum theory of gravity will be
associated with some type of deformation of the space-time structure. This opens
up possibilities of q-deformed theories of gravitation [23, 24]. Besides giving rise
to new paradigms in statistical mechanics [25, 26] and quantum optics [27, 28],
q-deformed oscillators find phenomenological applications in areas like molecular
physics [23, 24], condensed matter physics [29, 30] and nuclear physics [31, 32]. A
large collection of publications in these areas are available in the literature and for
brevity, I cite only a few. The study of q-deformed quantum mechanics will enhance
our knowledge of the standard quantum mechanics.
The purpose of this article is to introduce the reader to one of the fast developing
fields in theoretical physics and to mention Prof. K Babu Joseph’s contribution in this
field. His contribution is mainly in the application of the ideas of quantum groups
and q-deformations to physical problems. It includes the application of q-oscillator
to quantum optics [33], formulation of q-deformed quantum mechanics [22] and
phenomenological applications in condensed matter physics [34].
References
1. P. Kulish, Y. Reshetikhin, J. Sov. Math. 23, 2435 (1983)
2. E.K. Sklyanin, Func. Anal. Appl. 16, 263 (1982)
3. Y. Reshetikhin, L.A. Takhtajan, L.D. Faddeev, Leningrad Math. J. 1, 193 (1990)
4. V.G. Drinfeld, Proceedings of the International Congress of Mathematicians (Berkeley, 1986)
5. M. Jimbo, Lett. Math. Phys. 10(63), 247 (1986)
6. V.F.R. Jones, Bull. Am. Mathemat. Soc. 12, 103 (1985)
7. Y.I. Mannin, Ann. Inst. Fourier 37, 191 (1987)
8. L.C. Biedenharn, Int. J. Theoret. Phys. 32, 1789 (1990)
9. S. Majid, Int. J. Modern Phys. A 5, 1 (1990)
10. V. Chari, D. Takur, Current Sci. 59, 1297 (1990)
11. T. Tjin, Int. J. Modern Phys. A 8, 231 (1992)
12. E. Calegini, R. Giachetti, E. Sorace, M. Tarlini, J. Math. Phys. 31, 11 (1990)
13. M. Chaichian, R.G. Felipe, C. Montonen, J. Phys. A Math. Gen. 26, 4017 (1993)
14. E. Heine, J. Math. 34, 285 (1846)
15. H. Exton, q-Hypergeometric Functions and Applications (Ellis Horwood, Chichester, 1983)
16. L.C. Biedenharn, J. Phys. A Math. Gen. 22, L873 (1989)
17. M.J. Macfarlane, J. Phys. A Math. Gen. 22, 4581 (1989)
18. M. Chaichan, P. Kulish, Phys. Lett. 234B, 72 (1990)
19. O.W. Greenberg, Phys. Rev. Lett. 64, 705 (1990)
20. S. Chaturvedi, A.K. Kapoor, R. Sandhya, V. Srinivasan, R. Simon, Phys. Rev. A 43, 4555
(1991)