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Configuration ray representations in non-Abelian quantum kinematics and dynamics

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Abstract

Ray extensions of the regular representation of noncompact non-Abelian Lie groups are examined as generalizations of the Cartesian coordinate representation of ordinary quantum mechanics to the case of generalized non-Cartesian coordinates and generalized noncommuting momenta. (The momenta are in fact the generators of the representation, and so they satisfy the Lie algebra of the group.) The concept of configuration ray representation is introduced within this new kinematic formalism as subrepresentations of the regular representation which are embossed with the “relativity theory” of a given system. The main features of the mathematical formalism leading to these representations in configuration spacetime are discussed, and their importance for non-Abelian quantum kinematics and dynamics is emphasized. Two miscellaneous examples on the calculus of phase functions for configuration ray representations are given.

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References

  1. Bargmann, V. (1954).Annals of Mathematics,59, 1.

  2. Chaichian, M., and Nelipa, N. F. (1984).Introduction to Gauge Field Theories, Springer, Berlin.

  3. Houard, C. J. (1977).Journal of Mathematical Physics,18, 502.

  4. Komar, A. (1971). InProblems in the Foundation of Physics, M. Bunge, ed., Springer, Berlin.

  5. Krause, J. (1985).Journal of Physics A: Mathematical and General,18, 1309.

  6. Krause, J. (1986).Journal of Mathematical Physics,27, 2922.

  7. Krause, J. (1987).Journal of Mathematical Physics,28, 2268.

  8. Krause, J. (1988).Journal of Mathematical Physics,29, 1309.

  9. Krause, J. (1991).Journal of Mathematical Physics,32, 348.

  10. Krause, J. (1993a).Journal of Physics A: Mathematical and General,26, 6285.

  11. Krause, J. (1993b).International Journal of Theoretical Physics,32, 1363.

  12. Krause, J. (1993c). Quantum kinematic theory and dynamics of noncompact non-Abelian Lie groups, preprint PUCCH.

  13. Mariwalla, K. (1975).Physics Reports,20, 287.

  14. Messiah, A. (1961).Quantum Mechanics, North-Holland, Amsterdam.

  15. Michel, L. (1964). InGroup Theoretical Concepts and Methods in Elementary Particle Physics, F. Gürsey, ed., Gordon and Breach, New York.

  16. Naimark, M. A., and Stern, A. I. (1982).Theory of Group Representations, Springer, Berlin.

  17. Olver, P. J. (1986).Applications of Lie Groups to Differential Equations, Springer, Berlin.

  18. Racah, G. (1965).Ergebnisse der Exakten Naturwissenschaften,37, 38.

  19. Trümper, M. (1983).Annals of Physics,149, 203.

  20. Weyl, H. (1931).The Theory of Groups of Quantum Mechanics, Dover, New York.

  21. Wigner, E. P. (1959).Group Theory, Academic Press, New York.

  22. Yamada, K. (1982).Physical Review D,25, 3256.

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Krause, J. Configuration ray representations in non-Abelian quantum kinematics and dynamics. Int J Theor Phys 33, 1617–1637 (1994). https://doi.org/10.1007/BF00672687

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Keywords

  • Field Theory
  • Elementary Particle
  • Quantum Field Theory
  • Quantum Mechanic
  • Mathematical Formalism