Bohr-Sommerfeld quantization rule in noncommutative space

Abstract:
There is no possibility to measure coordinates more accurate than some minimal length in the string theory [1]. To reproduce this property of the string theory Kempf proposed to modify commutation relation between the coordinate and the momentum operator in the following way [2]: [X,P]=if(P). Value of the minimal length can be determined from this relation.

We developed the WKB approximation for the case of the above modification and on its basis we derived the Bohr-Sommerfeld quantization rule [3]. We also generalized the Bohr-Sommerfeld quantization rule for the case [X,P]=if(X,P). Such a modification is characterized by minimal momentum uncertainty as well as the minimal length.

We compared the spectra obtained with the help of the Bohr-Sommerfeld quantization rule with exact ones for several examples.

[1] D.J. Gross and P.F. Mende, Nucl. Phys. B303, 407 (1988).
[2] A. Kempf, G. Mangano and R.B. Mann, Phys. Rev. D52, 1108 (1995).
[3] T.V. Fityo, I.O. Vakarchuk and V.M. Tkachuk, J. Phys. A39, 379 (2006).

Joint work with I.O. Vakarchuk and V.M. Tkachuk.