Towards an Observable Test of Noncommutative Quantum Mechanics


  • Liang Shi-Dong School of Physics, Sun Yat-sen University, China
  • T. Harko Department of Physics, Babes-Bolyai University



noncommutative quantum mechanics, Seiberg–Witten map, Aharonov–Bohm effect, persistent current


The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.


M.R. Douglas, N.A. Nekrasov. Noncommutative field theory. Rev. Mod. Phys. 73, 977 (2001).

R.J. Szabo. Quantum field theory on noncommutative spaces. Phys. Rep. 378, 207 (2003)

H. Snyder. Quantized space-time. Phys. Rev. 71, 38 (1947).

A. Connes. Noncommutative Geometry (Academic, 1994).

L. Gouba. A comparative review of four formulations of noncommutative quantum mechanics. Intern. J. Modern Phys. A 31 (19), 1630025 (2016).

O. Bertolami, J.G. Rosa, C.M.L. de Arag?ao, P. Castorina, D. Zappal'a. Noncommutative gravitational quantum well. Phys. Rev. D 72, 025010 (2005).

F. Delduc, Q. Duret, F. Gieres, M. Lefrancois. Magnetic fields in noncommutative quantum mechanics. J. Phys.: Conf. Series 103, 012020 (2008).

J. Gamboa, M. Loewe, J.C. Rojas. Noncommutative quantum mechanics. Phys. Rev. D 64, 067901 (2001).

M. Chaichian, M.M. Sheikh-Jabbari, A. Tureanua. Hydrogen atom spectrum and the Lamb-shift in noncommutative QED. Phys. Rev. Lett. 86, 2716 (2001).

N. Seiberg, E.Witten. Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19 (1994)

N. Seiberg, E. Witten. String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999).

C. Bastos, O. Bertolami. Berry phase in the gravitational quantum well and the Seiberg-Witten map. Phys. Lett. A 372 5556 (2008).

A. Das, H. Falomir, M. Nieto, J. Gamboa, F. Mendez. Aharonov-Bohm effect in a class of noncommutative theories. Phys. Rev. D 84, 045002 (2011).

Shi-Dong Liang, Haoqi Li, Guang-Yao Huang. Detecting noncommutative phase space by the Aharonov-Bohm effect. Phys. Rev. A 90, 010102 (2014); Shi-Dong Liang. Poster presentation on Conference of 90 Years of Quantum Mechanics (NTU, 2017).

M. Chaichian, P. Presnajder, M.M. Sheikh-Jabbari, A. Tureanu. Aharonov-Bohm effect in noncommutative spaces. Phys. Lett. B 527, 149 (2002).

M. Chaichian, A. Demichev, P. Presnajder, M.M. Sheikh-Jabbari, A. Tureanu. Quantum theories on noncommutative spaces with nontrivial topology: Aharonov-Bohm and Casimir effects. Nucl. Phys. B 611, 383 (2001).

A. Kokado, T. Okamura, T. Saito. Noncommutative quantum mechanics and the Seiberg-Witten map. Phys. Rev. D 69, 125007 (2004).

S. Kovacik, P. Presnajder. Magnetic monopoles in noncommutative quantum mechanics 2. J. Math. Phys. 59, 082107 (2018).

T. Harko, S.-D. Liang. Energy-dependent noncommutative quantum mechanics. E. Phys. J. C 79, (2019) 300.

D. Loss, P. Goldbart. Period and amplitude halving in mesoscopic rings with spin. Phys. Rev. B 43, 13762 (1991).

G.-Y. Huang, S.-D. Liang. Quantum phase transitions in mesoscopic Rashba rings. Phys. Lett. A 375, 738 (2011).

M. Buttiker, Y. Imry, R. Landauer. Josephson behavior in small normal one-dimensional rings. Phys. Lett. A 96, 365 (1983).

L.P. Levy, G. Dolan, J. Dunsmuir, H. Bouchiat. Magnetization of mesoscopic copper rings: Evidence for persistent currents. Phys. Rev. Lett. 64, 2074 (1990).

D. Mailly, C. Chapelier, A. Benoit. Experimental observation of persistent currents in GaAs-AlGaAs single loop. Phys. Rev. Lett. 70, 2020 (1993).

V. Chandrasekhar, R.A. Webb, M.J. Brady, M.B. Ketchen, W.J. Gallagher, A. Kleinsasser. Magnetic response of a single, isolated gold loop. Phys. Rev. Lett. 67, 3578 (1991).

S. Carroll, J. Harvey, V.A. Kostelecky, C.D. Lane, T. Okamoto. Noncommutative field theory and Lorentz violation. Phys. Rev. Lett. 87, 141601(2001).

H. Falomir, J. Gamboa, M. Loewe, F. Mendez, J.C. Rojas. Testing spatial noncommutativity via the Aharonov-Bohm effect. Phys. Rev. D 66, 045018 (2002).

P.-M. Ho, H.-C. Kao. Noncommutative quantum mechanics from noncommutative quantum field theory. Phys. Rev. Lett. 88, 151602 (2002).

B. Basu, D. Chowdhury, S. Ghosh. Inertial spin Hall effect in noncommutative space. Phys. Lett. A 377, 1661 (2013).




How to Cite

Shi-Dong, L., & Harko, T. (2019). Towards an Observable Test of Noncommutative Quantum Mechanics. Ukrainian Journal of Physics, 64(11), 983.



Fields and elementary particles