Conventional and Enhanced Canonical Quantizations, Application to Some Simple Manifolds

Affiliation(s)

Unité de Recherche en Physique Théorique (URPT), Institut de Mathématiques et de Sciences Physiques (IMSP), Porto-Novo, Bénin.

Unité de Recherche en Physique Théorique (URPT), Institut de Mathématiques et de Sciences Physiques (IMSP), Porto-Novo, Bénin.

ABSTRACT

It is well known that the representations over an arbitrary configuration space related to a physical system of the Heisenberg algebra allow to distinguish the simply and non simply-connected manifolds [arXiv:quant-ph/9908.014, arXiv:hep-th/0608.023]. In the light of this classification, the dynamics of a quantum particle on the line is studied in the framework of the conventional quantization scheme as well as that of the enhanced quantization recently introduced by J. R. Klauder [arXiv:quant-ph/1204.2870]. The quantum action functional restricted to the phase space coherent states is obtained from the enhanced quantization procedure, showing the coexistence of classical and quantum theories, a fundamental advantage offered by this new approach. The example of the one dimensional harmonic oscillator is given. Next, the spectrum of a free particle on the two-sphere is recognized from the covariant diffeomorphic representations of the momentum operator in the configuration space. Our results based on simple models also point out the already-known link between interaction and topology at quantum level.

Cite this paper

G. Avossevou, J. Hounguevou and D. Takou, "Conventional and Enhanced Canonical Quantizations, Application to Some Simple Manifolds,"*Journal of Modern Physics*, Vol. 4 No. 11, 2013, pp. 1476-1485. doi: 10.4236/jmp.2013.411177.

G. Avossevou, J. Hounguevou and D. Takou, "Conventional and Enhanced Canonical Quantizations, Application to Some Simple Manifolds,"

References

[1] J. Govaerts and V. Villanueva, International Journal of Modern Physics A, Vol. 15, 2000, p. 4903. arXiv:quantph/9908.014.

http://dx.doi.org/10.1142/S0217751X00002846

[2] J. Govaerts and F. Payen, Modern Physics Letters A, Vol. 22, 2007, pp. 119-130. arXiv:hep-th/0608.023.

http://dx.doi.org/10.1142/S0217732307022335

[3] J. R. Klauder, Annals of Physics, Vol. 188, 1998, pp. 120-141.

http://dx.doi.org/10.1016/0003-4916(88)90092-9

[4] J. Govaerts, “Quantisation, Topology and Interactions,” Lecture Notes, Department of Theoretical Physics, School of Physics, The University of New South Wales, Sydney, 2006.

[5] G. Y. H. Avossevou, “From the Flat Circular Annulus Holonomy to the Aharonov-Bohm Effect: An Application of the von Neumann’s Self-Adjoint Extensions Theory,” Preprint, IMSP, 2012.

[6] J. R. Klauder, “Enhanced Quantization: A Primer,” ar-Xiv:quant-ph/1204.2870.

[7] J. R. Klauder, “Enhanced Quantum Procedures That Resolve Difficult Problems,” Lecture Notes for the Advanced Scholar Seminar, Interaction of Mathematics and Physics: New Perspectives, Mosccow, Russia, 2012, ar-Xiv:quant-ph/1204.2870.

[8] J. von Neumann, “Mathematische Grundlagen der Quantenmechanik,” Springer, Berlin, 1932.

[9] M. Reed and B. Simon, “Methods of Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness,” Academic Press, Waltham, 1972.

[10] G. Bonneau, J. Faraut and G. Valent, American Journal of Physics, Vol. 69, 2001, p. 322. arXiv:quant-ph/0103.153.

http://dx.doi.org/10.1119/1.1328351

[11] M. H. Al-Hashimi, M. Salman, A. Shalaby and U.-J. Wiese, “Supersymmetric Descendants of Self-Adjointly Extended Quantum Mechanical Hamiltonians,” 2013. Ar-Xiv:hep-th/1303.2343v1.

[12] S. T. Ali, J. P. Antoine and J.-P. Gazeau, “Coherent States, Wavelets, and Their Generalizations,” Springer-Verlag, Berlin, 2000.

http://dx.doi.org/10.1007/978-1-4612-1258-4

[13] J. R. Klauder, “Phase Space Geometry in Classical and Quantum Mechanics,” In Contemporary Problems in Mathematical Physics, World Scientific, Singapore, 2002, pp. 395-408. arXiv:quant-ph/0112.010.

[14] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover, New York, 1972.

[15] J. B. Geloun and J. R. Klauder, “Enhanced Quantization on the Circle,” arXiv:math-ph/1206.1180v1. (See the referencies therein)

[16] J. R. Klauder, “Enhanced Quantum Procedures That Resolve Difficult Problems,” 2012. arXiv:hep-ph/12064017v5.

[17] I. M. Green and S. A. Moszowski, Physical Review, Vol. 139, 1965, p. B790.

http://dx.doi.org/10.1103/PhysRev.139.B790

[18] K. Kowalski and J. Rembielinski, “Coherent States for a Particle on a Sphere,” 1999. arXiv:quant-ph/9912094v1.

[19] K. Kowalski, J. Rembielinski and L. C. Papaloucas, “Coherent States for a Quantum Particle on a Circle,” 1998. arXiv:quant-ph/9801029v1.

[1] J. Govaerts and V. Villanueva, International Journal of Modern Physics A, Vol. 15, 2000, p. 4903. arXiv:quantph/9908.014.

http://dx.doi.org/10.1142/S0217751X00002846

[2] J. Govaerts and F. Payen, Modern Physics Letters A, Vol. 22, 2007, pp. 119-130. arXiv:hep-th/0608.023.

http://dx.doi.org/10.1142/S0217732307022335

[3] J. R. Klauder, Annals of Physics, Vol. 188, 1998, pp. 120-141.

http://dx.doi.org/10.1016/0003-4916(88)90092-9

[4] J. Govaerts, “Quantisation, Topology and Interactions,” Lecture Notes, Department of Theoretical Physics, School of Physics, The University of New South Wales, Sydney, 2006.

[5] G. Y. H. Avossevou, “From the Flat Circular Annulus Holonomy to the Aharonov-Bohm Effect: An Application of the von Neumann’s Self-Adjoint Extensions Theory,” Preprint, IMSP, 2012.

[6] J. R. Klauder, “Enhanced Quantization: A Primer,” ar-Xiv:quant-ph/1204.2870.

[7] J. R. Klauder, “Enhanced Quantum Procedures That Resolve Difficult Problems,” Lecture Notes for the Advanced Scholar Seminar, Interaction of Mathematics and Physics: New Perspectives, Mosccow, Russia, 2012, ar-Xiv:quant-ph/1204.2870.

[8] J. von Neumann, “Mathematische Grundlagen der Quantenmechanik,” Springer, Berlin, 1932.

[9] M. Reed and B. Simon, “Methods of Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness,” Academic Press, Waltham, 1972.

[10] G. Bonneau, J. Faraut and G. Valent, American Journal of Physics, Vol. 69, 2001, p. 322. arXiv:quant-ph/0103.153.

http://dx.doi.org/10.1119/1.1328351

[11] M. H. Al-Hashimi, M. Salman, A. Shalaby and U.-J. Wiese, “Supersymmetric Descendants of Self-Adjointly Extended Quantum Mechanical Hamiltonians,” 2013. Ar-Xiv:hep-th/1303.2343v1.

[12] S. T. Ali, J. P. Antoine and J.-P. Gazeau, “Coherent States, Wavelets, and Their Generalizations,” Springer-Verlag, Berlin, 2000.

http://dx.doi.org/10.1007/978-1-4612-1258-4

[13] J. R. Klauder, “Phase Space Geometry in Classical and Quantum Mechanics,” In Contemporary Problems in Mathematical Physics, World Scientific, Singapore, 2002, pp. 395-408. arXiv:quant-ph/0112.010.

[14] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions,” Dover, New York, 1972.

[15] J. B. Geloun and J. R. Klauder, “Enhanced Quantization on the Circle,” arXiv:math-ph/1206.1180v1. (See the referencies therein)

[16] J. R. Klauder, “Enhanced Quantum Procedures That Resolve Difficult Problems,” 2012. arXiv:hep-ph/12064017v5.

[17] I. M. Green and S. A. Moszowski, Physical Review, Vol. 139, 1965, p. B790.

http://dx.doi.org/10.1103/PhysRev.139.B790

[18] K. Kowalski and J. Rembielinski, “Coherent States for a Particle on a Sphere,” 1999. arXiv:quant-ph/9912094v1.

[19] K. Kowalski, J. Rembielinski and L. C. Papaloucas, “Coherent States for a Quantum Particle on a Circle,” 1998. arXiv:quant-ph/9801029v1.