Relativistic partial-wave analysis using the velocity basis of the Poincaré group

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 60, Issue 6, 1999, Pages 4606-4615

  Bohm, A ^a   Kaldass, H ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0011061922     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.60.4606     Document Type: Article

References (21)

1. A. Bohm. Quantum Mechanics, 1st ed. (Springer, 1979);in particular Chapter XXI of the 3rd ed. (Springer, 1994);A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors, and Gel’fand Triplets, Lect. Notes in Phys., Vol. 348 (Springer, Berlin, 1989)
2. A. Bohm, S. Maxson, M. Loewe, and M. Gadella, Physica A 236, 485 (1997).
3. A. S. Wightman, Lectures on Invariance in Relativistic Quantum Mechanics (Les Houches, 1960).
4. H. Joos, Fortschr. Phys. 10, 65 (1962).
5. A. J. Macfarlane, Rev. Mod. Phys. 34, 41 (1962).
6. E. G. Beltrametti and G. Luzzatto, Nuovo Cimento 36, 1217 (1965);A. O. Barut in Lect. in Theor. Phys. Vol. 7 A, edited by W. E. Brittin and A. O. Barut (The University Of Colorado Press, Boulder, 1965), p. 121;
7. S. S. Sannikov, Phys. At. Nucl. (Yad. Fiz.) 3, 587 (1966) [Sov. J. Nucl. Phys. 4, 416 (1967)];
8. M. Comi, L. Lanz, L. A. Lugiato, and G. Ramella, J. Math. Phys. 16, 910 (1975).
9. The use of velocity eigenvectors has been suggested as early as 1965 by J. Werle, On a Symmetry Scheme Described by Non-Lie Algebra (International Center for Theoretical Physics, Trieste, 1965).
10. It has been incorporated in the spectrum-generating group approach for the mass and spin spectrum, A. Bohm, Phys. Rev. 175, 1767 (1968).
11. It has been used to relate the form factors for different decays to universal form factors for an (Formula presented) multiplet in A. Bohm, Phys. Rev. D 13, 2110 (1976)
12. and A. Bohm and J. Werle, Nucl. Phys. B 106, 165 (1976).
13. As “dynamical stability group of (Formula presented)” the same idea has been suggested by H. van Dam and L. C. Biedenharn, Phys. Rev. D 14, 405 (1976).
14. Velocity eigenvectors were reintroduced around 1990 as states of the Heavy Quark Effective Theory, again to reduce the number of independent form factors: A. F. Falk, H. Georgi, B. Grinstein, and M. B. Wise, Nucl. Phys. B 343, 1 (1990);H. Georgi Proceedings of the Theoretical Advanced Study Institute, edited by R. K. Ellis et al. (World Scientific, 1992) and reference thereof. In this approach one often gets the impression that the use of velocity basis vectors leads to approximate results. This, however is not the case if one takes into consideration that velocity eigenkets do not represent physical states, but that physical state vectors are obtained from the kets as “continuous superpositions” using the right measure in the integration. If this is done, the values of observable quantities do not depend upon whether one uses the velocity basis or the momentum basis, only that the use of the velocity basis often provides a more practical means of computation, by leading to form factors that do not depend upon the mass.
15. S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, 1995).
16. E. P. Wigner, Ann. Math. 40, 149 (1939).
17. A. Bohm in Studies in Mathematical Physics, edited by A. O. Barut (Reidel, 1973), p. 197;
18. A. Bohm, J. Math. Phys. 8, 1557 (1967) (Appendix B);B. Nagel in Studies in Mathematical Physics, edited by A. O. Barut (Reidel, 1973), p. 135;A. Bohm, M. Gadella, and S. Wickramasekara in Generalized Functions, Operator Theory, and Dynamical Systems, edited by I. Antoniou and G. Lumer (Chapman and Hall/CRC, London, 1999), p. 202.
19. E. Nelson, Ann. Math. 70, 572 (1959)
20. E. Nelson and W. F. Stinespring, Amer. J. Math. 81, 547 (1959).
21. See e.g., A. Bohm. Quantum Mechanics, 3rd ed. (Springer, 1993): Section XVI.1.