Advances in Applied Clifford Algebras

, Volume 27, Issue 3, pp 2445–2456 | Cite as

Quaternionic Aharonov–Bohm Effect



A quaternionic analog of the Aharonov–Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics. A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, S.L.: Does the Peres experiment using photons test for hyper-complex (quaternionic) quantum theories? (2016) arXiv:1604.04950 [quant-ph]
  2. 2.
    Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  3. 3.
    Atiyah, M.F.: Geometry of Yang-Mills fields. Publications of the Scuola Normale Superiore, Pisa (1979)MATHGoogle Scholar
  4. 4.
    Brumby, S.P., Joshi, G.C.: Experimental status of quaternionic quantum mechanics. Chaos Solitons Fractals 7, 747–752 (1996). arXiv:quant-ph/9610009 ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Davies, A.J.: Quaternionic Dirac equation. Phys. Rev. D 41, 2628–2630 (1990)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Davies, A.J., McKellar, B.H.J.: Nonrelativistic quaternionic quantum mechanics in one dimension. Phys. Rev. A 40, 4209–4214 (1989)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Davies, A.J., McKellar, B.H.J.: Observability of quaternionic quantum mechanics. Phys. Rev. A 46, 3671–3675 (1989)ADSCrossRefGoogle Scholar
  8. 8.
    De Leo, S., Ducati, G., Giardino, S.: Quaternioninc Dirac Scattering. J. Phys. Math. 6:1000130 (2015) arXiv:1505.01807 [math-ph]
  9. 9.
    De Leo, S., Giardino, S.: Dirac solutions for quaternionic potentials. J. Math. Phys., 55:022301–10 (2014) arXiv: 1311.6673 [math-ph]
  10. 10.
    De Leo, S., Ducati, G.: Quaternionic differential operators. J. Math. Phys. 42, 2236–2265 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    De Leo, S., Ducati, G.: Quaternionic potentials in non-relativistic quantum mechanics. J. Phys. A35, 5411–5426 (2002)ADSMathSciNetMATHGoogle Scholar
  12. 12.
    De Leo, S., Ducati, G.: Quaternionic bound states. J. Phys. A35, 3443–3454 (2005)MathSciNetMATHGoogle Scholar
  13. 13.
    De Leo, S., Ducati, G.: Quaternionic wave packets. J. Math. Phys. 48, 052111 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    De Leo, S., Rotelli, P.: The Quaternion scalar field. Phys. Rev. D 45, 575–579 (1992)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Devchand, C., Ogievetsky, V.: Four-dimensional integrable theories. Lect. Notes Phys. 447, 169 (1995). arXiv:hep-th/9410147 ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Evans, M., Gursey, F., Ogievetsky, V.: From 2-D conformal to 4-D selfdual theories: quaternionic analyticity. Phys. Rev. D 47, 3496–3508 (1993)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Giardino, S.: Quaternionic particle in a relativistic box. Found. Phys. 46(4):473–483 (2016) arXiv:1504.00643 [quant-ph]
  18. 18.
    Maia, M.D., Bezerra, V.B.: Geometric phase in quaternionic quantum mechanics. Int. J. Theor. Phys. 40, 1283–1294 (2001). arXiv:hep-th/0107107 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Procopio, L.M., Rozema, L.A., Dakić, B., Walther, P.: Comment on Adler’s. Does the Peres experiment using photons test for hyper-complex (quaternionic) quantum theories?. arXiv:1607.01648 [quant-ph] (2016)
  20. 20.
    Procopio, L.M., Rozema, L.A., Wong Z.J., Hamel, D.R., O’Brien, K., Zhang, X., Dakic, B., Walther P.: Experimental Test of Hyper-Complex Quantum Theories. (2016) arXiv:1602.01624 [quant-ph]
  21. 21.
    Vaz, J., da Rocha, R.: An Introduction to Clifford Algebras and Spinors. Oxford University Press, Oxford (2016)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Departamento de Física and Centro de Matemática e AplicaçõesUniversidade da Beira InteriorCovilhãPortugal

Personalised recommendations