Advances in Applied Clifford Algebras

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

Quaternionic Aharonov–Bohm Effect

  • Sergio Giardino


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.


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  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)zbMATHGoogle Scholar
  3. 3.
    Atiyah, M.F.: Geometry of Yang-Mills fields. Publications of the Scuola Normale Superiore, Pisa (1979)zbMATHGoogle 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 ADSMathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    De Leo, S., Ducati, G.: Quaternionic potentials in non-relativistic quantum mechanics. J. Phys. A35, 5411–5426 (2002)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    De Leo, S., Ducati, G.: Quaternionic bound states. J. Phys. A35, 3443–3454 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    De Leo, S., Ducati, G.: Quaternionic wave packets. J. Math. Phys. 48, 052111 (2007)ADSMathSciNetCrossRefzbMATHGoogle 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 MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

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

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