Length in a Noncommutative Phase Space

Authors

  • Kh. P. Gnatenko Department for Theoretical Physics, Ivan Franko National University of Lviv
  • V. M. Tkachuk Department for Theoretical Physics, Ivan Franko National University of Lviv

DOI:

https://doi.org/10.15407/ujpe63.2.102

Keywords:

noncommutative phase space, minimal length, uncertainty relations

Abstract

We study restrictions on the length in a noncommutative phase space caused by noncommutativity. The uncertainty relations for coordinates and momenta are considered, and the lower bound of the length is found. We also consider the eigenvalue problem for the squared length operator and find the expression for the minimal length in the noncommutative phase space.

References

<ol>
<li>N. Seiberg, E. Witten. String theory and noncommutative geometry. J. High Energy Phys. 9909, 032 (1999).
</li>
<li>S. Doplicher, K. Fredenhagen, J.E. Roberts. Spacetime quantization induced by classical gravity. Phys. Lett. B 331, 39 (1994).
<a href="https://doi.org/10.1016/0370-2693(94)90940-7">https://doi.org/10.1016/0370-2693(94)90940-7</a>
</li>
<li>J. Gamboa, M. Loewe, J.C. Rojas. Noncommutative quantum mechanics. Phys. Rev. D 64, 067901 (2001).
<a href="https://doi.org/10.1103/PhysRevD.64.067901">https://doi.org/10.1103/PhysRevD.64.067901</a>
</li>
<li>V.P. Nair, A.P. Polychronakos. Quantum mechanics on the noncommutative plane and sphere. Phys. Lett. B 505, 267 (2001).
<a href="https://doi.org/10.1016/S0370-2693(01)00339-2">https://doi.org/10.1016/S0370-2693(01)00339-2</a>
</li>
<li>K. Bolonek, P. Kosinski. On uncertainty relations in noncommutative quantum mechanics. Phys. Lett. B 547, 51 (2002).
<a href="https://doi.org/10.1016/S0370-2693(02)02731-4">https://doi.org/10.1016/S0370-2693(02)02731-4</a>
</li>
<li>C. Duval, P.A. Horvathy. Exotic Galilean symmetry in the noncommutative plane and the Hall effect. J. Phys. A 34, 10097 (2001).
<a href="https://doi.org/10.1088/0305-4470/34/47/314">https://doi.org/10.1088/0305-4470/34/47/314</a>
</li>
<li>M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanu. Hydrogen atom spectrum and the lamb shift in noncommutative QED. Phys. Rev. Lett. 86, 2716 (2001).
<a href="https://doi.org/10.1103/PhysRevLett.86.2716">https://doi.org/10.1103/PhysRevLett.86.2716</a>
</li>
<li>T.C. Adorno, M.C. Baldiotti, M. Chaichian, D.M. Gitman, A. Tureanu. Dirac equation in noncommutative space for hydrogen atom. Phys. Lett. B 682, 235 (2009).
<a href="https://doi.org/10.1016/j.physletb.2009.11.003">https://doi.org/10.1016/j.physletb.2009.11.003</a>
</li>
<li>J.M. Romero, J.D. Vergara. The Kepler problem and noncommutativity. Mod. Phys. Lett. A 18, 1673 (2003).
<a href="https://doi.org/10.1142/S0217732303011472">https://doi.org/10.1142/S0217732303011472</a>
</li>
<li> B. Mirza, M. Dehghani. Noncommutative geometry and classical orbits of particles in a central force potential. Commun. Theor. Phys. 42, 183 (2004).
<a href="https://doi.org/10.1088/0253-6102/42/2/183">https://doi.org/10.1088/0253-6102/42/2/183</a>
</li>
<li> O. Bertolami, P. Leal. Aspects of phase-space noncommutative quantum mechanics. Phys. Lett. B 750, 6 (2015).
<a href="https://doi.org/10.1016/j.physletb.2015.08.024">https://doi.org/10.1016/j.physletb.2015.08.024</a>
</li>
<li> O. Bertolami, J.G. Rosa, C.M.L. de Aragao, P. Castorina, D. Zappala. Scaling of variables and the relation between noncommutative parameters in noncommutative quantum mechanics. Mod. Phys. Lett. A 21, 795 (2006).
<a href="https://doi.org/10.1142/S0217732306019840">https://doi.org/10.1142/S0217732306019840</a>
</li>
<li> Kh.P. Gnatenko, V.M. Tkachuk. Effect of coordinate noncommutativity on the mass of a particle in a uniform field and the equivalence principle. Mod. Phys. Lett. A 31, 1650026 (2016).
<a href="https://doi.org/10.1142/S0217732316500267">https://doi.org/10.1142/S0217732316500267</a>
</li>
<li> Kh.P. Gnatenko. Estimating the upper bound of the parameter of noncommutativity on the basis of the equivalence principle. J. Phys. Stud. 17, 4001 (2013).
</li>
<li> Kh.P. Gnatenko. Physical systems in a space with noncommutativity of coordinates. J. Phys.: Conf. Ser. 670, 012023 (2016).
<a href="https://doi.org/10.1088/1742-6596/670/1/012023">https://doi.org/10.1088/1742-6596/670/1/012023</a>
</li>
<li> H. Snyder. Quantized space-time. Phys. Rev. 71, 38 (1947).
<a href="https://doi.org/10.1103/PhysRev.71.38">https://doi.org/10.1103/PhysRev.71.38</a>
</li>
<li> A.E.F. Djemai, H. Smail. On quantum mechanics on noncommutative quantum phase space. Commun. Theor. Phys. 41, 837 (2004).
<a href="https://doi.org/10.1088/0253-6102/41/6/837">https://doi.org/10.1088/0253-6102/41/6/837</a>
</li>
<li> Li Kang, Chamoun Nidal. Hydrogen atom spectrum in noncommutative phase space. Chin. Phys. Lett. 23, 1122 (2006).
<a href="https://doi.org/10.1088/0256-307X/23/5/016">https://doi.org/10.1088/0256-307X/23/5/016</a>
</li>
<li> S.A. Alavi. Lamb shift and Stark effect in simultaneous space-space and momentum-momentum noncommutative quantum mechanics and O. Mod. Phys. Lett. A 22, 377 (2007).
<a href="https://doi.org/10.1142/S0217732307018579">https://doi.org/10.1142/S0217732307018579</a>
</li>
<li> O. Bertolami, R. Queiroz. Phase-space noncommutativity and the Dirac equation. Phys. Lett. A 375, 4116 (2011).
<a href="https://doi.org/10.1016/j.physleta.2011.09.053">https://doi.org/10.1016/j.physleta.2011.09.053</a>
</li>
<li> A. Smailagic, E. Spallucci. Isotropic representation of the noncommutative 2D harmonic oscillator. Phys. Rev. D 65, 107701 (2002).
<a href="https://doi.org/10.1103/PhysRevD.65.107701">https://doi.org/10.1103/PhysRevD.65.107701</a>
</li>
<li> A. Smailagic, E. Spallucci. Noncommutative 3D harmonic oscillator. J. Phys. A 35, 363 (2002).
<a href="https://doi.org/10.1088/0305-4470/35/26/103">https://doi.org/10.1088/0305-4470/35/26/103</a>
</li>
<li> A. Hatzinikitas, I. Smyrnakis. The noncommutative harmonic oscillator in more than one dimension. J. Math. Phys. 43, 113 (2002).
<a href="https://doi.org/10.1063/1.1416196">https://doi.org/10.1063/1.1416196</a>
</li>
<li> Li Kang, Wang Jianhua, Chen Chiyi. Representation of noncommutative phase space. Mod. Phys. Lett. A 20, 2165 (2005).
<a href="https://doi.org/10.1142/S0217732305017421">https://doi.org/10.1142/S0217732305017421</a>
</li>
<li> C. Acatrinei. Path integral formulation of noncommutative quantum mechanics. J. of High Energy Phys. 9, 007 (2001).
</li>
<li> P.R. Giri, P. Roy. The non-commutative oscillator, symmetry and the Landau problem. Eur. Phys. J. C 57, 835 (2008).
<a href="https://doi.org/10.1140/epjc/s10052-008-0705-4">https://doi.org/10.1140/epjc/s10052-008-0705-4</a>
</li>
<li> J. Ben Geloun, S. Gangopadhyay, F.G. Scholtz. Harmonic oscillator in a background magnetic field in noncommutative quantum phase-space. EPL 86, 51001 (2009).
<a href="https://doi.org/10.1209/0295-5075/86/51001">https://doi.org/10.1209/0295-5075/86/51001</a>
</li>
<li> O. Bertolami, J.G. Rosa, C.M.L. de Aragao, P. Castorina, D. Zappala. Noncommutative gravitational quantum well. Phys. Rev. D 72, 025010 (2005).
<a href="https://doi.org/10.1103/PhysRevD.72.025010">https://doi.org/10.1103/PhysRevD.72.025010</a>
</li>
<li> C. Bastos, O. Bertolami. Berry phase in the gravitational quantum well and the Seiberg–Witten map. Phys. Lett. A 372, 5556 (2008).
<a href="https://doi.org/10.1016/j.physleta.2008.06.073">https://doi.org/10.1016/j.physleta.2008.06.073</a>
</li>
<li> Kh.P. Gnatenko, V. M. Tkachuk. Weak equivalence principle in noncommutative phase space and the parameters of noncommutativity. Phys. Lett. A 381, 2463 (2017).
<a href="https://doi.org/10.1016/j.physleta.2017.05.056">https://doi.org/10.1016/j.physleta.2017.05.056</a>
</li>
<li> J.M. Romero, J.A. Santiago, J.D. Vergara. Note about the quantum of area in a noncommutative space. Phys. Rev. D 68, 067503 (2003).
<a href="https://doi.org/10.1103/PhysRevD.68.067503">https://doi.org/10.1103/PhysRevD.68.067503</a>
</li>
<li> A. Kijanka and P. Kosinski. Noncommutative isotropic harmonic oscillator. Phys. Rev. D 70, 127702 (2004).
<a href="https://doi.org/10.1103/PhysRevD.70.127702">https://doi.org/10.1103/PhysRevD.70.127702</a>
</li>
<li> Kh.P. Gnatenko, V.M. Tkachuk. Minimal length, area, and volume in a space with noncommutativity of coordinates. J. Phys. Stud. 20, 1001 (2016).
</li>
<li> A.E.F. Djemai, H. Smail. On quantum mechanics on noncommutative quantum phase space. Commun. Theor. Phys. 41, 6 (2004).
<a href="https://doi.org/10.1088/0253-6102/41/6/837">https://doi.org/10.1088/0253-6102/41/6/837</a>
</li></ol>

Downloads

Published

2018-03-10

How to Cite

Gnatenko, K. P., & Tkachuk, V. M. (2018). Length in a Noncommutative Phase Space. Ukrainian Journal of Physics, 63(2), 102. https://doi.org/10.15407/ujpe63.2.102

Issue

Section

General physics