Abstract
This paper is an attempt to simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM). In our quaternionic approach the choice of “complex” geometries allows an appropriate definition of momentum operator and gives the possibility to obtain consistent formulations of standard theories. Barred operators represent the key to realizing a set of translation rules between quaternionic and complex quantum mechanics (QM). These translations enable us to obtain a rapid quaternionic counterpart of standard quantum mechanical results.
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References
Adler, S. L. (1994).Physics Letters,332B, 358.
Adler, S. L. (1995).Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, Oxford.
Albert, A. A. (1947).Annals of Mathematics,48, 495.
Altmann, S. L. (1986).Rotations, Quaternions, and Double Groups, Clarendon Press, Oxford.
Amaldi, U., de Boer, W., Frampton, P. H.,et al. (1992).Physics Letters,281B, 374.
De Leo, S. (1995).Progress of Theoretical Physics,94, 1109.
De Leo, S. (1996a).International Journal of Modern Physics A,11, 3973.
De Leo, S. (1996b).Journal of Mathematical Physics,37, 2955.
De Leo, S. (1996c).International Journal of Theoretical Physics,35, 1821.
De Leo, S. (1997).International Journal of Theoretical Physics,36, 1165.
De Leo, S. (n.d.). Hypercomplex group theory,Journal of Mathematical Physics, submitted [physics/9703033].
De Leo, S., and Rodrigues, W. A. (n.d.). Quaternionic Dirac equation, in preparation.
De Leo, S., and Rotelli, P. (1992).Physical Review D,45, 575.
De Leo, S., and Rotelli, P. (1994).Progress of Theoretical Physics,92, 917.
De Leo, S., and Rotelli, P. (1995).International Journal of Modern Physics A,10, 4359.
De Leo, S., and Rotelli, P. (1996a).Journal of Physics G,22, 1137.
De Leo, S., and Rotelli, P. (1996b).Modern Physics Letters A,11, 357.
Duffin, R. J. (1938).Physical Review 54, 1114.
Finkelstein, D., Jauch, J. M., Schiminovich, S., Speiser, D. (1962).Journal of Mathematical Physics,3, 207
Finkelstein, D., Jauch, J. M., and Speiser, D. (1963a).Journal of Mathematical Physics,4, 136.
Finkelstein, D., Jauch, J. M., Schiminovich, S., and Speiser, D. (1963b).Journal of Mathematical Physics,4, 788.
Finkelstein, D., Jauch, J. M., and Speiser, D. (1979).Notes on quaternion quantum mechanics, in Logico-Algebraic Approach to Quantum Mechanics II, C. A. Hooker, ed., Reidel, Dordrecht, pp. 367–421.
Fishbach, E., Nieto, M. M., and Scott, C. K. (1972).Progress of Theoretical Physics,48, 574.
Géhéniau, J. (1938).Academie Royale de Belgique.Memoires de la Classe des Sciences. Collection in-8,18, 1.
Georgi, H., and Glashow, S. L. (1974).Physical Review Letters,32, 438.
Glashow, S. L. (1961).Nuclear Physics,22, 579.
Hamilton, W. R. (1943). Quaternion Century Celebration,Proceedings of the Royal Irish Academy, Section A,50, 11.
Hamilton, W. R. (1969).Elements of Quaternions, Chelsea, New York.
Harari, H. (1979).Physics Letters,86B, 83.
Hestenes, D. (1966).Space-Time Algebra, Gordon and Breach, New York.
Hestenes, D. (1967).Journal of Mathematical Physics,8, 789.
Hestenes, D. (1975).Journal of Mathematical Physics,16, 556.
Hestenes, D. (1991).Foundations of Physics,20, 1213.
Horwitz, L. P., and Biedenharn, L. C. (1984).Annals of Physics,157, 432.
Itzykson, S., Zuber, J. B. (1985).Quantum Field Theory, McGraw-Hill, New York.
Kemmer, N. (1938).Proceedings of the Royal Society,166, 127.
Krajcik, R. A., and Nieto, M. M. (1974).Physical Review D,10, 4049.
Pati, J. C., and Salam, A. (1973).Physical Review D,8, 1240.
Petiau, G. (1936). Academie Royale de Belgique.Memoires de la Classe des Sciences. Collection in-8,16, 2.
Rembieliński, J. (1978).Journal of Physics A,11, 2323.
Rodrigues, O. (1840).Journal de Mathémetiques Pures et Appliquées,5, 380.
Rodrigues, W. A., Jr., and Capelas de Oliveira, E. (1990).International Journal of Theoretical Physics,29, 397.
Rotelli, P. (1989a).Modern Physics Letters A,4, 933.
Rotelli, P. (1989b).Modern Physics Letters A,4, 1763.
Salam, A. (1968). InProceedings of the 8th Nobel Symposium, Weak and Electromagnetic Interactions, N. Svartholm, ed., Almqvist & Wiksell, Stockholm, and Wiley, New York, p. 367.
Shupe, M. A. (1979).Physics Letters,86B, 87.
Weinberg, S. (1967).Physical Review Letters.19, 1264.
Zeni, J. R., and Rodrigues, W. A., Jr. (1992).International Journal of Modern Physics A,7, 1793.
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De Leo, S., Rodrigues, W.A. Quantum mechanics: From complex to complexified quaternions. Int J Theor Phys 36, 2725–2757 (1997). https://doi.org/10.1007/BF02435708
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DOI: https://doi.org/10.1007/BF02435708