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International Journal of Theoretical Physics

, Volume 44, Issue 11, pp 1889–1904 | Cite as

Perspectives: Quantum Mechanics on Phase Space

  • J. A. Brooke
  • F. E. Schroeck
Article

Abstract

The basic ideas in the theory of quantum mechanics on phase space are illustrated through an introduction of generalities, which seem to underlie most if not all such formulations and follow with examples taken primarily from kinematical particle model descriptions exhibiting either Galileian or Lorentzian symmetry. The structures of fundamental importance are the relevant (Lie) groups of symmetries and their homogeneous (and associated) spaces that, in the situations of interest, also possess Hamiltonian structures. Comments are made on the relation between the theory outlined and a recent paper by Carmeli, Cassinelli, Toigo, and Vacchini.

Key Words

phase space quantum theory quantization SIC Heyting effect algebra 

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REFERENCES

  1. Ali, S. T., Brooke, J. A., Busch, P., Gagnon, R., and Schroeck, Jr., F. E. (1988). Canadian Journal of Physics 66, 238–244.ADSMathSciNetGoogle Scholar
  2. Ali, S. T. and Prugovečki, E. (1986). Acta Applicandae Mathematicae 6, 19–45.CrossRefMathSciNetGoogle Scholar
  3. Ali, S. T. and Prugovečki, E. (1986). Acta Applicandae Mathematicae 6, 47–62.CrossRefMathSciNetGoogle Scholar
  4. Bargmann, V. (1954). Annals of Mathematics 59, 1–17.CrossRefMATHMathSciNetGoogle Scholar
  5. Brooke, J. A. (1987). The Physics of Phase Space, In Y. S. Kim and W. W. Zachary, eds., Lecture Notes in Physics, vol. 278, Springer, New York, pp. 366–368.Google Scholar
  6. Brooke, J. A. and Schroeck, Jr., F. E. (1996). Journal of Mathematical Physics 37, 5958–5986.CrossRefADSMathSciNetGoogle Scholar
  7. Brooke, J. A. and Schroeck, Jr., F. E. (1989). Nuclear Physics B (Proceedings Supplement) 6, 104–106.CrossRefADSMathSciNetGoogle Scholar
  8. Brooke, J. A. and Schroeck, Jr., F. E. (in preparation). Phase Space Representations of Relativistic, Massive Spinning Particles.Google Scholar
  9. Busch, P. (1991). International Journal of Theoretical Physics 30, 1217–1227.CrossRefMathSciNetGoogle Scholar
  10. Busch, P., Grabowski, M., and Lahti, P. J. (1995). Operational Quantum Physics, Lecture Notes in Physics, vol. 31, Springer, Berlin.Google Scholar
  11. Carmeli, C., Cassinelli, G., Toigo, A., and Vacchini, B. (2004). Journal of Physics A: Mathematical and General 37, 5057–5066.CrossRefADSMathSciNetGoogle Scholar
  12. Einstein, A. (1905a). Annalen der Physik 17, 891–921.ADSGoogle Scholar
  13. Einstein, A. (1905b). Annalen der Physik 18, 639–641.ADSGoogle Scholar
  14. Einstein, A. (1911). Annalen der Physik 35, 898–908.ADSMATHGoogle Scholar
  15. Einstein, A. (1916). Annalen der Physik 49, 769–822.ADSMATHGoogle Scholar
  16. Feynman, R., Leighton, R. B., and Sands, M. (1963). The Feynman Lectures in Physics, vol. 1, Addison-Wesley, Reading. Section 37–8.Google Scholar
  17. Finkelstein, D. (1997). Address given at the I.Q.S.A. workshop in Atlanta.Google Scholar
  18. FitzGerald, G. F. (1891). Nature XLVI, 165.Google Scholar
  19. Guillemin, V. and Sternberg, S. (1991). Symplectic Techniques in Physics, Cambridge University Press, New York.Google Scholar
  20. Healy, Jr., D. M. and Schroeck, Jr., F. E. (1995). Journal of Mathematical Physics 36, 453–507.CrossRefADSMathSciNetGoogle Scholar
  21. Hertz, H. (1889). Archives des Sciences Physiques et Naturelles 21, 281–308.Google Scholar
  22. Kostant, B. (1970). Lecture Notes in Mathematics, vol. 170, Springer, Berlin.Google Scholar
  23. vy-Leblond, J.-M. (1963). Journal of Mathematical Physics 4, 776–788.CrossRefGoogle Scholar
  24. Lorentz, H. A. (1886). Archives Néederlandaises, Haarlem, 2, 21; (1892), Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 1, 74.Google Scholar
  25. Mackey, G. W. (1952). Annals of Mathematics 55, 101–139.CrossRefMATHMathSciNetGoogle Scholar
  26. Mackey, G. W. (1953). Annals of Mathematics 58, 193–221.CrossRefMATHMathSciNetGoogle Scholar
  27. Michaelson, A. A. and Morley, E. W. (1886). American Journal of SciencetextbfXXXI, 377–386.Google Scholar
  28. Michaelson, A. A. and Morley, E. W. (1887). American Journal of Science XXXIV, 338–339.Google Scholar
  29. Minkowski, H. (1908). Göttingen Nachrichten, 53.Google Scholar
  30. Minkowski, H. (1909). Physikalisches Zeitschrift. 10, 104–111.Google Scholar
  31. Minkowski, H. (1910). Mathematische Annalen 68, 472–525.CrossRefMATHMathSciNetGoogle Scholar
  32. Minkowski, H. (1915). Annales de Physique 47, 927–938.MATHGoogle Scholar
  33. Poincaré, H. (1905). Comptes Rendus 140, 1504–1508.MATHGoogle Scholar
  34. , H. (1906). Rendiconti del Circolo Matemático di Palermo 21, 129.MATHGoogle Scholar
  35. ki, E. (1977). International Journal of Theoretical Physics 16, 321–331.CrossRefGoogle Scholar
  36. ki, E. (1978). Journal of Mathematical Physics 19, 2260–2271.CrossRefMathSciNetGoogle Scholar
  37. ki, E. (1986). Stochastic Quantum Mechanics and Quantum Space-Time, 2nd edn., D. Reidel, Dordrecht.Google Scholar
  38. Schroeck, Jr., F. E. (1996). Quantum Mechanics on Phase Space, Kluwer, Dordrecht.Google Scholar
  39. Schroeck, Jr., F. E. (2001). An algebra of effects in the formalism of quantum mechanics on phase space. International Journal of Theoretical Physics, in press.Google Scholar
  40. Schroeck, Jr., F. E. (2004). Algebra of effects in the formalism of quantum mechanics on phase space as an M.V. and a Heyting effect algebra. International Journal of Theoretical Physics, in press.Google Scholar
  41. Souriau, J.-M. (1970). Structure des systèmes dynamiques, Dunod, Paris; translation: Structure of dynamical systems. Progress in Mathematics 149, Birkhäuser, Boston, 1997.Google Scholar
  42. Voigt, W. (1887). Nachr. v.d. Königl, N. Gesells. der Wissenschaften und der Georg-Augusts-Univ. zu Göttingen, no. 2, pp. 41–51.Google Scholar
  43. Weyl, H. (1950). The Theory of Groups and Quantum Mechanics, Dover, New York; originally Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig, 1928.Google Scholar
  44. Wigner, E. P. (1932). Physical Reviews 40, 749–759.CrossRefADSMATHGoogle Scholar
  45. Wigner, E. P. (1939). Annals of Mathematics 40, 149–204. CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.University of SaskatchewanSaskatoonCanada
  2. 2.University of DenverDenver
  3. 3.Florida Atlantic UniversityBoca Raton
  4. 4.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

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